[ { "task_name": "SPP", "problem_type": "SPP", "instruction": "I’ve been handed the job of choosing which collection routes the neighborhood’s bin trucks will run so every street gets picked up by one, and only one, route. The idea is to pick a group of those candidate routes that together cover every street exactly once, and to keep the overall operating bill as small as possible — that bill is just the sum of the costs of the routes that get used. The exact streets, routes, and their costs are shown below.\n\n# num_streets=24\n# num_candidate_routes=114\nroute_id,route_operating_cost,streets_covered\nS1,24,D F G I J L\nS2,252,C F H\nS3,264,H L N O P Q\nS4,180,D E F K\nS5,405,B C D E G\nS6,55,H J K L N\nS7,616,B C D E F G H\nS8,534,Q R S T U X\nS9,260,R U V W X\nS10,256,D G H J\nS11,136,S T U V\nS12,42,J L M O Q U\nS13,497,A C D E F G L\nS14,147,A B C D E F G\nS15,405,T U V W X\nS16,400,R S T U\nS17,217,A B C D E F H\nS18,445,I J K M N\nS19,201,H I K\nS20,50,E F G H I\nS21,168,A B C D E F\nS22,282,C D E\nS23,132,R T X\nS24,679,I K L N O Q R\nS25,300,A C E F I J\nS26,340,N P Q R\nS27,42,A B C D E G\nS28,45,M P Q S T\nS29,70,D E F G H I L\nS30,20,W X\nS31,105,O P S T U\nS32,355,L R S T W\nS33,384,S T W X\nS34,325,G H I K M\nS35,410,A B C D F\nS36,490,A B E F H I J\nS37,120,I J L\nS38,400,A B C E\nS39,115,K L O Q S\nS40,87,T W X\nS41,87,N Q R\nS42,297,C E F\nS43,16,S U V W\nS44,42,G H M\nS45,370,J K L M N\nS46,207,A D F\nS47,510,S T U V W X\nS48,237,V W X\nS49,420,E K M N O\nS50,380,M O P S\nS51,36,B D E\nS52,594,C E F G I J\nS53,150,N O Q\nS54,500,F I J K L\nS55,64,E F G H\nS56,395,O P R S U\nS57,582,N O P Q S T\nS58,228,R S U W\nS59,33,H I M\nS60,162,E F G\nS61,396,N Q R U\nS62,280,L N P Q\nS63,558,F G H I K L\nS64,153,O R T\nS65,245,Q R S U V\nS66,216,N O Q R\nS67,116,A C E H\nS68,366,P Q S T V W\nS69,552,Q R S U W X\nS70,120,B C J\nS71,70,I L M N O\nS72,330,G I J L M P\nS73,492,A B C E F I\nS74,434,M N O P Q R T\nS75,290,S U V W X\nS76,168,M N O P\nS77,360,F G H I J\nS78,192,Q T U V W X\nS79,284,Q R T X\nS80,164,T U V X\nS81,285,P S V\nS82,28,G J K M\nS83,185,P R V W X\nS84,45,L M N R T\nS85,156,A D E F\nS86,240,E H J\nS87,228,O P T\nS88,77,F G H J K L N\nS89,204,N O P Q R S\nS90,210,Q R S T U W\nS91,10000,A\nS92,10000,B\nS93,10000,C\nS94,10000,D\nS95,10000,E\nS96,10000,F\nS97,10000,G\nS98,10000,H\nS99,10000,I\nS100,10000,J\nS101,10000,K\nS102,10000,L\nS103,10000,M\nS104,10000,N\nS105,10000,O\nS106,10000,P\nS107,10000,Q\nS108,10000,R\nS109,10000,S\nS110,10000,T\nS111,10000,U\nS112,10000,V\nS113,10000,W\nS114,10000,X\n\nIf you want the answer in a machine-friendly way, just stick to this tiny JSON shape when you reply:\n\n{\n \"solution\": [\"route_id\", ...]\n}\n\nThis just means \"solution\" should be an array of the route IDs you pick (one entry per chosen route). Think of it like writing down which collection routes get used — nothing fancy, just the list. This is just a sketch of the shape I need, not the actual selection.\n\nPlease make sure you use the exact route identifiers from the instance input — don't rename them or invent new ones. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 24, "num_sets": 114, "density": 0.16337719298245615, "sets": [ { "id": 1, "elements": [ 4, 6, 7, 9, 10, 12 ], "cost": 24 }, { "id": 2, "elements": [ 3, 6, 8 ], "cost": 252 }, { "id": 3, "elements": [ 8, 12, 14, 15, 16, 17 ], "cost": 264 }, { "id": 4, "elements": [ 4, 5, 6, 11 ], "cost": 180 }, { "id": 5, "elements": [ 2, 3, 4, 5, 7 ], "cost": 405 }, { "id": 6, "elements": [ 8, 10, 11, 12, 14 ], "cost": 55 }, { "id": 7, "elements": [ 2, 3, 4, 5, 6, 7, 8 ], "cost": 616 }, { "id": 8, "elements": [ 17, 18, 19, 20, 21, 24 ], "cost": 534 }, { "id": 9, "elements": [ 18, 21, 22, 23, 24 ], "cost": 260 }, { "id": 10, "elements": [ 4, 7, 8, 10 ], "cost": 256 }, { "id": 11, "elements": [ 19, 20, 21, 22 ], "cost": 136 }, { "id": 12, "elements": [ 10, 12, 13, 15, 17, 21 ], "cost": 42 }, { "id": 13, "elements": [ 1, 3, 4, 5, 6, 7, 12 ], "cost": 497 }, { "id": 14, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 147 }, { "id": 15, "elements": [ 20, 21, 22, 23, 24 ], "cost": 405 }, { "id": 16, "elements": [ 18, 19, 20, 21 ], "cost": 400 }, { "id": 17, "elements": [ 1, 2, 3, 4, 5, 6, 8 ], "cost": 217 }, { "id": 18, "elements": [ 9, 10, 11, 13, 14 ], "cost": 445 }, { "id": 19, "elements": [ 8, 9, 11 ], "cost": 201 }, { "id": 20, "elements": [ 5, 6, 7, 8, 9 ], "cost": 50 }, { "id": 21, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 168 }, { "id": 22, "elements": [ 3, 4, 5 ], "cost": 282 }, { "id": 23, "elements": [ 18, 20, 24 ], "cost": 132 }, { "id": 24, "elements": [ 9, 11, 12, 14, 15, 17, 18 ], "cost": 679 }, { "id": 25, "elements": [ 1, 3, 5, 6, 9, 10 ], "cost": 300 }, { "id": 26, "elements": [ 14, 16, 17, 18 ], "cost": 340 }, { "id": 27, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 42 }, { "id": 28, "elements": [ 13, 16, 17, 19, 20 ], "cost": 45 }, { "id": 29, "elements": [ 4, 5, 6, 7, 8, 9, 12 ], "cost": 70 }, { "id": 30, "elements": [ 23, 24 ], "cost": 20 }, { "id": 31, "elements": [ 15, 16, 19, 20, 21 ], "cost": 105 }, { "id": 32, "elements": [ 12, 18, 19, 20, 23 ], "cost": 355 }, { "id": 33, "elements": [ 19, 20, 23, 24 ], "cost": 384 }, { "id": 34, "elements": [ 7, 8, 9, 11, 13 ], "cost": 325 }, { "id": 35, "elements": [ 1, 2, 3, 4, 6 ], "cost": 410 }, { "id": 36, "elements": [ 1, 2, 5, 6, 8, 9, 10 ], "cost": 490 }, { "id": 37, "elements": [ 9, 10, 12 ], "cost": 120 }, { "id": 38, "elements": [ 1, 2, 3, 5 ], "cost": 400 }, { "id": 39, "elements": [ 11, 12, 15, 17, 19 ], "cost": 115 }, { "id": 40, "elements": [ 20, 23, 24 ], "cost": 87 }, { "id": 41, "elements": [ 14, 17, 18 ], "cost": 87 }, { "id": 42, "elements": [ 3, 5, 6 ], "cost": 297 }, { "id": 43, "elements": [ 19, 21, 22, 23 ], "cost": 16 }, { "id": 44, "elements": [ 7, 8, 13 ], "cost": 42 }, { "id": 45, "elements": [ 10, 11, 12, 13, 14 ], "cost": 370 }, { "id": 46, "elements": [ 1, 4, 6 ], "cost": 207 }, { "id": 47, "elements": [ 19, 20, 21, 22, 23, 24 ], "cost": 510 }, { "id": 48, "elements": [ 22, 23, 24 ], "cost": 237 }, { "id": 49, "elements": [ 5, 11, 13, 14, 15 ], "cost": 420 }, { "id": 50, "elements": [ 13, 15, 16, 19 ], "cost": 380 }, { "id": 51, "elements": [ 2, 4, 5 ], "cost": 36 }, { "id": 52, "elements": [ 3, 5, 6, 7, 9, 10 ], "cost": 594 }, { "id": 53, "elements": [ 14, 15, 17 ], "cost": 150 }, { "id": 54, "elements": [ 6, 9, 10, 11, 12 ], "cost": 500 }, { "id": 55, "elements": [ 5, 6, 7, 8 ], "cost": 64 }, { "id": 56, "elements": [ 15, 16, 18, 19, 21 ], "cost": 395 }, { "id": 57, "elements": [ 14, 15, 16, 17, 19, 20 ], "cost": 582 }, { "id": 58, "elements": [ 18, 19, 21, 23 ], "cost": 228 }, { "id": 59, "elements": [ 8, 9, 13 ], "cost": 33 }, { "id": 60, "elements": [ 5, 6, 7 ], "cost": 162 }, { "id": 61, "elements": [ 14, 17, 18, 21 ], "cost": 396 }, { "id": 62, "elements": [ 12, 14, 16, 17 ], "cost": 280 }, { "id": 63, "elements": [ 6, 7, 8, 9, 11, 12 ], "cost": 558 }, { "id": 64, "elements": [ 15, 18, 20 ], "cost": 153 }, { "id": 65, "elements": [ 17, 18, 19, 21, 22 ], "cost": 245 }, { "id": 66, "elements": [ 14, 15, 17, 18 ], "cost": 216 }, { "id": 67, "elements": [ 1, 3, 5, 8 ], "cost": 116 }, { "id": 68, "elements": [ 16, 17, 19, 20, 22, 23 ], "cost": 366 }, { "id": 69, "elements": [ 17, 18, 19, 21, 23, 24 ], "cost": 552 }, { "id": 70, "elements": [ 2, 3, 10 ], "cost": 120 }, { "id": 71, "elements": [ 9, 12, 13, 14, 15 ], "cost": 70 }, { "id": 72, "elements": [ 7, 9, 10, 12, 13, 16 ], "cost": 330 }, { "id": 73, "elements": [ 1, 2, 3, 5, 6, 9 ], "cost": 492 }, { "id": 74, "elements": [ 13, 14, 15, 16, 17, 18, 20 ], "cost": 434 }, { "id": 75, "elements": [ 19, 21, 22, 23, 24 ], "cost": 290 }, { "id": 76, "elements": [ 13, 14, 15, 16 ], "cost": 168 }, { "id": 77, "elements": [ 6, 7, 8, 9, 10 ], "cost": 360 }, { "id": 78, "elements": [ 17, 20, 21, 22, 23, 24 ], "cost": 192 }, { "id": 79, "elements": [ 17, 18, 20, 24 ], "cost": 284 }, { "id": 80, "elements": [ 20, 21, 22, 24 ], "cost": 164 }, { "id": 81, "elements": [ 16, 19, 22 ], "cost": 285 }, { "id": 82, "elements": [ 7, 10, 11, 13 ], "cost": 28 }, { "id": 83, "elements": [ 16, 18, 22, 23, 24 ], "cost": 185 }, { "id": 84, "elements": [ 12, 13, 14, 18, 20 ], "cost": 45 }, { "id": 85, "elements": [ 1, 4, 5, 6 ], "cost": 156 }, { "id": 86, "elements": [ 5, 8, 10 ], "cost": 240 }, { "id": 87, "elements": [ 15, 16, 20 ], "cost": 228 }, { "id": 88, "elements": [ 6, 7, 8, 10, 11, 12, 14 ], "cost": 77 }, { "id": 89, "elements": [ 14, 15, 16, 17, 18, 19 ], "cost": 204 }, { "id": 90, "elements": [ 17, 18, 19, 20, 21, 23 ], "cost": 210 }, { "id": 91, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 92, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 93, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 94, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 95, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 96, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 97, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 98, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 23 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 24 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0000_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0000_bag_stats.png" }, "solution": [ 34, 62, 64, 70, 75, 85 ], "obj": 1324.0, "instance_variant": { "num_elements": 24, "num_sets": 114, "sets": [ { "id": "S1", "elements": [ "D", "F", "G", "I", "J", "L" ], "cost": 24 }, { "id": "S2", "elements": [ "C", "F", "H" ], "cost": 252 }, { "id": "S3", "elements": [ "H", "L", "N", "O", "P", "Q" ], "cost": 264 }, { "id": "S4", "elements": [ "D", "E", "F", "K" ], "cost": 180 }, { "id": "S5", "elements": [ "B", "C", "D", "E", "G" ], "cost": 405 }, { "id": "S6", "elements": [ "H", "J", "K", "L", "N" ], "cost": 55 }, { "id": "S7", "elements": [ "B", "C", "D", "E", "F", "G", "H" ], "cost": 616 }, { "id": "S8", "elements": [ "Q", "R", "S", "T", "U", "X" ], "cost": 534 }, { "id": "S9", "elements": [ "R", "U", "V", "W", "X" ], "cost": 260 }, { "id": "S10", "elements": [ "D", "G", "H", "J" ], "cost": 256 }, { "id": "S11", "elements": [ "S", "T", "U", "V" ], "cost": 136 }, { "id": "S12", "elements": [ "J", "L", "M", "O", "Q", "U" ], "cost": 42 }, { "id": "S13", "elements": [ "A", "C", "D", "E", "F", "G", "L" ], "cost": 497 }, { "id": "S14", "elements": [ "A", "B", "C", "D", "E", "F", "G" ], "cost": 147 }, { "id": "S15", "elements": [ "T", "U", "V", "W", "X" ], "cost": 405 }, { "id": "S16", "elements": [ "R", "S", "T", "U" ], "cost": 400 }, { "id": "S17", "elements": [ "A", "B", "C", "D", "E", "F", "H" ], "cost": 217 }, { "id": "S18", "elements": [ "I", "J", "K", "M", "N" ], "cost": 445 }, { "id": "S19", "elements": [ "H", "I", "K" ], "cost": 201 }, { "id": "S20", "elements": [ "E", "F", "G", "H", "I" ], "cost": 50 }, { "id": "S21", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 168 }, { "id": "S22", "elements": [ "C", "D", "E" ], "cost": 282 }, { "id": "S23", "elements": [ "R", "T", "X" ], "cost": 132 }, { "id": "S24", "elements": [ "I", "K", "L", "N", "O", "Q", "R" ], "cost": 679 }, { "id": "S25", "elements": [ "A", "C", "E", "F", "I", "J" ], "cost": 300 }, { "id": "S26", "elements": [ "N", "P", "Q", "R" ], "cost": 340 }, { "id": "S27", "elements": [ "A", "B", "C", "D", "E", "G" ], "cost": 42 }, { "id": "S28", "elements": [ "M", "P", "Q", "S", "T" ], "cost": 45 }, { "id": "S29", "elements": [ "D", "E", "F", "G", "H", "I", "L" ], "cost": 70 }, { "id": "S30", "elements": [ "W", "X" ], "cost": 20 }, { "id": "S31", "elements": [ "O", "P", "S", "T", "U" ], "cost": 105 }, { "id": "S32", "elements": [ "L", "R", "S", "T", "W" ], "cost": 355 }, { "id": "S33", "elements": [ "S", "T", "W", "X" ], "cost": 384 }, { "id": "S34", "elements": [ "G", "H", "I", "K", "M" ], "cost": 325 }, { "id": "S35", "elements": [ "A", "B", "C", "D", "F" ], "cost": 410 }, { "id": "S36", "elements": [ "A", "B", "E", "F", "H", "I", "J" ], "cost": 490 }, { "id": "S37", "elements": [ "I", "J", "L" ], "cost": 120 }, { "id": "S38", "elements": [ "A", "B", "C", "E" ], "cost": 400 }, { "id": "S39", "elements": [ "K", "L", "O", "Q", "S" ], "cost": 115 }, { "id": "S40", "elements": [ "T", "W", "X" ], "cost": 87 }, { "id": "S41", "elements": [ "N", "Q", "R" ], "cost": 87 }, { "id": "S42", "elements": [ "C", "E", "F" ], "cost": 297 }, { "id": "S43", "elements": [ "S", "U", "V", "W" ], "cost": 16 }, { "id": "S44", "elements": [ "G", "H", "M" ], "cost": 42 }, { "id": "S45", "elements": [ "J", "K", "L", "M", "N" ], "cost": 370 }, { "id": "S46", "elements": [ "A", "D", "F" ], "cost": 207 }, { "id": "S47", "elements": [ "S", "T", "U", "V", "W", "X" ], "cost": 510 }, { "id": "S48", "elements": [ "V", "W", "X" ], "cost": 237 }, { "id": "S49", "elements": [ "E", "K", "M", "N", "O" ], "cost": 420 }, { "id": "S50", "elements": [ "M", "O", "P", "S" ], "cost": 380 }, { "id": "S51", "elements": [ "B", "D", "E" ], "cost": 36 }, { "id": "S52", "elements": [ "C", "E", "F", "G", "I", "J" ], "cost": 594 }, { "id": "S53", "elements": [ "N", "O", "Q" ], "cost": 150 }, { "id": "S54", "elements": [ "F", "I", "J", "K", "L" ], "cost": 500 }, { "id": "S55", "elements": [ "E", "F", "G", "H" ], "cost": 64 }, { "id": "S56", "elements": [ "O", "P", "R", "S", "U" ], "cost": 395 }, { "id": "S57", "elements": [ "N", "O", "P", "Q", "S", "T" ], "cost": 582 }, { "id": "S58", "elements": [ "R", "S", "U", "W" ], "cost": 228 }, { "id": "S59", "elements": [ "H", "I", "M" ], "cost": 33 }, { "id": "S60", "elements": [ "E", "F", "G" ], "cost": 162 }, { "id": "S61", "elements": [ "N", "Q", "R", "U" ], "cost": 396 }, { "id": "S62", "elements": [ "L", "N", "P", "Q" ], "cost": 280 }, { "id": "S63", "elements": [ "F", "G", "H", "I", "K", "L" ], "cost": 558 }, { "id": "S64", "elements": [ "O", "R", "T" ], "cost": 153 }, { "id": "S65", "elements": [ "Q", "R", "S", "U", "V" ], "cost": 245 }, { "id": "S66", "elements": [ "N", "O", "Q", "R" ], "cost": 216 }, { "id": "S67", "elements": [ "A", "C", "E", "H" ], "cost": 116 }, { "id": "S68", "elements": [ "P", "Q", "S", "T", "V", "W" ], "cost": 366 }, { "id": "S69", "elements": [ "Q", "R", "S", "U", "W", "X" ], "cost": 552 }, { "id": "S70", "elements": [ "B", "C", "J" ], "cost": 120 }, { "id": "S71", "elements": [ "I", "L", "M", "N", "O" ], "cost": 70 }, { "id": "S72", "elements": [ "G", "I", "J", "L", "M", "P" ], "cost": 330 }, { "id": "S73", "elements": [ "A", "B", "C", "E", "F", "I" ], "cost": 492 }, { "id": "S74", "elements": [ "M", "N", "O", "P", "Q", "R", "T" ], "cost": 434 }, { "id": "S75", "elements": [ "S", "U", "V", "W", "X" ], "cost": 290 }, { "id": "S76", "elements": [ "M", "N", "O", "P" ], "cost": 168 }, { "id": "S77", "elements": [ "F", "G", "H", "I", "J" ], "cost": 360 }, { "id": "S78", "elements": [ "Q", "T", "U", "V", "W", "X" ], "cost": 192 }, { "id": "S79", "elements": [ "Q", "R", "T", "X" ], "cost": 284 }, { "id": "S80", "elements": [ "T", "U", "V", "X" ], "cost": 164 }, { "id": "S81", "elements": [ "P", "S", "V" ], "cost": 285 }, { "id": "S82", "elements": [ "G", "J", "K", "M" ], "cost": 28 }, { "id": "S83", "elements": [ "P", "R", "V", "W", "X" ], "cost": 185 }, { "id": "S84", "elements": [ "L", "M", "N", "R", "T" ], "cost": 45 }, { "id": "S85", "elements": [ "A", "D", "E", "F" ], "cost": 156 }, { "id": "S86", "elements": [ "E", "H", "J" ], "cost": 240 }, { "id": "S87", "elements": [ "O", "P", "T" ], "cost": 228 }, { "id": "S88", "elements": [ "F", "G", "H", "J", "K", "L", "N" ], "cost": 77 }, { "id": "S89", "elements": [ "N", "O", "P", "Q", "R", "S" ], "cost": 204 }, { "id": "S90", "elements": [ "Q", "R", "S", "T", "U", "W" ], "cost": 210 }, { "id": "S91", "elements": [ "A" ], "cost": 10000 }, { "id": "S92", "elements": [ "B" ], "cost": 10000 }, { "id": "S93", "elements": [ "C" ], "cost": 10000 }, { "id": "S94", "elements": [ "D" ], "cost": 10000 }, { "id": "S95", "elements": [ "E" ], "cost": 10000 }, { "id": "S96", "elements": [ "F" ], "cost": 10000 }, { "id": "S97", "elements": [ "G" ], "cost": 10000 }, { "id": "S98", "elements": [ "H" ], "cost": 10000 }, { "id": "S99", "elements": [ "I" ], "cost": 10000 }, { "id": "S100", "elements": [ "J" ], "cost": 10000 }, { "id": "S101", "elements": [ "K" ], "cost": 10000 }, { "id": "S102", "elements": [ "L" ], "cost": 10000 }, { "id": "S103", "elements": [ "M" ], "cost": 10000 }, { "id": "S104", "elements": [ "N" ], "cost": 10000 }, { "id": "S105", "elements": [ "O" ], "cost": 10000 }, { "id": "S106", "elements": [ "P" ], "cost": 10000 }, { "id": "S107", "elements": [ "Q" ], "cost": 10000 }, { "id": "S108", "elements": [ "R" ], "cost": 10000 }, { "id": "S109", "elements": [ "S" ], "cost": 10000 }, { "id": "S110", "elements": [ "T" ], "cost": 10000 }, { "id": "S111", "elements": [ "U" ], "cost": 10000 }, { "id": "S112", "elements": [ "V" ], "cost": 10000 }, { "id": "S113", "elements": [ "W" ], "cost": 10000 }, { "id": "S114", "elements": [ "X" ], "cost": 10000 } ] }, "solution_variant": [ "S34", "S62", "S64", "S70", "S75", "S85" ], "context_index": 1, "input_format": "csv", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "Someone’s put together a list of possible buffet stations for an event, and the practical question is which of those stations to run so every dish on the menu gets served by one—and only one—station (no omissions, no duplicates). The measure of a better plan is simple: the lower the combined cost of the stations you pick, the better — just add their costs to get the total. The full details about dishes, stations, and prices appear below.\n\n- **num_dishes**: 25\n- **num_stations**: 127\n\n| station_id | station_cost | station_dishes |\n|---|---|---|\n| S1 | 171 | K M N |\n| S2 | 210 | U V W |\n| S3 | 150 | A C D E G |\n| S4 | 469 | N O R S T U V |\n| S5 | 258 | E G H I J K |\n| S6 | 208 | B C E G |\n| S7 | 7 | N O P Q T U V |\n| S8 | 42 | N P Q R S T |\n| S9 | 16 | O Q R U |\n| S10 | 105 | N Q R S U V X |\n| S11 | 270 | B C E F H |\n| S12 | 252 | N O P Q R S T |\n| S13 | 450 | G L M O P R |\n| S14 | 175 | M N O P R S U |\n| S15 | 350 | C F H I J K L |\n| S16 | 399 | P T U V W X Y |\n| S17 | 60 | J M R |\n| S18 | 129 | H J K |\n| S19 | 567 | M N P Q T U V |\n| S20 | 342 | F I K L N Q |\n| S21 | 114 | F L M |\n| S22 | 88 | K N O Q |\n| S23 | 225 | R T W |\n| S24 | 180 | N P Q |\n| S25 | 413 | R T U V W X Y |\n| S26 | 21 | D E F G I J O |\n| S27 | 693 | G H I J L M N |\n| S28 | 564 | C D E F G I |\n| S29 | 210 | G I J |\n| S30 | 672 | J K L M N O P |\n| S31 | 504 | A B C D E F G |\n| S32 | 348 | H L M O |\n| S33 | 16 | J K O Q |\n| S34 | 504 | P Q R S T U |\n| S35 | 345 | A B C E L |\n| S36 | 210 | N P S |\n| S37 | 180 | U V W X Y |\n| S38 | 141 | J K L |\n| S39 | 24 | U W Y |\n| S40 | 112 | C I K L |\n| S41 | 588 | P R S V X Y |\n| S42 | 5 | P Q R V W |\n| S43 | 16 | B E G K |\n| S44 | 69 | J K N |\n| S45 | 400 | D E F G |\n| S46 | 340 | N P S W |\n| S47 | 70 | J N P R W |\n| S48 | 410 | K L M N P |\n| S49 | 213 | T W Y |\n| S50 | 246 | E F G H I J |\n| S51 | 55 | R T U W Y |\n| S52 | 24 | B C F |\n| S53 | 275 | Q R S V W |\n| S54 | 98 | S T U V W X Y |\n| S55 | 288 | Q R T U |\n| S56 | 630 | O Q R T U V X |\n| S57 | 24 | G H I J M N |\n| S58 | 144 | O P Q |\n| S59 | 84 | G H K M |\n| S60 | 582 | A B C D E F |\n| S61 | 30 | J K L N O |\n| S62 | 392 | P R V X |\n| S63 | 276 | H I J K |\n| S64 | 354 | N O P Q R T |\n| S65 | 560 | E G I J K L M |\n| S66 | 340 | S U V X Y |\n| S67 | 30 | H K L |\n| S68 | 415 | B C D G I |\n| S69 | 567 | C E F G H K M |\n| S70 | 56 | G H I J K L N |\n| S71 | 40 | A B D G |\n| S72 | 348 | B C D E G H |\n| S73 | 308 | N O P Q S T W |\n| S74 | 324 | E I K L |\n| S75 | 468 | A B C D G I |\n| S76 | 637 | G H J K L M O |\n| S77 | 476 | F H J K L O R |\n| S78 | 72 | T U V |\n| S79 | 693 | G H I J K N O |\n| S80 | 245 | S T V X Y |\n| S81 | 66 | G H I |\n| S82 | 348 | A B C D E G |\n| S83 | 192 | G H J K |\n| S84 | 156 | Q R S T W X |\n| S85 | 112 | B D E F G H J |\n| S86 | 48 | V W X Y |\n| S87 | 77 | E F G H J K L |\n| S88 | 76 | A B C E |\n| S89 | 294 | J L M |\n| S90 | 468 | J L M P R T |\n| S91 | 400 | B C D F G |\n| S92 | 460 | C D E F G |\n| S93 | 72 | G M O |\n| S94 | 18 | B C D E F H |\n| S95 | 184 | L O P R |\n| S96 | 276 | H I J L M O |\n| S97 | 90 | S T U V W |\n| S98 | 78 | T V Y |\n| S99 | 322 | G I K L M O Q |\n| S100 | 44 | E F G J |\n| S101 | 228 | A B C D E J |\n| S102 | 376 | Q S T U |\n| S103 | 10000 | A |\n| S104 | 10000 | B |\n| S105 | 10000 | C |\n| S106 | 10000 | D |\n| S107 | 10000 | E |\n| S108 | 10000 | F |\n| S109 | 10000 | G |\n| S110 | 10000 | H |\n| S111 | 10000 | I |\n| S112 | 10000 | J |\n| S113 | 10000 | K |\n| S114 | 10000 | L |\n| S115 | 10000 | M |\n| S116 | 10000 | N |\n| S117 | 10000 | O |\n| S118 | 10000 | P |\n| S119 | 10000 | Q |\n| S120 | 10000 | R |\n| S121 | 10000 | S |\n| S122 | 10000 | T |\n| S123 | 10000 | U |\n| S124 | 10000 | V |\n| S125 | 10000 | W |\n| S126 | 10000 | X |\n| S127 | 10000 | Y |\n\nAlso, if you're sending back the chosen stations, a neat little JSON snippet like this makes it easy to check:\n\n{\n \"solution\": [\"station_id\", ...]\n}\n\nHere \"solution\" is just the list of station IDs you plan to run — one entry per station. The placeholder \"station_id\" is where you'd drop the actual identifier for a buffet station from the instance (so replace that placeholder with the real ID). This is only a sketch of the expected shape, not the actual answer.\n\nPlease make sure you use the exact identifiers from the instance input — don't rename them or invent new labels. Valid identifiers look like plain numbers such as \"1\" or \"23\", single capital letters like \"A\" or \"B\", or a capital letter followed by digits like \"A1\" or \"X7\".", "instance": { "problem_type": "SPP", "num_elements": 25, "num_sets": 127, "density": 0.17039370078740157, "sets": [ { "id": 1, "elements": [ 11, 13, 14 ], "cost": 171 }, { "id": 2, "elements": [ 21, 22, 23 ], "cost": 210 }, { "id": 3, "elements": [ 1, 3, 4, 5, 7 ], "cost": 150 }, { "id": 4, "elements": [ 14, 15, 18, 19, 20, 21, 22 ], "cost": 469 }, { "id": 5, "elements": [ 5, 7, 8, 9, 10, 11 ], "cost": 258 }, { "id": 6, "elements": [ 2, 3, 5, 7 ], "cost": 208 }, { "id": 7, "elements": [ 14, 15, 16, 17, 20, 21, 22 ], "cost": 7 }, { "id": 8, "elements": [ 14, 16, 17, 18, 19, 20 ], "cost": 42 }, { "id": 9, "elements": [ 15, 17, 18, 21 ], "cost": 16 }, { "id": 10, "elements": [ 14, 17, 18, 19, 21, 22, 24 ], "cost": 105 }, { "id": 11, "elements": [ 2, 3, 5, 6, 8 ], "cost": 270 }, { "id": 12, "elements": [ 14, 15, 16, 17, 18, 19, 20 ], "cost": 252 }, { "id": 13, "elements": [ 7, 12, 13, 15, 16, 18 ], "cost": 450 }, { "id": 14, "elements": [ 13, 14, 15, 16, 18, 19, 21 ], "cost": 175 }, { "id": 15, "elements": [ 3, 6, 8, 9, 10, 11, 12 ], "cost": 350 }, { "id": 16, "elements": [ 16, 20, 21, 22, 23, 24, 25 ], "cost": 399 }, { "id": 17, "elements": [ 10, 13, 18 ], "cost": 60 }, { "id": 18, "elements": [ 8, 10, 11 ], "cost": 129 }, { "id": 19, "elements": [ 13, 14, 16, 17, 20, 21, 22 ], "cost": 567 }, { "id": 20, "elements": [ 6, 9, 11, 12, 14, 17 ], "cost": 342 }, { "id": 21, "elements": [ 6, 12, 13 ], "cost": 114 }, { "id": 22, "elements": [ 11, 14, 15, 17 ], "cost": 88 }, { "id": 23, "elements": [ 18, 20, 23 ], "cost": 225 }, { "id": 24, "elements": [ 14, 16, 17 ], "cost": 180 }, { "id": 25, "elements": [ 18, 20, 21, 22, 23, 24, 25 ], "cost": 413 }, { "id": 26, "elements": [ 4, 5, 6, 7, 9, 10, 15 ], "cost": 21 }, { "id": 27, "elements": [ 7, 8, 9, 10, 12, 13, 14 ], "cost": 693 }, { "id": 28, "elements": [ 3, 4, 5, 6, 7, 9 ], "cost": 564 }, { "id": 29, "elements": [ 7, 9, 10 ], "cost": 210 }, { "id": 30, "elements": [ 10, 11, 12, 13, 14, 15, 16 ], "cost": 672 }, { "id": 31, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 504 }, { "id": 32, "elements": [ 8, 12, 13, 15 ], "cost": 348 }, { "id": 33, "elements": [ 10, 11, 15, 17 ], "cost": 16 }, { "id": 34, "elements": [ 16, 17, 18, 19, 20, 21 ], "cost": 504 }, { "id": 35, "elements": [ 1, 2, 3, 5, 12 ], "cost": 345 }, { "id": 36, "elements": [ 14, 16, 19 ], "cost": 210 }, { "id": 37, "elements": [ 21, 22, 23, 24, 25 ], "cost": 180 }, { "id": 38, "elements": [ 10, 11, 12 ], "cost": 141 }, { "id": 39, "elements": [ 21, 23, 25 ], "cost": 24 }, { "id": 40, "elements": [ 3, 9, 11, 12 ], "cost": 112 }, { "id": 41, "elements": [ 16, 18, 19, 22, 24, 25 ], "cost": 588 }, { "id": 42, "elements": [ 16, 17, 18, 22, 23 ], "cost": 5 }, { "id": 43, "elements": [ 2, 5, 7, 11 ], "cost": 16 }, { "id": 44, "elements": [ 10, 11, 14 ], "cost": 69 }, { "id": 45, "elements": [ 4, 5, 6, 7 ], "cost": 400 }, { "id": 46, "elements": [ 14, 16, 19, 23 ], "cost": 340 }, { "id": 47, "elements": [ 10, 14, 16, 18, 23 ], "cost": 70 }, { "id": 48, "elements": [ 11, 12, 13, 14, 16 ], "cost": 410 }, { "id": 49, "elements": [ 20, 23, 25 ], "cost": 213 }, { "id": 50, "elements": [ 5, 6, 7, 8, 9, 10 ], "cost": 246 }, { "id": 51, "elements": [ 18, 20, 21, 23, 25 ], "cost": 55 }, { "id": 52, "elements": [ 2, 3, 6 ], "cost": 24 }, { "id": 53, "elements": [ 17, 18, 19, 22, 23 ], "cost": 275 }, { "id": 54, "elements": [ 19, 20, 21, 22, 23, 24, 25 ], "cost": 98 }, { "id": 55, "elements": [ 17, 18, 20, 21 ], "cost": 288 }, { "id": 56, "elements": [ 15, 17, 18, 20, 21, 22, 24 ], "cost": 630 }, { "id": 57, "elements": [ 7, 8, 9, 10, 13, 14 ], "cost": 24 }, { "id": 58, "elements": [ 15, 16, 17 ], "cost": 144 }, { "id": 59, "elements": [ 7, 8, 11, 13 ], "cost": 84 }, { "id": 60, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 582 }, { "id": 61, "elements": [ 10, 11, 12, 14, 15 ], "cost": 30 }, { "id": 62, "elements": [ 16, 18, 22, 24 ], "cost": 392 }, { "id": 63, "elements": [ 8, 9, 10, 11 ], "cost": 276 }, { "id": 64, "elements": [ 14, 15, 16, 17, 18, 20 ], "cost": 354 }, { "id": 65, "elements": [ 5, 7, 9, 10, 11, 12, 13 ], "cost": 560 }, { "id": 66, "elements": [ 19, 21, 22, 24, 25 ], "cost": 340 }, { "id": 67, "elements": [ 8, 11, 12 ], "cost": 30 }, { "id": 68, "elements": [ 2, 3, 4, 7, 9 ], "cost": 415 }, { "id": 69, "elements": [ 3, 5, 6, 7, 8, 11, 13 ], "cost": 567 }, { "id": 70, "elements": [ 7, 8, 9, 10, 11, 12, 14 ], "cost": 56 }, { "id": 71, "elements": [ 1, 2, 4, 7 ], "cost": 40 }, { "id": 72, "elements": [ 2, 3, 4, 5, 7, 8 ], "cost": 348 }, { "id": 73, "elements": [ 14, 15, 16, 17, 19, 20, 23 ], "cost": 308 }, { "id": 74, "elements": [ 5, 9, 11, 12 ], "cost": 324 }, { "id": 75, "elements": [ 1, 2, 3, 4, 7, 9 ], "cost": 468 }, { "id": 76, "elements": [ 7, 8, 10, 11, 12, 13, 15 ], "cost": 637 }, { "id": 77, "elements": [ 6, 8, 10, 11, 12, 15, 18 ], "cost": 476 }, { "id": 78, "elements": [ 20, 21, 22 ], "cost": 72 }, { "id": 79, "elements": [ 7, 8, 9, 10, 11, 14, 15 ], "cost": 693 }, { "id": 80, "elements": [ 19, 20, 22, 24, 25 ], "cost": 245 }, { "id": 81, "elements": [ 7, 8, 9 ], "cost": 66 }, { "id": 82, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 348 }, { "id": 83, "elements": [ 7, 8, 10, 11 ], "cost": 192 }, { "id": 84, "elements": [ 17, 18, 19, 20, 23, 24 ], "cost": 156 }, { "id": 85, "elements": [ 2, 4, 5, 6, 7, 8, 10 ], "cost": 112 }, { "id": 86, "elements": [ 22, 23, 24, 25 ], "cost": 48 }, { "id": 87, "elements": [ 5, 6, 7, 8, 10, 11, 12 ], "cost": 77 }, { "id": 88, "elements": [ 1, 2, 3, 5 ], "cost": 76 }, { "id": 89, "elements": [ 10, 12, 13 ], "cost": 294 }, { "id": 90, "elements": [ 10, 12, 13, 16, 18, 20 ], "cost": 468 }, { "id": 91, "elements": [ 2, 3, 4, 6, 7 ], "cost": 400 }, { "id": 92, "elements": [ 3, 4, 5, 6, 7 ], "cost": 460 }, { "id": 93, "elements": [ 7, 13, 15 ], "cost": 72 }, { "id": 94, "elements": [ 2, 3, 4, 5, 6, 8 ], "cost": 18 }, { "id": 95, "elements": [ 12, 15, 16, 18 ], "cost": 184 }, { "id": 96, "elements": [ 8, 9, 10, 12, 13, 15 ], "cost": 276 }, { "id": 97, "elements": [ 19, 20, 21, 22, 23 ], "cost": 90 }, { "id": 98, "elements": [ 20, 22, 25 ], "cost": 78 }, { "id": 99, "elements": [ 7, 9, 11, 12, 13, 15, 17 ], "cost": 322 }, { "id": 100, "elements": [ 5, 6, 7, 10 ], "cost": 44 }, { "id": 101, "elements": [ 1, 2, 3, 4, 5, 10 ], "cost": 228 }, { "id": 102, "elements": [ 17, 19, 20, 21 ], "cost": 376 }, { "id": 103, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 118, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 119, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 120, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 121, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 122, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 123, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 124, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 125, "elements": [ 23 ], "cost": 10000, "singleton": true }, { "id": 126, "elements": [ 24 ], "cost": 10000, "singleton": true }, { "id": 127, "elements": [ 25 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0001_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0001_bag_stats.png" }, "solution": [ 12, 21, 37, 63, 82 ], "obj": 1170.0, "instance_variant": { "num_elements": 25, "num_sets": 127, "sets": [ { "id": "S1", "elements": [ "K", "M", "N" ], "cost": 171 }, { "id": "S2", "elements": [ "U", "V", "W" ], "cost": 210 }, { "id": "S3", "elements": [ "A", "C", "D", "E", "G" ], "cost": 150 }, { "id": "S4", "elements": [ "N", "O", "R", "S", "T", "U", "V" ], "cost": 469 }, { "id": "S5", "elements": [ "E", "G", "H", "I", "J", "K" ], "cost": 258 }, { "id": "S6", "elements": [ "B", "C", "E", "G" ], "cost": 208 }, { "id": "S7", "elements": [ "N", "O", "P", "Q", "T", "U", "V" ], "cost": 7 }, { "id": "S8", "elements": [ "N", "P", "Q", "R", "S", "T" ], "cost": 42 }, { "id": "S9", "elements": [ "O", "Q", "R", "U" ], "cost": 16 }, { "id": "S10", "elements": [ "N", "Q", "R", "S", "U", "V", "X" ], "cost": 105 }, { "id": "S11", "elements": [ "B", "C", "E", "F", "H" ], "cost": 270 }, { "id": "S12", "elements": [ "N", "O", "P", "Q", "R", "S", "T" ], "cost": 252 }, { "id": "S13", "elements": [ "G", "L", "M", "O", "P", "R" ], "cost": 450 }, { "id": "S14", "elements": [ "M", "N", "O", "P", "R", "S", "U" ], "cost": 175 }, { "id": "S15", "elements": [ "C", "F", "H", "I", "J", "K", "L" ], "cost": 350 }, { "id": "S16", "elements": [ "P", "T", "U", "V", "W", "X", "Y" ], "cost": 399 }, { "id": "S17", "elements": [ "J", "M", "R" ], "cost": 60 }, { "id": "S18", "elements": [ "H", "J", "K" ], "cost": 129 }, { "id": "S19", "elements": [ "M", "N", "P", "Q", "T", "U", "V" ], "cost": 567 }, { "id": "S20", "elements": [ "F", "I", "K", "L", "N", "Q" ], "cost": 342 }, { "id": "S21", "elements": [ "F", "L", "M" ], "cost": 114 }, { "id": "S22", "elements": [ "K", "N", "O", "Q" ], "cost": 88 }, { "id": "S23", "elements": [ "R", "T", "W" ], "cost": 225 }, { "id": "S24", "elements": [ "N", "P", "Q" ], "cost": 180 }, { "id": "S25", "elements": [ "R", "T", "U", "V", "W", "X", "Y" ], "cost": 413 }, { "id": "S26", "elements": [ "D", "E", "F", "G", "I", "J", "O" ], "cost": 21 }, { "id": "S27", "elements": [ "G", "H", "I", "J", "L", "M", "N" ], "cost": 693 }, { "id": "S28", "elements": [ "C", "D", "E", "F", "G", "I" ], "cost": 564 }, { "id": "S29", "elements": [ "G", "I", "J" ], "cost": 210 }, { "id": "S30", "elements": [ "J", "K", "L", "M", "N", "O", "P" ], "cost": 672 }, { "id": "S31", "elements": [ "A", "B", "C", "D", "E", "F", "G" ], "cost": 504 }, { "id": "S32", "elements": [ "H", "L", "M", "O" ], "cost": 348 }, { "id": "S33", "elements": [ "J", "K", "O", "Q" ], "cost": 16 }, { "id": "S34", "elements": [ "P", "Q", "R", "S", "T", "U" ], "cost": 504 }, { "id": "S35", "elements": [ "A", "B", "C", "E", "L" ], "cost": 345 }, { "id": "S36", "elements": [ "N", "P", "S" ], "cost": 210 }, { "id": "S37", "elements": [ "U", "V", "W", "X", "Y" ], "cost": 180 }, { "id": "S38", "elements": [ "J", "K", "L" ], "cost": 141 }, { "id": "S39", "elements": [ "U", "W", "Y" ], "cost": 24 }, { "id": "S40", "elements": [ "C", "I", "K", "L" ], "cost": 112 }, { "id": "S41", "elements": [ "P", "R", "S", "V", "X", "Y" ], "cost": 588 }, { "id": "S42", "elements": [ "P", "Q", "R", "V", "W" ], "cost": 5 }, { "id": "S43", "elements": [ "B", "E", "G", "K" ], "cost": 16 }, { "id": "S44", "elements": [ "J", "K", "N" ], "cost": 69 }, { "id": "S45", "elements": [ "D", "E", "F", "G" ], "cost": 400 }, { "id": "S46", "elements": [ "N", "P", "S", "W" ], "cost": 340 }, { "id": "S47", "elements": [ "J", "N", "P", "R", "W" ], "cost": 70 }, { "id": "S48", "elements": [ "K", "L", "M", "N", "P" ], "cost": 410 }, { "id": "S49", "elements": [ "T", "W", "Y" ], "cost": 213 }, { "id": "S50", "elements": [ "E", "F", "G", "H", "I", "J" ], "cost": 246 }, { "id": "S51", "elements": [ "R", "T", "U", "W", "Y" ], "cost": 55 }, { "id": "S52", "elements": [ "B", "C", "F" ], "cost": 24 }, { "id": "S53", "elements": [ "Q", "R", "S", "V", "W" ], "cost": 275 }, { "id": "S54", "elements": [ "S", "T", "U", "V", "W", "X", "Y" ], "cost": 98 }, { "id": "S55", "elements": [ "Q", "R", "T", "U" ], "cost": 288 }, { "id": "S56", "elements": [ "O", "Q", "R", "T", "U", "V", "X" ], "cost": 630 }, { "id": "S57", "elements": [ "G", "H", "I", "J", "M", "N" ], "cost": 24 }, { "id": "S58", "elements": [ "O", "P", "Q" ], "cost": 144 }, { "id": "S59", "elements": [ "G", "H", "K", "M" ], "cost": 84 }, { "id": "S60", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 582 }, { "id": "S61", "elements": [ "J", "K", "L", "N", "O" ], "cost": 30 }, { "id": "S62", "elements": [ "P", "R", "V", "X" ], "cost": 392 }, { "id": "S63", "elements": [ "H", "I", "J", "K" ], "cost": 276 }, { "id": "S64", "elements": [ "N", "O", "P", "Q", "R", "T" ], "cost": 354 }, { "id": "S65", "elements": [ "E", "G", "I", "J", "K", "L", "M" ], "cost": 560 }, { "id": "S66", "elements": [ "S", "U", "V", "X", "Y" ], "cost": 340 }, { "id": "S67", "elements": [ "H", "K", "L" ], "cost": 30 }, { "id": "S68", "elements": [ "B", "C", "D", "G", "I" ], "cost": 415 }, { "id": "S69", "elements": [ "C", "E", "F", "G", "H", "K", "M" ], "cost": 567 }, { "id": "S70", "elements": [ "G", "H", "I", "J", "K", "L", "N" ], "cost": 56 }, { "id": "S71", "elements": [ "A", "B", "D", "G" ], "cost": 40 }, { "id": "S72", "elements": [ "B", "C", "D", "E", "G", "H" ], "cost": 348 }, { "id": "S73", "elements": [ "N", "O", "P", "Q", "S", "T", "W" ], "cost": 308 }, { "id": "S74", "elements": [ "E", "I", "K", "L" ], "cost": 324 }, { "id": "S75", "elements": [ "A", "B", "C", "D", "G", "I" ], "cost": 468 }, { "id": "S76", "elements": [ "G", "H", "J", "K", "L", "M", "O" ], "cost": 637 }, { "id": "S77", "elements": [ "F", "H", "J", "K", "L", "O", "R" ], "cost": 476 }, { "id": "S78", "elements": [ "T", "U", "V" ], "cost": 72 }, { "id": "S79", "elements": [ "G", "H", "I", "J", "K", "N", "O" ], "cost": 693 }, { "id": "S80", "elements": [ "S", "T", "V", "X", "Y" ], "cost": 245 }, { "id": "S81", "elements": [ "G", "H", "I" ], "cost": 66 }, { "id": "S82", "elements": [ "A", "B", "C", "D", "E", "G" ], "cost": 348 }, { "id": "S83", "elements": [ "G", "H", "J", "K" ], "cost": 192 }, { "id": "S84", "elements": [ "Q", "R", "S", "T", "W", "X" ], "cost": 156 }, { "id": "S85", "elements": [ "B", "D", "E", "F", "G", "H", "J" ], "cost": 112 }, { "id": "S86", "elements": [ "V", "W", "X", "Y" ], "cost": 48 }, { "id": "S87", "elements": [ "E", "F", "G", "H", "J", "K", "L" ], "cost": 77 }, { "id": "S88", "elements": [ "A", "B", "C", "E" ], "cost": 76 }, { "id": "S89", "elements": [ "J", "L", "M" ], "cost": 294 }, { "id": "S90", "elements": [ "J", "L", "M", "P", "R", "T" ], "cost": 468 }, { "id": "S91", "elements": [ "B", "C", "D", "F", "G" ], "cost": 400 }, { "id": "S92", "elements": [ "C", "D", "E", "F", "G" ], "cost": 460 }, { "id": "S93", "elements": [ "G", "M", "O" ], "cost": 72 }, { "id": "S94", "elements": [ "B", "C", "D", "E", "F", "H" ], "cost": 18 }, { "id": "S95", "elements": [ "L", "O", "P", "R" ], "cost": 184 }, { "id": "S96", "elements": [ "H", "I", "J", "L", "M", "O" ], "cost": 276 }, { "id": "S97", "elements": [ "S", "T", "U", "V", "W" ], "cost": 90 }, { "id": "S98", "elements": [ "T", "V", "Y" ], "cost": 78 }, { "id": "S99", "elements": [ "G", "I", "K", "L", "M", "O", "Q" ], "cost": 322 }, { "id": "S100", "elements": [ "E", "F", "G", "J" ], "cost": 44 }, { "id": "S101", "elements": [ "A", "B", "C", "D", "E", "J" ], "cost": 228 }, { "id": "S102", "elements": [ "Q", "S", "T", "U" ], "cost": 376 }, { "id": "S103", "elements": [ "A" ], "cost": 10000 }, { "id": "S104", "elements": [ "B" ], "cost": 10000 }, { "id": "S105", "elements": [ "C" ], "cost": 10000 }, { "id": "S106", "elements": [ "D" ], "cost": 10000 }, { "id": "S107", "elements": [ "E" ], "cost": 10000 }, { "id": "S108", "elements": [ "F" ], "cost": 10000 }, { "id": "S109", "elements": [ "G" ], "cost": 10000 }, { "id": "S110", "elements": [ "H" ], "cost": 10000 }, { "id": "S111", "elements": [ "I" ], "cost": 10000 }, { "id": "S112", "elements": [ "J" ], "cost": 10000 }, { "id": "S113", "elements": [ "K" ], "cost": 10000 }, { "id": "S114", "elements": [ "L" ], "cost": 10000 }, { "id": "S115", "elements": [ "M" ], "cost": 10000 }, { "id": "S116", "elements": [ "N" ], "cost": 10000 }, { "id": "S117", "elements": [ "O" ], "cost": 10000 }, { "id": "S118", "elements": [ "P" ], "cost": 10000 }, { "id": "S119", "elements": [ "Q" ], "cost": 10000 }, { "id": "S120", "elements": [ "R" ], "cost": 10000 }, { "id": "S121", "elements": [ "S" ], "cost": 10000 }, { "id": "S122", "elements": [ "T" ], "cost": 10000 }, { "id": "S123", "elements": [ "U" ], "cost": 10000 }, { "id": "S124", "elements": [ "V" ], "cost": 10000 }, { "id": "S125", "elements": [ "W" ], "cost": 10000 }, { "id": "S126", "elements": [ "X" ], "cost": 10000 }, { "id": "S127", "elements": [ "Y" ], "cost": 10000 } ] }, "solution_variant": [ "S12", "S21", "S37", "S63", "S82" ], "context_index": 2, "input_format": "markdown_table", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "We’ve got a stack of classes that need spaces, and the job is to choose rooms so each class is assigned to one room only (no class in two rooms, no class left without a room). Every room that’s used carries a rental fee, and the total cost is just the sum of those room fees — the goal is to make that sum as small as it can be. The exact rooms, class lists and prices follow below.\n\n{\n \"num_classes\": 25,\n \"num_rooms\": 161,\n \"sets\": [\n {\n \"room_id\": \"S1\",\n \"classes_in_room\": [\n \"J\",\n \"L\",\n \"M\",\n \"O\",\n \"R\",\n \"S\"\n ],\n \"rental_cost\": 576\n },\n {\n \"room_id\": \"S2\",\n \"classes_in_room\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"G\"\n ],\n \"rental_cost\": 282\n },\n {\n \"room_id\": \"S3\",\n \"classes_in_room\": [\n \"N\",\n \"Q\",\n \"R\",\n \"S\"\n ],\n \"rental_cost\": 228\n },\n {\n \"room_id\": \"S4\",\n \"classes_in_room\": [\n \"M\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"X\"\n ],\n \"rental_cost\": 144\n },\n {\n \"room_id\": \"S5\",\n \"classes_in_room\": [\n \"J\",\n \"K\",\n \"N\",\n \"O\",\n \"P\"\n ],\n \"rental_cost\": 395\n },\n {\n \"room_id\": \"S6\",\n \"classes_in_room\": [\n \"L\",\n \"M\",\n \"N\",\n \"O\"\n ],\n \"rental_cost\": 368\n },\n {\n \"room_id\": \"S7\",\n \"classes_in_room\": [\n \"C\",\n \"D\",\n \"E\",\n \"H\",\n \"I\"\n ],\n \"rental_cost\": 355\n },\n {\n \"room_id\": \"S8\",\n \"classes_in_room\": [\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\"\n ],\n \"rental_cost\": 330\n },\n {\n \"room_id\": \"S9\",\n \"classes_in_room\": [\n \"E\",\n \"F\",\n \"G\",\n \"I\"\n ],\n \"rental_cost\": 296\n },\n {\n \"room_id\": \"S10\",\n \"classes_in_room\": [\n \"N\",\n \"O\",\n \"R\",\n \"U\"\n ],\n \"rental_cost\": 368\n },\n {\n \"room_id\": \"S11\",\n \"classes_in_room\": [\n \"L\",\n \"M\",\n \"O\",\n \"P\",\n \"Q\",\n \"R\"\n ],\n \"rental_cost\": 360\n },\n {\n \"room_id\": \"S12\",\n \"classes_in_room\": [\n \"H\",\n \"I\",\n \"L\",\n \"M\",\n \"N\",\n \"O\"\n ],\n \"rental_cost\": 390\n },\n {\n \"room_id\": \"S13\",\n \"classes_in_room\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"H\",\n \"I\"\n ],\n \"rental_cost\": 469\n },\n {\n \"room_id\": \"S14\",\n \"classes_in_room\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\"\n ],\n \"rental_cost\": 154\n },\n {\n \"room_id\": \"S15\",\n \"classes_in_room\": [\n \"P\",\n \"S\",\n \"T\",\n \"U\",\n \"V\"\n ],\n \"rental_cost\": 365\n },\n {\n \"room_id\": \"S16\",\n \"classes_in_room\": [\n \"A\",\n \"D\",\n \"F\"\n ],\n \"rental_cost\": 102\n },\n {\n \"room_id\": \"S17\",\n \"classes_in_room\": [\n \"S\",\n \"U\",\n \"V\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"rental_cost\": 234\n },\n {\n \"room_id\": \"S18\",\n \"classes_in_room\": [\n \"R\",\n \"T\",\n \"U\",\n \"V\",\n \"X\"\n ],\n \"rental_cost\": 300\n },\n {\n \"room_id\": \"S19\",\n \"classes_in_room\": [\n \"Q\",\n \"S\",\n \"W\",\n \"Y\"\n ],\n \"rental_cost\": 180\n },\n {\n \"room_id\": \"S20\",\n \"classes_in_room\": [\n \"L\",\n \"P\",\n \"R\"\n ],\n \"rental_cost\": 63\n },\n {\n \"room_id\": \"S21\",\n \"classes_in_room\": [\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"W\",\n \"Y\"\n ],\n \"rental_cost\": 84\n },\n {\n \"room_id\": \"S22\",\n \"classes_in_room\": [\n \"L\",\n \"O\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"U\"\n ],\n \"rental_cost\": 609\n },\n {\n \"room_id\": \"S23\",\n \"classes_in_room\": [\n \"L\",\n \"O\",\n \"P\",\n \"Q\",\n \"S\"\n ],\n \"rental_cost\": 260\n },\n {\n \"room_id\": \"S24\",\n \"classes_in_room\": [\n \"I\",\n \"K\",\n \"L\",\n \"N\"\n ],\n \"rental_cost\": 200\n },\n {\n \"room_id\": \"S25\",\n \"classes_in_room\": [\n \"S\",\n \"V\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"rental_cost\": 65\n },\n {\n \"room_id\": \"S26\",\n \"classes_in_room\": [\n \"H\",\n \"J\",\n \"K\",\n \"M\"\n ],\n \"rental_cost\": 208\n },\n {\n \"room_id\": \"S27\",\n \"classes_in_room\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\"\n ],\n \"rental_cost\": 330\n },\n {\n \"room_id\": \"S28\",\n \"classes_in_room\": [\n \"S\",\n \"T\",\n \"U\"\n ],\n \"rental_cost\": 117\n },\n {\n \"room_id\": \"S29\",\n \"classes_in_room\": [\n \"O\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\"\n ],\n \"rental_cost\": 240\n },\n {\n \"room_id\": \"S30\",\n \"classes_in_room\": [\n \"D\",\n \"E\",\n \"F\",\n \"I\"\n ],\n \"rental_cost\": 356\n },\n {\n \"room_id\": \"S31\",\n \"classes_in_room\": [\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"X\"\n ],\n \"rental_cost\": 450\n },\n {\n \"room_id\": \"S32\",\n \"classes_in_room\": [\n \"N\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\"\n ],\n \"rental_cost\": 25\n },\n {\n \"room_id\": \"S33\",\n \"classes_in_room\": [\n \"U\",\n \"V\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"rental_cost\": 335\n },\n {\n \"room_id\": \"S34\",\n \"classes_in_room\": [\n \"J\",\n \"K\",\n \"P\",\n \"R\"\n ],\n \"rental_cost\": 236\n },\n {\n \"room_id\": \"S35\",\n \"classes_in_room\": [\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\"\n ],\n \"rental_cost\": 224\n },\n {\n \"room_id\": \"S36\",\n \"classes_in_room\": [\n \"H\",\n \"I\",\n \"J\",\n \"O\"\n ],\n \"rental_cost\": 108\n },\n {\n \"room_id\": \"S37\",\n \"classes_in_room\": [\n \"L\",\n \"M\",\n \"O\",\n \"P\",\n \"R\",\n \"T\"\n ],\n \"rental_cost\": 522\n },\n {\n \"room_id\": \"S38\",\n \"classes_in_room\": [\n \"K\",\n \"L\",\n \"N\",\n \"O\",\n \"Q\"\n ],\n \"rental_cost\": 460\n },\n {\n \"room_id\": \"S39\",\n \"classes_in_room\": [\n \"M\",\n \"P\",\n \"Q\"\n ],\n \"rental_cost\": 207\n },\n {\n \"room_id\": \"S40\",\n \"classes_in_room\": [\n \"O\",\n \"R\",\n \"S\",\n \"T\"\n ],\n \"rental_cost\": 128\n },\n {\n \"room_id\": \"S41\",\n \"classes_in_room\": [\n \"M\",\n \"O\",\n \"P\"\n ],\n \"rental_cost\": 129\n },\n {\n \"room_id\": \"S42\",\n \"classes_in_room\": [\n \"B\",\n \"C\",\n \"E\",\n \"F\",\n \"H\",\n \"I\"\n ],\n \"rental_cost\": 366\n },\n {\n \"room_id\": \"S43\",\n \"classes_in_room\": [\n \"C\",\n \"E\",\n \"F\",\n \"I\",\n \"J\"\n ],\n \"rental_cost\": 420\n },\n {\n \"room_id\": \"S44\",\n \"classes_in_room\": [\n \"M\",\n \"Q\",\n \"T\",\n \"U\",\n \"V\"\n ],\n \"rental_cost\": 115\n },\n {\n \"room_id\": \"S45\",\n \"classes_in_room\": [\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"I\"\n ],\n \"rental_cost\": 14\n },\n {\n \"room_id\": \"S46\",\n \"classes_in_room\": [\n \"Q\",\n \"S\",\n \"X\",\n \"Y\"\n ],\n \"rental_cost\": 280\n },\n {\n \"room_id\": \"S47\",\n \"classes_in_room\": [\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"H\"\n ],\n \"rental_cost\": 295\n },\n {\n \"room_id\": \"S48\",\n \"classes_in_room\": [\n \"R\",\n \"T\",\n \"V\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"rental_cost\": 492\n },\n {\n \"room_id\": \"S49\",\n \"classes_in_room\": [\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"P\"\n ],\n \"rental_cost\": 238\n },\n {\n \"room_id\": \"S50\",\n \"classes_in_room\": [\n \"M\",\n \"P\",\n \"Q\",\n \"S\"\n ],\n \"rental_cost\": 8\n },\n {\n \"room_id\": \"S51\",\n \"classes_in_room\": [\n \"H\",\n \"I\",\n \"K\",\n \"M\"\n ],\n \"rental_cost\": 208\n },\n {\n \"room_id\": \"S52\",\n \"classes_in_room\": [\n \"S\",\n \"T\",\n \"V\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"rental_cost\": 108\n },\n {\n \"room_id\": \"S53\",\n \"classes_in_room\": [\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"Q\"\n ],\n \"rental_cost\": 345\n },\n {\n \"room_id\": \"S54\",\n \"classes_in_room\": [\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"J\"\n ],\n \"rental_cost\": 511\n },\n {\n \"room_id\": \"S55\",\n \"classes_in_room\": [\n \"P\",\n \"R\",\n \"U\"\n ],\n \"rental_cost\": 66\n },\n {\n \"room_id\": \"S56\",\n \"classes_in_room\": [\n \"P\",\n \"Q\",\n \"R\",\n \"S\"\n ],\n \"rental_cost\": 264\n },\n {\n \"room_id\": \"S57\",\n \"classes_in_room\": [\n \"Q\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"X\",\n \"Y\"\n ],\n \"rental_cost\": 490\n },\n {\n \"room_id\": \"S58\",\n \"classes_in_room\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"F\",\n \"G\",\n \"I\"\n ],\n \"rental_cost\": 378\n },\n {\n \"room_id\": \"S59\",\n \"classes_in_room\": [\n \"I\",\n \"L\",\n \"M\",\n \"Q\"\n ],\n \"rental_cost\": 340\n },\n {\n \"room_id\": \"S60\",\n \"classes_in_room\": [\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"H\"\n ],\n \"rental_cost\": 7\n },\n {\n \"room_id\": \"S61\",\n \"classes_in_room\": [\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"L\"\n ],\n \"rental_cost\": 120\n },\n {\n \"room_id\": \"S62\",\n \"classes_in_room\": [\n \"S\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"rental_cost\": 12\n },\n {\n \"room_id\": \"S63\",\n \"classes_in_room\": [\n \"A\",\n \"B\",\n \"E\",\n \"F\"\n ],\n \"rental_cost\": 28\n },\n {\n \"room_id\": \"S64\",\n \"classes_in_room\": [\n \"A\",\n \"B\",\n \"C\",\n \"F\"\n ],\n \"rental_cost\": 32\n },\n {\n \"room_id\": \"S65\",\n \"classes_in_room\": [\n \"H\",\n \"K\",\n \"N\"\n ],\n \"rental_cost\": 222\n },\n {\n \"room_id\": \"S66\",\n \"classes_in_room\": [\n \"T\",\n \"U\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"rental_cost\": 265\n },\n {\n \"room_id\": \"S67\",\n \"classes_in_room\": [\n \"A\",\n \"C\",\n \"D\",\n \"E\"\n ],\n \"rental_cost\": 348\n },\n {\n \"room_id\": \"S68\",\n \"classes_in_room\": [\n \"L\",\n \"O\",\n \"P\",\n \"Q\",\n \"R\"\n ],\n \"rental_cost\": 450\n },\n {\n \"room_id\": \"S69\",\n \"classes_in_room\": [\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\"\n ],\n \"rental_cost\": 455\n },\n {\n \"room_id\": \"S70\",\n \"classes_in_room\": [\n \"O\",\n \"R\",\n \"T\"\n ],\n \"rental_cost\": 216\n },\n {\n \"room_id\": \"S71\",\n \"classes_in_room\": [\n \"K\",\n \"L\",\n \"O\",\n \"P\",\n \"Q\"\n ],\n \"rental_cost\": 60\n },\n {\n \"room_id\": \"S72\",\n \"classes_in_room\": [\n \"B\",\n \"C\",\n \"D\",\n \"F\",\n \"H\"\n ],\n \"rental_cost\": 490\n },\n {\n \"room_id\": \"S73\",\n \"classes_in_room\": [\n \"T\",\n \"U\",\n \"V\",\n \"X\",\n \"Y\"\n ],\n \"rental_cost\": 10\n },\n {\n \"room_id\": \"S74\",\n \"classes_in_room\": [\n \"D\",\n \"G\",\n \"K\",\n \"O\"\n ],\n \"rental_cost\": 128\n },\n {\n \"room_id\": \"S75\",\n \"classes_in_room\": [\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"rental_cost\": 276\n },\n {\n \"room_id\": \"S76\",\n \"classes_in_room\": [\n \"H\",\n \"K\",\n \"L\",\n \"R\"\n ],\n \"rental_cost\": 244\n },\n {\n \"room_id\": \"S77\",\n \"classes_in_room\": [\n \"G\",\n \"H\",\n \"J\",\n \"K\",\n \"L\",\n \"N\"\n ],\n \"rental_cost\": 204\n },\n {\n \"room_id\": \"S78\",\n \"classes_in_room\": [\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\"\n ],\n \"rental_cost\": 160\n },\n {\n \"room_id\": \"S79\",\n \"classes_in_room\": [\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"rental_cost\": 475\n },\n {\n \"room_id\": \"S80\",\n \"classes_in_room\": [\n \"C\",\n \"D\",\n \"E\",\n \"G\",\n \"H\"\n ],\n \"rental_cost\": 350\n },\n {\n \"room_id\": \"S81\",\n \"classes_in_room\": [\n \"A\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\"\n ],\n \"rental_cost\": 474\n },\n {\n \"room_id\": \"S82\",\n \"classes_in_room\": [\n \"P\",\n \"Q\",\n \"S\",\n \"U\",\n \"V\",\n \"W\",\n \"Y\"\n ],\n \"rental_cost\": 21\n },\n {\n \"room_id\": \"S83\",\n \"classes_in_room\": [\n \"S\",\n \"T\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"rental_cost\": 100\n },\n {\n \"room_id\": \"S84\",\n \"classes_in_room\": [\n \"R\",\n \"S\",\n \"T\",\n \"V\"\n ],\n \"rental_cost\": 192\n },\n {\n \"room_id\": \"S85\",\n \"classes_in_room\": [\n \"M\",\n \"O\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"U\"\n ],\n \"rental_cost\": 686\n },\n {\n \"room_id\": \"S86\",\n \"classes_in_room\": [\n \"I\",\n \"L\",\n \"N\"\n ],\n \"rental_cost\": 6\n },\n {\n \"room_id\": \"S87\",\n \"classes_in_room\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"F\",\n \"G\"\n ],\n \"rental_cost\": 408\n },\n {\n \"room_id\": \"S88\",\n \"classes_in_room\": [\n \"C\",\n \"D\",\n \"F\",\n \"H\",\n \"J\",\n \"K\",\n \"M\"\n ],\n \"rental_cost\": 490\n },\n {\n \"room_id\": \"S89\",\n \"classes_in_room\": [\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"Q\"\n ],\n \"rental_cost\": 350\n },\n {\n \"room_id\": \"S90\",\n \"classes_in_room\": [\n \"A\",\n \"B\",\n \"D\",\n \"E\",\n \"F\",\n \"G\"\n ],\n \"rental_cost\": 180\n },\n {\n \"room_id\": \"S91\",\n \"classes_in_room\": [\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"P\"\n ],\n \"rental_cost\": 406\n },\n {\n \"room_id\": \"S92\",\n \"classes_in_room\": [\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"N\"\n ],\n \"rental_cost\": 495\n },\n {\n \"room_id\": \"S93\",\n \"classes_in_room\": [\n \"D\",\n \"F\",\n \"H\",\n \"I\",\n \"J\",\n \"K\"\n ],\n \"rental_cost\": 246\n },\n {\n \"room_id\": \"S94\",\n \"classes_in_room\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\"\n ],\n \"rental_cost\": 68\n },\n {\n \"room_id\": \"S95\",\n \"classes_in_room\": [\n \"M\",\n \"N\",\n \"O\"\n ],\n \"rental_cost\": 12\n },\n {\n \"room_id\": \"S96\",\n \"classes_in_room\": [\n \"O\",\n \"Q\",\n \"S\"\n ],\n \"rental_cost\": 150\n },\n {\n \"room_id\": \"S97\",\n \"classes_in_room\": [\n \"J\",\n \"M\",\n \"Q\"\n ],\n \"rental_cost\": 96\n },\n {\n \"room_id\": \"S98\",\n \"classes_in_room\": [\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\"\n ],\n \"rental_cost\": 110\n },\n {\n \"room_id\": \"S99\",\n \"classes_in_room\": [\n \"L\",\n \"M\",\n \"O\",\n \"P\",\n \"S\"\n ],\n \"rental_cost\": 320\n },\n {\n \"room_id\": \"S100\",\n \"classes_in_room\": [\n \"A\",\n \"D\",\n \"F\",\n \"G\",\n \"J\"\n ],\n \"rental_cost\": 245\n },\n {\n \"room_id\": \"S101\",\n \"classes_in_room\": [\n \"N\",\n \"O\",\n \"P\",\n \"R\",\n \"T\",\n \"U\"\n ],\n \"rental_cost\": 444\n },\n {\n \"room_id\": \"S102\",\n \"classes_in_room\": [\n \"G\",\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\"\n ],\n \"rental_cost\": 420\n },\n {\n \"room_id\": \"S103\",\n \"classes_in_room\": [\n \"V\",\n \"X\",\n \"Y\"\n ],\n \"rental_cost\": 273\n },\n {\n \"room_id\": \"S104\",\n \"classes_in_room\": [\n \"F\",\n \"K\",\n \"L\",\n \"M\"\n ],\n \"rental_cost\": 348\n },\n {\n \"room_id\": \"S105\",\n \"classes_in_room\": [\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\"\n ],\n \"rental_cost\": 290\n },\n {\n \"room_id\": \"S106\",\n \"classes_in_room\": [\n \"K\",\n \"N\",\n \"P\",\n \"Q\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"rental_cost\": 630\n },\n {\n \"room_id\": \"S107\",\n \"classes_in_room\": [\n \"O\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"V\"\n ],\n \"rental_cost\": 511\n },\n {\n \"room_id\": \"S108\",\n \"classes_in_room\": [\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"Y\"\n ],\n \"rental_cost\": 672\n },\n {\n \"room_id\": \"S109\",\n \"classes_in_room\": [\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"V\"\n ],\n \"rental_cost\": 445\n },\n {\n \"room_id\": \"S110\",\n \"classes_in_room\": [\n \"H\",\n \"J\",\n \"L\",\n \"P\",\n \"S\"\n ],\n \"rental_cost\": 470\n },\n {\n \"room_id\": \"S111\",\n \"classes_in_room\": [\n \"I\",\n \"J\",\n \"L\"\n ],\n \"rental_cost\": 135\n },\n {\n \"room_id\": \"S112\",\n \"classes_in_room\": [\n \"J\",\n \"K\",\n \"N\",\n \"P\",\n \"Q\"\n ],\n \"rental_cost\": 15\n },\n {\n \"room_id\": \"S113\",\n \"classes_in_room\": [\n \"Q\",\n \"R\",\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"rental_cost\": 385\n },\n {\n \"room_id\": \"S114\",\n \"classes_in_room\": [\n \"E\",\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"M\"\n ],\n \"rental_cost\": 406\n },\n {\n \"room_id\": \"S115\",\n \"classes_in_room\": [\n \"G\",\n \"K\",\n \"L\",\n \"O\",\n \"P\"\n ],\n \"rental_cost\": 370\n },\n {\n \"room_id\": \"S116\",\n \"classes_in_room\": [\n \"P\",\n \"Q\",\n \"S\"\n ],\n \"rental_cost\": 102\n },\n {\n \"room_id\": \"S117\",\n \"classes_in_room\": [\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"Q\",\n \"R\"\n ],\n \"rental_cost\": 354\n },\n {\n \"room_id\": \"S118\",\n \"classes_in_room\": [\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\"\n ],\n \"rental_cost\": 240\n },\n {\n \"room_id\": \"S119\",\n \"classes_in_room\": [\n \"K\",\n \"L\",\n \"M\"\n ],\n \"rental_cost\": 54\n },\n {\n \"room_id\": \"S120\",\n \"classes_in_room\": [\n \"L\",\n \"N\",\n \"O\",\n \"P\",\n \"X\"\n ],\n \"rental_cost\": 360\n },\n {\n \"room_id\": \"S121\",\n \"classes_in_room\": [\n \"G\",\n \"I\",\n \"J\"\n ],\n \"rental_cost\": 21\n },\n {\n \"room_id\": \"S122\",\n \"classes_in_room\": [\n \"J\",\n \"K\",\n \"M\",\n \"N\",\n \"O\",\n \"Q\"\n ],\n \"rental_cost\": 414\n },\n {\n \"room_id\": \"S123\",\n \"classes_in_room\": [\n \"V\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"rental_cost\": 60\n },\n {\n \"room_id\": \"S124\",\n \"classes_in_room\": [\n \"D\",\n \"H\",\n \"J\",\n \"K\",\n \"L\"\n ],\n \"rental_cost\": 465\n },\n {\n \"room_id\": \"S125\",\n \"classes_in_room\": [\n \"D\",\n \"F\",\n \"H\",\n \"I\",\n \"M\"\n ],\n \"rental_cost\": 25\n },\n {\n \"room_id\": \"S126\",\n \"classes_in_room\": [\n \"I\",\n \"J\",\n \"M\",\n \"N\",\n \"O\"\n ],\n \"rental_cost\": 135\n },\n {\n \"room_id\": \"S127\",\n \"classes_in_room\": [\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\"\n ],\n \"rental_cost\": 56\n },\n {\n \"room_id\": \"S128\",\n \"classes_in_room\": [\n \"O\",\n \"P\",\n \"Q\",\n \"S\",\n \"T\"\n ],\n \"rental_cost\": 175\n },\n {\n \"room_id\": \"S129\",\n \"classes_in_room\": [\n \"M\",\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\"\n ],\n \"rental_cost\": 588\n },\n {\n \"room_id\": \"S130\",\n \"classes_in_room\": [\n \"D\",\n \"G\",\n \"H\"\n ],\n \"rental_cost\": 294\n },\n {\n \"room_id\": \"S131\",\n \"classes_in_room\": [\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"R\"\n ],\n \"rental_cost\": 84\n },\n {\n \"room_id\": \"S132\",\n \"classes_in_room\": [\n \"T\",\n \"V\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"rental_cost\": 5\n },\n {\n \"room_id\": \"S133\",\n \"classes_in_room\": [\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"rental_cost\": 385\n },\n {\n \"room_id\": \"S134\",\n \"classes_in_room\": [\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\"\n ],\n \"rental_cost\": 616\n },\n {\n \"room_id\": \"S135\",\n \"classes_in_room\": [\n \"K\",\n \"L\",\n \"O\"\n ],\n \"rental_cost\": 252\n },\n {\n \"room_id\": \"S136\",\n \"classes_in_room\": [\n \"E\",\n \"F\",\n \"H\",\n \"K\"\n ],\n \"rental_cost\": 336\n },\n {\n \"room_id\": \"S137\",\n \"classes_in_room\": [\n \"A\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S138\",\n \"classes_in_room\": [\n \"B\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S139\",\n \"classes_in_room\": [\n \"C\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S140\",\n \"classes_in_room\": [\n \"D\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S141\",\n \"classes_in_room\": [\n \"E\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S142\",\n \"classes_in_room\": [\n \"F\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S143\",\n \"classes_in_room\": [\n \"G\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S144\",\n \"classes_in_room\": [\n \"H\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S145\",\n \"classes_in_room\": [\n \"I\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S146\",\n \"classes_in_room\": [\n \"J\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S147\",\n \"classes_in_room\": [\n \"K\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S148\",\n \"classes_in_room\": [\n \"L\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S149\",\n \"classes_in_room\": [\n \"M\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S150\",\n \"classes_in_room\": [\n \"N\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S151\",\n \"classes_in_room\": [\n \"O\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S152\",\n \"classes_in_room\": [\n \"P\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S153\",\n \"classes_in_room\": [\n \"Q\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S154\",\n \"classes_in_room\": [\n \"R\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S155\",\n \"classes_in_room\": [\n \"S\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S156\",\n \"classes_in_room\": [\n \"T\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S157\",\n \"classes_in_room\": [\n \"U\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S158\",\n \"classes_in_room\": [\n \"V\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S159\",\n \"classes_in_room\": [\n \"W\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S160\",\n \"classes_in_room\": [\n \"X\"\n ],\n \"rental_cost\": 10000\n },\n {\n \"room_id\": \"S161\",\n \"classes_in_room\": [\n \"Y\"\n ],\n \"rental_cost\": 10000\n }\n ]\n}\n\nWhen you tell me which rooms you've picked, just drop them into this simple JSON shape so it's easy to parse:\n\n{\n \"solution\": [\"room_id\", ...]\n}\n\nJust to keep it friendly: \"solution\" is the list of rooms you're going to rent (one entry per chosen room). Each \"room_id\" in that list should be the exact identifier from the input — it's just a placeholder here to show the shape, not the actual answer.\n\nPlease use the identifiers exactly as they appear in the instance input — do not rename them or invent new labels. For example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 25, "num_sets": 161, "density": 0.17763975155279504, "sets": [ { "id": 1, "elements": [ 10, 12, 13, 15, 18, 19 ], "cost": 576 }, { "id": 2, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 282 }, { "id": 3, "elements": [ 14, 17, 18, 19 ], "cost": 228 }, { "id": 4, "elements": [ 13, 16, 17, 18, 19, 24 ], "cost": 144 }, { "id": 5, "elements": [ 10, 11, 14, 15, 16 ], "cost": 395 }, { "id": 6, "elements": [ 12, 13, 14, 15 ], "cost": 368 }, { "id": 7, "elements": [ 3, 4, 5, 8, 9 ], "cost": 355 }, { "id": 8, "elements": [ 10, 11, 12, 13, 14 ], "cost": 330 }, { "id": 9, "elements": [ 5, 6, 7, 9 ], "cost": 296 }, { "id": 10, "elements": [ 14, 15, 18, 21 ], "cost": 368 }, { "id": 11, "elements": [ 12, 13, 15, 16, 17, 18 ], "cost": 360 }, { "id": 12, "elements": [ 8, 9, 12, 13, 14, 15 ], "cost": 390 }, { "id": 13, "elements": [ 1, 2, 3, 4, 5, 8, 9 ], "cost": 469 }, { "id": 14, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 154 }, { "id": 15, "elements": [ 16, 19, 20, 21, 22 ], "cost": 365 }, { "id": 16, "elements": [ 1, 4, 6 ], "cost": 102 }, { "id": 17, "elements": [ 19, 21, 22, 23, 24, 25 ], "cost": 234 }, { "id": 18, "elements": [ 18, 20, 21, 22, 24 ], "cost": 300 }, { "id": 19, "elements": [ 17, 19, 23, 25 ], "cost": 180 }, { "id": 20, "elements": [ 12, 16, 18 ], "cost": 63 }, { "id": 21, "elements": [ 16, 17, 18, 19, 20, 23, 25 ], "cost": 84 }, { "id": 22, "elements": [ 12, 15, 16, 17, 18, 19, 21 ], "cost": 609 }, { "id": 23, "elements": [ 12, 15, 16, 17, 19 ], "cost": 260 }, { "id": 24, "elements": [ 9, 11, 12, 14 ], "cost": 200 }, { "id": 25, "elements": [ 19, 22, 23, 24, 25 ], "cost": 65 }, { "id": 26, "elements": [ 8, 10, 11, 13 ], "cost": 208 }, { "id": 27, "elements": [ 1, 2, 3, 4, 5 ], "cost": 330 }, { "id": 28, "elements": [ 19, 20, 21 ], "cost": 117 }, { "id": 29, "elements": [ 15, 17, 18, 19, 20 ], "cost": 240 }, { "id": 30, "elements": [ 4, 5, 6, 9 ], "cost": 356 }, { "id": 31, "elements": [ 18, 19, 20, 21, 22, 24 ], "cost": 450 }, { "id": 32, "elements": [ 14, 17, 18, 19, 20 ], "cost": 25 }, { "id": 33, "elements": [ 21, 22, 23, 24, 25 ], "cost": 335 }, { "id": 34, "elements": [ 10, 11, 16, 18 ], "cost": 236 }, { "id": 35, "elements": [ 8, 9, 10, 11, 12, 13, 14 ], "cost": 224 }, { "id": 36, "elements": [ 8, 9, 10, 15 ], "cost": 108 }, { "id": 37, "elements": [ 12, 13, 15, 16, 18, 20 ], "cost": 522 }, { "id": 38, "elements": [ 11, 12, 14, 15, 17 ], "cost": 460 }, { "id": 39, "elements": [ 13, 16, 17 ], "cost": 207 }, { "id": 40, "elements": [ 15, 18, 19, 20 ], "cost": 128 }, { "id": 41, "elements": [ 13, 15, 16 ], "cost": 129 }, { "id": 42, "elements": [ 2, 3, 5, 6, 8, 9 ], "cost": 366 }, { "id": 43, "elements": [ 3, 5, 6, 9, 10 ], "cost": 420 }, { "id": 44, "elements": [ 13, 17, 20, 21, 22 ], "cost": 115 }, { "id": 45, "elements": [ 2, 3, 4, 5, 6, 7, 9 ], "cost": 14 }, { "id": 46, "elements": [ 17, 19, 24, 25 ], "cost": 280 }, { "id": 47, "elements": [ 2, 3, 4, 5, 8 ], "cost": 295 }, { "id": 48, "elements": [ 18, 20, 22, 23, 24, 25 ], "cost": 492 }, { "id": 49, "elements": [ 10, 11, 12, 13, 14, 15, 16 ], "cost": 238 }, { "id": 50, "elements": [ 13, 16, 17, 19 ], "cost": 8 }, { "id": 51, "elements": [ 8, 9, 11, 13 ], "cost": 208 }, { "id": 52, "elements": [ 19, 20, 22, 23, 24, 25 ], "cost": 108 }, { "id": 53, "elements": [ 13, 14, 15, 16, 17 ], "cost": 345 }, { "id": 54, "elements": [ 4, 5, 6, 7, 8, 9, 10 ], "cost": 511 }, { "id": 55, "elements": [ 16, 18, 21 ], "cost": 66 }, { "id": 56, "elements": [ 16, 17, 18, 19 ], "cost": 264 }, { "id": 57, "elements": [ 17, 19, 20, 21, 22, 24, 25 ], "cost": 490 }, { "id": 58, "elements": [ 1, 2, 3, 4, 6, 7, 9 ], "cost": 378 }, { "id": 59, "elements": [ 9, 12, 13, 17 ], "cost": 340 }, { "id": 60, "elements": [ 2, 3, 4, 5, 6, 7, 8 ], "cost": 7 }, { "id": 61, "elements": [ 7, 8, 9, 10, 11, 12 ], "cost": 120 }, { "id": 62, "elements": [ 19, 22, 23, 24 ], "cost": 12 }, { "id": 63, "elements": [ 1, 2, 5, 6 ], "cost": 28 }, { "id": 64, "elements": [ 1, 2, 3, 6 ], "cost": 32 }, { "id": 65, "elements": [ 8, 11, 14 ], "cost": 222 }, { "id": 66, "elements": [ 20, 21, 23, 24, 25 ], "cost": 265 }, { "id": 67, "elements": [ 1, 3, 4, 5 ], "cost": 348 }, { "id": 68, "elements": [ 12, 15, 16, 17, 18 ], "cost": 450 }, { "id": 69, "elements": [ 2, 3, 4, 5, 6 ], "cost": 455 }, { "id": 70, "elements": [ 15, 18, 20 ], "cost": 216 }, { "id": 71, "elements": [ 11, 12, 15, 16, 17 ], "cost": 60 }, { "id": 72, "elements": [ 2, 3, 4, 6, 8 ], "cost": 490 }, { "id": 73, "elements": [ 20, 21, 22, 24, 25 ], "cost": 10 }, { "id": 74, "elements": [ 4, 7, 11, 15 ], "cost": 128 }, { "id": 75, "elements": [ 20, 21, 22, 23, 24, 25 ], "cost": 276 }, { "id": 76, "elements": [ 8, 11, 12, 18 ], "cost": 244 }, { "id": 77, "elements": [ 7, 8, 10, 11, 12, 14 ], "cost": 204 }, { "id": 78, "elements": [ 16, 17, 18, 19, 20 ], "cost": 160 }, { "id": 79, "elements": [ 20, 21, 22, 23, 24 ], "cost": 475 }, { "id": 80, "elements": [ 3, 4, 5, 7, 8 ], "cost": 350 }, { "id": 81, "elements": [ 1, 3, 4, 5, 6, 7 ], "cost": 474 }, { "id": 82, "elements": [ 16, 17, 19, 21, 22, 23, 25 ], "cost": 21 }, { "id": 83, "elements": [ 19, 20, 23, 24, 25 ], "cost": 100 }, { "id": 84, "elements": [ 18, 19, 20, 22 ], "cost": 192 }, { "id": 85, "elements": [ 13, 15, 16, 17, 18, 19, 21 ], "cost": 686 }, { "id": 86, "elements": [ 9, 12, 14 ], "cost": 6 }, { "id": 87, "elements": [ 1, 2, 3, 4, 6, 7 ], "cost": 408 }, { "id": 88, "elements": [ 3, 4, 6, 8, 10, 11, 13 ], "cost": 490 }, { "id": 89, "elements": [ 11, 12, 13, 14, 15, 16, 17 ], "cost": 350 }, { "id": 90, "elements": [ 1, 2, 4, 5, 6, 7 ], "cost": 180 }, { "id": 91, "elements": [ 9, 10, 11, 12, 13, 14, 16 ], "cost": 406 }, { "id": 92, "elements": [ 9, 10, 11, 12, 14 ], "cost": 495 }, { "id": 93, "elements": [ 4, 6, 8, 9, 10, 11 ], "cost": 246 }, { "id": 94, "elements": [ 1, 2, 3, 4 ], "cost": 68 }, { "id": 95, "elements": [ 13, 14, 15 ], "cost": 12 }, { "id": 96, "elements": [ 15, 17, 19 ], "cost": 150 }, { "id": 97, "elements": [ 10, 13, 17 ], "cost": 96 }, { "id": 98, "elements": [ 3, 4, 5, 6, 7 ], "cost": 110 }, { "id": 99, "elements": [ 12, 13, 15, 16, 19 ], "cost": 320 }, { "id": 100, "elements": [ 1, 4, 6, 7, 10 ], "cost": 245 }, { "id": 101, "elements": [ 14, 15, 16, 18, 20, 21 ], "cost": 444 }, { "id": 102, "elements": [ 7, 10, 11, 12, 13, 14 ], "cost": 420 }, { "id": 103, "elements": [ 22, 24, 25 ], "cost": 273 }, { "id": 104, "elements": [ 6, 11, 12, 13 ], "cost": 348 }, { "id": 105, "elements": [ 18, 19, 20, 21, 22 ], "cost": 290 }, { "id": 106, "elements": [ 11, 14, 16, 17, 19, 20, 21 ], "cost": 630 }, { "id": 107, "elements": [ 15, 16, 17, 18, 19, 20, 22 ], "cost": 511 }, { "id": 108, "elements": [ 18, 19, 20, 21, 22, 23, 25 ], "cost": 672 }, { "id": 109, "elements": [ 17, 18, 19, 20, 22 ], "cost": 445 }, { "id": 110, "elements": [ 8, 10, 12, 16, 19 ], "cost": 470 }, { "id": 111, "elements": [ 9, 10, 12 ], "cost": 135 }, { "id": 112, "elements": [ 10, 11, 14, 16, 17 ], "cost": 15 }, { "id": 113, "elements": [ 17, 18, 20, 21, 22, 23, 24 ], "cost": 385 }, { "id": 114, "elements": [ 5, 7, 8, 9, 10, 11, 13 ], "cost": 406 }, { "id": 115, "elements": [ 7, 11, 12, 15, 16 ], "cost": 370 }, { "id": 116, "elements": [ 16, 17, 19 ], "cost": 102 }, { "id": 117, "elements": [ 13, 14, 15, 16, 17, 18 ], "cost": 354 }, { "id": 118, "elements": [ 18, 19, 20, 21, 22, 23 ], "cost": 240 }, { "id": 119, "elements": [ 11, 12, 13 ], "cost": 54 }, { "id": 120, "elements": [ 12, 14, 15, 16, 24 ], "cost": 360 }, { "id": 121, "elements": [ 7, 9, 10 ], "cost": 21 }, { "id": 122, "elements": [ 10, 11, 13, 14, 15, 17 ], "cost": 414 }, { "id": 123, "elements": [ 22, 23, 24, 25 ], "cost": 60 }, { "id": 124, "elements": [ 4, 8, 10, 11, 12 ], "cost": 465 }, { "id": 125, "elements": [ 4, 6, 8, 9, 13 ], "cost": 25 }, { "id": 126, "elements": [ 9, 10, 13, 14, 15 ], "cost": 135 }, { "id": 127, "elements": [ 17, 18, 19, 20, 21, 22, 23 ], "cost": 56 }, { "id": 128, "elements": [ 15, 16, 17, 19, 20 ], "cost": 175 }, { "id": 129, "elements": [ 13, 18, 19, 20, 21, 22, 23 ], "cost": 588 }, { "id": 130, "elements": [ 4, 7, 8 ], "cost": 294 }, { "id": 131, "elements": [ 11, 12, 13, 14, 15, 16, 18 ], "cost": 84 }, { "id": 132, "elements": [ 20, 22, 23, 24, 25 ], "cost": 5 }, { "id": 133, "elements": [ 19, 20, 21, 22, 23, 24, 25 ], "cost": 385 }, { "id": 134, "elements": [ 13, 14, 15, 16, 17, 18, 19 ], "cost": 616 }, { "id": 135, "elements": [ 11, 12, 15 ], "cost": 252 }, { "id": 136, "elements": [ 5, 6, 8, 11 ], "cost": 336 }, { "id": 137, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 138, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 139, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 140, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 141, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 142, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 143, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 144, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 145, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 146, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 147, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 148, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 149, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 150, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 151, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 152, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 153, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 154, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 155, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 156, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 157, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 158, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 159, "elements": [ 23 ], "cost": 10000, "singleton": true }, { "id": 160, "elements": [ 24 ], "cost": 10000, "singleton": true }, { "id": 161, "elements": [ 25 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0002_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0002_bag_stats.png" }, "solution": [ 14, 26, 55, 86, 96, 132 ], "obj": 589.0, "instance_variant": { "num_elements": 25, "num_sets": 161, "sets": [ { "id": "S1", "elements": [ "J", "L", "M", "O", "R", "S" ], "cost": 576 }, { "id": "S2", "elements": [ "A", "B", "C", "D", "E", "G" ], "cost": 282 }, { "id": "S3", "elements": [ "N", "Q", "R", "S" ], "cost": 228 }, { "id": "S4", "elements": [ "M", "P", "Q", "R", "S", "X" ], "cost": 144 }, { "id": "S5", "elements": [ "J", "K", "N", "O", "P" ], "cost": 395 }, { "id": "S6", "elements": [ "L", "M", "N", "O" ], "cost": 368 }, { "id": "S7", "elements": [ "C", "D", "E", "H", "I" ], "cost": 355 }, { "id": "S8", "elements": [ "J", "K", "L", "M", "N" ], "cost": 330 }, { "id": "S9", "elements": [ "E", "F", "G", "I" ], "cost": 296 }, { "id": "S10", "elements": [ "N", "O", "R", "U" ], "cost": 368 }, { "id": "S11", "elements": [ "L", "M", "O", "P", "Q", "R" ], "cost": 360 }, { "id": "S12", "elements": [ "H", "I", "L", "M", "N", "O" ], "cost": 390 }, { "id": "S13", "elements": [ "A", "B", "C", "D", "E", "H", "I" ], "cost": 469 }, { "id": "S14", "elements": [ "A", "B", "C", "D", "E", "F", "G" ], "cost": 154 }, { "id": "S15", "elements": [ "P", "S", "T", "U", "V" ], "cost": 365 }, { "id": "S16", "elements": [ "A", "D", "F" ], "cost": 102 }, { "id": "S17", "elements": [ "S", "U", "V", "W", "X", "Y" ], "cost": 234 }, { "id": "S18", "elements": [ "R", "T", "U", "V", "X" ], "cost": 300 }, { "id": "S19", "elements": [ "Q", "S", "W", "Y" ], "cost": 180 }, { "id": "S20", "elements": [ "L", "P", "R" ], "cost": 63 }, { "id": "S21", "elements": [ "P", "Q", "R", "S", "T", "W", "Y" ], "cost": 84 }, { "id": "S22", "elements": [ "L", "O", "P", "Q", "R", "S", "U" ], "cost": 609 }, { "id": "S23", "elements": [ "L", "O", "P", "Q", "S" ], "cost": 260 }, { "id": "S24", "elements": [ "I", "K", "L", "N" ], "cost": 200 }, { "id": "S25", "elements": [ "S", "V", "W", "X", "Y" ], "cost": 65 }, { "id": "S26", "elements": [ "H", "J", "K", "M" ], "cost": 208 }, { "id": "S27", "elements": [ "A", "B", "C", "D", "E" ], "cost": 330 }, { "id": "S28", "elements": [ "S", "T", "U" ], "cost": 117 }, { "id": "S29", "elements": [ "O", "Q", "R", "S", "T" ], "cost": 240 }, { "id": "S30", "elements": [ "D", "E", "F", "I" ], "cost": 356 }, { "id": "S31", "elements": [ "R", "S", "T", "U", "V", "X" ], "cost": 450 }, { "id": "S32", "elements": [ "N", "Q", "R", "S", "T" ], "cost": 25 }, { "id": "S33", "elements": [ "U", "V", "W", "X", "Y" ], "cost": 335 }, { "id": "S34", "elements": [ "J", "K", "P", "R" ], "cost": 236 }, { "id": "S35", "elements": [ "H", "I", "J", "K", "L", "M", "N" ], "cost": 224 }, { "id": "S36", "elements": [ "H", "I", "J", "O" ], "cost": 108 }, { "id": "S37", "elements": [ "L", "M", "O", "P", "R", "T" ], "cost": 522 }, { "id": "S38", "elements": [ "K", "L", "N", "O", "Q" ], "cost": 460 }, { "id": "S39", "elements": [ "M", "P", "Q" ], "cost": 207 }, { "id": "S40", "elements": [ "O", "R", "S", "T" ], "cost": 128 }, { "id": "S41", "elements": [ "M", "O", "P" ], "cost": 129 }, { "id": "S42", "elements": [ "B", "C", "E", "F", "H", "I" ], "cost": 366 }, { "id": "S43", "elements": [ "C", "E", "F", "I", "J" ], "cost": 420 }, { "id": "S44", "elements": [ "M", "Q", "T", "U", "V" ], "cost": 115 }, { "id": "S45", "elements": [ "B", "C", "D", "E", "F", "G", "I" ], "cost": 14 }, { "id": "S46", "elements": [ "Q", "S", "X", "Y" ], "cost": 280 }, { "id": "S47", "elements": [ "B", "C", "D", "E", "H" ], "cost": 295 }, { "id": "S48", "elements": [ "R", "T", "V", "W", "X", "Y" ], "cost": 492 }, { "id": "S49", "elements": [ "J", "K", "L", "M", "N", "O", "P" ], "cost": 238 }, { "id": "S50", "elements": [ "M", "P", "Q", "S" ], "cost": 8 }, { "id": "S51", "elements": [ "H", "I", "K", "M" ], "cost": 208 }, { "id": "S52", "elements": [ "S", "T", "V", "W", "X", "Y" ], "cost": 108 }, { "id": "S53", "elements": [ "M", "N", "O", "P", "Q" ], "cost": 345 }, { "id": "S54", "elements": [ "D", "E", "F", "G", "H", "I", "J" ], "cost": 511 }, { "id": "S55", "elements": [ "P", "R", "U" ], "cost": 66 }, { "id": "S56", "elements": [ "P", "Q", "R", "S" ], "cost": 264 }, { "id": "S57", "elements": [ "Q", "S", "T", "U", "V", "X", "Y" ], "cost": 490 }, { "id": "S58", "elements": [ "A", "B", "C", "D", "F", "G", "I" ], "cost": 378 }, { "id": "S59", "elements": [ "I", "L", "M", "Q" ], "cost": 340 }, { "id": "S60", "elements": [ "B", "C", "D", "E", "F", "G", "H" ], "cost": 7 }, { "id": "S61", "elements": [ "G", "H", "I", "J", "K", "L" ], "cost": 120 }, { "id": "S62", "elements": [ "S", "V", "W", "X" ], "cost": 12 }, { "id": "S63", "elements": [ "A", "B", "E", "F" ], "cost": 28 }, { "id": "S64", "elements": [ "A", "B", "C", "F" ], "cost": 32 }, { "id": "S65", "elements": [ "H", "K", "N" ], "cost": 222 }, { "id": "S66", "elements": [ "T", "U", "W", "X", "Y" ], "cost": 265 }, { "id": "S67", "elements": [ "A", "C", "D", "E" ], "cost": 348 }, { "id": "S68", "elements": [ "L", "O", "P", "Q", "R" ], "cost": 450 }, { "id": "S69", "elements": [ "B", "C", "D", "E", "F" ], "cost": 455 }, { "id": "S70", "elements": [ "O", "R", "T" ], "cost": 216 }, { "id": "S71", "elements": [ "K", "L", "O", "P", "Q" ], "cost": 60 }, { "id": "S72", "elements": [ "B", "C", "D", "F", "H" ], "cost": 490 }, { "id": "S73", "elements": [ "T", "U", "V", "X", "Y" ], "cost": 10 }, { "id": "S74", "elements": [ "D", "G", "K", "O" ], "cost": 128 }, { "id": "S75", "elements": [ "T", "U", "V", "W", "X", "Y" ], "cost": 276 }, { "id": "S76", "elements": [ "H", "K", "L", "R" ], "cost": 244 }, { "id": "S77", "elements": [ "G", "H", "J", "K", "L", "N" ], "cost": 204 }, { "id": "S78", "elements": [ "P", "Q", "R", "S", "T" ], "cost": 160 }, { "id": "S79", "elements": [ "T", "U", "V", "W", "X" ], "cost": 475 }, { "id": "S80", "elements": [ "C", "D", "E", "G", "H" ], "cost": 350 }, { "id": "S81", "elements": [ "A", "C", "D", "E", "F", "G" ], "cost": 474 }, { "id": "S82", "elements": [ "P", "Q", "S", "U", "V", "W", "Y" ], "cost": 21 }, { "id": "S83", "elements": [ "S", "T", "W", "X", "Y" ], "cost": 100 }, { "id": "S84", "elements": [ "R", "S", "T", "V" ], "cost": 192 }, { "id": "S85", "elements": [ "M", "O", "P", "Q", "R", "S", "U" ], "cost": 686 }, { "id": "S86", "elements": [ "I", "L", "N" ], "cost": 6 }, { "id": "S87", "elements": [ "A", "B", "C", "D", "F", "G" ], "cost": 408 }, { "id": "S88", "elements": [ "C", "D", "F", "H", "J", "K", "M" ], "cost": 490 }, { "id": "S89", "elements": [ "K", "L", "M", "N", "O", "P", "Q" ], "cost": 350 }, { "id": "S90", "elements": [ "A", "B", "D", "E", "F", "G" ], "cost": 180 }, { "id": "S91", "elements": [ "I", "J", "K", "L", "M", "N", "P" ], "cost": 406 }, { "id": "S92", "elements": [ "I", "J", "K", "L", "N" ], "cost": 495 }, { "id": "S93", "elements": [ "D", "F", "H", "I", "J", "K" ], "cost": 246 }, { "id": "S94", "elements": [ "A", "B", "C", "D" ], "cost": 68 }, { "id": "S95", "elements": [ "M", "N", "O" ], "cost": 12 }, { "id": "S96", "elements": [ "O", "Q", "S" ], "cost": 150 }, { "id": "S97", "elements": [ "J", "M", "Q" ], "cost": 96 }, { "id": "S98", "elements": [ "C", "D", "E", "F", "G" ], "cost": 110 }, { "id": "S99", "elements": [ "L", "M", "O", "P", "S" ], "cost": 320 }, { "id": "S100", "elements": [ "A", "D", "F", "G", "J" ], "cost": 245 }, { "id": "S101", "elements": [ "N", "O", "P", "R", "T", "U" ], "cost": 444 }, { "id": "S102", "elements": [ "G", "J", "K", "L", "M", "N" ], "cost": 420 }, { "id": "S103", "elements": [ "V", "X", "Y" ], "cost": 273 }, { "id": "S104", "elements": [ "F", "K", "L", "M" ], "cost": 348 }, { "id": "S105", "elements": [ "R", "S", "T", "U", "V" ], "cost": 290 }, { "id": "S106", "elements": [ "K", "N", "P", "Q", "S", "T", "U" ], "cost": 630 }, { "id": "S107", "elements": [ "O", "P", "Q", "R", "S", "T", "V" ], "cost": 511 }, { "id": "S108", "elements": [ "R", "S", "T", "U", "V", "W", "Y" ], "cost": 672 }, { "id": "S109", "elements": [ "Q", "R", "S", "T", "V" ], "cost": 445 }, { "id": "S110", "elements": [ "H", "J", "L", "P", "S" ], "cost": 470 }, { "id": "S111", "elements": [ "I", "J", "L" ], "cost": 135 }, { "id": "S112", "elements": [ "J", "K", "N", "P", "Q" ], "cost": 15 }, { "id": "S113", "elements": [ "Q", "R", "T", "U", "V", "W", "X" ], "cost": 385 }, { "id": "S114", "elements": [ "E", "G", "H", "I", "J", "K", "M" ], "cost": 406 }, { "id": "S115", "elements": [ "G", "K", "L", "O", "P" ], "cost": 370 }, { "id": "S116", "elements": [ "P", "Q", "S" ], "cost": 102 }, { "id": "S117", "elements": [ "M", "N", "O", "P", "Q", "R" ], "cost": 354 }, { "id": "S118", "elements": [ "R", "S", "T", "U", "V", "W" ], "cost": 240 }, { "id": "S119", "elements": [ "K", "L", "M" ], "cost": 54 }, { "id": "S120", "elements": [ "L", "N", "O", "P", "X" ], "cost": 360 }, { "id": "S121", "elements": [ "G", "I", "J" ], "cost": 21 }, { "id": "S122", "elements": [ "J", "K", "M", "N", "O", "Q" ], "cost": 414 }, { "id": "S123", "elements": [ "V", "W", "X", "Y" ], "cost": 60 }, { "id": "S124", "elements": [ "D", "H", "J", "K", "L" ], "cost": 465 }, { "id": "S125", "elements": [ "D", "F", "H", "I", "M" ], "cost": 25 }, { "id": "S126", "elements": [ "I", "J", "M", "N", "O" ], "cost": 135 }, { "id": "S127", "elements": [ "Q", "R", "S", "T", "U", "V", "W" ], "cost": 56 }, { "id": "S128", "elements": [ "O", "P", "Q", "S", "T" ], "cost": 175 }, { "id": "S129", "elements": [ "M", "R", "S", "T", "U", "V", "W" ], "cost": 588 }, { "id": "S130", "elements": [ "D", "G", "H" ], "cost": 294 }, { "id": "S131", "elements": [ "K", "L", "M", "N", "O", "P", "R" ], "cost": 84 }, { "id": "S132", "elements": [ "T", "V", "W", "X", "Y" ], "cost": 5 }, { "id": "S133", "elements": [ "S", "T", "U", "V", "W", "X", "Y" ], "cost": 385 }, { "id": "S134", "elements": [ "M", "N", "O", "P", "Q", "R", "S" ], "cost": 616 }, { "id": "S135", "elements": [ "K", "L", "O" ], "cost": 252 }, { "id": "S136", "elements": [ "E", "F", "H", "K" ], "cost": 336 }, { "id": "S137", "elements": [ "A" ], "cost": 10000 }, { "id": "S138", "elements": [ "B" ], "cost": 10000 }, { "id": "S139", "elements": [ "C" ], "cost": 10000 }, { "id": "S140", "elements": [ "D" ], "cost": 10000 }, { "id": "S141", "elements": [ "E" ], "cost": 10000 }, { "id": "S142", "elements": [ "F" ], "cost": 10000 }, { "id": "S143", "elements": [ "G" ], "cost": 10000 }, { "id": "S144", "elements": [ "H" ], "cost": 10000 }, { "id": "S145", "elements": [ "I" ], "cost": 10000 }, { "id": "S146", "elements": [ "J" ], "cost": 10000 }, { "id": "S147", "elements": [ "K" ], "cost": 10000 }, { "id": "S148", "elements": [ "L" ], "cost": 10000 }, { "id": "S149", "elements": [ "M" ], "cost": 10000 }, { "id": "S150", "elements": [ "N" ], "cost": 10000 }, { "id": "S151", "elements": [ "O" ], "cost": 10000 }, { "id": "S152", "elements": [ "P" ], "cost": 10000 }, { "id": "S153", "elements": [ "Q" ], "cost": 10000 }, { "id": "S154", "elements": [ "R" ], "cost": 10000 }, { "id": "S155", "elements": [ "S" ], "cost": 10000 }, { "id": "S156", "elements": [ "T" ], "cost": 10000 }, { "id": "S157", "elements": [ "U" ], "cost": 10000 }, { "id": "S158", "elements": [ "V" ], "cost": 10000 }, { "id": "S159", "elements": [ "W" ], "cost": 10000 }, { "id": "S160", "elements": [ "X" ], "cost": 10000 }, { "id": "S161", "elements": [ "Y" ], "cost": 10000 } ] }, "solution_variant": [ "S14", "S26", "S55", "S86", "S96", "S132" ], "context_index": 3, "input_format": "json", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "We’re sorting out which warehouse zones to power up so each item gets stored in exactly one chosen zone and the total cost stays as low as it can be — simply add up the activation cost of each zone we pick. Nothing can be left unassigned or placed in two spots. Concrete details about items, zones and prices follow below.\n\n# total_products=24\n# candidate_zones_count=133\nzone_id,activation_cost,products\nS1,110,12 16\nS2,108,11 12 14 15 17 19\nS3,316,1 4 5 6\nS4,168,12 13 14 15 16 18\nS5,135,1 2 5\nS6,623,6 8 9 10 11 12 13\nS7,75,19 21 23\nS8,492,6 7 8 9 10 11\nS9,365,11 14 15 16 19\nS10,12,17 18 19 21\nS11,405,0 1 2 3 4\nS12,66,9 10 12 13 14 15\nS13,420,4 5 7 8 11\nS14,90,0 1 2 3 6 7\nS15,90,13 15 17 19 20\nS16,576,2 3 4 5 6 8\nS17,204,5 7 8 9 10 11\nS18,126,13 15 17 18 19 21\nS19,100,0 2\nS20,378,2 3 5 6 7 8 9\nS21,42,6 7 9 10 11 13 14\nS22,266,0 1 2 3 4 5 6\nS23,500,7 8 11 14 16\nS24,159,11 15 19\nS25,129,16 19 20\nS26,80,20 22\nS27,57,15 20 21\nS28,576,0 1 2 3 5 6\nS29,228,2 3 4\nS30,255,10 12 14 16 17\nS31,84,17 19 20 21 22 23\nS32,138,2 4 5\nS33,546,2 4 5 6 7 8 9\nS34,153,8 12 15\nS35,360,18 19 21 22 23\nS36,174,3 4 5 6 9 10\nS37,225,11 13 14\nS38,215,0 1 2 3 6\nS39,279,10 14 17\nS40,144,0 1 2 5\nS41,212,0 2 3 5\nS42,490,11 13 14 15 16 17 19\nS43,300,10 11 12 13 14 16\nS44,32,0 1 3 5\nS45,210,11 15 16 17 20 23\nS46,230,19 20 21 22 23\nS47,511,0 3 5 6 7 8 9\nS48,115,17 19 20 21 22\nS49,95,0 2 3 6 7\nS50,425,2 5 6 7 9\nS51,224,2 3 5 8\nS52,174,13 15 16 17 19 20\nS53,444,0 2 3 4 6 7\nS54,116,14 19 21 22\nS55,110,1 2 5 7 9\nS56,3,18 19 20\nS57,192,15 17 19\nS58,70,7 8 10 11 12 13 14\nS59,564,0 1 2 3 4 6\nS60,273,2 5 8\nS61,39,17 18 22\nS62,340,5 6 7 10\nS63,28,4 6 7 8\nS64,52,10 12 13 15\nS65,45,4 6 10\nS66,240,9 10 11 14\nS67,344,4 5 8 10\nS68,200,13 15 16 19\nS69,306,10 11 13 14 16 17\nS70,406,17 18 19 20 21 22 23\nS71,165,2 7 10\nS72,231,20 21 23\nS73,168,18 20 21 23\nS74,133,13 14 15 16 17 18 19\nS75,185,14 18 20 21 22\nS76,560,7 9 10 11 12 13 14\nS77,305,16 20 21 22 23\nS78,399,0 2 3 4 5 6 7\nS79,24,14 15 16\nS80,365,2 4 5 6 8\nS81,15,2 4 9\nS82,80,9 11 12 13 14\nS83,126,4 5 6 7 8 10\nS84,168,5 6 7 8 9 10\nS85,200,0 2 4 5 7\nS86,522,18 19 20 21 22 23\nS87,240,16 18 20 21 22\nS88,410,15 16 17 20 23\nS89,80,6 7 10 11\nS90,440,6 7 8 9 10\nS91,162,0 2 7\nS92,132,4 5 6 7 8 11\nS93,144,14 15 16 17 19 20\nS94,525,7 8 9 10 11 13 14\nS95,40,3 4 6 7\nS96,100,1 4 5 6 9\nS97,177,5 9 11\nS98,69,18 19 21\nS99,392,0 1 2 4\nS100,165,0 2 3\nS101,21,6 10 12 13 14 15 16\nS102,240,16 18 19 20 21 22\nS103,231,1 2 4\nS104,125,11 12 15 16 19\nS105,84,7 10 17\nS106,594,15 17 18 19 20 21\nS107,75,0 1 2 3 5\nS108,84,0 1 2 3 4 5\nS109,170,16 17 18 19 20\nS110,10000,0\nS111,10000,1\nS112,10000,2\nS113,10000,3\nS114,10000,4\nS115,10000,5\nS116,10000,6\nS117,10000,7\nS118,10000,8\nS119,10000,9\nS120,10000,10\nS121,10000,11\nS122,10000,12\nS123,10000,13\nS124,10000,14\nS125,10000,15\nS126,10000,16\nS127,10000,17\nS128,10000,18\nS129,10000,19\nS130,10000,20\nS131,10000,21\nS132,10000,22\nS133,10000,23\n\nAlso, when you send back the chosen zones, do it in this relaxed little JSON shape — just a list of the zone identifiers you picked:\n\n{\n \"solution\": [\"zone_id\", ...]\n}\n\n\"solution\" is the array where you put the ids of the zones to power up. The placeholder \"zone_id\" is just showing the expected form — replace it with the actual ids from the instance. This JSON is only a sketch of the shape I expect, not the final answer itself.\n\nPlease use the identifiers exactly as they appear in the instance input — no renaming, no new labels. \n- for example: \"Valid identifiers look like plain numbers such as \"1\" or \"23\", single capital letters like \"A\" or \"B\", or a capital letter followed by digits like \"A1\" or \"X7\".\"", "instance": { "problem_type": "SPP", "num_elements": 24, "num_sets": 133, "density": 0.17105263157894737, "sets": [ { "id": 1, "elements": [ 13, 17 ], "cost": 110 }, { "id": 2, "elements": [ 12, 13, 15, 16, 18, 20 ], "cost": 108 }, { "id": 3, "elements": [ 2, 5, 6, 7 ], "cost": 316 }, { "id": 4, "elements": [ 13, 14, 15, 16, 17, 19 ], "cost": 168 }, { "id": 5, "elements": [ 2, 3, 6 ], "cost": 135 }, { "id": 6, "elements": [ 7, 9, 10, 11, 12, 13, 14 ], "cost": 623 }, { "id": 7, "elements": [ 20, 22, 24 ], "cost": 75 }, { "id": 8, "elements": [ 7, 8, 9, 10, 11, 12 ], "cost": 492 }, { "id": 9, "elements": [ 12, 15, 16, 17, 20 ], "cost": 365 }, { "id": 10, "elements": [ 18, 19, 20, 22 ], "cost": 12 }, { "id": 11, "elements": [ 1, 2, 3, 4, 5 ], "cost": 405 }, { "id": 12, "elements": [ 10, 11, 13, 14, 15, 16 ], "cost": 66 }, { "id": 13, "elements": [ 5, 6, 8, 9, 12 ], "cost": 420 }, { "id": 14, "elements": [ 1, 2, 3, 4, 7, 8 ], "cost": 90 }, { "id": 15, "elements": [ 14, 16, 18, 20, 21 ], "cost": 90 }, { "id": 16, "elements": [ 3, 4, 5, 6, 7, 9 ], "cost": 576 }, { "id": 17, "elements": [ 6, 8, 9, 10, 11, 12 ], "cost": 204 }, { "id": 18, "elements": [ 14, 16, 18, 19, 20, 22 ], "cost": 126 }, { "id": 19, "elements": [ 1, 3 ], "cost": 100 }, { "id": 20, "elements": [ 3, 4, 6, 7, 8, 9, 10 ], "cost": 378 }, { "id": 21, "elements": [ 7, 8, 10, 11, 12, 14, 15 ], "cost": 42 }, { "id": 22, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 266 }, { "id": 23, "elements": [ 8, 9, 12, 15, 17 ], "cost": 500 }, { "id": 24, "elements": [ 12, 16, 20 ], "cost": 159 }, { "id": 25, "elements": [ 17, 20, 21 ], "cost": 129 }, { "id": 26, "elements": [ 21, 23 ], "cost": 80 }, { "id": 27, "elements": [ 16, 21, 22 ], "cost": 57 }, { "id": 28, "elements": [ 1, 2, 3, 4, 6, 7 ], "cost": 576 }, { "id": 29, "elements": [ 3, 4, 5 ], "cost": 228 }, { "id": 30, "elements": [ 11, 13, 15, 17, 18 ], "cost": 255 }, { "id": 31, "elements": [ 18, 20, 21, 22, 23, 24 ], "cost": 84 }, { "id": 32, "elements": [ 3, 5, 6 ], "cost": 138 }, { "id": 33, "elements": [ 3, 5, 6, 7, 8, 9, 10 ], "cost": 546 }, { "id": 34, "elements": [ 9, 13, 16 ], "cost": 153 }, { "id": 35, "elements": [ 19, 20, 22, 23, 24 ], "cost": 360 }, { "id": 36, "elements": [ 4, 5, 6, 7, 10, 11 ], "cost": 174 }, { "id": 37, "elements": [ 12, 14, 15 ], "cost": 225 }, { "id": 38, "elements": [ 1, 2, 3, 4, 7 ], "cost": 215 }, { "id": 39, "elements": [ 11, 15, 18 ], "cost": 279 }, { "id": 40, "elements": [ 1, 2, 3, 6 ], "cost": 144 }, { "id": 41, "elements": [ 1, 3, 4, 6 ], "cost": 212 }, { "id": 42, "elements": [ 12, 14, 15, 16, 17, 18, 20 ], "cost": 490 }, { "id": 43, "elements": [ 11, 12, 13, 14, 15, 17 ], "cost": 300 }, { "id": 44, "elements": [ 1, 2, 4, 6 ], "cost": 32 }, { "id": 45, "elements": [ 12, 16, 17, 18, 21, 24 ], "cost": 210 }, { "id": 46, "elements": [ 20, 21, 22, 23, 24 ], "cost": 230 }, { "id": 47, "elements": [ 1, 4, 6, 7, 8, 9, 10 ], "cost": 511 }, { "id": 48, "elements": [ 18, 20, 21, 22, 23 ], "cost": 115 }, { "id": 49, "elements": [ 1, 3, 4, 7, 8 ], "cost": 95 }, { "id": 50, "elements": [ 3, 6, 7, 8, 10 ], "cost": 425 }, { "id": 51, "elements": [ 3, 4, 6, 9 ], "cost": 224 }, { "id": 52, "elements": [ 14, 16, 17, 18, 20, 21 ], "cost": 174 }, { "id": 53, "elements": [ 1, 3, 4, 5, 7, 8 ], "cost": 444 }, { "id": 54, "elements": [ 15, 20, 22, 23 ], "cost": 116 }, { "id": 55, "elements": [ 2, 3, 6, 8, 10 ], "cost": 110 }, { "id": 56, "elements": [ 19, 20, 21 ], "cost": 3 }, { "id": 57, "elements": [ 16, 18, 20 ], "cost": 192 }, { "id": 58, "elements": [ 8, 9, 11, 12, 13, 14, 15 ], "cost": 70 }, { "id": 59, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 564 }, { "id": 60, "elements": [ 3, 6, 9 ], "cost": 273 }, { "id": 61, "elements": [ 18, 19, 23 ], "cost": 39 }, { "id": 62, "elements": [ 6, 7, 8, 11 ], "cost": 340 }, { "id": 63, "elements": [ 5, 7, 8, 9 ], "cost": 28 }, { "id": 64, "elements": [ 11, 13, 14, 16 ], "cost": 52 }, { "id": 65, "elements": [ 5, 7, 11 ], "cost": 45 }, { "id": 66, "elements": [ 10, 11, 12, 15 ], "cost": 240 }, { "id": 67, "elements": [ 5, 6, 9, 11 ], "cost": 344 }, { "id": 68, "elements": [ 14, 16, 17, 20 ], "cost": 200 }, { "id": 69, "elements": [ 11, 12, 14, 15, 17, 18 ], "cost": 306 }, { "id": 70, "elements": [ 18, 19, 20, 21, 22, 23, 24 ], "cost": 406 }, { "id": 71, "elements": [ 3, 8, 11 ], "cost": 165 }, { "id": 72, "elements": [ 21, 22, 24 ], "cost": 231 }, { "id": 73, "elements": [ 19, 21, 22, 24 ], "cost": 168 }, { "id": 74, "elements": [ 14, 15, 16, 17, 18, 19, 20 ], "cost": 133 }, { "id": 75, "elements": [ 15, 19, 21, 22, 23 ], "cost": 185 }, { "id": 76, "elements": [ 8, 10, 11, 12, 13, 14, 15 ], "cost": 560 }, { "id": 77, "elements": [ 17, 21, 22, 23, 24 ], "cost": 305 }, { "id": 78, "elements": [ 1, 3, 4, 5, 6, 7, 8 ], "cost": 399 }, { "id": 79, "elements": [ 15, 16, 17 ], "cost": 24 }, { "id": 80, "elements": [ 3, 5, 6, 7, 9 ], "cost": 365 }, { "id": 81, "elements": [ 3, 5, 10 ], "cost": 15 }, { "id": 82, "elements": [ 10, 12, 13, 14, 15 ], "cost": 80 }, { "id": 83, "elements": [ 5, 6, 7, 8, 9, 11 ], "cost": 126 }, { "id": 84, "elements": [ 6, 7, 8, 9, 10, 11 ], "cost": 168 }, { "id": 85, "elements": [ 1, 3, 5, 6, 8 ], "cost": 200 }, { "id": 86, "elements": [ 19, 20, 21, 22, 23, 24 ], "cost": 522 }, { "id": 87, "elements": [ 17, 19, 21, 22, 23 ], "cost": 240 }, { "id": 88, "elements": [ 16, 17, 18, 21, 24 ], "cost": 410 }, { "id": 89, "elements": [ 7, 8, 11, 12 ], "cost": 80 }, { "id": 90, "elements": [ 7, 8, 9, 10, 11 ], "cost": 440 }, { "id": 91, "elements": [ 1, 3, 8 ], "cost": 162 }, { "id": 92, "elements": [ 5, 6, 7, 8, 9, 12 ], "cost": 132 }, { "id": 93, "elements": [ 15, 16, 17, 18, 20, 21 ], "cost": 144 }, { "id": 94, "elements": [ 8, 9, 10, 11, 12, 14, 15 ], "cost": 525 }, { "id": 95, "elements": [ 4, 5, 7, 8 ], "cost": 40 }, { "id": 96, "elements": [ 2, 5, 6, 7, 10 ], "cost": 100 }, { "id": 97, "elements": [ 6, 10, 12 ], "cost": 177 }, { "id": 98, "elements": [ 19, 20, 22 ], "cost": 69 }, { "id": 99, "elements": [ 1, 2, 3, 5 ], "cost": 392 }, { "id": 100, "elements": [ 1, 3, 4 ], "cost": 165 }, { "id": 101, "elements": [ 7, 11, 13, 14, 15, 16, 17 ], "cost": 21 }, { "id": 102, "elements": [ 17, 19, 20, 21, 22, 23 ], "cost": 240 }, { "id": 103, "elements": [ 2, 3, 5 ], "cost": 231 }, { "id": 104, "elements": [ 12, 13, 16, 17, 20 ], "cost": 125 }, { "id": 105, "elements": [ 8, 11, 18 ], "cost": 84 }, { "id": 106, "elements": [ 16, 18, 19, 20, 21, 22 ], "cost": 594 }, { "id": 107, "elements": [ 1, 2, 3, 4, 6 ], "cost": 75 }, { "id": 108, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 84 }, { "id": 109, "elements": [ 17, 18, 19, 20, 21 ], "cost": 170 }, { "id": 110, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 118, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 119, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 120, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 121, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 122, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 123, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 124, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 125, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 126, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 127, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 128, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 129, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 130, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 131, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 132, "elements": [ 23 ], "cost": 10000, "singleton": true }, { "id": 133, "elements": [ 24 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0003_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0003_bag_stats.png" }, "solution": [ 4, 8, 31, 108 ], "obj": 828.0, "instance_variant": { "num_elements": 24, "num_sets": 133, "sets": [ { "id": "S1", "elements": [ 12, 16 ], "cost": 110 }, { "id": "S2", "elements": [ 11, 12, 14, 15, 17, 19 ], "cost": 108 }, { "id": "S3", "elements": [ 1, 4, 5, 6 ], "cost": 316 }, { "id": "S4", "elements": [ 12, 13, 14, 15, 16, 18 ], "cost": 168 }, { "id": "S5", "elements": [ 1, 2, 5 ], "cost": 135 }, { "id": "S6", "elements": [ 6, 8, 9, 10, 11, 12, 13 ], "cost": 623 }, { "id": "S7", "elements": [ 19, 21, 23 ], "cost": 75 }, { "id": "S8", "elements": [ 6, 7, 8, 9, 10, 11 ], "cost": 492 }, { "id": "S9", "elements": [ 11, 14, 15, 16, 19 ], "cost": 365 }, { "id": "S10", "elements": [ 17, 18, 19, 21 ], "cost": 12 }, { "id": "S11", "elements": [ 0, 1, 2, 3, 4 ], "cost": 405 }, { "id": "S12", "elements": [ 9, 10, 12, 13, 14, 15 ], "cost": 66 }, { "id": "S13", "elements": [ 4, 5, 7, 8, 11 ], "cost": 420 }, { "id": "S14", "elements": [ 0, 1, 2, 3, 6, 7 ], "cost": 90 }, { "id": "S15", "elements": [ 13, 15, 17, 19, 20 ], "cost": 90 }, { "id": "S16", "elements": [ 2, 3, 4, 5, 6, 8 ], "cost": 576 }, { "id": "S17", "elements": [ 5, 7, 8, 9, 10, 11 ], "cost": 204 }, { "id": "S18", "elements": [ 13, 15, 17, 18, 19, 21 ], "cost": 126 }, { "id": "S19", "elements": [ 0, 2 ], "cost": 100 }, { "id": "S20", "elements": [ 2, 3, 5, 6, 7, 8, 9 ], "cost": 378 }, { "id": "S21", "elements": [ 6, 7, 9, 10, 11, 13, 14 ], "cost": 42 }, { "id": "S22", "elements": [ 0, 1, 2, 3, 4, 5, 6 ], "cost": 266 }, { "id": "S23", "elements": [ 7, 8, 11, 14, 16 ], "cost": 500 }, { "id": "S24", "elements": [ 11, 15, 19 ], "cost": 159 }, { "id": "S25", "elements": [ 16, 19, 20 ], "cost": 129 }, { "id": "S26", "elements": [ 20, 22 ], "cost": 80 }, { "id": "S27", "elements": [ 15, 20, 21 ], "cost": 57 }, { "id": "S28", "elements": [ 0, 1, 2, 3, 5, 6 ], "cost": 576 }, { "id": "S29", "elements": [ 2, 3, 4 ], "cost": 228 }, { "id": "S30", "elements": [ 10, 12, 14, 16, 17 ], "cost": 255 }, { "id": "S31", "elements": [ 17, 19, 20, 21, 22, 23 ], "cost": 84 }, { "id": "S32", "elements": [ 2, 4, 5 ], "cost": 138 }, { "id": "S33", "elements": [ 2, 4, 5, 6, 7, 8, 9 ], "cost": 546 }, { "id": "S34", "elements": [ 8, 12, 15 ], "cost": 153 }, { "id": "S35", "elements": [ 18, 19, 21, 22, 23 ], "cost": 360 }, { "id": "S36", "elements": [ 3, 4, 5, 6, 9, 10 ], "cost": 174 }, { "id": "S37", "elements": [ 11, 13, 14 ], "cost": 225 }, { "id": "S38", "elements": [ 0, 1, 2, 3, 6 ], "cost": 215 }, { "id": "S39", "elements": [ 10, 14, 17 ], "cost": 279 }, { "id": "S40", "elements": [ 0, 1, 2, 5 ], "cost": 144 }, { "id": "S41", "elements": [ 0, 2, 3, 5 ], "cost": 212 }, { "id": "S42", "elements": [ 11, 13, 14, 15, 16, 17, 19 ], "cost": 490 }, { "id": "S43", "elements": [ 10, 11, 12, 13, 14, 16 ], "cost": 300 }, { "id": "S44", "elements": [ 0, 1, 3, 5 ], "cost": 32 }, { "id": "S45", "elements": [ 11, 15, 16, 17, 20, 23 ], "cost": 210 }, { "id": "S46", "elements": [ 19, 20, 21, 22, 23 ], "cost": 230 }, { "id": "S47", "elements": [ 0, 3, 5, 6, 7, 8, 9 ], "cost": 511 }, { "id": "S48", "elements": [ 17, 19, 20, 21, 22 ], "cost": 115 }, { "id": "S49", "elements": [ 0, 2, 3, 6, 7 ], "cost": 95 }, { "id": "S50", "elements": [ 2, 5, 6, 7, 9 ], "cost": 425 }, { "id": "S51", "elements": [ 2, 3, 5, 8 ], "cost": 224 }, { "id": "S52", "elements": [ 13, 15, 16, 17, 19, 20 ], "cost": 174 }, { "id": "S53", "elements": [ 0, 2, 3, 4, 6, 7 ], "cost": 444 }, { "id": "S54", "elements": [ 14, 19, 21, 22 ], "cost": 116 }, { "id": "S55", "elements": [ 1, 2, 5, 7, 9 ], "cost": 110 }, { "id": "S56", "elements": [ 18, 19, 20 ], "cost": 3 }, { "id": "S57", "elements": [ 15, 17, 19 ], "cost": 192 }, { "id": "S58", "elements": [ 7, 8, 10, 11, 12, 13, 14 ], "cost": 70 }, { "id": "S59", "elements": [ 0, 1, 2, 3, 4, 6 ], "cost": 564 }, { "id": "S60", "elements": [ 2, 5, 8 ], "cost": 273 }, { "id": "S61", "elements": [ 17, 18, 22 ], "cost": 39 }, { "id": "S62", "elements": [ 5, 6, 7, 10 ], "cost": 340 }, { "id": "S63", "elements": [ 4, 6, 7, 8 ], "cost": 28 }, { "id": "S64", "elements": [ 10, 12, 13, 15 ], "cost": 52 }, { "id": "S65", "elements": [ 4, 6, 10 ], "cost": 45 }, { "id": "S66", "elements": [ 9, 10, 11, 14 ], "cost": 240 }, { "id": "S67", "elements": [ 4, 5, 8, 10 ], "cost": 344 }, { "id": "S68", "elements": [ 13, 15, 16, 19 ], "cost": 200 }, { "id": "S69", "elements": [ 10, 11, 13, 14, 16, 17 ], "cost": 306 }, { "id": "S70", "elements": [ 17, 18, 19, 20, 21, 22, 23 ], "cost": 406 }, { "id": "S71", "elements": [ 2, 7, 10 ], "cost": 165 }, { "id": "S72", "elements": [ 20, 21, 23 ], "cost": 231 }, { "id": "S73", "elements": [ 18, 20, 21, 23 ], "cost": 168 }, { "id": "S74", "elements": [ 13, 14, 15, 16, 17, 18, 19 ], "cost": 133 }, { "id": "S75", "elements": [ 14, 18, 20, 21, 22 ], "cost": 185 }, { "id": "S76", "elements": [ 7, 9, 10, 11, 12, 13, 14 ], "cost": 560 }, { "id": "S77", "elements": [ 16, 20, 21, 22, 23 ], "cost": 305 }, { "id": "S78", "elements": [ 0, 2, 3, 4, 5, 6, 7 ], "cost": 399 }, { "id": "S79", "elements": [ 14, 15, 16 ], "cost": 24 }, { "id": "S80", "elements": [ 2, 4, 5, 6, 8 ], "cost": 365 }, { "id": "S81", "elements": [ 2, 4, 9 ], "cost": 15 }, { "id": "S82", "elements": [ 9, 11, 12, 13, 14 ], "cost": 80 }, { "id": "S83", "elements": [ 4, 5, 6, 7, 8, 10 ], "cost": 126 }, { "id": "S84", "elements": [ 5, 6, 7, 8, 9, 10 ], "cost": 168 }, { "id": "S85", "elements": [ 0, 2, 4, 5, 7 ], "cost": 200 }, { "id": "S86", "elements": [ 18, 19, 20, 21, 22, 23 ], "cost": 522 }, { "id": "S87", "elements": [ 16, 18, 20, 21, 22 ], "cost": 240 }, { "id": "S88", "elements": [ 15, 16, 17, 20, 23 ], "cost": 410 }, { "id": "S89", "elements": [ 6, 7, 10, 11 ], "cost": 80 }, { "id": "S90", "elements": [ 6, 7, 8, 9, 10 ], "cost": 440 }, { "id": "S91", "elements": [ 0, 2, 7 ], "cost": 162 }, { "id": "S92", "elements": [ 4, 5, 6, 7, 8, 11 ], "cost": 132 }, { "id": "S93", "elements": [ 14, 15, 16, 17, 19, 20 ], "cost": 144 }, { "id": "S94", "elements": [ 7, 8, 9, 10, 11, 13, 14 ], "cost": 525 }, { "id": "S95", "elements": [ 3, 4, 6, 7 ], "cost": 40 }, { "id": "S96", "elements": [ 1, 4, 5, 6, 9 ], "cost": 100 }, { "id": "S97", "elements": [ 5, 9, 11 ], "cost": 177 }, { "id": "S98", "elements": [ 18, 19, 21 ], "cost": 69 }, { "id": "S99", "elements": [ 0, 1, 2, 4 ], "cost": 392 }, { "id": "S100", "elements": [ 0, 2, 3 ], "cost": 165 }, { "id": "S101", "elements": [ 6, 10, 12, 13, 14, 15, 16 ], "cost": 21 }, { "id": "S102", "elements": [ 16, 18, 19, 20, 21, 22 ], "cost": 240 }, { "id": "S103", "elements": [ 1, 2, 4 ], "cost": 231 }, { "id": "S104", "elements": [ 11, 12, 15, 16, 19 ], "cost": 125 }, { "id": "S105", "elements": [ 7, 10, 17 ], "cost": 84 }, { "id": "S106", "elements": [ 15, 17, 18, 19, 20, 21 ], "cost": 594 }, { "id": "S107", "elements": [ 0, 1, 2, 3, 5 ], "cost": 75 }, { "id": "S108", "elements": [ 0, 1, 2, 3, 4, 5 ], "cost": 84 }, { "id": "S109", "elements": [ 16, 17, 18, 19, 20 ], "cost": 170 }, { "id": "S110", "elements": [ 0 ], "cost": 10000 }, { "id": "S111", "elements": [ 1 ], "cost": 10000 }, { "id": "S112", "elements": [ 2 ], "cost": 10000 }, { "id": "S113", "elements": [ 3 ], "cost": 10000 }, { "id": "S114", "elements": [ 4 ], "cost": 10000 }, { "id": "S115", "elements": [ 5 ], "cost": 10000 }, { "id": "S116", "elements": [ 6 ], "cost": 10000 }, { "id": "S117", "elements": [ 7 ], "cost": 10000 }, { "id": "S118", "elements": [ 8 ], "cost": 10000 }, { "id": "S119", "elements": [ 9 ], "cost": 10000 }, { "id": "S120", "elements": [ 10 ], "cost": 10000 }, { "id": "S121", "elements": [ 11 ], "cost": 10000 }, { "id": "S122", "elements": [ 12 ], "cost": 10000 }, { "id": "S123", "elements": [ 13 ], "cost": 10000 }, { "id": "S124", "elements": [ 14 ], "cost": 10000 }, { "id": "S125", "elements": [ 15 ], "cost": 10000 }, { "id": "S126", "elements": [ 16 ], "cost": 10000 }, { "id": "S127", "elements": [ 17 ], "cost": 10000 }, { "id": "S128", "elements": [ 18 ], "cost": 10000 }, { "id": "S129", "elements": [ 19 ], "cost": 10000 }, { "id": "S130", "elements": [ 20 ], "cost": 10000 }, { "id": "S131", "elements": [ 21 ], "cost": 10000 }, { "id": "S132", "elements": [ 22 ], "cost": 10000 }, { "id": "S133", "elements": [ 23 ], "cost": 10000 } ] }, "solution_variant": [ "S4", "S8", "S31", "S108" ], "context_index": 4, "input_format": "csv", "input_index_base": 0 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "Recently the shipping team was handed a set of pre-made parcel groupings; the task now is to pick which groupings to send so that every package is assigned to a single chosen grouping and nothing is missing or duplicated. Each grouping carries a delivery cost, and the final bill is calculated by adding up the costs of the groupings actually dispatched — the objective is to keep that final number as small as possible. The full list of groupings, which parcels they cover, and their costs is shown below.\n\n- **num_packages**: 25\n- **num_batches**: 145\n\n| batch_id | delivery_cost | package_ids |\n|---|---|---|\n| S1 | 356 | 20 22 23 25 |\n| S2 | 700 | 5 8 9 10 12 13 16 |\n| S3 | 170 | 21 22 23 24 25 |\n| S4 | 275 | 3 4 5 6 7 |\n| S5 | 342 | 1 2 3 5 6 7 |\n| S6 | 350 | 6 7 8 9 11 12 14 |\n| S7 | 126 | 11 12 16 17 18 19 |\n| S8 | 237 | 18 19 20 |\n| S9 | 68 | 12 13 14 15 |\n| S10 | 455 | 12 13 15 16 17 |\n| S11 | 630 | 14 15 16 17 18 19 23 |\n| S12 | 432 | 12 13 14 15 16 18 |\n| S13 | 441 | 15 16 18 21 22 23 24 |\n| S14 | 208 | 7 9 12 16 |\n| S15 | 588 | 1 2 3 4 5 6 7 |\n| S16 | 36 | 4 12 14 |\n| S17 | 42 | 17 18 19 20 21 23 |\n| S18 | 595 | 10 12 13 14 15 16 17 |\n| S19 | 192 | 10 11 12 14 |\n| S20 | 156 | 1 2 3 4 5 6 |\n| S21 | 132 | 10 11 16 |\n| S22 | 48 | 10 15 16 |\n| S23 | 426 | 8 14 15 16 17 20 |\n| S24 | 350 | 15 16 18 19 20 21 23 |\n| S25 | 216 | 19 20 21 22 23 25 |\n| S26 | 360 | 8 10 11 16 |\n| S27 | 560 | 13 16 17 18 19 20 22 |\n| S28 | 534 | 19 21 22 23 24 25 |\n| S29 | 576 | 20 21 22 23 24 25 |\n| S30 | 182 | 12 15 17 18 19 20 21 |\n| S31 | 152 | 2 3 7 11 |\n| S32 | 380 | 13 15 20 21 |\n| S33 | 185 | 6 8 9 10 11 |\n| S34 | 12 | 2 3 4 6 7 8 |\n| S35 | 498 | 6 7 8 9 10 11 |\n| S36 | 152 | 19 20 21 24 |\n| S37 | 180 | 17 18 19 23 25 |\n| S38 | 30 | 1 3 5 8 9 |\n| S39 | 255 | 11 12 13 |\n| S40 | 96 | 15 18 19 21 |\n| S41 | 425 | 15 16 17 18 21 |\n| S42 | 288 | 11 12 13 16 |\n| S43 | 616 | 15 19 20 21 22 24 25 |\n| S44 | 336 | 11 12 13 14 15 16 17 |\n| S45 | 65 | 13 15 17 18 20 |\n| S46 | 258 | 18 20 24 |\n| S47 | 600 | 4 6 8 9 10 13 |\n| S48 | 35 | 1 2 3 4 5 |\n| S49 | 102 | 7 9 10 |\n| S50 | 415 | 9 14 15 16 17 |\n| S51 | 528 | 1 2 3 4 5 8 |\n| S52 | 16 | 2 3 6 7 |\n| S53 | 410 | 16 17 19 20 21 |\n| S54 | 249 | 23 24 25 |\n| S55 | 308 | 1 2 3 4 |\n| S56 | 180 | 17 20 21 22 |\n| S57 | 132 | 18 19 22 23 |\n| S58 | 291 | 2 6 7 |\n| S59 | 301 | 14 15 16 18 19 20 21 |\n| S60 | 164 | 18 22 23 25 |\n| S61 | 205 | 9 10 11 12 13 |\n| S62 | 658 | 1 3 4 5 6 8 10 |\n| S63 | 92 | 14 15 16 18 |\n| S64 | 60 | 1 2 3 5 6 8 |\n| S65 | 300 | 8 9 17 |\n| S66 | 114 | 11 14 15 |\n| S67 | 255 | 16 17 18 |\n| S68 | 292 | 17 18 21 22 |\n| S69 | 238 | 17 19 20 22 23 24 25 |\n| S70 | 236 | 2 3 4 5 |\n| S71 | 165 | 2 5 6 7 10 |\n| S72 | 248 | 7 8 9 11 |\n| S73 | 60 | 20 21 23 24 |\n| S74 | 85 | 13 14 16 17 18 |\n| S75 | 120 | 3 4 5 6 9 |\n| S76 | 185 | 7 12 13 14 15 |\n| S77 | 693 | 4 6 7 8 9 10 13 |\n| S78 | 177 | 19 22 23 |\n| S79 | 325 | 1 2 3 4 7 |\n| S80 | 582 | 3 4 5 6 8 9 |\n| S81 | 3 | 15 17 19 |\n| S82 | 434 | 19 20 21 22 23 24 25 |\n| S83 | 198 | 11 12 13 14 15 17 |\n| S84 | 5 | 4 6 8 9 11 |\n| S85 | 292 | 3 5 6 7 |\n| S86 | 288 | 7 8 10 15 |\n| S87 | 255 | 11 12 13 15 18 |\n| S88 | 264 | 18 19 20 22 24 25 |\n| S89 | 539 | 13 16 17 18 21 22 24 |\n| S90 | 260 | 2 3 5 6 8 |\n| S91 | 12 | 13 15 16 18 19 22 |\n| S92 | 264 | 19 22 23 25 |\n| S93 | 60 | 6 8 9 10 14 |\n| S94 | 497 | 18 20 21 22 23 24 25 |\n| S95 | 312 | 16 17 21 23 |\n| S96 | 105 | 11 12 13 14 15 16 19 |\n| S97 | 679 | 7 8 9 10 11 12 14 |\n| S98 | 60 | 18 23 24 25 |\n| S99 | 164 | 11 13 14 19 |\n| S100 | 435 | 6 8 10 12 15 |\n| S101 | 576 | 7 9 10 11 14 15 |\n| S102 | 192 | 4 5 6 7 9 11 |\n| S103 | 490 | 9 10 12 13 14 15 16 |\n| S104 | 686 | 12 13 14 15 17 19 20 |\n| S105 | 315 | 3 6 8 9 10 11 14 |\n| S106 | 164 | 18 21 22 23 |\n| S107 | 204 | 1 2 3 4 5 7 |\n| S108 | 216 | 13 15 16 20 |\n| S109 | 78 | 11 12 13 14 15 18 |\n| S110 | 12 | 5 7 8 9 10 11 |\n| S111 | 207 | 13 17 19 |\n| S112 | 198 | 11 14 15 16 17 19 |\n| S113 | 165 | 18 20 21 22 23 |\n| S114 | 296 | 11 12 17 19 |\n| S115 | 66 | 16 17 18 19 20 21 |\n| S116 | 176 | 10 11 13 14 |\n| S117 | 205 | 16 18 19 20 23 |\n| S118 | 480 | 10 12 14 16 17 |\n| S119 | 240 | 13 14 15 18 19 21 |\n| S120 | 16 | 20 22 24 25 |\n| S121 | 10000 | 1 |\n| S122 | 10000 | 2 |\n| S123 | 10000 | 3 |\n| S124 | 10000 | 4 |\n| S125 | 10000 | 5 |\n| S126 | 10000 | 6 |\n| S127 | 10000 | 7 |\n| S128 | 10000 | 8 |\n| S129 | 10000 | 9 |\n| S130 | 10000 | 10 |\n| S131 | 10000 | 11 |\n| S132 | 10000 | 12 |\n| S133 | 10000 | 13 |\n| S134 | 10000 | 14 |\n| S135 | 10000 | 15 |\n| S136 | 10000 | 16 |\n| S137 | 10000 | 17 |\n| S138 | 10000 | 18 |\n| S139 | 10000 | 19 |\n| S140 | 10000 | 20 |\n| S141 | 10000 | 21 |\n| S142 | 10000 | 22 |\n| S143 | 10000 | 23 |\n| S144 | 10000 | 24 |\n| S145 | 10000 | 25 |\n\nAlso, when you send back which groupings we should actually dispatch, please use this simple JSON layout so it's easy to parse:\n\n{\n \"solution\": [\"grouping_id\", ...]\n}\n\n\"solution\" is just a list of the chosen parcel groupings (use the exact identifiers from the instance). This block is just a sketch of the expected shape — not the final answer itself — and each entry in the array should be the identifier of a grouping you want to pick.\n\nPlease make sure all identifiers are used exactly as they appear in the instance input — no renaming and no new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.", "instance": { "problem_type": "SPP", "num_elements": 25, "num_sets": 145, "density": 0.17544827586206896, "sets": [ { "id": 1, "elements": [ 20, 22, 23, 25 ], "cost": 356 }, { "id": 2, "elements": [ 5, 8, 9, 10, 12, 13, 16 ], "cost": 700 }, { "id": 3, "elements": [ 21, 22, 23, 24, 25 ], "cost": 170 }, { "id": 4, "elements": [ 3, 4, 5, 6, 7 ], "cost": 275 }, { "id": 5, "elements": [ 1, 2, 3, 5, 6, 7 ], "cost": 342 }, { "id": 6, "elements": [ 6, 7, 8, 9, 11, 12, 14 ], "cost": 350 }, { "id": 7, "elements": [ 11, 12, 16, 17, 18, 19 ], "cost": 126 }, { "id": 8, "elements": [ 18, 19, 20 ], "cost": 237 }, { "id": 9, "elements": [ 12, 13, 14, 15 ], "cost": 68 }, { "id": 10, "elements": [ 12, 13, 15, 16, 17 ], "cost": 455 }, { "id": 11, "elements": [ 14, 15, 16, 17, 18, 19, 23 ], "cost": 630 }, { "id": 12, "elements": [ 12, 13, 14, 15, 16, 18 ], "cost": 432 }, { "id": 13, "elements": [ 15, 16, 18, 21, 22, 23, 24 ], "cost": 441 }, { "id": 14, "elements": [ 7, 9, 12, 16 ], "cost": 208 }, { "id": 15, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 588 }, { "id": 16, "elements": [ 4, 12, 14 ], "cost": 36 }, { "id": 17, "elements": [ 17, 18, 19, 20, 21, 23 ], "cost": 42 }, { "id": 18, "elements": [ 10, 12, 13, 14, 15, 16, 17 ], "cost": 595 }, { "id": 19, "elements": [ 10, 11, 12, 14 ], "cost": 192 }, { "id": 20, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 156 }, { "id": 21, "elements": [ 10, 11, 16 ], "cost": 132 }, { "id": 22, "elements": [ 10, 15, 16 ], "cost": 48 }, { "id": 23, "elements": [ 8, 14, 15, 16, 17, 20 ], "cost": 426 }, { "id": 24, "elements": [ 15, 16, 18, 19, 20, 21, 23 ], "cost": 350 }, { "id": 25, "elements": [ 19, 20, 21, 22, 23, 25 ], "cost": 216 }, { "id": 26, "elements": [ 8, 10, 11, 16 ], "cost": 360 }, { "id": 27, "elements": [ 13, 16, 17, 18, 19, 20, 22 ], "cost": 560 }, { "id": 28, "elements": [ 19, 21, 22, 23, 24, 25 ], "cost": 534 }, { "id": 29, "elements": [ 20, 21, 22, 23, 24, 25 ], "cost": 576 }, { "id": 30, "elements": [ 12, 15, 17, 18, 19, 20, 21 ], "cost": 182 }, { "id": 31, "elements": [ 2, 3, 7, 11 ], "cost": 152 }, { "id": 32, "elements": [ 13, 15, 20, 21 ], "cost": 380 }, { "id": 33, "elements": [ 6, 8, 9, 10, 11 ], "cost": 185 }, { "id": 34, "elements": [ 2, 3, 4, 6, 7, 8 ], "cost": 12 }, { "id": 35, "elements": [ 6, 7, 8, 9, 10, 11 ], "cost": 498 }, { "id": 36, "elements": [ 19, 20, 21, 24 ], "cost": 152 }, { "id": 37, "elements": [ 17, 18, 19, 23, 25 ], "cost": 180 }, { "id": 38, "elements": [ 1, 3, 5, 8, 9 ], "cost": 30 }, { "id": 39, "elements": [ 11, 12, 13 ], "cost": 255 }, { "id": 40, "elements": [ 15, 18, 19, 21 ], "cost": 96 }, { "id": 41, "elements": [ 15, 16, 17, 18, 21 ], "cost": 425 }, { "id": 42, "elements": [ 11, 12, 13, 16 ], "cost": 288 }, { "id": 43, "elements": [ 15, 19, 20, 21, 22, 24, 25 ], "cost": 616 }, { "id": 44, "elements": [ 11, 12, 13, 14, 15, 16, 17 ], "cost": 336 }, { "id": 45, "elements": [ 13, 15, 17, 18, 20 ], "cost": 65 }, { "id": 46, "elements": [ 18, 20, 24 ], "cost": 258 }, { "id": 47, "elements": [ 4, 6, 8, 9, 10, 13 ], "cost": 600 }, { "id": 48, "elements": [ 1, 2, 3, 4, 5 ], "cost": 35 }, { "id": 49, "elements": [ 7, 9, 10 ], "cost": 102 }, { "id": 50, "elements": [ 9, 14, 15, 16, 17 ], "cost": 415 }, { "id": 51, "elements": [ 1, 2, 3, 4, 5, 8 ], "cost": 528 }, { "id": 52, "elements": [ 2, 3, 6, 7 ], "cost": 16 }, { "id": 53, "elements": [ 16, 17, 19, 20, 21 ], "cost": 410 }, { "id": 54, "elements": [ 23, 24, 25 ], "cost": 249 }, { "id": 55, "elements": [ 1, 2, 3, 4 ], "cost": 308 }, { "id": 56, "elements": [ 17, 20, 21, 22 ], "cost": 180 }, { "id": 57, "elements": [ 18, 19, 22, 23 ], "cost": 132 }, { "id": 58, "elements": [ 2, 6, 7 ], "cost": 291 }, { "id": 59, "elements": [ 14, 15, 16, 18, 19, 20, 21 ], "cost": 301 }, { "id": 60, "elements": [ 18, 22, 23, 25 ], "cost": 164 }, { "id": 61, "elements": [ 9, 10, 11, 12, 13 ], "cost": 205 }, { "id": 62, "elements": [ 1, 3, 4, 5, 6, 8, 10 ], "cost": 658 }, { "id": 63, "elements": [ 14, 15, 16, 18 ], "cost": 92 }, { "id": 64, "elements": [ 1, 2, 3, 5, 6, 8 ], "cost": 60 }, { "id": 65, "elements": [ 8, 9, 17 ], "cost": 300 }, { "id": 66, "elements": [ 11, 14, 15 ], "cost": 114 }, { "id": 67, "elements": [ 16, 17, 18 ], "cost": 255 }, { "id": 68, "elements": [ 17, 18, 21, 22 ], "cost": 292 }, { "id": 69, "elements": [ 17, 19, 20, 22, 23, 24, 25 ], "cost": 238 }, { "id": 70, "elements": [ 2, 3, 4, 5 ], "cost": 236 }, { "id": 71, "elements": [ 2, 5, 6, 7, 10 ], "cost": 165 }, { "id": 72, "elements": [ 7, 8, 9, 11 ], "cost": 248 }, { "id": 73, "elements": [ 20, 21, 23, 24 ], "cost": 60 }, { "id": 74, "elements": [ 13, 14, 16, 17, 18 ], "cost": 85 }, { "id": 75, "elements": [ 3, 4, 5, 6, 9 ], "cost": 120 }, { "id": 76, "elements": [ 7, 12, 13, 14, 15 ], "cost": 185 }, { "id": 77, "elements": [ 4, 6, 7, 8, 9, 10, 13 ], "cost": 693 }, { "id": 78, "elements": [ 19, 22, 23 ], "cost": 177 }, { "id": 79, "elements": [ 1, 2, 3, 4, 7 ], "cost": 325 }, { "id": 80, "elements": [ 3, 4, 5, 6, 8, 9 ], "cost": 582 }, { "id": 81, "elements": [ 15, 17, 19 ], "cost": 3 }, { "id": 82, "elements": [ 19, 20, 21, 22, 23, 24, 25 ], "cost": 434 }, { "id": 83, "elements": [ 11, 12, 13, 14, 15, 17 ], "cost": 198 }, { "id": 84, "elements": [ 4, 6, 8, 9, 11 ], "cost": 5 }, { "id": 85, "elements": [ 3, 5, 6, 7 ], "cost": 292 }, { "id": 86, "elements": [ 7, 8, 10, 15 ], "cost": 288 }, { "id": 87, "elements": [ 11, 12, 13, 15, 18 ], "cost": 255 }, { "id": 88, "elements": [ 18, 19, 20, 22, 24, 25 ], "cost": 264 }, { "id": 89, "elements": [ 13, 16, 17, 18, 21, 22, 24 ], "cost": 539 }, { "id": 90, "elements": [ 2, 3, 5, 6, 8 ], "cost": 260 }, { "id": 91, "elements": [ 13, 15, 16, 18, 19, 22 ], "cost": 12 }, { "id": 92, "elements": [ 19, 22, 23, 25 ], "cost": 264 }, { "id": 93, "elements": [ 6, 8, 9, 10, 14 ], "cost": 60 }, { "id": 94, "elements": [ 18, 20, 21, 22, 23, 24, 25 ], "cost": 497 }, { "id": 95, "elements": [ 16, 17, 21, 23 ], "cost": 312 }, { "id": 96, "elements": [ 11, 12, 13, 14, 15, 16, 19 ], "cost": 105 }, { "id": 97, "elements": [ 7, 8, 9, 10, 11, 12, 14 ], "cost": 679 }, { "id": 98, "elements": [ 18, 23, 24, 25 ], "cost": 60 }, { "id": 99, "elements": [ 11, 13, 14, 19 ], "cost": 164 }, { "id": 100, "elements": [ 6, 8, 10, 12, 15 ], "cost": 435 }, { "id": 101, "elements": [ 7, 9, 10, 11, 14, 15 ], "cost": 576 }, { "id": 102, "elements": [ 4, 5, 6, 7, 9, 11 ], "cost": 192 }, { "id": 103, "elements": [ 9, 10, 12, 13, 14, 15, 16 ], "cost": 490 }, { "id": 104, "elements": [ 12, 13, 14, 15, 17, 19, 20 ], "cost": 686 }, { "id": 105, "elements": [ 3, 6, 8, 9, 10, 11, 14 ], "cost": 315 }, { "id": 106, "elements": [ 18, 21, 22, 23 ], "cost": 164 }, { "id": 107, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 204 }, { "id": 108, "elements": [ 13, 15, 16, 20 ], "cost": 216 }, { "id": 109, "elements": [ 11, 12, 13, 14, 15, 18 ], "cost": 78 }, { "id": 110, "elements": [ 5, 7, 8, 9, 10, 11 ], "cost": 12 }, { "id": 111, "elements": [ 13, 17, 19 ], "cost": 207 }, { "id": 112, "elements": [ 11, 14, 15, 16, 17, 19 ], "cost": 198 }, { "id": 113, "elements": [ 18, 20, 21, 22, 23 ], "cost": 165 }, { "id": 114, "elements": [ 11, 12, 17, 19 ], "cost": 296 }, { "id": 115, "elements": [ 16, 17, 18, 19, 20, 21 ], "cost": 66 }, { "id": 116, "elements": [ 10, 11, 13, 14 ], "cost": 176 }, { "id": 117, "elements": [ 16, 18, 19, 20, 23 ], "cost": 205 }, { "id": 118, "elements": [ 10, 12, 14, 16, 17 ], "cost": 480 }, { "id": 119, "elements": [ 13, 14, 15, 18, 19, 21 ], "cost": 240 }, { "id": 120, "elements": [ 20, 22, 24, 25 ], "cost": 16 }, { "id": 121, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 122, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 123, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 124, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 125, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 126, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 127, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 128, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 129, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 130, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 131, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 132, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 133, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 134, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 135, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 136, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 137, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 138, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 139, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 140, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 141, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 142, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 143, "elements": [ 23 ], "cost": 10000, "singleton": true }, { "id": 144, "elements": [ 24 ], "cost": 10000, "singleton": true }, { "id": 145, "elements": [ 25 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0004_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0004_bag_stats.png" }, "solution": [ 33, 48, 76, 88, 95 ], "obj": 981.0, "instance_variant": { "num_elements": 25, "num_sets": 145, "sets": [ { "id": "S1", "elements": [ 20, 22, 23, 25 ], "cost": 356 }, { "id": "S2", "elements": [ 5, 8, 9, 10, 12, 13, 16 ], "cost": 700 }, { "id": "S3", "elements": [ 21, 22, 23, 24, 25 ], "cost": 170 }, { "id": "S4", "elements": [ 3, 4, 5, 6, 7 ], "cost": 275 }, { "id": "S5", "elements": [ 1, 2, 3, 5, 6, 7 ], "cost": 342 }, { "id": "S6", "elements": [ 6, 7, 8, 9, 11, 12, 14 ], "cost": 350 }, { "id": "S7", "elements": [ 11, 12, 16, 17, 18, 19 ], "cost": 126 }, { "id": "S8", "elements": [ 18, 19, 20 ], "cost": 237 }, { "id": "S9", "elements": [ 12, 13, 14, 15 ], "cost": 68 }, { "id": "S10", "elements": [ 12, 13, 15, 16, 17 ], "cost": 455 }, { "id": "S11", "elements": [ 14, 15, 16, 17, 18, 19, 23 ], "cost": 630 }, { "id": "S12", "elements": [ 12, 13, 14, 15, 16, 18 ], "cost": 432 }, { "id": "S13", "elements": [ 15, 16, 18, 21, 22, 23, 24 ], "cost": 441 }, { "id": "S14", "elements": [ 7, 9, 12, 16 ], "cost": 208 }, { "id": "S15", "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 588 }, { "id": "S16", "elements": [ 4, 12, 14 ], "cost": 36 }, { "id": "S17", "elements": [ 17, 18, 19, 20, 21, 23 ], "cost": 42 }, { "id": "S18", "elements": [ 10, 12, 13, 14, 15, 16, 17 ], "cost": 595 }, { "id": "S19", "elements": [ 10, 11, 12, 14 ], "cost": 192 }, { "id": "S20", "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 156 }, { "id": "S21", "elements": [ 10, 11, 16 ], "cost": 132 }, { "id": "S22", "elements": [ 10, 15, 16 ], "cost": 48 }, { "id": "S23", "elements": [ 8, 14, 15, 16, 17, 20 ], "cost": 426 }, { "id": "S24", "elements": [ 15, 16, 18, 19, 20, 21, 23 ], "cost": 350 }, { "id": "S25", "elements": [ 19, 20, 21, 22, 23, 25 ], "cost": 216 }, { "id": "S26", "elements": [ 8, 10, 11, 16 ], "cost": 360 }, { "id": "S27", "elements": [ 13, 16, 17, 18, 19, 20, 22 ], "cost": 560 }, { "id": "S28", "elements": [ 19, 21, 22, 23, 24, 25 ], "cost": 534 }, { "id": "S29", "elements": [ 20, 21, 22, 23, 24, 25 ], "cost": 576 }, { "id": "S30", "elements": [ 12, 15, 17, 18, 19, 20, 21 ], "cost": 182 }, { "id": "S31", "elements": [ 2, 3, 7, 11 ], "cost": 152 }, { "id": "S32", "elements": [ 13, 15, 20, 21 ], "cost": 380 }, { "id": "S33", "elements": [ 6, 8, 9, 10, 11 ], "cost": 185 }, { "id": "S34", "elements": [ 2, 3, 4, 6, 7, 8 ], "cost": 12 }, { "id": "S35", "elements": [ 6, 7, 8, 9, 10, 11 ], "cost": 498 }, { "id": "S36", "elements": [ 19, 20, 21, 24 ], "cost": 152 }, { "id": "S37", "elements": [ 17, 18, 19, 23, 25 ], "cost": 180 }, { "id": "S38", "elements": [ 1, 3, 5, 8, 9 ], "cost": 30 }, { "id": "S39", "elements": [ 11, 12, 13 ], "cost": 255 }, { "id": "S40", "elements": [ 15, 18, 19, 21 ], "cost": 96 }, { "id": "S41", "elements": [ 15, 16, 17, 18, 21 ], "cost": 425 }, { "id": "S42", "elements": [ 11, 12, 13, 16 ], "cost": 288 }, { "id": "S43", "elements": [ 15, 19, 20, 21, 22, 24, 25 ], "cost": 616 }, { "id": "S44", "elements": [ 11, 12, 13, 14, 15, 16, 17 ], "cost": 336 }, { "id": "S45", "elements": [ 13, 15, 17, 18, 20 ], "cost": 65 }, { "id": "S46", "elements": [ 18, 20, 24 ], "cost": 258 }, { "id": "S47", "elements": [ 4, 6, 8, 9, 10, 13 ], "cost": 600 }, { "id": "S48", "elements": [ 1, 2, 3, 4, 5 ], "cost": 35 }, { "id": "S49", "elements": [ 7, 9, 10 ], "cost": 102 }, { "id": "S50", "elements": [ 9, 14, 15, 16, 17 ], "cost": 415 }, { "id": "S51", "elements": [ 1, 2, 3, 4, 5, 8 ], "cost": 528 }, { "id": "S52", "elements": [ 2, 3, 6, 7 ], "cost": 16 }, { "id": "S53", "elements": [ 16, 17, 19, 20, 21 ], "cost": 410 }, { "id": "S54", "elements": [ 23, 24, 25 ], "cost": 249 }, { "id": "S55", "elements": [ 1, 2, 3, 4 ], "cost": 308 }, { "id": "S56", "elements": [ 17, 20, 21, 22 ], "cost": 180 }, { "id": "S57", "elements": [ 18, 19, 22, 23 ], "cost": 132 }, { "id": "S58", "elements": [ 2, 6, 7 ], "cost": 291 }, { "id": "S59", "elements": [ 14, 15, 16, 18, 19, 20, 21 ], "cost": 301 }, { "id": "S60", "elements": [ 18, 22, 23, 25 ], "cost": 164 }, { "id": "S61", "elements": [ 9, 10, 11, 12, 13 ], "cost": 205 }, { "id": "S62", "elements": [ 1, 3, 4, 5, 6, 8, 10 ], "cost": 658 }, { "id": "S63", "elements": [ 14, 15, 16, 18 ], "cost": 92 }, { "id": "S64", "elements": [ 1, 2, 3, 5, 6, 8 ], "cost": 60 }, { "id": "S65", "elements": [ 8, 9, 17 ], "cost": 300 }, { "id": "S66", "elements": [ 11, 14, 15 ], "cost": 114 }, { "id": "S67", "elements": [ 16, 17, 18 ], "cost": 255 }, { "id": "S68", "elements": [ 17, 18, 21, 22 ], "cost": 292 }, { "id": "S69", "elements": [ 17, 19, 20, 22, 23, 24, 25 ], "cost": 238 }, { "id": "S70", "elements": [ 2, 3, 4, 5 ], "cost": 236 }, { "id": "S71", "elements": [ 2, 5, 6, 7, 10 ], "cost": 165 }, { "id": "S72", "elements": [ 7, 8, 9, 11 ], "cost": 248 }, { "id": "S73", "elements": [ 20, 21, 23, 24 ], "cost": 60 }, { "id": "S74", "elements": [ 13, 14, 16, 17, 18 ], "cost": 85 }, { "id": "S75", "elements": [ 3, 4, 5, 6, 9 ], "cost": 120 }, { "id": "S76", "elements": [ 7, 12, 13, 14, 15 ], "cost": 185 }, { "id": "S77", "elements": [ 4, 6, 7, 8, 9, 10, 13 ], "cost": 693 }, { "id": "S78", "elements": [ 19, 22, 23 ], "cost": 177 }, { "id": "S79", "elements": [ 1, 2, 3, 4, 7 ], "cost": 325 }, { "id": "S80", "elements": [ 3, 4, 5, 6, 8, 9 ], "cost": 582 }, { "id": "S81", "elements": [ 15, 17, 19 ], "cost": 3 }, { "id": "S82", "elements": [ 19, 20, 21, 22, 23, 24, 25 ], "cost": 434 }, { "id": "S83", "elements": [ 11, 12, 13, 14, 15, 17 ], "cost": 198 }, { "id": "S84", "elements": [ 4, 6, 8, 9, 11 ], "cost": 5 }, { "id": "S85", "elements": [ 3, 5, 6, 7 ], "cost": 292 }, { "id": "S86", "elements": [ 7, 8, 10, 15 ], "cost": 288 }, { "id": "S87", "elements": [ 11, 12, 13, 15, 18 ], "cost": 255 }, { "id": "S88", "elements": [ 18, 19, 20, 22, 24, 25 ], "cost": 264 }, { "id": "S89", "elements": [ 13, 16, 17, 18, 21, 22, 24 ], "cost": 539 }, { "id": "S90", "elements": [ 2, 3, 5, 6, 8 ], "cost": 260 }, { "id": "S91", "elements": [ 13, 15, 16, 18, 19, 22 ], "cost": 12 }, { "id": "S92", "elements": [ 19, 22, 23, 25 ], "cost": 264 }, { "id": "S93", "elements": [ 6, 8, 9, 10, 14 ], "cost": 60 }, { "id": "S94", "elements": [ 18, 20, 21, 22, 23, 24, 25 ], "cost": 497 }, { "id": "S95", "elements": [ 16, 17, 21, 23 ], "cost": 312 }, { "id": "S96", "elements": [ 11, 12, 13, 14, 15, 16, 19 ], "cost": 105 }, { "id": "S97", "elements": [ 7, 8, 9, 10, 11, 12, 14 ], "cost": 679 }, { "id": "S98", "elements": [ 18, 23, 24, 25 ], "cost": 60 }, { "id": "S99", "elements": [ 11, 13, 14, 19 ], "cost": 164 }, { "id": "S100", "elements": [ 6, 8, 10, 12, 15 ], "cost": 435 }, { "id": "S101", "elements": [ 7, 9, 10, 11, 14, 15 ], "cost": 576 }, { "id": "S102", "elements": [ 4, 5, 6, 7, 9, 11 ], "cost": 192 }, { "id": "S103", "elements": [ 9, 10, 12, 13, 14, 15, 16 ], "cost": 490 }, { "id": "S104", "elements": [ 12, 13, 14, 15, 17, 19, 20 ], "cost": 686 }, { "id": "S105", "elements": [ 3, 6, 8, 9, 10, 11, 14 ], "cost": 315 }, { "id": "S106", "elements": [ 18, 21, 22, 23 ], "cost": 164 }, { "id": "S107", "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 204 }, { "id": "S108", "elements": [ 13, 15, 16, 20 ], "cost": 216 }, { "id": "S109", "elements": [ 11, 12, 13, 14, 15, 18 ], "cost": 78 }, { "id": "S110", "elements": [ 5, 7, 8, 9, 10, 11 ], "cost": 12 }, { "id": "S111", "elements": [ 13, 17, 19 ], "cost": 207 }, { "id": "S112", "elements": [ 11, 14, 15, 16, 17, 19 ], "cost": 198 }, { "id": "S113", "elements": [ 18, 20, 21, 22, 23 ], "cost": 165 }, { "id": "S114", "elements": [ 11, 12, 17, 19 ], "cost": 296 }, { "id": "S115", "elements": [ 16, 17, 18, 19, 20, 21 ], "cost": 66 }, { "id": "S116", "elements": [ 10, 11, 13, 14 ], "cost": 176 }, { "id": "S117", "elements": [ 16, 18, 19, 20, 23 ], "cost": 205 }, { "id": "S118", "elements": [ 10, 12, 14, 16, 17 ], "cost": 480 }, { "id": "S119", "elements": [ 13, 14, 15, 18, 19, 21 ], "cost": 240 }, { "id": "S120", "elements": [ 20, 22, 24, 25 ], "cost": 16 }, { "id": "S121", "elements": [ 1 ], "cost": 10000 }, { "id": "S122", "elements": [ 2 ], "cost": 10000 }, { "id": "S123", "elements": [ 3 ], "cost": 10000 }, { "id": "S124", "elements": [ 4 ], "cost": 10000 }, { "id": "S125", "elements": [ 5 ], "cost": 10000 }, { "id": "S126", "elements": [ 6 ], "cost": 10000 }, { "id": "S127", "elements": [ 7 ], "cost": 10000 }, { "id": "S128", "elements": [ 8 ], "cost": 10000 }, { "id": "S129", "elements": [ 9 ], "cost": 10000 }, { "id": "S130", "elements": [ 10 ], "cost": 10000 }, { "id": "S131", "elements": [ 11 ], "cost": 10000 }, { "id": "S132", "elements": [ 12 ], "cost": 10000 }, { "id": "S133", "elements": [ 13 ], "cost": 10000 }, { "id": "S134", "elements": [ 14 ], "cost": 10000 }, { "id": "S135", "elements": [ 15 ], "cost": 10000 }, { "id": "S136", "elements": [ 16 ], "cost": 10000 }, { "id": "S137", "elements": [ 17 ], "cost": 10000 }, { "id": "S138", "elements": [ 18 ], "cost": 10000 }, { "id": "S139", "elements": [ 19 ], "cost": 10000 }, { "id": "S140", "elements": [ 20 ], "cost": 10000 }, { "id": "S141", "elements": [ 21 ], "cost": 10000 }, { "id": "S142", "elements": [ 22 ], "cost": 10000 }, { "id": "S143", "elements": [ 23 ], "cost": 10000 }, { "id": "S144", "elements": [ 24 ], "cost": 10000 }, { "id": "S145", "elements": [ 25 ], "cost": 10000 } ] }, "solution_variant": [ "S33", "S48", "S76", "S88", "S95" ], "context_index": 5, "input_format": "markdown_table", "input_index_base": 1 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "We’ve got a patch of land with a bunch of species to place and a handful of ready-made beds to choose from, each with its own cost to prepare. The choice to make is which beds to prepare so every species is assigned to one and only one of the beds we pick — no species missing and no doubling up. The way to judge different choices is by adding up the setup fees for the beds we end up using and preferring the plan with the smallest total. The exact bed options, what they cover, and their prices are listed below.\n\n# num_species=19\n# num_beds=98\nbed_id,bed_setup_cost,species_in_bed\nS1,90,10 11 12 13 14\nS2,492,14 15 16 17 18 19\nS3,212,9 10 11 12\nS4,54,6 10 12\nS5,291,15 16 17\nS6,284,1 2 3 5\nS7,276,12 13 16\nS8,395,10 11 13 14 15\nS9,120,7 8 10 11 12\nS10,70,4 5\nS11,70,12 13 14 15 17\nS12,60,3 5 6\nS13,128,6 7\nS14,306,11 12 13 14 16 17\nS15,66,14 18\nS16,475,1 2 3 4 5\nS17,240,5 6 7 8 9 11\nS18,410,3 4 6 7 9\nS19,45,10 13 14\nS20,48,12 13 14\nS21,183,4 5 10\nS22,490,2 3 4 5 7\nS23,252,1 2 3 4\nS24,171,17 18 19\nS25,213,14 17 18\nS26,9,2 3 4\nS27,276,10 12 13 14\nS28,415,5 6 7 8 9\nS29,51,1 4 6\nS30,222,1 2 3\nS31,188,11 13 14 15\nS32,240,2 3 4 6 7\nS33,160,14 15 16 17 18\nS34,296,5 6 7 8\nS35,72,12 14 16\nS36,184,15 16 17 19\nS37,465,15 16 17 18 19\nS38,288,6 8 10\nS39,111,14 15 17\nS40,112,12 15 18 19\nS41,188,15 16 17 18\nS42,470,5 7 8 10 11\nS43,178,9 13\nS44,40,1 3 4 5 6\nS45,148,14 17\nS46,244,6 7 8 10\nS47,475,3 4 5 6 7\nS48,48,5 7 10 11\nS49,84,13 14\nS50,100,13 14 16 17 18\nS51,185,1 2 5 6 7\nS52,86,3 4\nS53,213,16 18 19\nS54,272,4 5 6 8\nS55,180,3 5\nS56,340,1 2 4 5\nS57,245,8 10 12 13 14\nS58,104,1 4 5 6\nS59,160,11 15 16 18\nS60,138,9 13 14\nS61,132,5 7 12 13\nS62,290,6 7 8 9 10\nS63,72,5 7 8 9\nS64,205,14 16 17 18 19\nS65,285,3 4 6 7 8\nS66,395,5 7 8 9 14\nS67,342,6 7 8 9 10 12\nS68,276,12 13 14 16\nS69,150,13 15 16 17 18 19\nS70,66,11 14 15\nS71,48,2 3 4 5\nS72,78,1 3 4\nS73,80,2 3 4 5 6\nS74,182,12 16\nS75,18,15 17 18\nS76,460,8 9 11 12 14\nS77,116,9 12\nS78,240,4 5 6\nS79,155,1 2 3 4 6\nS80,10000,1\nS81,10000,2\nS82,10000,3\nS83,10000,4\nS84,10000,5\nS85,10000,6\nS86,10000,7\nS87,10000,8\nS88,10000,9\nS89,10000,10\nS90,10000,11\nS91,10000,12\nS92,10000,13\nS93,10000,14\nS94,10000,15\nS95,10000,16\nS96,10000,17\nS97,10000,18\nS98,10000,19\n\nIf you want to hand me the chosen beds in a simple, machine-friendly way, just return them in a tiny JSON snippet like this:\n\n{\n \"solution\": [\"bed_id\", ...]\n}\n\n\"solution\" is the list of beds to prepare; each \"bed_id\" is a placeholder for one of the bed identifiers from the instance (replace it with the actual bed ID). This is just the expected shape — a sketch of how I want the answer formatted, not the plan itself.\n\nPlease use the exact identifiers from the instance input — don't rename them or invent new labels. \nValid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.", "instance": { "problem_type": "SPP", "num_elements": 19, "num_sets": 98, "density": 0.1761546723952739, "sets": [ { "id": 1, "elements": [ 10, 11, 12, 13, 14 ], "cost": 90 }, { "id": 2, "elements": [ 14, 15, 16, 17, 18, 19 ], "cost": 492 }, { "id": 3, "elements": [ 9, 10, 11, 12 ], "cost": 212 }, { "id": 4, "elements": [ 6, 10, 12 ], "cost": 54 }, { "id": 5, "elements": [ 15, 16, 17 ], "cost": 291 }, { "id": 6, "elements": [ 1, 2, 3, 5 ], "cost": 284 }, { "id": 7, "elements": [ 12, 13, 16 ], "cost": 276 }, { "id": 8, "elements": [ 10, 11, 13, 14, 15 ], "cost": 395 }, { "id": 9, "elements": [ 7, 8, 10, 11, 12 ], "cost": 120 }, { "id": 10, "elements": [ 4, 5 ], "cost": 70 }, { "id": 11, "elements": [ 12, 13, 14, 15, 17 ], "cost": 70 }, { "id": 12, "elements": [ 3, 5, 6 ], "cost": 60 }, { "id": 13, "elements": [ 6, 7 ], "cost": 128 }, { "id": 14, "elements": [ 11, 12, 13, 14, 16, 17 ], "cost": 306 }, { "id": 15, "elements": [ 14, 18 ], "cost": 66 }, { "id": 16, "elements": [ 1, 2, 3, 4, 5 ], "cost": 475 }, { "id": 17, "elements": [ 5, 6, 7, 8, 9, 11 ], "cost": 240 }, { "id": 18, "elements": [ 3, 4, 6, 7, 9 ], "cost": 410 }, { "id": 19, "elements": [ 10, 13, 14 ], "cost": 45 }, { "id": 20, "elements": [ 12, 13, 14 ], "cost": 48 }, { "id": 21, "elements": [ 4, 5, 10 ], "cost": 183 }, { "id": 22, "elements": [ 2, 3, 4, 5, 7 ], "cost": 490 }, { "id": 23, "elements": [ 1, 2, 3, 4 ], "cost": 252 }, { "id": 24, "elements": [ 17, 18, 19 ], "cost": 171 }, { "id": 25, "elements": [ 14, 17, 18 ], "cost": 213 }, { "id": 26, "elements": [ 2, 3, 4 ], "cost": 9 }, { "id": 27, "elements": [ 10, 12, 13, 14 ], "cost": 276 }, { "id": 28, "elements": [ 5, 6, 7, 8, 9 ], "cost": 415 }, { "id": 29, "elements": [ 1, 4, 6 ], "cost": 51 }, { "id": 30, "elements": [ 1, 2, 3 ], "cost": 222 }, { "id": 31, "elements": [ 11, 13, 14, 15 ], "cost": 188 }, { "id": 32, "elements": [ 2, 3, 4, 6, 7 ], "cost": 240 }, { "id": 33, "elements": [ 14, 15, 16, 17, 18 ], "cost": 160 }, { "id": 34, "elements": [ 5, 6, 7, 8 ], "cost": 296 }, { "id": 35, "elements": [ 12, 14, 16 ], "cost": 72 }, { "id": 36, "elements": [ 15, 16, 17, 19 ], "cost": 184 }, { "id": 37, "elements": [ 15, 16, 17, 18, 19 ], "cost": 465 }, { "id": 38, "elements": [ 6, 8, 10 ], "cost": 288 }, { "id": 39, "elements": [ 14, 15, 17 ], "cost": 111 }, { "id": 40, "elements": [ 12, 15, 18, 19 ], "cost": 112 }, { "id": 41, "elements": [ 15, 16, 17, 18 ], "cost": 188 }, { "id": 42, "elements": [ 5, 7, 8, 10, 11 ], "cost": 470 }, { "id": 43, "elements": [ 9, 13 ], "cost": 178 }, { "id": 44, "elements": [ 1, 3, 4, 5, 6 ], "cost": 40 }, { "id": 45, "elements": [ 14, 17 ], "cost": 148 }, { "id": 46, "elements": [ 6, 7, 8, 10 ], "cost": 244 }, { "id": 47, "elements": [ 3, 4, 5, 6, 7 ], "cost": 475 }, { "id": 48, "elements": [ 5, 7, 10, 11 ], "cost": 48 }, { "id": 49, "elements": [ 13, 14 ], "cost": 84 }, { "id": 50, "elements": [ 13, 14, 16, 17, 18 ], "cost": 100 }, { "id": 51, "elements": [ 1, 2, 5, 6, 7 ], "cost": 185 }, { "id": 52, "elements": [ 3, 4 ], "cost": 86 }, { "id": 53, "elements": [ 16, 18, 19 ], "cost": 213 }, { "id": 54, "elements": [ 4, 5, 6, 8 ], "cost": 272 }, { "id": 55, "elements": [ 3, 5 ], "cost": 180 }, { "id": 56, "elements": [ 1, 2, 4, 5 ], "cost": 340 }, { "id": 57, "elements": [ 8, 10, 12, 13, 14 ], "cost": 245 }, { "id": 58, "elements": [ 1, 4, 5, 6 ], "cost": 104 }, { "id": 59, "elements": [ 11, 15, 16, 18 ], "cost": 160 }, { "id": 60, "elements": [ 9, 13, 14 ], "cost": 138 }, { "id": 61, "elements": [ 5, 7, 12, 13 ], "cost": 132 }, { "id": 62, "elements": [ 6, 7, 8, 9, 10 ], "cost": 290 }, { "id": 63, "elements": [ 5, 7, 8, 9 ], "cost": 72 }, { "id": 64, "elements": [ 14, 16, 17, 18, 19 ], "cost": 205 }, { "id": 65, "elements": [ 3, 4, 6, 7, 8 ], "cost": 285 }, { "id": 66, "elements": [ 5, 7, 8, 9, 14 ], "cost": 395 }, { "id": 67, "elements": [ 6, 7, 8, 9, 10, 12 ], "cost": 342 }, { "id": 68, "elements": [ 12, 13, 14, 16 ], "cost": 276 }, { "id": 69, "elements": [ 13, 15, 16, 17, 18, 19 ], "cost": 150 }, { "id": 70, "elements": [ 11, 14, 15 ], "cost": 66 }, { "id": 71, "elements": [ 2, 3, 4, 5 ], "cost": 48 }, { "id": 72, "elements": [ 1, 3, 4 ], "cost": 78 }, { "id": 73, "elements": [ 2, 3, 4, 5, 6 ], "cost": 80 }, { "id": 74, "elements": [ 12, 16 ], "cost": 182 }, { "id": 75, "elements": [ 15, 17, 18 ], "cost": 18 }, { "id": 76, "elements": [ 8, 9, 11, 12, 14 ], "cost": 460 }, { "id": 77, "elements": [ 9, 12 ], "cost": 116 }, { "id": 78, "elements": [ 4, 5, 6 ], "cost": 240 }, { "id": 79, "elements": [ 1, 2, 3, 4, 6 ], "cost": 155 }, { "id": 80, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 81, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 82, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 83, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 84, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 85, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 86, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 87, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 88, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 89, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 90, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 91, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 92, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 93, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 94, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 95, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 96, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 97, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 98, "elements": [ 19 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0005_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0005_bag_stats.png" }, "solution": [ 1, 37, 63, 79 ], "obj": 782.0, "instance_variant": { "num_elements": 19, "num_sets": 98, "sets": [ { "id": "S1", "elements": [ 10, 11, 12, 13, 14 ], "cost": 90 }, { "id": "S2", "elements": [ 14, 15, 16, 17, 18, 19 ], "cost": 492 }, { "id": "S3", "elements": [ 9, 10, 11, 12 ], "cost": 212 }, { "id": "S4", "elements": [ 6, 10, 12 ], "cost": 54 }, { "id": "S5", "elements": [ 15, 16, 17 ], "cost": 291 }, { "id": "S6", "elements": [ 1, 2, 3, 5 ], "cost": 284 }, { "id": "S7", "elements": [ 12, 13, 16 ], "cost": 276 }, { "id": "S8", "elements": [ 10, 11, 13, 14, 15 ], "cost": 395 }, { "id": "S9", "elements": [ 7, 8, 10, 11, 12 ], "cost": 120 }, { "id": "S10", "elements": [ 4, 5 ], "cost": 70 }, { "id": "S11", "elements": [ 12, 13, 14, 15, 17 ], "cost": 70 }, { "id": "S12", "elements": [ 3, 5, 6 ], "cost": 60 }, { "id": "S13", "elements": [ 6, 7 ], "cost": 128 }, { "id": "S14", "elements": [ 11, 12, 13, 14, 16, 17 ], "cost": 306 }, { "id": "S15", "elements": [ 14, 18 ], "cost": 66 }, { "id": "S16", "elements": [ 1, 2, 3, 4, 5 ], "cost": 475 }, { "id": "S17", "elements": [ 5, 6, 7, 8, 9, 11 ], "cost": 240 }, { "id": "S18", "elements": [ 3, 4, 6, 7, 9 ], "cost": 410 }, { "id": "S19", "elements": [ 10, 13, 14 ], "cost": 45 }, { "id": "S20", "elements": [ 12, 13, 14 ], "cost": 48 }, { "id": "S21", "elements": [ 4, 5, 10 ], "cost": 183 }, { "id": "S22", "elements": [ 2, 3, 4, 5, 7 ], "cost": 490 }, { "id": "S23", "elements": [ 1, 2, 3, 4 ], "cost": 252 }, { "id": "S24", "elements": [ 17, 18, 19 ], "cost": 171 }, { "id": "S25", "elements": [ 14, 17, 18 ], "cost": 213 }, { "id": "S26", "elements": [ 2, 3, 4 ], "cost": 9 }, { "id": "S27", "elements": [ 10, 12, 13, 14 ], "cost": 276 }, { "id": "S28", "elements": [ 5, 6, 7, 8, 9 ], "cost": 415 }, { "id": "S29", "elements": [ 1, 4, 6 ], "cost": 51 }, { "id": "S30", "elements": [ 1, 2, 3 ], "cost": 222 }, { "id": "S31", "elements": [ 11, 13, 14, 15 ], "cost": 188 }, { "id": "S32", "elements": [ 2, 3, 4, 6, 7 ], "cost": 240 }, { "id": "S33", "elements": [ 14, 15, 16, 17, 18 ], "cost": 160 }, { "id": "S34", "elements": [ 5, 6, 7, 8 ], "cost": 296 }, { "id": "S35", "elements": [ 12, 14, 16 ], "cost": 72 }, { "id": "S36", "elements": [ 15, 16, 17, 19 ], "cost": 184 }, { "id": "S37", "elements": [ 15, 16, 17, 18, 19 ], "cost": 465 }, { "id": "S38", "elements": [ 6, 8, 10 ], "cost": 288 }, { "id": "S39", "elements": [ 14, 15, 17 ], "cost": 111 }, { "id": "S40", "elements": [ 12, 15, 18, 19 ], "cost": 112 }, { "id": "S41", "elements": [ 15, 16, 17, 18 ], "cost": 188 }, { "id": "S42", "elements": [ 5, 7, 8, 10, 11 ], "cost": 470 }, { "id": "S43", "elements": [ 9, 13 ], "cost": 178 }, { "id": "S44", "elements": [ 1, 3, 4, 5, 6 ], "cost": 40 }, { "id": "S45", "elements": [ 14, 17 ], "cost": 148 }, { "id": "S46", "elements": [ 6, 7, 8, 10 ], "cost": 244 }, { "id": "S47", "elements": [ 3, 4, 5, 6, 7 ], "cost": 475 }, { "id": "S48", "elements": [ 5, 7, 10, 11 ], "cost": 48 }, { "id": "S49", "elements": [ 13, 14 ], "cost": 84 }, { "id": "S50", "elements": [ 13, 14, 16, 17, 18 ], "cost": 100 }, { "id": "S51", "elements": [ 1, 2, 5, 6, 7 ], "cost": 185 }, { "id": "S52", "elements": [ 3, 4 ], "cost": 86 }, { "id": "S53", "elements": [ 16, 18, 19 ], "cost": 213 }, { "id": "S54", "elements": [ 4, 5, 6, 8 ], "cost": 272 }, { "id": "S55", "elements": [ 3, 5 ], "cost": 180 }, { "id": "S56", "elements": [ 1, 2, 4, 5 ], "cost": 340 }, { "id": "S57", "elements": [ 8, 10, 12, 13, 14 ], "cost": 245 }, { "id": "S58", "elements": [ 1, 4, 5, 6 ], "cost": 104 }, { "id": "S59", "elements": [ 11, 15, 16, 18 ], "cost": 160 }, { "id": "S60", "elements": [ 9, 13, 14 ], "cost": 138 }, { "id": "S61", "elements": [ 5, 7, 12, 13 ], "cost": 132 }, { "id": "S62", "elements": [ 6, 7, 8, 9, 10 ], "cost": 290 }, { "id": "S63", "elements": [ 5, 7, 8, 9 ], "cost": 72 }, { "id": "S64", "elements": [ 14, 16, 17, 18, 19 ], "cost": 205 }, { "id": "S65", "elements": [ 3, 4, 6, 7, 8 ], "cost": 285 }, { "id": "S66", "elements": [ 5, 7, 8, 9, 14 ], "cost": 395 }, { "id": "S67", "elements": [ 6, 7, 8, 9, 10, 12 ], "cost": 342 }, { "id": "S68", "elements": [ 12, 13, 14, 16 ], "cost": 276 }, { "id": "S69", "elements": [ 13, 15, 16, 17, 18, 19 ], "cost": 150 }, { "id": "S70", "elements": [ 11, 14, 15 ], "cost": 66 }, { "id": "S71", "elements": [ 2, 3, 4, 5 ], "cost": 48 }, { "id": "S72", "elements": [ 1, 3, 4 ], "cost": 78 }, { "id": "S73", "elements": [ 2, 3, 4, 5, 6 ], "cost": 80 }, { "id": "S74", "elements": [ 12, 16 ], "cost": 182 }, { "id": "S75", "elements": [ 15, 17, 18 ], "cost": 18 }, { "id": "S76", "elements": [ 8, 9, 11, 12, 14 ], "cost": 460 }, { "id": "S77", "elements": [ 9, 12 ], "cost": 116 }, { "id": "S78", "elements": [ 4, 5, 6 ], "cost": 240 }, { "id": "S79", "elements": [ 1, 2, 3, 4, 6 ], "cost": 155 }, { "id": "S80", "elements": [ 1 ], "cost": 10000 }, { "id": "S81", "elements": [ 2 ], "cost": 10000 }, { "id": "S82", "elements": [ 3 ], "cost": 10000 }, { "id": "S83", "elements": [ 4 ], "cost": 10000 }, { "id": "S84", "elements": [ 5 ], "cost": 10000 }, { "id": "S85", "elements": [ 6 ], "cost": 10000 }, { "id": "S86", "elements": [ 7 ], "cost": 10000 }, { "id": "S87", "elements": [ 8 ], "cost": 10000 }, { "id": "S88", "elements": [ 9 ], "cost": 10000 }, { "id": "S89", "elements": [ 10 ], "cost": 10000 }, { "id": "S90", "elements": [ 11 ], "cost": 10000 }, { "id": "S91", "elements": [ 12 ], "cost": 10000 }, { "id": "S92", "elements": [ 13 ], "cost": 10000 }, { "id": "S93", "elements": [ 14 ], "cost": 10000 }, { "id": "S94", "elements": [ 15 ], "cost": 10000 }, { "id": "S95", "elements": [ 16 ], "cost": 10000 }, { "id": "S96", "elements": [ 17 ], "cost": 10000 }, { "id": "S97", "elements": [ 18 ], "cost": 10000 }, { "id": "S98", "elements": [ 19 ], "cost": 10000 } ] }, "solution_variant": [ "S1", "S37", "S63", "S79" ], "context_index": 6, "input_format": "csv", "input_index_base": 1 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "There’s a seasonal promotion being planned where a bunch of garments need to be grouped into preset outfit bundles. The decision is which bundles to actually offer so that each garment is covered by one, and only one, chosen bundle. The winning plan is simply the one with the smallest combined price — just sum the prices of the bundles that are chosen — and every item must be included exactly once. The full list of items and bundle choices is shown below.\n\n# total_garments=19\n# total_bundles_available=94\nbundle_id,bundle_price,garments_in_bundle\nS1,180,13 14 15 18\nS2,78,2 3\nS3,261,0 1 2\nS4,225,15 17 18\nS5,183,0 3 4\nS6,54,7 8\nS7,445,0 1 2 3 6\nS8,320,7 9 10 11\nS9,192,4 5 6\nS10,64,3 4 5 7\nS11,158,1 4\nS12,198,10 11 12 13 14 15\nS13,255,6 7 8 10 11\nS14,305,0 1 2 3 4\nS15,265,2 3 4 5 7\nS16,60,9 11 12\nS17,120,10 11 14\nS18,195,7 9 10 11 13\nS19,78,5 7 9\nS20,310,12 13 14 16 17\nS21,335,13 15 16 17 18\nS22,60,1 2 4 7\nS23,24,1 2\nS24,9,0 2 3\nS25,276,7 8 9 10\nS26,45,12 13 15\nS27,90,15 17\nS28,40,6 7 8 9\nS29,261,6 7 10\nS30,46,9 12\nS31,130,14 15 16 17 18\nS32,450,13 14 16 17 18\nS33,116,13 14 15 16\nS34,180,7 11\nS35,225,6 9 10 11 12\nS36,208,4 5 6 7\nS37,265,1 2 3 4 5\nS38,132,6 7 11\nS39,132,16 17 18\nS40,252,5 7 8 9\nS41,114,11 13\nS42,297,9 10 14\nS43,485,7 10 11 12 13\nS44,316,0 1 2 3\nS45,348,4 5 7 8\nS46,225,14 16 17\nS47,76,14 15\nS48,144,11 18\nS49,405,7 9 10 12 13\nS50,69,10 11 12\nS51,148,4 5 6 8\nS52,108,0 1 2 3 4 5\nS53,145,11 12 13 14 16\nS54,195,0 1 3\nS55,186,13 14 17\nS56,485,0 2 3 5 7\nS57,425,12 13 14 15 16\nS58,12,0 1 2 3 4 6\nS59,152,14 15 16 18\nS60,475,5 6 7 8 13\nS61,132,6 7 9 11\nS62,190,6 7\nS63,170,6 8\nS64,485,10 11 12 13 14\nS65,498,9 10 11 12 13 14\nS66,272,14 15 16 17\nS67,258,2 3 4 5 6 7\nS68,261,2 3 4\nS69,26,0 2\nS70,320,10 12 13 15\nS71,182,6 9\nS72,90,0 2 3 4 5 7\nS73,495,5 7 8 9 13\nS74,200,7 8 9 12\nS75,126,4 7\nS76,10000,0\nS77,10000,1\nS78,10000,2\nS79,10000,3\nS80,10000,4\nS81,10000,5\nS82,10000,6\nS83,10000,7\nS84,10000,8\nS85,10000,9\nS86,10000,10\nS87,10000,11\nS88,10000,12\nS89,10000,13\nS90,10000,14\nS91,10000,15\nS92,10000,16\nS93,10000,17\nS94,10000,18\n\nAlso, when you send the final selection, please stick to a tiny JSON layout like this:\n\n{\n \"solution\": [\"bundle_id\", ...]\n}\n\nHere \"solution\" is the list of the outfit bundle IDs you choose — each entry should be the exact bundle identifier from the instance (the placeholder \"bundle_id\" just shows the kind of label to use). The ellipsis means you can list as many bundle IDs as needed. This is just a sketch of the shape I want, not the actual answer.\n\nPlease make sure you use the identifiers exactly as they appear in the instance input — no renaming and no new labels.\n\nFor example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 19, "num_sets": 94, "density": 0.1690929451287794, "sets": [ { "id": 1, "elements": [ 14, 15, 16, 19 ], "cost": 180 }, { "id": 2, "elements": [ 3, 4 ], "cost": 78 }, { "id": 3, "elements": [ 1, 2, 3 ], "cost": 261 }, { "id": 4, "elements": [ 16, 18, 19 ], "cost": 225 }, { "id": 5, "elements": [ 1, 4, 5 ], "cost": 183 }, { "id": 6, "elements": [ 8, 9 ], "cost": 54 }, { "id": 7, "elements": [ 1, 2, 3, 4, 7 ], "cost": 445 }, { "id": 8, "elements": [ 8, 10, 11, 12 ], "cost": 320 }, { "id": 9, "elements": [ 5, 6, 7 ], "cost": 192 }, { "id": 10, "elements": [ 4, 5, 6, 8 ], "cost": 64 }, { "id": 11, "elements": [ 2, 5 ], "cost": 158 }, { "id": 12, "elements": [ 11, 12, 13, 14, 15, 16 ], "cost": 198 }, { "id": 13, "elements": [ 7, 8, 9, 11, 12 ], "cost": 255 }, { "id": 14, "elements": [ 1, 2, 3, 4, 5 ], "cost": 305 }, { "id": 15, "elements": [ 3, 4, 5, 6, 8 ], "cost": 265 }, { "id": 16, "elements": [ 10, 12, 13 ], "cost": 60 }, { "id": 17, "elements": [ 11, 12, 15 ], "cost": 120 }, { "id": 18, "elements": [ 8, 10, 11, 12, 14 ], "cost": 195 }, { "id": 19, "elements": [ 6, 8, 10 ], "cost": 78 }, { "id": 20, "elements": [ 13, 14, 15, 17, 18 ], "cost": 310 }, { "id": 21, "elements": [ 14, 16, 17, 18, 19 ], "cost": 335 }, { "id": 22, "elements": [ 2, 3, 5, 8 ], "cost": 60 }, { "id": 23, "elements": [ 2, 3 ], "cost": 24 }, { "id": 24, "elements": [ 1, 3, 4 ], "cost": 9 }, { "id": 25, "elements": [ 8, 9, 10, 11 ], "cost": 276 }, { "id": 26, "elements": [ 13, 14, 16 ], "cost": 45 }, { "id": 27, "elements": [ 16, 18 ], "cost": 90 }, { "id": 28, "elements": [ 7, 8, 9, 10 ], "cost": 40 }, { "id": 29, "elements": [ 7, 8, 11 ], "cost": 261 }, { "id": 30, "elements": [ 10, 13 ], "cost": 46 }, { "id": 31, "elements": [ 15, 16, 17, 18, 19 ], "cost": 130 }, { "id": 32, "elements": [ 14, 15, 17, 18, 19 ], "cost": 450 }, { "id": 33, "elements": [ 14, 15, 16, 17 ], "cost": 116 }, { "id": 34, "elements": [ 8, 12 ], "cost": 180 }, { "id": 35, "elements": [ 7, 10, 11, 12, 13 ], "cost": 225 }, { "id": 36, "elements": [ 5, 6, 7, 8 ], "cost": 208 }, { "id": 37, "elements": [ 2, 3, 4, 5, 6 ], "cost": 265 }, { "id": 38, "elements": [ 7, 8, 12 ], "cost": 132 }, { "id": 39, "elements": [ 17, 18, 19 ], "cost": 132 }, { "id": 40, "elements": [ 6, 8, 9, 10 ], "cost": 252 }, { "id": 41, "elements": [ 12, 14 ], "cost": 114 }, { "id": 42, "elements": [ 10, 11, 15 ], "cost": 297 }, { "id": 43, "elements": [ 8, 11, 12, 13, 14 ], "cost": 485 }, { "id": 44, "elements": [ 1, 2, 3, 4 ], "cost": 316 }, { "id": 45, "elements": [ 5, 6, 8, 9 ], "cost": 348 }, { "id": 46, "elements": [ 15, 17, 18 ], "cost": 225 }, { "id": 47, "elements": [ 15, 16 ], "cost": 76 }, { "id": 48, "elements": [ 12, 19 ], "cost": 144 }, { "id": 49, "elements": [ 8, 10, 11, 13, 14 ], "cost": 405 }, { "id": 50, "elements": [ 11, 12, 13 ], "cost": 69 }, { "id": 51, "elements": [ 5, 6, 7, 9 ], "cost": 148 }, { "id": 52, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 108 }, { "id": 53, "elements": [ 12, 13, 14, 15, 17 ], "cost": 145 }, { "id": 54, "elements": [ 1, 2, 4 ], "cost": 195 }, { "id": 55, "elements": [ 14, 15, 18 ], "cost": 186 }, { "id": 56, "elements": [ 1, 3, 4, 6, 8 ], "cost": 485 }, { "id": 57, "elements": [ 13, 14, 15, 16, 17 ], "cost": 425 }, { "id": 58, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 12 }, { "id": 59, "elements": [ 15, 16, 17, 19 ], "cost": 152 }, { "id": 60, "elements": [ 6, 7, 8, 9, 14 ], "cost": 475 }, { "id": 61, "elements": [ 7, 8, 10, 12 ], "cost": 132 }, { "id": 62, "elements": [ 7, 8 ], "cost": 190 }, { "id": 63, "elements": [ 7, 9 ], "cost": 170 }, { "id": 64, "elements": [ 11, 12, 13, 14, 15 ], "cost": 485 }, { "id": 65, "elements": [ 10, 11, 12, 13, 14, 15 ], "cost": 498 }, { "id": 66, "elements": [ 15, 16, 17, 18 ], "cost": 272 }, { "id": 67, "elements": [ 3, 4, 5, 6, 7, 8 ], "cost": 258 }, { "id": 68, "elements": [ 3, 4, 5 ], "cost": 261 }, { "id": 69, "elements": [ 1, 3 ], "cost": 26 }, { "id": 70, "elements": [ 11, 13, 14, 16 ], "cost": 320 }, { "id": 71, "elements": [ 7, 10 ], "cost": 182 }, { "id": 72, "elements": [ 1, 3, 4, 5, 6, 8 ], "cost": 90 }, { "id": 73, "elements": [ 6, 8, 9, 10, 14 ], "cost": 495 }, { "id": 74, "elements": [ 8, 9, 10, 13 ], "cost": 200 }, { "id": 75, "elements": [ 5, 8 ], "cost": 126 }, { "id": 76, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 77, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 78, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 79, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 80, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 81, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 82, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 83, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 84, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 85, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 86, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 87, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 88, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 89, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 90, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 91, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 92, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 93, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 94, "elements": [ 19 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0006_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0006_bag_stats.png" }, "solution": [ 17, 26, 28, 39, 52 ], "obj": 445.0, "instance_variant": { "num_elements": 19, "num_sets": 94, "sets": [ { "id": "S1", "elements": [ 13, 14, 15, 18 ], "cost": 180 }, { "id": "S2", "elements": [ 2, 3 ], "cost": 78 }, { "id": "S3", "elements": [ 0, 1, 2 ], "cost": 261 }, { "id": "S4", "elements": [ 15, 17, 18 ], "cost": 225 }, { "id": "S5", "elements": [ 0, 3, 4 ], "cost": 183 }, { "id": "S6", "elements": [ 7, 8 ], "cost": 54 }, { "id": "S7", "elements": [ 0, 1, 2, 3, 6 ], "cost": 445 }, { "id": "S8", "elements": [ 7, 9, 10, 11 ], "cost": 320 }, { "id": "S9", "elements": [ 4, 5, 6 ], "cost": 192 }, { "id": "S10", "elements": [ 3, 4, 5, 7 ], "cost": 64 }, { "id": "S11", "elements": [ 1, 4 ], "cost": 158 }, { "id": "S12", "elements": [ 10, 11, 12, 13, 14, 15 ], "cost": 198 }, { "id": "S13", "elements": [ 6, 7, 8, 10, 11 ], "cost": 255 }, { "id": "S14", "elements": [ 0, 1, 2, 3, 4 ], "cost": 305 }, { "id": "S15", "elements": [ 2, 3, 4, 5, 7 ], "cost": 265 }, { "id": "S16", "elements": [ 9, 11, 12 ], "cost": 60 }, { "id": "S17", "elements": [ 10, 11, 14 ], "cost": 120 }, { "id": "S18", "elements": [ 7, 9, 10, 11, 13 ], "cost": 195 }, { "id": "S19", "elements": [ 5, 7, 9 ], "cost": 78 }, { "id": "S20", "elements": [ 12, 13, 14, 16, 17 ], "cost": 310 }, { "id": "S21", "elements": [ 13, 15, 16, 17, 18 ], "cost": 335 }, { "id": "S22", "elements": [ 1, 2, 4, 7 ], "cost": 60 }, { "id": "S23", "elements": [ 1, 2 ], "cost": 24 }, { "id": "S24", "elements": [ 0, 2, 3 ], "cost": 9 }, { "id": "S25", "elements": [ 7, 8, 9, 10 ], "cost": 276 }, { "id": "S26", "elements": [ 12, 13, 15 ], "cost": 45 }, { "id": "S27", "elements": [ 15, 17 ], "cost": 90 }, { "id": "S28", "elements": [ 6, 7, 8, 9 ], "cost": 40 }, { "id": "S29", "elements": [ 6, 7, 10 ], "cost": 261 }, { "id": "S30", "elements": [ 9, 12 ], "cost": 46 }, { "id": "S31", "elements": [ 14, 15, 16, 17, 18 ], "cost": 130 }, { "id": "S32", "elements": [ 13, 14, 16, 17, 18 ], "cost": 450 }, { "id": "S33", "elements": [ 13, 14, 15, 16 ], "cost": 116 }, { "id": "S34", "elements": [ 7, 11 ], "cost": 180 }, { "id": "S35", "elements": [ 6, 9, 10, 11, 12 ], "cost": 225 }, { "id": "S36", "elements": [ 4, 5, 6, 7 ], "cost": 208 }, { "id": "S37", "elements": [ 1, 2, 3, 4, 5 ], "cost": 265 }, { "id": "S38", "elements": [ 6, 7, 11 ], "cost": 132 }, { "id": "S39", "elements": [ 16, 17, 18 ], "cost": 132 }, { "id": "S40", "elements": [ 5, 7, 8, 9 ], "cost": 252 }, { "id": "S41", "elements": [ 11, 13 ], "cost": 114 }, { "id": "S42", "elements": [ 9, 10, 14 ], "cost": 297 }, { "id": "S43", "elements": [ 7, 10, 11, 12, 13 ], "cost": 485 }, { "id": "S44", "elements": [ 0, 1, 2, 3 ], "cost": 316 }, { "id": "S45", "elements": [ 4, 5, 7, 8 ], "cost": 348 }, { "id": "S46", "elements": [ 14, 16, 17 ], "cost": 225 }, { "id": "S47", "elements": [ 14, 15 ], "cost": 76 }, { "id": "S48", "elements": [ 11, 18 ], "cost": 144 }, { "id": "S49", "elements": [ 7, 9, 10, 12, 13 ], "cost": 405 }, { "id": "S50", "elements": [ 10, 11, 12 ], "cost": 69 }, { "id": "S51", "elements": [ 4, 5, 6, 8 ], "cost": 148 }, { "id": "S52", "elements": [ 0, 1, 2, 3, 4, 5 ], "cost": 108 }, { "id": "S53", "elements": [ 11, 12, 13, 14, 16 ], "cost": 145 }, { "id": "S54", "elements": [ 0, 1, 3 ], "cost": 195 }, { "id": "S55", "elements": [ 13, 14, 17 ], "cost": 186 }, { "id": "S56", "elements": [ 0, 2, 3, 5, 7 ], "cost": 485 }, { "id": "S57", "elements": [ 12, 13, 14, 15, 16 ], "cost": 425 }, { "id": "S58", "elements": [ 0, 1, 2, 3, 4, 6 ], "cost": 12 }, { "id": "S59", "elements": [ 14, 15, 16, 18 ], "cost": 152 }, { "id": "S60", "elements": [ 5, 6, 7, 8, 13 ], "cost": 475 }, { "id": "S61", "elements": [ 6, 7, 9, 11 ], "cost": 132 }, { "id": "S62", "elements": [ 6, 7 ], "cost": 190 }, { "id": "S63", "elements": [ 6, 8 ], "cost": 170 }, { "id": "S64", "elements": [ 10, 11, 12, 13, 14 ], "cost": 485 }, { "id": "S65", "elements": [ 9, 10, 11, 12, 13, 14 ], "cost": 498 }, { "id": "S66", "elements": [ 14, 15, 16, 17 ], "cost": 272 }, { "id": "S67", "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 258 }, { "id": "S68", "elements": [ 2, 3, 4 ], "cost": 261 }, { "id": "S69", "elements": [ 0, 2 ], "cost": 26 }, { "id": "S70", "elements": [ 10, 12, 13, 15 ], "cost": 320 }, { "id": "S71", "elements": [ 6, 9 ], "cost": 182 }, { "id": "S72", "elements": [ 0, 2, 3, 4, 5, 7 ], "cost": 90 }, { "id": "S73", "elements": [ 5, 7, 8, 9, 13 ], "cost": 495 }, { "id": "S74", "elements": [ 7, 8, 9, 12 ], "cost": 200 }, { "id": "S75", "elements": [ 4, 7 ], "cost": 126 }, { "id": "S76", "elements": [ 0 ], "cost": 10000 }, { "id": "S77", "elements": [ 1 ], "cost": 10000 }, { "id": "S78", "elements": [ 2 ], "cost": 10000 }, { "id": "S79", "elements": [ 3 ], "cost": 10000 }, { "id": "S80", "elements": [ 4 ], "cost": 10000 }, { "id": "S81", "elements": [ 5 ], "cost": 10000 }, { "id": "S82", "elements": [ 6 ], "cost": 10000 }, { "id": "S83", "elements": [ 7 ], "cost": 10000 }, { "id": "S84", "elements": [ 8 ], "cost": 10000 }, { "id": "S85", "elements": [ 9 ], "cost": 10000 }, { "id": "S86", "elements": [ 10 ], "cost": 10000 }, { "id": "S87", "elements": [ 11 ], "cost": 10000 }, { "id": "S88", "elements": [ 12 ], "cost": 10000 }, { "id": "S89", "elements": [ 13 ], "cost": 10000 }, { "id": "S90", "elements": [ 14 ], "cost": 10000 }, { "id": "S91", "elements": [ 15 ], "cost": 10000 }, { "id": "S92", "elements": [ 16 ], "cost": 10000 }, { "id": "S93", "elements": [ 17 ], "cost": 10000 }, { "id": "S94", "elements": [ 18 ], "cost": 10000 } ] }, "solution_variant": [ "S17", "S26", "S28", "S39", "S52" ], "context_index": 7, "input_format": "csv", "input_index_base": 0 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "Someone at the branch wants the shelves reorganized so each book sits in a single selected shelf group and no title ends up unassigned or assigned twice. The trick is choosing which groups to use so the combined cost is as small as it can be — simply add together the costs of the groups you pick to see the total. The detailed options and prices follow below.\n\n- **num_titles**: 19\n- **num_shelf_groups_available**: 85\n\n| shelf_group_id | shelf_group_cost | titles_in_group |\n|---|---|---|\n| S1 | 50 | 3 4 5 6 8 |\n| S2 | 4 | 5 9 10 11 |\n| S3 | 495 | 11 12 13 14 16 |\n| S4 | 116 | 11 14 15 16 |\n| S5 | 92 | 14 16 |\n| S6 | 36 | 10 11 14 |\n| S7 | 249 | 8 11 13 |\n| S8 | 270 | 14 16 17 |\n| S9 | 243 | 1 3 5 |\n| S10 | 252 | 5 6 8 9 |\n| S11 | 174 | 17 18 19 |\n| S12 | 516 | 1 2 3 4 5 7 |\n| S13 | 76 | 14 16 18 19 |\n| S14 | 68 | 1 2 3 4 |\n| S15 | 26 | 4 5 |\n| S16 | 308 | 3 4 7 8 |\n| S17 | 260 | 5 7 8 10 11 |\n| S18 | 57 | 1 2 3 |\n| S19 | 65 | 2 5 6 7 9 |\n| S20 | 60 | 8 9 11 13 14 |\n| S21 | 135 | 15 16 17 18 19 |\n| S22 | 46 | 8 9 |\n| S23 | 65 | 5 7 8 9 11 |\n| S24 | 28 | 15 17 18 19 |\n| S25 | 69 | 4 6 9 |\n| S26 | 69 | 12 16 18 |\n| S27 | 279 | 1 2 5 |\n| S28 | 50 | 9 10 11 12 14 |\n| S29 | 92 | 13 14 16 17 |\n| S30 | 105 | 3 4 5 |\n| S31 | 45 | 2 4 5 |\n| S32 | 410 | 12 13 14 15 17 |\n| S33 | 168 | 4 5 6 7 8 10 |\n| S34 | 275 | 1 2 3 4 6 |\n| S35 | 44 | 13 15 16 17 |\n| S36 | 177 | 1 2 4 |\n| S37 | 445 | 6 10 11 12 13 |\n| S38 | 237 | 2 3 4 |\n| S39 | 32 | 4 7 8 10 |\n| S40 | 188 | 10 12 13 14 |\n| S41 | 124 | 9 10 |\n| S42 | 340 | 14 15 16 17 18 |\n| S43 | 486 | 1 2 3 4 5 6 |\n| S44 | 34 | 14 17 |\n| S45 | 120 | 10 12 14 15 |\n| S46 | 160 | 14 15 16 18 |\n| S47 | 21 | 7 8 9 |\n| S48 | 42 | 8 10 |\n| S49 | 105 | 8 9 11 12 13 |\n| S50 | 144 | 2 3 7 |\n| S51 | 78 | 13 15 |\n| S52 | 38 | 2 3 |\n| S53 | 228 | 9 10 11 12 13 15 |\n| S54 | 372 | 13 15 16 17 18 19 |\n| S55 | 230 | 1 3 4 5 6 |\n| S56 | 136 | 1 3 |\n| S57 | 261 | 6 7 11 |\n| S58 | 264 | 2 3 4 7 |\n| S59 | 225 | 7 10 11 12 13 |\n| S60 | 171 | 12 16 17 |\n| S61 | 270 | 12 13 15 16 18 |\n| S62 | 78 | 16 18 |\n| S63 | 48 | 14 17 18 19 |\n| S64 | 195 | 3 4 5 7 8 |\n| S65 | 195 | 7 8 10 11 12 |\n| S66 | 24 | 13 14 15 16 17 19 |\n| S67 | 10000 | 1 |\n| S68 | 10000 | 2 |\n| S69 | 10000 | 3 |\n| S70 | 10000 | 4 |\n| S71 | 10000 | 5 |\n| S72 | 10000 | 6 |\n| S73 | 10000 | 7 |\n| S74 | 10000 | 8 |\n| S75 | 10000 | 9 |\n| S76 | 10000 | 10 |\n| S77 | 10000 | 11 |\n| S78 | 10000 | 12 |\n| S79 | 10000 | 13 |\n| S80 | 10000 | 14 |\n| S81 | 10000 | 15 |\n| S82 | 10000 | 16 |\n| S83 | 10000 | 17 |\n| S84 | 10000 | 18 |\n| S85 | 10000 | 19 |\n\nIf you want to tell me which shelf groups to pick, just send it back in this little JSON shape so I can read it cleanly:\n\n{\n \"solution\": [\"shelf_group_id\", ...]\n}\n\nThe \"solution\" array is where you list the shelf group identifiers you choose. Each entry in that list is a placeholder for a single selected shelf group (think of it like the group name or code for the shelves where each book will live). This is just a sketch of the shape I expect, not the actual answer — replace those placeholders with the real IDs from the instance.\n\nPlease make sure to use the exact identifiers given in the instance input — do not rename them or invent new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 19, "num_sets": 85, "density": 0.17089783281733747, "sets": [ { "id": 1, "elements": [ 3, 4, 5, 6, 8 ], "cost": 50 }, { "id": 2, "elements": [ 5, 9, 10, 11 ], "cost": 4 }, { "id": 3, "elements": [ 11, 12, 13, 14, 16 ], "cost": 495 }, { "id": 4, "elements": [ 11, 14, 15, 16 ], "cost": 116 }, { "id": 5, "elements": [ 14, 16 ], "cost": 92 }, { "id": 6, "elements": [ 10, 11, 14 ], "cost": 36 }, { "id": 7, "elements": [ 8, 11, 13 ], "cost": 249 }, { "id": 8, "elements": [ 14, 16, 17 ], "cost": 270 }, { "id": 9, "elements": [ 1, 3, 5 ], "cost": 243 }, { "id": 10, "elements": [ 5, 6, 8, 9 ], "cost": 252 }, { "id": 11, "elements": [ 17, 18, 19 ], "cost": 174 }, { "id": 12, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 516 }, { "id": 13, "elements": [ 14, 16, 18, 19 ], "cost": 76 }, { "id": 14, "elements": [ 1, 2, 3, 4 ], "cost": 68 }, { "id": 15, "elements": [ 4, 5 ], "cost": 26 }, { "id": 16, "elements": [ 3, 4, 7, 8 ], "cost": 308 }, { "id": 17, "elements": [ 5, 7, 8, 10, 11 ], "cost": 260 }, { "id": 18, "elements": [ 1, 2, 3 ], "cost": 57 }, { "id": 19, "elements": [ 2, 5, 6, 7, 9 ], "cost": 65 }, { "id": 20, "elements": [ 8, 9, 11, 13, 14 ], "cost": 60 }, { "id": 21, "elements": [ 15, 16, 17, 18, 19 ], "cost": 135 }, { "id": 22, "elements": [ 8, 9 ], "cost": 46 }, { "id": 23, "elements": [ 5, 7, 8, 9, 11 ], "cost": 65 }, { "id": 24, "elements": [ 15, 17, 18, 19 ], "cost": 28 }, { "id": 25, "elements": [ 4, 6, 9 ], "cost": 69 }, { "id": 26, "elements": [ 12, 16, 18 ], "cost": 69 }, { "id": 27, "elements": [ 1, 2, 5 ], "cost": 279 }, { "id": 28, "elements": [ 9, 10, 11, 12, 14 ], "cost": 50 }, { "id": 29, "elements": [ 13, 14, 16, 17 ], "cost": 92 }, { "id": 30, "elements": [ 3, 4, 5 ], "cost": 105 }, { "id": 31, "elements": [ 2, 4, 5 ], "cost": 45 }, { "id": 32, "elements": [ 12, 13, 14, 15, 17 ], "cost": 410 }, { "id": 33, "elements": [ 4, 5, 6, 7, 8, 10 ], "cost": 168 }, { "id": 34, "elements": [ 1, 2, 3, 4, 6 ], "cost": 275 }, { "id": 35, "elements": [ 13, 15, 16, 17 ], "cost": 44 }, { "id": 36, "elements": [ 1, 2, 4 ], "cost": 177 }, { "id": 37, "elements": [ 6, 10, 11, 12, 13 ], "cost": 445 }, { "id": 38, "elements": [ 2, 3, 4 ], "cost": 237 }, { "id": 39, "elements": [ 4, 7, 8, 10 ], "cost": 32 }, { "id": 40, "elements": [ 10, 12, 13, 14 ], "cost": 188 }, { "id": 41, "elements": [ 9, 10 ], "cost": 124 }, { "id": 42, "elements": [ 14, 15, 16, 17, 18 ], "cost": 340 }, { "id": 43, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 486 }, { "id": 44, "elements": [ 14, 17 ], "cost": 34 }, { "id": 45, "elements": [ 10, 12, 14, 15 ], "cost": 120 }, { "id": 46, "elements": [ 14, 15, 16, 18 ], "cost": 160 }, { "id": 47, "elements": [ 7, 8, 9 ], "cost": 21 }, { "id": 48, "elements": [ 8, 10 ], "cost": 42 }, { "id": 49, "elements": [ 8, 9, 11, 12, 13 ], "cost": 105 }, { "id": 50, "elements": [ 2, 3, 7 ], "cost": 144 }, { "id": 51, "elements": [ 13, 15 ], "cost": 78 }, { "id": 52, "elements": [ 2, 3 ], "cost": 38 }, { "id": 53, "elements": [ 9, 10, 11, 12, 13, 15 ], "cost": 228 }, { "id": 54, "elements": [ 13, 15, 16, 17, 18, 19 ], "cost": 372 }, { "id": 55, "elements": [ 1, 3, 4, 5, 6 ], "cost": 230 }, { "id": 56, "elements": [ 1, 3 ], "cost": 136 }, { "id": 57, "elements": [ 6, 7, 11 ], "cost": 261 }, { "id": 58, "elements": [ 2, 3, 4, 7 ], "cost": 264 }, { "id": 59, "elements": [ 7, 10, 11, 12, 13 ], "cost": 225 }, { "id": 60, "elements": [ 12, 16, 17 ], "cost": 171 }, { "id": 61, "elements": [ 12, 13, 15, 16, 18 ], "cost": 270 }, { "id": 62, "elements": [ 16, 18 ], "cost": 78 }, { "id": 63, "elements": [ 14, 17, 18, 19 ], "cost": 48 }, { "id": 64, "elements": [ 3, 4, 5, 7, 8 ], "cost": 195 }, { "id": 65, "elements": [ 7, 8, 10, 11, 12 ], "cost": 195 }, { "id": 66, "elements": [ 13, 14, 15, 16, 17, 19 ], "cost": 24 }, { "id": 67, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 68, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 69, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 70, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 71, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 72, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 73, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 74, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 75, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 76, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 77, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 78, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 79, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 80, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 81, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 82, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 83, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 84, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 85, "elements": [ 19 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0007_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0007_bag_stats.png" }, "solution": [ 21, 23, 34, 40 ], "obj": 663.0, "instance_variant": { "num_elements": 19, "num_sets": 85, "sets": [ { "id": "S1", "elements": [ 3, 4, 5, 6, 8 ], "cost": 50 }, { "id": "S2", "elements": [ 5, 9, 10, 11 ], "cost": 4 }, { "id": "S3", "elements": [ 11, 12, 13, 14, 16 ], "cost": 495 }, { "id": "S4", "elements": [ 11, 14, 15, 16 ], "cost": 116 }, { "id": "S5", "elements": [ 14, 16 ], "cost": 92 }, { "id": "S6", "elements": [ 10, 11, 14 ], "cost": 36 }, { "id": "S7", "elements": [ 8, 11, 13 ], "cost": 249 }, { "id": "S8", "elements": [ 14, 16, 17 ], "cost": 270 }, { "id": "S9", "elements": [ 1, 3, 5 ], "cost": 243 }, { "id": "S10", "elements": [ 5, 6, 8, 9 ], "cost": 252 }, { "id": "S11", "elements": [ 17, 18, 19 ], "cost": 174 }, { "id": "S12", "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 516 }, { "id": "S13", "elements": [ 14, 16, 18, 19 ], "cost": 76 }, { "id": "S14", "elements": [ 1, 2, 3, 4 ], "cost": 68 }, { "id": "S15", "elements": [ 4, 5 ], "cost": 26 }, { "id": "S16", "elements": [ 3, 4, 7, 8 ], "cost": 308 }, { "id": "S17", "elements": [ 5, 7, 8, 10, 11 ], "cost": 260 }, { "id": "S18", "elements": [ 1, 2, 3 ], "cost": 57 }, { "id": "S19", "elements": [ 2, 5, 6, 7, 9 ], "cost": 65 }, { "id": "S20", "elements": [ 8, 9, 11, 13, 14 ], "cost": 60 }, { "id": "S21", "elements": [ 15, 16, 17, 18, 19 ], "cost": 135 }, { "id": "S22", "elements": [ 8, 9 ], "cost": 46 }, { "id": "S23", "elements": [ 5, 7, 8, 9, 11 ], "cost": 65 }, { "id": "S24", "elements": [ 15, 17, 18, 19 ], "cost": 28 }, { "id": "S25", "elements": [ 4, 6, 9 ], "cost": 69 }, { "id": "S26", "elements": [ 12, 16, 18 ], "cost": 69 }, { "id": "S27", "elements": [ 1, 2, 5 ], "cost": 279 }, { "id": "S28", "elements": [ 9, 10, 11, 12, 14 ], "cost": 50 }, { "id": "S29", "elements": [ 13, 14, 16, 17 ], "cost": 92 }, { "id": "S30", "elements": [ 3, 4, 5 ], "cost": 105 }, { "id": "S31", "elements": [ 2, 4, 5 ], "cost": 45 }, { "id": "S32", "elements": [ 12, 13, 14, 15, 17 ], "cost": 410 }, { "id": "S33", "elements": [ 4, 5, 6, 7, 8, 10 ], "cost": 168 }, { "id": "S34", "elements": [ 1, 2, 3, 4, 6 ], "cost": 275 }, { "id": "S35", "elements": [ 13, 15, 16, 17 ], "cost": 44 }, { "id": "S36", "elements": [ 1, 2, 4 ], "cost": 177 }, { "id": "S37", "elements": [ 6, 10, 11, 12, 13 ], "cost": 445 }, { "id": "S38", "elements": [ 2, 3, 4 ], "cost": 237 }, { "id": "S39", "elements": [ 4, 7, 8, 10 ], "cost": 32 }, { "id": "S40", "elements": [ 10, 12, 13, 14 ], "cost": 188 }, { "id": "S41", "elements": [ 9, 10 ], "cost": 124 }, { "id": "S42", "elements": [ 14, 15, 16, 17, 18 ], "cost": 340 }, { "id": "S43", "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 486 }, { "id": "S44", "elements": [ 14, 17 ], "cost": 34 }, { "id": "S45", "elements": [ 10, 12, 14, 15 ], "cost": 120 }, { "id": "S46", "elements": [ 14, 15, 16, 18 ], "cost": 160 }, { "id": "S47", "elements": [ 7, 8, 9 ], "cost": 21 }, { "id": "S48", "elements": [ 8, 10 ], "cost": 42 }, { "id": "S49", "elements": [ 8, 9, 11, 12, 13 ], "cost": 105 }, { "id": "S50", "elements": [ 2, 3, 7 ], "cost": 144 }, { "id": "S51", "elements": [ 13, 15 ], "cost": 78 }, { "id": "S52", "elements": [ 2, 3 ], "cost": 38 }, { "id": "S53", "elements": [ 9, 10, 11, 12, 13, 15 ], "cost": 228 }, { "id": "S54", "elements": [ 13, 15, 16, 17, 18, 19 ], "cost": 372 }, { "id": "S55", "elements": [ 1, 3, 4, 5, 6 ], "cost": 230 }, { "id": "S56", "elements": [ 1, 3 ], "cost": 136 }, { "id": "S57", "elements": [ 6, 7, 11 ], "cost": 261 }, { "id": "S58", "elements": [ 2, 3, 4, 7 ], "cost": 264 }, { "id": "S59", "elements": [ 7, 10, 11, 12, 13 ], "cost": 225 }, { "id": "S60", "elements": [ 12, 16, 17 ], "cost": 171 }, { "id": "S61", "elements": [ 12, 13, 15, 16, 18 ], "cost": 270 }, { "id": "S62", "elements": [ 16, 18 ], "cost": 78 }, { "id": "S63", "elements": [ 14, 17, 18, 19 ], "cost": 48 }, { "id": "S64", "elements": [ 3, 4, 5, 7, 8 ], "cost": 195 }, { "id": "S65", "elements": [ 7, 8, 10, 11, 12 ], "cost": 195 }, { "id": "S66", "elements": [ 13, 14, 15, 16, 17, 19 ], "cost": 24 }, { "id": "S67", "elements": [ 1 ], "cost": 10000 }, { "id": "S68", "elements": [ 2 ], "cost": 10000 }, { "id": "S69", "elements": [ 3 ], "cost": 10000 }, { "id": "S70", "elements": [ 4 ], "cost": 10000 }, { "id": "S71", "elements": [ 5 ], "cost": 10000 }, { "id": "S72", "elements": [ 6 ], "cost": 10000 }, { "id": "S73", "elements": [ 7 ], "cost": 10000 }, { "id": "S74", "elements": [ 8 ], "cost": 10000 }, { "id": "S75", "elements": [ 9 ], "cost": 10000 }, { "id": "S76", "elements": [ 10 ], "cost": 10000 }, { "id": "S77", "elements": [ 11 ], "cost": 10000 }, { "id": "S78", "elements": [ 12 ], "cost": 10000 }, { "id": "S79", "elements": [ 13 ], "cost": 10000 }, { "id": "S80", "elements": [ 14 ], "cost": 10000 }, { "id": "S81", "elements": [ 15 ], "cost": 10000 }, { "id": "S82", "elements": [ 16 ], "cost": 10000 }, { "id": "S83", "elements": [ 17 ], "cost": 10000 }, { "id": "S84", "elements": [ 18 ], "cost": 10000 }, { "id": "S85", "elements": [ 19 ], "cost": 10000 } ] }, "solution_variant": [ "S21", "S23", "S34", "S40" ], "context_index": 8, "input_format": "markdown_table", "input_index_base": 1 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "Back at the office the plan was sketched on a whiteboard: choose some clusters to host the company’s services, making sure each service ends up on exactly one chosen cluster — nothing missing and nothing running in two places. What makes one plan preferable is the final hosting charge; it’s calculated by summing the cost of every cluster we include, and the smallest sum wins. The concrete list of services, available clusters, and their costs appears below.\n\n- **total_services**: 19\n- **total_clusters**: 114\n\n| cluster_id | cluster_cost | services |\n|---|---|---|\n| S1 | 136 | B C D E |\n| S2 | 12 | O P Q S |\n| S3 | 35 | D E F G H |\n| S4 | 114 | K O Q |\n| S5 | 388 | A B C E |\n| S6 | 117 | K L M |\n| S7 | 6 | J M N |\n| S8 | 88 | I J |\n| S9 | 395 | D E F G J |\n| S10 | 50 | N P Q R S |\n| S11 | 80 | E H |\n| S12 | 455 | C D E F G |\n| S13 | 90 | B C F |\n| S14 | 45 | D E H |\n| S15 | 28 | M Q |\n| S16 | 66 | P Q |\n| S17 | 32 | K L |\n| S18 | 480 | C D E G H |\n| S19 | 148 | C D E F |\n| S20 | 195 | M R S |\n| S21 | 230 | M O P Q S |\n| S22 | 108 | K L M N O R |\n| S23 | 60 | H K L M |\n| S24 | 171 | N O S |\n| S25 | 27 | L M P |\n| S26 | 255 | L O P Q S |\n| S27 | 380 | A C E G |\n| S28 | 80 | C E G J K |\n| S29 | 225 | M O P |\n| S30 | 264 | L N P R |\n| S31 | 243 | M N P |\n| S32 | 245 | A B C E F |\n| S33 | 388 | H I J L |\n| S34 | 136 | N O P Q |\n| S35 | 64 | N P R S |\n| S36 | 69 | G J M |\n| S37 | 210 | C D E |\n| S38 | 304 | M N O P |\n| S39 | 570 | I J K L M N |\n| S40 | 32 | J K L P |\n| S41 | 160 | E G H J |\n| S42 | 118 | B E |\n| S43 | 304 | L N O Q |\n| S44 | 148 | J K |\n| S45 | 88 | N Q |\n| S46 | 328 | I J K O |\n| S47 | 260 | A B C E I |\n| S48 | 340 | A B C D E |\n| S49 | 156 | D F G I |\n| S50 | 344 | P Q R S |\n| S51 | 224 | N P Q R |\n| S52 | 195 | A B C D F |\n| S53 | 185 | K L M N P |\n| S54 | 16 | F G H J |\n| S55 | 249 | I J K |\n| S56 | 115 | E F G H I |\n| S57 | 44 | A B E F |\n| S58 | 141 | I L M |\n| S59 | 296 | B C E F |\n| S60 | 60 | B D E F G |\n| S61 | 285 | C E F |\n| S62 | 84 | E F K |\n| S63 | 420 | K L N O P Q |\n| S64 | 368 | G I J K |\n| S65 | 285 | F J M |\n| S66 | 160 | A B C D |\n| S67 | 228 | J K L N |\n| S68 | 291 | L O P |\n| S69 | 123 | E G I |\n| S70 | 475 | K M N O P |\n| S71 | 295 | H I J K L |\n| S72 | 201 | G H I |\n| S73 | 240 | E G H I J |\n| S74 | 9 | K O P |\n| S75 | 260 | F G I J L |\n| S76 | 8 | G H J L |\n| S77 | 135 | L M O P Q |\n| S78 | 81 | F H I |\n| S79 | 16 | H I K L |\n| S80 | 204 | L O Q S |\n| S81 | 215 | I J K M N |\n| S82 | 480 | O P Q R S |\n| S83 | 72 | N O P Q R S |\n| S84 | 51 | O P Q |\n| S85 | 316 | H J K L |\n| S86 | 100 | G H I J K |\n| S87 | 136 | K L M N |\n| S88 | 18 | O R S |\n| S89 | 44 | B C |\n| S90 | 495 | D G I J O |\n| S91 | 172 | K M N Q |\n| S92 | 28 | D E |\n| S93 | 174 | A B D |\n| S94 | 111 | P R S |\n| S95 | 36 | B C D |\n| S96 | 10000 | A |\n| S97 | 10000 | B |\n| S98 | 10000 | C |\n| S99 | 10000 | D |\n| S100 | 10000 | E |\n| S101 | 10000 | F |\n| S102 | 10000 | G |\n| S103 | 10000 | H |\n| S104 | 10000 | I |\n| S105 | 10000 | J |\n| S106 | 10000 | K |\n| S107 | 10000 | L |\n| S108 | 10000 | M |\n| S109 | 10000 | N |\n| S110 | 10000 | O |\n| S111 | 10000 | P |\n| S112 | 10000 | Q |\n| S113 | 10000 | R |\n| S114 | 10000 | S |\n\nIf you want the plan in a neat, machine-friendly shape, just return a small JSON object like this:\n\n{\n \"solution\": [\"cluster_id\", ...]\n}\n\n\"solution\" should list the clusters you picked to host each service — each entry is a cluster identifier placeholder (so in the real reply you'll replace \"cluster_id\" with the actual IDs from the instance). This is just a sketch of the shape I expect, not the actual answer.\n\nPlease make sure to use the exact identifiers from the input — do not rename them or invent new ones. For example:\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 19, "num_sets": 114, "density": 0.1772853185595568, "sets": [ { "id": 1, "elements": [ 2, 3, 4, 5 ], "cost": 136 }, { "id": 2, "elements": [ 15, 16, 17, 19 ], "cost": 12 }, { "id": 3, "elements": [ 4, 5, 6, 7, 8 ], "cost": 35 }, { "id": 4, "elements": [ 11, 15, 17 ], "cost": 114 }, { "id": 5, "elements": [ 1, 2, 3, 5 ], "cost": 388 }, { "id": 6, "elements": [ 11, 12, 13 ], "cost": 117 }, { "id": 7, "elements": [ 10, 13, 14 ], "cost": 6 }, { "id": 8, "elements": [ 9, 10 ], "cost": 88 }, { "id": 9, "elements": [ 4, 5, 6, 7, 10 ], "cost": 395 }, { "id": 10, "elements": [ 14, 16, 17, 18, 19 ], "cost": 50 }, { "id": 11, "elements": [ 5, 8 ], "cost": 80 }, { "id": 12, "elements": [ 3, 4, 5, 6, 7 ], "cost": 455 }, { "id": 13, "elements": [ 2, 3, 6 ], "cost": 90 }, { "id": 14, "elements": [ 4, 5, 8 ], "cost": 45 }, { "id": 15, "elements": [ 13, 17 ], "cost": 28 }, { "id": 16, "elements": [ 16, 17 ], "cost": 66 }, { "id": 17, "elements": [ 11, 12 ], "cost": 32 }, { "id": 18, "elements": [ 3, 4, 5, 7, 8 ], "cost": 480 }, { "id": 19, "elements": [ 3, 4, 5, 6 ], "cost": 148 }, { "id": 20, "elements": [ 13, 18, 19 ], "cost": 195 }, { "id": 21, "elements": [ 13, 15, 16, 17, 19 ], "cost": 230 }, { "id": 22, "elements": [ 11, 12, 13, 14, 15, 18 ], "cost": 108 }, { "id": 23, "elements": [ 8, 11, 12, 13 ], "cost": 60 }, { "id": 24, "elements": [ 14, 15, 19 ], "cost": 171 }, { "id": 25, "elements": [ 12, 13, 16 ], "cost": 27 }, { "id": 26, "elements": [ 12, 15, 16, 17, 19 ], "cost": 255 }, { "id": 27, "elements": [ 1, 3, 5, 7 ], "cost": 380 }, { "id": 28, "elements": [ 3, 5, 7, 10, 11 ], "cost": 80 }, { "id": 29, "elements": [ 13, 15, 16 ], "cost": 225 }, { "id": 30, "elements": [ 12, 14, 16, 18 ], "cost": 264 }, { "id": 31, "elements": [ 13, 14, 16 ], "cost": 243 }, { "id": 32, "elements": [ 1, 2, 3, 5, 6 ], "cost": 245 }, { "id": 33, "elements": [ 8, 9, 10, 12 ], "cost": 388 }, { "id": 34, "elements": [ 14, 15, 16, 17 ], "cost": 136 }, { "id": 35, "elements": [ 14, 16, 18, 19 ], "cost": 64 }, { "id": 36, "elements": [ 7, 10, 13 ], "cost": 69 }, { "id": 37, "elements": [ 3, 4, 5 ], "cost": 210 }, { "id": 38, "elements": [ 13, 14, 15, 16 ], "cost": 304 }, { "id": 39, "elements": [ 9, 10, 11, 12, 13, 14 ], "cost": 570 }, { "id": 40, "elements": [ 10, 11, 12, 16 ], "cost": 32 }, { "id": 41, "elements": [ 5, 7, 8, 10 ], "cost": 160 }, { "id": 42, "elements": [ 2, 5 ], "cost": 118 }, { "id": 43, "elements": [ 12, 14, 15, 17 ], "cost": 304 }, { "id": 44, "elements": [ 10, 11 ], "cost": 148 }, { "id": 45, "elements": [ 14, 17 ], "cost": 88 }, { "id": 46, "elements": [ 9, 10, 11, 15 ], "cost": 328 }, { "id": 47, "elements": [ 1, 2, 3, 5, 9 ], "cost": 260 }, { "id": 48, "elements": [ 1, 2, 3, 4, 5 ], "cost": 340 }, { "id": 49, "elements": [ 4, 6, 7, 9 ], "cost": 156 }, { "id": 50, "elements": [ 16, 17, 18, 19 ], "cost": 344 }, { "id": 51, "elements": [ 14, 16, 17, 18 ], "cost": 224 }, { "id": 52, "elements": [ 1, 2, 3, 4, 6 ], "cost": 195 }, { "id": 53, "elements": [ 11, 12, 13, 14, 16 ], "cost": 185 }, { "id": 54, "elements": [ 6, 7, 8, 10 ], "cost": 16 }, { "id": 55, "elements": [ 9, 10, 11 ], "cost": 249 }, { "id": 56, "elements": [ 5, 6, 7, 8, 9 ], "cost": 115 }, { "id": 57, "elements": [ 1, 2, 5, 6 ], "cost": 44 }, { "id": 58, "elements": [ 9, 12, 13 ], "cost": 141 }, { "id": 59, "elements": [ 2, 3, 5, 6 ], "cost": 296 }, { "id": 60, "elements": [ 2, 4, 5, 6, 7 ], "cost": 60 }, { "id": 61, "elements": [ 3, 5, 6 ], "cost": 285 }, { "id": 62, "elements": [ 5, 6, 11 ], "cost": 84 }, { "id": 63, "elements": [ 11, 12, 14, 15, 16, 17 ], "cost": 420 }, { "id": 64, "elements": [ 7, 9, 10, 11 ], "cost": 368 }, { "id": 65, "elements": [ 6, 10, 13 ], "cost": 285 }, { "id": 66, "elements": [ 1, 2, 3, 4 ], "cost": 160 }, { "id": 67, "elements": [ 10, 11, 12, 14 ], "cost": 228 }, { "id": 68, "elements": [ 12, 15, 16 ], "cost": 291 }, { "id": 69, "elements": [ 5, 7, 9 ], "cost": 123 }, { "id": 70, "elements": [ 11, 13, 14, 15, 16 ], "cost": 475 }, { "id": 71, "elements": [ 8, 9, 10, 11, 12 ], "cost": 295 }, { "id": 72, "elements": [ 7, 8, 9 ], "cost": 201 }, { "id": 73, "elements": [ 5, 7, 8, 9, 10 ], "cost": 240 }, { "id": 74, "elements": [ 11, 15, 16 ], "cost": 9 }, { "id": 75, "elements": [ 6, 7, 9, 10, 12 ], "cost": 260 }, { "id": 76, "elements": [ 7, 8, 10, 12 ], "cost": 8 }, { "id": 77, "elements": [ 12, 13, 15, 16, 17 ], "cost": 135 }, { "id": 78, "elements": [ 6, 8, 9 ], "cost": 81 }, { "id": 79, "elements": [ 8, 9, 11, 12 ], "cost": 16 }, { "id": 80, "elements": [ 12, 15, 17, 19 ], "cost": 204 }, { "id": 81, "elements": [ 9, 10, 11, 13, 14 ], "cost": 215 }, { "id": 82, "elements": [ 15, 16, 17, 18, 19 ], "cost": 480 }, { "id": 83, "elements": [ 14, 15, 16, 17, 18, 19 ], "cost": 72 }, { "id": 84, "elements": [ 15, 16, 17 ], "cost": 51 }, { "id": 85, "elements": [ 8, 10, 11, 12 ], "cost": 316 }, { "id": 86, "elements": [ 7, 8, 9, 10, 11 ], "cost": 100 }, { "id": 87, "elements": [ 11, 12, 13, 14 ], "cost": 136 }, { "id": 88, "elements": [ 15, 18, 19 ], "cost": 18 }, { "id": 89, "elements": [ 2, 3 ], "cost": 44 }, { "id": 90, "elements": [ 4, 7, 9, 10, 15 ], "cost": 495 }, { "id": 91, "elements": [ 11, 13, 14, 17 ], "cost": 172 }, { "id": 92, "elements": [ 4, 5 ], "cost": 28 }, { "id": 93, "elements": [ 1, 2, 4 ], "cost": 174 }, { "id": 94, "elements": [ 16, 18, 19 ], "cost": 111 }, { "id": 95, "elements": [ 2, 3, 4 ], "cost": 36 }, { "id": 96, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 97, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 98, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 19 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0008_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0008_bag_stats.png" }, "solution": [ 7, 16, 17, 56, 66, 88 ], "obj": 397.0, "instance_variant": { "num_elements": 19, "num_sets": 114, "sets": [ { "id": "S1", "elements": [ "B", "C", "D", "E" ], "cost": 136 }, { "id": "S2", "elements": [ "O", "P", "Q", "S" ], "cost": 12 }, { "id": "S3", "elements": [ "D", "E", "F", "G", "H" ], "cost": 35 }, { "id": "S4", "elements": [ "K", "O", "Q" ], "cost": 114 }, { "id": "S5", "elements": [ "A", "B", "C", "E" ], "cost": 388 }, { "id": "S6", "elements": [ "K", "L", "M" ], "cost": 117 }, { "id": "S7", "elements": [ "J", "M", "N" ], "cost": 6 }, { "id": "S8", "elements": [ "I", "J" ], "cost": 88 }, { "id": "S9", "elements": [ "D", "E", "F", "G", "J" ], "cost": 395 }, { "id": "S10", "elements": [ "N", "P", "Q", "R", "S" ], "cost": 50 }, { "id": "S11", "elements": [ "E", "H" ], "cost": 80 }, { "id": "S12", "elements": [ "C", "D", "E", "F", "G" ], "cost": 455 }, { "id": "S13", "elements": [ "B", "C", "F" ], "cost": 90 }, { "id": "S14", "elements": [ "D", "E", "H" ], "cost": 45 }, { "id": "S15", "elements": [ "M", "Q" ], "cost": 28 }, { "id": "S16", "elements": [ "P", "Q" ], "cost": 66 }, { "id": "S17", "elements": [ "K", "L" ], "cost": 32 }, { "id": "S18", "elements": [ "C", "D", "E", "G", "H" ], "cost": 480 }, { "id": "S19", "elements": [ "C", "D", "E", "F" ], "cost": 148 }, { "id": "S20", "elements": [ "M", "R", "S" ], "cost": 195 }, { "id": "S21", "elements": [ "M", "O", "P", "Q", "S" ], "cost": 230 }, { "id": "S22", "elements": [ "K", "L", "M", "N", "O", "R" ], "cost": 108 }, { "id": "S23", "elements": [ "H", "K", "L", "M" ], "cost": 60 }, { "id": "S24", "elements": [ "N", "O", "S" ], "cost": 171 }, { "id": "S25", "elements": [ "L", "M", "P" ], "cost": 27 }, { "id": "S26", "elements": [ "L", "O", "P", "Q", "S" ], "cost": 255 }, { "id": "S27", "elements": [ "A", "C", "E", "G" ], "cost": 380 }, { "id": "S28", "elements": [ "C", "E", "G", "J", "K" ], "cost": 80 }, { "id": "S29", "elements": [ "M", "O", "P" ], "cost": 225 }, { "id": "S30", "elements": [ "L", "N", "P", "R" ], "cost": 264 }, { "id": "S31", "elements": [ "M", "N", "P" ], "cost": 243 }, { "id": "S32", "elements": [ "A", "B", "C", "E", "F" ], "cost": 245 }, { "id": "S33", "elements": [ "H", "I", "J", "L" ], "cost": 388 }, { "id": "S34", "elements": [ "N", "O", "P", "Q" ], "cost": 136 }, { "id": "S35", "elements": [ "N", "P", "R", "S" ], "cost": 64 }, { "id": "S36", "elements": [ "G", "J", "M" ], "cost": 69 }, { "id": "S37", "elements": [ "C", "D", "E" ], "cost": 210 }, { "id": "S38", "elements": [ "M", "N", "O", "P" ], "cost": 304 }, { "id": "S39", "elements": [ "I", "J", "K", "L", "M", "N" ], "cost": 570 }, { "id": "S40", "elements": [ "J", "K", "L", "P" ], "cost": 32 }, { "id": "S41", "elements": [ "E", "G", "H", "J" ], "cost": 160 }, { "id": "S42", "elements": [ "B", "E" ], "cost": 118 }, { "id": "S43", "elements": [ "L", "N", "O", "Q" ], "cost": 304 }, { "id": "S44", "elements": [ "J", "K" ], "cost": 148 }, { "id": "S45", "elements": [ "N", "Q" ], "cost": 88 }, { "id": "S46", "elements": [ "I", "J", "K", "O" ], "cost": 328 }, { "id": "S47", "elements": [ "A", "B", "C", "E", "I" ], "cost": 260 }, { "id": "S48", "elements": [ "A", "B", "C", "D", "E" ], "cost": 340 }, { "id": "S49", "elements": [ "D", "F", "G", "I" ], "cost": 156 }, { "id": "S50", "elements": [ "P", "Q", "R", "S" ], "cost": 344 }, { "id": "S51", "elements": [ "N", "P", "Q", "R" ], "cost": 224 }, { "id": "S52", "elements": [ "A", "B", "C", "D", "F" ], "cost": 195 }, { "id": "S53", "elements": [ "K", "L", "M", "N", "P" ], "cost": 185 }, { "id": "S54", "elements": [ "F", "G", "H", "J" ], "cost": 16 }, { "id": "S55", "elements": [ "I", "J", "K" ], "cost": 249 }, { "id": "S56", "elements": [ "E", "F", "G", "H", "I" ], "cost": 115 }, { "id": "S57", "elements": [ "A", "B", "E", "F" ], "cost": 44 }, { "id": "S58", "elements": [ "I", "L", "M" ], "cost": 141 }, { "id": "S59", "elements": [ "B", "C", "E", "F" ], "cost": 296 }, { "id": "S60", "elements": [ "B", "D", "E", "F", "G" ], "cost": 60 }, { "id": "S61", "elements": [ "C", "E", "F" ], "cost": 285 }, { "id": "S62", "elements": [ "E", "F", "K" ], "cost": 84 }, { "id": "S63", "elements": [ "K", "L", "N", "O", "P", "Q" ], "cost": 420 }, { "id": "S64", "elements": [ "G", "I", "J", "K" ], "cost": 368 }, { "id": "S65", "elements": [ "F", "J", "M" ], "cost": 285 }, { "id": "S66", "elements": [ "A", "B", "C", "D" ], "cost": 160 }, { "id": "S67", "elements": [ "J", "K", "L", "N" ], "cost": 228 }, { "id": "S68", "elements": [ "L", "O", "P" ], "cost": 291 }, { "id": "S69", "elements": [ "E", "G", "I" ], "cost": 123 }, { "id": "S70", "elements": [ "K", "M", "N", "O", "P" ], "cost": 475 }, { "id": "S71", "elements": [ "H", "I", "J", "K", "L" ], "cost": 295 }, { "id": "S72", "elements": [ "G", "H", "I" ], "cost": 201 }, { "id": "S73", "elements": [ "E", "G", "H", "I", "J" ], "cost": 240 }, { "id": "S74", "elements": [ "K", "O", "P" ], "cost": 9 }, { "id": "S75", "elements": [ "F", "G", "I", "J", "L" ], "cost": 260 }, { "id": "S76", "elements": [ "G", "H", "J", "L" ], "cost": 8 }, { "id": "S77", "elements": [ "L", "M", "O", "P", "Q" ], "cost": 135 }, { "id": "S78", "elements": [ "F", "H", "I" ], "cost": 81 }, { "id": "S79", "elements": [ "H", "I", "K", "L" ], "cost": 16 }, { "id": "S80", "elements": [ "L", "O", "Q", "S" ], "cost": 204 }, { "id": "S81", "elements": [ "I", "J", "K", "M", "N" ], "cost": 215 }, { "id": "S82", "elements": [ "O", "P", "Q", "R", "S" ], "cost": 480 }, { "id": "S83", "elements": [ "N", "O", "P", "Q", "R", "S" ], "cost": 72 }, { "id": "S84", "elements": [ "O", "P", "Q" ], "cost": 51 }, { "id": "S85", "elements": [ "H", "J", "K", "L" ], "cost": 316 }, { "id": "S86", "elements": [ "G", "H", "I", "J", "K" ], "cost": 100 }, { "id": "S87", "elements": [ "K", "L", "M", "N" ], "cost": 136 }, { "id": "S88", "elements": [ "O", "R", "S" ], "cost": 18 }, { "id": "S89", "elements": [ "B", "C" ], "cost": 44 }, { "id": "S90", "elements": [ "D", "G", "I", "J", "O" ], "cost": 495 }, { "id": "S91", "elements": [ "K", "M", "N", "Q" ], "cost": 172 }, { "id": "S92", "elements": [ "D", "E" ], "cost": 28 }, { "id": "S93", "elements": [ "A", "B", "D" ], "cost": 174 }, { "id": "S94", "elements": [ "P", "R", "S" ], "cost": 111 }, { "id": "S95", "elements": [ "B", "C", "D" ], "cost": 36 }, { "id": "S96", "elements": [ "A" ], "cost": 10000 }, { "id": "S97", "elements": [ "B" ], "cost": 10000 }, { "id": "S98", "elements": [ "C" ], "cost": 10000 }, { "id": "S99", "elements": [ "D" ], "cost": 10000 }, { "id": "S100", "elements": [ "E" ], "cost": 10000 }, { "id": "S101", "elements": [ "F" ], "cost": 10000 }, { "id": "S102", "elements": [ "G" ], "cost": 10000 }, { "id": "S103", "elements": [ "H" ], "cost": 10000 }, { "id": "S104", "elements": [ "I" ], "cost": 10000 }, { "id": "S105", "elements": [ "J" ], "cost": 10000 }, { "id": "S106", "elements": [ "K" ], "cost": 10000 }, { "id": "S107", "elements": [ "L" ], "cost": 10000 }, { "id": "S108", "elements": [ "M" ], "cost": 10000 }, { "id": "S109", "elements": [ "N" ], "cost": 10000 }, { "id": "S110", "elements": [ "O" ], "cost": 10000 }, { "id": "S111", "elements": [ "P" ], "cost": 10000 }, { "id": "S112", "elements": [ "Q" ], "cost": 10000 }, { "id": "S113", "elements": [ "R" ], "cost": 10000 }, { "id": "S114", "elements": [ "S" ], "cost": 10000 } ] }, "solution_variant": [ "S7", "S16", "S17", "S56", "S66", "S88" ], "context_index": 9, "input_format": "markdown_table", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "There’s a community event coming up where every little program needs its own covered spot, so the task is to choose a set of tents to rent that will host all activities. Each activity must be hosted in one, and only one, of the tents we pick (no splitting and no leaving anyone out), and the goal is to make the rental bill as low as possible by adding up the costs of the tents selected. The exact tent options and the activity list are shown below.\n\nWe have 22 activities to cover and 111 candidate tents to consider:\nTent S1 hosts activities 9 10 14 and costs 300 to rent.\nTent S2 hosts activities 16 18 19 20 and costs 224 to rent.\nTent S3 hosts activities 6 8 9 10 12 and costs 130 to rent.\nTent S4 hosts activities 11 12 13 15 and costs 192 to rent.\nTent S5 hosts activities 10 13 14 15 18 and costs 415 to rent.\nTent S6 hosts activities 1 3 4 5 7 and costs 55 to rent.\nTent S7 hosts activities 13 14 15 16 17 20 and costs 438 to rent.\nTent S8 hosts activities 2 5 and costs 132 to rent.\nTent S9 hosts activities 2 3 5 6 and costs 300 to rent.\nTent S10 hosts activities 11 15 17 and costs 72 to rent.\nTent S11 hosts activities 0 1 2 3 4 6 and costs 480 to rent.\nTent S12 hosts activities 8 9 10 11 12 14 and costs 522 to rent.\nTent S13 hosts activities 12 13 16 and costs 252 to rent.\nTent S14 hosts activities 11 12 14 16 and costs 320 to rent.\nTent S15 hosts activities 15 16 17 18 19 20 and costs 12 to rent.\nTent S16 hosts activities 0 1 2 and costs 186 to rent.\nTent S17 hosts activities 15 18 19 20 21 and costs 495 to rent.\nTent S18 hosts activities 12 15 16 18 and costs 228 to rent.\nTent S19 hosts activities 11 13 14 15 16 18 and costs 258 to rent.\nTent S20 hosts activities 19 21 and costs 108 to rent.\nTent S21 hosts activities 15 16 18 20 21 and costs 370 to rent.\nTent S22 hosts activities 0 1 3 5 and costs 320 to rent.\nTent S23 hosts activities 8 9 10 11 12 13 and costs 390 to rent.\nTent S24 hosts activities 19 20 21 and costs 255 to rent.\nTent S25 hosts activities 8 9 10 13 and costs 336 to rent.\nTent S26 hosts activities 13 15 16 17 18 and costs 430 to rent.\nTent S27 hosts activities 5 7 8 9 10 12 16 and costs 455 to rent.\nTent S28 hosts activities 0 2 and costs 60 to rent.\nTent S29 hosts activities 7 9 13 and costs 297 to rent.\nTent S30 hosts activities 14 16 and costs 178 to rent.\nTent S31 hosts activities 16 17 18 19 20 21 and costs 90 to rent.\nTent S32 hosts activities 15 16 17 18 19 21 and costs 96 to rent.\nTent S33 hosts activities 7 9 10 11 12 14 and costs 498 to rent.\nTent S34 hosts activities 2 3 6 8 and costs 4 to rent.\nTent S35 hosts activities 12 14 15 and costs 144 to rent.\nTent S36 hosts activities 10 12 13 14 15 and costs 150 to rent.\nTent S37 hosts activities 12 13 14 and costs 201 to rent.\nTent S38 hosts activities 7 9 11 12 13 14 and costs 264 to rent.\nTent S39 hosts activities 1 4 5 9 and costs 80 to rent.\nTent S40 hosts activities 0 2 3 4 5 6 and costs 528 to rent.\nTent S41 hosts activities 7 8 10 12 13 and costs 395 to rent.\nTent S42 hosts activities 10 11 12 13 14 15 and costs 504 to rent.\nTent S43 hosts activities 3 5 6 9 and costs 372 to rent.\nTent S44 hosts activities 2 3 4 5 6 and costs 220 to rent.\nTent S45 hosts activities 1 3 4 5 7 8 and costs 330 to rent.\nTent S46 hosts activities 0 1 3 4 5 6 8 and costs 406 to rent.\nTent S47 hosts activities 1 3 5 6 and costs 292 to rent.\nTent S48 hosts activities 17 18 19 20 21 and costs 460 to rent.\nTent S49 hosts activities 7 8 10 11 12 13 and costs 102 to rent.\nTent S50 hosts activities 11 13 14 15 and costs 400 to rent.\nTent S51 hosts activities 8 9 10 14 and costs 252 to rent.\nTent S52 hosts activities 10 11 12 14 15 and costs 455 to rent.\nTent S53 hosts activities 13 15 17 18 and costs 288 to rent.\nTent S54 hosts activities 1 2 4 5 and costs 24 to rent.\nTent S55 hosts activities 2 3 4 6 8 and costs 95 to rent.\nTent S56 hosts activities 9 11 12 14 15 16 and costs 96 to rent.\nTent S57 hosts activities 1 2 3 5 6 and costs 245 to rent.\nTent S58 hosts activities 16 17 19 20 and costs 344 to rent.\nTent S59 hosts activities 7 9 10 11 and costs 204 to rent.\nTent S60 hosts activities 0 1 2 3 4 5 and costs 150 to rent.\nTent S61 hosts activities 13 15 16 17 18 21 and costs 96 to rent.\nTent S62 hosts activities 12 15 16 18 19 and costs 375 to rent.\nTent S63 hosts activities 13 16 18 20 and costs 12 to rent.\nTent S64 hosts activities 3 4 5 8 and costs 60 to rent.\nTent S65 hosts activities 13 14 15 16 17 18 19 and costs 553 to rent.\nTent S66 hosts activities 1 2 3 4 5 6 and costs 294 to rent.\nTent S67 hosts activities 5 6 7 8 9 10 and costs 18 to rent.\nTent S68 hosts activities 16 18 19 and costs 51 to rent.\nTent S69 hosts activities 12 13 15 16 and costs 76 to rent.\nTent S70 hosts activities 4 5 7 9 and costs 232 to rent.\nTent S71 hosts activities 15 18 and costs 168 to rent.\nTent S72 hosts activities 0 1 2 3 4 9 and costs 486 to rent.\nTent S73 hosts activities 14 15 17 and costs 27 to rent.\nTent S74 hosts activities 12 13 14 19 and costs 224 to rent.\nTent S75 hosts activities 15 17 20 and costs 132 to rent.\nTent S76 hosts activities 11 12 13 15 18 and costs 215 to rent.\nTent S77 hosts activities 0 5 and costs 122 to rent.\nTent S78 hosts activities 1 2 and costs 180 to rent.\nTent S79 hosts activities 7 8 12 13 14 and costs 430 to rent.\nTent S80 hosts activities 3 4 5 7 8 and costs 320 to rent.\nTent S81 hosts activities 7 8 9 10 and costs 292 to rent.\nTent S82 hosts activities 15 16 17 18 19 and costs 365 to rent.\nTent S83 hosts activities 11 14 16 17 19 20 and costs 474 to rent.\nTent S84 hosts activities 11 16 19 and costs 261 to rent.\nTent S85 hosts activities 14 15 18 19 and costs 216 to rent.\nTent S86 hosts activities 4 6 7 8 11 and costs 475 to rent.\nTent S87 hosts activities 11 16 17 and costs 129 to rent.\nTent S88 hosts activities 15 16 19 20 and costs 324 to rent.\nTent S89 hosts activities 6 7 8 10 11 12 and costs 186 to rent.\nTent S90 hosts activities 0 and costs 10000 to rent.\nTent S91 hosts activities 1 and costs 10000 to rent.\nTent S92 hosts activities 2 and costs 10000 to rent.\nTent S93 hosts activities 3 and costs 10000 to rent.\nTent S94 hosts activities 4 and costs 10000 to rent.\nTent S95 hosts activities 5 and costs 10000 to rent.\nTent S96 hosts activities 6 and costs 10000 to rent.\nTent S97 hosts activities 7 and costs 10000 to rent.\nTent S98 hosts activities 8 and costs 10000 to rent.\nTent S99 hosts activities 9 and costs 10000 to rent.\nTent S100 hosts activities 10 and costs 10000 to rent.\nTent S101 hosts activities 11 and costs 10000 to rent.\nTent S102 hosts activities 12 and costs 10000 to rent.\nTent S103 hosts activities 13 and costs 10000 to rent.\nTent S104 hosts activities 14 and costs 10000 to rent.\nTent S105 hosts activities 15 and costs 10000 to rent.\nTent S106 hosts activities 16 and costs 10000 to rent.\nTent S107 hosts activities 17 and costs 10000 to rent.\nTent S108 hosts activities 18 and costs 10000 to rent.\nTent S109 hosts activities 19 and costs 10000 to rent.\nTent S110 hosts activities 20 and costs 10000 to rent.\nTent S111 hosts activities 21 and costs 10000 to rent.\nLet's pick tents so all 22 activities are covered while keeping the total rental bill as low as possible.\n\nAlso, when you send back your choice, please use this simple JSON layout so it's easy to read by both people and tools:\n\n{\n \"solution\": [\"tent_id\", ...]\n}\n\nThis is just a sketch of the shape I expect:\n- \"solution\" is the list of the tents you want to rent (one entry per tent).\n- The \"tent_id\" placeholder is where you'll put the actual tent identifiers from the instance (replace the placeholder with the real IDs).\n- The JSON itself is only an outline — fill in the real tent IDs when you submit the answer.\n\nPlease make sure you use the exact identifiers from the instance input — do not rename them and do not invent new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 22, "num_sets": 111, "density": 0.1732186732186732, "sets": [ { "id": 1, "elements": [ 10, 11, 15 ], "cost": 300 }, { "id": 2, "elements": [ 17, 19, 20, 21 ], "cost": 224 }, { "id": 3, "elements": [ 7, 9, 10, 11, 13 ], "cost": 130 }, { "id": 4, "elements": [ 12, 13, 14, 16 ], "cost": 192 }, { "id": 5, "elements": [ 11, 14, 15, 16, 19 ], "cost": 415 }, { "id": 6, "elements": [ 2, 4, 5, 6, 8 ], "cost": 55 }, { "id": 7, "elements": [ 14, 15, 16, 17, 18, 21 ], "cost": 438 }, { "id": 8, "elements": [ 3, 6 ], "cost": 132 }, { "id": 9, "elements": [ 3, 4, 6, 7 ], "cost": 300 }, { "id": 10, "elements": [ 12, 16, 18 ], "cost": 72 }, { "id": 11, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 480 }, { "id": 12, "elements": [ 9, 10, 11, 12, 13, 15 ], "cost": 522 }, { "id": 13, "elements": [ 13, 14, 17 ], "cost": 252 }, { "id": 14, "elements": [ 12, 13, 15, 17 ], "cost": 320 }, { "id": 15, "elements": [ 16, 17, 18, 19, 20, 21 ], "cost": 12 }, { "id": 16, "elements": [ 1, 2, 3 ], "cost": 186 }, { "id": 17, "elements": [ 16, 19, 20, 21, 22 ], "cost": 495 }, { "id": 18, "elements": [ 13, 16, 17, 19 ], "cost": 228 }, { "id": 19, "elements": [ 12, 14, 15, 16, 17, 19 ], "cost": 258 }, { "id": 20, "elements": [ 20, 22 ], "cost": 108 }, { "id": 21, "elements": [ 16, 17, 19, 21, 22 ], "cost": 370 }, { "id": 22, "elements": [ 1, 2, 4, 6 ], "cost": 320 }, { "id": 23, "elements": [ 9, 10, 11, 12, 13, 14 ], "cost": 390 }, { "id": 24, "elements": [ 20, 21, 22 ], "cost": 255 }, { "id": 25, "elements": [ 9, 10, 11, 14 ], "cost": 336 }, { "id": 26, "elements": [ 14, 16, 17, 18, 19 ], "cost": 430 }, { "id": 27, "elements": [ 6, 8, 9, 10, 11, 13, 17 ], "cost": 455 }, { "id": 28, "elements": [ 1, 3 ], "cost": 60 }, { "id": 29, "elements": [ 8, 10, 14 ], "cost": 297 }, { "id": 30, "elements": [ 15, 17 ], "cost": 178 }, { "id": 31, "elements": [ 17, 18, 19, 20, 21, 22 ], "cost": 90 }, { "id": 32, "elements": [ 16, 17, 18, 19, 20, 22 ], "cost": 96 }, { "id": 33, "elements": [ 8, 10, 11, 12, 13, 15 ], "cost": 498 }, { "id": 34, "elements": [ 3, 4, 7, 9 ], "cost": 4 }, { "id": 35, "elements": [ 13, 15, 16 ], "cost": 144 }, { "id": 36, "elements": [ 11, 13, 14, 15, 16 ], "cost": 150 }, { "id": 37, "elements": [ 13, 14, 15 ], "cost": 201 }, { "id": 38, "elements": [ 8, 10, 12, 13, 14, 15 ], "cost": 264 }, { "id": 39, "elements": [ 2, 5, 6, 10 ], "cost": 80 }, { "id": 40, "elements": [ 1, 3, 4, 5, 6, 7 ], "cost": 528 }, { "id": 41, "elements": [ 8, 9, 11, 13, 14 ], "cost": 395 }, { "id": 42, "elements": [ 11, 12, 13, 14, 15, 16 ], "cost": 504 }, { "id": 43, "elements": [ 4, 6, 7, 10 ], "cost": 372 }, { "id": 44, "elements": [ 3, 4, 5, 6, 7 ], "cost": 220 }, { "id": 45, "elements": [ 2, 4, 5, 6, 8, 9 ], "cost": 330 }, { "id": 46, "elements": [ 1, 2, 4, 5, 6, 7, 9 ], "cost": 406 }, { "id": 47, "elements": [ 2, 4, 6, 7 ], "cost": 292 }, { "id": 48, "elements": [ 18, 19, 20, 21, 22 ], "cost": 460 }, { "id": 49, "elements": [ 8, 9, 11, 12, 13, 14 ], "cost": 102 }, { "id": 50, "elements": [ 12, 14, 15, 16 ], "cost": 400 }, { "id": 51, "elements": [ 9, 10, 11, 15 ], "cost": 252 }, { "id": 52, "elements": [ 11, 12, 13, 15, 16 ], "cost": 455 }, { "id": 53, "elements": [ 14, 16, 18, 19 ], "cost": 288 }, { "id": 54, "elements": [ 2, 3, 5, 6 ], "cost": 24 }, { "id": 55, "elements": [ 3, 4, 5, 7, 9 ], "cost": 95 }, { "id": 56, "elements": [ 10, 12, 13, 15, 16, 17 ], "cost": 96 }, { "id": 57, "elements": [ 2, 3, 4, 6, 7 ], "cost": 245 }, { "id": 58, "elements": [ 17, 18, 20, 21 ], "cost": 344 }, { "id": 59, "elements": [ 8, 10, 11, 12 ], "cost": 204 }, { "id": 60, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 150 }, { "id": 61, "elements": [ 14, 16, 17, 18, 19, 22 ], "cost": 96 }, { "id": 62, "elements": [ 13, 16, 17, 19, 20 ], "cost": 375 }, { "id": 63, "elements": [ 14, 17, 19, 21 ], "cost": 12 }, { "id": 64, "elements": [ 4, 5, 6, 9 ], "cost": 60 }, { "id": 65, "elements": [ 14, 15, 16, 17, 18, 19, 20 ], "cost": 553 }, { "id": 66, "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 294 }, { "id": 67, "elements": [ 6, 7, 8, 9, 10, 11 ], "cost": 18 }, { "id": 68, "elements": [ 17, 19, 20 ], "cost": 51 }, { "id": 69, "elements": [ 13, 14, 16, 17 ], "cost": 76 }, { "id": 70, "elements": [ 5, 6, 8, 10 ], "cost": 232 }, { "id": 71, "elements": [ 16, 19 ], "cost": 168 }, { "id": 72, "elements": [ 1, 2, 3, 4, 5, 10 ], "cost": 486 }, { "id": 73, "elements": [ 15, 16, 18 ], "cost": 27 }, { "id": 74, "elements": [ 13, 14, 15, 20 ], "cost": 224 }, { "id": 75, "elements": [ 16, 18, 21 ], "cost": 132 }, { "id": 76, "elements": [ 12, 13, 14, 16, 19 ], "cost": 215 }, { "id": 77, "elements": [ 1, 6 ], "cost": 122 }, { "id": 78, "elements": [ 2, 3 ], "cost": 180 }, { "id": 79, "elements": [ 8, 9, 13, 14, 15 ], "cost": 430 }, { "id": 80, "elements": [ 4, 5, 6, 8, 9 ], "cost": 320 }, { "id": 81, "elements": [ 8, 9, 10, 11 ], "cost": 292 }, { "id": 82, "elements": [ 16, 17, 18, 19, 20 ], "cost": 365 }, { "id": 83, "elements": [ 12, 15, 17, 18, 20, 21 ], "cost": 474 }, { "id": 84, "elements": [ 12, 17, 20 ], "cost": 261 }, { "id": 85, "elements": [ 15, 16, 19, 20 ], "cost": 216 }, { "id": 86, "elements": [ 5, 7, 8, 9, 12 ], "cost": 475 }, { "id": 87, "elements": [ 12, 17, 18 ], "cost": 129 }, { "id": 88, "elements": [ 16, 17, 20, 21 ], "cost": 324 }, { "id": 89, "elements": [ 7, 8, 9, 11, 12, 13 ], "cost": 186 }, { "id": 90, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 91, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 92, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 93, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 94, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 95, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 96, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 97, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 98, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 22 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0009_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0009_bag_stats.png" }, "solution": [ 3, 6, 28, 31, 50 ], "obj": 735.0, "instance_variant": { "num_elements": 22, "num_sets": 111, "sets": [ { "id": "S1", "elements": [ 9, 10, 14 ], "cost": 300 }, { "id": "S2", "elements": [ 16, 18, 19, 20 ], "cost": 224 }, { "id": "S3", "elements": [ 6, 8, 9, 10, 12 ], "cost": 130 }, { "id": "S4", "elements": [ 11, 12, 13, 15 ], "cost": 192 }, { "id": "S5", "elements": [ 10, 13, 14, 15, 18 ], "cost": 415 }, { "id": "S6", "elements": [ 1, 3, 4, 5, 7 ], "cost": 55 }, { "id": "S7", "elements": [ 13, 14, 15, 16, 17, 20 ], "cost": 438 }, { "id": "S8", "elements": [ 2, 5 ], "cost": 132 }, { "id": "S9", "elements": [ 2, 3, 5, 6 ], "cost": 300 }, { "id": "S10", "elements": [ 11, 15, 17 ], "cost": 72 }, { "id": "S11", "elements": [ 0, 1, 2, 3, 4, 6 ], "cost": 480 }, { "id": "S12", "elements": [ 8, 9, 10, 11, 12, 14 ], "cost": 522 }, { "id": "S13", "elements": [ 12, 13, 16 ], "cost": 252 }, { "id": "S14", "elements": [ 11, 12, 14, 16 ], "cost": 320 }, { "id": "S15", "elements": [ 15, 16, 17, 18, 19, 20 ], "cost": 12 }, { "id": "S16", "elements": [ 0, 1, 2 ], "cost": 186 }, { "id": "S17", "elements": [ 15, 18, 19, 20, 21 ], "cost": 495 }, { "id": "S18", "elements": [ 12, 15, 16, 18 ], "cost": 228 }, { "id": "S19", "elements": [ 11, 13, 14, 15, 16, 18 ], "cost": 258 }, { "id": "S20", "elements": [ 19, 21 ], "cost": 108 }, { "id": "S21", "elements": [ 15, 16, 18, 20, 21 ], "cost": 370 }, { "id": "S22", "elements": [ 0, 1, 3, 5 ], "cost": 320 }, { "id": "S23", "elements": [ 8, 9, 10, 11, 12, 13 ], "cost": 390 }, { "id": "S24", "elements": [ 19, 20, 21 ], "cost": 255 }, { "id": "S25", "elements": [ 8, 9, 10, 13 ], "cost": 336 }, { "id": "S26", "elements": [ 13, 15, 16, 17, 18 ], "cost": 430 }, { "id": "S27", "elements": [ 5, 7, 8, 9, 10, 12, 16 ], "cost": 455 }, { "id": "S28", "elements": [ 0, 2 ], "cost": 60 }, { "id": "S29", "elements": [ 7, 9, 13 ], "cost": 297 }, { "id": "S30", "elements": [ 14, 16 ], "cost": 178 }, { "id": "S31", "elements": [ 16, 17, 18, 19, 20, 21 ], "cost": 90 }, { "id": "S32", "elements": [ 15, 16, 17, 18, 19, 21 ], "cost": 96 }, { "id": "S33", "elements": [ 7, 9, 10, 11, 12, 14 ], "cost": 498 }, { "id": "S34", "elements": [ 2, 3, 6, 8 ], "cost": 4 }, { "id": "S35", "elements": [ 12, 14, 15 ], "cost": 144 }, { "id": "S36", "elements": [ 10, 12, 13, 14, 15 ], "cost": 150 }, { "id": "S37", "elements": [ 12, 13, 14 ], "cost": 201 }, { "id": "S38", "elements": [ 7, 9, 11, 12, 13, 14 ], "cost": 264 }, { "id": "S39", "elements": [ 1, 4, 5, 9 ], "cost": 80 }, { "id": "S40", "elements": [ 0, 2, 3, 4, 5, 6 ], "cost": 528 }, { "id": "S41", "elements": [ 7, 8, 10, 12, 13 ], "cost": 395 }, { "id": "S42", "elements": [ 10, 11, 12, 13, 14, 15 ], "cost": 504 }, { "id": "S43", "elements": [ 3, 5, 6, 9 ], "cost": 372 }, { "id": "S44", "elements": [ 2, 3, 4, 5, 6 ], "cost": 220 }, { "id": "S45", "elements": [ 1, 3, 4, 5, 7, 8 ], "cost": 330 }, { "id": "S46", "elements": [ 0, 1, 3, 4, 5, 6, 8 ], "cost": 406 }, { "id": "S47", "elements": [ 1, 3, 5, 6 ], "cost": 292 }, { "id": "S48", "elements": [ 17, 18, 19, 20, 21 ], "cost": 460 }, { "id": "S49", "elements": [ 7, 8, 10, 11, 12, 13 ], "cost": 102 }, { "id": "S50", "elements": [ 11, 13, 14, 15 ], "cost": 400 }, { "id": "S51", "elements": [ 8, 9, 10, 14 ], "cost": 252 }, { "id": "S52", "elements": [ 10, 11, 12, 14, 15 ], "cost": 455 }, { "id": "S53", "elements": [ 13, 15, 17, 18 ], "cost": 288 }, { "id": "S54", "elements": [ 1, 2, 4, 5 ], "cost": 24 }, { "id": "S55", "elements": [ 2, 3, 4, 6, 8 ], "cost": 95 }, { "id": "S56", "elements": [ 9, 11, 12, 14, 15, 16 ], "cost": 96 }, { "id": "S57", "elements": [ 1, 2, 3, 5, 6 ], "cost": 245 }, { "id": "S58", "elements": [ 16, 17, 19, 20 ], "cost": 344 }, { "id": "S59", "elements": [ 7, 9, 10, 11 ], "cost": 204 }, { "id": "S60", "elements": [ 0, 1, 2, 3, 4, 5 ], "cost": 150 }, { "id": "S61", "elements": [ 13, 15, 16, 17, 18, 21 ], "cost": 96 }, { "id": "S62", "elements": [ 12, 15, 16, 18, 19 ], "cost": 375 }, { "id": "S63", "elements": [ 13, 16, 18, 20 ], "cost": 12 }, { "id": "S64", "elements": [ 3, 4, 5, 8 ], "cost": 60 }, { "id": "S65", "elements": [ 13, 14, 15, 16, 17, 18, 19 ], "cost": 553 }, { "id": "S66", "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 294 }, { "id": "S67", "elements": [ 5, 6, 7, 8, 9, 10 ], "cost": 18 }, { "id": "S68", "elements": [ 16, 18, 19 ], "cost": 51 }, { "id": "S69", "elements": [ 12, 13, 15, 16 ], "cost": 76 }, { "id": "S70", "elements": [ 4, 5, 7, 9 ], "cost": 232 }, { "id": "S71", "elements": [ 15, 18 ], "cost": 168 }, { "id": "S72", "elements": [ 0, 1, 2, 3, 4, 9 ], "cost": 486 }, { "id": "S73", "elements": [ 14, 15, 17 ], "cost": 27 }, { "id": "S74", "elements": [ 12, 13, 14, 19 ], "cost": 224 }, { "id": "S75", "elements": [ 15, 17, 20 ], "cost": 132 }, { "id": "S76", "elements": [ 11, 12, 13, 15, 18 ], "cost": 215 }, { "id": "S77", "elements": [ 0, 5 ], "cost": 122 }, { "id": "S78", "elements": [ 1, 2 ], "cost": 180 }, { "id": "S79", "elements": [ 7, 8, 12, 13, 14 ], "cost": 430 }, { "id": "S80", "elements": [ 3, 4, 5, 7, 8 ], "cost": 320 }, { "id": "S81", "elements": [ 7, 8, 9, 10 ], "cost": 292 }, { "id": "S82", "elements": [ 15, 16, 17, 18, 19 ], "cost": 365 }, { "id": "S83", "elements": [ 11, 14, 16, 17, 19, 20 ], "cost": 474 }, { "id": "S84", "elements": [ 11, 16, 19 ], "cost": 261 }, { "id": "S85", "elements": [ 14, 15, 18, 19 ], "cost": 216 }, { "id": "S86", "elements": [ 4, 6, 7, 8, 11 ], "cost": 475 }, { "id": "S87", "elements": [ 11, 16, 17 ], "cost": 129 }, { "id": "S88", "elements": [ 15, 16, 19, 20 ], "cost": 324 }, { "id": "S89", "elements": [ 6, 7, 8, 10, 11, 12 ], "cost": 186 }, { "id": "S90", "elements": [ 0 ], "cost": 10000 }, { "id": "S91", "elements": [ 1 ], "cost": 10000 }, { "id": "S92", "elements": [ 2 ], "cost": 10000 }, { "id": "S93", "elements": [ 3 ], "cost": 10000 }, { "id": "S94", "elements": [ 4 ], "cost": 10000 }, { "id": "S95", "elements": [ 5 ], "cost": 10000 }, { "id": "S96", "elements": [ 6 ], "cost": 10000 }, { "id": "S97", "elements": [ 7 ], "cost": 10000 }, { "id": "S98", "elements": [ 8 ], "cost": 10000 }, { "id": "S99", "elements": [ 9 ], "cost": 10000 }, { "id": "S100", "elements": [ 10 ], "cost": 10000 }, { "id": "S101", "elements": [ 11 ], "cost": 10000 }, { "id": "S102", "elements": [ 12 ], "cost": 10000 }, { "id": "S103", "elements": [ 13 ], "cost": 10000 }, { "id": "S104", "elements": [ 14 ], "cost": 10000 }, { "id": "S105", "elements": [ 15 ], "cost": 10000 }, { "id": "S106", "elements": [ 16 ], "cost": 10000 }, { "id": "S107", "elements": [ 17 ], "cost": 10000 }, { "id": "S108", "elements": [ 18 ], "cost": 10000 }, { "id": "S109", "elements": [ 19 ], "cost": 10000 }, { "id": "S110", "elements": [ 20 ], "cost": 10000 }, { "id": "S111", "elements": [ 21 ], "cost": 10000 } ] }, "solution_variant": [ "S3", "S6", "S28", "S31", "S50" ], "context_index": 10, "input_format": "nl", "input_index_base": 0 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "A hospital manager has to decide which combination of care teams to call in so that each patient is assigned to exactly one team. What makes one combination better is its total cost — you calculate it by summing the staffing cost of all the teams chosen, and the goal is to keep that sum as small as possible. The constraint is practical and simple: every patient must be covered, and no patient can be assigned to more than one team. The concrete details are provided below.\n\n{\n \"total_patients\": 19,\n \"num_candidate_teams\": 117,\n \"sets\": [\n {\n \"team_id\": \"S1\",\n \"covered_patient_ids\": [\n 1,\n 2,\n 4,\n 5\n ],\n \"team_cost\": 20\n },\n {\n \"team_id\": \"S2\",\n \"covered_patient_ids\": [\n 1,\n 4,\n 5\n ],\n \"team_cost\": 24\n },\n {\n \"team_id\": \"S3\",\n \"covered_patient_ids\": [\n 11,\n 14,\n 15\n ],\n \"team_cost\": 114\n },\n {\n \"team_id\": \"S4\",\n \"covered_patient_ids\": [\n 11,\n 12,\n 13,\n 14,\n 15\n ],\n \"team_cost\": 10\n },\n {\n \"team_id\": \"S5\",\n \"covered_patient_ids\": [\n 13,\n 15,\n 16,\n 17\n ],\n \"team_cost\": 348\n },\n {\n \"team_id\": \"S6\",\n \"covered_patient_ids\": [\n 10,\n 12,\n 15,\n 16,\n 17\n ],\n \"team_cost\": 380\n },\n {\n \"team_id\": \"S7\",\n \"covered_patient_ids\": [\n 5,\n 6,\n 7,\n 8,\n 9\n ],\n \"team_cost\": 400\n },\n {\n \"team_id\": \"S8\",\n \"covered_patient_ids\": [\n 5,\n 6,\n 7,\n 9\n ],\n \"team_cost\": 264\n },\n {\n \"team_id\": \"S9\",\n \"covered_patient_ids\": [\n 9,\n 10,\n 11,\n 12,\n 13,\n 14\n ],\n \"team_cost\": 288\n },\n {\n \"team_id\": \"S10\",\n \"covered_patient_ids\": [\n 13,\n 15,\n 16,\n 17,\n 18,\n 19\n ],\n \"team_cost\": 330\n },\n {\n \"team_id\": \"S11\",\n \"covered_patient_ids\": [\n 4,\n 5,\n 8\n ],\n \"team_cost\": 75\n },\n {\n \"team_id\": \"S12\",\n \"covered_patient_ids\": [\n 14,\n 15,\n 16,\n 19\n ],\n \"team_cost\": 316\n },\n {\n \"team_id\": \"S13\",\n \"covered_patient_ids\": [\n 9,\n 11,\n 13\n ],\n \"team_cost\": 18\n },\n {\n \"team_id\": \"S14\",\n \"covered_patient_ids\": [\n 1,\n 2,\n 3,\n 4\n ],\n \"team_cost\": 248\n },\n {\n \"team_id\": \"S15\",\n \"covered_patient_ids\": [\n 6,\n 10,\n 11,\n 13,\n 14\n ],\n \"team_cost\": 10\n },\n {\n \"team_id\": \"S16\",\n \"covered_patient_ids\": [\n 1,\n 2\n ],\n \"team_cost\": 114\n },\n {\n \"team_id\": \"S17\",\n \"covered_patient_ids\": [\n 17,\n 18,\n 19\n ],\n \"team_cost\": 78\n },\n {\n \"team_id\": \"S18\",\n \"covered_patient_ids\": [\n 4,\n 5,\n 6\n ],\n \"team_cost\": 213\n },\n {\n \"team_id\": \"S19\",\n \"covered_patient_ids\": [\n 3,\n 9,\n 10\n ],\n \"team_cost\": 9\n },\n {\n \"team_id\": \"S20\",\n \"covered_patient_ids\": [\n 2,\n 4,\n 5,\n 7,\n 8\n ],\n \"team_cost\": 155\n },\n {\n \"team_id\": \"S21\",\n \"covered_patient_ids\": [\n 13,\n 15,\n 16\n ],\n \"team_cost\": 240\n },\n {\n \"team_id\": \"S22\",\n \"covered_patient_ids\": [\n 1,\n 2,\n 3,\n 4,\n 5\n ],\n \"team_cost\": 445\n },\n {\n \"team_id\": \"S23\",\n \"covered_patient_ids\": [\n 11,\n 13,\n 18\n ],\n \"team_cost\": 237\n },\n {\n \"team_id\": \"S24\",\n \"covered_patient_ids\": [\n 15,\n 16,\n 17,\n 19\n ],\n \"team_cost\": 144\n },\n {\n \"team_id\": \"S25\",\n \"covered_patient_ids\": [\n 14,\n 15,\n 16,\n 18,\n 19\n ],\n \"team_cost\": 425\n },\n {\n \"team_id\": \"S26\",\n \"covered_patient_ids\": [\n 5,\n 6,\n 7,\n 8\n ],\n \"team_cost\": 256\n },\n {\n \"team_id\": \"S27\",\n \"covered_patient_ids\": [\n 8,\n 9,\n 10,\n 11\n ],\n \"team_cost\": 108\n },\n {\n \"team_id\": \"S28\",\n \"covered_patient_ids\": [\n 5,\n 6,\n 7\n ],\n \"team_cost\": 165\n },\n {\n \"team_id\": \"S29\",\n \"covered_patient_ids\": [\n 15,\n 16,\n 17,\n 18,\n 19\n ],\n \"team_cost\": 190\n },\n {\n \"team_id\": \"S30\",\n \"covered_patient_ids\": [\n 9,\n 11,\n 12,\n 13\n ],\n \"team_cost\": 232\n },\n {\n \"team_id\": \"S31\",\n \"covered_patient_ids\": [\n 10,\n 12,\n 13,\n 14\n ],\n \"team_cost\": 228\n },\n {\n \"team_id\": \"S32\",\n \"covered_patient_ids\": [\n 1,\n 3,\n 6\n ],\n \"team_cost\": 198\n },\n {\n \"team_id\": \"S33\",\n \"covered_patient_ids\": [\n 1,\n 2,\n 5,\n 6\n ],\n \"team_cost\": 20\n },\n {\n \"team_id\": \"S34\",\n \"covered_patient_ids\": [\n 9,\n 10,\n 11,\n 14\n ],\n \"team_cost\": 352\n },\n {\n \"team_id\": \"S35\",\n \"covered_patient_ids\": [\n 7,\n 8,\n 9,\n 12,\n 13\n ],\n \"team_cost\": 170\n },\n {\n \"team_id\": \"S36\",\n \"covered_patient_ids\": [\n 9,\n 10\n ],\n \"team_cost\": 104\n },\n {\n \"team_id\": \"S37\",\n \"covered_patient_ids\": [\n 14,\n 16,\n 17,\n 18,\n 19\n ],\n \"team_cost\": 255\n },\n {\n \"team_id\": \"S38\",\n \"covered_patient_ids\": [\n 7,\n 8,\n 9\n ],\n \"team_cost\": 291\n },\n {\n \"team_id\": \"S39\",\n \"covered_patient_ids\": [\n 12,\n 14,\n 15,\n 17\n ],\n \"team_cost\": 360\n },\n {\n \"team_id\": \"S40\",\n \"covered_patient_ids\": [\n 14,\n 15,\n 16\n ],\n \"team_cost\": 222\n },\n {\n \"team_id\": \"S41\",\n \"covered_patient_ids\": [\n 6,\n 9,\n 11,\n 12,\n 13\n ],\n \"team_cost\": 270\n },\n {\n \"team_id\": \"S42\",\n \"covered_patient_ids\": [\n 11,\n 13,\n 14,\n 15\n ],\n \"team_cost\": 292\n },\n {\n \"team_id\": \"S43\",\n \"covered_patient_ids\": [\n 3,\n 4,\n 5,\n 6,\n 8\n ],\n \"team_cost\": 390\n },\n {\n \"team_id\": \"S44\",\n \"covered_patient_ids\": [\n 11,\n 13,\n 14,\n 17\n ],\n \"team_cost\": 100\n },\n {\n \"team_id\": \"S45\",\n \"covered_patient_ids\": [\n 8,\n 10,\n 11,\n 14\n ],\n \"team_cost\": 56\n },\n {\n \"team_id\": \"S46\",\n \"covered_patient_ids\": [\n 3,\n 4,\n 6,\n 7\n ],\n \"team_cost\": 36\n },\n {\n \"team_id\": \"S47\",\n \"covered_patient_ids\": [\n 9,\n 10,\n 11,\n 12,\n 14\n ],\n \"team_cost\": 335\n },\n {\n \"team_id\": \"S48\",\n \"covered_patient_ids\": [\n 10,\n 13\n ],\n \"team_cost\": 42\n },\n {\n \"team_id\": \"S49\",\n \"covered_patient_ids\": [\n 2,\n 3,\n 4,\n 5,\n 7,\n 9\n ],\n \"team_cost\": 162\n },\n {\n \"team_id\": \"S50\",\n \"covered_patient_ids\": [\n 14,\n 15,\n 17\n ],\n \"team_cost\": 78\n },\n {\n \"team_id\": \"S51\",\n \"covered_patient_ids\": [\n 8,\n 10,\n 11\n ],\n \"team_cost\": 177\n },\n {\n \"team_id\": \"S52\",\n \"covered_patient_ids\": [\n 7,\n 9,\n 10,\n 11,\n 12\n ],\n \"team_cost\": 400\n },\n {\n \"team_id\": \"S53\",\n \"covered_patient_ids\": [\n 12,\n 14,\n 15,\n 16,\n 17\n ],\n \"team_cost\": 30\n },\n {\n \"team_id\": \"S54\",\n \"covered_patient_ids\": [\n 4,\n 5,\n 7\n ],\n \"team_cost\": 27\n },\n {\n \"team_id\": \"S55\",\n \"covered_patient_ids\": [\n 4,\n 6,\n 7,\n 8,\n 10\n ],\n \"team_cost\": 485\n },\n {\n \"team_id\": \"S56\",\n \"covered_patient_ids\": [\n 9,\n 11\n ],\n \"team_cost\": 54\n },\n {\n \"team_id\": \"S57\",\n \"covered_patient_ids\": [\n 11,\n 14,\n 15,\n 16\n ],\n \"team_cost\": 176\n },\n {\n \"team_id\": \"S58\",\n \"covered_patient_ids\": [\n 3,\n 5,\n 6,\n 7,\n 10\n ],\n \"team_cost\": 480\n },\n {\n \"team_id\": \"S59\",\n \"covered_patient_ids\": [\n 12,\n 14,\n 15,\n 18\n ],\n \"team_cost\": 72\n },\n {\n \"team_id\": \"S60\",\n \"covered_patient_ids\": [\n 5,\n 7,\n 8,\n 11\n ],\n \"team_cost\": 152\n },\n {\n \"team_id\": \"S61\",\n \"covered_patient_ids\": [\n 11,\n 14\n ],\n \"team_cost\": 4\n },\n {\n \"team_id\": \"S62\",\n \"covered_patient_ids\": [\n 3,\n 5,\n 7,\n 8\n ],\n \"team_cost\": 276\n },\n {\n \"team_id\": \"S63\",\n \"covered_patient_ids\": [\n 2,\n 3,\n 4\n ],\n \"team_cost\": 276\n },\n {\n \"team_id\": \"S64\",\n \"covered_patient_ids\": [\n 4,\n 5,\n 6,\n 8,\n 10\n ],\n \"team_cost\": 10\n },\n {\n \"team_id\": \"S65\",\n \"covered_patient_ids\": [\n 4,\n 8,\n 10\n ],\n \"team_cost\": 39\n },\n {\n \"team_id\": \"S66\",\n \"covered_patient_ids\": [\n 3,\n 4,\n 6,\n 7,\n 8\n ],\n \"team_cost\": 405\n },\n {\n \"team_id\": \"S67\",\n \"covered_patient_ids\": [\n 6,\n 7,\n 10\n ],\n \"team_cost\": 90\n },\n {\n \"team_id\": \"S68\",\n \"covered_patient_ids\": [\n 8,\n 11,\n 13\n ],\n \"team_cost\": 6\n },\n {\n \"team_id\": \"S69\",\n \"covered_patient_ids\": [\n 13,\n 14,\n 16\n ],\n \"team_cost\": 240\n },\n {\n \"team_id\": \"S70\",\n \"covered_patient_ids\": [\n 3,\n 4,\n 5\n ],\n \"team_cost\": 192\n },\n {\n \"team_id\": \"S71\",\n \"covered_patient_ids\": [\n 1,\n 2,\n 3,\n 5,\n 6,\n 7\n ],\n \"team_cost\": 252\n },\n {\n \"team_id\": \"S72\",\n \"covered_patient_ids\": [\n 11,\n 13,\n 14,\n 15,\n 16,\n 17\n ],\n \"team_cost\": 246\n },\n {\n \"team_id\": \"S73\",\n \"covered_patient_ids\": [\n 12,\n 13,\n 15,\n 16,\n 17,\n 18\n ],\n \"team_cost\": 390\n },\n {\n \"team_id\": \"S74\",\n \"covered_patient_ids\": [\n 1,\n 2,\n 3,\n 5\n ],\n \"team_cost\": 380\n },\n {\n \"team_id\": \"S75\",\n \"covered_patient_ids\": [\n 5,\n 7,\n 8,\n 9\n ],\n \"team_cost\": 80\n },\n {\n \"team_id\": \"S76\",\n \"covered_patient_ids\": [\n 2,\n 4,\n 7\n ],\n \"team_cost\": 171\n },\n {\n \"team_id\": \"S77\",\n \"covered_patient_ids\": [\n 7,\n 9,\n 11,\n 12,\n 13\n ],\n \"team_cost\": 270\n },\n {\n \"team_id\": \"S78\",\n \"covered_patient_ids\": [\n 8,\n 9,\n 11,\n 12\n ],\n \"team_cost\": 108\n },\n {\n \"team_id\": \"S79\",\n \"covered_patient_ids\": [\n 6,\n 9,\n 10,\n 11\n ],\n \"team_cost\": 268\n },\n {\n \"team_id\": \"S80\",\n \"covered_patient_ids\": [\n 12,\n 15,\n 16\n ],\n \"team_cost\": 75\n },\n {\n \"team_id\": \"S81\",\n \"covered_patient_ids\": [\n 3,\n 8\n ],\n \"team_cost\": 46\n },\n {\n \"team_id\": \"S82\",\n \"covered_patient_ids\": [\n 4,\n 6,\n 7,\n 9\n ],\n \"team_cost\": 372\n },\n {\n \"team_id\": \"S83\",\n \"covered_patient_ids\": [\n 12,\n 13,\n 14\n ],\n \"team_cost\": 174\n },\n {\n \"team_id\": \"S84\",\n \"covered_patient_ids\": [\n 4,\n 5,\n 6,\n 7,\n 9,\n 11\n ],\n \"team_cost\": 342\n },\n {\n \"team_id\": \"S85\",\n \"covered_patient_ids\": [\n 10,\n 14,\n 15,\n 16\n ],\n \"team_cost\": 196\n },\n {\n \"team_id\": \"S86\",\n \"covered_patient_ids\": [\n 1,\n 3,\n 4,\n 6\n ],\n \"team_cost\": 376\n },\n {\n \"team_id\": \"S87\",\n \"covered_patient_ids\": [\n 16,\n 18\n ],\n \"team_cost\": 104\n },\n {\n \"team_id\": \"S88\",\n \"covered_patient_ids\": [\n 12,\n 13,\n 14,\n 15,\n 17\n ],\n \"team_cost\": 265\n },\n {\n \"team_id\": \"S89\",\n \"covered_patient_ids\": [\n 2,\n 3,\n 4,\n 5,\n 6\n ],\n \"team_cost\": 485\n },\n {\n \"team_id\": \"S90\",\n \"covered_patient_ids\": [\n 13,\n 15,\n 16,\n 17,\n 18\n ],\n \"team_cost\": 275\n },\n {\n \"team_id\": \"S91\",\n \"covered_patient_ids\": [\n 4,\n 7,\n 8,\n 9,\n 11\n ],\n \"team_cost\": 430\n },\n {\n \"team_id\": \"S92\",\n \"covered_patient_ids\": [\n 7,\n 8,\n 10,\n 12\n ],\n \"team_cost\": 300\n },\n {\n \"team_id\": \"S93\",\n \"covered_patient_ids\": [\n 2,\n 5,\n 6,\n 7,\n 8,\n 9\n ],\n \"team_cost\": 66\n },\n {\n \"team_id\": \"S94\",\n \"covered_patient_ids\": [\n 16,\n 17,\n 18,\n 19\n ],\n \"team_cost\": 376\n },\n {\n \"team_id\": \"S95\",\n \"covered_patient_ids\": [\n 2,\n 3,\n 5,\n 6,\n 7\n ],\n \"team_cost\": 480\n },\n {\n \"team_id\": \"S96\",\n \"covered_patient_ids\": [\n 5,\n 8,\n 10\n ],\n \"team_cost\": 39\n },\n {\n \"team_id\": \"S97\",\n \"covered_patient_ids\": [\n 14,\n 15\n ],\n \"team_cost\": 188\n },\n {\n \"team_id\": \"S98\",\n \"covered_patient_ids\": [\n 4,\n 5,\n 6,\n 7\n ],\n \"team_cost\": 60\n },\n {\n \"team_id\": \"S99\",\n \"covered_patient_ids\": [\n 1\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S100\",\n \"covered_patient_ids\": [\n 2\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S101\",\n \"covered_patient_ids\": [\n 3\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S102\",\n \"covered_patient_ids\": [\n 4\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S103\",\n \"covered_patient_ids\": [\n 5\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S104\",\n \"covered_patient_ids\": [\n 6\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S105\",\n \"covered_patient_ids\": [\n 7\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S106\",\n \"covered_patient_ids\": [\n 8\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S107\",\n \"covered_patient_ids\": [\n 9\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S108\",\n \"covered_patient_ids\": [\n 10\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S109\",\n \"covered_patient_ids\": [\n 11\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S110\",\n \"covered_patient_ids\": [\n 12\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S111\",\n \"covered_patient_ids\": [\n 13\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S112\",\n \"covered_patient_ids\": [\n 14\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S113\",\n \"covered_patient_ids\": [\n 15\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S114\",\n \"covered_patient_ids\": [\n 16\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S115\",\n \"covered_patient_ids\": [\n 17\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S116\",\n \"covered_patient_ids\": [\n 18\n ],\n \"team_cost\": 10000\n },\n {\n \"team_id\": \"S117\",\n \"covered_patient_ids\": [\n 19\n ],\n \"team_cost\": 10000\n }\n ]\n}\n\nAlso, when you reply, please put the chosen teams into a little JSON snippet like this so it's easy to read and process:\n\n{\n \"solution\": [\"team_id\", ...]\n}\n\n\"solution\" is the top-level field holding the list of care teams to call in. Each item in that list is a team identifier (I'm showing the placeholder \"team_id\" here to match the hospital story) — in your actual submission replace those placeholders with the exact identifiers from the instance. This JSON is just a sketch of the shape I expect, not the final answer itself.\n\nPlease make sure you use the identifiers exactly as they appear in the instance input — no renaming and no new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.", "instance": { "problem_type": "SPP", "num_elements": 19, "num_sets": 117, "density": 0.18443544759334232, "sets": [ { "id": 1, "elements": [ 1, 2, 4, 5 ], "cost": 20 }, { "id": 2, "elements": [ 1, 4, 5 ], "cost": 24 }, { "id": 3, "elements": [ 11, 14, 15 ], "cost": 114 }, { "id": 4, "elements": [ 11, 12, 13, 14, 15 ], "cost": 10 }, { "id": 5, "elements": [ 13, 15, 16, 17 ], "cost": 348 }, { "id": 6, "elements": [ 10, 12, 15, 16, 17 ], "cost": 380 }, { "id": 7, "elements": [ 5, 6, 7, 8, 9 ], "cost": 400 }, { "id": 8, "elements": [ 5, 6, 7, 9 ], "cost": 264 }, { "id": 9, "elements": [ 9, 10, 11, 12, 13, 14 ], "cost": 288 }, { "id": 10, "elements": [ 13, 15, 16, 17, 18, 19 ], "cost": 330 }, { "id": 11, "elements": [ 4, 5, 8 ], "cost": 75 }, { "id": 12, "elements": [ 14, 15, 16, 19 ], "cost": 316 }, { "id": 13, "elements": [ 9, 11, 13 ], "cost": 18 }, { "id": 14, "elements": [ 1, 2, 3, 4 ], "cost": 248 }, { "id": 15, "elements": [ 6, 10, 11, 13, 14 ], "cost": 10 }, { "id": 16, "elements": [ 1, 2 ], "cost": 114 }, { "id": 17, "elements": [ 17, 18, 19 ], "cost": 78 }, { "id": 18, "elements": [ 4, 5, 6 ], "cost": 213 }, { "id": 19, "elements": [ 3, 9, 10 ], "cost": 9 }, { "id": 20, "elements": [ 2, 4, 5, 7, 8 ], "cost": 155 }, { "id": 21, "elements": [ 13, 15, 16 ], "cost": 240 }, { "id": 22, "elements": [ 1, 2, 3, 4, 5 ], "cost": 445 }, { "id": 23, "elements": [ 11, 13, 18 ], "cost": 237 }, { "id": 24, "elements": [ 15, 16, 17, 19 ], "cost": 144 }, { "id": 25, "elements": [ 14, 15, 16, 18, 19 ], "cost": 425 }, { "id": 26, "elements": [ 5, 6, 7, 8 ], "cost": 256 }, { "id": 27, "elements": [ 8, 9, 10, 11 ], "cost": 108 }, { "id": 28, "elements": [ 5, 6, 7 ], "cost": 165 }, { "id": 29, "elements": [ 15, 16, 17, 18, 19 ], "cost": 190 }, { "id": 30, "elements": [ 9, 11, 12, 13 ], "cost": 232 }, { "id": 31, "elements": [ 10, 12, 13, 14 ], "cost": 228 }, { "id": 32, "elements": [ 1, 3, 6 ], "cost": 198 }, { "id": 33, "elements": [ 1, 2, 5, 6 ], "cost": 20 }, { "id": 34, "elements": [ 9, 10, 11, 14 ], "cost": 352 }, { "id": 35, "elements": [ 7, 8, 9, 12, 13 ], "cost": 170 }, { "id": 36, "elements": [ 9, 10 ], "cost": 104 }, { "id": 37, "elements": [ 14, 16, 17, 18, 19 ], "cost": 255 }, { "id": 38, "elements": [ 7, 8, 9 ], "cost": 291 }, { "id": 39, "elements": [ 12, 14, 15, 17 ], "cost": 360 }, { "id": 40, "elements": [ 14, 15, 16 ], "cost": 222 }, { "id": 41, "elements": [ 6, 9, 11, 12, 13 ], "cost": 270 }, { "id": 42, "elements": [ 11, 13, 14, 15 ], "cost": 292 }, { "id": 43, "elements": [ 3, 4, 5, 6, 8 ], "cost": 390 }, { "id": 44, "elements": [ 11, 13, 14, 17 ], "cost": 100 }, { "id": 45, "elements": [ 8, 10, 11, 14 ], "cost": 56 }, { "id": 46, "elements": [ 3, 4, 6, 7 ], "cost": 36 }, { "id": 47, "elements": [ 9, 10, 11, 12, 14 ], "cost": 335 }, { "id": 48, "elements": [ 10, 13 ], "cost": 42 }, { "id": 49, "elements": [ 2, 3, 4, 5, 7, 9 ], "cost": 162 }, { "id": 50, "elements": [ 14, 15, 17 ], "cost": 78 }, { "id": 51, "elements": [ 8, 10, 11 ], "cost": 177 }, { "id": 52, "elements": [ 7, 9, 10, 11, 12 ], "cost": 400 }, { "id": 53, "elements": [ 12, 14, 15, 16, 17 ], "cost": 30 }, { "id": 54, "elements": [ 4, 5, 7 ], "cost": 27 }, { "id": 55, "elements": [ 4, 6, 7, 8, 10 ], "cost": 485 }, { "id": 56, "elements": [ 9, 11 ], "cost": 54 }, { "id": 57, "elements": [ 11, 14, 15, 16 ], "cost": 176 }, { "id": 58, "elements": [ 3, 5, 6, 7, 10 ], "cost": 480 }, { "id": 59, "elements": [ 12, 14, 15, 18 ], "cost": 72 }, { "id": 60, "elements": [ 5, 7, 8, 11 ], "cost": 152 }, { "id": 61, "elements": [ 11, 14 ], "cost": 4 }, { "id": 62, "elements": [ 3, 5, 7, 8 ], "cost": 276 }, { "id": 63, "elements": [ 2, 3, 4 ], "cost": 276 }, { "id": 64, "elements": [ 4, 5, 6, 8, 10 ], "cost": 10 }, { "id": 65, "elements": [ 4, 8, 10 ], "cost": 39 }, { "id": 66, "elements": [ 3, 4, 6, 7, 8 ], "cost": 405 }, { "id": 67, "elements": [ 6, 7, 10 ], "cost": 90 }, { "id": 68, "elements": [ 8, 11, 13 ], "cost": 6 }, { "id": 69, "elements": [ 13, 14, 16 ], "cost": 240 }, { "id": 70, "elements": [ 3, 4, 5 ], "cost": 192 }, { "id": 71, "elements": [ 1, 2, 3, 5, 6, 7 ], "cost": 252 }, { "id": 72, "elements": [ 11, 13, 14, 15, 16, 17 ], "cost": 246 }, { "id": 73, "elements": [ 12, 13, 15, 16, 17, 18 ], "cost": 390 }, { "id": 74, "elements": [ 1, 2, 3, 5 ], "cost": 380 }, { "id": 75, "elements": [ 5, 7, 8, 9 ], "cost": 80 }, { "id": 76, "elements": [ 2, 4, 7 ], "cost": 171 }, { "id": 77, "elements": [ 7, 9, 11, 12, 13 ], "cost": 270 }, { "id": 78, "elements": [ 8, 9, 11, 12 ], "cost": 108 }, { "id": 79, "elements": [ 6, 9, 10, 11 ], "cost": 268 }, { "id": 80, "elements": [ 12, 15, 16 ], "cost": 75 }, { "id": 81, "elements": [ 3, 8 ], "cost": 46 }, { "id": 82, "elements": [ 4, 6, 7, 9 ], "cost": 372 }, { "id": 83, "elements": [ 12, 13, 14 ], "cost": 174 }, { "id": 84, "elements": [ 4, 5, 6, 7, 9, 11 ], "cost": 342 }, { "id": 85, "elements": [ 10, 14, 15, 16 ], "cost": 196 }, { "id": 86, "elements": [ 1, 3, 4, 6 ], "cost": 376 }, { "id": 87, "elements": [ 16, 18 ], "cost": 104 }, { "id": 88, "elements": [ 12, 13, 14, 15, 17 ], "cost": 265 }, { "id": 89, "elements": [ 2, 3, 4, 5, 6 ], "cost": 485 }, { "id": 90, "elements": [ 13, 15, 16, 17, 18 ], "cost": 275 }, { "id": 91, "elements": [ 4, 7, 8, 9, 11 ], "cost": 430 }, { "id": 92, "elements": [ 7, 8, 10, 12 ], "cost": 300 }, { "id": 93, "elements": [ 2, 5, 6, 7, 8, 9 ], "cost": 66 }, { "id": 94, "elements": [ 16, 17, 18, 19 ], "cost": 376 }, { "id": 95, "elements": [ 2, 3, 5, 6, 7 ], "cost": 480 }, { "id": 96, "elements": [ 5, 8, 10 ], "cost": 39 }, { "id": 97, "elements": [ 14, 15 ], "cost": 188 }, { "id": 98, "elements": [ 4, 5, 6, 7 ], "cost": 60 }, { "id": 99, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 19 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0010_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0010_bag_stats.png" }, "solution": [ 14, 15, 17, 75, 80 ], "obj": 491.0, "instance_variant": { "num_elements": 19, "num_sets": 117, "sets": [ { "id": "S1", "elements": [ 1, 2, 4, 5 ], "cost": 20 }, { "id": "S2", "elements": [ 1, 4, 5 ], "cost": 24 }, { "id": "S3", "elements": [ 11, 14, 15 ], "cost": 114 }, { "id": "S4", "elements": [ 11, 12, 13, 14, 15 ], "cost": 10 }, { "id": "S5", "elements": [ 13, 15, 16, 17 ], "cost": 348 }, { "id": "S6", "elements": [ 10, 12, 15, 16, 17 ], "cost": 380 }, { "id": "S7", "elements": [ 5, 6, 7, 8, 9 ], "cost": 400 }, { "id": "S8", "elements": [ 5, 6, 7, 9 ], "cost": 264 }, { "id": "S9", "elements": [ 9, 10, 11, 12, 13, 14 ], "cost": 288 }, { "id": "S10", "elements": [ 13, 15, 16, 17, 18, 19 ], "cost": 330 }, { "id": "S11", "elements": [ 4, 5, 8 ], "cost": 75 }, { "id": "S12", "elements": [ 14, 15, 16, 19 ], "cost": 316 }, { "id": "S13", "elements": [ 9, 11, 13 ], "cost": 18 }, { "id": "S14", "elements": [ 1, 2, 3, 4 ], "cost": 248 }, { "id": "S15", "elements": [ 6, 10, 11, 13, 14 ], "cost": 10 }, { "id": "S16", "elements": [ 1, 2 ], "cost": 114 }, { "id": "S17", "elements": [ 17, 18, 19 ], "cost": 78 }, { "id": "S18", "elements": [ 4, 5, 6 ], "cost": 213 }, { "id": "S19", "elements": [ 3, 9, 10 ], "cost": 9 }, { "id": "S20", "elements": [ 2, 4, 5, 7, 8 ], "cost": 155 }, { "id": "S21", "elements": [ 13, 15, 16 ], "cost": 240 }, { "id": "S22", "elements": [ 1, 2, 3, 4, 5 ], "cost": 445 }, { "id": "S23", "elements": [ 11, 13, 18 ], "cost": 237 }, { "id": "S24", "elements": [ 15, 16, 17, 19 ], "cost": 144 }, { "id": "S25", "elements": [ 14, 15, 16, 18, 19 ], "cost": 425 }, { "id": "S26", "elements": [ 5, 6, 7, 8 ], "cost": 256 }, { "id": "S27", "elements": [ 8, 9, 10, 11 ], "cost": 108 }, { "id": "S28", "elements": [ 5, 6, 7 ], "cost": 165 }, { "id": "S29", "elements": [ 15, 16, 17, 18, 19 ], "cost": 190 }, { "id": "S30", "elements": [ 9, 11, 12, 13 ], "cost": 232 }, { "id": "S31", "elements": [ 10, 12, 13, 14 ], "cost": 228 }, { "id": "S32", "elements": [ 1, 3, 6 ], "cost": 198 }, { "id": "S33", "elements": [ 1, 2, 5, 6 ], "cost": 20 }, { "id": "S34", "elements": [ 9, 10, 11, 14 ], "cost": 352 }, { "id": "S35", "elements": [ 7, 8, 9, 12, 13 ], "cost": 170 }, { "id": "S36", "elements": [ 9, 10 ], "cost": 104 }, { "id": "S37", "elements": [ 14, 16, 17, 18, 19 ], "cost": 255 }, { "id": "S38", "elements": [ 7, 8, 9 ], "cost": 291 }, { "id": "S39", "elements": [ 12, 14, 15, 17 ], "cost": 360 }, { "id": "S40", "elements": [ 14, 15, 16 ], "cost": 222 }, { "id": "S41", "elements": [ 6, 9, 11, 12, 13 ], "cost": 270 }, { "id": "S42", "elements": [ 11, 13, 14, 15 ], "cost": 292 }, { "id": "S43", "elements": [ 3, 4, 5, 6, 8 ], "cost": 390 }, { "id": "S44", "elements": [ 11, 13, 14, 17 ], "cost": 100 }, { "id": "S45", "elements": [ 8, 10, 11, 14 ], "cost": 56 }, { "id": "S46", "elements": [ 3, 4, 6, 7 ], "cost": 36 }, { "id": "S47", "elements": [ 9, 10, 11, 12, 14 ], "cost": 335 }, { "id": "S48", "elements": [ 10, 13 ], "cost": 42 }, { "id": "S49", "elements": [ 2, 3, 4, 5, 7, 9 ], "cost": 162 }, { "id": "S50", "elements": [ 14, 15, 17 ], "cost": 78 }, { "id": "S51", "elements": [ 8, 10, 11 ], "cost": 177 }, { "id": "S52", "elements": [ 7, 9, 10, 11, 12 ], "cost": 400 }, { "id": "S53", "elements": [ 12, 14, 15, 16, 17 ], "cost": 30 }, { "id": "S54", "elements": [ 4, 5, 7 ], "cost": 27 }, { "id": "S55", "elements": [ 4, 6, 7, 8, 10 ], "cost": 485 }, { "id": "S56", "elements": [ 9, 11 ], "cost": 54 }, { "id": "S57", "elements": [ 11, 14, 15, 16 ], "cost": 176 }, { "id": "S58", "elements": [ 3, 5, 6, 7, 10 ], "cost": 480 }, { "id": "S59", "elements": [ 12, 14, 15, 18 ], "cost": 72 }, { "id": "S60", "elements": [ 5, 7, 8, 11 ], "cost": 152 }, { "id": "S61", "elements": [ 11, 14 ], "cost": 4 }, { "id": "S62", "elements": [ 3, 5, 7, 8 ], "cost": 276 }, { "id": "S63", "elements": [ 2, 3, 4 ], "cost": 276 }, { "id": "S64", "elements": [ 4, 5, 6, 8, 10 ], "cost": 10 }, { "id": "S65", "elements": [ 4, 8, 10 ], "cost": 39 }, { "id": "S66", "elements": [ 3, 4, 6, 7, 8 ], "cost": 405 }, { "id": "S67", "elements": [ 6, 7, 10 ], "cost": 90 }, { "id": "S68", "elements": [ 8, 11, 13 ], "cost": 6 }, { "id": "S69", "elements": [ 13, 14, 16 ], "cost": 240 }, { "id": "S70", "elements": [ 3, 4, 5 ], "cost": 192 }, { "id": "S71", "elements": [ 1, 2, 3, 5, 6, 7 ], "cost": 252 }, { "id": "S72", "elements": [ 11, 13, 14, 15, 16, 17 ], "cost": 246 }, { "id": "S73", "elements": [ 12, 13, 15, 16, 17, 18 ], "cost": 390 }, { "id": "S74", "elements": [ 1, 2, 3, 5 ], "cost": 380 }, { "id": "S75", "elements": [ 5, 7, 8, 9 ], "cost": 80 }, { "id": "S76", "elements": [ 2, 4, 7 ], "cost": 171 }, { "id": "S77", "elements": [ 7, 9, 11, 12, 13 ], "cost": 270 }, { "id": "S78", "elements": [ 8, 9, 11, 12 ], "cost": 108 }, { "id": "S79", "elements": [ 6, 9, 10, 11 ], "cost": 268 }, { "id": "S80", "elements": [ 12, 15, 16 ], "cost": 75 }, { "id": "S81", "elements": [ 3, 8 ], "cost": 46 }, { "id": "S82", "elements": [ 4, 6, 7, 9 ], "cost": 372 }, { "id": "S83", "elements": [ 12, 13, 14 ], "cost": 174 }, { "id": "S84", "elements": [ 4, 5, 6, 7, 9, 11 ], "cost": 342 }, { "id": "S85", "elements": [ 10, 14, 15, 16 ], "cost": 196 }, { "id": "S86", "elements": [ 1, 3, 4, 6 ], "cost": 376 }, { "id": "S87", "elements": [ 16, 18 ], "cost": 104 }, { "id": "S88", "elements": [ 12, 13, 14, 15, 17 ], "cost": 265 }, { "id": "S89", "elements": [ 2, 3, 4, 5, 6 ], "cost": 485 }, { "id": "S90", "elements": [ 13, 15, 16, 17, 18 ], "cost": 275 }, { "id": "S91", "elements": [ 4, 7, 8, 9, 11 ], "cost": 430 }, { "id": "S92", "elements": [ 7, 8, 10, 12 ], "cost": 300 }, { "id": "S93", "elements": [ 2, 5, 6, 7, 8, 9 ], "cost": 66 }, { "id": "S94", "elements": [ 16, 17, 18, 19 ], "cost": 376 }, { "id": "S95", "elements": [ 2, 3, 5, 6, 7 ], "cost": 480 }, { "id": "S96", "elements": [ 5, 8, 10 ], "cost": 39 }, { "id": "S97", "elements": [ 14, 15 ], "cost": 188 }, { "id": "S98", "elements": [ 4, 5, 6, 7 ], "cost": 60 }, { "id": "S99", "elements": [ 1 ], "cost": 10000 }, { "id": "S100", "elements": [ 2 ], "cost": 10000 }, { "id": "S101", "elements": [ 3 ], "cost": 10000 }, { "id": "S102", "elements": [ 4 ], "cost": 10000 }, { "id": "S103", "elements": [ 5 ], "cost": 10000 }, { "id": "S104", "elements": [ 6 ], "cost": 10000 }, { "id": "S105", "elements": [ 7 ], "cost": 10000 }, { "id": "S106", "elements": [ 8 ], "cost": 10000 }, { "id": "S107", "elements": [ 9 ], "cost": 10000 }, { "id": "S108", "elements": [ 10 ], "cost": 10000 }, { "id": "S109", "elements": [ 11 ], "cost": 10000 }, { "id": "S110", "elements": [ 12 ], "cost": 10000 }, { "id": "S111", "elements": [ 13 ], "cost": 10000 }, { "id": "S112", "elements": [ 14 ], "cost": 10000 }, { "id": "S113", "elements": [ 15 ], "cost": 10000 }, { "id": "S114", "elements": [ 16 ], "cost": 10000 }, { "id": "S115", "elements": [ 17 ], "cost": 10000 }, { "id": "S116", "elements": [ 18 ], "cost": 10000 }, { "id": "S117", "elements": [ 19 ], "cost": 10000 } ] }, "solution_variant": [ "S14", "S15", "S17", "S75", "S80" ], "context_index": 11, "input_format": "json", "input_index_base": 1 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "Someone on the crew is lining up performers and needs to decide which stages to book so every act has one assigned spot and nobody gets assigned twice. Once the stage selections are made, the total expense is just the sum of the costs for those booked stages, and the aim is to keep that sum as low as possible. The full list of acts, stage options, and their costs appears below.\n\n{\n \"total_performers\": 22,\n \"total_stage_options\": 127,\n \"sets\": [\n {\n \"stage_id\": \"S1\",\n \"stage_performers\": [\n \"D\",\n \"E\",\n \"I\"\n ],\n \"stage_cost\": 102\n },\n {\n \"stage_id\": \"S2\",\n \"stage_performers\": [\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"Q\"\n ],\n \"stage_cost\": 63\n },\n {\n \"stage_id\": \"S3\",\n \"stage_performers\": [\n \"M\",\n \"Q\",\n \"S\",\n \"U\"\n ],\n \"stage_cost\": 140\n },\n {\n \"stage_id\": \"S4\",\n \"stage_performers\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\"\n ],\n \"stage_cost\": 402\n },\n {\n \"stage_id\": \"S5\",\n \"stage_performers\": [\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"L\",\n \"M\",\n \"Q\"\n ],\n \"stage_cost\": 217\n },\n {\n \"stage_id\": \"S6\",\n \"stage_performers\": [\n \"N\",\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\"\n ],\n \"stage_cost\": 540\n },\n {\n \"stage_id\": \"S7\",\n \"stage_performers\": [\n \"A\",\n \"C\",\n \"F\",\n \"I\"\n ],\n \"stage_cost\": 388\n },\n {\n \"stage_id\": \"S8\",\n \"stage_performers\": [\n \"E\",\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"O\"\n ],\n \"stage_cost\": 492\n },\n {\n \"stage_id\": \"S9\",\n \"stage_performers\": [\n \"F\",\n \"G\",\n \"J\",\n \"K\",\n \"L\",\n \"M\"\n ],\n \"stage_cost\": 468\n },\n {\n \"stage_id\": \"S10\",\n \"stage_performers\": [\n \"S\",\n \"T\",\n \"U\",\n \"V\"\n ],\n \"stage_cost\": 244\n },\n {\n \"stage_id\": \"S11\",\n \"stage_performers\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"I\"\n ],\n \"stage_cost\": 455\n },\n {\n \"stage_id\": \"S12\",\n \"stage_performers\": [\n \"O\",\n \"R\"\n ],\n \"stage_cost\": 174\n },\n {\n \"stage_id\": \"S13\",\n \"stage_performers\": [\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"Q\",\n \"S\"\n ],\n \"stage_cost\": 595\n },\n {\n \"stage_id\": \"S14\",\n \"stage_performers\": [\n \"B\",\n \"D\",\n \"F\",\n \"G\",\n \"H\"\n ],\n \"stage_cost\": 495\n },\n {\n \"stage_id\": \"S15\",\n \"stage_performers\": [\n \"I\",\n \"J\",\n \"M\"\n ],\n \"stage_cost\": 258\n },\n {\n \"stage_id\": \"S16\",\n \"stage_performers\": [\n \"Q\",\n \"S\",\n \"T\"\n ],\n \"stage_cost\": 129\n },\n {\n \"stage_id\": \"S17\",\n \"stage_performers\": [\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\"\n ],\n \"stage_cost\": 294\n },\n {\n \"stage_id\": \"S18\",\n \"stage_performers\": [\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"V\"\n ],\n \"stage_cost\": 75\n },\n {\n \"stage_id\": \"S19\",\n \"stage_performers\": [\n \"P\",\n \"T\",\n \"V\"\n ],\n \"stage_cost\": 144\n },\n {\n \"stage_id\": \"S20\",\n \"stage_performers\": [\n \"D\",\n \"K\",\n \"O\"\n ],\n \"stage_cost\": 24\n },\n {\n \"stage_id\": \"S21\",\n \"stage_performers\": [\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"L\"\n ],\n \"stage_cost\": 85\n },\n {\n \"stage_id\": \"S22\",\n \"stage_performers\": [\n \"N\",\n \"Q\",\n \"T\",\n \"V\"\n ],\n \"stage_cost\": 164\n },\n {\n \"stage_id\": \"S23\",\n \"stage_performers\": [\n \"I\",\n \"J\",\n \"L\"\n ],\n \"stage_cost\": 231\n },\n {\n \"stage_id\": \"S24\",\n \"stage_performers\": [\n \"Q\",\n \"U\",\n \"V\"\n ],\n \"stage_cost\": 123\n },\n {\n \"stage_id\": \"S25\",\n \"stage_performers\": [\n \"C\",\n \"D\",\n \"E\",\n \"G\"\n ],\n \"stage_cost\": 112\n },\n {\n \"stage_id\": \"S26\",\n \"stage_performers\": [\n \"J\",\n \"K\",\n \"N\",\n \"O\"\n ],\n \"stage_cost\": 376\n },\n {\n \"stage_id\": \"S27\",\n \"stage_performers\": [\n \"D\",\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"K\"\n ],\n \"stage_cost\": 510\n },\n {\n \"stage_id\": \"S28\",\n \"stage_performers\": [\n \"E\",\n \"F\",\n \"G\",\n \"H\",\n \"K\",\n \"L\"\n ],\n \"stage_cost\": 306\n },\n {\n \"stage_id\": \"S29\",\n \"stage_performers\": [\n \"I\",\n \"K\",\n \"M\"\n ],\n \"stage_cost\": 213\n },\n {\n \"stage_id\": \"S30\",\n \"stage_performers\": [\n \"A\",\n \"D\",\n \"G\",\n \"H\"\n ],\n \"stage_cost\": 364\n },\n {\n \"stage_id\": \"S31\",\n \"stage_performers\": [\n \"R\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"stage_cost\": 76\n },\n {\n \"stage_id\": \"S32\",\n \"stage_performers\": [\n \"B\",\n \"H\",\n \"I\"\n ],\n \"stage_cost\": 222\n },\n {\n \"stage_id\": \"S33\",\n \"stage_performers\": [\n \"D\",\n \"F\",\n \"J\"\n ],\n \"stage_cost\": 66\n },\n {\n \"stage_id\": \"S34\",\n \"stage_performers\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"H\"\n ],\n \"stage_cost\": 336\n },\n {\n \"stage_id\": \"S35\",\n \"stage_performers\": [\n \"M\",\n \"O\",\n \"R\"\n ],\n \"stage_cost\": 36\n },\n {\n \"stage_id\": \"S36\",\n \"stage_performers\": [\n \"O\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"stage_cost\": 498\n },\n {\n \"stage_id\": \"S37\",\n \"stage_performers\": [\n \"O\",\n \"Q\",\n \"R\",\n \"S\",\n \"V\"\n ],\n \"stage_cost\": 65\n },\n {\n \"stage_id\": \"S38\",\n \"stage_performers\": [\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"O\"\n ],\n \"stage_cost\": 552\n },\n {\n \"stage_id\": \"S39\",\n \"stage_performers\": [\n \"M\",\n \"P\",\n \"R\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"stage_cost\": 378\n },\n {\n \"stage_id\": \"S40\",\n \"stage_performers\": [\n \"S\",\n \"U\"\n ],\n \"stage_cost\": 168\n },\n {\n \"stage_id\": \"S41\",\n \"stage_performers\": [\n \"E\",\n \"I\",\n \"K\"\n ],\n \"stage_cost\": 231\n },\n {\n \"stage_id\": \"S42\",\n \"stage_performers\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\"\n ],\n \"stage_cost\": 105\n },\n {\n \"stage_id\": \"S43\",\n \"stage_performers\": [\n \"E\",\n \"K\",\n \"N\"\n ],\n \"stage_cost\": 195\n },\n {\n \"stage_id\": \"S44\",\n \"stage_performers\": [\n \"P\",\n \"Q\",\n \"S\",\n \"V\"\n ],\n \"stage_cost\": 24\n },\n {\n \"stage_id\": \"S45\",\n \"stage_performers\": [\n \"N\",\n \"O\",\n \"P\",\n \"Q\",\n \"R\",\n \"T\"\n ],\n \"stage_cost\": 276\n },\n {\n \"stage_id\": \"S46\",\n \"stage_performers\": [\n \"N\",\n \"O\",\n \"P\",\n \"Q\",\n \"S\"\n ],\n \"stage_cost\": 195\n },\n {\n \"stage_id\": \"S47\",\n \"stage_performers\": [\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"I\",\n \"K\"\n ],\n \"stage_cost\": 60\n },\n {\n \"stage_id\": \"S48\",\n \"stage_performers\": [\n \"E\",\n \"G\",\n \"H\",\n \"I\",\n \"L\"\n ],\n \"stage_cost\": 275\n },\n {\n \"stage_id\": \"S49\",\n \"stage_performers\": [\n \"B\",\n \"C\",\n \"E\",\n \"F\",\n \"G\",\n \"K\"\n ],\n \"stage_cost\": 288\n },\n {\n \"stage_id\": \"S50\",\n \"stage_performers\": [\n \"Q\",\n \"S\",\n \"T\",\n \"U\",\n \"V\"\n ],\n \"stage_cost\": 280\n },\n {\n \"stage_id\": \"S51\",\n \"stage_performers\": [\n \"I\",\n \"J\",\n \"K\",\n \"L\"\n ],\n \"stage_cost\": 116\n },\n {\n \"stage_id\": \"S52\",\n \"stage_performers\": [\n \"C\",\n \"D\",\n \"H\"\n ],\n \"stage_cost\": 42\n },\n {\n \"stage_id\": \"S53\",\n \"stage_performers\": [\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\"\n ],\n \"stage_cost\": 540\n },\n {\n \"stage_id\": \"S54\",\n \"stage_performers\": [\n \"G\",\n \"J\",\n \"K\",\n \"M\",\n \"N\"\n ],\n \"stage_cost\": 250\n },\n {\n \"stage_id\": \"S55\",\n \"stage_performers\": [\n \"T\",\n \"U\",\n \"V\"\n ],\n \"stage_cost\": 156\n },\n {\n \"stage_id\": \"S56\",\n \"stage_performers\": [\n \"D\",\n \"E\",\n \"F\",\n \"G\"\n ],\n \"stage_cost\": 76\n },\n {\n \"stage_id\": \"S57\",\n \"stage_performers\": [\n \"E\",\n \"I\",\n \"J\",\n \"M\"\n ],\n \"stage_cost\": 128\n },\n {\n \"stage_id\": \"S58\",\n \"stage_performers\": [\n \"L\",\n \"N\",\n \"O\",\n \"P\",\n \"S\"\n ],\n \"stage_cost\": 180\n },\n {\n \"stage_id\": \"S59\",\n \"stage_performers\": [\n \"G\",\n \"I\",\n \"K\",\n \"M\"\n ],\n \"stage_cost\": 48\n },\n {\n \"stage_id\": \"S60\",\n \"stage_performers\": [\n \"A\",\n \"B\",\n \"F\"\n ],\n \"stage_cost\": 6\n },\n {\n \"stage_id\": \"S61\",\n \"stage_performers\": [\n \"K\",\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"Q\"\n ],\n \"stage_cost\": 414\n },\n {\n \"stage_id\": \"S62\",\n \"stage_performers\": [\n \"G\",\n \"H\",\n \"K\",\n \"M\",\n \"O\"\n ],\n \"stage_cost\": 345\n },\n {\n \"stage_id\": \"S63\",\n \"stage_performers\": [\n \"I\",\n \"K\",\n \"L\",\n \"N\",\n \"O\"\n ],\n \"stage_cost\": 440\n },\n {\n \"stage_id\": \"S64\",\n \"stage_performers\": [\n \"L\",\n \"N\",\n \"P\"\n ],\n \"stage_cost\": 165\n },\n {\n \"stage_id\": \"S65\",\n \"stage_performers\": [\n \"C\",\n \"E\",\n \"F\",\n \"I\"\n ],\n \"stage_cost\": 280\n },\n {\n \"stage_id\": \"S66\",\n \"stage_performers\": [\n \"O\",\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\"\n ],\n \"stage_cost\": 6\n },\n {\n \"stage_id\": \"S67\",\n \"stage_performers\": [\n \"H\",\n \"J\",\n \"K\",\n \"L\",\n \"M\"\n ],\n \"stage_cost\": 10\n },\n {\n \"stage_id\": \"S68\",\n \"stage_performers\": [\n \"N\",\n \"P\",\n \"S\"\n ],\n \"stage_cost\": 159\n },\n {\n \"stage_id\": \"S69\",\n \"stage_performers\": [\n \"K\",\n \"N\",\n \"R\"\n ],\n \"stage_cost\": 183\n },\n {\n \"stage_id\": \"S70\",\n \"stage_performers\": [\n \"H\",\n \"I\",\n \"J\",\n \"M\"\n ],\n \"stage_cost\": 228\n },\n {\n \"stage_id\": \"S71\",\n \"stage_performers\": [\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"S\"\n ],\n \"stage_cost\": 130\n },\n {\n \"stage_id\": \"S72\",\n \"stage_performers\": [\n \"F\",\n \"J\",\n \"L\"\n ],\n \"stage_cost\": 15\n },\n {\n \"stage_id\": \"S73\",\n \"stage_performers\": [\n \"J\",\n \"K\",\n \"L\"\n ],\n \"stage_cost\": 48\n },\n {\n \"stage_id\": \"S74\",\n \"stage_performers\": [\n \"O\",\n \"R\",\n \"S\",\n \"T\"\n ],\n \"stage_cost\": 400\n },\n {\n \"stage_id\": \"S75\",\n \"stage_performers\": [\n \"N\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"U\"\n ],\n \"stage_cost\": 102\n },\n {\n \"stage_id\": \"S76\",\n \"stage_performers\": [\n \"L\",\n \"N\",\n \"R\"\n ],\n \"stage_cost\": 132\n },\n {\n \"stage_id\": \"S77\",\n \"stage_performers\": [\n \"D\",\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"J\"\n ],\n \"stage_cost\": 540\n },\n {\n \"stage_id\": \"S78\",\n \"stage_performers\": [\n \"L\",\n \"O\",\n \"P\"\n ],\n \"stage_cost\": 270\n },\n {\n \"stage_id\": \"S79\",\n \"stage_performers\": [\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"V\"\n ],\n \"stage_cost\": 85\n },\n {\n \"stage_id\": \"S80\",\n \"stage_performers\": [\n \"H\",\n \"J\",\n \"L\",\n \"N\"\n ],\n \"stage_cost\": 48\n },\n {\n \"stage_id\": \"S81\",\n \"stage_performers\": [\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"O\"\n ],\n \"stage_cost\": 396\n },\n {\n \"stage_id\": \"S82\",\n \"stage_performers\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\"\n ],\n \"stage_cost\": 216\n },\n {\n \"stage_id\": \"S83\",\n \"stage_performers\": [\n \"A\",\n \"B\",\n \"D\"\n ],\n \"stage_cost\": 126\n },\n {\n \"stage_id\": \"S84\",\n \"stage_performers\": [\n \"O\",\n \"P\",\n \"Q\"\n ],\n \"stage_cost\": 99\n },\n {\n \"stage_id\": \"S85\",\n \"stage_performers\": [\n \"H\",\n \"I\",\n \"K\",\n \"M\",\n \"N\",\n \"O\"\n ],\n \"stage_cost\": 282\n },\n {\n \"stage_id\": \"S86\",\n \"stage_performers\": [\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"M\"\n ],\n \"stage_cost\": 375\n },\n {\n \"stage_id\": \"S87\",\n \"stage_performers\": [\n \"B\",\n \"C\",\n \"E\",\n \"F\"\n ],\n \"stage_cost\": 336\n },\n {\n \"stage_id\": \"S88\",\n \"stage_performers\": [\n \"D\",\n \"E\",\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\"\n ],\n \"stage_cost\": 504\n },\n {\n \"stage_id\": \"S89\",\n \"stage_performers\": [\n \"H\",\n \"K\",\n \"L\",\n \"N\",\n \"O\",\n \"P\"\n ],\n \"stage_cost\": 390\n },\n {\n \"stage_id\": \"S90\",\n \"stage_performers\": [\n \"B\",\n \"C\",\n \"F\",\n \"G\",\n \"H\",\n \"I\"\n ],\n \"stage_cost\": 534\n },\n {\n \"stage_id\": \"S91\",\n \"stage_performers\": [\n \"L\",\n \"M\",\n \"N\"\n ],\n \"stage_cost\": 30\n },\n {\n \"stage_id\": \"S92\",\n \"stage_performers\": [\n \"P\",\n \"Q\",\n \"S\"\n ],\n \"stage_cost\": 105\n },\n {\n \"stage_id\": \"S93\",\n \"stage_performers\": [\n \"N\",\n \"O\",\n \"P\",\n \"Q\",\n \"S\",\n \"T\"\n ],\n \"stage_cost\": 426\n },\n {\n \"stage_id\": \"S94\",\n \"stage_performers\": [\n \"D\",\n \"E\",\n \"G\"\n ],\n \"stage_cost\": 57\n },\n {\n \"stage_id\": \"S95\",\n \"stage_performers\": [\n \"F\",\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"L\"\n ],\n \"stage_cost\": 414\n },\n {\n \"stage_id\": \"S96\",\n \"stage_performers\": [\n \"S\",\n \"T\",\n \"U\"\n ],\n \"stage_cost\": 33\n },\n {\n \"stage_id\": \"S97\",\n \"stage_performers\": [\n \"O\",\n \"Q\",\n \"R\",\n \"S\",\n \"U\",\n \"V\"\n ],\n \"stage_cost\": 12\n },\n {\n \"stage_id\": \"S98\",\n \"stage_performers\": [\n \"C\",\n \"F\",\n \"G\",\n \"I\",\n \"J\",\n \"K\"\n ],\n \"stage_cost\": 600\n },\n {\n \"stage_id\": \"S99\",\n \"stage_performers\": [\n \"J\",\n \"K\",\n \"M\",\n \"N\",\n \"P\"\n ],\n \"stage_cost\": 285\n },\n {\n \"stage_id\": \"S100\",\n \"stage_performers\": [\n \"G\",\n \"J\",\n \"L\"\n ],\n \"stage_cost\": 153\n },\n {\n \"stage_id\": \"S101\",\n \"stage_performers\": [\n \"E\",\n \"G\",\n \"I\"\n ],\n \"stage_cost\": 186\n },\n {\n \"stage_id\": \"S102\",\n \"stage_performers\": [\n \"D\",\n \"F\",\n \"H\"\n ],\n \"stage_cost\": 219\n },\n {\n \"stage_id\": \"S103\",\n \"stage_performers\": [\n \"B\",\n \"D\",\n \"E\",\n \"G\"\n ],\n \"stage_cost\": 204\n },\n {\n \"stage_id\": \"S104\",\n \"stage_performers\": [\n \"D\",\n \"E\",\n \"F\",\n \"J\",\n \"L\"\n ],\n \"stage_cost\": 65\n },\n {\n \"stage_id\": \"S105\",\n \"stage_performers\": [\n \"A\",\n \"C\",\n \"E\",\n \"F\",\n \"G\"\n ],\n \"stage_cost\": 10\n },\n {\n \"stage_id\": \"S106\",\n \"stage_performers\": [\n \"A\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S107\",\n \"stage_performers\": [\n \"B\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S108\",\n \"stage_performers\": [\n \"C\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S109\",\n \"stage_performers\": [\n \"D\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S110\",\n \"stage_performers\": [\n \"E\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S111\",\n \"stage_performers\": [\n \"F\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S112\",\n \"stage_performers\": [\n \"G\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S113\",\n \"stage_performers\": [\n \"H\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S114\",\n \"stage_performers\": [\n \"I\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S115\",\n \"stage_performers\": [\n \"J\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S116\",\n \"stage_performers\": [\n \"K\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S117\",\n \"stage_performers\": [\n \"L\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S118\",\n \"stage_performers\": [\n \"M\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S119\",\n \"stage_performers\": [\n \"N\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S120\",\n \"stage_performers\": [\n \"O\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S121\",\n \"stage_performers\": [\n \"P\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S122\",\n \"stage_performers\": [\n \"Q\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S123\",\n \"stage_performers\": [\n \"R\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S124\",\n \"stage_performers\": [\n \"S\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S125\",\n \"stage_performers\": [\n \"T\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S126\",\n \"stage_performers\": [\n \"U\"\n ],\n \"stage_cost\": 10000\n },\n {\n \"stage_id\": \"S127\",\n \"stage_performers\": [\n \"V\"\n ],\n \"stage_cost\": 10000\n }\n ]\n}\n\nWhen you send back which stages you want to book, just use a little JSON with this shape so it's easy to read and check:\n\n{\n \"solution\": [\"stage_id\", ...]\n}\n\nThink of it like a simple form: \"solution\" is a list of the stage IDs you choose (one per act). The \"stage_id\" placeholder stands in for whatever exact stage identifiers appear in the instance — replace those placeholders with the real IDs when you answer. This block is just a sketch of the shape I expect, not the actual selection.\n\nPlease use the identifiers exactly as they appear in the instance input — no renaming and no new labels.\n- \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 22, "num_sets": 127, "density": 0.17680744452397995, "sets": [ { "id": 1, "elements": [ 4, 5, 9 ], "cost": 102 }, { "id": 2, "elements": [ 10, 11, 12, 13, 14, 15, 17 ], "cost": 63 }, { "id": 3, "elements": [ 13, 17, 19, 21 ], "cost": 140 }, { "id": 4, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 402 }, { "id": 5, "elements": [ 7, 8, 9, 10, 12, 13, 17 ], "cost": 217 }, { "id": 6, "elements": [ 14, 18, 19, 20, 21, 22 ], "cost": 540 }, { "id": 7, "elements": [ 1, 3, 6, 9 ], "cost": 388 }, { "id": 8, "elements": [ 5, 11, 12, 13, 14, 15 ], "cost": 492 }, { "id": 9, "elements": [ 6, 7, 10, 11, 12, 13 ], "cost": 468 }, { "id": 10, "elements": [ 19, 20, 21, 22 ], "cost": 244 }, { "id": 11, "elements": [ 1, 2, 3, 4, 5, 6, 9 ], "cost": 455 }, { "id": 12, "elements": [ 15, 18 ], "cost": 174 }, { "id": 13, "elements": [ 12, 13, 14, 15, 16, 17, 19 ], "cost": 595 }, { "id": 14, "elements": [ 2, 4, 6, 7, 8 ], "cost": 495 }, { "id": 15, "elements": [ 9, 10, 13 ], "cost": 258 }, { "id": 16, "elements": [ 17, 19, 20 ], "cost": 129 }, { "id": 17, "elements": [ 17, 18, 19, 20, 21, 22 ], "cost": 294 }, { "id": 18, "elements": [ 16, 17, 18, 19, 22 ], "cost": 75 }, { "id": 19, "elements": [ 16, 20, 22 ], "cost": 144 }, { "id": 20, "elements": [ 4, 11, 15 ], "cost": 24 }, { "id": 21, "elements": [ 6, 7, 8, 9, 12 ], "cost": 85 }, { "id": 22, "elements": [ 14, 17, 20, 22 ], "cost": 164 }, { "id": 23, "elements": [ 9, 10, 12 ], "cost": 231 }, { "id": 24, "elements": [ 17, 21, 22 ], "cost": 123 }, { "id": 25, "elements": [ 3, 4, 5, 7 ], "cost": 112 }, { "id": 26, "elements": [ 10, 11, 14, 15 ], "cost": 376 }, { "id": 27, "elements": [ 4, 6, 7, 8, 9, 11 ], "cost": 510 }, { "id": 28, "elements": [ 5, 6, 7, 8, 11, 12 ], "cost": 306 }, { "id": 29, "elements": [ 9, 11, 13 ], "cost": 213 }, { "id": 30, "elements": [ 1, 4, 7, 8 ], "cost": 364 }, { "id": 31, "elements": [ 18, 19, 20, 21 ], "cost": 76 }, { "id": 32, "elements": [ 2, 8, 9 ], "cost": 222 }, { "id": 33, "elements": [ 4, 6, 10 ], "cost": 66 }, { "id": 34, "elements": [ 1, 2, 3, 4, 5, 8 ], "cost": 336 }, { "id": 35, "elements": [ 13, 15, 18 ], "cost": 36 }, { "id": 36, "elements": [ 15, 17, 18, 19, 20, 21 ], "cost": 498 }, { "id": 37, "elements": [ 15, 17, 18, 19, 22 ], "cost": 65 }, { "id": 38, "elements": [ 3, 4, 5, 6, 7, 15 ], "cost": 552 }, { "id": 39, "elements": [ 13, 16, 18, 19, 20, 21 ], "cost": 378 }, { "id": 40, "elements": [ 19, 21 ], "cost": 168 }, { "id": 41, "elements": [ 5, 9, 11 ], "cost": 231 }, { "id": 42, "elements": [ 1, 2, 3, 4, 5 ], "cost": 105 }, { "id": 43, "elements": [ 5, 11, 14 ], "cost": 195 }, { "id": 44, "elements": [ 16, 17, 19, 22 ], "cost": 24 }, { "id": 45, "elements": [ 14, 15, 16, 17, 18, 20 ], "cost": 276 }, { "id": 46, "elements": [ 14, 15, 16, 17, 19 ], "cost": 195 }, { "id": 47, "elements": [ 4, 5, 6, 7, 9, 11 ], "cost": 60 }, { "id": 48, "elements": [ 5, 7, 8, 9, 12 ], "cost": 275 }, { "id": 49, "elements": [ 2, 3, 5, 6, 7, 11 ], "cost": 288 }, { "id": 50, "elements": [ 17, 19, 20, 21, 22 ], "cost": 280 }, { "id": 51, "elements": [ 9, 10, 11, 12 ], "cost": 116 }, { "id": 52, "elements": [ 3, 4, 8 ], "cost": 42 }, { "id": 53, "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 540 }, { "id": 54, "elements": [ 7, 10, 11, 13, 14 ], "cost": 250 }, { "id": 55, "elements": [ 20, 21, 22 ], "cost": 156 }, { "id": 56, "elements": [ 4, 5, 6, 7 ], "cost": 76 }, { "id": 57, "elements": [ 5, 9, 10, 13 ], "cost": 128 }, { "id": 58, "elements": [ 12, 14, 15, 16, 19 ], "cost": 180 }, { "id": 59, "elements": [ 7, 9, 11, 13 ], "cost": 48 }, { "id": 60, "elements": [ 1, 2, 6 ], "cost": 6 }, { "id": 61, "elements": [ 11, 13, 14, 15, 16, 17 ], "cost": 414 }, { "id": 62, "elements": [ 7, 8, 11, 13, 15 ], "cost": 345 }, { "id": 63, "elements": [ 9, 11, 12, 14, 15 ], "cost": 440 }, { "id": 64, "elements": [ 12, 14, 16 ], "cost": 165 }, { "id": 65, "elements": [ 3, 5, 6, 9 ], "cost": 280 }, { "id": 66, "elements": [ 15, 18, 19, 20, 21, 22 ], "cost": 6 }, { "id": 67, "elements": [ 8, 10, 11, 12, 13 ], "cost": 10 }, { "id": 68, "elements": [ 14, 16, 19 ], "cost": 159 }, { "id": 69, "elements": [ 11, 14, 18 ], "cost": 183 }, { "id": 70, "elements": [ 8, 9, 10, 13 ], "cost": 228 }, { "id": 71, "elements": [ 13, 14, 15, 16, 19 ], "cost": 130 }, { "id": 72, "elements": [ 6, 10, 12 ], "cost": 15 }, { "id": 73, "elements": [ 10, 11, 12 ], "cost": 48 }, { "id": 74, "elements": [ 15, 18, 19, 20 ], "cost": 400 }, { "id": 75, "elements": [ 14, 16, 17, 18, 19, 21 ], "cost": 102 }, { "id": 76, "elements": [ 12, 14, 18 ], "cost": 132 }, { "id": 77, "elements": [ 4, 6, 7, 8, 9, 10 ], "cost": 540 }, { "id": 78, "elements": [ 12, 15, 16 ], "cost": 270 }, { "id": 79, "elements": [ 17, 18, 19, 20, 22 ], "cost": 85 }, { "id": 80, "elements": [ 8, 10, 12, 14 ], "cost": 48 }, { "id": 81, "elements": [ 10, 11, 12, 13, 14, 15 ], "cost": 396 }, { "id": 82, "elements": [ 1, 2, 3, 4 ], "cost": 216 }, { "id": 83, "elements": [ 1, 2, 4 ], "cost": 126 }, { "id": 84, "elements": [ 15, 16, 17 ], "cost": 99 }, { "id": 85, "elements": [ 8, 9, 11, 13, 14, 15 ], "cost": 282 }, { "id": 86, "elements": [ 9, 10, 11, 12, 13 ], "cost": 375 }, { "id": 87, "elements": [ 2, 3, 5, 6 ], "cost": 336 }, { "id": 88, "elements": [ 4, 5, 7, 8, 9, 10, 11 ], "cost": 504 }, { "id": 89, "elements": [ 8, 11, 12, 14, 15, 16 ], "cost": 390 }, { "id": 90, "elements": [ 2, 3, 6, 7, 8, 9 ], "cost": 534 }, { "id": 91, "elements": [ 12, 13, 14 ], "cost": 30 }, { "id": 92, "elements": [ 16, 17, 19 ], "cost": 105 }, { "id": 93, "elements": [ 14, 15, 16, 17, 19, 20 ], "cost": 426 }, { "id": 94, "elements": [ 4, 5, 7 ], "cost": 57 }, { "id": 95, "elements": [ 6, 8, 9, 10, 11, 12 ], "cost": 414 }, { "id": 96, "elements": [ 19, 20, 21 ], "cost": 33 }, { "id": 97, "elements": [ 15, 17, 18, 19, 21, 22 ], "cost": 12 }, { "id": 98, "elements": [ 3, 6, 7, 9, 10, 11 ], "cost": 600 }, { "id": 99, "elements": [ 10, 11, 13, 14, 16 ], "cost": 285 }, { "id": 100, "elements": [ 7, 10, 12 ], "cost": 153 }, { "id": 101, "elements": [ 5, 7, 9 ], "cost": 186 }, { "id": 102, "elements": [ 4, 6, 8 ], "cost": 219 }, { "id": 103, "elements": [ 2, 4, 5, 7 ], "cost": 204 }, { "id": 104, "elements": [ 4, 5, 6, 10, 12 ], "cost": 65 }, { "id": 105, "elements": [ 1, 3, 5, 6, 7 ], "cost": 10 }, { "id": 106, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 118, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 119, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 120, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 121, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 122, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 123, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 124, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 125, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 126, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 127, "elements": [ 22 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0011_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0011_bag_stats.png" }, "solution": [ 12, 25, 29, 55, 60, 80, 92 ], "obj": 814.0, "instance_variant": { "num_elements": 22, "num_sets": 127, "sets": [ { "id": "S1", "elements": [ "D", "E", "I" ], "cost": 102 }, { "id": "S2", "elements": [ "J", "K", "L", "M", "N", "O", "Q" ], "cost": 63 }, { "id": "S3", "elements": [ "M", "Q", "S", "U" ], "cost": 140 }, { "id": "S4", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 402 }, { "id": "S5", "elements": [ "G", "H", "I", "J", "L", "M", "Q" ], "cost": 217 }, { "id": "S6", "elements": [ "N", "R", "S", "T", "U", "V" ], "cost": 540 }, { "id": "S7", "elements": [ "A", "C", "F", "I" ], "cost": 388 }, { "id": "S8", "elements": [ "E", "K", "L", "M", "N", "O" ], "cost": 492 }, { "id": "S9", "elements": [ "F", "G", "J", "K", "L", "M" ], "cost": 468 }, { "id": "S10", "elements": [ "S", "T", "U", "V" ], "cost": 244 }, { "id": "S11", "elements": [ "A", "B", "C", "D", "E", "F", "I" ], "cost": 455 }, { "id": "S12", "elements": [ "O", "R" ], "cost": 174 }, { "id": "S13", "elements": [ "L", "M", "N", "O", "P", "Q", "S" ], "cost": 595 }, { "id": "S14", "elements": [ "B", "D", "F", "G", "H" ], "cost": 495 }, { "id": "S15", "elements": [ "I", "J", "M" ], "cost": 258 }, { "id": "S16", "elements": [ "Q", "S", "T" ], "cost": 129 }, { "id": "S17", "elements": [ "Q", "R", "S", "T", "U", "V" ], "cost": 294 }, { "id": "S18", "elements": [ "P", "Q", "R", "S", "V" ], "cost": 75 }, { "id": "S19", "elements": [ "P", "T", "V" ], "cost": 144 }, { "id": "S20", "elements": [ "D", "K", "O" ], "cost": 24 }, { "id": "S21", "elements": [ "F", "G", "H", "I", "L" ], "cost": 85 }, { "id": "S22", "elements": [ "N", "Q", "T", "V" ], "cost": 164 }, { "id": "S23", "elements": [ "I", "J", "L" ], "cost": 231 }, { "id": "S24", "elements": [ "Q", "U", "V" ], "cost": 123 }, { "id": "S25", "elements": [ "C", "D", "E", "G" ], "cost": 112 }, { "id": "S26", "elements": [ "J", "K", "N", "O" ], "cost": 376 }, { "id": "S27", "elements": [ "D", "F", "G", "H", "I", "K" ], "cost": 510 }, { "id": "S28", "elements": [ "E", "F", "G", "H", "K", "L" ], "cost": 306 }, { "id": "S29", "elements": [ "I", "K", "M" ], "cost": 213 }, { "id": "S30", "elements": [ "A", "D", "G", "H" ], "cost": 364 }, { "id": "S31", "elements": [ "R", "S", "T", "U" ], "cost": 76 }, { "id": "S32", "elements": [ "B", "H", "I" ], "cost": 222 }, { "id": "S33", "elements": [ "D", "F", "J" ], "cost": 66 }, { "id": "S34", "elements": [ "A", "B", "C", "D", "E", "H" ], "cost": 336 }, { "id": "S35", "elements": [ "M", "O", "R" ], "cost": 36 }, { "id": "S36", "elements": [ "O", "Q", "R", "S", "T", "U" ], "cost": 498 }, { "id": "S37", "elements": [ "O", "Q", "R", "S", "V" ], "cost": 65 }, { "id": "S38", "elements": [ "C", "D", "E", "F", "G", "O" ], "cost": 552 }, { "id": "S39", "elements": [ "M", "P", "R", "S", "T", "U" ], "cost": 378 }, { "id": "S40", "elements": [ "S", "U" ], "cost": 168 }, { "id": "S41", "elements": [ "E", "I", "K" ], "cost": 231 }, { "id": "S42", "elements": [ "A", "B", "C", "D", "E" ], "cost": 105 }, { "id": "S43", "elements": [ "E", "K", "N" ], "cost": 195 }, { "id": "S44", "elements": [ "P", "Q", "S", "V" ], "cost": 24 }, { "id": "S45", "elements": [ "N", "O", "P", "Q", "R", "T" ], "cost": 276 }, { "id": "S46", "elements": [ "N", "O", "P", "Q", "S" ], "cost": 195 }, { "id": "S47", "elements": [ "D", "E", "F", "G", "I", "K" ], "cost": 60 }, { "id": "S48", "elements": [ "E", "G", "H", "I", "L" ], "cost": 275 }, { "id": "S49", "elements": [ "B", "C", "E", "F", "G", "K" ], "cost": 288 }, { "id": "S50", "elements": [ "Q", "S", "T", "U", "V" ], "cost": 280 }, { "id": "S51", "elements": [ "I", "J", "K", "L" ], "cost": 116 }, { "id": "S52", "elements": [ "C", "D", "H" ], "cost": 42 }, { "id": "S53", "elements": [ "B", "C", "D", "E", "F", "G" ], "cost": 540 }, { "id": "S54", "elements": [ "G", "J", "K", "M", "N" ], "cost": 250 }, { "id": "S55", "elements": [ "T", "U", "V" ], "cost": 156 }, { "id": "S56", "elements": [ "D", "E", "F", "G" ], "cost": 76 }, { "id": "S57", "elements": [ "E", "I", "J", "M" ], "cost": 128 }, { "id": "S58", "elements": [ "L", "N", "O", "P", "S" ], "cost": 180 }, { "id": "S59", "elements": [ "G", "I", "K", "M" ], "cost": 48 }, { "id": "S60", "elements": [ "A", "B", "F" ], "cost": 6 }, { "id": "S61", "elements": [ "K", "M", "N", "O", "P", "Q" ], "cost": 414 }, { "id": "S62", "elements": [ "G", "H", "K", "M", "O" ], "cost": 345 }, { "id": "S63", "elements": [ "I", "K", "L", "N", "O" ], "cost": 440 }, { "id": "S64", "elements": [ "L", "N", "P" ], "cost": 165 }, { "id": "S65", "elements": [ "C", "E", "F", "I" ], "cost": 280 }, { "id": "S66", "elements": [ "O", "R", "S", "T", "U", "V" ], "cost": 6 }, { "id": "S67", "elements": [ "H", "J", "K", "L", "M" ], "cost": 10 }, { "id": "S68", "elements": [ "N", "P", "S" ], "cost": 159 }, { "id": "S69", "elements": [ "K", "N", "R" ], "cost": 183 }, { "id": "S70", "elements": [ "H", "I", "J", "M" ], "cost": 228 }, { "id": "S71", "elements": [ "M", "N", "O", "P", "S" ], "cost": 130 }, { "id": "S72", "elements": [ "F", "J", "L" ], "cost": 15 }, { "id": "S73", "elements": [ "J", "K", "L" ], "cost": 48 }, { "id": "S74", "elements": [ "O", "R", "S", "T" ], "cost": 400 }, { "id": "S75", "elements": [ "N", "P", "Q", "R", "S", "U" ], "cost": 102 }, { "id": "S76", "elements": [ "L", "N", "R" ], "cost": 132 }, { "id": "S77", "elements": [ "D", "F", "G", "H", "I", "J" ], "cost": 540 }, { "id": "S78", "elements": [ "L", "O", "P" ], "cost": 270 }, { "id": "S79", "elements": [ "Q", "R", "S", "T", "V" ], "cost": 85 }, { "id": "S80", "elements": [ "H", "J", "L", "N" ], "cost": 48 }, { "id": "S81", "elements": [ "J", "K", "L", "M", "N", "O" ], "cost": 396 }, { "id": "S82", "elements": [ "A", "B", "C", "D" ], "cost": 216 }, { "id": "S83", "elements": [ "A", "B", "D" ], "cost": 126 }, { "id": "S84", "elements": [ "O", "P", "Q" ], "cost": 99 }, { "id": "S85", "elements": [ "H", "I", "K", "M", "N", "O" ], "cost": 282 }, { "id": "S86", "elements": [ "I", "J", "K", "L", "M" ], "cost": 375 }, { "id": "S87", "elements": [ "B", "C", "E", "F" ], "cost": 336 }, { "id": "S88", "elements": [ "D", "E", "G", "H", "I", "J", "K" ], "cost": 504 }, { "id": "S89", "elements": [ "H", "K", "L", "N", "O", "P" ], "cost": 390 }, { "id": "S90", "elements": [ "B", "C", "F", "G", "H", "I" ], "cost": 534 }, { "id": "S91", "elements": [ "L", "M", "N" ], "cost": 30 }, { "id": "S92", "elements": [ "P", "Q", "S" ], "cost": 105 }, { "id": "S93", "elements": [ "N", "O", "P", "Q", "S", "T" ], "cost": 426 }, { "id": "S94", "elements": [ "D", "E", "G" ], "cost": 57 }, { "id": "S95", "elements": [ "F", "H", "I", "J", "K", "L" ], "cost": 414 }, { "id": "S96", "elements": [ "S", "T", "U" ], "cost": 33 }, { "id": "S97", "elements": [ "O", "Q", "R", "S", "U", "V" ], "cost": 12 }, { "id": "S98", "elements": [ "C", "F", "G", "I", "J", "K" ], "cost": 600 }, { "id": "S99", "elements": [ "J", "K", "M", "N", "P" ], "cost": 285 }, { "id": "S100", "elements": [ "G", "J", "L" ], "cost": 153 }, { "id": "S101", "elements": [ "E", "G", "I" ], "cost": 186 }, { "id": "S102", "elements": [ "D", "F", "H" ], "cost": 219 }, { "id": "S103", "elements": [ "B", "D", "E", "G" ], "cost": 204 }, { "id": "S104", "elements": [ "D", "E", "F", "J", "L" ], "cost": 65 }, { "id": "S105", "elements": [ "A", "C", "E", "F", "G" ], "cost": 10 }, { "id": "S106", "elements": [ "A" ], "cost": 10000 }, { "id": "S107", "elements": [ "B" ], "cost": 10000 }, { "id": "S108", "elements": [ "C" ], "cost": 10000 }, { "id": "S109", "elements": [ "D" ], "cost": 10000 }, { "id": "S110", "elements": [ "E" ], "cost": 10000 }, { "id": "S111", "elements": [ "F" ], "cost": 10000 }, { "id": "S112", "elements": [ "G" ], "cost": 10000 }, { "id": "S113", "elements": [ "H" ], "cost": 10000 }, { "id": "S114", "elements": [ "I" ], "cost": 10000 }, { "id": "S115", "elements": [ "J" ], "cost": 10000 }, { "id": "S116", "elements": [ "K" ], "cost": 10000 }, { "id": "S117", "elements": [ "L" ], "cost": 10000 }, { "id": "S118", "elements": [ "M" ], "cost": 10000 }, { "id": "S119", "elements": [ "N" ], "cost": 10000 }, { "id": "S120", "elements": [ "O" ], "cost": 10000 }, { "id": "S121", "elements": [ "P" ], "cost": 10000 }, { "id": "S122", "elements": [ "Q" ], "cost": 10000 }, { "id": "S123", "elements": [ "R" ], "cost": 10000 }, { "id": "S124", "elements": [ "S" ], "cost": 10000 }, { "id": "S125", "elements": [ "T" ], "cost": 10000 }, { "id": "S126", "elements": [ "U" ], "cost": 10000 }, { "id": "S127", "elements": [ "V" ], "cost": 10000 } ] }, "solution_variant": [ "S12", "S25", "S29", "S55", "S60", "S80", "S92" ], "context_index": 12, "input_format": "json", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "Recently the school looked at a bunch of possible study groups and asked which ones to run so every student ends up in exactly one chosen group; the clear aim is to keep the total coaching cost as low as possible — that total comes from summing the cost of each group that’s used. No student can be omitted or assigned to more than one group. The exact lists and costs are provided below.\n\n{\n \"total_students\": 25,\n \"candidate_groups\": 160,\n \"sets\": [\n {\n \"group_id\": \"S1\",\n \"student_ids\": [\n 0,\n 1,\n 2,\n 3,\n 6,\n 8,\n 9\n ],\n \"coaching_cost\": 98\n },\n {\n \"group_id\": \"S2\",\n \"student_ids\": [\n 6,\n 8,\n 12,\n 13,\n 14\n ],\n \"coaching_cost\": 105\n },\n {\n \"group_id\": \"S3\",\n \"student_ids\": [\n 11,\n 12,\n 16,\n 17\n ],\n \"coaching_cost\": 64\n },\n {\n \"group_id\": \"S4\",\n \"student_ids\": [\n 0,\n 2,\n 3\n ],\n \"coaching_cost\": 162\n },\n {\n \"group_id\": \"S5\",\n \"student_ids\": [\n 2,\n 4,\n 7\n ],\n \"coaching_cost\": 252\n },\n {\n \"group_id\": \"S6\",\n \"student_ids\": [\n 17,\n 20,\n 22,\n 23\n ],\n \"coaching_cost\": 88\n },\n {\n \"group_id\": \"S7\",\n \"student_ids\": [\n 3,\n 4,\n 6,\n 8,\n 9,\n 10,\n 11\n ],\n \"coaching_cost\": 42\n },\n {\n \"group_id\": \"S8\",\n \"student_ids\": [\n 2,\n 4,\n 5,\n 6,\n 7,\n 9\n ],\n \"coaching_cost\": 480\n },\n {\n \"group_id\": \"S9\",\n \"student_ids\": [\n 8,\n 9,\n 10,\n 11,\n 13,\n 14,\n 15\n ],\n \"coaching_cost\": 7\n },\n {\n \"group_id\": \"S10\",\n \"student_ids\": [\n 1,\n 2,\n 4,\n 5,\n 6,\n 7,\n 10\n ],\n \"coaching_cost\": 105\n },\n {\n \"group_id\": \"S11\",\n \"student_ids\": [\n 19,\n 20,\n 21,\n 22,\n 24\n ],\n \"coaching_cost\": 205\n },\n {\n \"group_id\": \"S12\",\n \"student_ids\": [\n 13,\n 14,\n 16,\n 17,\n 18\n ],\n \"coaching_cost\": 420\n },\n {\n \"group_id\": \"S13\",\n \"student_ids\": [\n 9,\n 10,\n 11,\n 12,\n 13,\n 14,\n 16\n ],\n \"coaching_cost\": 378\n },\n {\n \"group_id\": \"S14\",\n \"student_ids\": [\n 16,\n 20,\n 22,\n 23,\n 24\n ],\n \"coaching_cost\": 15\n },\n {\n \"group_id\": \"S15\",\n \"student_ids\": [\n 11,\n 14,\n 18,\n 19\n ],\n \"coaching_cost\": 400\n },\n {\n \"group_id\": \"S16\",\n \"student_ids\": [\n 6,\n 7,\n 8,\n 9,\n 10,\n 12,\n 13\n ],\n \"coaching_cost\": 672\n },\n {\n \"group_id\": \"S17\",\n \"student_ids\": [\n 17,\n 20,\n 23\n ],\n \"coaching_cost\": 165\n },\n {\n \"group_id\": \"S18\",\n \"student_ids\": [\n 19,\n 21,\n 22,\n 23\n ],\n \"coaching_cost\": 84\n },\n {\n \"group_id\": \"S19\",\n \"student_ids\": [\n 5,\n 8,\n 10\n ],\n \"coaching_cost\": 162\n },\n {\n \"group_id\": \"S20\",\n \"student_ids\": [\n 2,\n 5,\n 6\n ],\n \"coaching_cost\": 117\n },\n {\n \"group_id\": \"S21\",\n \"student_ids\": [\n 19,\n 20,\n 21,\n 23,\n 24\n ],\n \"coaching_cost\": 415\n },\n {\n \"group_id\": \"S22\",\n \"student_ids\": [\n 21,\n 22,\n 23\n ],\n \"coaching_cost\": 165\n },\n {\n \"group_id\": \"S23\",\n \"student_ids\": [\n 12,\n 13,\n 14,\n 15,\n 17,\n 18\n ],\n \"coaching_cost\": 126\n },\n {\n \"group_id\": \"S24\",\n \"student_ids\": [\n 0,\n 1,\n 3,\n 7\n ],\n \"coaching_cost\": 320\n },\n {\n \"group_id\": \"S25\",\n \"student_ids\": [\n 10,\n 12,\n 14\n ],\n \"coaching_cost\": 30\n },\n {\n \"group_id\": \"S26\",\n \"student_ids\": [\n 9,\n 11,\n 12,\n 13,\n 14,\n 15,\n 16\n ],\n \"coaching_cost\": 245\n },\n {\n \"group_id\": \"S27\",\n \"student_ids\": [\n 1,\n 2,\n 5\n ],\n \"coaching_cost\": 93\n },\n {\n \"group_id\": \"S28\",\n \"student_ids\": [\n 11,\n 14,\n 15,\n 21\n ],\n \"coaching_cost\": 100\n },\n {\n \"group_id\": \"S29\",\n \"student_ids\": [\n 15,\n 16,\n 18,\n 20,\n 23\n ],\n \"coaching_cost\": 310\n },\n {\n \"group_id\": \"S30\",\n \"student_ids\": [\n 7,\n 9,\n 15\n ],\n \"coaching_cost\": 291\n },\n {\n \"group_id\": \"S31\",\n \"student_ids\": [\n 10,\n 11,\n 12,\n 13\n ],\n \"coaching_cost\": 68\n },\n {\n \"group_id\": \"S32\",\n \"student_ids\": [\n 0,\n 2,\n 3,\n 4,\n 5,\n 6\n ],\n \"coaching_cost\": 294\n },\n {\n \"group_id\": \"S33\",\n \"student_ids\": [\n 0,\n 1,\n 2,\n 3,\n 6\n ],\n \"coaching_cost\": 460\n },\n {\n \"group_id\": \"S34\",\n \"student_ids\": [\n 3,\n 6,\n 7,\n 8\n ],\n \"coaching_cost\": 260\n },\n {\n \"group_id\": \"S35\",\n \"student_ids\": [\n 18,\n 20,\n 22,\n 23,\n 24\n ],\n \"coaching_cost\": 190\n },\n {\n \"group_id\": \"S36\",\n \"student_ids\": [\n 9,\n 11,\n 14,\n 15\n ],\n \"coaching_cost\": 316\n },\n {\n \"group_id\": \"S37\",\n \"student_ids\": [\n 0,\n 1,\n 3,\n 5,\n 6,\n 10\n ],\n \"coaching_cost\": 72\n },\n {\n \"group_id\": \"S38\",\n \"student_ids\": [\n 11,\n 13,\n 14,\n 15,\n 18\n ],\n \"coaching_cost\": 60\n },\n {\n \"group_id\": \"S39\",\n \"student_ids\": [\n 20,\n 21,\n 23\n ],\n \"coaching_cost\": 99\n },\n {\n \"group_id\": \"S40\",\n \"student_ids\": [\n 9,\n 11,\n 12,\n 14,\n 15,\n 18\n ],\n \"coaching_cost\": 312\n },\n {\n \"group_id\": \"S41\",\n \"student_ids\": [\n 16,\n 18,\n 21,\n 22,\n 23\n ],\n \"coaching_cost\": 240\n },\n {\n \"group_id\": \"S42\",\n \"student_ids\": [\n 11,\n 13,\n 16\n ],\n \"coaching_cost\": 264\n },\n {\n \"group_id\": \"S43\",\n \"student_ids\": [\n 16,\n 21,\n 22\n ],\n \"coaching_cost\": 216\n },\n {\n \"group_id\": \"S44\",\n \"student_ids\": [\n 1,\n 2,\n 4\n ],\n \"coaching_cost\": 30\n },\n {\n \"group_id\": \"S45\",\n \"student_ids\": [\n 13,\n 14,\n 15,\n 16\n ],\n \"coaching_cost\": 240\n },\n {\n \"group_id\": \"S46\",\n \"student_ids\": [\n 7,\n 8,\n 9\n ],\n \"coaching_cost\": 297\n },\n {\n \"group_id\": \"S47\",\n \"student_ids\": [\n 10,\n 11,\n 13,\n 14,\n 19\n ],\n \"coaching_cost\": 305\n },\n {\n \"group_id\": \"S48\",\n \"student_ids\": [\n 6,\n 10,\n 11,\n 12,\n 13,\n 16\n ],\n \"coaching_cost\": 366\n },\n {\n \"group_id\": \"S49\",\n \"student_ids\": [\n 4,\n 7,\n 9,\n 10\n ],\n \"coaching_cost\": 328\n },\n {\n \"group_id\": \"S50\",\n \"student_ids\": [\n 1,\n 3,\n 4\n ],\n \"coaching_cost\": 276\n },\n {\n \"group_id\": \"S51\",\n \"student_ids\": [\n 9,\n 11,\n 13,\n 14,\n 16\n ],\n \"coaching_cost\": 275\n },\n {\n \"group_id\": \"S52\",\n \"student_ids\": [\n 21,\n 22,\n 23,\n 24\n ],\n \"coaching_cost\": 276\n },\n {\n \"group_id\": \"S53\",\n \"student_ids\": [\n 4,\n 5,\n 6,\n 7,\n 8,\n 10\n ],\n \"coaching_cost\": 462\n },\n {\n \"group_id\": \"S54\",\n \"student_ids\": [\n 20,\n 22,\n 23\n ],\n \"coaching_cost\": 282\n },\n {\n \"group_id\": \"S55\",\n \"student_ids\": [\n 17,\n 19,\n 20,\n 22,\n 23,\n 24\n ],\n \"coaching_cost\": 522\n },\n {\n \"group_id\": \"S56\",\n \"student_ids\": [\n 16,\n 17,\n 18,\n 20,\n 22\n ],\n \"coaching_cost\": 410\n },\n {\n \"group_id\": \"S57\",\n \"student_ids\": [\n 0,\n 1,\n 2,\n 3,\n 4,\n 5,\n 6\n ],\n \"coaching_cost\": 189\n },\n {\n \"group_id\": \"S58\",\n \"student_ids\": [\n 15,\n 16,\n 20,\n 21,\n 22,\n 23,\n 24\n ],\n \"coaching_cost\": 630\n },\n {\n \"group_id\": \"S59\",\n \"student_ids\": [\n 12,\n 13,\n 14,\n 17,\n 20\n ],\n \"coaching_cost\": 215\n },\n {\n \"group_id\": \"S60\",\n \"student_ids\": [\n 4,\n 5,\n 6\n ],\n \"coaching_cost\": 63\n },\n {\n \"group_id\": \"S61\",\n \"student_ids\": [\n 11,\n 12,\n 13,\n 14,\n 15\n ],\n \"coaching_cost\": 65\n },\n {\n \"group_id\": \"S62\",\n \"student_ids\": [\n 0,\n 1,\n 2,\n 3,\n 4,\n 5,\n 7\n ],\n \"coaching_cost\": 574\n },\n {\n \"group_id\": \"S63\",\n \"student_ids\": [\n 15,\n 18,\n 19,\n 20,\n 21\n ],\n \"coaching_cost\": 355\n },\n {\n \"group_id\": \"S64\",\n \"student_ids\": [\n 5,\n 8,\n 9,\n 10\n ],\n \"coaching_cost\": 240\n },\n {\n \"group_id\": \"S65\",\n \"student_ids\": [\n 14,\n 17,\n 18,\n 21,\n 22\n ],\n \"coaching_cost\": 185\n },\n {\n \"group_id\": \"S66\",\n \"student_ids\": [\n 19,\n 20,\n 22,\n 23\n ],\n \"coaching_cost\": 124\n },\n {\n \"group_id\": \"S67\",\n \"student_ids\": [\n 8,\n 9,\n 15\n ],\n \"coaching_cost\": 201\n },\n {\n \"group_id\": \"S68\",\n \"student_ids\": [\n 16,\n 17,\n 18,\n 19,\n 20,\n 21,\n 22\n ],\n \"coaching_cost\": 357\n },\n {\n \"group_id\": \"S69\",\n \"student_ids\": [\n 14,\n 15,\n 16,\n 17,\n 18,\n 20,\n 22\n ],\n \"coaching_cost\": 210\n },\n {\n \"group_id\": \"S70\",\n \"student_ids\": [\n 4,\n 5,\n 7\n ],\n \"coaching_cost\": 294\n },\n {\n \"group_id\": \"S71\",\n \"student_ids\": [\n 2,\n 3,\n 4,\n 5,\n 7,\n 10\n ],\n \"coaching_cost\": 594\n },\n {\n \"group_id\": \"S72\",\n \"student_ids\": [\n 10,\n 11,\n 12,\n 14,\n 15,\n 16\n ],\n \"coaching_cost\": 192\n },\n {\n \"group_id\": \"S73\",\n \"student_ids\": [\n 10,\n 11,\n 12,\n 13,\n 14,\n 16\n ],\n \"coaching_cost\": 450\n },\n {\n \"group_id\": \"S74\",\n \"student_ids\": [\n 1,\n 2,\n 5,\n 6\n ],\n \"coaching_cost\": 324\n },\n {\n \"group_id\": \"S75\",\n \"student_ids\": [\n 0,\n 3,\n 4,\n 6,\n 7\n ],\n \"coaching_cost\": 25\n },\n {\n \"group_id\": \"S76\",\n \"student_ids\": [\n 10,\n 11,\n 16\n ],\n \"coaching_cost\": 210\n },\n {\n \"group_id\": \"S77\",\n \"student_ids\": [\n 14,\n 18,\n 20\n ],\n \"coaching_cost\": 12\n },\n {\n \"group_id\": \"S78\",\n \"student_ids\": [\n 12,\n 13,\n 14,\n 16,\n 19\n ],\n \"coaching_cost\": 210\n },\n {\n \"group_id\": \"S79\",\n \"student_ids\": [\n 1,\n 4,\n 6\n ],\n \"coaching_cost\": 186\n },\n {\n \"group_id\": \"S80\",\n \"student_ids\": [\n 1,\n 2,\n 3\n ],\n \"coaching_cost\": 288\n },\n {\n \"group_id\": \"S81\",\n \"student_ids\": [\n 10,\n 11,\n 12,\n 13,\n 14,\n 16,\n 17\n ],\n \"coaching_cost\": 357\n },\n {\n \"group_id\": \"S82\",\n \"student_ids\": [\n 11,\n 12,\n 13,\n 14,\n 15,\n 16\n ],\n \"coaching_cost\": 330\n },\n {\n \"group_id\": \"S83\",\n \"student_ids\": [\n 7,\n 10,\n 12\n ],\n \"coaching_cost\": 66\n },\n {\n \"group_id\": \"S84\",\n \"student_ids\": [\n 13,\n 14,\n 15,\n 16,\n 17,\n 20,\n 21\n ],\n \"coaching_cost\": 105\n },\n {\n \"group_id\": \"S85\",\n \"student_ids\": [\n 7,\n 8,\n 10\n ],\n \"coaching_cost\": 84\n },\n {\n \"group_id\": \"S86\",\n \"student_ids\": [\n 5,\n 8,\n 9,\n 10,\n 12,\n 13\n ],\n \"coaching_cost\": 438\n },\n {\n \"group_id\": \"S87\",\n \"student_ids\": [\n 3,\n 4,\n 6,\n 7,\n 8,\n 9\n ],\n \"coaching_cost\": 468\n },\n {\n \"group_id\": \"S88\",\n \"student_ids\": [\n 17,\n 18,\n 19,\n 20,\n 21,\n 22,\n 23\n ],\n \"coaching_cost\": 448\n },\n {\n \"group_id\": \"S89\",\n \"student_ids\": [\n 5,\n 6,\n 7,\n 8,\n 9\n ],\n \"coaching_cost\": 370\n },\n {\n \"group_id\": \"S90\",\n \"student_ids\": [\n 2,\n 4,\n 6\n ],\n \"coaching_cost\": 180\n },\n {\n \"group_id\": \"S91\",\n \"student_ids\": [\n 8,\n 10,\n 12,\n 13\n ],\n \"coaching_cost\": 12\n },\n {\n \"group_id\": \"S92\",\n \"student_ids\": [\n 2,\n 3,\n 6\n ],\n \"coaching_cost\": 123\n },\n {\n \"group_id\": \"S93\",\n \"student_ids\": [\n 7,\n 9,\n 11,\n 12,\n 14,\n 18\n ],\n \"coaching_cost\": 384\n },\n {\n \"group_id\": \"S94\",\n \"student_ids\": [\n 0,\n 1,\n 2,\n 5,\n 6,\n 7,\n 10\n ],\n \"coaching_cost\": 623\n },\n {\n \"group_id\": \"S95\",\n \"student_ids\": [\n 5,\n 6,\n 7,\n 10,\n 11,\n 13\n ],\n \"coaching_cost\": 534\n },\n {\n \"group_id\": \"S96\",\n \"student_ids\": [\n 4,\n 5,\n 7,\n 8,\n 10,\n 11,\n 12\n ],\n \"coaching_cost\": 343\n },\n {\n \"group_id\": \"S97\",\n \"student_ids\": [\n 2,\n 4,\n 5,\n 9\n ],\n \"coaching_cost\": 76\n },\n {\n \"group_id\": \"S98\",\n \"student_ids\": [\n 10,\n 11,\n 12,\n 13,\n 14,\n 15\n ],\n \"coaching_cost\": 216\n },\n {\n \"group_id\": \"S99\",\n \"student_ids\": [\n 19,\n 22,\n 23,\n 24\n ],\n \"coaching_cost\": 184\n },\n {\n \"group_id\": \"S100\",\n \"student_ids\": [\n 3,\n 4,\n 8,\n 9,\n 12\n ],\n \"coaching_cost\": 40\n },\n {\n \"group_id\": \"S101\",\n \"student_ids\": [\n 17,\n 18,\n 19,\n 23\n ],\n \"coaching_cost\": 108\n },\n {\n \"group_id\": \"S102\",\n \"student_ids\": [\n 18,\n 19,\n 20,\n 21,\n 22,\n 23,\n 24\n ],\n \"coaching_cost\": 119\n },\n {\n \"group_id\": \"S103\",\n \"student_ids\": [\n 8,\n 10,\n 11,\n 13\n ],\n \"coaching_cost\": 208\n },\n {\n \"group_id\": \"S104\",\n \"student_ids\": [\n 8,\n 10,\n 12\n ],\n \"coaching_cost\": 222\n },\n {\n \"group_id\": \"S105\",\n \"student_ids\": [\n 0,\n 1,\n 2,\n 3,\n 4,\n 5\n ],\n \"coaching_cost\": 366\n },\n {\n \"group_id\": \"S106\",\n \"student_ids\": [\n 6,\n 7,\n 9,\n 11,\n 12\n ],\n \"coaching_cost\": 275\n },\n {\n \"group_id\": \"S107\",\n \"student_ids\": [\n 5,\n 6,\n 7\n ],\n \"coaching_cost\": 192\n },\n {\n \"group_id\": \"S108\",\n \"student_ids\": [\n 0,\n 1,\n 3,\n 4,\n 8\n ],\n \"coaching_cost\": 25\n },\n {\n \"group_id\": \"S109\",\n \"student_ids\": [\n 3,\n 4,\n 6,\n 8,\n 9\n ],\n \"coaching_cost\": 255\n },\n {\n \"group_id\": \"S110\",\n \"student_ids\": [\n 4,\n 5,\n 9,\n 10\n ],\n \"coaching_cost\": 108\n },\n {\n \"group_id\": \"S111\",\n \"student_ids\": [\n 2,\n 6,\n 7,\n 10\n ],\n \"coaching_cost\": 360\n },\n {\n \"group_id\": \"S112\",\n \"student_ids\": [\n 18,\n 19,\n 20,\n 21,\n 22,\n 24\n ],\n \"coaching_cost\": 354\n },\n {\n \"group_id\": \"S113\",\n \"student_ids\": [\n 0,\n 1,\n 2\n ],\n \"coaching_cost\": 102\n },\n {\n \"group_id\": \"S114\",\n \"student_ids\": [\n 2,\n 3,\n 4,\n 5,\n 6,\n 8,\n 10\n ],\n \"coaching_cost\": 105\n },\n {\n \"group_id\": \"S115\",\n \"student_ids\": [\n 2,\n 4,\n 6,\n 7,\n 8\n ],\n \"coaching_cost\": 315\n },\n {\n \"group_id\": \"S116\",\n \"student_ids\": [\n 11,\n 15,\n 16,\n 17,\n 18,\n 20\n ],\n \"coaching_cost\": 402\n },\n {\n \"group_id\": \"S117\",\n \"student_ids\": [\n 15,\n 17,\n 18,\n 19,\n 20,\n 21\n ],\n \"coaching_cost\": 174\n },\n {\n \"group_id\": \"S118\",\n \"student_ids\": [\n 0,\n 1,\n 3,\n 4,\n 5,\n 6\n ],\n \"coaching_cost\": 72\n },\n {\n \"group_id\": \"S119\",\n \"student_ids\": [\n 13,\n 15,\n 17\n ],\n \"coaching_cost\": 261\n },\n {\n \"group_id\": \"S120\",\n \"student_ids\": [\n 12,\n 13,\n 16,\n 17,\n 20\n ],\n \"coaching_cost\": 215\n },\n {\n \"group_id\": \"S121\",\n \"student_ids\": [\n 11,\n 12,\n 14,\n 17\n ],\n \"coaching_cost\": 360\n },\n {\n \"group_id\": \"S122\",\n \"student_ids\": [\n 16,\n 19,\n 21,\n 22\n ],\n \"coaching_cost\": 180\n },\n {\n \"group_id\": \"S123\",\n \"student_ids\": [\n 0,\n 1,\n 2,\n 3\n ],\n \"coaching_cost\": 152\n },\n {\n \"group_id\": \"S124\",\n \"student_ids\": [\n 17,\n 18,\n 19,\n 20,\n 21,\n 22\n ],\n \"coaching_cost\": 456\n },\n {\n \"group_id\": \"S125\",\n \"student_ids\": [\n 14,\n 15,\n 17\n ],\n \"coaching_cost\": 3\n },\n {\n \"group_id\": \"S126\",\n \"student_ids\": [\n 2,\n 3,\n 7\n ],\n \"coaching_cost\": 258\n },\n {\n \"group_id\": \"S127\",\n \"student_ids\": [\n 13,\n 14,\n 15,\n 16,\n 17,\n 19,\n 24\n ],\n \"coaching_cost\": 203\n },\n {\n \"group_id\": \"S128\",\n \"student_ids\": [\n 9,\n 10,\n 12\n ],\n \"coaching_cost\": 156\n },\n {\n \"group_id\": \"S129\",\n \"student_ids\": [\n 1,\n 2,\n 3,\n 4,\n 5,\n 6,\n 7\n ],\n \"coaching_cost\": 126\n },\n {\n \"group_id\": \"S130\",\n \"student_ids\": [\n 3,\n 4,\n 5,\n 6,\n 7,\n 11\n ],\n \"coaching_cost\": 360\n },\n {\n \"group_id\": \"S131\",\n \"student_ids\": [\n 8,\n 12,\n 13,\n 14,\n 15\n ],\n \"coaching_cost\": 325\n },\n {\n \"group_id\": \"S132\",\n \"student_ids\": [\n 15,\n 19,\n 20,\n 21,\n 22,\n 23,\n 24\n ],\n \"coaching_cost\": 693\n },\n {\n \"group_id\": \"S133\",\n \"student_ids\": [\n 19,\n 21,\n 22,\n 23,\n 24\n ],\n \"coaching_cost\": 400\n },\n {\n \"group_id\": \"S134\",\n \"student_ids\": [\n 14,\n 15,\n 18,\n 20,\n 21\n ],\n \"coaching_cost\": 300\n },\n {\n \"group_id\": \"S135\",\n \"student_ids\": [\n 11,\n 12,\n 13,\n 15,\n 16,\n 17,\n 19\n ],\n \"coaching_cost\": 266\n },\n {\n \"group_id\": \"S136\",\n \"student_ids\": [\n 0\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S137\",\n \"student_ids\": [\n 1\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S138\",\n \"student_ids\": [\n 2\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S139\",\n \"student_ids\": [\n 3\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S140\",\n \"student_ids\": [\n 4\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S141\",\n \"student_ids\": [\n 5\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S142\",\n \"student_ids\": [\n 6\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S143\",\n \"student_ids\": [\n 7\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S144\",\n \"student_ids\": [\n 8\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S145\",\n \"student_ids\": [\n 9\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S146\",\n \"student_ids\": [\n 10\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S147\",\n \"student_ids\": [\n 11\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S148\",\n \"student_ids\": [\n 12\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S149\",\n \"student_ids\": [\n 13\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S150\",\n \"student_ids\": [\n 14\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S151\",\n \"student_ids\": [\n 15\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S152\",\n \"student_ids\": [\n 16\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S153\",\n \"student_ids\": [\n 17\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S154\",\n \"student_ids\": [\n 18\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S155\",\n \"student_ids\": [\n 19\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S156\",\n \"student_ids\": [\n 20\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S157\",\n \"student_ids\": [\n 21\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S158\",\n \"student_ids\": [\n 22\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S159\",\n \"student_ids\": [\n 23\n ],\n \"coaching_cost\": 10000\n },\n {\n \"group_id\": \"S160\",\n \"student_ids\": [\n 24\n ],\n \"coaching_cost\": 10000\n }\n ]\n}\n\nWhen you send the final choice, just paste it in that simple JSON shape so it's easy to read and check.\n\n{\n \"solution\": [\"group_id\", ...]\n}\n\n\"solution\" is just the list of chosen study-group ids — one entry per selected group. Think of it like filling in a short form: replace the placeholder with the actual group ids from the instance. This JSON is only a sketch of the expected shape, not the real answer.\n\nPlease make sure to use the exact identifiers from the instance input — do not rename them or invent new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 25, "num_sets": 160, "density": 0.16925, "sets": [ { "id": 1, "elements": [ 1, 2, 3, 4, 7, 9, 10 ], "cost": 98 }, { "id": 2, "elements": [ 7, 9, 13, 14, 15 ], "cost": 105 }, { "id": 3, "elements": [ 12, 13, 17, 18 ], "cost": 64 }, { "id": 4, "elements": [ 1, 3, 4 ], "cost": 162 }, { "id": 5, "elements": [ 3, 5, 8 ], "cost": 252 }, { "id": 6, "elements": [ 18, 21, 23, 24 ], "cost": 88 }, { "id": 7, "elements": [ 4, 5, 7, 9, 10, 11, 12 ], "cost": 42 }, { "id": 8, "elements": [ 3, 5, 6, 7, 8, 10 ], "cost": 480 }, { "id": 9, "elements": [ 9, 10, 11, 12, 14, 15, 16 ], "cost": 7 }, { "id": 10, "elements": [ 2, 3, 5, 6, 7, 8, 11 ], "cost": 105 }, { "id": 11, "elements": [ 20, 21, 22, 23, 25 ], "cost": 205 }, { "id": 12, "elements": [ 14, 15, 17, 18, 19 ], "cost": 420 }, { "id": 13, "elements": [ 10, 11, 12, 13, 14, 15, 17 ], "cost": 378 }, { "id": 14, "elements": [ 17, 21, 23, 24, 25 ], "cost": 15 }, { "id": 15, "elements": [ 12, 15, 19, 20 ], "cost": 400 }, { "id": 16, "elements": [ 7, 8, 9, 10, 11, 13, 14 ], "cost": 672 }, { "id": 17, "elements": [ 18, 21, 24 ], "cost": 165 }, { "id": 18, "elements": [ 20, 22, 23, 24 ], "cost": 84 }, { "id": 19, "elements": [ 6, 9, 11 ], "cost": 162 }, { "id": 20, "elements": [ 3, 6, 7 ], "cost": 117 }, { "id": 21, "elements": [ 20, 21, 22, 24, 25 ], "cost": 415 }, { "id": 22, "elements": [ 22, 23, 24 ], "cost": 165 }, { "id": 23, "elements": [ 13, 14, 15, 16, 18, 19 ], "cost": 126 }, { "id": 24, "elements": [ 1, 2, 4, 8 ], "cost": 320 }, { "id": 25, "elements": [ 11, 13, 15 ], "cost": 30 }, { "id": 26, "elements": [ 10, 12, 13, 14, 15, 16, 17 ], "cost": 245 }, { "id": 27, "elements": [ 2, 3, 6 ], "cost": 93 }, { "id": 28, "elements": [ 12, 15, 16, 22 ], "cost": 100 }, { "id": 29, "elements": [ 16, 17, 19, 21, 24 ], "cost": 310 }, { "id": 30, "elements": [ 8, 10, 16 ], "cost": 291 }, { "id": 31, "elements": [ 11, 12, 13, 14 ], "cost": 68 }, { "id": 32, "elements": [ 1, 3, 4, 5, 6, 7 ], "cost": 294 }, { "id": 33, "elements": [ 1, 2, 3, 4, 7 ], "cost": 460 }, { "id": 34, "elements": [ 4, 7, 8, 9 ], "cost": 260 }, { "id": 35, "elements": [ 19, 21, 23, 24, 25 ], "cost": 190 }, { "id": 36, "elements": [ 10, 12, 15, 16 ], "cost": 316 }, { "id": 37, "elements": [ 1, 2, 4, 6, 7, 11 ], "cost": 72 }, { "id": 38, "elements": [ 12, 14, 15, 16, 19 ], "cost": 60 }, { "id": 39, "elements": [ 21, 22, 24 ], "cost": 99 }, { "id": 40, "elements": [ 10, 12, 13, 15, 16, 19 ], "cost": 312 }, { "id": 41, "elements": [ 17, 19, 22, 23, 24 ], "cost": 240 }, { "id": 42, "elements": [ 12, 14, 17 ], "cost": 264 }, { "id": 43, "elements": [ 17, 22, 23 ], "cost": 216 }, { "id": 44, "elements": [ 2, 3, 5 ], "cost": 30 }, { "id": 45, "elements": [ 14, 15, 16, 17 ], "cost": 240 }, { "id": 46, "elements": [ 8, 9, 10 ], "cost": 297 }, { "id": 47, "elements": [ 11, 12, 14, 15, 20 ], "cost": 305 }, { "id": 48, "elements": [ 7, 11, 12, 13, 14, 17 ], "cost": 366 }, { "id": 49, "elements": [ 5, 8, 10, 11 ], "cost": 328 }, { "id": 50, "elements": [ 2, 4, 5 ], "cost": 276 }, { "id": 51, "elements": [ 10, 12, 14, 15, 17 ], "cost": 275 }, { "id": 52, "elements": [ 22, 23, 24, 25 ], "cost": 276 }, { "id": 53, "elements": [ 5, 6, 7, 8, 9, 11 ], "cost": 462 }, { "id": 54, "elements": [ 21, 23, 24 ], "cost": 282 }, { "id": 55, "elements": [ 18, 20, 21, 23, 24, 25 ], "cost": 522 }, { "id": 56, "elements": [ 17, 18, 19, 21, 23 ], "cost": 410 }, { "id": 57, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 189 }, { "id": 58, "elements": [ 16, 17, 21, 22, 23, 24, 25 ], "cost": 630 }, { "id": 59, "elements": [ 13, 14, 15, 18, 21 ], "cost": 215 }, { "id": 60, "elements": [ 5, 6, 7 ], "cost": 63 }, { "id": 61, "elements": [ 12, 13, 14, 15, 16 ], "cost": 65 }, { "id": 62, "elements": [ 1, 2, 3, 4, 5, 6, 8 ], "cost": 574 }, { "id": 63, "elements": [ 16, 19, 20, 21, 22 ], "cost": 355 }, { "id": 64, "elements": [ 6, 9, 10, 11 ], "cost": 240 }, { "id": 65, "elements": [ 15, 18, 19, 22, 23 ], "cost": 185 }, { "id": 66, "elements": [ 20, 21, 23, 24 ], "cost": 124 }, { "id": 67, "elements": [ 9, 10, 16 ], "cost": 201 }, { "id": 68, "elements": [ 17, 18, 19, 20, 21, 22, 23 ], "cost": 357 }, { "id": 69, "elements": [ 15, 16, 17, 18, 19, 21, 23 ], "cost": 210 }, { "id": 70, "elements": [ 5, 6, 8 ], "cost": 294 }, { "id": 71, "elements": [ 3, 4, 5, 6, 8, 11 ], "cost": 594 }, { "id": 72, "elements": [ 11, 12, 13, 15, 16, 17 ], "cost": 192 }, { "id": 73, "elements": [ 11, 12, 13, 14, 15, 17 ], "cost": 450 }, { "id": 74, "elements": [ 2, 3, 6, 7 ], "cost": 324 }, { "id": 75, "elements": [ 1, 4, 5, 7, 8 ], "cost": 25 }, { "id": 76, "elements": [ 11, 12, 17 ], "cost": 210 }, { "id": 77, "elements": [ 15, 19, 21 ], "cost": 12 }, { "id": 78, "elements": [ 13, 14, 15, 17, 20 ], "cost": 210 }, { "id": 79, "elements": [ 2, 5, 7 ], "cost": 186 }, { "id": 80, "elements": [ 2, 3, 4 ], "cost": 288 }, { "id": 81, "elements": [ 11, 12, 13, 14, 15, 17, 18 ], "cost": 357 }, { "id": 82, "elements": [ 12, 13, 14, 15, 16, 17 ], "cost": 330 }, { "id": 83, "elements": [ 8, 11, 13 ], "cost": 66 }, { "id": 84, "elements": [ 14, 15, 16, 17, 18, 21, 22 ], "cost": 105 }, { "id": 85, "elements": [ 8, 9, 11 ], "cost": 84 }, { "id": 86, "elements": [ 6, 9, 10, 11, 13, 14 ], "cost": 438 }, { "id": 87, "elements": [ 4, 5, 7, 8, 9, 10 ], "cost": 468 }, { "id": 88, "elements": [ 18, 19, 20, 21, 22, 23, 24 ], "cost": 448 }, { "id": 89, "elements": [ 6, 7, 8, 9, 10 ], "cost": 370 }, { "id": 90, "elements": [ 3, 5, 7 ], "cost": 180 }, { "id": 91, "elements": [ 9, 11, 13, 14 ], "cost": 12 }, { "id": 92, "elements": [ 3, 4, 7 ], "cost": 123 }, { "id": 93, "elements": [ 8, 10, 12, 13, 15, 19 ], "cost": 384 }, { "id": 94, "elements": [ 1, 2, 3, 6, 7, 8, 11 ], "cost": 623 }, { "id": 95, "elements": [ 6, 7, 8, 11, 12, 14 ], "cost": 534 }, { "id": 96, "elements": [ 5, 6, 8, 9, 11, 12, 13 ], "cost": 343 }, { "id": 97, "elements": [ 3, 5, 6, 10 ], "cost": 76 }, { "id": 98, "elements": [ 11, 12, 13, 14, 15, 16 ], "cost": 216 }, { "id": 99, "elements": [ 20, 23, 24, 25 ], "cost": 184 }, { "id": 100, "elements": [ 4, 5, 9, 10, 13 ], "cost": 40 }, { "id": 101, "elements": [ 18, 19, 20, 24 ], "cost": 108 }, { "id": 102, "elements": [ 19, 20, 21, 22, 23, 24, 25 ], "cost": 119 }, { "id": 103, "elements": [ 9, 11, 12, 14 ], "cost": 208 }, { "id": 104, "elements": [ 9, 11, 13 ], "cost": 222 }, { "id": 105, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 366 }, { "id": 106, "elements": [ 7, 8, 10, 12, 13 ], "cost": 275 }, { "id": 107, "elements": [ 6, 7, 8 ], "cost": 192 }, { "id": 108, "elements": [ 1, 2, 4, 5, 9 ], "cost": 25 }, { "id": 109, "elements": [ 4, 5, 7, 9, 10 ], "cost": 255 }, { "id": 110, "elements": [ 5, 6, 10, 11 ], "cost": 108 }, { "id": 111, "elements": [ 3, 7, 8, 11 ], "cost": 360 }, { "id": 112, "elements": [ 19, 20, 21, 22, 23, 25 ], "cost": 354 }, { "id": 113, "elements": [ 1, 2, 3 ], "cost": 102 }, { "id": 114, "elements": [ 3, 4, 5, 6, 7, 9, 11 ], "cost": 105 }, { "id": 115, "elements": [ 3, 5, 7, 8, 9 ], "cost": 315 }, { "id": 116, "elements": [ 12, 16, 17, 18, 19, 21 ], "cost": 402 }, { "id": 117, "elements": [ 16, 18, 19, 20, 21, 22 ], "cost": 174 }, { "id": 118, "elements": [ 1, 2, 4, 5, 6, 7 ], "cost": 72 }, { "id": 119, "elements": [ 14, 16, 18 ], "cost": 261 }, { "id": 120, "elements": [ 13, 14, 17, 18, 21 ], "cost": 215 }, { "id": 121, "elements": [ 12, 13, 15, 18 ], "cost": 360 }, { "id": 122, "elements": [ 17, 20, 22, 23 ], "cost": 180 }, { "id": 123, "elements": [ 1, 2, 3, 4 ], "cost": 152 }, { "id": 124, "elements": [ 18, 19, 20, 21, 22, 23 ], "cost": 456 }, { "id": 125, "elements": [ 15, 16, 18 ], "cost": 3 }, { "id": 126, "elements": [ 3, 4, 8 ], "cost": 258 }, { "id": 127, "elements": [ 14, 15, 16, 17, 18, 20, 25 ], "cost": 203 }, { "id": 128, "elements": [ 10, 11, 13 ], "cost": 156 }, { "id": 129, "elements": [ 2, 3, 4, 5, 6, 7, 8 ], "cost": 126 }, { "id": 130, "elements": [ 4, 5, 6, 7, 8, 12 ], "cost": 360 }, { "id": 131, "elements": [ 9, 13, 14, 15, 16 ], "cost": 325 }, { "id": 132, "elements": [ 16, 20, 21, 22, 23, 24, 25 ], "cost": 693 }, { "id": 133, "elements": [ 20, 22, 23, 24, 25 ], "cost": 400 }, { "id": 134, "elements": [ 15, 16, 19, 21, 22 ], "cost": 300 }, { "id": 135, "elements": [ 12, 13, 14, 16, 17, 18, 20 ], "cost": 266 }, { "id": 136, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 137, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 138, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 139, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 140, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 141, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 142, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 143, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 144, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 145, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 146, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 147, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 148, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 149, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 150, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 151, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 152, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 153, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 154, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 155, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 156, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 157, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 158, "elements": [ 23 ], "cost": 10000, "singleton": true }, { "id": 159, "elements": [ 24 ], "cost": 10000, "singleton": true }, { "id": 160, "elements": [ 25 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0012_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0012_bag_stats.png" }, "solution": [ 27, 67, 75, 81, 102 ], "obj": 795.0, "instance_variant": { "num_elements": 25, "num_sets": 160, "sets": [ { "id": "S1", "elements": [ 0, 1, 2, 3, 6, 8, 9 ], "cost": 98 }, { "id": "S2", "elements": [ 6, 8, 12, 13, 14 ], "cost": 105 }, { "id": "S3", "elements": [ 11, 12, 16, 17 ], "cost": 64 }, { "id": "S4", "elements": [ 0, 2, 3 ], "cost": 162 }, { "id": "S5", "elements": [ 2, 4, 7 ], "cost": 252 }, { "id": "S6", "elements": [ 17, 20, 22, 23 ], "cost": 88 }, { "id": "S7", "elements": [ 3, 4, 6, 8, 9, 10, 11 ], "cost": 42 }, { "id": "S8", "elements": [ 2, 4, 5, 6, 7, 9 ], "cost": 480 }, { "id": "S9", "elements": [ 8, 9, 10, 11, 13, 14, 15 ], "cost": 7 }, { "id": "S10", "elements": [ 1, 2, 4, 5, 6, 7, 10 ], "cost": 105 }, { "id": "S11", "elements": [ 19, 20, 21, 22, 24 ], "cost": 205 }, { "id": "S12", "elements": [ 13, 14, 16, 17, 18 ], "cost": 420 }, { "id": "S13", "elements": [ 9, 10, 11, 12, 13, 14, 16 ], "cost": 378 }, { "id": "S14", "elements": [ 16, 20, 22, 23, 24 ], "cost": 15 }, { "id": "S15", "elements": [ 11, 14, 18, 19 ], "cost": 400 }, { "id": "S16", "elements": [ 6, 7, 8, 9, 10, 12, 13 ], "cost": 672 }, { "id": "S17", "elements": [ 17, 20, 23 ], "cost": 165 }, { "id": "S18", "elements": [ 19, 21, 22, 23 ], "cost": 84 }, { "id": "S19", "elements": [ 5, 8, 10 ], "cost": 162 }, { "id": "S20", "elements": [ 2, 5, 6 ], "cost": 117 }, { "id": "S21", "elements": [ 19, 20, 21, 23, 24 ], "cost": 415 }, { "id": "S22", "elements": [ 21, 22, 23 ], "cost": 165 }, { "id": "S23", "elements": [ 12, 13, 14, 15, 17, 18 ], "cost": 126 }, { "id": "S24", "elements": [ 0, 1, 3, 7 ], "cost": 320 }, { "id": "S25", "elements": [ 10, 12, 14 ], "cost": 30 }, { "id": "S26", "elements": [ 9, 11, 12, 13, 14, 15, 16 ], "cost": 245 }, { "id": "S27", "elements": [ 1, 2, 5 ], "cost": 93 }, { "id": "S28", "elements": [ 11, 14, 15, 21 ], "cost": 100 }, { "id": "S29", "elements": [ 15, 16, 18, 20, 23 ], "cost": 310 }, { "id": "S30", "elements": [ 7, 9, 15 ], "cost": 291 }, { "id": "S31", "elements": [ 10, 11, 12, 13 ], "cost": 68 }, { "id": "S32", "elements": [ 0, 2, 3, 4, 5, 6 ], "cost": 294 }, { "id": "S33", "elements": [ 0, 1, 2, 3, 6 ], "cost": 460 }, { "id": "S34", "elements": [ 3, 6, 7, 8 ], "cost": 260 }, { "id": "S35", "elements": [ 18, 20, 22, 23, 24 ], "cost": 190 }, { "id": "S36", "elements": [ 9, 11, 14, 15 ], "cost": 316 }, { "id": "S37", "elements": [ 0, 1, 3, 5, 6, 10 ], "cost": 72 }, { "id": "S38", "elements": [ 11, 13, 14, 15, 18 ], "cost": 60 }, { "id": "S39", "elements": [ 20, 21, 23 ], "cost": 99 }, { "id": "S40", "elements": [ 9, 11, 12, 14, 15, 18 ], "cost": 312 }, { "id": "S41", "elements": [ 16, 18, 21, 22, 23 ], "cost": 240 }, { "id": "S42", "elements": [ 11, 13, 16 ], "cost": 264 }, { "id": "S43", "elements": [ 16, 21, 22 ], "cost": 216 }, { "id": "S44", "elements": [ 1, 2, 4 ], "cost": 30 }, { "id": "S45", "elements": [ 13, 14, 15, 16 ], "cost": 240 }, { "id": "S46", "elements": [ 7, 8, 9 ], "cost": 297 }, { "id": "S47", "elements": [ 10, 11, 13, 14, 19 ], "cost": 305 }, { "id": "S48", "elements": [ 6, 10, 11, 12, 13, 16 ], "cost": 366 }, { "id": "S49", "elements": [ 4, 7, 9, 10 ], "cost": 328 }, { "id": "S50", "elements": [ 1, 3, 4 ], "cost": 276 }, { "id": "S51", "elements": [ 9, 11, 13, 14, 16 ], "cost": 275 }, { "id": "S52", "elements": [ 21, 22, 23, 24 ], "cost": 276 }, { "id": "S53", "elements": [ 4, 5, 6, 7, 8, 10 ], "cost": 462 }, { "id": "S54", "elements": [ 20, 22, 23 ], "cost": 282 }, { "id": "S55", "elements": [ 17, 19, 20, 22, 23, 24 ], "cost": 522 }, { "id": "S56", "elements": [ 16, 17, 18, 20, 22 ], "cost": 410 }, { "id": "S57", "elements": [ 0, 1, 2, 3, 4, 5, 6 ], "cost": 189 }, { "id": "S58", "elements": [ 15, 16, 20, 21, 22, 23, 24 ], "cost": 630 }, { "id": "S59", "elements": [ 12, 13, 14, 17, 20 ], "cost": 215 }, { "id": "S60", "elements": [ 4, 5, 6 ], "cost": 63 }, { "id": "S61", "elements": [ 11, 12, 13, 14, 15 ], "cost": 65 }, { "id": "S62", "elements": [ 0, 1, 2, 3, 4, 5, 7 ], "cost": 574 }, { "id": "S63", "elements": [ 15, 18, 19, 20, 21 ], "cost": 355 }, { "id": "S64", "elements": [ 5, 8, 9, 10 ], "cost": 240 }, { "id": "S65", "elements": [ 14, 17, 18, 21, 22 ], "cost": 185 }, { "id": "S66", "elements": [ 19, 20, 22, 23 ], "cost": 124 }, { "id": "S67", "elements": [ 8, 9, 15 ], "cost": 201 }, { "id": "S68", "elements": [ 16, 17, 18, 19, 20, 21, 22 ], "cost": 357 }, { "id": "S69", "elements": [ 14, 15, 16, 17, 18, 20, 22 ], "cost": 210 }, { "id": "S70", "elements": [ 4, 5, 7 ], "cost": 294 }, { "id": "S71", "elements": [ 2, 3, 4, 5, 7, 10 ], "cost": 594 }, { "id": "S72", "elements": [ 10, 11, 12, 14, 15, 16 ], "cost": 192 }, { "id": "S73", "elements": [ 10, 11, 12, 13, 14, 16 ], "cost": 450 }, { "id": "S74", "elements": [ 1, 2, 5, 6 ], "cost": 324 }, { "id": "S75", "elements": [ 0, 3, 4, 6, 7 ], "cost": 25 }, { "id": "S76", "elements": [ 10, 11, 16 ], "cost": 210 }, { "id": "S77", "elements": [ 14, 18, 20 ], "cost": 12 }, { "id": "S78", "elements": [ 12, 13, 14, 16, 19 ], "cost": 210 }, { "id": "S79", "elements": [ 1, 4, 6 ], "cost": 186 }, { "id": "S80", "elements": [ 1, 2, 3 ], "cost": 288 }, { "id": "S81", "elements": [ 10, 11, 12, 13, 14, 16, 17 ], "cost": 357 }, { "id": "S82", "elements": [ 11, 12, 13, 14, 15, 16 ], "cost": 330 }, { "id": "S83", "elements": [ 7, 10, 12 ], "cost": 66 }, { "id": "S84", "elements": [ 13, 14, 15, 16, 17, 20, 21 ], "cost": 105 }, { "id": "S85", "elements": [ 7, 8, 10 ], "cost": 84 }, { "id": "S86", "elements": [ 5, 8, 9, 10, 12, 13 ], "cost": 438 }, { "id": "S87", "elements": [ 3, 4, 6, 7, 8, 9 ], "cost": 468 }, { "id": "S88", "elements": [ 17, 18, 19, 20, 21, 22, 23 ], "cost": 448 }, { "id": "S89", "elements": [ 5, 6, 7, 8, 9 ], "cost": 370 }, { "id": "S90", "elements": [ 2, 4, 6 ], "cost": 180 }, { "id": "S91", "elements": [ 8, 10, 12, 13 ], "cost": 12 }, { "id": "S92", "elements": [ 2, 3, 6 ], "cost": 123 }, { "id": "S93", "elements": [ 7, 9, 11, 12, 14, 18 ], "cost": 384 }, { "id": "S94", "elements": [ 0, 1, 2, 5, 6, 7, 10 ], "cost": 623 }, { "id": "S95", "elements": [ 5, 6, 7, 10, 11, 13 ], "cost": 534 }, { "id": "S96", "elements": [ 4, 5, 7, 8, 10, 11, 12 ], "cost": 343 }, { "id": "S97", "elements": [ 2, 4, 5, 9 ], "cost": 76 }, { "id": "S98", "elements": [ 10, 11, 12, 13, 14, 15 ], "cost": 216 }, { "id": "S99", "elements": [ 19, 22, 23, 24 ], "cost": 184 }, { "id": "S100", "elements": [ 3, 4, 8, 9, 12 ], "cost": 40 }, { "id": "S101", "elements": [ 17, 18, 19, 23 ], "cost": 108 }, { "id": "S102", "elements": [ 18, 19, 20, 21, 22, 23, 24 ], "cost": 119 }, { "id": "S103", "elements": [ 8, 10, 11, 13 ], "cost": 208 }, { "id": "S104", "elements": [ 8, 10, 12 ], "cost": 222 }, { "id": "S105", "elements": [ 0, 1, 2, 3, 4, 5 ], "cost": 366 }, { "id": "S106", "elements": [ 6, 7, 9, 11, 12 ], "cost": 275 }, { "id": "S107", "elements": [ 5, 6, 7 ], "cost": 192 }, { "id": "S108", "elements": [ 0, 1, 3, 4, 8 ], "cost": 25 }, { "id": "S109", "elements": [ 3, 4, 6, 8, 9 ], "cost": 255 }, { "id": "S110", "elements": [ 4, 5, 9, 10 ], "cost": 108 }, { "id": "S111", "elements": [ 2, 6, 7, 10 ], "cost": 360 }, { "id": "S112", "elements": [ 18, 19, 20, 21, 22, 24 ], "cost": 354 }, { "id": "S113", "elements": [ 0, 1, 2 ], "cost": 102 }, { "id": "S114", "elements": [ 2, 3, 4, 5, 6, 8, 10 ], "cost": 105 }, { "id": "S115", "elements": [ 2, 4, 6, 7, 8 ], "cost": 315 }, { "id": "S116", "elements": [ 11, 15, 16, 17, 18, 20 ], "cost": 402 }, { "id": "S117", "elements": [ 15, 17, 18, 19, 20, 21 ], "cost": 174 }, { "id": "S118", "elements": [ 0, 1, 3, 4, 5, 6 ], "cost": 72 }, { "id": "S119", "elements": [ 13, 15, 17 ], "cost": 261 }, { "id": "S120", "elements": [ 12, 13, 16, 17, 20 ], "cost": 215 }, { "id": "S121", "elements": [ 11, 12, 14, 17 ], "cost": 360 }, { "id": "S122", "elements": [ 16, 19, 21, 22 ], "cost": 180 }, { "id": "S123", "elements": [ 0, 1, 2, 3 ], "cost": 152 }, { "id": "S124", "elements": [ 17, 18, 19, 20, 21, 22 ], "cost": 456 }, { "id": "S125", "elements": [ 14, 15, 17 ], "cost": 3 }, { "id": "S126", "elements": [ 2, 3, 7 ], "cost": 258 }, { "id": "S127", "elements": [ 13, 14, 15, 16, 17, 19, 24 ], "cost": 203 }, { "id": "S128", "elements": [ 9, 10, 12 ], "cost": 156 }, { "id": "S129", "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 126 }, { "id": "S130", "elements": [ 3, 4, 5, 6, 7, 11 ], "cost": 360 }, { "id": "S131", "elements": [ 8, 12, 13, 14, 15 ], "cost": 325 }, { "id": "S132", "elements": [ 15, 19, 20, 21, 22, 23, 24 ], "cost": 693 }, { "id": "S133", "elements": [ 19, 21, 22, 23, 24 ], "cost": 400 }, { "id": "S134", "elements": [ 14, 15, 18, 20, 21 ], "cost": 300 }, { "id": "S135", "elements": [ 11, 12, 13, 15, 16, 17, 19 ], "cost": 266 }, { "id": "S136", "elements": [ 0 ], "cost": 10000 }, { "id": "S137", "elements": [ 1 ], "cost": 10000 }, { "id": "S138", "elements": [ 2 ], "cost": 10000 }, { "id": "S139", "elements": [ 3 ], "cost": 10000 }, { "id": "S140", "elements": [ 4 ], "cost": 10000 }, { "id": "S141", "elements": [ 5 ], "cost": 10000 }, { "id": "S142", "elements": [ 6 ], "cost": 10000 }, { "id": "S143", "elements": [ 7 ], "cost": 10000 }, { "id": "S144", "elements": [ 8 ], "cost": 10000 }, { "id": "S145", "elements": [ 9 ], "cost": 10000 }, { "id": "S146", "elements": [ 10 ], "cost": 10000 }, { "id": "S147", "elements": [ 11 ], "cost": 10000 }, { "id": "S148", "elements": [ 12 ], "cost": 10000 }, { "id": "S149", "elements": [ 13 ], "cost": 10000 }, { "id": "S150", "elements": [ 14 ], "cost": 10000 }, { "id": "S151", "elements": [ 15 ], "cost": 10000 }, { "id": "S152", "elements": [ 16 ], "cost": 10000 }, { "id": "S153", "elements": [ 17 ], "cost": 10000 }, { "id": "S154", "elements": [ 18 ], "cost": 10000 }, { "id": "S155", "elements": [ 19 ], "cost": 10000 }, { "id": "S156", "elements": [ 20 ], "cost": 10000 }, { "id": "S157", "elements": [ 21 ], "cost": 10000 }, { "id": "S158", "elements": [ 22 ], "cost": 10000 }, { "id": "S159", "elements": [ 23 ], "cost": 10000 }, { "id": "S160", "elements": [ 24 ], "cost": 10000 } ] }, "solution_variant": [ "S27", "S67", "S75", "S81", "S102" ], "context_index": 13, "input_format": "json", "input_index_base": 0 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "At a small house renovation, the owner is choosing from a set of contractor deals where each deal takes care of some rooms and has a fee. The job is to assign exactly one deal to every room (so no room is left out or covered more than once) and then compute the total by adding the fees of the selected deals — the best assignment is the one with the lowest combined cost. The particular packages and their room coverage are given below.\n\n- **total_rooms**: 23\n- **total_packages**: 129\n\n| package_id | package_fee | rooms_covered |\n|---|---|---|\n| S1 | 360 | A C D E |\n| S2 | 261 | B E I |\n| S3 | 445 | A D E F K |\n| S4 | 39 | A B D |\n| S5 | 24 | C D G |\n| S6 | 595 | H I J K L M O |\n| S7 | 330 | I L M N P |\n| S8 | 216 | N P Q R S T |\n| S9 | 148 | R U V W |\n| S10 | 340 | N R T U |\n| S11 | 10 | A B C E F |\n| S12 | 140 | H J K L M O P |\n| S13 | 392 | Q R T U |\n| S14 | 120 | J M N O R |\n| S15 | 225 | N P Q S V |\n| S16 | 320 | I J L N O |\n| S17 | 658 | L N O Q R S T |\n| S18 | 288 | J K M N P Q |\n| S19 | 370 | A B C E H |\n| S20 | 340 | D E F G H |\n| S21 | 6 | C D E F G K |\n| S22 | 376 | M N O Q |\n| S23 | 360 | N O P Q S |\n| S24 | 78 | A B C D E F |\n| S25 | 114 | C D H |\n| S26 | 469 | E F G H I J L |\n| S27 | 588 | P Q R S T U W |\n| S28 | 196 | I K |\n| S29 | 255 | G I K |\n| S30 | 30 | A B D E F G |\n| S31 | 284 | A C D G |\n| S32 | 280 | A B D F G |\n| S33 | 276 | C E I J |\n| S34 | 450 | I L N O P |\n| S35 | 237 | T U V |\n| S36 | 679 | F H J K L M P |\n| S37 | 102 | M N R S T W |\n| S38 | 152 | F H I K |\n| S39 | 99 | I J M |\n| S40 | 700 | D E G H I K L |\n| S41 | 225 | Q S T U W |\n| S42 | 240 | R S T U V W |\n| S43 | 588 | C D F G H I |\n| S44 | 156 | A C E F |\n| S45 | 120 | I K M |\n| S46 | 194 | N O |\n| S47 | 330 | G H I J K L |\n| S48 | 414 | A C D F G H |\n| S49 | 240 | Q T U W |\n| S50 | 420 | C E F G J L |\n| S51 | 18 | O R S T U V |\n| S52 | 212 | K N O P |\n| S53 | 27 | B C D |\n| S54 | 324 | F G K L |\n| S55 | 72 | R T U |\n| S56 | 170 | K L O P Q |\n| S57 | 399 | A B C D E F G |\n| S58 | 318 | I K M N O P |\n| S59 | 456 | H I L M P Q |\n| S60 | 184 | R S |\n| S61 | 120 | O P Q R S U |\n| S62 | 260 | Q R S W |\n| S63 | 476 | P R S T U V W |\n| S64 | 495 | H I J K M |\n| S65 | 216 | N O P R |\n| S66 | 261 | C G H |\n| S67 | 325 | B D E F G |\n| S68 | 686 | P Q R S T U V |\n| S69 | 50 | L M P Q R |\n| S70 | 156 | P S T |\n| S71 | 558 | E G H I J K |\n| S72 | 285 | L O P |\n| S73 | 396 | H I J K L N |\n| S74 | 147 | A C D |\n| S75 | 156 | C F G J |\n| S76 | 376 | G M N O |\n| S77 | 135 | O P T |\n| S78 | 136 | B C D E |\n| S79 | 399 | J L N O P Q R |\n| S80 | 66 | M O P Q S V |\n| S81 | 600 | F G H I J K |\n| S82 | 273 | T U W |\n| S83 | 192 | I J K L N P |\n| S84 | 183 | K L P |\n| S85 | 36 | D E G |\n| S86 | 343 | A D E F G H I |\n| S87 | 150 | E F G I J |\n| S88 | 90 | P S T U V W |\n| S89 | 315 | G H I J K M O |\n| S90 | 106 | I L |\n| S91 | 171 | J K L |\n| S92 | 24 | G H J L |\n| S93 | 60 | L N P Q |\n| S94 | 301 | O Q S T U V W |\n| S95 | 306 | N O Q S U W |\n| S96 | 12 | E F G H I J |\n| S97 | 130 | B C |\n| S98 | 356 | K L O Q |\n| S99 | 156 | H J K |\n| S100 | 273 | F I K |\n| S101 | 306 | I K N O P R |\n| S102 | 288 | S U V |\n| S103 | 168 | F I |\n| S104 | 450 | J L M N Q |\n| S105 | 460 | G I J L N |\n| S106 | 174 | C H J |\n| S107 | 10000 | A |\n| S108 | 10000 | B |\n| S109 | 10000 | C |\n| S110 | 10000 | D |\n| S111 | 10000 | E |\n| S112 | 10000 | F |\n| S113 | 10000 | G |\n| S114 | 10000 | H |\n| S115 | 10000 | I |\n| S116 | 10000 | J |\n| S117 | 10000 | K |\n| S118 | 10000 | L |\n| S119 | 10000 | M |\n| S120 | 10000 | N |\n| S121 | 10000 | O |\n| S122 | 10000 | P |\n| S123 | 10000 | Q |\n| S124 | 10000 | R |\n| S125 | 10000 | S |\n| S126 | 10000 | T |\n| S127 | 10000 | U |\n| S128 | 10000 | V |\n| S129 | 10000 | W |\n\nIf you want to return the chosen contractor deals, just send it in this simple JSON shape.\n\n{\n \"solution\": [\"deal_id\", ...]\n}\n\nThis says: put the IDs of the selected deals in the solution list (one entry per chosen package). It's just a sketch of the shape I expect — not the actual answer itself.\n\nPlease make sure you use the exact identifiers from the problem instance, without renaming or inventing new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 23, "num_sets": 129, "density": 0.17559824738793395, "sets": [ { "id": 1, "elements": [ 1, 3, 4, 5 ], "cost": 360 }, { "id": 2, "elements": [ 2, 5, 9 ], "cost": 261 }, { "id": 3, "elements": [ 1, 4, 5, 6, 11 ], "cost": 445 }, { "id": 4, "elements": [ 1, 2, 4 ], "cost": 39 }, { "id": 5, "elements": [ 3, 4, 7 ], "cost": 24 }, { "id": 6, "elements": [ 8, 9, 10, 11, 12, 13, 15 ], "cost": 595 }, { "id": 7, "elements": [ 9, 12, 13, 14, 16 ], "cost": 330 }, { "id": 8, "elements": [ 14, 16, 17, 18, 19, 20 ], "cost": 216 }, { "id": 9, "elements": [ 18, 21, 22, 23 ], "cost": 148 }, { "id": 10, "elements": [ 14, 18, 20, 21 ], "cost": 340 }, { "id": 11, "elements": [ 1, 2, 3, 5, 6 ], "cost": 10 }, { "id": 12, "elements": [ 8, 10, 11, 12, 13, 15, 16 ], "cost": 140 }, { "id": 13, "elements": [ 17, 18, 20, 21 ], "cost": 392 }, { "id": 14, "elements": [ 10, 13, 14, 15, 18 ], "cost": 120 }, { "id": 15, "elements": [ 14, 16, 17, 19, 22 ], "cost": 225 }, { "id": 16, "elements": [ 9, 10, 12, 14, 15 ], "cost": 320 }, { "id": 17, "elements": [ 12, 14, 15, 17, 18, 19, 20 ], "cost": 658 }, { "id": 18, "elements": [ 10, 11, 13, 14, 16, 17 ], "cost": 288 }, { "id": 19, "elements": [ 1, 2, 3, 5, 8 ], "cost": 370 }, { "id": 20, "elements": [ 4, 5, 6, 7, 8 ], "cost": 340 }, { "id": 21, "elements": [ 3, 4, 5, 6, 7, 11 ], "cost": 6 }, { "id": 22, "elements": [ 13, 14, 15, 17 ], "cost": 376 }, { "id": 23, "elements": [ 14, 15, 16, 17, 19 ], "cost": 360 }, { "id": 24, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 78 }, { "id": 25, "elements": [ 3, 4, 8 ], "cost": 114 }, { "id": 26, "elements": [ 5, 6, 7, 8, 9, 10, 12 ], "cost": 469 }, { "id": 27, "elements": [ 16, 17, 18, 19, 20, 21, 23 ], "cost": 588 }, { "id": 28, "elements": [ 9, 11 ], "cost": 196 }, { "id": 29, "elements": [ 7, 9, 11 ], "cost": 255 }, { "id": 30, "elements": [ 1, 2, 4, 5, 6, 7 ], "cost": 30 }, { "id": 31, "elements": [ 1, 3, 4, 7 ], "cost": 284 }, { "id": 32, "elements": [ 1, 2, 4, 6, 7 ], "cost": 280 }, { "id": 33, "elements": [ 3, 5, 9, 10 ], "cost": 276 }, { "id": 34, "elements": [ 9, 12, 14, 15, 16 ], "cost": 450 }, { "id": 35, "elements": [ 20, 21, 22 ], "cost": 237 }, { "id": 36, "elements": [ 6, 8, 10, 11, 12, 13, 16 ], "cost": 679 }, { "id": 37, "elements": [ 13, 14, 18, 19, 20, 23 ], "cost": 102 }, { "id": 38, "elements": [ 6, 8, 9, 11 ], "cost": 152 }, { "id": 39, "elements": [ 9, 10, 13 ], "cost": 99 }, { "id": 40, "elements": [ 4, 5, 7, 8, 9, 11, 12 ], "cost": 700 }, { "id": 41, "elements": [ 17, 19, 20, 21, 23 ], "cost": 225 }, { "id": 42, "elements": [ 18, 19, 20, 21, 22, 23 ], "cost": 240 }, { "id": 43, "elements": [ 3, 4, 6, 7, 8, 9 ], "cost": 588 }, { "id": 44, "elements": [ 1, 3, 5, 6 ], "cost": 156 }, { "id": 45, "elements": [ 9, 11, 13 ], "cost": 120 }, { "id": 46, "elements": [ 14, 15 ], "cost": 194 }, { "id": 47, "elements": [ 7, 8, 9, 10, 11, 12 ], "cost": 330 }, { "id": 48, "elements": [ 1, 3, 4, 6, 7, 8 ], "cost": 414 }, { "id": 49, "elements": [ 17, 20, 21, 23 ], "cost": 240 }, { "id": 50, "elements": [ 3, 5, 6, 7, 10, 12 ], "cost": 420 }, { "id": 51, "elements": [ 15, 18, 19, 20, 21, 22 ], "cost": 18 }, { "id": 52, "elements": [ 11, 14, 15, 16 ], "cost": 212 }, { "id": 53, "elements": [ 2, 3, 4 ], "cost": 27 }, { "id": 54, "elements": [ 6, 7, 11, 12 ], "cost": 324 }, { "id": 55, "elements": [ 18, 20, 21 ], "cost": 72 }, { "id": 56, "elements": [ 11, 12, 15, 16, 17 ], "cost": 170 }, { "id": 57, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 399 }, { "id": 58, "elements": [ 9, 11, 13, 14, 15, 16 ], "cost": 318 }, { "id": 59, "elements": [ 8, 9, 12, 13, 16, 17 ], "cost": 456 }, { "id": 60, "elements": [ 18, 19 ], "cost": 184 }, { "id": 61, "elements": [ 15, 16, 17, 18, 19, 21 ], "cost": 120 }, { "id": 62, "elements": [ 17, 18, 19, 23 ], "cost": 260 }, { "id": 63, "elements": [ 16, 18, 19, 20, 21, 22, 23 ], "cost": 476 }, { "id": 64, "elements": [ 8, 9, 10, 11, 13 ], "cost": 495 }, { "id": 65, "elements": [ 14, 15, 16, 18 ], "cost": 216 }, { "id": 66, "elements": [ 3, 7, 8 ], "cost": 261 }, { "id": 67, "elements": [ 2, 4, 5, 6, 7 ], "cost": 325 }, { "id": 68, "elements": [ 16, 17, 18, 19, 20, 21, 22 ], "cost": 686 }, { "id": 69, "elements": [ 12, 13, 16, 17, 18 ], "cost": 50 }, { "id": 70, "elements": [ 16, 19, 20 ], "cost": 156 }, { "id": 71, "elements": [ 5, 7, 8, 9, 10, 11 ], "cost": 558 }, { "id": 72, "elements": [ 12, 15, 16 ], "cost": 285 }, { "id": 73, "elements": [ 8, 9, 10, 11, 12, 14 ], "cost": 396 }, { "id": 74, "elements": [ 1, 3, 4 ], "cost": 147 }, { "id": 75, "elements": [ 3, 6, 7, 10 ], "cost": 156 }, { "id": 76, "elements": [ 7, 13, 14, 15 ], "cost": 376 }, { "id": 77, "elements": [ 15, 16, 20 ], "cost": 135 }, { "id": 78, "elements": [ 2, 3, 4, 5 ], "cost": 136 }, { "id": 79, "elements": [ 10, 12, 14, 15, 16, 17, 18 ], "cost": 399 }, { "id": 80, "elements": [ 13, 15, 16, 17, 19, 22 ], "cost": 66 }, { "id": 81, "elements": [ 6, 7, 8, 9, 10, 11 ], "cost": 600 }, { "id": 82, "elements": [ 20, 21, 23 ], "cost": 273 }, { "id": 83, "elements": [ 9, 10, 11, 12, 14, 16 ], "cost": 192 }, { "id": 84, "elements": [ 11, 12, 16 ], "cost": 183 }, { "id": 85, "elements": [ 4, 5, 7 ], "cost": 36 }, { "id": 86, "elements": [ 1, 4, 5, 6, 7, 8, 9 ], "cost": 343 }, { "id": 87, "elements": [ 5, 6, 7, 9, 10 ], "cost": 150 }, { "id": 88, "elements": [ 16, 19, 20, 21, 22, 23 ], "cost": 90 }, { "id": 89, "elements": [ 7, 8, 9, 10, 11, 13, 15 ], "cost": 315 }, { "id": 90, "elements": [ 9, 12 ], "cost": 106 }, { "id": 91, "elements": [ 10, 11, 12 ], "cost": 171 }, { "id": 92, "elements": [ 7, 8, 10, 12 ], "cost": 24 }, { "id": 93, "elements": [ 12, 14, 16, 17 ], "cost": 60 }, { "id": 94, "elements": [ 15, 17, 19, 20, 21, 22, 23 ], "cost": 301 }, { "id": 95, "elements": [ 14, 15, 17, 19, 21, 23 ], "cost": 306 }, { "id": 96, "elements": [ 5, 6, 7, 8, 9, 10 ], "cost": 12 }, { "id": 97, "elements": [ 2, 3 ], "cost": 130 }, { "id": 98, "elements": [ 11, 12, 15, 17 ], "cost": 356 }, { "id": 99, "elements": [ 8, 10, 11 ], "cost": 156 }, { "id": 100, "elements": [ 6, 9, 11 ], "cost": 273 }, { "id": 101, "elements": [ 9, 11, 14, 15, 16, 18 ], "cost": 306 }, { "id": 102, "elements": [ 19, 21, 22 ], "cost": 288 }, { "id": 103, "elements": [ 6, 9 ], "cost": 168 }, { "id": 104, "elements": [ 10, 12, 13, 14, 17 ], "cost": 450 }, { "id": 105, "elements": [ 7, 9, 10, 12, 14 ], "cost": 460 }, { "id": 106, "elements": [ 3, 8, 10 ], "cost": 174 }, { "id": 107, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 118, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 119, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 120, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 121, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 122, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 123, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 124, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 125, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 126, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 127, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 128, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 129, "elements": [ 23 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0013_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0013_bag_stats.png" }, "solution": [ 24, 42, 89, 93 ], "obj": 693.0, "instance_variant": { "num_elements": 23, "num_sets": 129, "sets": [ { "id": "S1", "elements": [ "A", "C", "D", "E" ], "cost": 360 }, { "id": "S2", "elements": [ "B", "E", "I" ], "cost": 261 }, { "id": "S3", "elements": [ "A", "D", "E", "F", "K" ], "cost": 445 }, { "id": "S4", "elements": [ "A", "B", "D" ], "cost": 39 }, { "id": "S5", "elements": [ "C", "D", "G" ], "cost": 24 }, { "id": "S6", "elements": [ "H", "I", "J", "K", "L", "M", "O" ], "cost": 595 }, { "id": "S7", "elements": [ "I", "L", "M", "N", "P" ], "cost": 330 }, { "id": "S8", "elements": [ "N", "P", "Q", "R", "S", "T" ], "cost": 216 }, { "id": "S9", "elements": [ "R", "U", "V", "W" ], "cost": 148 }, { "id": "S10", "elements": [ "N", "R", "T", "U" ], "cost": 340 }, { "id": "S11", "elements": [ "A", "B", "C", "E", "F" ], "cost": 10 }, { "id": "S12", "elements": [ "H", "J", "K", "L", "M", "O", "P" ], "cost": 140 }, { "id": "S13", "elements": [ "Q", "R", "T", "U" ], "cost": 392 }, { "id": "S14", "elements": [ "J", "M", "N", "O", "R" ], "cost": 120 }, { "id": "S15", "elements": [ "N", "P", "Q", "S", "V" ], "cost": 225 }, { "id": "S16", "elements": [ "I", "J", "L", "N", "O" ], "cost": 320 }, { "id": "S17", "elements": [ "L", "N", "O", "Q", "R", "S", "T" ], "cost": 658 }, { "id": "S18", "elements": [ "J", "K", "M", "N", "P", "Q" ], "cost": 288 }, { "id": "S19", "elements": [ "A", "B", "C", "E", "H" ], "cost": 370 }, { "id": "S20", "elements": [ "D", "E", "F", "G", "H" ], "cost": 340 }, { "id": "S21", "elements": [ "C", "D", "E", "F", "G", "K" ], "cost": 6 }, { "id": "S22", "elements": [ "M", "N", "O", "Q" ], "cost": 376 }, { "id": "S23", "elements": [ "N", "O", "P", "Q", "S" ], "cost": 360 }, { "id": "S24", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 78 }, { "id": "S25", "elements": [ "C", "D", "H" ], "cost": 114 }, { "id": "S26", "elements": [ "E", "F", "G", "H", "I", "J", "L" ], "cost": 469 }, { "id": "S27", "elements": [ "P", "Q", "R", "S", "T", "U", "W" ], "cost": 588 }, { "id": "S28", "elements": [ "I", "K" ], "cost": 196 }, { "id": "S29", "elements": [ "G", "I", "K" ], "cost": 255 }, { "id": "S30", "elements": [ "A", "B", "D", "E", "F", "G" ], "cost": 30 }, { "id": "S31", "elements": [ "A", "C", "D", "G" ], "cost": 284 }, { "id": "S32", "elements": [ "A", "B", "D", "F", "G" ], "cost": 280 }, { "id": "S33", "elements": [ "C", "E", "I", "J" ], "cost": 276 }, { "id": "S34", "elements": [ "I", "L", "N", "O", "P" ], "cost": 450 }, { "id": "S35", "elements": [ "T", "U", "V" ], "cost": 237 }, { "id": "S36", "elements": [ "F", "H", "J", "K", "L", "M", "P" ], "cost": 679 }, { "id": "S37", "elements": [ "M", "N", "R", "S", "T", "W" ], "cost": 102 }, { "id": "S38", "elements": [ "F", "H", "I", "K" ], "cost": 152 }, { "id": "S39", "elements": [ "I", "J", "M" ], "cost": 99 }, { "id": "S40", "elements": [ "D", "E", "G", "H", "I", "K", "L" ], "cost": 700 }, { "id": "S41", "elements": [ "Q", "S", "T", "U", "W" ], "cost": 225 }, { "id": "S42", "elements": [ "R", "S", "T", "U", "V", "W" ], "cost": 240 }, { "id": "S43", "elements": [ "C", "D", "F", "G", "H", "I" ], "cost": 588 }, { "id": "S44", "elements": [ "A", "C", "E", "F" ], "cost": 156 }, { "id": "S45", "elements": [ "I", "K", "M" ], "cost": 120 }, { "id": "S46", "elements": [ "N", "O" ], "cost": 194 }, { "id": "S47", "elements": [ "G", "H", "I", "J", "K", "L" ], "cost": 330 }, { "id": "S48", "elements": [ "A", "C", "D", "F", "G", "H" ], "cost": 414 }, { "id": "S49", "elements": [ "Q", "T", "U", "W" ], "cost": 240 }, { "id": "S50", "elements": [ "C", "E", "F", "G", "J", "L" ], "cost": 420 }, { "id": "S51", "elements": [ "O", "R", "S", "T", "U", "V" ], "cost": 18 }, { "id": "S52", "elements": [ "K", "N", "O", "P" ], "cost": 212 }, { "id": "S53", "elements": [ "B", "C", "D" ], "cost": 27 }, { "id": "S54", "elements": [ "F", "G", "K", "L" ], "cost": 324 }, { "id": "S55", "elements": [ "R", "T", "U" ], "cost": 72 }, { "id": "S56", "elements": [ "K", "L", "O", "P", "Q" ], "cost": 170 }, { "id": "S57", "elements": [ "A", "B", "C", "D", "E", "F", "G" ], "cost": 399 }, { "id": "S58", "elements": [ "I", "K", "M", "N", "O", "P" ], "cost": 318 }, { "id": "S59", "elements": [ "H", "I", "L", "M", "P", "Q" ], "cost": 456 }, { "id": "S60", "elements": [ "R", "S" ], "cost": 184 }, { "id": "S61", "elements": [ "O", "P", "Q", "R", "S", "U" ], "cost": 120 }, { "id": "S62", "elements": [ "Q", "R", "S", "W" ], "cost": 260 }, { "id": "S63", "elements": [ "P", "R", "S", "T", "U", "V", "W" ], "cost": 476 }, { "id": "S64", "elements": [ "H", "I", "J", "K", "M" ], "cost": 495 }, { "id": "S65", "elements": [ "N", "O", "P", "R" ], "cost": 216 }, { "id": "S66", "elements": [ "C", "G", "H" ], "cost": 261 }, { "id": "S67", "elements": [ "B", "D", "E", "F", "G" ], "cost": 325 }, { "id": "S68", "elements": [ "P", "Q", "R", "S", "T", "U", "V" ], "cost": 686 }, { "id": "S69", "elements": [ "L", "M", "P", "Q", "R" ], "cost": 50 }, { "id": "S70", "elements": [ "P", "S", "T" ], "cost": 156 }, { "id": "S71", "elements": [ "E", "G", "H", "I", "J", "K" ], "cost": 558 }, { "id": "S72", "elements": [ "L", "O", "P" ], "cost": 285 }, { "id": "S73", "elements": [ "H", "I", "J", "K", "L", "N" ], "cost": 396 }, { "id": "S74", "elements": [ "A", "C", "D" ], "cost": 147 }, { "id": "S75", "elements": [ "C", "F", "G", "J" ], "cost": 156 }, { "id": "S76", "elements": [ "G", "M", "N", "O" ], "cost": 376 }, { "id": "S77", "elements": [ "O", "P", "T" ], "cost": 135 }, { "id": "S78", "elements": [ "B", "C", "D", "E" ], "cost": 136 }, { "id": "S79", "elements": [ "J", "L", "N", "O", "P", "Q", "R" ], "cost": 399 }, { "id": "S80", "elements": [ "M", "O", "P", "Q", "S", "V" ], "cost": 66 }, { "id": "S81", "elements": [ "F", "G", "H", "I", "J", "K" ], "cost": 600 }, { "id": "S82", "elements": [ "T", "U", "W" ], "cost": 273 }, { "id": "S83", "elements": [ "I", "J", "K", "L", "N", "P" ], "cost": 192 }, { "id": "S84", "elements": [ "K", "L", "P" ], "cost": 183 }, { "id": "S85", "elements": [ "D", "E", "G" ], "cost": 36 }, { "id": "S86", "elements": [ "A", "D", "E", "F", "G", "H", "I" ], "cost": 343 }, { "id": "S87", "elements": [ "E", "F", "G", "I", "J" ], "cost": 150 }, { "id": "S88", "elements": [ "P", "S", "T", "U", "V", "W" ], "cost": 90 }, { "id": "S89", "elements": [ "G", "H", "I", "J", "K", "M", "O" ], "cost": 315 }, { "id": "S90", "elements": [ "I", "L" ], "cost": 106 }, { "id": "S91", "elements": [ "J", "K", "L" ], "cost": 171 }, { "id": "S92", "elements": [ "G", "H", "J", "L" ], "cost": 24 }, { "id": "S93", "elements": [ "L", "N", "P", "Q" ], "cost": 60 }, { "id": "S94", "elements": [ "O", "Q", "S", "T", "U", "V", "W" ], "cost": 301 }, { "id": "S95", "elements": [ "N", "O", "Q", "S", "U", "W" ], "cost": 306 }, { "id": "S96", "elements": [ "E", "F", "G", "H", "I", "J" ], "cost": 12 }, { "id": "S97", "elements": [ "B", "C" ], "cost": 130 }, { "id": "S98", "elements": [ "K", "L", "O", "Q" ], "cost": 356 }, { "id": "S99", "elements": [ "H", "J", "K" ], "cost": 156 }, { "id": "S100", "elements": [ "F", "I", "K" ], "cost": 273 }, { "id": "S101", "elements": [ "I", "K", "N", "O", "P", "R" ], "cost": 306 }, { "id": "S102", "elements": [ "S", "U", "V" ], "cost": 288 }, { "id": "S103", "elements": [ "F", "I" ], "cost": 168 }, { "id": "S104", "elements": [ "J", "L", "M", "N", "Q" ], "cost": 450 }, { "id": "S105", "elements": [ "G", "I", "J", "L", "N" ], "cost": 460 }, { "id": "S106", "elements": [ "C", "H", "J" ], "cost": 174 }, { "id": "S107", "elements": [ "A" ], "cost": 10000 }, { "id": "S108", "elements": [ "B" ], "cost": 10000 }, { "id": "S109", "elements": [ "C" ], "cost": 10000 }, { "id": "S110", "elements": [ "D" ], "cost": 10000 }, { "id": "S111", "elements": [ "E" ], "cost": 10000 }, { "id": "S112", "elements": [ "F" ], "cost": 10000 }, { "id": "S113", "elements": [ "G" ], "cost": 10000 }, { "id": "S114", "elements": [ "H" ], "cost": 10000 }, { "id": "S115", "elements": [ "I" ], "cost": 10000 }, { "id": "S116", "elements": [ "J" ], "cost": 10000 }, { "id": "S117", "elements": [ "K" ], "cost": 10000 }, { "id": "S118", "elements": [ "L" ], "cost": 10000 }, { "id": "S119", "elements": [ "M" ], "cost": 10000 }, { "id": "S120", "elements": [ "N" ], "cost": 10000 }, { "id": "S121", "elements": [ "O" ], "cost": 10000 }, { "id": "S122", "elements": [ "P" ], "cost": 10000 }, { "id": "S123", "elements": [ "Q" ], "cost": 10000 }, { "id": "S124", "elements": [ "R" ], "cost": 10000 }, { "id": "S125", "elements": [ "S" ], "cost": 10000 }, { "id": "S126", "elements": [ "T" ], "cost": 10000 }, { "id": "S127", "elements": [ "U" ], "cost": 10000 }, { "id": "S128", "elements": [ "V" ], "cost": 10000 }, { "id": "S129", "elements": [ "W" ], "cost": 10000 } ] }, "solution_variant": [ "S24", "S42", "S89", "S93" ], "context_index": 14, "input_format": "markdown_table", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "Many people don’t realize that choosing which bundles to publish is a bit like solving a puzzle: each bundle covers some shows and has a licensing fee. The task is to pick a set of bundles so every show is in exactly one published bundle — no show missing, no show appearing twice. The best pick is the one with the smallest total licensing bill, which you find by adding the fees of the bundles you chose. The detailed list of bundles and costs appears below.\n\n{\n \"num_shows_total\": 21,\n \"num_bundles_available\": 113,\n \"sets\": [\n {\n \"bundle_id\": \"S1\",\n \"shows_in_bundle\": [\n \"I\",\n \"L\",\n \"N\"\n ],\n \"licensing_fee\": 39\n },\n {\n \"bundle_id\": \"S2\",\n \"shows_in_bundle\": [\n \"D\",\n \"E\",\n \"G\",\n \"I\"\n ],\n \"licensing_fee\": 316\n },\n {\n \"bundle_id\": \"S3\",\n \"shows_in_bundle\": [\n \"N\",\n \"O\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\"\n ],\n \"licensing_fee\": 264\n },\n {\n \"bundle_id\": \"S4\",\n \"shows_in_bundle\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\"\n ],\n \"licensing_fee\": 378\n },\n {\n \"bundle_id\": \"S5\",\n \"shows_in_bundle\": [\n \"I\",\n \"K\",\n \"L\",\n \"O\"\n ],\n \"licensing_fee\": 288\n },\n {\n \"bundle_id\": \"S6\",\n \"shows_in_bundle\": [\n \"C\",\n \"G\",\n \"K\"\n ],\n \"licensing_fee\": 165\n },\n {\n \"bundle_id\": \"S7\",\n \"shows_in_bundle\": [\n \"R\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"licensing_fee\": 208\n },\n {\n \"bundle_id\": \"S8\",\n \"shows_in_bundle\": [\n \"C\",\n \"D\"\n ],\n \"licensing_fee\": 174\n },\n {\n \"bundle_id\": \"S9\",\n \"shows_in_bundle\": [\n \"J\",\n \"L\",\n \"M\"\n ],\n \"licensing_fee\": 57\n },\n {\n \"bundle_id\": \"S10\",\n \"shows_in_bundle\": [\n \"F\",\n \"I\",\n \"K\"\n ],\n \"licensing_fee\": 234\n },\n {\n \"bundle_id\": \"S11\",\n \"shows_in_bundle\": [\n \"M\",\n \"O\",\n \"Q\"\n ],\n \"licensing_fee\": 54\n },\n {\n \"bundle_id\": \"S12\",\n \"shows_in_bundle\": [\n \"K\",\n \"N\",\n \"P\"\n ],\n \"licensing_fee\": 27\n },\n {\n \"bundle_id\": \"S13\",\n \"shows_in_bundle\": [\n \"C\",\n \"E\",\n \"G\",\n \"I\"\n ],\n \"licensing_fee\": 8\n },\n {\n \"bundle_id\": \"S14\",\n \"shows_in_bundle\": [\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"L\"\n ],\n \"licensing_fee\": 198\n },\n {\n \"bundle_id\": \"S15\",\n \"shows_in_bundle\": [\n \"Q\",\n \"R\",\n \"T\",\n \"U\"\n ],\n \"licensing_fee\": 144\n },\n {\n \"bundle_id\": \"S16\",\n \"shows_in_bundle\": [\n \"P\",\n \"S\",\n \"T\"\n ],\n \"licensing_fee\": 297\n },\n {\n \"bundle_id\": \"S17\",\n \"shows_in_bundle\": [\n \"Q\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"licensing_fee\": 316\n },\n {\n \"bundle_id\": \"S18\",\n \"shows_in_bundle\": [\n \"J\",\n \"N\",\n \"O\",\n \"P\",\n \"T\"\n ],\n \"licensing_fee\": 395\n },\n {\n \"bundle_id\": \"S19\",\n \"shows_in_bundle\": [\n \"A\",\n \"C\",\n \"F\"\n ],\n \"licensing_fee\": 21\n },\n {\n \"bundle_id\": \"S20\",\n \"shows_in_bundle\": [\n \"M\",\n \"N\",\n \"O\"\n ],\n \"licensing_fee\": 294\n },\n {\n \"bundle_id\": \"S21\",\n \"shows_in_bundle\": [\n \"Q\",\n \"S\",\n \"T\"\n ],\n \"licensing_fee\": 138\n },\n {\n \"bundle_id\": \"S22\",\n \"shows_in_bundle\": [\n \"N\",\n \"P\",\n \"R\"\n ],\n \"licensing_fee\": 84\n },\n {\n \"bundle_id\": \"S23\",\n \"shows_in_bundle\": [\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"licensing_fee\": 528\n },\n {\n \"bundle_id\": \"S24\",\n \"shows_in_bundle\": [\n \"L\",\n \"O\",\n \"Q\"\n ],\n \"licensing_fee\": 180\n },\n {\n \"bundle_id\": \"S25\",\n \"shows_in_bundle\": [\n \"J\",\n \"L\",\n \"M\",\n \"P\",\n \"R\"\n ],\n \"licensing_fee\": 245\n },\n {\n \"bundle_id\": \"S26\",\n \"shows_in_bundle\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\"\n ],\n \"licensing_fee\": 172\n },\n {\n \"bundle_id\": \"S27\",\n \"shows_in_bundle\": [\n \"L\",\n \"O\"\n ],\n \"licensing_fee\": 4\n },\n {\n \"bundle_id\": \"S28\",\n \"shows_in_bundle\": [\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"H\"\n ],\n \"licensing_fee\": 180\n },\n {\n \"bundle_id\": \"S29\",\n \"shows_in_bundle\": [\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"licensing_fee\": 315\n },\n {\n \"bundle_id\": \"S30\",\n \"shows_in_bundle\": [\n \"P\",\n \"T\",\n \"U\"\n ],\n \"licensing_fee\": 12\n },\n {\n \"bundle_id\": \"S31\",\n \"shows_in_bundle\": [\n \"B\",\n \"C\",\n \"F\",\n \"G\",\n \"H\"\n ],\n \"licensing_fee\": 80\n },\n {\n \"bundle_id\": \"S32\",\n \"shows_in_bundle\": [\n \"S\",\n \"U\"\n ],\n \"licensing_fee\": 140\n },\n {\n \"bundle_id\": \"S33\",\n \"shows_in_bundle\": [\n \"F\",\n \"H\",\n \"I\"\n ],\n \"licensing_fee\": 261\n },\n {\n \"bundle_id\": \"S34\",\n \"shows_in_bundle\": [\n \"F\",\n \"I\",\n \"J\",\n \"L\",\n \"N\",\n \"P\"\n ],\n \"licensing_fee\": 516\n },\n {\n \"bundle_id\": \"S35\",\n \"shows_in_bundle\": [\n \"A\",\n \"C\",\n \"D\",\n \"E\"\n ],\n \"licensing_fee\": 44\n },\n {\n \"bundle_id\": \"S36\",\n \"shows_in_bundle\": [\n \"A\",\n \"B\",\n \"D\"\n ],\n \"licensing_fee\": 69\n },\n {\n \"bundle_id\": \"S37\",\n \"shows_in_bundle\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"F\"\n ],\n \"licensing_fee\": 60\n },\n {\n \"bundle_id\": \"S38\",\n \"shows_in_bundle\": [\n \"L\",\n \"M\",\n \"N\",\n \"R\"\n ],\n \"licensing_fee\": 164\n },\n {\n \"bundle_id\": \"S39\",\n \"shows_in_bundle\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\"\n ],\n \"licensing_fee\": 465\n },\n {\n \"bundle_id\": \"S40\",\n \"shows_in_bundle\": [\n \"F\",\n \"H\",\n \"J\",\n \"K\",\n \"L\",\n \"M\"\n ],\n \"licensing_fee\": 42\n },\n {\n \"bundle_id\": \"S41\",\n \"shows_in_bundle\": [\n \"H\",\n \"I\",\n \"J\",\n \"K\"\n ],\n \"licensing_fee\": 100\n },\n {\n \"bundle_id\": \"S42\",\n \"shows_in_bundle\": [\n \"E\",\n \"F\",\n \"I\"\n ],\n \"licensing_fee\": 102\n },\n {\n \"bundle_id\": \"S43\",\n \"shows_in_bundle\": [\n \"K\",\n \"L\"\n ],\n \"licensing_fee\": 166\n },\n {\n \"bundle_id\": \"S44\",\n \"shows_in_bundle\": [\n \"Q\",\n \"R\",\n \"T\"\n ],\n \"licensing_fee\": 264\n },\n {\n \"bundle_id\": \"S45\",\n \"shows_in_bundle\": [\n \"E\",\n \"F\",\n \"H\",\n \"I\",\n \"K\"\n ],\n \"licensing_fee\": 330\n },\n {\n \"bundle_id\": \"S46\",\n \"shows_in_bundle\": [\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"L\"\n ],\n \"licensing_fee\": 305\n },\n {\n \"bundle_id\": \"S47\",\n \"shows_in_bundle\": [\n \"R\",\n \"S\",\n \"T\"\n ],\n \"licensing_fee\": 15\n },\n {\n \"bundle_id\": \"S48\",\n \"shows_in_bundle\": [\n \"A\",\n \"E\",\n \"G\"\n ],\n \"licensing_fee\": 162\n },\n {\n \"bundle_id\": \"S49\",\n \"shows_in_bundle\": [\n \"H\",\n \"J\",\n \"L\",\n \"N\"\n ],\n \"licensing_fee\": 240\n },\n {\n \"bundle_id\": \"S50\",\n \"shows_in_bundle\": [\n \"A\",\n \"B\",\n \"C\",\n \"E\"\n ],\n \"licensing_fee\": 376\n },\n {\n \"bundle_id\": \"S51\",\n \"shows_in_bundle\": [\n \"F\",\n \"G\",\n \"H\",\n \"J\"\n ],\n \"licensing_fee\": 156\n },\n {\n \"bundle_id\": \"S52\",\n \"shows_in_bundle\": [\n \"I\",\n \"K\"\n ],\n \"licensing_fee\": 130\n },\n {\n \"bundle_id\": \"S53\",\n \"shows_in_bundle\": [\n \"M\",\n \"P\",\n \"R\"\n ],\n \"licensing_fee\": 12\n },\n {\n \"bundle_id\": \"S54\",\n \"shows_in_bundle\": [\n \"H\",\n \"I\",\n \"J\"\n ],\n \"licensing_fee\": 204\n },\n {\n \"bundle_id\": \"S55\",\n \"shows_in_bundle\": [\n \"P\",\n \"R\",\n \"T\"\n ],\n \"licensing_fee\": 21\n },\n {\n \"bundle_id\": \"S56\",\n \"shows_in_bundle\": [\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"M\"\n ],\n \"licensing_fee\": 156\n },\n {\n \"bundle_id\": \"S57\",\n \"shows_in_bundle\": [\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"L\",\n \"M\"\n ],\n \"licensing_fee\": 432\n },\n {\n \"bundle_id\": \"S58\",\n \"shows_in_bundle\": [\n \"G\",\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"M\"\n ],\n \"licensing_fee\": 366\n },\n {\n \"bundle_id\": \"S59\",\n \"shows_in_bundle\": [\n \"L\",\n \"M\",\n \"O\",\n \"P\",\n \"Q\"\n ],\n \"licensing_fee\": 90\n },\n {\n \"bundle_id\": \"S60\",\n \"shows_in_bundle\": [\n \"J\",\n \"K\",\n \"L\",\n \"N\"\n ],\n \"licensing_fee\": 204\n },\n {\n \"bundle_id\": \"S61\",\n \"shows_in_bundle\": [\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\"\n ],\n \"licensing_fee\": 50\n },\n {\n \"bundle_id\": \"S62\",\n \"shows_in_bundle\": [\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\"\n ],\n \"licensing_fee\": 120\n },\n {\n \"bundle_id\": \"S63\",\n \"shows_in_bundle\": [\n \"B\",\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"H\"\n ],\n \"licensing_fee\": 78\n },\n {\n \"bundle_id\": \"S64\",\n \"shows_in_bundle\": [\n \"H\",\n \"K\",\n \"L\",\n \"M\",\n \"O\"\n ],\n \"licensing_fee\": 295\n },\n {\n \"bundle_id\": \"S65\",\n \"shows_in_bundle\": [\n \"G\",\n \"J\",\n \"K\",\n \"L\",\n \"N\"\n ],\n \"licensing_fee\": 325\n },\n {\n \"bundle_id\": \"S66\",\n \"shows_in_bundle\": [\n \"M\",\n \"Q\",\n \"R\",\n \"T\"\n ],\n \"licensing_fee\": 92\n },\n {\n \"bundle_id\": \"S67\",\n \"shows_in_bundle\": [\n \"M\",\n \"O\",\n \"R\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"licensing_fee\": 204\n },\n {\n \"bundle_id\": \"S68\",\n \"shows_in_bundle\": [\n \"A\",\n \"D\",\n \"F\"\n ],\n \"licensing_fee\": 231\n },\n {\n \"bundle_id\": \"S69\",\n \"shows_in_bundle\": [\n \"D\",\n \"G\",\n \"I\",\n \"J\",\n \"K\"\n ],\n \"licensing_fee\": 190\n },\n {\n \"bundle_id\": \"S70\",\n \"shows_in_bundle\": [\n \"O\",\n \"P\",\n \"Q\",\n \"S\",\n \"T\"\n ],\n \"licensing_fee\": 410\n },\n {\n \"bundle_id\": \"S71\",\n \"shows_in_bundle\": [\n \"H\",\n \"I\",\n \"J\",\n \"M\"\n ],\n \"licensing_fee\": 376\n },\n {\n \"bundle_id\": \"S72\",\n \"shows_in_bundle\": [\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"O\"\n ],\n \"licensing_fee\": 144\n },\n {\n \"bundle_id\": \"S73\",\n \"shows_in_bundle\": [\n \"F\",\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"M\"\n ],\n \"licensing_fee\": 444\n },\n {\n \"bundle_id\": \"S74\",\n \"shows_in_bundle\": [\n \"J\",\n \"K\",\n \"L\"\n ],\n \"licensing_fee\": 267\n },\n {\n \"bundle_id\": \"S75\",\n \"shows_in_bundle\": [\n \"J\",\n \"K\",\n \"L\",\n \"N\",\n \"P\"\n ],\n \"licensing_fee\": 120\n },\n {\n \"bundle_id\": \"S76\",\n \"shows_in_bundle\": [\n \"B\",\n \"C\",\n \"E\"\n ],\n \"licensing_fee\": 105\n },\n {\n \"bundle_id\": \"S77\",\n \"shows_in_bundle\": [\n \"C\",\n \"G\"\n ],\n \"licensing_fee\": 34\n },\n {\n \"bundle_id\": \"S78\",\n \"shows_in_bundle\": [\n \"N\",\n \"P\",\n \"S\"\n ],\n \"licensing_fee\": 87\n },\n {\n \"bundle_id\": \"S79\",\n \"shows_in_bundle\": [\n \"I\",\n \"L\",\n \"M\"\n ],\n \"licensing_fee\": 234\n },\n {\n \"bundle_id\": \"S80\",\n \"shows_in_bundle\": [\n \"O\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"licensing_fee\": 246\n },\n {\n \"bundle_id\": \"S81\",\n \"shows_in_bundle\": [\n \"M\",\n \"O\",\n \"P\",\n \"Q\"\n ],\n \"licensing_fee\": 88\n },\n {\n \"bundle_id\": \"S82\",\n \"shows_in_bundle\": [\n \"E\",\n \"H\",\n \"I\",\n \"J\",\n \"K\"\n ],\n \"licensing_fee\": 225\n },\n {\n \"bundle_id\": \"S83\",\n \"shows_in_bundle\": [\n \"P\",\n \"Q\"\n ],\n \"licensing_fee\": 82\n },\n {\n \"bundle_id\": \"S84\",\n \"shows_in_bundle\": [\n \"K\",\n \"M\",\n \"N\",\n \"P\"\n ],\n \"licensing_fee\": 224\n },\n {\n \"bundle_id\": \"S85\",\n \"shows_in_bundle\": [\n \"A\",\n \"C\",\n \"D\",\n \"G\"\n ],\n \"licensing_fee\": 196\n },\n {\n \"bundle_id\": \"S86\",\n \"shows_in_bundle\": [\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"I\"\n ],\n \"licensing_fee\": 228\n },\n {\n \"bundle_id\": \"S87\",\n \"shows_in_bundle\": [\n \"H\",\n \"J\",\n \"K\"\n ],\n \"licensing_fee\": 213\n },\n {\n \"bundle_id\": \"S88\",\n \"shows_in_bundle\": [\n \"E\",\n \"F\",\n \"G\",\n \"H\"\n ],\n \"licensing_fee\": 56\n },\n {\n \"bundle_id\": \"S89\",\n \"shows_in_bundle\": [\n \"M\",\n \"N\",\n \"Q\",\n \"R\"\n ],\n \"licensing_fee\": 208\n },\n {\n \"bundle_id\": \"S90\",\n \"shows_in_bundle\": [\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"H\"\n ],\n \"licensing_fee\": 400\n },\n {\n \"bundle_id\": \"S91\",\n \"shows_in_bundle\": [\n \"S\",\n \"T\",\n \"U\"\n ],\n \"licensing_fee\": 282\n },\n {\n \"bundle_id\": \"S92\",\n \"shows_in_bundle\": [\n \"H\",\n \"M\"\n ],\n \"licensing_fee\": 168\n },\n {\n \"bundle_id\": \"S93\",\n \"shows_in_bundle\": [\n \"A\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S94\",\n \"shows_in_bundle\": [\n \"B\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S95\",\n \"shows_in_bundle\": [\n \"C\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S96\",\n \"shows_in_bundle\": [\n \"D\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S97\",\n \"shows_in_bundle\": [\n \"E\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S98\",\n \"shows_in_bundle\": [\n \"F\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S99\",\n \"shows_in_bundle\": [\n \"G\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S100\",\n \"shows_in_bundle\": [\n \"H\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S101\",\n \"shows_in_bundle\": [\n \"I\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S102\",\n \"shows_in_bundle\": [\n \"J\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S103\",\n \"shows_in_bundle\": [\n \"K\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S104\",\n \"shows_in_bundle\": [\n \"L\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S105\",\n \"shows_in_bundle\": [\n \"M\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S106\",\n \"shows_in_bundle\": [\n \"N\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S107\",\n \"shows_in_bundle\": [\n \"O\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S108\",\n \"shows_in_bundle\": [\n \"P\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S109\",\n \"shows_in_bundle\": [\n \"Q\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S110\",\n \"shows_in_bundle\": [\n \"R\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S111\",\n \"shows_in_bundle\": [\n \"S\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S112\",\n \"shows_in_bundle\": [\n \"T\"\n ],\n \"licensing_fee\": 10000\n },\n {\n \"bundle_id\": \"S113\",\n \"shows_in_bundle\": [\n \"U\"\n ],\n \"licensing_fee\": 10000\n }\n ]\n}\n\nIf you could return your final choice in a simple JSON form, that makes it super easy for me to read. Something like this:\n\n{\n \"solution\": [\"bundle_id\", ...]\n}\n\nThis just means \"solution\" is a list of the bundle IDs you want to publish — each string in the array is the ID of one chosen bundle. It's a sketch of the shape I expect, not your actual answer; replace the placeholder entries with the real bundle identifiers from the instance.\n\nPlease be careful to use the identifiers exactly as they appear in the input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 21, "num_sets": 113, "density": 0.16561314791403287, "sets": [ { "id": 1, "elements": [ 9, 12, 14 ], "cost": 39 }, { "id": 2, "elements": [ 4, 5, 7, 9 ], "cost": 316 }, { "id": 3, "elements": [ 14, 15, 16, 17, 18, 19 ], "cost": 264 }, { "id": 4, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 378 }, { "id": 5, "elements": [ 9, 11, 12, 15 ], "cost": 288 }, { "id": 6, "elements": [ 3, 7, 11 ], "cost": 165 }, { "id": 7, "elements": [ 18, 19, 20, 21 ], "cost": 208 }, { "id": 8, "elements": [ 3, 4 ], "cost": 174 }, { "id": 9, "elements": [ 10, 12, 13 ], "cost": 57 }, { "id": 10, "elements": [ 6, 9, 11 ], "cost": 234 }, { "id": 11, "elements": [ 13, 15, 17 ], "cost": 54 }, { "id": 12, "elements": [ 11, 14, 16 ], "cost": 27 }, { "id": 13, "elements": [ 3, 5, 7, 9 ], "cost": 8 }, { "id": 14, "elements": [ 7, 8, 9, 10, 11, 12 ], "cost": 198 }, { "id": 15, "elements": [ 17, 18, 20, 21 ], "cost": 144 }, { "id": 16, "elements": [ 16, 19, 20 ], "cost": 297 }, { "id": 17, "elements": [ 17, 19, 20, 21 ], "cost": 316 }, { "id": 18, "elements": [ 10, 14, 15, 16, 20 ], "cost": 395 }, { "id": 19, "elements": [ 1, 3, 6 ], "cost": 21 }, { "id": 20, "elements": [ 13, 14, 15 ], "cost": 294 }, { "id": 21, "elements": [ 17, 19, 20 ], "cost": 138 }, { "id": 22, "elements": [ 14, 16, 18 ], "cost": 84 }, { "id": 23, "elements": [ 16, 17, 18, 19, 20, 21 ], "cost": 528 }, { "id": 24, "elements": [ 12, 15, 17 ], "cost": 180 }, { "id": 25, "elements": [ 10, 12, 13, 16, 18 ], "cost": 245 }, { "id": 26, "elements": [ 1, 2, 3, 4 ], "cost": 172 }, { "id": 27, "elements": [ 12, 15 ], "cost": 4 }, { "id": 28, "elements": [ 2, 3, 4, 5, 8 ], "cost": 180 }, { "id": 29, "elements": [ 17, 18, 19, 20, 21 ], "cost": 315 }, { "id": 30, "elements": [ 16, 20, 21 ], "cost": 12 }, { "id": 31, "elements": [ 2, 3, 6, 7, 8 ], "cost": 80 }, { "id": 32, "elements": [ 19, 21 ], "cost": 140 }, { "id": 33, "elements": [ 6, 8, 9 ], "cost": 261 }, { "id": 34, "elements": [ 6, 9, 10, 12, 14, 16 ], "cost": 516 }, { "id": 35, "elements": [ 1, 3, 4, 5 ], "cost": 44 }, { "id": 36, "elements": [ 1, 2, 4 ], "cost": 69 }, { "id": 37, "elements": [ 1, 2, 3, 4, 6 ], "cost": 60 }, { "id": 38, "elements": [ 12, 13, 14, 18 ], "cost": 164 }, { "id": 39, "elements": [ 1, 2, 3, 4, 5 ], "cost": 465 }, { "id": 40, "elements": [ 6, 8, 10, 11, 12, 13 ], "cost": 42 }, { "id": 41, "elements": [ 8, 9, 10, 11 ], "cost": 100 }, { "id": 42, "elements": [ 5, 6, 9 ], "cost": 102 }, { "id": 43, "elements": [ 11, 12 ], "cost": 166 }, { "id": 44, "elements": [ 17, 18, 20 ], "cost": 264 }, { "id": 45, "elements": [ 5, 6, 8, 9, 11 ], "cost": 330 }, { "id": 46, "elements": [ 7, 8, 9, 10, 12 ], "cost": 305 }, { "id": 47, "elements": [ 18, 19, 20 ], "cost": 15 }, { "id": 48, "elements": [ 1, 5, 7 ], "cost": 162 }, { "id": 49, "elements": [ 8, 10, 12, 14 ], "cost": 240 }, { "id": 50, "elements": [ 1, 2, 3, 5 ], "cost": 376 }, { "id": 51, "elements": [ 6, 7, 8, 10 ], "cost": 156 }, { "id": 52, "elements": [ 9, 11 ], "cost": 130 }, { "id": 53, "elements": [ 13, 16, 18 ], "cost": 12 }, { "id": 54, "elements": [ 8, 9, 10 ], "cost": 204 }, { "id": 55, "elements": [ 16, 18, 20 ], "cost": 21 }, { "id": 56, "elements": [ 8, 9, 10, 11, 12, 13 ], "cost": 156 }, { "id": 57, "elements": [ 7, 8, 9, 10, 12, 13 ], "cost": 432 }, { "id": 58, "elements": [ 7, 9, 10, 11, 12, 13 ], "cost": 366 }, { "id": 59, "elements": [ 12, 13, 15, 16, 17 ], "cost": 90 }, { "id": 60, "elements": [ 10, 11, 12, 14 ], "cost": 204 }, { "id": 61, "elements": [ 7, 8, 9, 10, 11 ], "cost": 50 }, { "id": 62, "elements": [ 2, 3, 4, 5, 6 ], "cost": 120 }, { "id": 63, "elements": [ 2, 4, 5, 6, 7, 8 ], "cost": 78 }, { "id": 64, "elements": [ 8, 11, 12, 13, 15 ], "cost": 295 }, { "id": 65, "elements": [ 7, 10, 11, 12, 14 ], "cost": 325 }, { "id": 66, "elements": [ 13, 17, 18, 20 ], "cost": 92 }, { "id": 67, "elements": [ 13, 15, 18, 19, 20, 21 ], "cost": 204 }, { "id": 68, "elements": [ 1, 4, 6 ], "cost": 231 }, { "id": 69, "elements": [ 4, 7, 9, 10, 11 ], "cost": 190 }, { "id": 70, "elements": [ 15, 16, 17, 19, 20 ], "cost": 410 }, { "id": 71, "elements": [ 8, 9, 10, 13 ], "cost": 376 }, { "id": 72, "elements": [ 10, 11, 12, 13, 14, 15 ], "cost": 144 }, { "id": 73, "elements": [ 6, 9, 10, 11, 12, 13 ], "cost": 444 }, { "id": 74, "elements": [ 10, 11, 12 ], "cost": 267 }, { "id": 75, "elements": [ 10, 11, 12, 14, 16 ], "cost": 120 }, { "id": 76, "elements": [ 2, 3, 5 ], "cost": 105 }, { "id": 77, "elements": [ 3, 7 ], "cost": 34 }, { "id": 78, "elements": [ 14, 16, 19 ], "cost": 87 }, { "id": 79, "elements": [ 9, 12, 13 ], "cost": 234 }, { "id": 80, "elements": [ 15, 17, 18, 19, 20, 21 ], "cost": 246 }, { "id": 81, "elements": [ 13, 15, 16, 17 ], "cost": 88 }, { "id": 82, "elements": [ 5, 8, 9, 10, 11 ], "cost": 225 }, { "id": 83, "elements": [ 16, 17 ], "cost": 82 }, { "id": 84, "elements": [ 11, 13, 14, 16 ], "cost": 224 }, { "id": 85, "elements": [ 1, 3, 4, 7 ], "cost": 196 }, { "id": 86, "elements": [ 2, 3, 4, 5, 6, 9 ], "cost": 228 }, { "id": 87, "elements": [ 8, 10, 11 ], "cost": 213 }, { "id": 88, "elements": [ 5, 6, 7, 8 ], "cost": 56 }, { "id": 89, "elements": [ 13, 14, 17, 18 ], "cost": 208 }, { "id": 90, "elements": [ 3, 4, 5, 6, 8 ], "cost": 400 }, { "id": 91, "elements": [ 19, 20, 21 ], "cost": 282 }, { "id": 92, "elements": [ 8, 13 ], "cost": 168 }, { "id": 93, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 94, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 95, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 96, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 97, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 98, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 21 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0014_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0014_bag_stats.png" }, "solution": [ 36, 42, 75, 77, 80, 92 ], "obj": 739.0, "instance_variant": { "num_elements": 21, "num_sets": 113, "sets": [ { "id": "S1", "elements": [ "I", "L", "N" ], "cost": 39 }, { "id": "S2", "elements": [ "D", "E", "G", "I" ], "cost": 316 }, { "id": "S3", "elements": [ "N", "O", "P", "Q", "R", "S" ], "cost": 264 }, { "id": "S4", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 378 }, { "id": "S5", "elements": [ "I", "K", "L", "O" ], "cost": 288 }, { "id": "S6", "elements": [ "C", "G", "K" ], "cost": 165 }, { "id": "S7", "elements": [ "R", "S", "T", "U" ], "cost": 208 }, { "id": "S8", "elements": [ "C", "D" ], "cost": 174 }, { "id": "S9", "elements": [ "J", "L", "M" ], "cost": 57 }, { "id": "S10", "elements": [ "F", "I", "K" ], "cost": 234 }, { "id": "S11", "elements": [ "M", "O", "Q" ], "cost": 54 }, { "id": "S12", "elements": [ "K", "N", "P" ], "cost": 27 }, { "id": "S13", "elements": [ "C", "E", "G", "I" ], "cost": 8 }, { "id": "S14", "elements": [ "G", "H", "I", "J", "K", "L" ], "cost": 198 }, { "id": "S15", "elements": [ "Q", "R", "T", "U" ], "cost": 144 }, { "id": "S16", "elements": [ "P", "S", "T" ], "cost": 297 }, { "id": "S17", "elements": [ "Q", "S", "T", "U" ], "cost": 316 }, { "id": "S18", "elements": [ "J", "N", "O", "P", "T" ], "cost": 395 }, { "id": "S19", "elements": [ "A", "C", "F" ], "cost": 21 }, { "id": "S20", "elements": [ "M", "N", "O" ], "cost": 294 }, { "id": "S21", "elements": [ "Q", "S", "T" ], "cost": 138 }, { "id": "S22", "elements": [ "N", "P", "R" ], "cost": 84 }, { "id": "S23", "elements": [ "P", "Q", "R", "S", "T", "U" ], "cost": 528 }, { "id": "S24", "elements": [ "L", "O", "Q" ], "cost": 180 }, { "id": "S25", "elements": [ "J", "L", "M", "P", "R" ], "cost": 245 }, { "id": "S26", "elements": [ "A", "B", "C", "D" ], "cost": 172 }, { "id": "S27", "elements": [ "L", "O" ], "cost": 4 }, { "id": "S28", "elements": [ "B", "C", "D", "E", "H" ], "cost": 180 }, { "id": "S29", "elements": [ "Q", "R", "S", "T", "U" ], "cost": 315 }, { "id": "S30", "elements": [ "P", "T", "U" ], "cost": 12 }, { "id": "S31", "elements": [ "B", "C", "F", "G", "H" ], "cost": 80 }, { "id": "S32", "elements": [ "S", "U" ], "cost": 140 }, { "id": "S33", "elements": [ "F", "H", "I" ], "cost": 261 }, { "id": "S34", "elements": [ "F", "I", "J", "L", "N", "P" ], "cost": 516 }, { "id": "S35", "elements": [ "A", "C", "D", "E" ], "cost": 44 }, { "id": "S36", "elements": [ "A", "B", "D" ], "cost": 69 }, { "id": "S37", "elements": [ "A", "B", "C", "D", "F" ], "cost": 60 }, { "id": "S38", "elements": [ "L", "M", "N", "R" ], "cost": 164 }, { "id": "S39", "elements": [ "A", "B", "C", "D", "E" ], "cost": 465 }, { "id": "S40", "elements": [ "F", "H", "J", "K", "L", "M" ], "cost": 42 }, { "id": "S41", "elements": [ "H", "I", "J", "K" ], "cost": 100 }, { "id": "S42", "elements": [ "E", "F", "I" ], "cost": 102 }, { "id": "S43", "elements": [ "K", "L" ], "cost": 166 }, { "id": "S44", "elements": [ "Q", "R", "T" ], "cost": 264 }, { "id": "S45", "elements": [ "E", "F", "H", "I", "K" ], "cost": 330 }, { "id": "S46", "elements": [ "G", "H", "I", "J", "L" ], "cost": 305 }, { "id": "S47", "elements": [ "R", "S", "T" ], "cost": 15 }, { "id": "S48", "elements": [ "A", "E", "G" ], "cost": 162 }, { "id": "S49", "elements": [ "H", "J", "L", "N" ], "cost": 240 }, { "id": "S50", "elements": [ "A", "B", "C", "E" ], "cost": 376 }, { "id": "S51", "elements": [ "F", "G", "H", "J" ], "cost": 156 }, { "id": "S52", "elements": [ "I", "K" ], "cost": 130 }, { "id": "S53", "elements": [ "M", "P", "R" ], "cost": 12 }, { "id": "S54", "elements": [ "H", "I", "J" ], "cost": 204 }, { "id": "S55", "elements": [ "P", "R", "T" ], "cost": 21 }, { "id": "S56", "elements": [ "H", "I", "J", "K", "L", "M" ], "cost": 156 }, { "id": "S57", "elements": [ "G", "H", "I", "J", "L", "M" ], "cost": 432 }, { "id": "S58", "elements": [ "G", "I", "J", "K", "L", "M" ], "cost": 366 }, { "id": "S59", "elements": [ "L", "M", "O", "P", "Q" ], "cost": 90 }, { "id": "S60", "elements": [ "J", "K", "L", "N" ], "cost": 204 }, { "id": "S61", "elements": [ "G", "H", "I", "J", "K" ], "cost": 50 }, { "id": "S62", "elements": [ "B", "C", "D", "E", "F" ], "cost": 120 }, { "id": "S63", "elements": [ "B", "D", "E", "F", "G", "H" ], "cost": 78 }, { "id": "S64", "elements": [ "H", "K", "L", "M", "O" ], "cost": 295 }, { "id": "S65", "elements": [ "G", "J", "K", "L", "N" ], "cost": 325 }, { "id": "S66", "elements": [ "M", "Q", "R", "T" ], "cost": 92 }, { "id": "S67", "elements": [ "M", "O", "R", "S", "T", "U" ], "cost": 204 }, { "id": "S68", "elements": [ "A", "D", "F" ], "cost": 231 }, { "id": "S69", "elements": [ "D", "G", "I", "J", "K" ], "cost": 190 }, { "id": "S70", "elements": [ "O", "P", "Q", "S", "T" ], "cost": 410 }, { "id": "S71", "elements": [ "H", "I", "J", "M" ], "cost": 376 }, { "id": "S72", "elements": [ "J", "K", "L", "M", "N", "O" ], "cost": 144 }, { "id": "S73", "elements": [ "F", "I", "J", "K", "L", "M" ], "cost": 444 }, { "id": "S74", "elements": [ "J", "K", "L" ], "cost": 267 }, { "id": "S75", "elements": [ "J", "K", "L", "N", "P" ], "cost": 120 }, { "id": "S76", "elements": [ "B", "C", "E" ], "cost": 105 }, { "id": "S77", "elements": [ "C", "G" ], "cost": 34 }, { "id": "S78", "elements": [ "N", "P", "S" ], "cost": 87 }, { "id": "S79", "elements": [ "I", "L", "M" ], "cost": 234 }, { "id": "S80", "elements": [ "O", "Q", "R", "S", "T", "U" ], "cost": 246 }, { "id": "S81", "elements": [ "M", "O", "P", "Q" ], "cost": 88 }, { "id": "S82", "elements": [ "E", "H", "I", "J", "K" ], "cost": 225 }, { "id": "S83", "elements": [ "P", "Q" ], "cost": 82 }, { "id": "S84", "elements": [ "K", "M", "N", "P" ], "cost": 224 }, { "id": "S85", "elements": [ "A", "C", "D", "G" ], "cost": 196 }, { "id": "S86", "elements": [ "B", "C", "D", "E", "F", "I" ], "cost": 228 }, { "id": "S87", "elements": [ "H", "J", "K" ], "cost": 213 }, { "id": "S88", "elements": [ "E", "F", "G", "H" ], "cost": 56 }, { "id": "S89", "elements": [ "M", "N", "Q", "R" ], "cost": 208 }, { "id": "S90", "elements": [ "C", "D", "E", "F", "H" ], "cost": 400 }, { "id": "S91", "elements": [ "S", "T", "U" ], "cost": 282 }, { "id": "S92", "elements": [ "H", "M" ], "cost": 168 }, { "id": "S93", "elements": [ "A" ], "cost": 10000 }, { "id": "S94", "elements": [ "B" ], "cost": 10000 }, { "id": "S95", "elements": [ "C" ], "cost": 10000 }, { "id": "S96", "elements": [ "D" ], "cost": 10000 }, { "id": "S97", "elements": [ "E" ], "cost": 10000 }, { "id": "S98", "elements": [ "F" ], "cost": 10000 }, { "id": "S99", "elements": [ "G" ], "cost": 10000 }, { "id": "S100", "elements": [ "H" ], "cost": 10000 }, { "id": "S101", "elements": [ "I" ], "cost": 10000 }, { "id": "S102", "elements": [ "J" ], "cost": 10000 }, { "id": "S103", "elements": [ "K" ], "cost": 10000 }, { "id": "S104", "elements": [ "L" ], "cost": 10000 }, { "id": "S105", "elements": [ "M" ], "cost": 10000 }, { "id": "S106", "elements": [ "N" ], "cost": 10000 }, { "id": "S107", "elements": [ "O" ], "cost": 10000 }, { "id": "S108", "elements": [ "P" ], "cost": 10000 }, { "id": "S109", "elements": [ "Q" ], "cost": 10000 }, { "id": "S110", "elements": [ "R" ], "cost": 10000 }, { "id": "S111", "elements": [ "S" ], "cost": 10000 }, { "id": "S112", "elements": [ "T" ], "cost": 10000 }, { "id": "S113", "elements": [ "U" ], "cost": 10000 } ] }, "solution_variant": [ "S36", "S42", "S75", "S77", "S80", "S92" ], "context_index": 15, "input_format": "json", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "At the prep center the team has to choose which box combinations will go on the menu; the constraint is that each ingredient must be contained in one and only one chosen box (nothing missing, nothing doubled), and the best lineup is the one with the lowest total procurement charge, computed by adding the costs of the picked boxes. The detailed options and their prices follow below.\n\nThe team must cover all 22 ingredients by selecting from the 125 candidate boxes.\nBox S1 contains 3 4 5 6 8 9 and has a procurement charge of 102.\nBox S2 contains 1 2 3 and has a procurement charge of 234.\nBox S3 contains 5 8 10 12 and has a procurement charge of 312.\nBox S4 contains 3 4 5 6 7 8 and has a procurement charge of 402.\nBox S5 contains 10 11 12 13 14 15 16 and has a procurement charge of 483.\nBox S6 contains 13 14 15 17 18 and has a procurement charge of 495.\nBox S7 contains 13 14 and has a procurement charge of 168.\nBox S8 contains 2 4 5 6 and has a procurement charge of 60.\nBox S9 contains 9 11 12 14 15 and has a procurement charge of 65.\nBox S10 contains 3 4 5 6 and has a procurement charge of 176.\nBox S11 contains 13 14 15 and has a procurement charge of 108.\nBox S12 contains 3 4 5 6 7 and has a procurement charge of 190.\nBox S13 contains 5 6 7 8 and has a procurement charge of 80.\nBox S14 contains 1 3 4 5 7 and has a procurement charge of 140.\nBox S15 contains 3 5 6 7 11 and has a procurement charge of 260.\nBox S16 contains 5 6 7 10 and has a procurement charge of 268.\nBox S17 contains 15 16 17 19 and has a procurement charge of 196.\nBox S18 contains 2 3 and has a procurement charge of 58.\nBox S19 contains 19 20 21 and has a procurement charge of 138.\nBox S20 contains 18 19 20 21 and has a procurement charge of 272.\nBox S21 contains 3 5 6 7 8 11 and has a procurement charge of 414.\nBox S22 contains 14 16 17 18 20 22 and has a procurement charge of 570.\nBox S23 contains 19 21 22 and has a procurement charge of 69.\nBox S24 contains 18 19 and has a procurement charge of 192.\nBox S25 contains 1 3 6 and has a procurement charge of 81.\nBox S26 contains 16 18 20 and has a procurement charge of 21.\nBox S27 contains 17 19 20 and has a procurement charge of 225.\nBox S28 contains 13 14 15 17 20 and has a procurement charge of 215.\nBox S29 contains 15 16 17 18 and has a procurement charge of 176.\nBox S30 contains 15 18 19 and has a procurement charge of 291.\nBox S31 contains 11 13 15 16 18 19 and has a procurement charge of 522.\nBox S32 contains 12 14 16 18 and has a procurement charge of 56.\nBox S33 contains 6 7 8 9 10 11 and has a procurement charge of 162.\nBox S34 contains 2 3 4 5 6 and has a procurement charge of 425.\nBox S35 contains 18 19 21 and has a procurement charge of 300.\nBox S36 contains 5 6 8 10 11 and has a procurement charge of 215.\nBox S37 contains 9 10 11 12 14 and has a procurement charge of 260.\nBox S38 contains 13 16 and has a procurement charge of 118.\nBox S39 contains 1 2 3 4 5 6 and has a procurement charge of 330.\nBox S40 contains 8 9 10 11 12 13 and has a procurement charge of 60.\nBox S41 contains 11 12 15 and has a procurement charge of 18.\nBox S42 contains 10 11 12 13 14 16 and has a procurement charge of 78.\nBox S43 contains 10 11 and has a procurement charge of 74.\nBox S44 contains 2 8 10 and has a procurement charge of 282.\nBox S45 contains 4 5 6 8 and has a procurement charge of 228.\nBox S46 contains 18 19 20 and has a procurement charge of 114.\nBox S47 contains 2 3 5 6 7 and has a procurement charge of 235.\nBox S48 contains 12 13 14 15 16 17 and has a procurement charge of 546.\nBox S49 contains 15 17 21 and has a procurement charge of 159.\nBox S50 contains 12 17 18 21 and has a procurement charge of 192.\nBox S51 contains 18 20 and has a procurement charge of 84.\nBox S52 contains 12 13 14 15 16 18 and has a procurement charge of 90.\nBox S53 contains 14 17 19 20 22 and has a procurement charge of 485.\nBox S54 contains 2 4 5 6 7 8 and has a procurement charge of 336.\nBox S55 contains 7 8 11 12 13 15 and has a procurement charge of 204.\nBox S56 contains 16 17 18 19 20 21 and has a procurement charge of 276.\nBox S57 contains 14 16 17 18 20 21 and has a procurement charge of 36.\nBox S58 contains 15 16 17 19 20 21 and has a procurement charge of 126.\nBox S59 contains 4 7 10 12 and has a procurement charge of 228.\nBox S60 contains 9 15 16 and has a procurement charge of 81.\nBox S61 contains 13 15 16 17 and has a procurement charge of 64.\nBox S62 contains 5 6 7 and has a procurement charge of 63.\nBox S63 contains 2 3 4 5 7 10 and has a procurement charge of 72.\nBox S64 contains 7 10 11 12 13 16 and has a procurement charge of 78.\nBox S65 contains 8 9 12 13 and has a procurement charge of 360.\nBox S66 contains 16 21 22 and has a procurement charge of 108.\nBox S67 contains 6 7 9 11 13 and has a procurement charge of 90.\nBox S68 contains 1 2 3 4 5 and has a procurement charge of 270.\nBox S69 contains 11 12 15 16 17 18 and has a procurement charge of 90.\nBox S70 contains 7 9 10 11 12 13 and has a procurement charge of 588.\nBox S71 contains 14 15 16 17 18 and has a procurement charge of 45.\nBox S72 contains 15 16 19 and has a procurement charge of 171.\nBox S73 contains 13 19 21 and has a procurement charge of 99.\nBox S74 contains 20 21 22 and has a procurement charge of 39.\nBox S75 contains 16 17 18 22 and has a procurement charge of 116.\nBox S76 contains 16 17 18 19 20 22 and has a procurement charge of 126.\nBox S77 contains 1 2 3 5 and has a procurement charge of 296.\nBox S78 contains 5 7 8 9 and has a procurement charge of 336.\nBox S79 contains 9 10 11 12 13 14 and has a procurement charge of 510.\nBox S80 contains 9 12 13 and has a procurement charge of 165.\nBox S81 contains 15 16 17 18 20 and has a procurement charge of 190.\nBox S82 contains 11 12 17 and has a procurement charge of 33.\nBox S83 contains 14 15 17 18 19 20 and has a procurement charge of 600.\nBox S84 contains 12 14 and has a procurement charge of 146.\nBox S85 contains 13 14 17 18 19 22 and has a procurement charge of 126.\nBox S86 contains 4 7 9 and has a procurement charge of 291.\nBox S87 contains 16 18 20 21 22 and has a procurement charge of 260.\nBox S88 contains 14 16 19 21 and has a procurement charge of 156.\nBox S89 contains 9 10 and has a procurement charge of 54.\nBox S90 contains 5 6 7 8 9 10 and has a procurement charge of 174.\nBox S91 contains 5 7 8 9 10 11 and has a procurement charge of 114.\nBox S92 contains 6 8 9 10 13 and has a procurement charge of 305.\nBox S93 contains 12 14 17 and has a procurement charge of 177.\nBox S94 contains 7 14 15 and has a procurement charge of 99.\nBox S95 contains 13 15 17 and has a procurement charge of 189.\nBox S96 contains 5 6 8 10 11 12 and has a procurement charge of 516.\nBox S97 contains 12 14 15 17 and has a procurement charge of 372.\nBox S98 contains 7 8 9 10 11 and has a procurement charge of 10.\nBox S99 contains 14 16 17 and has a procurement charge of 258.\nBox S100 contains 15 16 18 and has a procurement charge of 201.\nBox S101 contains 17 18 19 20 21 22 and has a procurement charge of 96.\nBox S102 contains 14 15 17 19 and has a procurement charge of 324.\nBox S103 contains 9 10 14 15 18 and has a procurement charge of 65.\nBox S104 contains 1 and has a procurement charge of 10000.\nBox S105 contains 2 and has a procurement charge of 10000.\nBox S106 contains 3 and has a procurement charge of 10000.\nBox S107 contains 4 and has a procurement charge of 10000.\nBox S108 contains 5 and has a procurement charge of 10000.\nBox S109 contains 6 and has a procurement charge of 10000.\nBox S110 contains 7 and has a procurement charge of 10000.\nBox S111 contains 8 and has a procurement charge of 10000.\nBox S112 contains 9 and has a procurement charge of 10000.\nBox S113 contains 10 and has a procurement charge of 10000.\nBox S114 contains 11 and has a procurement charge of 10000.\nBox S115 contains 12 and has a procurement charge of 10000.\nBox S116 contains 13 and has a procurement charge of 10000.\nBox S117 contains 14 and has a procurement charge of 10000.\nBox S118 contains 15 and has a procurement charge of 10000.\nBox S119 contains 16 and has a procurement charge of 10000.\nBox S120 contains 17 and has a procurement charge of 10000.\nBox S121 contains 18 and has a procurement charge of 10000.\nBox S122 contains 19 and has a procurement charge of 10000.\nBox S123 contains 20 and has a procurement charge of 10000.\nBox S124 contains 21 and has a procurement charge of 10000.\nBox S125 contains 22 and has a procurement charge of 10000.\nThe team will pick boxes that cover every ingredient exactly once while minimizing the total procurement charge.\n\nAlso, when you send back the chosen lineup, please use this simple JSON shape so it's easy to parse:\n\n{\n \"solution\": [\"box_id\", ...]\n}\n\nThis just says: put the ids of the boxes you picked into the solution list (one id per chosen box). Think of \"box_id\" as a placeholder for whatever box identifier shows up in the instance — the JSON above is just a sketch of the expected shape, not the actual answer.\n\nA quick reminder: all identifiers must be used exactly as they appear in the instance input — no renaming and no new labels. \n- \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 22, "num_sets": 125, "density": 0.17054545454545456, "sets": [ { "id": 1, "elements": [ 3, 4, 5, 6, 8, 9 ], "cost": 102 }, { "id": 2, "elements": [ 1, 2, 3 ], "cost": 234 }, { "id": 3, "elements": [ 5, 8, 10, 12 ], "cost": 312 }, { "id": 4, "elements": [ 3, 4, 5, 6, 7, 8 ], "cost": 402 }, { "id": 5, "elements": [ 10, 11, 12, 13, 14, 15, 16 ], "cost": 483 }, { "id": 6, "elements": [ 13, 14, 15, 17, 18 ], "cost": 495 }, { "id": 7, "elements": [ 13, 14 ], "cost": 168 }, { "id": 8, "elements": [ 2, 4, 5, 6 ], "cost": 60 }, { "id": 9, "elements": [ 9, 11, 12, 14, 15 ], "cost": 65 }, { "id": 10, "elements": [ 3, 4, 5, 6 ], "cost": 176 }, { "id": 11, "elements": [ 13, 14, 15 ], "cost": 108 }, { "id": 12, "elements": [ 3, 4, 5, 6, 7 ], "cost": 190 }, { "id": 13, "elements": [ 5, 6, 7, 8 ], "cost": 80 }, { "id": 14, "elements": [ 1, 3, 4, 5, 7 ], "cost": 140 }, { "id": 15, "elements": [ 3, 5, 6, 7, 11 ], "cost": 260 }, { "id": 16, "elements": [ 5, 6, 7, 10 ], "cost": 268 }, { "id": 17, "elements": [ 15, 16, 17, 19 ], "cost": 196 }, { "id": 18, "elements": [ 2, 3 ], "cost": 58 }, { "id": 19, "elements": [ 19, 20, 21 ], "cost": 138 }, { "id": 20, "elements": [ 18, 19, 20, 21 ], "cost": 272 }, { "id": 21, "elements": [ 3, 5, 6, 7, 8, 11 ], "cost": 414 }, { "id": 22, "elements": [ 14, 16, 17, 18, 20, 22 ], "cost": 570 }, { "id": 23, "elements": [ 19, 21, 22 ], "cost": 69 }, { "id": 24, "elements": [ 18, 19 ], "cost": 192 }, { "id": 25, "elements": [ 1, 3, 6 ], "cost": 81 }, { "id": 26, "elements": [ 16, 18, 20 ], "cost": 21 }, { "id": 27, "elements": [ 17, 19, 20 ], "cost": 225 }, { "id": 28, "elements": [ 13, 14, 15, 17, 20 ], "cost": 215 }, { "id": 29, "elements": [ 15, 16, 17, 18 ], "cost": 176 }, { "id": 30, "elements": [ 15, 18, 19 ], "cost": 291 }, { "id": 31, "elements": [ 11, 13, 15, 16, 18, 19 ], "cost": 522 }, { "id": 32, "elements": [ 12, 14, 16, 18 ], "cost": 56 }, { "id": 33, "elements": [ 6, 7, 8, 9, 10, 11 ], "cost": 162 }, { "id": 34, "elements": [ 2, 3, 4, 5, 6 ], "cost": 425 }, { "id": 35, "elements": [ 18, 19, 21 ], "cost": 300 }, { "id": 36, "elements": [ 5, 6, 8, 10, 11 ], "cost": 215 }, { "id": 37, "elements": [ 9, 10, 11, 12, 14 ], "cost": 260 }, { "id": 38, "elements": [ 13, 16 ], "cost": 118 }, { "id": 39, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 330 }, { "id": 40, "elements": [ 8, 9, 10, 11, 12, 13 ], "cost": 60 }, { "id": 41, "elements": [ 11, 12, 15 ], "cost": 18 }, { "id": 42, "elements": [ 10, 11, 12, 13, 14, 16 ], "cost": 78 }, { "id": 43, "elements": [ 10, 11 ], "cost": 74 }, { "id": 44, "elements": [ 2, 8, 10 ], "cost": 282 }, { "id": 45, "elements": [ 4, 5, 6, 8 ], "cost": 228 }, { "id": 46, "elements": [ 18, 19, 20 ], "cost": 114 }, { "id": 47, "elements": [ 2, 3, 5, 6, 7 ], "cost": 235 }, { "id": 48, "elements": [ 12, 13, 14, 15, 16, 17 ], "cost": 546 }, { "id": 49, "elements": [ 15, 17, 21 ], "cost": 159 }, { "id": 50, "elements": [ 12, 17, 18, 21 ], "cost": 192 }, { "id": 51, "elements": [ 18, 20 ], "cost": 84 }, { "id": 52, "elements": [ 12, 13, 14, 15, 16, 18 ], "cost": 90 }, { "id": 53, "elements": [ 14, 17, 19, 20, 22 ], "cost": 485 }, { "id": 54, "elements": [ 2, 4, 5, 6, 7, 8 ], "cost": 336 }, { "id": 55, "elements": [ 7, 8, 11, 12, 13, 15 ], "cost": 204 }, { "id": 56, "elements": [ 16, 17, 18, 19, 20, 21 ], "cost": 276 }, { "id": 57, "elements": [ 14, 16, 17, 18, 20, 21 ], "cost": 36 }, { "id": 58, "elements": [ 15, 16, 17, 19, 20, 21 ], "cost": 126 }, { "id": 59, "elements": [ 4, 7, 10, 12 ], "cost": 228 }, { "id": 60, "elements": [ 9, 15, 16 ], "cost": 81 }, { "id": 61, "elements": [ 13, 15, 16, 17 ], "cost": 64 }, { "id": 62, "elements": [ 5, 6, 7 ], "cost": 63 }, { "id": 63, "elements": [ 2, 3, 4, 5, 7, 10 ], "cost": 72 }, { "id": 64, "elements": [ 7, 10, 11, 12, 13, 16 ], "cost": 78 }, { "id": 65, "elements": [ 8, 9, 12, 13 ], "cost": 360 }, { "id": 66, "elements": [ 16, 21, 22 ], "cost": 108 }, { "id": 67, "elements": [ 6, 7, 9, 11, 13 ], "cost": 90 }, { "id": 68, "elements": [ 1, 2, 3, 4, 5 ], "cost": 270 }, { "id": 69, "elements": [ 11, 12, 15, 16, 17, 18 ], "cost": 90 }, { "id": 70, "elements": [ 7, 9, 10, 11, 12, 13 ], "cost": 588 }, { "id": 71, "elements": [ 14, 15, 16, 17, 18 ], "cost": 45 }, { "id": 72, "elements": [ 15, 16, 19 ], "cost": 171 }, { "id": 73, "elements": [ 13, 19, 21 ], "cost": 99 }, { "id": 74, "elements": [ 20, 21, 22 ], "cost": 39 }, { "id": 75, "elements": [ 16, 17, 18, 22 ], "cost": 116 }, { "id": 76, "elements": [ 16, 17, 18, 19, 20, 22 ], "cost": 126 }, { "id": 77, "elements": [ 1, 2, 3, 5 ], "cost": 296 }, { "id": 78, "elements": [ 5, 7, 8, 9 ], "cost": 336 }, { "id": 79, "elements": [ 9, 10, 11, 12, 13, 14 ], "cost": 510 }, { "id": 80, "elements": [ 9, 12, 13 ], "cost": 165 }, { "id": 81, "elements": [ 15, 16, 17, 18, 20 ], "cost": 190 }, { "id": 82, "elements": [ 11, 12, 17 ], "cost": 33 }, { "id": 83, "elements": [ 14, 15, 17, 18, 19, 20 ], "cost": 600 }, { "id": 84, "elements": [ 12, 14 ], "cost": 146 }, { "id": 85, "elements": [ 13, 14, 17, 18, 19, 22 ], "cost": 126 }, { "id": 86, "elements": [ 4, 7, 9 ], "cost": 291 }, { "id": 87, "elements": [ 16, 18, 20, 21, 22 ], "cost": 260 }, { "id": 88, "elements": [ 14, 16, 19, 21 ], "cost": 156 }, { "id": 89, "elements": [ 9, 10 ], "cost": 54 }, { "id": 90, "elements": [ 5, 6, 7, 8, 9, 10 ], "cost": 174 }, { "id": 91, "elements": [ 5, 7, 8, 9, 10, 11 ], "cost": 114 }, { "id": 92, "elements": [ 6, 8, 9, 10, 13 ], "cost": 305 }, { "id": 93, "elements": [ 12, 14, 17 ], "cost": 177 }, { "id": 94, "elements": [ 7, 14, 15 ], "cost": 99 }, { "id": 95, "elements": [ 13, 15, 17 ], "cost": 189 }, { "id": 96, "elements": [ 5, 6, 8, 10, 11, 12 ], "cost": 516 }, { "id": 97, "elements": [ 12, 14, 15, 17 ], "cost": 372 }, { "id": 98, "elements": [ 7, 8, 9, 10, 11 ], "cost": 10 }, { "id": 99, "elements": [ 14, 16, 17 ], "cost": 258 }, { "id": 100, "elements": [ 15, 16, 18 ], "cost": 201 }, { "id": 101, "elements": [ 17, 18, 19, 20, 21, 22 ], "cost": 96 }, { "id": 102, "elements": [ 14, 15, 17, 19 ], "cost": 324 }, { "id": 103, "elements": [ 9, 10, 14, 15, 18 ], "cost": 65 }, { "id": 104, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 118, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 119, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 120, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 121, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 122, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 123, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 124, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 125, "elements": [ 22 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0015_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0015_bag_stats.png" }, "solution": [ 23, 39, 51, 61, 84, 98 ], "obj": 703.0, "instance_variant": { "num_elements": 22, "num_sets": 125, "sets": [ { "id": "S1", "elements": [ 3, 4, 5, 6, 8, 9 ], "cost": 102 }, { "id": "S2", "elements": [ 1, 2, 3 ], "cost": 234 }, { "id": "S3", "elements": [ 5, 8, 10, 12 ], "cost": 312 }, { "id": "S4", "elements": [ 3, 4, 5, 6, 7, 8 ], "cost": 402 }, { "id": "S5", "elements": [ 10, 11, 12, 13, 14, 15, 16 ], "cost": 483 }, { "id": "S6", "elements": [ 13, 14, 15, 17, 18 ], "cost": 495 }, { "id": "S7", "elements": [ 13, 14 ], "cost": 168 }, { "id": "S8", "elements": [ 2, 4, 5, 6 ], "cost": 60 }, { "id": "S9", "elements": [ 9, 11, 12, 14, 15 ], "cost": 65 }, { "id": "S10", "elements": [ 3, 4, 5, 6 ], "cost": 176 }, { "id": "S11", "elements": [ 13, 14, 15 ], "cost": 108 }, { "id": "S12", "elements": [ 3, 4, 5, 6, 7 ], "cost": 190 }, { "id": "S13", "elements": [ 5, 6, 7, 8 ], "cost": 80 }, { "id": "S14", "elements": [ 1, 3, 4, 5, 7 ], "cost": 140 }, { "id": "S15", "elements": [ 3, 5, 6, 7, 11 ], "cost": 260 }, { "id": "S16", "elements": [ 5, 6, 7, 10 ], "cost": 268 }, { "id": "S17", "elements": [ 15, 16, 17, 19 ], "cost": 196 }, { "id": "S18", "elements": [ 2, 3 ], "cost": 58 }, { "id": "S19", "elements": [ 19, 20, 21 ], "cost": 138 }, { "id": "S20", "elements": [ 18, 19, 20, 21 ], "cost": 272 }, { "id": "S21", "elements": [ 3, 5, 6, 7, 8, 11 ], "cost": 414 }, { "id": "S22", "elements": [ 14, 16, 17, 18, 20, 22 ], "cost": 570 }, { "id": "S23", "elements": [ 19, 21, 22 ], "cost": 69 }, { "id": "S24", "elements": [ 18, 19 ], "cost": 192 }, { "id": "S25", "elements": [ 1, 3, 6 ], "cost": 81 }, { "id": "S26", "elements": [ 16, 18, 20 ], "cost": 21 }, { "id": "S27", "elements": [ 17, 19, 20 ], "cost": 225 }, { "id": "S28", "elements": [ 13, 14, 15, 17, 20 ], "cost": 215 }, { "id": "S29", "elements": [ 15, 16, 17, 18 ], "cost": 176 }, { "id": "S30", "elements": [ 15, 18, 19 ], "cost": 291 }, { "id": "S31", "elements": [ 11, 13, 15, 16, 18, 19 ], "cost": 522 }, { "id": "S32", "elements": [ 12, 14, 16, 18 ], "cost": 56 }, { "id": "S33", "elements": [ 6, 7, 8, 9, 10, 11 ], "cost": 162 }, { "id": "S34", "elements": [ 2, 3, 4, 5, 6 ], "cost": 425 }, { "id": "S35", "elements": [ 18, 19, 21 ], "cost": 300 }, { "id": "S36", "elements": [ 5, 6, 8, 10, 11 ], "cost": 215 }, { "id": "S37", "elements": [ 9, 10, 11, 12, 14 ], "cost": 260 }, { "id": "S38", "elements": [ 13, 16 ], "cost": 118 }, { "id": "S39", "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 330 }, { "id": "S40", "elements": [ 8, 9, 10, 11, 12, 13 ], "cost": 60 }, { "id": "S41", "elements": [ 11, 12, 15 ], "cost": 18 }, { "id": "S42", "elements": [ 10, 11, 12, 13, 14, 16 ], "cost": 78 }, { "id": "S43", "elements": [ 10, 11 ], "cost": 74 }, { "id": "S44", "elements": [ 2, 8, 10 ], "cost": 282 }, { "id": "S45", "elements": [ 4, 5, 6, 8 ], "cost": 228 }, { "id": "S46", "elements": [ 18, 19, 20 ], "cost": 114 }, { "id": "S47", "elements": [ 2, 3, 5, 6, 7 ], "cost": 235 }, { "id": "S48", "elements": [ 12, 13, 14, 15, 16, 17 ], "cost": 546 }, { "id": "S49", "elements": [ 15, 17, 21 ], "cost": 159 }, { "id": "S50", "elements": [ 12, 17, 18, 21 ], "cost": 192 }, { "id": "S51", "elements": [ 18, 20 ], "cost": 84 }, { "id": "S52", "elements": [ 12, 13, 14, 15, 16, 18 ], "cost": 90 }, { "id": "S53", "elements": [ 14, 17, 19, 20, 22 ], "cost": 485 }, { "id": "S54", "elements": [ 2, 4, 5, 6, 7, 8 ], "cost": 336 }, { "id": "S55", "elements": [ 7, 8, 11, 12, 13, 15 ], "cost": 204 }, { "id": "S56", "elements": [ 16, 17, 18, 19, 20, 21 ], "cost": 276 }, { "id": "S57", "elements": [ 14, 16, 17, 18, 20, 21 ], "cost": 36 }, { "id": "S58", "elements": [ 15, 16, 17, 19, 20, 21 ], "cost": 126 }, { "id": "S59", "elements": [ 4, 7, 10, 12 ], "cost": 228 }, { "id": "S60", "elements": [ 9, 15, 16 ], "cost": 81 }, { "id": "S61", "elements": [ 13, 15, 16, 17 ], "cost": 64 }, { "id": "S62", "elements": [ 5, 6, 7 ], "cost": 63 }, { "id": "S63", "elements": [ 2, 3, 4, 5, 7, 10 ], "cost": 72 }, { "id": "S64", "elements": [ 7, 10, 11, 12, 13, 16 ], "cost": 78 }, { "id": "S65", "elements": [ 8, 9, 12, 13 ], "cost": 360 }, { "id": "S66", "elements": [ 16, 21, 22 ], "cost": 108 }, { "id": "S67", "elements": [ 6, 7, 9, 11, 13 ], "cost": 90 }, { "id": "S68", "elements": [ 1, 2, 3, 4, 5 ], "cost": 270 }, { "id": "S69", "elements": [ 11, 12, 15, 16, 17, 18 ], "cost": 90 }, { "id": "S70", "elements": [ 7, 9, 10, 11, 12, 13 ], "cost": 588 }, { "id": "S71", "elements": [ 14, 15, 16, 17, 18 ], "cost": 45 }, { "id": "S72", "elements": [ 15, 16, 19 ], "cost": 171 }, { "id": "S73", "elements": [ 13, 19, 21 ], "cost": 99 }, { "id": "S74", "elements": [ 20, 21, 22 ], "cost": 39 }, { "id": "S75", "elements": [ 16, 17, 18, 22 ], "cost": 116 }, { "id": "S76", "elements": [ 16, 17, 18, 19, 20, 22 ], "cost": 126 }, { "id": "S77", "elements": [ 1, 2, 3, 5 ], "cost": 296 }, { "id": "S78", "elements": [ 5, 7, 8, 9 ], "cost": 336 }, { "id": "S79", "elements": [ 9, 10, 11, 12, 13, 14 ], "cost": 510 }, { "id": "S80", "elements": [ 9, 12, 13 ], "cost": 165 }, { "id": "S81", "elements": [ 15, 16, 17, 18, 20 ], "cost": 190 }, { "id": "S82", "elements": [ 11, 12, 17 ], "cost": 33 }, { "id": "S83", "elements": [ 14, 15, 17, 18, 19, 20 ], "cost": 600 }, { "id": "S84", "elements": [ 12, 14 ], "cost": 146 }, { "id": "S85", "elements": [ 13, 14, 17, 18, 19, 22 ], "cost": 126 }, { "id": "S86", "elements": [ 4, 7, 9 ], "cost": 291 }, { "id": "S87", "elements": [ 16, 18, 20, 21, 22 ], "cost": 260 }, { "id": "S88", "elements": [ 14, 16, 19, 21 ], "cost": 156 }, { "id": "S89", "elements": [ 9, 10 ], "cost": 54 }, { "id": "S90", "elements": [ 5, 6, 7, 8, 9, 10 ], "cost": 174 }, { "id": "S91", "elements": [ 5, 7, 8, 9, 10, 11 ], "cost": 114 }, { "id": "S92", "elements": [ 6, 8, 9, 10, 13 ], "cost": 305 }, { "id": "S93", "elements": [ 12, 14, 17 ], "cost": 177 }, { "id": "S94", "elements": [ 7, 14, 15 ], "cost": 99 }, { "id": "S95", "elements": [ 13, 15, 17 ], "cost": 189 }, { "id": "S96", "elements": [ 5, 6, 8, 10, 11, 12 ], "cost": 516 }, { "id": "S97", "elements": [ 12, 14, 15, 17 ], "cost": 372 }, { "id": "S98", "elements": [ 7, 8, 9, 10, 11 ], "cost": 10 }, { "id": "S99", "elements": [ 14, 16, 17 ], "cost": 258 }, { "id": "S100", "elements": [ 15, 16, 18 ], "cost": 201 }, { "id": "S101", "elements": [ 17, 18, 19, 20, 21, 22 ], "cost": 96 }, { "id": "S102", "elements": [ 14, 15, 17, 19 ], "cost": 324 }, { "id": "S103", "elements": [ 9, 10, 14, 15, 18 ], "cost": 65 }, { "id": "S104", "elements": [ 1 ], "cost": 10000 }, { "id": "S105", "elements": [ 2 ], "cost": 10000 }, { "id": "S106", "elements": [ 3 ], "cost": 10000 }, { "id": "S107", "elements": [ 4 ], "cost": 10000 }, { "id": "S108", "elements": [ 5 ], "cost": 10000 }, { "id": "S109", "elements": [ 6 ], "cost": 10000 }, { "id": "S110", "elements": [ 7 ], "cost": 10000 }, { "id": "S111", "elements": [ 8 ], "cost": 10000 }, { "id": "S112", "elements": [ 9 ], "cost": 10000 }, { "id": "S113", "elements": [ 10 ], "cost": 10000 }, { "id": "S114", "elements": [ 11 ], "cost": 10000 }, { "id": "S115", "elements": [ 12 ], "cost": 10000 }, { "id": "S116", "elements": [ 13 ], "cost": 10000 }, { "id": "S117", "elements": [ 14 ], "cost": 10000 }, { "id": "S118", "elements": [ 15 ], "cost": 10000 }, { "id": "S119", "elements": [ 16 ], "cost": 10000 }, { "id": "S120", "elements": [ 17 ], "cost": 10000 }, { "id": "S121", "elements": [ 18 ], "cost": 10000 }, { "id": "S122", "elements": [ 19 ], "cost": 10000 }, { "id": "S123", "elements": [ 20 ], "cost": 10000 }, { "id": "S124", "elements": [ 21 ], "cost": 10000 }, { "id": "S125", "elements": [ 22 ], "cost": 10000 } ] }, "solution_variant": [ "S23", "S39", "S51", "S61", "S84", "S98" ], "context_index": 16, "input_format": "nl", "input_index_base": 1 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "There’s a volunteer coordinator juggling multiple tasks and dozens of candidate teams. The coordinator needs to decide which teams to sign on so that every task is covered by exactly one chosen team — no task missing, no task handled by two teams — and each chosen team comes with a coordination cost. The quality of any choice is judged by adding up those team costs, and the preferable choices are the ones with the lowest total bill. The concrete details are shown below.\n\n- **num_tasks**: 21\n- **num_candidate_teams**: 132\n\n| team_id | coordination_cost | tasks_covered |\n|---|---|---|\n| S1 | 160 | M N P Q T |\n| S2 | 297 | B C D |\n| S3 | 500 | F G H I J |\n| S4 | 240 | M O P Q R S |\n| S5 | 132 | K M O |\n| S6 | 325 | I K L M P |\n| S7 | 60 | N O Q R S |\n| S8 | 68 | S U |\n| S9 | 42 | J K N |\n| S10 | 120 | G H I K L N |\n| S11 | 285 | E G H J L |\n| S12 | 285 | E F G I K |\n| S13 | 208 | P Q S U |\n| S14 | 240 | K M N O R |\n| S15 | 360 | E G H L |\n| S16 | 195 | O R U |\n| S17 | 415 | P Q R S T |\n| S18 | 132 | D F G H I J |\n| S19 | 90 | L O |\n| S20 | 132 | B D E I |\n| S21 | 348 | A B C D F G |\n| S22 | 159 | I K M |\n| S23 | 237 | R T U |\n| S24 | 20 | L N |\n| S25 | 115 | O P Q S U |\n| S26 | 528 | A B C D E G |\n| S27 | 160 | K M N Q |\n| S28 | 120 | L M N P |\n| S29 | 12 | R S T U |\n| S30 | 96 | J L N P |\n| S31 | 92 | I J L M |\n| S32 | 96 | E F G H I J |\n| S33 | 65 | G H J K M |\n| S34 | 95 | I K M O P |\n| S35 | 186 | N O P |\n| S36 | 414 | P Q R S T U |\n| S37 | 152 | O P S T |\n| S38 | 300 | K L M N O P |\n| S39 | 18 | I K L |\n| S40 | 210 | B D F |\n| S41 | 117 | G K L |\n| S42 | 380 | I M N P |\n| S43 | 236 | B D F G |\n| S44 | 294 | C E F G I J |\n| S45 | 129 | A C G |\n| S46 | 516 | I J K L N O |\n| S47 | 120 | N O Q |\n| S48 | 132 | L M O P |\n| S49 | 50 | Q R S T U |\n| S50 | 475 | A B C D F |\n| S51 | 102 | L M P |\n| S52 | 36 | B D G |\n| S53 | 60 | L N P Q |\n| S54 | 93 | K N O |\n| S55 | 260 | H J L M |\n| S56 | 165 | B C E H I |\n| S57 | 174 | H K L M O P |\n| S58 | 171 | C F I |\n| S59 | 42 | G I |\n| S60 | 207 | H J L |\n| S61 | 10 | E G |\n| S62 | 24 | A B E |\n| S63 | 366 | O P R S T U |\n| S64 | 207 | Q T U |\n| S65 | 288 | F G I K L O |\n| S66 | 150 | L M N O P |\n| S67 | 342 | C D E F G I |\n| S68 | 150 | L M N O P Q |\n| S69 | 156 | G H I L |\n| S70 | 384 | K M N P |\n| S71 | 160 | B C E F G |\n| S72 | 30 | C D E F G H |\n| S73 | 205 | A B C E F |\n| S74 | 450 | J K L M O P |\n| S75 | 120 | L M O P Q S |\n| S76 | 213 | B D E |\n| S77 | 285 | F I J L M |\n| S78 | 66 | A B C |\n| S79 | 15 | O R S T U |\n| S80 | 522 | O P Q R S U |\n| S81 | 328 | C D H I |\n| S82 | 240 | C D H |\n| S83 | 24 | L M O Q R S |\n| S84 | 224 | C E F G |\n| S85 | 284 | E H J K |\n| S86 | 78 | O T |\n| S87 | 172 | L O Q R |\n| S88 | 180 | O P R S |\n| S89 | 430 | O P Q S T |\n| S90 | 54 | M N O P R S |\n| S91 | 6 | C I J |\n| S92 | 295 | H K L M N |\n| S93 | 480 | H I J L M |\n| S94 | 66 | A B C E H J |\n| S95 | 325 | D F G H I |\n| S96 | 140 | E J |\n| S97 | 150 | D E F G H |\n| S98 | 246 | S T U |\n| S99 | 3 | L M N |\n| S100 | 184 | K P |\n| S101 | 312 | B D E F |\n| S102 | 240 | B C D E G |\n| S103 | 24 | N O Q U |\n| S104 | 300 | M N P Q S |\n| S105 | 246 | J L M N O Q |\n| S106 | 18 | B C D E F G |\n| S107 | 252 | D E G J |\n| S108 | 70 | L N P S T |\n| S109 | 38 | K L |\n| S110 | 120 | K O R |\n| S111 | 185 | H I J K L |\n| S112 | 10000 | A |\n| S113 | 10000 | B |\n| S114 | 10000 | C |\n| S115 | 10000 | D |\n| S116 | 10000 | E |\n| S117 | 10000 | F |\n| S118 | 10000 | G |\n| S119 | 10000 | H |\n| S120 | 10000 | I |\n| S121 | 10000 | J |\n| S122 | 10000 | K |\n| S123 | 10000 | L |\n| S124 | 10000 | M |\n| S125 | 10000 | N |\n| S126 | 10000 | O |\n| S127 | 10000 | P |\n| S128 | 10000 | Q |\n| S129 | 10000 | R |\n| S130 | 10000 | S |\n| S131 | 10000 | T |\n| S132 | 10000 | U |\n\nIf you want my pick, please send it back in this simple shape — nothing fancy, just a tiny JSON list of the chosen teams:\n\n{\n \"solution\": [\"team_id\", ...]\n}\n\nThis just means: put the IDs of the teams you want in the solution array (each entry is a team identifier). Think of it like ticking boxes on a form — the array holds the team IDs you’re signing on. The JSON above is just a sketch of the shape I expect, not the actual answer.\n\nPlease make sure you use the exact identifiers from the instance input — don’t rename them or invent new ones. Valid identifiers look like plain numbers such as \"1\" or \"23\", single capital letters like \"A\" or \"B\", or a capital letter followed by digits like \"A1\" or \"X7\".", "instance": { "problem_type": "SPP", "num_elements": 21, "num_sets": 132, "density": 0.17929292929292928, "sets": [ { "id": 1, "elements": [ 13, 14, 16, 17, 20 ], "cost": 160 }, { "id": 2, "elements": [ 2, 3, 4 ], "cost": 297 }, { "id": 3, "elements": [ 6, 7, 8, 9, 10 ], "cost": 500 }, { "id": 4, "elements": [ 13, 15, 16, 17, 18, 19 ], "cost": 240 }, { "id": 5, "elements": [ 11, 13, 15 ], "cost": 132 }, { "id": 6, "elements": [ 9, 11, 12, 13, 16 ], "cost": 325 }, { "id": 7, "elements": [ 14, 15, 17, 18, 19 ], "cost": 60 }, { "id": 8, "elements": [ 19, 21 ], "cost": 68 }, { "id": 9, "elements": [ 10, 11, 14 ], "cost": 42 }, { "id": 10, "elements": [ 7, 8, 9, 11, 12, 14 ], "cost": 120 }, { "id": 11, "elements": [ 5, 7, 8, 10, 12 ], "cost": 285 }, { "id": 12, "elements": [ 5, 6, 7, 9, 11 ], "cost": 285 }, { "id": 13, "elements": [ 16, 17, 19, 21 ], "cost": 208 }, { "id": 14, "elements": [ 11, 13, 14, 15, 18 ], "cost": 240 }, { "id": 15, "elements": [ 5, 7, 8, 12 ], "cost": 360 }, { "id": 16, "elements": [ 15, 18, 21 ], "cost": 195 }, { "id": 17, "elements": [ 16, 17, 18, 19, 20 ], "cost": 415 }, { "id": 18, "elements": [ 4, 6, 7, 8, 9, 10 ], "cost": 132 }, { "id": 19, "elements": [ 12, 15 ], "cost": 90 }, { "id": 20, "elements": [ 2, 4, 5, 9 ], "cost": 132 }, { "id": 21, "elements": [ 1, 2, 3, 4, 6, 7 ], "cost": 348 }, { "id": 22, "elements": [ 9, 11, 13 ], "cost": 159 }, { "id": 23, "elements": [ 18, 20, 21 ], "cost": 237 }, { "id": 24, "elements": [ 12, 14 ], "cost": 20 }, { "id": 25, "elements": [ 15, 16, 17, 19, 21 ], "cost": 115 }, { "id": 26, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 528 }, { "id": 27, "elements": [ 11, 13, 14, 17 ], "cost": 160 }, { "id": 28, "elements": [ 12, 13, 14, 16 ], "cost": 120 }, { "id": 29, "elements": [ 18, 19, 20, 21 ], "cost": 12 }, { "id": 30, "elements": [ 10, 12, 14, 16 ], "cost": 96 }, { "id": 31, "elements": [ 9, 10, 12, 13 ], "cost": 92 }, { "id": 32, "elements": [ 5, 6, 7, 8, 9, 10 ], "cost": 96 }, { "id": 33, "elements": [ 7, 8, 10, 11, 13 ], "cost": 65 }, { "id": 34, "elements": [ 9, 11, 13, 15, 16 ], "cost": 95 }, { "id": 35, "elements": [ 14, 15, 16 ], "cost": 186 }, { "id": 36, "elements": [ 16, 17, 18, 19, 20, 21 ], "cost": 414 }, { "id": 37, "elements": [ 15, 16, 19, 20 ], "cost": 152 }, { "id": 38, "elements": [ 11, 12, 13, 14, 15, 16 ], "cost": 300 }, { "id": 39, "elements": [ 9, 11, 12 ], "cost": 18 }, { "id": 40, "elements": [ 2, 4, 6 ], "cost": 210 }, { "id": 41, "elements": [ 7, 11, 12 ], "cost": 117 }, { "id": 42, "elements": [ 9, 13, 14, 16 ], "cost": 380 }, { "id": 43, "elements": [ 2, 4, 6, 7 ], "cost": 236 }, { "id": 44, "elements": [ 3, 5, 6, 7, 9, 10 ], "cost": 294 }, { "id": 45, "elements": [ 1, 3, 7 ], "cost": 129 }, { "id": 46, "elements": [ 9, 10, 11, 12, 14, 15 ], "cost": 516 }, { "id": 47, "elements": [ 14, 15, 17 ], "cost": 120 }, { "id": 48, "elements": [ 12, 13, 15, 16 ], "cost": 132 }, { "id": 49, "elements": [ 17, 18, 19, 20, 21 ], "cost": 50 }, { "id": 50, "elements": [ 1, 2, 3, 4, 6 ], "cost": 475 }, { "id": 51, "elements": [ 12, 13, 16 ], "cost": 102 }, { "id": 52, "elements": [ 2, 4, 7 ], "cost": 36 }, { "id": 53, "elements": [ 12, 14, 16, 17 ], "cost": 60 }, { "id": 54, "elements": [ 11, 14, 15 ], "cost": 93 }, { "id": 55, "elements": [ 8, 10, 12, 13 ], "cost": 260 }, { "id": 56, "elements": [ 2, 3, 5, 8, 9 ], "cost": 165 }, { "id": 57, "elements": [ 8, 11, 12, 13, 15, 16 ], "cost": 174 }, { "id": 58, "elements": [ 3, 6, 9 ], "cost": 171 }, { "id": 59, "elements": [ 7, 9 ], "cost": 42 }, { "id": 60, "elements": [ 8, 10, 12 ], "cost": 207 }, { "id": 61, "elements": [ 5, 7 ], "cost": 10 }, { "id": 62, "elements": [ 1, 2, 5 ], "cost": 24 }, { "id": 63, "elements": [ 15, 16, 18, 19, 20, 21 ], "cost": 366 }, { "id": 64, "elements": [ 17, 20, 21 ], "cost": 207 }, { "id": 65, "elements": [ 6, 7, 9, 11, 12, 15 ], "cost": 288 }, { "id": 66, "elements": [ 12, 13, 14, 15, 16 ], "cost": 150 }, { "id": 67, "elements": [ 3, 4, 5, 6, 7, 9 ], "cost": 342 }, { "id": 68, "elements": [ 12, 13, 14, 15, 16, 17 ], "cost": 150 }, { "id": 69, "elements": [ 7, 8, 9, 12 ], "cost": 156 }, { "id": 70, "elements": [ 11, 13, 14, 16 ], "cost": 384 }, { "id": 71, "elements": [ 2, 3, 5, 6, 7 ], "cost": 160 }, { "id": 72, "elements": [ 3, 4, 5, 6, 7, 8 ], "cost": 30 }, { "id": 73, "elements": [ 1, 2, 3, 5, 6 ], "cost": 205 }, { "id": 74, "elements": [ 10, 11, 12, 13, 15, 16 ], "cost": 450 }, { "id": 75, "elements": [ 12, 13, 15, 16, 17, 19 ], "cost": 120 }, { "id": 76, "elements": [ 2, 4, 5 ], "cost": 213 }, { "id": 77, "elements": [ 6, 9, 10, 12, 13 ], "cost": 285 }, { "id": 78, "elements": [ 1, 2, 3 ], "cost": 66 }, { "id": 79, "elements": [ 15, 18, 19, 20, 21 ], "cost": 15 }, { "id": 80, "elements": [ 15, 16, 17, 18, 19, 21 ], "cost": 522 }, { "id": 81, "elements": [ 3, 4, 8, 9 ], "cost": 328 }, { "id": 82, "elements": [ 3, 4, 8 ], "cost": 240 }, { "id": 83, "elements": [ 12, 13, 15, 17, 18, 19 ], "cost": 24 }, { "id": 84, "elements": [ 3, 5, 6, 7 ], "cost": 224 }, { "id": 85, "elements": [ 5, 8, 10, 11 ], "cost": 284 }, { "id": 86, "elements": [ 15, 20 ], "cost": 78 }, { "id": 87, "elements": [ 12, 15, 17, 18 ], "cost": 172 }, { "id": 88, "elements": [ 15, 16, 18, 19 ], "cost": 180 }, { "id": 89, "elements": [ 15, 16, 17, 19, 20 ], "cost": 430 }, { "id": 90, "elements": [ 13, 14, 15, 16, 18, 19 ], "cost": 54 }, { "id": 91, "elements": [ 3, 9, 10 ], "cost": 6 }, { "id": 92, "elements": [ 8, 11, 12, 13, 14 ], "cost": 295 }, { "id": 93, "elements": [ 8, 9, 10, 12, 13 ], "cost": 480 }, { "id": 94, "elements": [ 1, 2, 3, 5, 8, 10 ], "cost": 66 }, { "id": 95, "elements": [ 4, 6, 7, 8, 9 ], "cost": 325 }, { "id": 96, "elements": [ 5, 10 ], "cost": 140 }, { "id": 97, "elements": [ 4, 5, 6, 7, 8 ], "cost": 150 }, { "id": 98, "elements": [ 19, 20, 21 ], "cost": 246 }, { "id": 99, "elements": [ 12, 13, 14 ], "cost": 3 }, { "id": 100, "elements": [ 11, 16 ], "cost": 184 }, { "id": 101, "elements": [ 2, 4, 5, 6 ], "cost": 312 }, { "id": 102, "elements": [ 2, 3, 4, 5, 7 ], "cost": 240 }, { "id": 103, "elements": [ 14, 15, 17, 21 ], "cost": 24 }, { "id": 104, "elements": [ 13, 14, 16, 17, 19 ], "cost": 300 }, { "id": 105, "elements": [ 10, 12, 13, 14, 15, 17 ], "cost": 246 }, { "id": 106, "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 18 }, { "id": 107, "elements": [ 4, 5, 7, 10 ], "cost": 252 }, { "id": 108, "elements": [ 12, 14, 16, 19, 20 ], "cost": 70 }, { "id": 109, "elements": [ 11, 12 ], "cost": 38 }, { "id": 110, "elements": [ 11, 15, 18 ], "cost": 120 }, { "id": 111, "elements": [ 8, 9, 10, 11, 12 ], "cost": 185 }, { "id": 112, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 118, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 119, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 120, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 121, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 122, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 123, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 124, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 125, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 126, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 127, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 128, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 129, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 130, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 131, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 132, "elements": [ 21 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0016_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0016_bag_stats.png" }, "solution": [ 29, 31, 47, 78, 97, 100 ], "obj": 624.0, "instance_variant": { "num_elements": 21, "num_sets": 132, "sets": [ { "id": "S1", "elements": [ "M", "N", "P", "Q", "T" ], "cost": 160 }, { "id": "S2", "elements": [ "B", "C", "D" ], "cost": 297 }, { "id": "S3", "elements": [ "F", "G", "H", "I", "J" ], "cost": 500 }, { "id": "S4", "elements": [ "M", "O", "P", "Q", "R", "S" ], "cost": 240 }, { "id": "S5", "elements": [ "K", "M", "O" ], "cost": 132 }, { "id": "S6", "elements": [ "I", "K", "L", "M", "P" ], "cost": 325 }, { "id": "S7", "elements": [ "N", "O", "Q", "R", "S" ], "cost": 60 }, { "id": "S8", "elements": [ "S", "U" ], "cost": 68 }, { "id": "S9", "elements": [ "J", "K", "N" ], "cost": 42 }, { "id": "S10", "elements": [ "G", "H", "I", "K", "L", "N" ], "cost": 120 }, { "id": "S11", "elements": [ "E", "G", "H", "J", "L" ], "cost": 285 }, { "id": "S12", "elements": [ "E", "F", "G", "I", "K" ], "cost": 285 }, { "id": "S13", "elements": [ "P", "Q", "S", "U" ], "cost": 208 }, { "id": "S14", "elements": [ "K", "M", "N", "O", "R" ], "cost": 240 }, { "id": "S15", "elements": [ "E", "G", "H", "L" ], "cost": 360 }, { "id": "S16", "elements": [ "O", "R", "U" ], "cost": 195 }, { "id": "S17", "elements": [ "P", "Q", "R", "S", "T" ], "cost": 415 }, { "id": "S18", "elements": [ "D", "F", "G", "H", "I", "J" ], "cost": 132 }, { "id": "S19", "elements": [ "L", "O" ], "cost": 90 }, { "id": "S20", "elements": [ "B", "D", "E", "I" ], "cost": 132 }, { "id": "S21", "elements": [ "A", "B", "C", "D", "F", "G" ], "cost": 348 }, { "id": "S22", "elements": [ "I", "K", "M" ], "cost": 159 }, { "id": "S23", "elements": [ "R", "T", "U" ], "cost": 237 }, { "id": "S24", "elements": [ "L", "N" ], "cost": 20 }, { "id": "S25", "elements": [ "O", "P", "Q", "S", "U" ], "cost": 115 }, { "id": "S26", "elements": [ "A", "B", "C", "D", "E", "G" ], "cost": 528 }, { "id": "S27", "elements": [ "K", "M", "N", "Q" ], "cost": 160 }, { "id": "S28", "elements": [ "L", "M", "N", "P" ], "cost": 120 }, { "id": "S29", "elements": [ "R", "S", "T", "U" ], "cost": 12 }, { "id": "S30", "elements": [ "J", "L", "N", "P" ], "cost": 96 }, { "id": "S31", "elements": [ "I", "J", "L", "M" ], "cost": 92 }, { "id": "S32", "elements": [ "E", "F", "G", "H", "I", "J" ], "cost": 96 }, { "id": "S33", "elements": [ "G", "H", "J", "K", "M" ], "cost": 65 }, { "id": "S34", "elements": [ "I", "K", "M", "O", "P" ], "cost": 95 }, { "id": "S35", "elements": [ "N", "O", "P" ], "cost": 186 }, { "id": "S36", "elements": [ "P", "Q", "R", "S", "T", "U" ], "cost": 414 }, { "id": "S37", "elements": [ "O", "P", "S", "T" ], "cost": 152 }, { "id": "S38", "elements": [ "K", "L", "M", "N", "O", "P" ], "cost": 300 }, { "id": "S39", "elements": [ "I", "K", "L" ], "cost": 18 }, { "id": "S40", "elements": [ "B", "D", "F" ], "cost": 210 }, { "id": "S41", "elements": [ "G", "K", "L" ], "cost": 117 }, { "id": "S42", "elements": [ "I", "M", "N", "P" ], "cost": 380 }, { "id": "S43", "elements": [ "B", "D", "F", "G" ], "cost": 236 }, { "id": "S44", "elements": [ "C", "E", "F", "G", "I", "J" ], "cost": 294 }, { "id": "S45", "elements": [ "A", "C", "G" ], "cost": 129 }, { "id": "S46", "elements": [ "I", "J", "K", "L", "N", "O" ], "cost": 516 }, { "id": "S47", "elements": [ "N", "O", "Q" ], "cost": 120 }, { "id": "S48", "elements": [ "L", "M", "O", "P" ], "cost": 132 }, { "id": "S49", "elements": [ "Q", "R", "S", "T", "U" ], "cost": 50 }, { "id": "S50", "elements": [ "A", "B", "C", "D", "F" ], "cost": 475 }, { "id": "S51", "elements": [ "L", "M", "P" ], "cost": 102 }, { "id": "S52", "elements": [ "B", "D", "G" ], "cost": 36 }, { "id": "S53", "elements": [ "L", "N", "P", "Q" ], "cost": 60 }, { "id": "S54", "elements": [ "K", "N", "O" ], "cost": 93 }, { "id": "S55", "elements": [ "H", "J", "L", "M" ], "cost": 260 }, { "id": "S56", "elements": [ "B", "C", "E", "H", "I" ], "cost": 165 }, { "id": "S57", "elements": [ "H", "K", "L", "M", "O", "P" ], "cost": 174 }, { "id": "S58", "elements": [ "C", "F", "I" ], "cost": 171 }, { "id": "S59", "elements": [ "G", "I" ], "cost": 42 }, { "id": "S60", "elements": [ "H", "J", "L" ], "cost": 207 }, { "id": "S61", "elements": [ "E", "G" ], "cost": 10 }, { "id": "S62", "elements": [ "A", "B", "E" ], "cost": 24 }, { "id": "S63", "elements": [ "O", "P", "R", "S", "T", "U" ], "cost": 366 }, { "id": "S64", "elements": [ "Q", "T", "U" ], "cost": 207 }, { "id": "S65", "elements": [ "F", "G", "I", "K", "L", "O" ], "cost": 288 }, { "id": "S66", "elements": [ "L", "M", "N", "O", "P" ], "cost": 150 }, { "id": "S67", "elements": [ "C", "D", "E", "F", "G", "I" ], "cost": 342 }, { "id": "S68", "elements": [ "L", "M", "N", "O", "P", "Q" ], "cost": 150 }, { "id": "S69", "elements": [ "G", "H", "I", "L" ], "cost": 156 }, { "id": "S70", "elements": [ "K", "M", "N", "P" ], "cost": 384 }, { "id": "S71", "elements": [ "B", "C", "E", "F", "G" ], "cost": 160 }, { "id": "S72", "elements": [ "C", "D", "E", "F", "G", "H" ], "cost": 30 }, { "id": "S73", "elements": [ "A", "B", "C", "E", "F" ], "cost": 205 }, { "id": "S74", "elements": [ "J", "K", "L", "M", "O", "P" ], "cost": 450 }, { "id": "S75", "elements": [ "L", "M", "O", "P", "Q", "S" ], "cost": 120 }, { "id": "S76", "elements": [ "B", "D", "E" ], "cost": 213 }, { "id": "S77", "elements": [ "F", "I", "J", "L", "M" ], "cost": 285 }, { "id": "S78", "elements": [ "A", "B", "C" ], "cost": 66 }, { "id": "S79", "elements": [ "O", "R", "S", "T", "U" ], "cost": 15 }, { "id": "S80", "elements": [ "O", "P", "Q", "R", "S", "U" ], "cost": 522 }, { "id": "S81", "elements": [ "C", "D", "H", "I" ], "cost": 328 }, { "id": "S82", "elements": [ "C", "D", "H" ], "cost": 240 }, { "id": "S83", "elements": [ "L", "M", "O", "Q", "R", "S" ], "cost": 24 }, { "id": "S84", "elements": [ "C", "E", "F", "G" ], "cost": 224 }, { "id": "S85", "elements": [ "E", "H", "J", "K" ], "cost": 284 }, { "id": "S86", "elements": [ "O", "T" ], "cost": 78 }, { "id": "S87", "elements": [ "L", "O", "Q", "R" ], "cost": 172 }, { "id": "S88", "elements": [ "O", "P", "R", "S" ], "cost": 180 }, { "id": "S89", "elements": [ "O", "P", "Q", "S", "T" ], "cost": 430 }, { "id": "S90", "elements": [ "M", "N", "O", "P", "R", "S" ], "cost": 54 }, { "id": "S91", "elements": [ "C", "I", "J" ], "cost": 6 }, { "id": "S92", "elements": [ "H", "K", "L", "M", "N" ], "cost": 295 }, { "id": "S93", "elements": [ "H", "I", "J", "L", "M" ], "cost": 480 }, { "id": "S94", "elements": [ "A", "B", "C", "E", "H", "J" ], "cost": 66 }, { "id": "S95", "elements": [ "D", "F", "G", "H", "I" ], "cost": 325 }, { "id": "S96", "elements": [ "E", "J" ], "cost": 140 }, { "id": "S97", "elements": [ "D", "E", "F", "G", "H" ], "cost": 150 }, { "id": "S98", "elements": [ "S", "T", "U" ], "cost": 246 }, { "id": "S99", "elements": [ "L", "M", "N" ], "cost": 3 }, { "id": "S100", "elements": [ "K", "P" ], "cost": 184 }, { "id": "S101", "elements": [ "B", "D", "E", "F" ], "cost": 312 }, { "id": "S102", "elements": [ "B", "C", "D", "E", "G" ], "cost": 240 }, { "id": "S103", "elements": [ "N", "O", "Q", "U" ], "cost": 24 }, { "id": "S104", "elements": [ "M", "N", "P", "Q", "S" ], "cost": 300 }, { "id": "S105", "elements": [ "J", "L", "M", "N", "O", "Q" ], "cost": 246 }, { "id": "S106", "elements": [ "B", "C", "D", "E", "F", "G" ], "cost": 18 }, { "id": "S107", "elements": [ "D", "E", "G", "J" ], "cost": 252 }, { "id": "S108", "elements": [ "L", "N", "P", "S", "T" ], "cost": 70 }, { "id": "S109", "elements": [ "K", "L" ], "cost": 38 }, { "id": "S110", "elements": [ "K", "O", "R" ], "cost": 120 }, { "id": "S111", "elements": [ "H", "I", "J", "K", "L" ], "cost": 185 }, { "id": "S112", "elements": [ "A" ], "cost": 10000 }, { "id": "S113", "elements": [ "B" ], "cost": 10000 }, { "id": "S114", "elements": [ "C" ], "cost": 10000 }, { "id": "S115", "elements": [ "D" ], "cost": 10000 }, { "id": "S116", "elements": [ "E" ], "cost": 10000 }, { "id": "S117", "elements": [ "F" ], "cost": 10000 }, { "id": "S118", "elements": [ "G" ], "cost": 10000 }, { "id": "S119", "elements": [ "H" ], "cost": 10000 }, { "id": "S120", "elements": [ "I" ], "cost": 10000 }, { "id": "S121", "elements": [ "J" ], "cost": 10000 }, { "id": "S122", "elements": [ "K" ], "cost": 10000 }, { "id": "S123", "elements": [ "L" ], "cost": 10000 }, { "id": "S124", "elements": [ "M" ], "cost": 10000 }, { "id": "S125", "elements": [ "N" ], "cost": 10000 }, { "id": "S126", "elements": [ "O" ], "cost": 10000 }, { "id": "S127", "elements": [ "P" ], "cost": 10000 }, { "id": "S128", "elements": [ "Q" ], "cost": 10000 }, { "id": "S129", "elements": [ "R" ], "cost": 10000 }, { "id": "S130", "elements": [ "S" ], "cost": 10000 }, { "id": "S131", "elements": [ "T" ], "cost": 10000 }, { "id": "S132", "elements": [ "U" ], "cost": 10000 } ] }, "solution_variant": [ "S29", "S31", "S47", "S78", "S97", "S100" ], "context_index": 17, "input_format": "markdown_table", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "There’s a pile of ad placement bundles on the table and a grid of ad slots that all need attention. Each bundle covers some of the slots and has a sticker price, so the task is to pick bundles that together cover every slot exactly once, and to keep the total bill down by summing the prices of the bundles chosen. The specific bundles, which slots they cover, and their costs are shown below.\n\n# total_ad_slots=19\n# total_bundles_available=87\nbundle_id,bundle_price,slots_covered\nS1,123,F G H\nS2,100,N O P Q\nS3,160,J K L M N\nS4,76,N O Q R\nS5,376,C D F G\nS6,276,M N Q\nS7,2,A B\nS8,189,P R S\nS9,48,N O P Q R S\nS10,10,E F G H J\nS11,294,C E H\nS12,288,J L M O\nS13,72,L P\nS14,12,D E F H\nS15,82,J K\nS16,490,K L M N O\nS17,12,I L\nS18,70,A B C D F\nS19,176,E F G H\nS20,355,M O P Q R\nS21,415,D E F G J\nS22,116,A B C D\nS23,255,C D E F H\nS24,225,N O P Q R\nS25,5,N P Q R S\nS26,165,I J K L M\nS27,174,J L M\nS28,30,O R\nS29,52,E F\nS30,35,H J K L M\nS31,160,F H I K\nS32,186,D H\nS33,189,J K M\nS34,56,G J L M\nS35,245,A B C D E\nS36,240,D F G H J\nS37,10,M O\nS38,80,H J L M N\nS39,36,N Q R S\nS40,300,A C D E F\nS41,297,A B C\nS42,125,H I K L N\nS43,219,L Q S\nS44,185,M N O P Q\nS45,255,J K L\nS46,240,O P Q R S\nS47,172,M N O Q\nS48,90,B E F G H\nS49,402,F G H I J L\nS50,44,A C\nS51,335,I J K M O\nS52,56,L M N O\nS53,276,M P Q\nS54,20,P R\nS55,125,F I J K L\nS56,10,A B C E G\nS57,410,H I J L N\nS58,132,A B C F\nS59,440,L M N O P\nS60,132,A B C E\nS61,196,A B D E\nS62,290,F G H I K\nS63,261,L N O\nS64,252,O P R\nS65,150,J L M O P\nS66,285,B C E F H\nS67,56,J M N O\nS68,122,I K\nS69,10000,A\nS70,10000,B\nS71,10000,C\nS72,10000,D\nS73,10000,E\nS74,10000,F\nS75,10000,G\nS76,10000,H\nS77,10000,I\nS78,10000,J\nS79,10000,K\nS80,10000,L\nS81,10000,M\nS82,10000,N\nS83,10000,O\nS84,10000,P\nS85,10000,Q\nS86,10000,R\nS87,10000,S\n\nIf you want to reply with the chosen bundles, just follow this little JSON shape so it's easy to read and parse:\n\n{\n \"solution\": [\"bundle_id\", ...]\n}\n\n\"solution\" is the list of bundle IDs you pick (each entry is the ID of one ad bundle that will cover some slots). The \"bundle_id\" text is a placeholder showing the kind of values you should put there — replace each placeholder with the actual bundle identifier from the instance when you answer. This JSON is just the sketch of the shape I want you to use, not the final selection itself.\n\nPlease make sure to use the exact identifiers from the instance input — don't rename them or invent new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.", "instance": { "problem_type": "SPP", "num_elements": 19, "num_sets": 87, "density": 0.17422867513611615, "sets": [ { "id": 1, "elements": [ 6, 7, 8 ], "cost": 123 }, { "id": 2, "elements": [ 14, 15, 16, 17 ], "cost": 100 }, { "id": 3, "elements": [ 10, 11, 12, 13, 14 ], "cost": 160 }, { "id": 4, "elements": [ 14, 15, 17, 18 ], "cost": 76 }, { "id": 5, "elements": [ 3, 4, 6, 7 ], "cost": 376 }, { "id": 6, "elements": [ 13, 14, 17 ], "cost": 276 }, { "id": 7, "elements": [ 1, 2 ], "cost": 2 }, { "id": 8, "elements": [ 16, 18, 19 ], "cost": 189 }, { "id": 9, "elements": [ 14, 15, 16, 17, 18, 19 ], "cost": 48 }, { "id": 10, "elements": [ 5, 6, 7, 8, 10 ], "cost": 10 }, { "id": 11, "elements": [ 3, 5, 8 ], "cost": 294 }, { "id": 12, "elements": [ 10, 12, 13, 15 ], "cost": 288 }, { "id": 13, "elements": [ 12, 16 ], "cost": 72 }, { "id": 14, "elements": [ 4, 5, 6, 8 ], "cost": 12 }, { "id": 15, "elements": [ 10, 11 ], "cost": 82 }, { "id": 16, "elements": [ 11, 12, 13, 14, 15 ], "cost": 490 }, { "id": 17, "elements": [ 9, 12 ], "cost": 12 }, { "id": 18, "elements": [ 1, 2, 3, 4, 6 ], "cost": 70 }, { "id": 19, "elements": [ 5, 6, 7, 8 ], "cost": 176 }, { "id": 20, "elements": [ 13, 15, 16, 17, 18 ], "cost": 355 }, { "id": 21, "elements": [ 4, 5, 6, 7, 10 ], "cost": 415 }, { "id": 22, "elements": [ 1, 2, 3, 4 ], "cost": 116 }, { "id": 23, "elements": [ 3, 4, 5, 6, 8 ], "cost": 255 }, { "id": 24, "elements": [ 14, 15, 16, 17, 18 ], "cost": 225 }, { "id": 25, "elements": [ 14, 16, 17, 18, 19 ], "cost": 5 }, { "id": 26, "elements": [ 9, 10, 11, 12, 13 ], "cost": 165 }, { "id": 27, "elements": [ 10, 12, 13 ], "cost": 174 }, { "id": 28, "elements": [ 15, 18 ], "cost": 30 }, { "id": 29, "elements": [ 5, 6 ], "cost": 52 }, { "id": 30, "elements": [ 8, 10, 11, 12, 13 ], "cost": 35 }, { "id": 31, "elements": [ 6, 8, 9, 11 ], "cost": 160 }, { "id": 32, "elements": [ 4, 8 ], "cost": 186 }, { "id": 33, "elements": [ 10, 11, 13 ], "cost": 189 }, { "id": 34, "elements": [ 7, 10, 12, 13 ], "cost": 56 }, { "id": 35, "elements": [ 1, 2, 3, 4, 5 ], "cost": 245 }, { "id": 36, "elements": [ 4, 6, 7, 8, 10 ], "cost": 240 }, { "id": 37, "elements": [ 13, 15 ], "cost": 10 }, { "id": 38, "elements": [ 8, 10, 12, 13, 14 ], "cost": 80 }, { "id": 39, "elements": [ 14, 17, 18, 19 ], "cost": 36 }, { "id": 40, "elements": [ 1, 3, 4, 5, 6 ], "cost": 300 }, { "id": 41, "elements": [ 1, 2, 3 ], "cost": 297 }, { "id": 42, "elements": [ 8, 9, 11, 12, 14 ], "cost": 125 }, { "id": 43, "elements": [ 12, 17, 19 ], "cost": 219 }, { "id": 44, "elements": [ 13, 14, 15, 16, 17 ], "cost": 185 }, { "id": 45, "elements": [ 10, 11, 12 ], "cost": 255 }, { "id": 46, "elements": [ 15, 16, 17, 18, 19 ], "cost": 240 }, { "id": 47, "elements": [ 13, 14, 15, 17 ], "cost": 172 }, { "id": 48, "elements": [ 2, 5, 6, 7, 8 ], "cost": 90 }, { "id": 49, "elements": [ 6, 7, 8, 9, 10, 12 ], "cost": 402 }, { "id": 50, "elements": [ 1, 3 ], "cost": 44 }, { "id": 51, "elements": [ 9, 10, 11, 13, 15 ], "cost": 335 }, { "id": 52, "elements": [ 12, 13, 14, 15 ], "cost": 56 }, { "id": 53, "elements": [ 13, 16, 17 ], "cost": 276 }, { "id": 54, "elements": [ 16, 18 ], "cost": 20 }, { "id": 55, "elements": [ 6, 9, 10, 11, 12 ], "cost": 125 }, { "id": 56, "elements": [ 1, 2, 3, 5, 7 ], "cost": 10 }, { "id": 57, "elements": [ 8, 9, 10, 12, 14 ], "cost": 410 }, { "id": 58, "elements": [ 1, 2, 3, 6 ], "cost": 132 }, { "id": 59, "elements": [ 12, 13, 14, 15, 16 ], "cost": 440 }, { "id": 60, "elements": [ 1, 2, 3, 5 ], "cost": 132 }, { "id": 61, "elements": [ 1, 2, 4, 5 ], "cost": 196 }, { "id": 62, "elements": [ 6, 7, 8, 9, 11 ], "cost": 290 }, { "id": 63, "elements": [ 12, 14, 15 ], "cost": 261 }, { "id": 64, "elements": [ 15, 16, 18 ], "cost": 252 }, { "id": 65, "elements": [ 10, 12, 13, 15, 16 ], "cost": 150 }, { "id": 66, "elements": [ 2, 3, 5, 6, 8 ], "cost": 285 }, { "id": 67, "elements": [ 10, 13, 14, 15 ], "cost": 56 }, { "id": 68, "elements": [ 9, 11 ], "cost": 122 }, { "id": 69, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 70, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 71, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 72, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 73, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 74, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 75, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 76, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 77, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 78, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 79, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 80, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 81, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 82, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 83, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 84, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 85, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 86, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 87, "elements": [ 19 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0017_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0017_bag_stats.png" }, "solution": [ 25, 32, 37, 55, 56 ], "obj": 336.0, "instance_variant": { "num_elements": 19, "num_sets": 87, "sets": [ { "id": "S1", "elements": [ "F", "G", "H" ], "cost": 123 }, { "id": "S2", "elements": [ "N", "O", "P", "Q" ], "cost": 100 }, { "id": "S3", "elements": [ "J", "K", "L", "M", "N" ], "cost": 160 }, { "id": "S4", "elements": [ "N", "O", "Q", "R" ], "cost": 76 }, { "id": "S5", "elements": [ "C", "D", "F", "G" ], "cost": 376 }, { "id": "S6", "elements": [ "M", "N", "Q" ], "cost": 276 }, { "id": "S7", "elements": [ "A", "B" ], "cost": 2 }, { "id": "S8", "elements": [ "P", "R", "S" ], "cost": 189 }, { "id": "S9", "elements": [ "N", "O", "P", "Q", "R", "S" ], "cost": 48 }, { "id": "S10", "elements": [ "E", "F", "G", "H", "J" ], "cost": 10 }, { "id": "S11", "elements": [ "C", "E", "H" ], "cost": 294 }, { "id": "S12", "elements": [ "J", "L", "M", "O" ], "cost": 288 }, { "id": "S13", "elements": [ "L", "P" ], "cost": 72 }, { "id": "S14", "elements": [ "D", "E", "F", "H" ], "cost": 12 }, { "id": "S15", "elements": [ "J", "K" ], "cost": 82 }, { "id": "S16", "elements": [ "K", "L", "M", "N", "O" ], "cost": 490 }, { "id": "S17", "elements": [ "I", "L" ], "cost": 12 }, { "id": "S18", "elements": [ "A", "B", "C", "D", "F" ], "cost": 70 }, { "id": "S19", "elements": [ "E", "F", "G", "H" ], "cost": 176 }, { "id": "S20", "elements": [ "M", "O", "P", "Q", "R" ], "cost": 355 }, { "id": "S21", "elements": [ "D", "E", "F", "G", "J" ], "cost": 415 }, { "id": "S22", "elements": [ "A", "B", "C", "D" ], "cost": 116 }, { "id": "S23", "elements": [ "C", "D", "E", "F", "H" ], "cost": 255 }, { "id": "S24", "elements": [ "N", "O", "P", "Q", "R" ], "cost": 225 }, { "id": "S25", "elements": [ "N", "P", "Q", "R", "S" ], "cost": 5 }, { "id": "S26", "elements": [ "I", "J", "K", "L", "M" ], "cost": 165 }, { "id": "S27", "elements": [ "J", "L", "M" ], "cost": 174 }, { "id": "S28", "elements": [ "O", "R" ], "cost": 30 }, { "id": "S29", "elements": [ "E", "F" ], "cost": 52 }, { "id": "S30", "elements": [ "H", "J", "K", "L", "M" ], "cost": 35 }, { "id": "S31", "elements": [ "F", "H", "I", "K" ], "cost": 160 }, { "id": "S32", "elements": [ "D", "H" ], "cost": 186 }, { "id": "S33", "elements": [ "J", "K", "M" ], "cost": 189 }, { "id": "S34", "elements": [ "G", "J", "L", "M" ], "cost": 56 }, { "id": "S35", "elements": [ "A", "B", "C", "D", "E" ], "cost": 245 }, { "id": "S36", "elements": [ "D", "F", "G", "H", "J" ], "cost": 240 }, { "id": "S37", "elements": [ "M", "O" ], "cost": 10 }, { "id": "S38", "elements": [ "H", "J", "L", "M", "N" ], "cost": 80 }, { "id": "S39", "elements": [ "N", "Q", "R", "S" ], "cost": 36 }, { "id": "S40", "elements": [ "A", "C", "D", "E", "F" ], "cost": 300 }, { "id": "S41", "elements": [ "A", "B", "C" ], "cost": 297 }, { "id": "S42", "elements": [ "H", "I", "K", "L", "N" ], "cost": 125 }, { "id": "S43", "elements": [ "L", "Q", "S" ], "cost": 219 }, { "id": "S44", "elements": [ "M", "N", "O", "P", "Q" ], "cost": 185 }, { "id": "S45", "elements": [ "J", "K", "L" ], "cost": 255 }, { "id": "S46", "elements": [ "O", "P", "Q", "R", "S" ], "cost": 240 }, { "id": "S47", "elements": [ "M", "N", "O", "Q" ], "cost": 172 }, { "id": "S48", "elements": [ "B", "E", "F", "G", "H" ], "cost": 90 }, { "id": "S49", "elements": [ "F", "G", "H", "I", "J", "L" ], "cost": 402 }, { "id": "S50", "elements": [ "A", "C" ], "cost": 44 }, { "id": "S51", "elements": [ "I", "J", "K", "M", "O" ], "cost": 335 }, { "id": "S52", "elements": [ "L", "M", "N", "O" ], "cost": 56 }, { "id": "S53", "elements": [ "M", "P", "Q" ], "cost": 276 }, { "id": "S54", "elements": [ "P", "R" ], "cost": 20 }, { "id": "S55", "elements": [ "F", "I", "J", "K", "L" ], "cost": 125 }, { "id": "S56", "elements": [ "A", "B", "C", "E", "G" ], "cost": 10 }, { "id": "S57", "elements": [ "H", "I", "J", "L", "N" ], "cost": 410 }, { "id": "S58", "elements": [ "A", "B", "C", "F" ], "cost": 132 }, { "id": "S59", "elements": [ "L", "M", "N", "O", "P" ], "cost": 440 }, { "id": "S60", "elements": [ "A", "B", "C", "E" ], "cost": 132 }, { "id": "S61", "elements": [ "A", "B", "D", "E" ], "cost": 196 }, { "id": "S62", "elements": [ "F", "G", "H", "I", "K" ], "cost": 290 }, { "id": "S63", "elements": [ "L", "N", "O" ], "cost": 261 }, { "id": "S64", "elements": [ "O", "P", "R" ], "cost": 252 }, { "id": "S65", "elements": [ "J", "L", "M", "O", "P" ], "cost": 150 }, { "id": "S66", "elements": [ "B", "C", "E", "F", "H" ], "cost": 285 }, { "id": "S67", "elements": [ "J", "M", "N", "O" ], "cost": 56 }, { "id": "S68", "elements": [ "I", "K" ], "cost": 122 }, { "id": "S69", "elements": [ "A" ], "cost": 10000 }, { "id": "S70", "elements": [ "B" ], "cost": 10000 }, { "id": "S71", "elements": [ "C" ], "cost": 10000 }, { "id": "S72", "elements": [ "D" ], "cost": 10000 }, { "id": "S73", "elements": [ "E" ], "cost": 10000 }, { "id": "S74", "elements": [ "F" ], "cost": 10000 }, { "id": "S75", "elements": [ "G" ], "cost": 10000 }, { "id": "S76", "elements": [ "H" ], "cost": 10000 }, { "id": "S77", "elements": [ "I" ], "cost": 10000 }, { "id": "S78", "elements": [ "J" ], "cost": 10000 }, { "id": "S79", "elements": [ "K" ], "cost": 10000 }, { "id": "S80", "elements": [ "L" ], "cost": 10000 }, { "id": "S81", "elements": [ "M" ], "cost": 10000 }, { "id": "S82", "elements": [ "N" ], "cost": 10000 }, { "id": "S83", "elements": [ "O" ], "cost": 10000 }, { "id": "S84", "elements": [ "P" ], "cost": 10000 }, { "id": "S85", "elements": [ "Q" ], "cost": 10000 }, { "id": "S86", "elements": [ "R" ], "cost": 10000 }, { "id": "S87", "elements": [ "S" ], "cost": 10000 } ] }, "solution_variant": [ "S25", "S32", "S37", "S55", "S56" ], "context_index": 18, "input_format": "csv", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "There’s a production puzzle on the table: a stack of components that all need to be produced and a bunch of candidate batch runs that each make some of those components. The job is to pick a selection of those runs so every component is produced exactly once — no leftovers, no duplicates. Each run carries a cost, so once a selection is made the total expense is simply all those run costs added together, and the aim is to keep that total as small as possible. The specific parts, available batches, and their prices appear below.\n\n- **total_components**: 22\n- **total_candidate_batches**: 125\n\n| batch_id | batch_cost | produced_components |\n|---|---|---|\n| S1 | 258 | 3 4 6 |\n| S2 | 27 | 19 20 22 |\n| S3 | 540 | 16 18 19 20 21 22 |\n| S4 | 300 | 9 10 13 14 |\n| S5 | 125 | 8 10 11 12 13 |\n| S6 | 50 | 13 15 16 17 18 |\n| S7 | 186 | 3 6 7 |\n| S8 | 188 | 2 4 5 6 |\n| S9 | 558 | 5 6 7 8 9 11 |\n| S10 | 396 | 11 12 13 15 17 18 |\n| S11 | 462 | 15 18 19 20 21 22 |\n| S12 | 200 | 4 5 |\n| S13 | 252 | 17 19 20 |\n| S14 | 300 | 9 10 11 12 13 16 |\n| S15 | 72 | 16 17 18 19 20 22 |\n| S16 | 185 | 7 11 12 13 15 |\n| S17 | 200 | 6 7 8 10 11 |\n| S18 | 165 | 9 10 11 |\n| S19 | 440 | 4 5 8 9 10 |\n| S20 | 465 | 14 16 17 18 19 |\n| S21 | 288 | 17 18 19 20 21 22 |\n| S22 | 132 | 15 17 19 |\n| S23 | 213 | 17 18 20 |\n| S24 | 462 | 1 2 3 4 5 6 7 |\n| S25 | 280 | 3 6 7 8 12 |\n| S26 | 24 | 2 3 6 10 |\n| S27 | 36 | 6 7 8 9 10 12 |\n| S28 | 93 | 18 20 21 |\n| S29 | 150 | 15 18 20 |\n| S30 | 395 | 6 7 8 9 11 |\n| S31 | 258 | 1 5 9 |\n| S32 | 44 | 16 18 20 21 |\n| S33 | 196 | 14 15 16 17 |\n| S34 | 112 | 7 10 |\n| S35 | 114 | 12 14 15 16 17 18 |\n| S36 | 230 | 17 18 19 20 21 |\n| S37 | 16 | 7 9 10 11 |\n| S38 | 224 | 9 12 13 14 |\n| S39 | 63 | 6 7 11 |\n| S40 | 15 | 19 20 21 |\n| S41 | 476 | 8 9 10 11 12 13 14 |\n| S42 | 132 | 12 13 14 15 16 18 |\n| S43 | 380 | 7 8 10 11 |\n| S44 | 9 | 15 17 18 |\n| S45 | 600 | 4 6 7 8 9 10 |\n| S46 | 8 | 10 11 13 16 |\n| S47 | 32 | 2 5 6 8 |\n| S48 | 192 | 2 4 5 9 |\n| S49 | 390 | 6 8 9 10 11 |\n| S50 | 426 | 1 3 4 5 6 8 |\n| S51 | 370 | 12 13 14 15 16 |\n| S52 | 25 | 11 12 13 14 15 |\n| S53 | 159 | 17 21 22 |\n| S54 | 224 | 5 8 9 10 |\n| S55 | 114 | 2 3 5 6 7 8 |\n| S56 | 104 | 6 11 |\n| S57 | 128 | 12 13 14 19 |\n| S58 | 248 | 15 19 20 21 |\n| S59 | 44 | 9 14 |\n| S60 | 70 | 2 6 |\n| S61 | 8 | 15 17 |\n| S62 | 28 | 9 10 12 13 |\n| S63 | 564 | 14 16 18 19 20 21 |\n| S64 | 285 | 14 16 18 20 21 |\n| S65 | 255 | 17 18 19 20 22 |\n| S66 | 219 | 1 2 6 |\n| S67 | 81 | 5 6 10 |\n| S68 | 188 | 1 2 |\n| S69 | 60 | 1 2 5 6 |\n| S70 | 144 | 14 16 17 |\n| S71 | 120 | 6 8 9 10 11 12 |\n| S72 | 15 | 16 17 19 20 22 |\n| S73 | 208 | 8 9 10 11 |\n| S74 | 28 | 18 21 |\n| S75 | 540 | 16 17 19 20 21 22 |\n| S76 | 400 | 9 10 11 12 13 |\n| S77 | 224 | 18 20 21 22 |\n| S78 | 120 | 15 16 20 |\n| S79 | 168 | 14 15 19 |\n| S80 | 78 | 8 9 10 11 12 14 |\n| S81 | 270 | 1 3 5 |\n| S82 | 440 | 1 2 3 4 8 |\n| S83 | 66 | 8 9 10 11 13 18 |\n| S84 | 266 | 12 16 17 19 20 21 22 |\n| S85 | 105 | 14 15 17 18 20 |\n| S86 | 430 | 13 14 18 19 20 |\n| S87 | 114 | 15 16 18 |\n| S88 | 228 | 12 13 14 |\n| S89 | 450 | 2 3 4 5 7 8 |\n| S90 | 258 | 13 16 17 |\n| S91 | 96 | 8 9 11 14 |\n| S92 | 546 | 12 13 14 15 16 17 |\n| S93 | 88 | 8 12 13 15 |\n| S94 | 384 | 10 11 12 13 15 17 |\n| S95 | 288 | 5 6 7 8 9 10 |\n| S96 | 172 | 2 3 4 6 |\n| S97 | 130 | 16 17 18 19 21 |\n| S98 | 388 | 1 2 3 4 |\n| S99 | 164 | 12 14 16 17 |\n| S100 | 198 | 4 6 7 8 10 12 |\n| S101 | 204 | 1 2 3 6 |\n| S102 | 72 | 4 6 |\n| S103 | 200 | 13 18 19 20 21 |\n| S104 | 10000 | 1 |\n| S105 | 10000 | 2 |\n| S106 | 10000 | 3 |\n| S107 | 10000 | 4 |\n| S108 | 10000 | 5 |\n| S109 | 10000 | 6 |\n| S110 | 10000 | 7 |\n| S111 | 10000 | 8 |\n| S112 | 10000 | 9 |\n| S113 | 10000 | 10 |\n| S114 | 10000 | 11 |\n| S115 | 10000 | 12 |\n| S116 | 10000 | 13 |\n| S117 | 10000 | 14 |\n| S118 | 10000 | 15 |\n| S119 | 10000 | 16 |\n| S120 | 10000 | 17 |\n| S121 | 10000 | 18 |\n| S122 | 10000 | 19 |\n| S123 | 10000 | 20 |\n| S124 | 10000 | 21 |\n| S125 | 10000 | 22 |\n\nOh, and when you give the final pick, please stick to this little JSON layout so it's easy to read and check:\n\n{\n \"solution\": [\"batch_id\", ...]\n}\n\nThis is just a sketch of the shape I expect: \"solution\" holds a list of the batch/run identifiers you picked (one per produced component). Think of \"batch_id\" as a placeholder for whatever run label appears in the instance — it's just showing where each chosen run goes. The JSON is only the shape, not the actual answer.\n\nQuick heads-up: use the identifiers exactly as they appear in the instance input — no renaming, no made-up labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.", "instance": { "problem_type": "SPP", "num_elements": 22, "num_sets": 125, "density": 0.17054545454545456, "sets": [ { "id": 1, "elements": [ 3, 4, 6 ], "cost": 258 }, { "id": 2, "elements": [ 19, 20, 22 ], "cost": 27 }, { "id": 3, "elements": [ 16, 18, 19, 20, 21, 22 ], "cost": 540 }, { "id": 4, "elements": [ 9, 10, 13, 14 ], "cost": 300 }, { "id": 5, "elements": [ 8, 10, 11, 12, 13 ], "cost": 125 }, { "id": 6, "elements": [ 13, 15, 16, 17, 18 ], "cost": 50 }, { "id": 7, "elements": [ 3, 6, 7 ], "cost": 186 }, { "id": 8, "elements": [ 2, 4, 5, 6 ], "cost": 188 }, { "id": 9, "elements": [ 5, 6, 7, 8, 9, 11 ], "cost": 558 }, { "id": 10, "elements": [ 11, 12, 13, 15, 17, 18 ], "cost": 396 }, { "id": 11, "elements": [ 15, 18, 19, 20, 21, 22 ], "cost": 462 }, { "id": 12, "elements": [ 4, 5 ], "cost": 200 }, { "id": 13, "elements": [ 17, 19, 20 ], "cost": 252 }, { "id": 14, "elements": [ 9, 10, 11, 12, 13, 16 ], "cost": 300 }, { "id": 15, "elements": [ 16, 17, 18, 19, 20, 22 ], "cost": 72 }, { "id": 16, "elements": [ 7, 11, 12, 13, 15 ], "cost": 185 }, { "id": 17, "elements": [ 6, 7, 8, 10, 11 ], "cost": 200 }, { "id": 18, "elements": [ 9, 10, 11 ], "cost": 165 }, { "id": 19, "elements": [ 4, 5, 8, 9, 10 ], "cost": 440 }, { "id": 20, "elements": [ 14, 16, 17, 18, 19 ], "cost": 465 }, { "id": 21, "elements": [ 17, 18, 19, 20, 21, 22 ], "cost": 288 }, { "id": 22, "elements": [ 15, 17, 19 ], "cost": 132 }, { "id": 23, "elements": [ 17, 18, 20 ], "cost": 213 }, { "id": 24, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 462 }, { "id": 25, "elements": [ 3, 6, 7, 8, 12 ], "cost": 280 }, { "id": 26, "elements": [ 2, 3, 6, 10 ], "cost": 24 }, { "id": 27, "elements": [ 6, 7, 8, 9, 10, 12 ], "cost": 36 }, { "id": 28, "elements": [ 18, 20, 21 ], "cost": 93 }, { "id": 29, "elements": [ 15, 18, 20 ], "cost": 150 }, { "id": 30, "elements": [ 6, 7, 8, 9, 11 ], "cost": 395 }, { "id": 31, "elements": [ 1, 5, 9 ], "cost": 258 }, { "id": 32, "elements": [ 16, 18, 20, 21 ], "cost": 44 }, { "id": 33, "elements": [ 14, 15, 16, 17 ], "cost": 196 }, { "id": 34, "elements": [ 7, 10 ], "cost": 112 }, { "id": 35, "elements": [ 12, 14, 15, 16, 17, 18 ], "cost": 114 }, { "id": 36, "elements": [ 17, 18, 19, 20, 21 ], "cost": 230 }, { "id": 37, "elements": [ 7, 9, 10, 11 ], "cost": 16 }, { "id": 38, "elements": [ 9, 12, 13, 14 ], "cost": 224 }, { "id": 39, "elements": [ 6, 7, 11 ], "cost": 63 }, { "id": 40, "elements": [ 19, 20, 21 ], "cost": 15 }, { "id": 41, "elements": [ 8, 9, 10, 11, 12, 13, 14 ], "cost": 476 }, { "id": 42, "elements": [ 12, 13, 14, 15, 16, 18 ], "cost": 132 }, { "id": 43, "elements": [ 7, 8, 10, 11 ], "cost": 380 }, { "id": 44, "elements": [ 15, 17, 18 ], "cost": 9 }, { "id": 45, "elements": [ 4, 6, 7, 8, 9, 10 ], "cost": 600 }, { "id": 46, "elements": [ 10, 11, 13, 16 ], "cost": 8 }, { "id": 47, "elements": [ 2, 5, 6, 8 ], "cost": 32 }, { "id": 48, "elements": [ 2, 4, 5, 9 ], "cost": 192 }, { "id": 49, "elements": [ 6, 8, 9, 10, 11 ], "cost": 390 }, { "id": 50, "elements": [ 1, 3, 4, 5, 6, 8 ], "cost": 426 }, { "id": 51, "elements": [ 12, 13, 14, 15, 16 ], "cost": 370 }, { "id": 52, "elements": [ 11, 12, 13, 14, 15 ], "cost": 25 }, { "id": 53, "elements": [ 17, 21, 22 ], "cost": 159 }, { "id": 54, "elements": [ 5, 8, 9, 10 ], "cost": 224 }, { "id": 55, "elements": [ 2, 3, 5, 6, 7, 8 ], "cost": 114 }, { "id": 56, "elements": [ 6, 11 ], "cost": 104 }, { "id": 57, "elements": [ 12, 13, 14, 19 ], "cost": 128 }, { "id": 58, "elements": [ 15, 19, 20, 21 ], "cost": 248 }, { "id": 59, "elements": [ 9, 14 ], "cost": 44 }, { "id": 60, "elements": [ 2, 6 ], "cost": 70 }, { "id": 61, "elements": [ 15, 17 ], "cost": 8 }, { "id": 62, "elements": [ 9, 10, 12, 13 ], "cost": 28 }, { "id": 63, "elements": [ 14, 16, 18, 19, 20, 21 ], "cost": 564 }, { "id": 64, "elements": [ 14, 16, 18, 20, 21 ], "cost": 285 }, { "id": 65, "elements": [ 17, 18, 19, 20, 22 ], "cost": 255 }, { "id": 66, "elements": [ 1, 2, 6 ], "cost": 219 }, { "id": 67, "elements": [ 5, 6, 10 ], "cost": 81 }, { "id": 68, "elements": [ 1, 2 ], "cost": 188 }, { "id": 69, "elements": [ 1, 2, 5, 6 ], "cost": 60 }, { "id": 70, "elements": [ 14, 16, 17 ], "cost": 144 }, { "id": 71, "elements": [ 6, 8, 9, 10, 11, 12 ], "cost": 120 }, { "id": 72, "elements": [ 16, 17, 19, 20, 22 ], "cost": 15 }, { "id": 73, "elements": [ 8, 9, 10, 11 ], "cost": 208 }, { "id": 74, "elements": [ 18, 21 ], "cost": 28 }, { "id": 75, "elements": [ 16, 17, 19, 20, 21, 22 ], "cost": 540 }, { "id": 76, "elements": [ 9, 10, 11, 12, 13 ], "cost": 400 }, { "id": 77, "elements": [ 18, 20, 21, 22 ], "cost": 224 }, { "id": 78, "elements": [ 15, 16, 20 ], "cost": 120 }, { "id": 79, "elements": [ 14, 15, 19 ], "cost": 168 }, { "id": 80, "elements": [ 8, 9, 10, 11, 12, 14 ], "cost": 78 }, { "id": 81, "elements": [ 1, 3, 5 ], "cost": 270 }, { "id": 82, "elements": [ 1, 2, 3, 4, 8 ], "cost": 440 }, { "id": 83, "elements": [ 8, 9, 10, 11, 13, 18 ], "cost": 66 }, { "id": 84, "elements": [ 12, 16, 17, 19, 20, 21, 22 ], "cost": 266 }, { "id": 85, "elements": [ 14, 15, 17, 18, 20 ], "cost": 105 }, { "id": 86, "elements": [ 13, 14, 18, 19, 20 ], "cost": 430 }, { "id": 87, "elements": [ 15, 16, 18 ], "cost": 114 }, { "id": 88, "elements": [ 12, 13, 14 ], "cost": 228 }, { "id": 89, "elements": [ 2, 3, 4, 5, 7, 8 ], "cost": 450 }, { "id": 90, "elements": [ 13, 16, 17 ], "cost": 258 }, { "id": 91, "elements": [ 8, 9, 11, 14 ], "cost": 96 }, { "id": 92, "elements": [ 12, 13, 14, 15, 16, 17 ], "cost": 546 }, { "id": 93, "elements": [ 8, 12, 13, 15 ], "cost": 88 }, { "id": 94, "elements": [ 10, 11, 12, 13, 15, 17 ], "cost": 384 }, { "id": 95, "elements": [ 5, 6, 7, 8, 9, 10 ], "cost": 288 }, { "id": 96, "elements": [ 2, 3, 4, 6 ], "cost": 172 }, { "id": 97, "elements": [ 16, 17, 18, 19, 21 ], "cost": 130 }, { "id": 98, "elements": [ 1, 2, 3, 4 ], "cost": 388 }, { "id": 99, "elements": [ 12, 14, 16, 17 ], "cost": 164 }, { "id": 100, "elements": [ 4, 6, 7, 8, 10, 12 ], "cost": 198 }, { "id": 101, "elements": [ 1, 2, 3, 6 ], "cost": 204 }, { "id": 102, "elements": [ 4, 6 ], "cost": 72 }, { "id": 103, "elements": [ 13, 18, 19, 20, 21 ], "cost": 200 }, { "id": 104, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 118, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 119, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 120, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 121, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 122, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 123, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 124, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 125, "elements": [ 22 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0018_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0018_bag_stats.png" }, "solution": [ 2, 12, 37, 70, 74, 93, 101 ], "obj": 707.0, "instance_variant": { "num_elements": 22, "num_sets": 125, "sets": [ { "id": "S1", "elements": [ 3, 4, 6 ], "cost": 258 }, { "id": "S2", "elements": [ 19, 20, 22 ], "cost": 27 }, { "id": "S3", "elements": [ 16, 18, 19, 20, 21, 22 ], "cost": 540 }, { "id": "S4", "elements": [ 9, 10, 13, 14 ], "cost": 300 }, { "id": "S5", "elements": [ 8, 10, 11, 12, 13 ], "cost": 125 }, { "id": "S6", "elements": [ 13, 15, 16, 17, 18 ], "cost": 50 }, { "id": "S7", "elements": [ 3, 6, 7 ], "cost": 186 }, { "id": "S8", "elements": [ 2, 4, 5, 6 ], "cost": 188 }, { "id": "S9", "elements": [ 5, 6, 7, 8, 9, 11 ], "cost": 558 }, { "id": "S10", "elements": [ 11, 12, 13, 15, 17, 18 ], "cost": 396 }, { "id": "S11", "elements": [ 15, 18, 19, 20, 21, 22 ], "cost": 462 }, { "id": "S12", "elements": [ 4, 5 ], "cost": 200 }, { "id": "S13", "elements": [ 17, 19, 20 ], "cost": 252 }, { "id": "S14", "elements": [ 9, 10, 11, 12, 13, 16 ], "cost": 300 }, { "id": "S15", "elements": [ 16, 17, 18, 19, 20, 22 ], "cost": 72 }, { "id": "S16", "elements": [ 7, 11, 12, 13, 15 ], "cost": 185 }, { "id": "S17", "elements": [ 6, 7, 8, 10, 11 ], "cost": 200 }, { "id": "S18", "elements": [ 9, 10, 11 ], "cost": 165 }, { "id": "S19", "elements": [ 4, 5, 8, 9, 10 ], "cost": 440 }, { "id": "S20", "elements": [ 14, 16, 17, 18, 19 ], "cost": 465 }, { "id": "S21", "elements": [ 17, 18, 19, 20, 21, 22 ], "cost": 288 }, { "id": "S22", "elements": [ 15, 17, 19 ], "cost": 132 }, { "id": "S23", "elements": [ 17, 18, 20 ], "cost": 213 }, { "id": "S24", "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 462 }, { "id": "S25", "elements": [ 3, 6, 7, 8, 12 ], "cost": 280 }, { "id": "S26", "elements": [ 2, 3, 6, 10 ], "cost": 24 }, { "id": "S27", "elements": [ 6, 7, 8, 9, 10, 12 ], "cost": 36 }, { "id": "S28", "elements": [ 18, 20, 21 ], "cost": 93 }, { "id": "S29", "elements": [ 15, 18, 20 ], "cost": 150 }, { "id": "S30", "elements": [ 6, 7, 8, 9, 11 ], "cost": 395 }, { "id": "S31", "elements": [ 1, 5, 9 ], "cost": 258 }, { "id": "S32", "elements": [ 16, 18, 20, 21 ], "cost": 44 }, { "id": "S33", "elements": [ 14, 15, 16, 17 ], "cost": 196 }, { "id": "S34", "elements": [ 7, 10 ], "cost": 112 }, { "id": "S35", "elements": [ 12, 14, 15, 16, 17, 18 ], "cost": 114 }, { "id": "S36", "elements": [ 17, 18, 19, 20, 21 ], "cost": 230 }, { "id": "S37", "elements": [ 7, 9, 10, 11 ], "cost": 16 }, { "id": "S38", "elements": [ 9, 12, 13, 14 ], "cost": 224 }, { "id": "S39", "elements": [ 6, 7, 11 ], "cost": 63 }, { "id": "S40", "elements": [ 19, 20, 21 ], "cost": 15 }, { "id": "S41", "elements": [ 8, 9, 10, 11, 12, 13, 14 ], "cost": 476 }, { "id": "S42", "elements": [ 12, 13, 14, 15, 16, 18 ], "cost": 132 }, { "id": "S43", "elements": [ 7, 8, 10, 11 ], "cost": 380 }, { "id": "S44", "elements": [ 15, 17, 18 ], "cost": 9 }, { "id": "S45", "elements": [ 4, 6, 7, 8, 9, 10 ], "cost": 600 }, { "id": "S46", "elements": [ 10, 11, 13, 16 ], "cost": 8 }, { "id": "S47", "elements": [ 2, 5, 6, 8 ], "cost": 32 }, { "id": "S48", "elements": [ 2, 4, 5, 9 ], "cost": 192 }, { "id": "S49", "elements": [ 6, 8, 9, 10, 11 ], "cost": 390 }, { "id": "S50", "elements": [ 1, 3, 4, 5, 6, 8 ], "cost": 426 }, { "id": "S51", "elements": [ 12, 13, 14, 15, 16 ], "cost": 370 }, { "id": "S52", "elements": [ 11, 12, 13, 14, 15 ], "cost": 25 }, { "id": "S53", "elements": [ 17, 21, 22 ], "cost": 159 }, { "id": "S54", "elements": [ 5, 8, 9, 10 ], "cost": 224 }, { "id": "S55", "elements": [ 2, 3, 5, 6, 7, 8 ], "cost": 114 }, { "id": "S56", "elements": [ 6, 11 ], "cost": 104 }, { "id": "S57", "elements": [ 12, 13, 14, 19 ], "cost": 128 }, { "id": "S58", "elements": [ 15, 19, 20, 21 ], "cost": 248 }, { "id": "S59", "elements": [ 9, 14 ], "cost": 44 }, { "id": "S60", "elements": [ 2, 6 ], "cost": 70 }, { "id": "S61", "elements": [ 15, 17 ], "cost": 8 }, { "id": "S62", "elements": [ 9, 10, 12, 13 ], "cost": 28 }, { "id": "S63", "elements": [ 14, 16, 18, 19, 20, 21 ], "cost": 564 }, { "id": "S64", "elements": [ 14, 16, 18, 20, 21 ], "cost": 285 }, { "id": "S65", "elements": [ 17, 18, 19, 20, 22 ], "cost": 255 }, { "id": "S66", "elements": [ 1, 2, 6 ], "cost": 219 }, { "id": "S67", "elements": [ 5, 6, 10 ], "cost": 81 }, { "id": "S68", "elements": [ 1, 2 ], "cost": 188 }, { "id": "S69", "elements": [ 1, 2, 5, 6 ], "cost": 60 }, { "id": "S70", "elements": [ 14, 16, 17 ], "cost": 144 }, { "id": "S71", "elements": [ 6, 8, 9, 10, 11, 12 ], "cost": 120 }, { "id": "S72", "elements": [ 16, 17, 19, 20, 22 ], "cost": 15 }, { "id": "S73", "elements": [ 8, 9, 10, 11 ], "cost": 208 }, { "id": "S74", "elements": [ 18, 21 ], "cost": 28 }, { "id": "S75", "elements": [ 16, 17, 19, 20, 21, 22 ], "cost": 540 }, { "id": "S76", "elements": [ 9, 10, 11, 12, 13 ], "cost": 400 }, { "id": "S77", "elements": [ 18, 20, 21, 22 ], "cost": 224 }, { "id": "S78", "elements": [ 15, 16, 20 ], "cost": 120 }, { "id": "S79", "elements": [ 14, 15, 19 ], "cost": 168 }, { "id": "S80", "elements": [ 8, 9, 10, 11, 12, 14 ], "cost": 78 }, { "id": "S81", "elements": [ 1, 3, 5 ], "cost": 270 }, { "id": "S82", "elements": [ 1, 2, 3, 4, 8 ], "cost": 440 }, { "id": "S83", "elements": [ 8, 9, 10, 11, 13, 18 ], "cost": 66 }, { "id": "S84", "elements": [ 12, 16, 17, 19, 20, 21, 22 ], "cost": 266 }, { "id": "S85", "elements": [ 14, 15, 17, 18, 20 ], "cost": 105 }, { "id": "S86", "elements": [ 13, 14, 18, 19, 20 ], "cost": 430 }, { "id": "S87", "elements": [ 15, 16, 18 ], "cost": 114 }, { "id": "S88", "elements": [ 12, 13, 14 ], "cost": 228 }, { "id": "S89", "elements": [ 2, 3, 4, 5, 7, 8 ], "cost": 450 }, { "id": "S90", "elements": [ 13, 16, 17 ], "cost": 258 }, { "id": "S91", "elements": [ 8, 9, 11, 14 ], "cost": 96 }, { "id": "S92", "elements": [ 12, 13, 14, 15, 16, 17 ], "cost": 546 }, { "id": "S93", "elements": [ 8, 12, 13, 15 ], "cost": 88 }, { "id": "S94", "elements": [ 10, 11, 12, 13, 15, 17 ], "cost": 384 }, { "id": "S95", "elements": [ 5, 6, 7, 8, 9, 10 ], "cost": 288 }, { "id": "S96", "elements": [ 2, 3, 4, 6 ], "cost": 172 }, { "id": "S97", "elements": [ 16, 17, 18, 19, 21 ], "cost": 130 }, { "id": "S98", "elements": [ 1, 2, 3, 4 ], "cost": 388 }, { "id": "S99", "elements": [ 12, 14, 16, 17 ], "cost": 164 }, { "id": "S100", "elements": [ 4, 6, 7, 8, 10, 12 ], "cost": 198 }, { "id": "S101", "elements": [ 1, 2, 3, 6 ], "cost": 204 }, { "id": "S102", "elements": [ 4, 6 ], "cost": 72 }, { "id": "S103", "elements": [ 13, 18, 19, 20, 21 ], "cost": 200 }, { "id": "S104", "elements": [ 1 ], "cost": 10000 }, { "id": "S105", "elements": [ 2 ], "cost": 10000 }, { "id": "S106", "elements": [ 3 ], "cost": 10000 }, { "id": "S107", "elements": [ 4 ], "cost": 10000 }, { "id": "S108", "elements": [ 5 ], "cost": 10000 }, { "id": "S109", "elements": [ 6 ], "cost": 10000 }, { "id": "S110", "elements": [ 7 ], "cost": 10000 }, { "id": "S111", "elements": [ 8 ], "cost": 10000 }, { "id": "S112", "elements": [ 9 ], "cost": 10000 }, { "id": "S113", "elements": [ 10 ], "cost": 10000 }, { "id": "S114", "elements": [ 11 ], "cost": 10000 }, { "id": "S115", "elements": [ 12 ], "cost": 10000 }, { "id": "S116", "elements": [ 13 ], "cost": 10000 }, { "id": "S117", "elements": [ 14 ], "cost": 10000 }, { "id": "S118", "elements": [ 15 ], "cost": 10000 }, { "id": "S119", "elements": [ 16 ], "cost": 10000 }, { "id": "S120", "elements": [ 17 ], "cost": 10000 }, { "id": "S121", "elements": [ 18 ], "cost": 10000 }, { "id": "S122", "elements": [ 19 ], "cost": 10000 }, { "id": "S123", "elements": [ 20 ], "cost": 10000 }, { "id": "S124", "elements": [ 21 ], "cost": 10000 }, { "id": "S125", "elements": [ 22 ], "cost": 10000 } ] }, "solution_variant": [ "S2", "S12", "S37", "S70", "S74", "S93", "S101" ], "context_index": 19, "input_format": "markdown_table", "input_index_base": 1 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "Back when a couple downsized, they faced a pile of belongings and a handful of potential crates and had to pick which crates to use so every item got packed into one chosen crate with nothing omitted or doubled up. Since every crate costs money, the winning plan was the one with the lowest total cost, calculated by adding the prices of the selected crates. The item list and crate options appear below.\n\n# total_items_count=25\n# total_available_crates=143\ncrate_id,crate_cost,items_in_crate\nS1,45,16 17 19\nS2,434,7 9 11 12 13 14 15\nS3,114,2 5 8\nS4,420,10 11 12 13 15 16\nS5,600,13 16 18 21 22 24\nS6,600,19 20 21 22 23 24\nS7,230,12 13 15 17 19\nS8,240,1 4 6\nS9,679,18 19 20 21 22 23 24\nS10,168,0 1 2 3 4 6\nS11,27,19 20 21\nS12,9,0 1 3\nS13,539,7 10 11 12 13 14 16\nS14,220,20 21 22 23 24\nS15,207,0 2 4\nS16,140,5 6 10 11 12 13 17\nS17,380,12 13 17 18\nS18,72,0 1 2 3\nS19,518,15 16 17 18 20 21 24\nS20,280,1 2 3 5\nS21,364,0 1 2 3 4 5 6\nS22,78,0 1 2 3 4 5\nS23,504,11 14 15 16 17 20 21\nS24,196,1 4 5 7 8 9 11\nS25,402,0 2 4 7 8 10\nS26,175,13 15 16 18 19 20 21\nS27,112,6 7 8 9 10 12 13\nS28,102,5 6 7 8 9 13\nS29,160,17 18 20 22 24\nS30,162,15 18 19 20 22 24\nS31,288,16 19 20 21 23 24\nS32,328,11 12 14 15\nS33,255,11 14 15\nS34,84,13 14 15 19\nS35,45,0 2 3\nS36,273,0 2 3 4 5 6 7\nS37,342,12 13 14 16 17 19\nS38,245,0 1 2 3 4 5 7\nS39,54,13 14 15 16 17 19\nS40,360,2 3 4 5 6\nS41,693,12 14 15 17 18 19 20\nS42,330,14 16 17 18 19 20\nS43,292,8 9 10 13\nS44,230,1 3 4 6 7\nS45,70,6 7 8 9 10\nS46,50,2 3 4 7 11\nS47,84,14 17 19\nS48,332,1 2 5 8\nS49,165,10 11 14 17 18\nS50,582,14 15 16 17 19 21\nS51,390,0 1 2 3 4 7\nS52,450,16 17 18 19 20\nS53,108,17 19 23\nS54,81,1 3 4\nS55,54,6 9 10\nS56,294,20 22 24\nS57,196,18 20 22 23\nS58,162,7 9 14\nS59,168,20 21 23\nS60,304,4 5 6 7\nS61,385,15 18 19 21 22 23 24\nS62,282,12 13 14\nS63,504,1 2 4 5 7 9\nS64,155,18 20 22 23 24\nS65,238,11 12 13 15 16 17 19\nS66,72,0 4 6 7\nS67,686,2 3 5 6 7 8 10\nS68,8,1 4 6 11\nS69,492,18 19 21 22 23 24\nS70,415,1 2 4 5 6\nS71,164,20 21 22 24\nS72,400,12 13 15 17 18\nS73,204,3 5 6 9 10 12\nS74,217,9 10 11 12 13 14 15\nS75,110,3 6 7 8 14\nS76,468,7 9 10 11 12 14\nS77,476,16 17 18 19 20 22 24\nS78,153,0 5 6\nS79,324,9 13 16 17\nS80,588,18 19 20 21 23 24\nS81,27,16 18 19\nS82,222,9 10 13\nS83,288,18 19 20 21 22 24\nS84,300,11 13 14 16 17 21\nS85,24,19 20 21 22\nS86,72,6 7 8\nS87,12,9 10 13 14\nS88,48,0 1 4 5 6 7\nS89,287,6 7 8 9 11 12 13\nS90,168,0 1 3 4 5 6\nS91,15,18 19 23\nS92,120,9 15 16 17 19\nS93,60,0 1 4 7\nS94,320,13 17 19 20 21\nS95,4,9 11 12 14\nS96,112,16 17 18 19 20 21 23\nS97,532,17 18 19 20 21 22 24\nS98,84,10 11 15\nS99,30,12 13 14 15 16 17\nS100,78,8 9 10 12 14 15\nS101,75,4 7 8\nS102,69,10 12 18\nS103,378,5 7 8 9 10 11\nS104,228,14 15 16 18 19 21\nS105,264,0 2 3 4 5 8\nS106,212,14 16 17 18\nS107,21,11 12 13\nS108,460,14 17 18 19 22\nS109,518,5 7 8 9 10 11 12\nS110,301,0 1 2 3 4 5 8\nS111,125,4 8 9 11 14\nS112,511,3 4 5 6 7 8 9\nS113,80,0 1 2 3 7\nS114,465,8 11 13 15 18\nS115,128,17 18 19 20\nS116,371,13 16 17 18 19 20 22\nS117,364,0 2 3 4 5 6 8\nS118,370,7 8 11 12 13\nS119,10000,0\nS120,10000,1\nS121,10000,2\nS122,10000,3\nS123,10000,4\nS124,10000,5\nS125,10000,6\nS126,10000,7\nS127,10000,8\nS128,10000,9\nS129,10000,10\nS130,10000,11\nS131,10000,12\nS132,10000,13\nS133,10000,14\nS134,10000,15\nS135,10000,16\nS136,10000,17\nS137,10000,18\nS138,10000,19\nS139,10000,20\nS140,10000,21\nS141,10000,22\nS142,10000,23\nS143,10000,24\n\nIf you want to tell me which crates you’d pick, just use this little JSON shape so I can read the chosen crate IDs cleanly:\n\n{\n \"solution\": [\"crate_id\", ...]\n}\n\nThink of it like a checklist: \"solution\" is the list of crates you picked (put the exact crate identifiers from the instance in that array). I used the placeholder crate_id to match the packing story — swap that out for the real IDs when you reply. This JSON is just a sketch of the shape I expect, not the actual answer.\n\nPlease be sure to use the identifiers exactly as they appear in the instance input — no renaming and no new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 25, "num_sets": 143, "density": 0.1751048951048951, "sets": [ { "id": 1, "elements": [ 17, 18, 20 ], "cost": 45 }, { "id": 2, "elements": [ 8, 10, 12, 13, 14, 15, 16 ], "cost": 434 }, { "id": 3, "elements": [ 3, 6, 9 ], "cost": 114 }, { "id": 4, "elements": [ 11, 12, 13, 14, 16, 17 ], "cost": 420 }, { "id": 5, "elements": [ 14, 17, 19, 22, 23, 25 ], "cost": 600 }, { "id": 6, "elements": [ 20, 21, 22, 23, 24, 25 ], "cost": 600 }, { "id": 7, "elements": [ 13, 14, 16, 18, 20 ], "cost": 230 }, { "id": 8, "elements": [ 2, 5, 7 ], "cost": 240 }, { "id": 9, "elements": [ 19, 20, 21, 22, 23, 24, 25 ], "cost": 679 }, { "id": 10, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 168 }, { "id": 11, "elements": [ 20, 21, 22 ], "cost": 27 }, { "id": 12, "elements": [ 1, 2, 4 ], "cost": 9 }, { "id": 13, "elements": [ 8, 11, 12, 13, 14, 15, 17 ], "cost": 539 }, { "id": 14, "elements": [ 21, 22, 23, 24, 25 ], "cost": 220 }, { "id": 15, "elements": [ 1, 3, 5 ], "cost": 207 }, { "id": 16, "elements": [ 6, 7, 11, 12, 13, 14, 18 ], "cost": 140 }, { "id": 17, "elements": [ 13, 14, 18, 19 ], "cost": 380 }, { "id": 18, "elements": [ 1, 2, 3, 4 ], "cost": 72 }, { "id": 19, "elements": [ 16, 17, 18, 19, 21, 22, 25 ], "cost": 518 }, { "id": 20, "elements": [ 2, 3, 4, 6 ], "cost": 280 }, { "id": 21, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 364 }, { "id": 22, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 78 }, { "id": 23, "elements": [ 12, 15, 16, 17, 18, 21, 22 ], "cost": 504 }, { "id": 24, "elements": [ 2, 5, 6, 8, 9, 10, 12 ], "cost": 196 }, { "id": 25, "elements": [ 1, 3, 5, 8, 9, 11 ], "cost": 402 }, { "id": 26, "elements": [ 14, 16, 17, 19, 20, 21, 22 ], "cost": 175 }, { "id": 27, "elements": [ 7, 8, 9, 10, 11, 13, 14 ], "cost": 112 }, { "id": 28, "elements": [ 6, 7, 8, 9, 10, 14 ], "cost": 102 }, { "id": 29, "elements": [ 18, 19, 21, 23, 25 ], "cost": 160 }, { "id": 30, "elements": [ 16, 19, 20, 21, 23, 25 ], "cost": 162 }, { "id": 31, "elements": [ 17, 20, 21, 22, 24, 25 ], "cost": 288 }, { "id": 32, "elements": [ 12, 13, 15, 16 ], "cost": 328 }, { "id": 33, "elements": [ 12, 15, 16 ], "cost": 255 }, { "id": 34, "elements": [ 14, 15, 16, 20 ], "cost": 84 }, { "id": 35, "elements": [ 1, 3, 4 ], "cost": 45 }, { "id": 36, "elements": [ 1, 3, 4, 5, 6, 7, 8 ], "cost": 273 }, { "id": 37, "elements": [ 13, 14, 15, 17, 18, 20 ], "cost": 342 }, { "id": 38, "elements": [ 1, 2, 3, 4, 5, 6, 8 ], "cost": 245 }, { "id": 39, "elements": [ 14, 15, 16, 17, 18, 20 ], "cost": 54 }, { "id": 40, "elements": [ 3, 4, 5, 6, 7 ], "cost": 360 }, { "id": 41, "elements": [ 13, 15, 16, 18, 19, 20, 21 ], "cost": 693 }, { "id": 42, "elements": [ 15, 17, 18, 19, 20, 21 ], "cost": 330 }, { "id": 43, "elements": [ 9, 10, 11, 14 ], "cost": 292 }, { "id": 44, "elements": [ 2, 4, 5, 7, 8 ], "cost": 230 }, { "id": 45, "elements": [ 7, 8, 9, 10, 11 ], "cost": 70 }, { "id": 46, "elements": [ 3, 4, 5, 8, 12 ], "cost": 50 }, { "id": 47, "elements": [ 15, 18, 20 ], "cost": 84 }, { "id": 48, "elements": [ 2, 3, 6, 9 ], "cost": 332 }, { "id": 49, "elements": [ 11, 12, 15, 18, 19 ], "cost": 165 }, { "id": 50, "elements": [ 15, 16, 17, 18, 20, 22 ], "cost": 582 }, { "id": 51, "elements": [ 1, 2, 3, 4, 5, 8 ], "cost": 390 }, { "id": 52, "elements": [ 17, 18, 19, 20, 21 ], "cost": 450 }, { "id": 53, "elements": [ 18, 20, 24 ], "cost": 108 }, { "id": 54, "elements": [ 2, 4, 5 ], "cost": 81 }, { "id": 55, "elements": [ 7, 10, 11 ], "cost": 54 }, { "id": 56, "elements": [ 21, 23, 25 ], "cost": 294 }, { "id": 57, "elements": [ 19, 21, 23, 24 ], "cost": 196 }, { "id": 58, "elements": [ 8, 10, 15 ], "cost": 162 }, { "id": 59, "elements": [ 21, 22, 24 ], "cost": 168 }, { "id": 60, "elements": [ 5, 6, 7, 8 ], "cost": 304 }, { "id": 61, "elements": [ 16, 19, 20, 22, 23, 24, 25 ], "cost": 385 }, { "id": 62, "elements": [ 13, 14, 15 ], "cost": 282 }, { "id": 63, "elements": [ 2, 3, 5, 6, 8, 10 ], "cost": 504 }, { "id": 64, "elements": [ 19, 21, 23, 24, 25 ], "cost": 155 }, { "id": 65, "elements": [ 12, 13, 14, 16, 17, 18, 20 ], "cost": 238 }, { "id": 66, "elements": [ 1, 5, 7, 8 ], "cost": 72 }, { "id": 67, "elements": [ 3, 4, 6, 7, 8, 9, 11 ], "cost": 686 }, { "id": 68, "elements": [ 2, 5, 7, 12 ], "cost": 8 }, { "id": 69, "elements": [ 19, 20, 22, 23, 24, 25 ], "cost": 492 }, { "id": 70, "elements": [ 2, 3, 5, 6, 7 ], "cost": 415 }, { "id": 71, "elements": [ 21, 22, 23, 25 ], "cost": 164 }, { "id": 72, "elements": [ 13, 14, 16, 18, 19 ], "cost": 400 }, { "id": 73, "elements": [ 4, 6, 7, 10, 11, 13 ], "cost": 204 }, { "id": 74, "elements": [ 10, 11, 12, 13, 14, 15, 16 ], "cost": 217 }, { "id": 75, "elements": [ 4, 7, 8, 9, 15 ], "cost": 110 }, { "id": 76, "elements": [ 8, 10, 11, 12, 13, 15 ], "cost": 468 }, { "id": 77, "elements": [ 17, 18, 19, 20, 21, 23, 25 ], "cost": 476 }, { "id": 78, "elements": [ 1, 6, 7 ], "cost": 153 }, { "id": 79, "elements": [ 10, 14, 17, 18 ], "cost": 324 }, { "id": 80, "elements": [ 19, 20, 21, 22, 24, 25 ], "cost": 588 }, { "id": 81, "elements": [ 17, 19, 20 ], "cost": 27 }, { "id": 82, "elements": [ 10, 11, 14 ], "cost": 222 }, { "id": 83, "elements": [ 19, 20, 21, 22, 23, 25 ], "cost": 288 }, { "id": 84, "elements": [ 12, 14, 15, 17, 18, 22 ], "cost": 300 }, { "id": 85, "elements": [ 20, 21, 22, 23 ], "cost": 24 }, { "id": 86, "elements": [ 7, 8, 9 ], "cost": 72 }, { "id": 87, "elements": [ 10, 11, 14, 15 ], "cost": 12 }, { "id": 88, "elements": [ 1, 2, 5, 6, 7, 8 ], "cost": 48 }, { "id": 89, "elements": [ 7, 8, 9, 10, 12, 13, 14 ], "cost": 287 }, { "id": 90, "elements": [ 1, 2, 4, 5, 6, 7 ], "cost": 168 }, { "id": 91, "elements": [ 19, 20, 24 ], "cost": 15 }, { "id": 92, "elements": [ 10, 16, 17, 18, 20 ], "cost": 120 }, { "id": 93, "elements": [ 1, 2, 5, 8 ], "cost": 60 }, { "id": 94, "elements": [ 14, 18, 20, 21, 22 ], "cost": 320 }, { "id": 95, "elements": [ 10, 12, 13, 15 ], "cost": 4 }, { "id": 96, "elements": [ 17, 18, 19, 20, 21, 22, 24 ], "cost": 112 }, { "id": 97, "elements": [ 18, 19, 20, 21, 22, 23, 25 ], "cost": 532 }, { "id": 98, "elements": [ 11, 12, 16 ], "cost": 84 }, { "id": 99, "elements": [ 13, 14, 15, 16, 17, 18 ], "cost": 30 }, { "id": 100, "elements": [ 9, 10, 11, 13, 15, 16 ], "cost": 78 }, { "id": 101, "elements": [ 5, 8, 9 ], "cost": 75 }, { "id": 102, "elements": [ 11, 13, 19 ], "cost": 69 }, { "id": 103, "elements": [ 6, 8, 9, 10, 11, 12 ], "cost": 378 }, { "id": 104, "elements": [ 15, 16, 17, 19, 20, 22 ], "cost": 228 }, { "id": 105, "elements": [ 1, 3, 4, 5, 6, 9 ], "cost": 264 }, { "id": 106, "elements": [ 15, 17, 18, 19 ], "cost": 212 }, { "id": 107, "elements": [ 12, 13, 14 ], "cost": 21 }, { "id": 108, "elements": [ 15, 18, 19, 20, 23 ], "cost": 460 }, { "id": 109, "elements": [ 6, 8, 9, 10, 11, 12, 13 ], "cost": 518 }, { "id": 110, "elements": [ 1, 2, 3, 4, 5, 6, 9 ], "cost": 301 }, { "id": 111, "elements": [ 5, 9, 10, 12, 15 ], "cost": 125 }, { "id": 112, "elements": [ 4, 5, 6, 7, 8, 9, 10 ], "cost": 511 }, { "id": 113, "elements": [ 1, 2, 3, 4, 8 ], "cost": 80 }, { "id": 114, "elements": [ 9, 12, 14, 16, 19 ], "cost": 465 }, { "id": 115, "elements": [ 18, 19, 20, 21 ], "cost": 128 }, { "id": 116, "elements": [ 14, 17, 18, 19, 20, 21, 23 ], "cost": 371 }, { "id": 117, "elements": [ 1, 3, 4, 5, 6, 7, 9 ], "cost": 364 }, { "id": 118, "elements": [ 8, 9, 12, 13, 14 ], "cost": 370 }, { "id": 119, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 120, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 121, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 122, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 123, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 124, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 125, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 126, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 127, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 128, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 129, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 130, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 131, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 132, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 133, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 134, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 135, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 136, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 137, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 138, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 139, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 140, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 141, "elements": [ 23 ], "cost": 10000, "singleton": true }, { "id": 142, "elements": [ 24 ], "cost": 10000, "singleton": true }, { "id": 143, "elements": [ 25 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0019_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0019_bag_stats.png" }, "solution": [ 10, 71, 91, 99, 103 ], "obj": 755.0, "instance_variant": { "num_elements": 25, "num_sets": 143, "sets": [ { "id": "S1", "elements": [ 16, 17, 19 ], "cost": 45 }, { "id": "S2", "elements": [ 7, 9, 11, 12, 13, 14, 15 ], "cost": 434 }, { "id": "S3", "elements": [ 2, 5, 8 ], "cost": 114 }, { "id": "S4", "elements": [ 10, 11, 12, 13, 15, 16 ], "cost": 420 }, { "id": "S5", "elements": [ 13, 16, 18, 21, 22, 24 ], "cost": 600 }, { "id": "S6", "elements": [ 19, 20, 21, 22, 23, 24 ], "cost": 600 }, { "id": "S7", "elements": [ 12, 13, 15, 17, 19 ], "cost": 230 }, { "id": "S8", "elements": [ 1, 4, 6 ], "cost": 240 }, { "id": "S9", "elements": [ 18, 19, 20, 21, 22, 23, 24 ], "cost": 679 }, { "id": "S10", "elements": [ 0, 1, 2, 3, 4, 6 ], "cost": 168 }, { "id": "S11", "elements": [ 19, 20, 21 ], "cost": 27 }, { "id": "S12", "elements": [ 0, 1, 3 ], "cost": 9 }, { "id": "S13", "elements": [ 7, 10, 11, 12, 13, 14, 16 ], "cost": 539 }, { "id": "S14", "elements": [ 20, 21, 22, 23, 24 ], "cost": 220 }, { "id": "S15", "elements": [ 0, 2, 4 ], "cost": 207 }, { "id": "S16", "elements": [ 5, 6, 10, 11, 12, 13, 17 ], "cost": 140 }, { "id": "S17", "elements": [ 12, 13, 17, 18 ], "cost": 380 }, { "id": "S18", "elements": [ 0, 1, 2, 3 ], "cost": 72 }, { "id": "S19", "elements": [ 15, 16, 17, 18, 20, 21, 24 ], "cost": 518 }, { "id": "S20", "elements": [ 1, 2, 3, 5 ], "cost": 280 }, { "id": "S21", "elements": [ 0, 1, 2, 3, 4, 5, 6 ], "cost": 364 }, { "id": "S22", "elements": [ 0, 1, 2, 3, 4, 5 ], "cost": 78 }, { "id": "S23", "elements": [ 11, 14, 15, 16, 17, 20, 21 ], "cost": 504 }, { "id": "S24", "elements": [ 1, 4, 5, 7, 8, 9, 11 ], "cost": 196 }, { "id": "S25", "elements": [ 0, 2, 4, 7, 8, 10 ], "cost": 402 }, { "id": "S26", "elements": [ 13, 15, 16, 18, 19, 20, 21 ], "cost": 175 }, { "id": "S27", "elements": [ 6, 7, 8, 9, 10, 12, 13 ], "cost": 112 }, { "id": "S28", "elements": [ 5, 6, 7, 8, 9, 13 ], "cost": 102 }, { "id": "S29", "elements": [ 17, 18, 20, 22, 24 ], "cost": 160 }, { "id": "S30", "elements": [ 15, 18, 19, 20, 22, 24 ], "cost": 162 }, { "id": "S31", "elements": [ 16, 19, 20, 21, 23, 24 ], "cost": 288 }, { "id": "S32", "elements": [ 11, 12, 14, 15 ], "cost": 328 }, { "id": "S33", "elements": [ 11, 14, 15 ], "cost": 255 }, { "id": "S34", "elements": [ 13, 14, 15, 19 ], "cost": 84 }, { "id": "S35", "elements": [ 0, 2, 3 ], "cost": 45 }, { "id": "S36", "elements": [ 0, 2, 3, 4, 5, 6, 7 ], "cost": 273 }, { "id": "S37", "elements": [ 12, 13, 14, 16, 17, 19 ], "cost": 342 }, { "id": "S38", "elements": [ 0, 1, 2, 3, 4, 5, 7 ], "cost": 245 }, { "id": "S39", "elements": [ 13, 14, 15, 16, 17, 19 ], "cost": 54 }, { "id": "S40", "elements": [ 2, 3, 4, 5, 6 ], "cost": 360 }, { "id": "S41", "elements": [ 12, 14, 15, 17, 18, 19, 20 ], "cost": 693 }, { "id": "S42", "elements": [ 14, 16, 17, 18, 19, 20 ], "cost": 330 }, { "id": "S43", "elements": [ 8, 9, 10, 13 ], "cost": 292 }, { "id": "S44", "elements": [ 1, 3, 4, 6, 7 ], "cost": 230 }, { "id": "S45", "elements": [ 6, 7, 8, 9, 10 ], "cost": 70 }, { "id": "S46", "elements": [ 2, 3, 4, 7, 11 ], "cost": 50 }, { "id": "S47", "elements": [ 14, 17, 19 ], "cost": 84 }, { "id": "S48", "elements": [ 1, 2, 5, 8 ], "cost": 332 }, { "id": "S49", "elements": [ 10, 11, 14, 17, 18 ], "cost": 165 }, { "id": "S50", "elements": [ 14, 15, 16, 17, 19, 21 ], "cost": 582 }, { "id": "S51", "elements": [ 0, 1, 2, 3, 4, 7 ], "cost": 390 }, { "id": "S52", "elements": [ 16, 17, 18, 19, 20 ], "cost": 450 }, { "id": "S53", "elements": [ 17, 19, 23 ], "cost": 108 }, { "id": "S54", "elements": [ 1, 3, 4 ], "cost": 81 }, { "id": "S55", "elements": [ 6, 9, 10 ], "cost": 54 }, { "id": "S56", "elements": [ 20, 22, 24 ], "cost": 294 }, { "id": "S57", "elements": [ 18, 20, 22, 23 ], "cost": 196 }, { "id": "S58", "elements": [ 7, 9, 14 ], "cost": 162 }, { "id": "S59", "elements": [ 20, 21, 23 ], "cost": 168 }, { "id": "S60", "elements": [ 4, 5, 6, 7 ], "cost": 304 }, { "id": "S61", "elements": [ 15, 18, 19, 21, 22, 23, 24 ], "cost": 385 }, { "id": "S62", "elements": [ 12, 13, 14 ], "cost": 282 }, { "id": "S63", "elements": [ 1, 2, 4, 5, 7, 9 ], "cost": 504 }, { "id": "S64", "elements": [ 18, 20, 22, 23, 24 ], "cost": 155 }, { "id": "S65", "elements": [ 11, 12, 13, 15, 16, 17, 19 ], "cost": 238 }, { "id": "S66", "elements": [ 0, 4, 6, 7 ], "cost": 72 }, { "id": "S67", "elements": [ 2, 3, 5, 6, 7, 8, 10 ], "cost": 686 }, { "id": "S68", "elements": [ 1, 4, 6, 11 ], "cost": 8 }, { "id": "S69", "elements": [ 18, 19, 21, 22, 23, 24 ], "cost": 492 }, { "id": "S70", "elements": [ 1, 2, 4, 5, 6 ], "cost": 415 }, { "id": "S71", "elements": [ 20, 21, 22, 24 ], "cost": 164 }, { "id": "S72", "elements": [ 12, 13, 15, 17, 18 ], "cost": 400 }, { "id": "S73", "elements": [ 3, 5, 6, 9, 10, 12 ], "cost": 204 }, { "id": "S74", "elements": [ 9, 10, 11, 12, 13, 14, 15 ], "cost": 217 }, { "id": "S75", "elements": [ 3, 6, 7, 8, 14 ], "cost": 110 }, { "id": "S76", "elements": [ 7, 9, 10, 11, 12, 14 ], "cost": 468 }, { "id": "S77", "elements": [ 16, 17, 18, 19, 20, 22, 24 ], "cost": 476 }, { "id": "S78", "elements": [ 0, 5, 6 ], "cost": 153 }, { "id": "S79", "elements": [ 9, 13, 16, 17 ], "cost": 324 }, { "id": "S80", "elements": [ 18, 19, 20, 21, 23, 24 ], "cost": 588 }, { "id": "S81", "elements": [ 16, 18, 19 ], "cost": 27 }, { "id": "S82", "elements": [ 9, 10, 13 ], "cost": 222 }, { "id": "S83", "elements": [ 18, 19, 20, 21, 22, 24 ], "cost": 288 }, { "id": "S84", "elements": [ 11, 13, 14, 16, 17, 21 ], "cost": 300 }, { "id": "S85", "elements": [ 19, 20, 21, 22 ], "cost": 24 }, { "id": "S86", "elements": [ 6, 7, 8 ], "cost": 72 }, { "id": "S87", "elements": [ 9, 10, 13, 14 ], "cost": 12 }, { "id": "S88", "elements": [ 0, 1, 4, 5, 6, 7 ], "cost": 48 }, { "id": "S89", "elements": [ 6, 7, 8, 9, 11, 12, 13 ], "cost": 287 }, { "id": "S90", "elements": [ 0, 1, 3, 4, 5, 6 ], "cost": 168 }, { "id": "S91", "elements": [ 18, 19, 23 ], "cost": 15 }, { "id": "S92", "elements": [ 9, 15, 16, 17, 19 ], "cost": 120 }, { "id": "S93", "elements": [ 0, 1, 4, 7 ], "cost": 60 }, { "id": "S94", "elements": [ 13, 17, 19, 20, 21 ], "cost": 320 }, { "id": "S95", "elements": [ 9, 11, 12, 14 ], "cost": 4 }, { "id": "S96", "elements": [ 16, 17, 18, 19, 20, 21, 23 ], "cost": 112 }, { "id": "S97", "elements": [ 17, 18, 19, 20, 21, 22, 24 ], "cost": 532 }, { "id": "S98", "elements": [ 10, 11, 15 ], "cost": 84 }, { "id": "S99", "elements": [ 12, 13, 14, 15, 16, 17 ], "cost": 30 }, { "id": "S100", "elements": [ 8, 9, 10, 12, 14, 15 ], "cost": 78 }, { "id": "S101", "elements": [ 4, 7, 8 ], "cost": 75 }, { "id": "S102", "elements": [ 10, 12, 18 ], "cost": 69 }, { "id": "S103", "elements": [ 5, 7, 8, 9, 10, 11 ], "cost": 378 }, { "id": "S104", "elements": [ 14, 15, 16, 18, 19, 21 ], "cost": 228 }, { "id": "S105", "elements": [ 0, 2, 3, 4, 5, 8 ], "cost": 264 }, { "id": "S106", "elements": [ 14, 16, 17, 18 ], "cost": 212 }, { "id": "S107", "elements": [ 11, 12, 13 ], "cost": 21 }, { "id": "S108", "elements": [ 14, 17, 18, 19, 22 ], "cost": 460 }, { "id": "S109", "elements": [ 5, 7, 8, 9, 10, 11, 12 ], "cost": 518 }, { "id": "S110", "elements": [ 0, 1, 2, 3, 4, 5, 8 ], "cost": 301 }, { "id": "S111", "elements": [ 4, 8, 9, 11, 14 ], "cost": 125 }, { "id": "S112", "elements": [ 3, 4, 5, 6, 7, 8, 9 ], "cost": 511 }, { "id": "S113", "elements": [ 0, 1, 2, 3, 7 ], "cost": 80 }, { "id": "S114", "elements": [ 8, 11, 13, 15, 18 ], "cost": 465 }, { "id": "S115", "elements": [ 17, 18, 19, 20 ], "cost": 128 }, { "id": "S116", "elements": [ 13, 16, 17, 18, 19, 20, 22 ], "cost": 371 }, { "id": "S117", "elements": [ 0, 2, 3, 4, 5, 6, 8 ], "cost": 364 }, { "id": "S118", "elements": [ 7, 8, 11, 12, 13 ], "cost": 370 }, { "id": "S119", "elements": [ 0 ], "cost": 10000 }, { "id": "S120", "elements": [ 1 ], "cost": 10000 }, { "id": "S121", "elements": [ 2 ], "cost": 10000 }, { "id": "S122", "elements": [ 3 ], "cost": 10000 }, { "id": "S123", "elements": [ 4 ], "cost": 10000 }, { "id": "S124", "elements": [ 5 ], "cost": 10000 }, { "id": "S125", "elements": [ 6 ], "cost": 10000 }, { "id": "S126", "elements": [ 7 ], "cost": 10000 }, { "id": "S127", "elements": [ 8 ], "cost": 10000 }, { "id": "S128", "elements": [ 9 ], "cost": 10000 }, { "id": "S129", "elements": [ 10 ], "cost": 10000 }, { "id": "S130", "elements": [ 11 ], "cost": 10000 }, { "id": "S131", "elements": [ 12 ], "cost": 10000 }, { "id": "S132", "elements": [ 13 ], "cost": 10000 }, { "id": "S133", "elements": [ 14 ], "cost": 10000 }, { "id": "S134", "elements": [ 15 ], "cost": 10000 }, { "id": "S135", "elements": [ 16 ], "cost": 10000 }, { "id": "S136", "elements": [ 17 ], "cost": 10000 }, { "id": "S137", "elements": [ 18 ], "cost": 10000 }, { "id": "S138", "elements": [ 19 ], "cost": 10000 }, { "id": "S139", "elements": [ 20 ], "cost": 10000 }, { "id": "S140", "elements": [ 21 ], "cost": 10000 }, { "id": "S141", "elements": [ 22 ], "cost": 10000 }, { "id": "S142", "elements": [ 23 ], "cost": 10000 }, { "id": "S143", "elements": [ 24 ], "cost": 10000 } ] }, "solution_variant": [ "S10", "S71", "S91", "S99", "S103" ], "context_index": 20, "input_format": "csv", "input_index_base": 0 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "I was putting together an event and had a stack of vendor packages to choose from — each package covers some of the planned activities and has a price. The job is to pick a handful of those packages so every activity on the schedule is handled by one and only one package (nothing left out, nothing covered twice), and to keep the total bill as small as possible by adding up the prices of the chosen packages. The concrete packages and the activity list are shown below.\n\n{\n \"total_activities\": 22,\n \"total_packages\": 122,\n \"sets\": [\n {\n \"package_id\": \"S1\",\n \"covered_activities\": [\n \"B\",\n \"F\",\n \"G\",\n \"H\",\n \"I\"\n ],\n \"package_price\": 95\n },\n {\n \"package_id\": \"S2\",\n \"covered_activities\": [\n \"N\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\"\n ],\n \"package_price\": 312\n },\n {\n \"package_id\": \"S3\",\n \"covered_activities\": [\n \"F\",\n \"H\",\n \"J\",\n \"K\",\n \"L\"\n ],\n \"package_price\": 20\n },\n {\n \"package_id\": \"S4\",\n \"covered_activities\": [\n \"A\",\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"I\"\n ],\n \"package_price\": 300\n },\n {\n \"package_id\": \"S5\",\n \"covered_activities\": [\n \"A\",\n \"G\"\n ],\n \"package_price\": 100\n },\n {\n \"package_id\": \"S6\",\n \"covered_activities\": [\n \"A\",\n \"B\",\n \"E\",\n \"F\"\n ],\n \"package_price\": 16\n },\n {\n \"package_id\": \"S7\",\n \"covered_activities\": [\n \"A\",\n \"C\"\n ],\n \"package_price\": 166\n },\n {\n \"package_id\": \"S8\",\n \"covered_activities\": [\n \"G\",\n \"K\",\n \"L\"\n ],\n \"package_price\": 21\n },\n {\n \"package_id\": \"S9\",\n \"covered_activities\": [\n \"F\",\n \"H\",\n \"I\",\n \"J\"\n ],\n \"package_price\": 332\n },\n {\n \"package_id\": \"S10\",\n \"covered_activities\": [\n \"B\",\n \"C\",\n \"E\",\n \"F\"\n ],\n \"package_price\": 256\n },\n {\n \"package_id\": \"S11\",\n \"covered_activities\": [\n \"Q\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"package_price\": 72\n },\n {\n \"package_id\": \"S12\",\n \"covered_activities\": [\n \"J\",\n \"L\",\n \"N\",\n \"O\",\n \"P\"\n ],\n \"package_price\": 315\n },\n {\n \"package_id\": \"S13\",\n \"covered_activities\": [\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"U\"\n ],\n \"package_price\": 45\n },\n {\n \"package_id\": \"S14\",\n \"covered_activities\": [\n \"E\",\n \"G\",\n \"H\",\n \"J\",\n \"L\",\n \"N\"\n ],\n \"package_price\": 354\n },\n {\n \"package_id\": \"S15\",\n \"covered_activities\": [\n \"D\",\n \"F\",\n \"I\"\n ],\n \"package_price\": 66\n },\n {\n \"package_id\": \"S16\",\n \"covered_activities\": [\n \"J\",\n \"K\",\n \"M\"\n ],\n \"package_price\": 6\n },\n {\n \"package_id\": \"S17\",\n \"covered_activities\": [\n \"C\",\n \"D\",\n \"G\"\n ],\n \"package_price\": 147\n },\n {\n \"package_id\": \"S18\",\n \"covered_activities\": [\n \"L\",\n \"N\",\n \"P\"\n ],\n \"package_price\": 51\n },\n {\n \"package_id\": \"S19\",\n \"covered_activities\": [\n \"O\",\n \"P\",\n \"Q\",\n \"R\"\n ],\n \"package_price\": 40\n },\n {\n \"package_id\": \"S20\",\n \"covered_activities\": [\n \"N\",\n \"P\",\n \"Q\",\n \"R\",\n \"T\",\n \"V\"\n ],\n \"package_price\": 114\n },\n {\n \"package_id\": \"S21\",\n \"covered_activities\": [\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"N\",\n \"P\"\n ],\n \"package_price\": 162\n },\n {\n \"package_id\": \"S22\",\n \"covered_activities\": [\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"H\"\n ],\n \"package_price\": 320\n },\n {\n \"package_id\": \"S23\",\n \"covered_activities\": [\n \"C\",\n \"F\",\n \"G\",\n \"H\",\n \"I\"\n ],\n \"package_price\": 380\n },\n {\n \"package_id\": \"S24\",\n \"covered_activities\": [\n \"C\",\n \"D\",\n \"E\",\n \"G\",\n \"K\"\n ],\n \"package_price\": 350\n },\n {\n \"package_id\": \"S25\",\n \"covered_activities\": [\n \"E\",\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\"\n ],\n \"package_price\": 246\n },\n {\n \"package_id\": \"S26\",\n \"covered_activities\": [\n \"F\",\n \"I\",\n \"M\",\n \"O\"\n ],\n \"package_price\": 360\n },\n {\n \"package_id\": \"S27\",\n \"covered_activities\": [\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\"\n ],\n \"package_price\": 260\n },\n {\n \"package_id\": \"S28\",\n \"covered_activities\": [\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"R\",\n \"S\"\n ],\n \"package_price\": 6\n },\n {\n \"package_id\": \"S29\",\n \"covered_activities\": [\n \"B\",\n \"C\",\n \"F\",\n \"G\"\n ],\n \"package_price\": 168\n },\n {\n \"package_id\": \"S30\",\n \"covered_activities\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"H\"\n ],\n \"package_price\": 290\n },\n {\n \"package_id\": \"S31\",\n \"covered_activities\": [\n \"P\",\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\"\n ],\n \"package_price\": 444\n },\n {\n \"package_id\": \"S32\",\n \"covered_activities\": [\n \"R\",\n \"S\",\n \"T\"\n ],\n \"package_price\": 18\n },\n {\n \"package_id\": \"S33\",\n \"covered_activities\": [\n \"O\",\n \"P\",\n \"R\",\n \"S\",\n \"T\"\n ],\n \"package_price\": 350\n },\n {\n \"package_id\": \"S34\",\n \"covered_activities\": [\n \"D\",\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"J\"\n ],\n \"package_price\": 180\n },\n {\n \"package_id\": \"S35\",\n \"covered_activities\": [\n \"P\",\n \"R\",\n \"S\",\n \"T\",\n \"V\"\n ],\n \"package_price\": 415\n },\n {\n \"package_id\": \"S36\",\n \"covered_activities\": [\n \"S\",\n \"T\",\n \"U\"\n ],\n \"package_price\": 243\n },\n {\n \"package_id\": \"S37\",\n \"covered_activities\": [\n \"B\",\n \"D\"\n ],\n \"package_price\": 140\n },\n {\n \"package_id\": \"S38\",\n \"covered_activities\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\"\n ],\n \"package_price\": 280\n },\n {\n \"package_id\": \"S39\",\n \"covered_activities\": [\n \"J\",\n \"M\",\n \"N\",\n \"R\"\n ],\n \"package_price\": 316\n },\n {\n \"package_id\": \"S40\",\n \"covered_activities\": [\n \"I\",\n \"M\",\n \"O\",\n \"Q\"\n ],\n \"package_price\": 240\n },\n {\n \"package_id\": \"S41\",\n \"covered_activities\": [\n \"Q\",\n \"R\",\n \"S\",\n \"U\"\n ],\n \"package_price\": 224\n },\n {\n \"package_id\": \"S42\",\n \"covered_activities\": [\n \"C\",\n \"D\",\n \"E\",\n \"G\"\n ],\n \"package_price\": 384\n },\n {\n \"package_id\": \"S43\",\n \"covered_activities\": [\n \"T\",\n \"U\"\n ],\n \"package_price\": 2\n },\n {\n \"package_id\": \"S44\",\n \"covered_activities\": [\n \"E\",\n \"F\",\n \"G\",\n \"H\"\n ],\n \"package_price\": 16\n },\n {\n \"package_id\": \"S45\",\n \"covered_activities\": [\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\"\n ],\n \"package_price\": 25\n },\n {\n \"package_id\": \"S46\",\n \"covered_activities\": [\n \"B\",\n \"G\"\n ],\n \"package_price\": 46\n },\n {\n \"package_id\": \"S47\",\n \"covered_activities\": [\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"M\"\n ],\n \"package_price\": 108\n },\n {\n \"package_id\": \"S48\",\n \"covered_activities\": [\n \"P\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"package_price\": 276\n },\n {\n \"package_id\": \"S49\",\n \"covered_activities\": [\n \"G\",\n \"I\",\n \"J\",\n \"K\",\n \"M\"\n ],\n \"package_price\": 370\n },\n {\n \"package_id\": \"S50\",\n \"covered_activities\": [\n \"H\",\n \"I\",\n \"J\",\n \"L\",\n \"M\",\n \"O\"\n ],\n \"package_price\": 558\n },\n {\n \"package_id\": \"S51\",\n \"covered_activities\": [\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"R\",\n \"S\",\n \"T\"\n ],\n \"package_price\": 623\n },\n {\n \"package_id\": \"S52\",\n \"covered_activities\": [\n \"N\",\n \"Q\",\n \"R\"\n ],\n \"package_price\": 195\n },\n {\n \"package_id\": \"S53\",\n \"covered_activities\": [\n \"I\",\n \"L\",\n \"N\"\n ],\n \"package_price\": 201\n },\n {\n \"package_id\": \"S54\",\n \"covered_activities\": [\n \"H\",\n \"M\",\n \"Q\"\n ],\n \"package_price\": 126\n },\n {\n \"package_id\": \"S55\",\n \"covered_activities\": [\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"U\",\n \"V\"\n ],\n \"package_price\": 546\n },\n {\n \"package_id\": \"S56\",\n \"covered_activities\": [\n \"D\",\n \"F\",\n \"G\",\n \"J\"\n ],\n \"package_price\": 160\n },\n {\n \"package_id\": \"S57\",\n \"covered_activities\": [\n \"H\",\n \"I\",\n \"J\"\n ],\n \"package_price\": 42\n },\n {\n \"package_id\": \"S58\",\n \"covered_activities\": [\n \"E\",\n \"H\",\n \"I\"\n ],\n \"package_price\": 183\n },\n {\n \"package_id\": \"S59\",\n \"covered_activities\": [\n \"L\",\n \"N\",\n \"P\",\n \"Q\",\n \"S\"\n ],\n \"package_price\": 400\n },\n {\n \"package_id\": \"S60\",\n \"covered_activities\": [\n \"B\",\n \"E\",\n \"F\"\n ],\n \"package_price\": 297\n },\n {\n \"package_id\": \"S61\",\n \"covered_activities\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"F\",\n \"I\"\n ],\n \"package_price\": 408\n },\n {\n \"package_id\": \"S62\",\n \"covered_activities\": [\n \"A\",\n \"C\",\n \"D\",\n \"E\"\n ],\n \"package_price\": 188\n },\n {\n \"package_id\": \"S63\",\n \"covered_activities\": [\n \"P\",\n \"Q\",\n \"T\"\n ],\n \"package_price\": 63\n },\n {\n \"package_id\": \"S64\",\n \"covered_activities\": [\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"O\"\n ],\n \"package_price\": 450\n },\n {\n \"package_id\": \"S65\",\n \"covered_activities\": [\n \"L\",\n \"M\",\n \"O\",\n \"R\"\n ],\n \"package_price\": 364\n },\n {\n \"package_id\": \"S66\",\n \"covered_activities\": [\n \"J\",\n \"L\",\n \"N\",\n \"P\"\n ],\n \"package_price\": 376\n },\n {\n \"package_id\": \"S67\",\n \"covered_activities\": [\n \"I\",\n \"K\",\n \"L\",\n \"M\",\n \"N\"\n ],\n \"package_price\": 85\n },\n {\n \"package_id\": \"S68\",\n \"covered_activities\": [\n \"R\",\n \"T\",\n \"U\",\n \"V\"\n ],\n \"package_price\": 172\n },\n {\n \"package_id\": \"S69\",\n \"covered_activities\": [\n \"A\",\n \"D\"\n ],\n \"package_price\": 80\n },\n {\n \"package_id\": \"S70\",\n \"covered_activities\": [\n \"E\",\n \"F\",\n \"H\",\n \"K\"\n ],\n \"package_price\": 332\n },\n {\n \"package_id\": \"S71\",\n \"covered_activities\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"F\",\n \"G\"\n ],\n \"package_price\": 162\n },\n {\n \"package_id\": \"S72\",\n \"covered_activities\": [\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"Q\"\n ],\n \"package_price\": 340\n },\n {\n \"package_id\": \"S73\",\n \"covered_activities\": [\n \"L\",\n \"M\",\n \"N\",\n \"P\",\n \"S\"\n ],\n \"package_price\": 145\n },\n {\n \"package_id\": \"S74\",\n \"covered_activities\": [\n \"R\",\n \"S\",\n \"U\",\n \"V\"\n ],\n \"package_price\": 384\n },\n {\n \"package_id\": \"S75\",\n \"covered_activities\": [\n \"N\",\n \"Q\"\n ],\n \"package_price\": 182\n },\n {\n \"package_id\": \"S76\",\n \"covered_activities\": [\n \"I\",\n \"K\",\n \"P\"\n ],\n \"package_price\": 84\n },\n {\n \"package_id\": \"S77\",\n \"covered_activities\": [\n \"C\",\n \"D\",\n \"E\",\n \"G\",\n \"H\",\n \"I\"\n ],\n \"package_price\": 468\n },\n {\n \"package_id\": \"S78\",\n \"covered_activities\": [\n \"O\",\n \"R\",\n \"S\"\n ],\n \"package_price\": 147\n },\n {\n \"package_id\": \"S79\",\n \"covered_activities\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"F\"\n ],\n \"package_price\": 225\n },\n {\n \"package_id\": \"S80\",\n \"covered_activities\": [\n \"E\",\n \"H\"\n ],\n \"package_price\": 182\n },\n {\n \"package_id\": \"S81\",\n \"covered_activities\": [\n \"M\",\n \"N\",\n \"O\"\n ],\n \"package_price\": 180\n },\n {\n \"package_id\": \"S82\",\n \"covered_activities\": [\n \"I\",\n \"J\",\n \"K\",\n \"L\"\n ],\n \"package_price\": 296\n },\n {\n \"package_id\": \"S83\",\n \"covered_activities\": [\n \"H\",\n \"I\",\n \"J\",\n \"L\",\n \"M\",\n \"N\"\n ],\n \"package_price\": 510\n },\n {\n \"package_id\": \"S84\",\n \"covered_activities\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\"\n ],\n \"package_price\": 305\n },\n {\n \"package_id\": \"S85\",\n \"covered_activities\": [\n \"M\",\n \"P\",\n \"Q\"\n ],\n \"package_price\": 282\n },\n {\n \"package_id\": \"S86\",\n \"covered_activities\": [\n \"T\",\n \"U\",\n \"V\"\n ],\n \"package_price\": 12\n },\n {\n \"package_id\": \"S87\",\n \"covered_activities\": [\n \"D\",\n \"E\",\n \"I\",\n \"J\"\n ],\n \"package_price\": 212\n },\n {\n \"package_id\": \"S88\",\n \"covered_activities\": [\n \"A\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\"\n ],\n \"package_price\": 426\n },\n {\n \"package_id\": \"S89\",\n \"covered_activities\": [\n \"E\",\n \"F\",\n \"J\"\n ],\n \"package_price\": 222\n },\n {\n \"package_id\": \"S90\",\n \"covered_activities\": [\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"package_price\": 570\n },\n {\n \"package_id\": \"S91\",\n \"covered_activities\": [\n \"N\",\n \"O\",\n \"Q\",\n \"R\",\n \"T\"\n ],\n \"package_price\": 170\n },\n {\n \"package_id\": \"S92\",\n \"covered_activities\": [\n \"O\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\"\n ],\n \"package_price\": 462\n },\n {\n \"package_id\": \"S93\",\n \"covered_activities\": [\n \"O\",\n \"Q\",\n \"R\",\n \"T\"\n ],\n \"package_price\": 168\n },\n {\n \"package_id\": \"S94\",\n \"covered_activities\": [\n \"O\",\n \"R\",\n \"U\"\n ],\n \"package_price\": 12\n },\n {\n \"package_id\": \"S95\",\n \"covered_activities\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\"\n ],\n \"package_price\": 30\n },\n {\n \"package_id\": \"S96\",\n \"covered_activities\": [\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"L\"\n ],\n \"package_price\": 438\n },\n {\n \"package_id\": \"S97\",\n \"covered_activities\": [\n \"S\",\n \"T\",\n \"U\",\n \"V\"\n ],\n \"package_price\": 264\n },\n {\n \"package_id\": \"S98\",\n \"covered_activities\": [\n \"D\",\n \"G\",\n \"I\",\n \"K\"\n ],\n \"package_price\": 388\n },\n {\n \"package_id\": \"S99\",\n \"covered_activities\": [\n \"A\",\n \"B\",\n \"C\",\n \"E\",\n \"F\",\n \"G\"\n ],\n \"package_price\": 444\n },\n {\n \"package_id\": \"S100\",\n \"covered_activities\": [\n \"L\",\n \"M\",\n \"N\",\n \"Q\"\n ],\n \"package_price\": 376\n },\n {\n \"package_id\": \"S101\",\n \"covered_activities\": [\n \"A\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S102\",\n \"covered_activities\": [\n \"B\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S103\",\n \"covered_activities\": [\n \"C\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S104\",\n \"covered_activities\": [\n \"D\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S105\",\n \"covered_activities\": [\n \"E\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S106\",\n \"covered_activities\": [\n \"F\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S107\",\n \"covered_activities\": [\n \"G\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S108\",\n \"covered_activities\": [\n \"H\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S109\",\n \"covered_activities\": [\n \"I\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S110\",\n \"covered_activities\": [\n \"J\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S111\",\n \"covered_activities\": [\n \"K\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S112\",\n \"covered_activities\": [\n \"L\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S113\",\n \"covered_activities\": [\n \"M\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S114\",\n \"covered_activities\": [\n \"N\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S115\",\n \"covered_activities\": [\n \"O\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S116\",\n \"covered_activities\": [\n \"P\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S117\",\n \"covered_activities\": [\n \"Q\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S118\",\n \"covered_activities\": [\n \"R\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S119\",\n \"covered_activities\": [\n \"S\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S120\",\n \"covered_activities\": [\n \"T\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S121\",\n \"covered_activities\": [\n \"U\"\n ],\n \"package_price\": 10000\n },\n {\n \"package_id\": \"S122\",\n \"covered_activities\": [\n \"V\"\n ],\n \"package_price\": 10000\n }\n ]\n}\n\nIf you want to send the chosen packages back to me, just drop them in a little JSON snippet like this so it's easy to parse:\n\n{\n \"solution\": [\"package_id\", ...]\n}\n\nThink of \"solution\" as the list of package IDs you picked. \"package_id\" is just a placeholder showing the kind of entries I expect — replace it with the actual package identifiers from the problem (that could be numbers, letters, or mixes). This is just a sketch of the shape I want, not the final answer itself.\n\nPlease make sure to use the exact identifiers from the instance input — do not rename them or invent new labels. \nFor example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 22, "num_sets": 122, "density": 0.169150521609538, "sets": [ { "id": 1, "elements": [ 2, 6, 7, 8, 9 ], "cost": 95 }, { "id": 2, "elements": [ 14, 16, 17, 18, 19, 20 ], "cost": 312 }, { "id": 3, "elements": [ 6, 8, 10, 11, 12 ], "cost": 20 }, { "id": 4, "elements": [ 1, 4, 5, 6, 7, 9 ], "cost": 300 }, { "id": 5, "elements": [ 1, 7 ], "cost": 100 }, { "id": 6, "elements": [ 1, 2, 5, 6 ], "cost": 16 }, { "id": 7, "elements": [ 1, 3 ], "cost": 166 }, { "id": 8, "elements": [ 7, 11, 12 ], "cost": 21 }, { "id": 9, "elements": [ 6, 8, 9, 10 ], "cost": 332 }, { "id": 10, "elements": [ 2, 3, 5, 6 ], "cost": 256 }, { "id": 11, "elements": [ 17, 19, 20, 21 ], "cost": 72 }, { "id": 12, "elements": [ 10, 12, 14, 15, 16 ], "cost": 315 }, { "id": 13, "elements": [ 16, 17, 18, 19, 21 ], "cost": 45 }, { "id": 14, "elements": [ 5, 7, 8, 10, 12, 14 ], "cost": 354 }, { "id": 15, "elements": [ 4, 6, 9 ], "cost": 66 }, { "id": 16, "elements": [ 10, 11, 13 ], "cost": 6 }, { "id": 17, "elements": [ 3, 4, 7 ], "cost": 147 }, { "id": 18, "elements": [ 12, 14, 16 ], "cost": 51 }, { "id": 19, "elements": [ 15, 16, 17, 18 ], "cost": 40 }, { "id": 20, "elements": [ 14, 16, 17, 18, 20, 22 ], "cost": 114 }, { "id": 21, "elements": [ 7, 8, 9, 10, 14, 16 ], "cost": 162 }, { "id": 22, "elements": [ 4, 5, 6, 7, 8 ], "cost": 320 }, { "id": 23, "elements": [ 3, 6, 7, 8, 9 ], "cost": 380 }, { "id": 24, "elements": [ 3, 4, 5, 7, 11 ], "cost": 350 }, { "id": 25, "elements": [ 5, 7, 8, 9, 10, 11 ], "cost": 246 }, { "id": 26, "elements": [ 6, 9, 13, 15 ], "cost": 360 }, { "id": 27, "elements": [ 18, 19, 20, 21, 22 ], "cost": 260 }, { "id": 28, "elements": [ 13, 14, 15, 16, 18, 19 ], "cost": 6 }, { "id": 29, "elements": [ 2, 3, 6, 7 ], "cost": 168 }, { "id": 30, "elements": [ 1, 2, 3, 4, 8 ], "cost": 290 }, { "id": 31, "elements": [ 16, 18, 19, 20, 21, 22 ], "cost": 444 }, { "id": 32, "elements": [ 18, 19, 20 ], "cost": 18 }, { "id": 33, "elements": [ 15, 16, 18, 19, 20 ], "cost": 350 }, { "id": 34, "elements": [ 4, 6, 7, 8, 9, 10 ], "cost": 180 }, { "id": 35, "elements": [ 16, 18, 19, 20, 22 ], "cost": 415 }, { "id": 36, "elements": [ 19, 20, 21 ], "cost": 243 }, { "id": 37, "elements": [ 2, 4 ], "cost": 140 }, { "id": 38, "elements": [ 1, 2, 3, 4 ], "cost": 280 }, { "id": 39, "elements": [ 10, 13, 14, 18 ], "cost": 316 }, { "id": 40, "elements": [ 9, 13, 15, 17 ], "cost": 240 }, { "id": 41, "elements": [ 17, 18, 19, 21 ], "cost": 224 }, { "id": 42, "elements": [ 3, 4, 5, 7 ], "cost": 384 }, { "id": 43, "elements": [ 20, 21 ], "cost": 2 }, { "id": 44, "elements": [ 5, 6, 7, 8 ], "cost": 16 }, { "id": 45, "elements": [ 10, 11, 12, 13, 14 ], "cost": 25 }, { "id": 46, "elements": [ 2, 7 ], "cost": 46 }, { "id": 47, "elements": [ 8, 9, 10, 11, 12, 13 ], "cost": 108 }, { "id": 48, "elements": [ 16, 19, 20, 21 ], "cost": 276 }, { "id": 49, "elements": [ 7, 9, 10, 11, 13 ], "cost": 370 }, { "id": 50, "elements": [ 8, 9, 10, 12, 13, 15 ], "cost": 558 }, { "id": 51, "elements": [ 13, 14, 15, 16, 18, 19, 20 ], "cost": 623 }, { "id": 52, "elements": [ 14, 17, 18 ], "cost": 195 }, { "id": 53, "elements": [ 9, 12, 14 ], "cost": 201 }, { "id": 54, "elements": [ 8, 13, 17 ], "cost": 126 }, { "id": 55, "elements": [ 16, 17, 18, 19, 21, 22 ], "cost": 546 }, { "id": 56, "elements": [ 4, 6, 7, 10 ], "cost": 160 }, { "id": 57, "elements": [ 8, 9, 10 ], "cost": 42 }, { "id": 58, "elements": [ 5, 8, 9 ], "cost": 183 }, { "id": 59, "elements": [ 12, 14, 16, 17, 19 ], "cost": 400 }, { "id": 60, "elements": [ 2, 5, 6 ], "cost": 297 }, { "id": 61, "elements": [ 1, 2, 3, 4, 6, 9 ], "cost": 408 }, { "id": 62, "elements": [ 1, 3, 4, 5 ], "cost": 188 }, { "id": 63, "elements": [ 16, 17, 20 ], "cost": 63 }, { "id": 64, "elements": [ 10, 11, 12, 13, 14, 15 ], "cost": 450 }, { "id": 65, "elements": [ 12, 13, 15, 18 ], "cost": 364 }, { "id": 66, "elements": [ 10, 12, 14, 16 ], "cost": 376 }, { "id": 67, "elements": [ 9, 11, 12, 13, 14 ], "cost": 85 }, { "id": 68, "elements": [ 18, 20, 21, 22 ], "cost": 172 }, { "id": 69, "elements": [ 1, 4 ], "cost": 80 }, { "id": 70, "elements": [ 5, 6, 8, 11 ], "cost": 332 }, { "id": 71, "elements": [ 1, 2, 3, 4, 6, 7 ], "cost": 162 }, { "id": 72, "elements": [ 11, 12, 13, 14, 17 ], "cost": 340 }, { "id": 73, "elements": [ 12, 13, 14, 16, 19 ], "cost": 145 }, { "id": 74, "elements": [ 18, 19, 21, 22 ], "cost": 384 }, { "id": 75, "elements": [ 14, 17 ], "cost": 182 }, { "id": 76, "elements": [ 9, 11, 16 ], "cost": 84 }, { "id": 77, "elements": [ 3, 4, 5, 7, 8, 9 ], "cost": 468 }, { "id": 78, "elements": [ 15, 18, 19 ], "cost": 147 }, { "id": 79, "elements": [ 1, 2, 3, 4, 6 ], "cost": 225 }, { "id": 80, "elements": [ 5, 8 ], "cost": 182 }, { "id": 81, "elements": [ 13, 14, 15 ], "cost": 180 }, { "id": 82, "elements": [ 9, 10, 11, 12 ], "cost": 296 }, { "id": 83, "elements": [ 8, 9, 10, 12, 13, 14 ], "cost": 510 }, { "id": 84, "elements": [ 1, 2, 3, 4, 5 ], "cost": 305 }, { "id": 85, "elements": [ 13, 16, 17 ], "cost": 282 }, { "id": 86, "elements": [ 20, 21, 22 ], "cost": 12 }, { "id": 87, "elements": [ 4, 5, 9, 10 ], "cost": 212 }, { "id": 88, "elements": [ 1, 3, 4, 5, 6, 7 ], "cost": 426 }, { "id": 89, "elements": [ 5, 6, 10 ], "cost": 222 }, { "id": 90, "elements": [ 16, 17, 18, 19, 20, 21 ], "cost": 570 }, { "id": 91, "elements": [ 14, 15, 17, 18, 20 ], "cost": 170 }, { "id": 92, "elements": [ 15, 16, 17, 18, 19, 20 ], "cost": 462 }, { "id": 93, "elements": [ 15, 17, 18, 20 ], "cost": 168 }, { "id": 94, "elements": [ 15, 18, 21 ], "cost": 12 }, { "id": 95, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 30 }, { "id": 96, "elements": [ 7, 8, 9, 10, 11, 12 ], "cost": 438 }, { "id": 97, "elements": [ 19, 20, 21, 22 ], "cost": 264 }, { "id": 98, "elements": [ 4, 7, 9, 11 ], "cost": 388 }, { "id": 99, "elements": [ 1, 2, 3, 5, 6, 7 ], "cost": 444 }, { "id": 100, "elements": [ 12, 13, 14, 17 ], "cost": 376 }, { "id": 101, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 118, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 119, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 120, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 121, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 122, "elements": [ 22 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0020_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0020_bag_stats.png" }, "solution": [ 1, 19, 45, 62, 97 ], "obj": 612.0, "instance_variant": { "num_elements": 22, "num_sets": 122, "sets": [ { "id": "S1", "elements": [ "B", "F", "G", "H", "I" ], "cost": 95 }, { "id": "S2", "elements": [ "N", "P", "Q", "R", "S", "T" ], "cost": 312 }, { "id": "S3", "elements": [ "F", "H", "J", "K", "L" ], "cost": 20 }, { "id": "S4", "elements": [ "A", "D", "E", "F", "G", "I" ], "cost": 300 }, { "id": "S5", "elements": [ "A", "G" ], "cost": 100 }, { "id": "S6", "elements": [ "A", "B", "E", "F" ], "cost": 16 }, { "id": "S7", "elements": [ "A", "C" ], "cost": 166 }, { "id": "S8", "elements": [ "G", "K", "L" ], "cost": 21 }, { "id": "S9", "elements": [ "F", "H", "I", "J" ], "cost": 332 }, { "id": "S10", "elements": [ "B", "C", "E", "F" ], "cost": 256 }, { "id": "S11", "elements": [ "Q", "S", "T", "U" ], "cost": 72 }, { "id": "S12", "elements": [ "J", "L", "N", "O", "P" ], "cost": 315 }, { "id": "S13", "elements": [ "P", "Q", "R", "S", "U" ], "cost": 45 }, { "id": "S14", "elements": [ "E", "G", "H", "J", "L", "N" ], "cost": 354 }, { "id": "S15", "elements": [ "D", "F", "I" ], "cost": 66 }, { "id": "S16", "elements": [ "J", "K", "M" ], "cost": 6 }, { "id": "S17", "elements": [ "C", "D", "G" ], "cost": 147 }, { "id": "S18", "elements": [ "L", "N", "P" ], "cost": 51 }, { "id": "S19", "elements": [ "O", "P", "Q", "R" ], "cost": 40 }, { "id": "S20", "elements": [ "N", "P", "Q", "R", "T", "V" ], "cost": 114 }, { "id": "S21", "elements": [ "G", "H", "I", "J", "N", "P" ], "cost": 162 }, { "id": "S22", "elements": [ "D", "E", "F", "G", "H" ], "cost": 320 }, { "id": "S23", "elements": [ "C", "F", "G", "H", "I" ], "cost": 380 }, { "id": "S24", "elements": [ "C", "D", "E", "G", "K" ], "cost": 350 }, { "id": "S25", "elements": [ "E", "G", "H", "I", "J", "K" ], "cost": 246 }, { "id": "S26", "elements": [ "F", "I", "M", "O" ], "cost": 360 }, { "id": "S27", "elements": [ "R", "S", "T", "U", "V" ], "cost": 260 }, { "id": "S28", "elements": [ "M", "N", "O", "P", "R", "S" ], "cost": 6 }, { "id": "S29", "elements": [ "B", "C", "F", "G" ], "cost": 168 }, { "id": "S30", "elements": [ "A", "B", "C", "D", "H" ], "cost": 290 }, { "id": "S31", "elements": [ "P", "R", "S", "T", "U", "V" ], "cost": 444 }, { "id": "S32", "elements": [ "R", "S", "T" ], "cost": 18 }, { "id": "S33", "elements": [ "O", "P", "R", "S", "T" ], "cost": 350 }, { "id": "S34", "elements": [ "D", "F", "G", "H", "I", "J" ], "cost": 180 }, { "id": "S35", "elements": [ "P", "R", "S", "T", "V" ], "cost": 415 }, { "id": "S36", "elements": [ "S", "T", "U" ], "cost": 243 }, { "id": "S37", "elements": [ "B", "D" ], "cost": 140 }, { "id": "S38", "elements": [ "A", "B", "C", "D" ], "cost": 280 }, { "id": "S39", "elements": [ "J", "M", "N", "R" ], "cost": 316 }, { "id": "S40", "elements": [ "I", "M", "O", "Q" ], "cost": 240 }, { "id": "S41", "elements": [ "Q", "R", "S", "U" ], "cost": 224 }, { "id": "S42", "elements": [ "C", "D", "E", "G" ], "cost": 384 }, { "id": "S43", "elements": [ "T", "U" ], "cost": 2 }, { "id": "S44", "elements": [ "E", "F", "G", "H" ], "cost": 16 }, { "id": "S45", "elements": [ "J", "K", "L", "M", "N" ], "cost": 25 }, { "id": "S46", "elements": [ "B", "G" ], "cost": 46 }, { "id": "S47", "elements": [ "H", "I", "J", "K", "L", "M" ], "cost": 108 }, { "id": "S48", "elements": [ "P", "S", "T", "U" ], "cost": 276 }, { "id": "S49", "elements": [ "G", "I", "J", "K", "M" ], "cost": 370 }, { "id": "S50", "elements": [ "H", "I", "J", "L", "M", "O" ], "cost": 558 }, { "id": "S51", "elements": [ "M", "N", "O", "P", "R", "S", "T" ], "cost": 623 }, { "id": "S52", "elements": [ "N", "Q", "R" ], "cost": 195 }, { "id": "S53", "elements": [ "I", "L", "N" ], "cost": 201 }, { "id": "S54", "elements": [ "H", "M", "Q" ], "cost": 126 }, { "id": "S55", "elements": [ "P", "Q", "R", "S", "U", "V" ], "cost": 546 }, { "id": "S56", "elements": [ "D", "F", "G", "J" ], "cost": 160 }, { "id": "S57", "elements": [ "H", "I", "J" ], "cost": 42 }, { "id": "S58", "elements": [ "E", "H", "I" ], "cost": 183 }, { "id": "S59", "elements": [ "L", "N", "P", "Q", "S" ], "cost": 400 }, { "id": "S60", "elements": [ "B", "E", "F" ], "cost": 297 }, { "id": "S61", "elements": [ "A", "B", "C", "D", "F", "I" ], "cost": 408 }, { "id": "S62", "elements": [ "A", "C", "D", "E" ], "cost": 188 }, { "id": "S63", "elements": [ "P", "Q", "T" ], "cost": 63 }, { "id": "S64", "elements": [ "J", "K", "L", "M", "N", "O" ], "cost": 450 }, { "id": "S65", "elements": [ "L", "M", "O", "R" ], "cost": 364 }, { "id": "S66", "elements": [ "J", "L", "N", "P" ], "cost": 376 }, { "id": "S67", "elements": [ "I", "K", "L", "M", "N" ], "cost": 85 }, { "id": "S68", "elements": [ "R", "T", "U", "V" ], "cost": 172 }, { "id": "S69", "elements": [ "A", "D" ], "cost": 80 }, { "id": "S70", "elements": [ "E", "F", "H", "K" ], "cost": 332 }, { "id": "S71", "elements": [ "A", "B", "C", "D", "F", "G" ], "cost": 162 }, { "id": "S72", "elements": [ "K", "L", "M", "N", "Q" ], "cost": 340 }, { "id": "S73", "elements": [ "L", "M", "N", "P", "S" ], "cost": 145 }, { "id": "S74", "elements": [ "R", "S", "U", "V" ], "cost": 384 }, { "id": "S75", "elements": [ "N", "Q" ], "cost": 182 }, { "id": "S76", "elements": [ "I", "K", "P" ], "cost": 84 }, { "id": "S77", "elements": [ "C", "D", "E", "G", "H", "I" ], "cost": 468 }, { "id": "S78", "elements": [ "O", "R", "S" ], "cost": 147 }, { "id": "S79", "elements": [ "A", "B", "C", "D", "F" ], "cost": 225 }, { "id": "S80", "elements": [ "E", "H" ], "cost": 182 }, { "id": "S81", "elements": [ "M", "N", "O" ], "cost": 180 }, { "id": "S82", "elements": [ "I", "J", "K", "L" ], "cost": 296 }, { "id": "S83", "elements": [ "H", "I", "J", "L", "M", "N" ], "cost": 510 }, { "id": "S84", "elements": [ "A", "B", "C", "D", "E" ], "cost": 305 }, { "id": "S85", "elements": [ "M", "P", "Q" ], "cost": 282 }, { "id": "S86", "elements": [ "T", "U", "V" ], "cost": 12 }, { "id": "S87", "elements": [ "D", "E", "I", "J" ], "cost": 212 }, { "id": "S88", "elements": [ "A", "C", "D", "E", "F", "G" ], "cost": 426 }, { "id": "S89", "elements": [ "E", "F", "J" ], "cost": 222 }, { "id": "S90", "elements": [ "P", "Q", "R", "S", "T", "U" ], "cost": 570 }, { "id": "S91", "elements": [ "N", "O", "Q", "R", "T" ], "cost": 170 }, { "id": "S92", "elements": [ "O", "P", "Q", "R", "S", "T" ], "cost": 462 }, { "id": "S93", "elements": [ "O", "Q", "R", "T" ], "cost": 168 }, { "id": "S94", "elements": [ "O", "R", "U" ], "cost": 12 }, { "id": "S95", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 30 }, { "id": "S96", "elements": [ "G", "H", "I", "J", "K", "L" ], "cost": 438 }, { "id": "S97", "elements": [ "S", "T", "U", "V" ], "cost": 264 }, { "id": "S98", "elements": [ "D", "G", "I", "K" ], "cost": 388 }, { "id": "S99", "elements": [ "A", "B", "C", "E", "F", "G" ], "cost": 444 }, { "id": "S100", "elements": [ "L", "M", "N", "Q" ], "cost": 376 }, { "id": "S101", "elements": [ "A" ], "cost": 10000 }, { "id": "S102", "elements": [ "B" ], "cost": 10000 }, { "id": "S103", "elements": [ "C" ], "cost": 10000 }, { "id": "S104", "elements": [ "D" ], "cost": 10000 }, { "id": "S105", "elements": [ "E" ], "cost": 10000 }, { "id": "S106", "elements": [ "F" ], "cost": 10000 }, { "id": "S107", "elements": [ "G" ], "cost": 10000 }, { "id": "S108", "elements": [ "H" ], "cost": 10000 }, { "id": "S109", "elements": [ "I" ], "cost": 10000 }, { "id": "S110", "elements": [ "J" ], "cost": 10000 }, { "id": "S111", "elements": [ "K" ], "cost": 10000 }, { "id": "S112", "elements": [ "L" ], "cost": 10000 }, { "id": "S113", "elements": [ "M" ], "cost": 10000 }, { "id": "S114", "elements": [ "N" ], "cost": 10000 }, { "id": "S115", "elements": [ "O" ], "cost": 10000 }, { "id": "S116", "elements": [ "P" ], "cost": 10000 }, { "id": "S117", "elements": [ "Q" ], "cost": 10000 }, { "id": "S118", "elements": [ "R" ], "cost": 10000 }, { "id": "S119", "elements": [ "S" ], "cost": 10000 }, { "id": "S120", "elements": [ "T" ], "cost": 10000 }, { "id": "S121", "elements": [ "U" ], "cost": 10000 }, { "id": "S122", "elements": [ "V" ], "cost": 10000 } ] }, "solution_variant": [ "S1", "S19", "S45", "S62", "S97" ], "context_index": 21, "input_format": "json", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "Recently the PTA noticed multiple combo meal offers and wants to pick some so each kid’s lunch requirement is satisfied by a single selected combo — no duplicates, no missing lunches. The plan is to add up the prices of whatever combos are picked to see the total cost, and the aim is to keep that total as low as possible. The exact bundle options, which kids need what, and the prices are given below.\n\n- **total_children**: 23\n- **total_bundles**: 144\n\n| bundle_id | bundle_price | covered_children |\n|---|---|---|\n| S1 | 510 | A B C D E F |\n| S2 | 190 | H I |\n| S3 | 135 | A B C D E |\n| S4 | 100 | L M N O T |\n| S5 | 177 | C G H |\n| S6 | 60 | S T V W |\n| S7 | 240 | I J P |\n| S8 | 480 | L M N O P Q |\n| S9 | 354 | C F G H I J |\n| S10 | 65 | R S U V W |\n| S11 | 54 | B C D |\n| S12 | 175 | A B C D E F H |\n| S13 | 6 | K L M N O R |\n| S14 | 600 | E F G H J K |\n| S15 | 138 | L N O |\n| S16 | 50 | M O Q R V |\n| S17 | 180 | A C D E H J |\n| S18 | 280 | E F J K N |\n| S19 | 336 | N P Q S |\n| S20 | 637 | A B C D E F G |\n| S21 | 315 | Q R S T U V W |\n| S22 | 72 | L O P R |\n| S23 | 96 | C E F G I L |\n| S24 | 106 | C D |\n| S25 | 30 | B C D E H |\n| S26 | 465 | N O P R S |\n| S27 | 18 | H J L |\n| S28 | 28 | A B C D |\n| S29 | 546 | F H I J K M |\n| S30 | 204 | A B C |\n| S31 | 5 | J L M N O |\n| S32 | 228 | R T V W |\n| S33 | 427 | J K L M N O Q |\n| S34 | 110 | B G |\n| S35 | 45 | F H L |\n| S36 | 15 | R S V |\n| S37 | 198 | S T |\n| S38 | 384 | H I J K L N |\n| S39 | 390 | Q R T U V |\n| S40 | 35 | G H J M N |\n| S41 | 475 | P Q R S T |\n| S42 | 290 | P Q R U W |\n| S43 | 24 | A C D E F I |\n| S44 | 66 | B C E |\n| S45 | 12 | J L O |\n| S46 | 55 | R T U V W |\n| S47 | 195 | T U V |\n| S48 | 414 | P Q S T V W |\n| S49 | 350 | S T U V W |\n| S50 | 210 | F G H I J K M |\n| S51 | 20 | K N O R S |\n| S52 | 66 | B C E F G H |\n| S53 | 140 | N P Q R S T U |\n| S54 | 328 | L M Q T |\n| S55 | 138 | A B C E F G |\n| S56 | 249 | O P R |\n| S57 | 148 | J K M N |\n| S58 | 240 | H K L M N O |\n| S59 | 108 | J L M |\n| S60 | 260 | A C F H I |\n| S61 | 240 | P Q R |\n| S62 | 276 | B C D E F G |\n| S63 | 150 | Q R S U V |\n| S64 | 18 | M Q S |\n| S65 | 252 | F H I J K N O |\n| S66 | 228 | K M N O P Q |\n| S67 | 384 | R S T U |\n| S68 | 291 | N P R |\n| S69 | 355 | B C D E F |\n| S70 | 42 | F H J |\n| S71 | 414 | I J K L N O |\n| S72 | 325 | E G H I J |\n| S73 | 164 | H J K M |\n| S74 | 372 | A B C D E G |\n| S75 | 56 | D E F G I J K |\n| S76 | 39 | I L P |\n| S77 | 420 | C D H I J |\n| S78 | 245 | F H I L O |\n| S79 | 392 | O R S T U V W |\n| S80 | 252 | A C D E |\n| S81 | 150 | F J K M O P |\n| S82 | 102 | P R S U V W |\n| S83 | 270 | C D E F G H |\n| S84 | 72 | A C F |\n| S85 | 165 | D G I |\n| S86 | 161 | L M O P Q R S |\n| S87 | 132 | B E F G |\n| S88 | 70 | K P R T W |\n| S89 | 104 | P Q R V |\n| S90 | 210 | E F G H I J |\n| S91 | 40 | T U V W |\n| S92 | 90 | A D G |\n| S93 | 360 | N O P Q R T |\n| S94 | 430 | Q R S V W |\n| S95 | 35 | G H J K M |\n| S96 | 74 | L O |\n| S97 | 125 | G H L M N |\n| S98 | 294 | T V W |\n| S99 | 588 | L O P Q R S |\n| S100 | 305 | A B C D F |\n| S101 | 200 | A B C E |\n| S102 | 490 | A B C D E H I |\n| S103 | 25 | C D E F H |\n| S104 | 222 | R S T U V W |\n| S105 | 235 | A B C F G |\n| S106 | 356 | J N P Q |\n| S107 | 42 | S T V |\n| S108 | 186 | V W |\n| S109 | 165 | E G H I M |\n| S110 | 588 | I K L M N O |\n| S111 | 70 | B C D E G |\n| S112 | 376 | M N P R |\n| S113 | 532 | P Q R S T U V |\n| S114 | 332 | N O R V |\n| S115 | 115 | L N O R S |\n| S116 | 30 | N O Q R T |\n| S117 | 288 | B C D E |\n| S118 | 259 | P R S T U V W |\n| S119 | 96 | F G K |\n| S120 | 78 | P Q R S U V |\n| S121 | 118 | H M |\n| S122 | 10000 | A |\n| S123 | 10000 | B |\n| S124 | 10000 | C |\n| S125 | 10000 | D |\n| S126 | 10000 | E |\n| S127 | 10000 | F |\n| S128 | 10000 | G |\n| S129 | 10000 | H |\n| S130 | 10000 | I |\n| S131 | 10000 | J |\n| S132 | 10000 | K |\n| S133 | 10000 | L |\n| S134 | 10000 | M |\n| S135 | 10000 | N |\n| S136 | 10000 | O |\n| S137 | 10000 | P |\n| S138 | 10000 | Q |\n| S139 | 10000 | R |\n| S140 | 10000 | S |\n| S141 | 10000 | T |\n| S142 | 10000 | U |\n| S143 | 10000 | V |\n| S144 | 10000 | W |\n\nIf you want to tell me which combos the PTA will pick, a simple JSON sketch works great — something like this:\n\n{\n \"solution\": [\"combo_id\"]\n}\n\n\"solution\" is just the list of combo identifiers you’re choosing (one entry per selected combo). Think of it as a tiny form: drop the combo IDs in that array and that’s the set of combos you picked. This is only a sketch of the expected shape, not the actual answer.\n\nPlease make sure to use the identifiers exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 23, "num_sets": 144, "density": 0.1793478260869565, "sets": [ { "id": 1, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 510 }, { "id": 2, "elements": [ 8, 9 ], "cost": 190 }, { "id": 3, "elements": [ 1, 2, 3, 4, 5 ], "cost": 135 }, { "id": 4, "elements": [ 12, 13, 14, 15, 20 ], "cost": 100 }, { "id": 5, "elements": [ 3, 7, 8 ], "cost": 177 }, { "id": 6, "elements": [ 19, 20, 22, 23 ], "cost": 60 }, { "id": 7, "elements": [ 9, 10, 16 ], "cost": 240 }, { "id": 8, "elements": [ 12, 13, 14, 15, 16, 17 ], "cost": 480 }, { "id": 9, "elements": [ 3, 6, 7, 8, 9, 10 ], "cost": 354 }, { "id": 10, "elements": [ 18, 19, 21, 22, 23 ], "cost": 65 }, { "id": 11, "elements": [ 2, 3, 4 ], "cost": 54 }, { "id": 12, "elements": [ 1, 2, 3, 4, 5, 6, 8 ], "cost": 175 }, { "id": 13, "elements": [ 11, 12, 13, 14, 15, 18 ], "cost": 6 }, { "id": 14, "elements": [ 5, 6, 7, 8, 10, 11 ], "cost": 600 }, { "id": 15, "elements": [ 12, 14, 15 ], "cost": 138 }, { "id": 16, "elements": [ 13, 15, 17, 18, 22 ], "cost": 50 }, { "id": 17, "elements": [ 1, 3, 4, 5, 8, 10 ], "cost": 180 }, { "id": 18, "elements": [ 5, 6, 10, 11, 14 ], "cost": 280 }, { "id": 19, "elements": [ 14, 16, 17, 19 ], "cost": 336 }, { "id": 20, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 637 }, { "id": 21, "elements": [ 17, 18, 19, 20, 21, 22, 23 ], "cost": 315 }, { "id": 22, "elements": [ 12, 15, 16, 18 ], "cost": 72 }, { "id": 23, "elements": [ 3, 5, 6, 7, 9, 12 ], "cost": 96 }, { "id": 24, "elements": [ 3, 4 ], "cost": 106 }, { "id": 25, "elements": [ 2, 3, 4, 5, 8 ], "cost": 30 }, { "id": 26, "elements": [ 14, 15, 16, 18, 19 ], "cost": 465 }, { "id": 27, "elements": [ 8, 10, 12 ], "cost": 18 }, { "id": 28, "elements": [ 1, 2, 3, 4 ], "cost": 28 }, { "id": 29, "elements": [ 6, 8, 9, 10, 11, 13 ], "cost": 546 }, { "id": 30, "elements": [ 1, 2, 3 ], "cost": 204 }, { "id": 31, "elements": [ 10, 12, 13, 14, 15 ], "cost": 5 }, { "id": 32, "elements": [ 18, 20, 22, 23 ], "cost": 228 }, { "id": 33, "elements": [ 10, 11, 12, 13, 14, 15, 17 ], "cost": 427 }, { "id": 34, "elements": [ 2, 7 ], "cost": 110 }, { "id": 35, "elements": [ 6, 8, 12 ], "cost": 45 }, { "id": 36, "elements": [ 18, 19, 22 ], "cost": 15 }, { "id": 37, "elements": [ 19, 20 ], "cost": 198 }, { "id": 38, "elements": [ 8, 9, 10, 11, 12, 14 ], "cost": 384 }, { "id": 39, "elements": [ 17, 18, 20, 21, 22 ], "cost": 390 }, { "id": 40, "elements": [ 7, 8, 10, 13, 14 ], "cost": 35 }, { "id": 41, "elements": [ 16, 17, 18, 19, 20 ], "cost": 475 }, { "id": 42, "elements": [ 16, 17, 18, 21, 23 ], "cost": 290 }, { "id": 43, "elements": [ 1, 3, 4, 5, 6, 9 ], "cost": 24 }, { "id": 44, "elements": [ 2, 3, 5 ], "cost": 66 }, { "id": 45, "elements": [ 10, 12, 15 ], "cost": 12 }, { "id": 46, "elements": [ 18, 20, 21, 22, 23 ], "cost": 55 }, { "id": 47, "elements": [ 20, 21, 22 ], "cost": 195 }, { "id": 48, "elements": [ 16, 17, 19, 20, 22, 23 ], "cost": 414 }, { "id": 49, "elements": [ 19, 20, 21, 22, 23 ], "cost": 350 }, { "id": 50, "elements": [ 6, 7, 8, 9, 10, 11, 13 ], "cost": 210 }, { "id": 51, "elements": [ 11, 14, 15, 18, 19 ], "cost": 20 }, { "id": 52, "elements": [ 2, 3, 5, 6, 7, 8 ], "cost": 66 }, { "id": 53, "elements": [ 14, 16, 17, 18, 19, 20, 21 ], "cost": 140 }, { "id": 54, "elements": [ 12, 13, 17, 20 ], "cost": 328 }, { "id": 55, "elements": [ 1, 2, 3, 5, 6, 7 ], "cost": 138 }, { "id": 56, "elements": [ 15, 16, 18 ], "cost": 249 }, { "id": 57, "elements": [ 10, 11, 13, 14 ], "cost": 148 }, { "id": 58, "elements": [ 8, 11, 12, 13, 14, 15 ], "cost": 240 }, { "id": 59, "elements": [ 10, 12, 13 ], "cost": 108 }, { "id": 60, "elements": [ 1, 3, 6, 8, 9 ], "cost": 260 }, { "id": 61, "elements": [ 16, 17, 18 ], "cost": 240 }, { "id": 62, "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 276 }, { "id": 63, "elements": [ 17, 18, 19, 21, 22 ], "cost": 150 }, { "id": 64, "elements": [ 13, 17, 19 ], "cost": 18 }, { "id": 65, "elements": [ 6, 8, 9, 10, 11, 14, 15 ], "cost": 252 }, { "id": 66, "elements": [ 11, 13, 14, 15, 16, 17 ], "cost": 228 }, { "id": 67, "elements": [ 18, 19, 20, 21 ], "cost": 384 }, { "id": 68, "elements": [ 14, 16, 18 ], "cost": 291 }, { "id": 69, "elements": [ 2, 3, 4, 5, 6 ], "cost": 355 }, { "id": 70, "elements": [ 6, 8, 10 ], "cost": 42 }, { "id": 71, "elements": [ 9, 10, 11, 12, 14, 15 ], "cost": 414 }, { "id": 72, "elements": [ 5, 7, 8, 9, 10 ], "cost": 325 }, { "id": 73, "elements": [ 8, 10, 11, 13 ], "cost": 164 }, { "id": 74, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 372 }, { "id": 75, "elements": [ 4, 5, 6, 7, 9, 10, 11 ], "cost": 56 }, { "id": 76, "elements": [ 9, 12, 16 ], "cost": 39 }, { "id": 77, "elements": [ 3, 4, 8, 9, 10 ], "cost": 420 }, { "id": 78, "elements": [ 6, 8, 9, 12, 15 ], "cost": 245 }, { "id": 79, "elements": [ 15, 18, 19, 20, 21, 22, 23 ], "cost": 392 }, { "id": 80, "elements": [ 1, 3, 4, 5 ], "cost": 252 }, { "id": 81, "elements": [ 6, 10, 11, 13, 15, 16 ], "cost": 150 }, { "id": 82, "elements": [ 16, 18, 19, 21, 22, 23 ], "cost": 102 }, { "id": 83, "elements": [ 3, 4, 5, 6, 7, 8 ], "cost": 270 }, { "id": 84, "elements": [ 1, 3, 6 ], "cost": 72 }, { "id": 85, "elements": [ 4, 7, 9 ], "cost": 165 }, { "id": 86, "elements": [ 12, 13, 15, 16, 17, 18, 19 ], "cost": 161 }, { "id": 87, "elements": [ 2, 5, 6, 7 ], "cost": 132 }, { "id": 88, "elements": [ 11, 16, 18, 20, 23 ], "cost": 70 }, { "id": 89, "elements": [ 16, 17, 18, 22 ], "cost": 104 }, { "id": 90, "elements": [ 5, 6, 7, 8, 9, 10 ], "cost": 210 }, { "id": 91, "elements": [ 20, 21, 22, 23 ], "cost": 40 }, { "id": 92, "elements": [ 1, 4, 7 ], "cost": 90 }, { "id": 93, "elements": [ 14, 15, 16, 17, 18, 20 ], "cost": 360 }, { "id": 94, "elements": [ 17, 18, 19, 22, 23 ], "cost": 430 }, { "id": 95, "elements": [ 7, 8, 10, 11, 13 ], "cost": 35 }, { "id": 96, "elements": [ 12, 15 ], "cost": 74 }, { "id": 97, "elements": [ 7, 8, 12, 13, 14 ], "cost": 125 }, { "id": 98, "elements": [ 20, 22, 23 ], "cost": 294 }, { "id": 99, "elements": [ 12, 15, 16, 17, 18, 19 ], "cost": 588 }, { "id": 100, "elements": [ 1, 2, 3, 4, 6 ], "cost": 305 }, { "id": 101, "elements": [ 1, 2, 3, 5 ], "cost": 200 }, { "id": 102, "elements": [ 1, 2, 3, 4, 5, 8, 9 ], "cost": 490 }, { "id": 103, "elements": [ 3, 4, 5, 6, 8 ], "cost": 25 }, { "id": 104, "elements": [ 18, 19, 20, 21, 22, 23 ], "cost": 222 }, { "id": 105, "elements": [ 1, 2, 3, 6, 7 ], "cost": 235 }, { "id": 106, "elements": [ 10, 14, 16, 17 ], "cost": 356 }, { "id": 107, "elements": [ 19, 20, 22 ], "cost": 42 }, { "id": 108, "elements": [ 22, 23 ], "cost": 186 }, { "id": 109, "elements": [ 5, 7, 8, 9, 13 ], "cost": 165 }, { "id": 110, "elements": [ 9, 11, 12, 13, 14, 15 ], "cost": 588 }, { "id": 111, "elements": [ 2, 3, 4, 5, 7 ], "cost": 70 }, { "id": 112, "elements": [ 13, 14, 16, 18 ], "cost": 376 }, { "id": 113, "elements": [ 16, 17, 18, 19, 20, 21, 22 ], "cost": 532 }, { "id": 114, "elements": [ 14, 15, 18, 22 ], "cost": 332 }, { "id": 115, "elements": [ 12, 14, 15, 18, 19 ], "cost": 115 }, { "id": 116, "elements": [ 14, 15, 17, 18, 20 ], "cost": 30 }, { "id": 117, "elements": [ 2, 3, 4, 5 ], "cost": 288 }, { "id": 118, "elements": [ 16, 18, 19, 20, 21, 22, 23 ], "cost": 259 }, { "id": 119, "elements": [ 6, 7, 11 ], "cost": 96 }, { "id": 120, "elements": [ 16, 17, 18, 19, 21, 22 ], "cost": 78 }, { "id": 121, "elements": [ 8, 13 ], "cost": 118 }, { "id": 122, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 123, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 124, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 125, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 126, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 127, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 128, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 129, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 130, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 131, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 132, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 133, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 134, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 135, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 136, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 137, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 138, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 139, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 140, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 141, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 142, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 143, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 144, "elements": [ 23 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0021_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0021_bag_stats.png" }, "solution": [ 27, 34, 43, 66, 104 ], "obj": 602.0, "instance_variant": { "num_elements": 23, "num_sets": 144, "sets": [ { "id": "S1", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 510 }, { "id": "S2", "elements": [ "H", "I" ], "cost": 190 }, { "id": "S3", "elements": [ "A", "B", "C", "D", "E" ], "cost": 135 }, { "id": "S4", "elements": [ "L", "M", "N", "O", "T" ], "cost": 100 }, { "id": "S5", "elements": [ "C", "G", "H" ], "cost": 177 }, { "id": "S6", "elements": [ "S", "T", "V", "W" ], "cost": 60 }, { "id": "S7", "elements": [ "I", "J", "P" ], "cost": 240 }, { "id": "S8", "elements": [ "L", "M", "N", "O", "P", "Q" ], "cost": 480 }, { "id": "S9", "elements": [ "C", "F", "G", "H", "I", "J" ], "cost": 354 }, { "id": "S10", "elements": [ "R", "S", "U", "V", "W" ], "cost": 65 }, { "id": "S11", "elements": [ "B", "C", "D" ], "cost": 54 }, { "id": "S12", "elements": [ "A", "B", "C", "D", "E", "F", "H" ], "cost": 175 }, { "id": "S13", "elements": [ "K", "L", "M", "N", "O", "R" ], "cost": 6 }, { "id": "S14", "elements": [ "E", "F", "G", "H", "J", "K" ], "cost": 600 }, { "id": "S15", "elements": [ "L", "N", "O" ], "cost": 138 }, { "id": "S16", "elements": [ "M", "O", "Q", "R", "V" ], "cost": 50 }, { "id": "S17", "elements": [ "A", "C", "D", "E", "H", "J" ], "cost": 180 }, { "id": "S18", "elements": [ "E", "F", "J", "K", "N" ], "cost": 280 }, { "id": "S19", "elements": [ "N", "P", "Q", "S" ], "cost": 336 }, { "id": "S20", "elements": [ "A", "B", "C", "D", "E", "F", "G" ], "cost": 637 }, { "id": "S21", "elements": [ "Q", "R", "S", "T", "U", "V", "W" ], "cost": 315 }, { "id": "S22", "elements": [ "L", "O", "P", "R" ], "cost": 72 }, { "id": "S23", "elements": [ "C", "E", "F", "G", "I", "L" ], "cost": 96 }, { "id": "S24", "elements": [ "C", "D" ], "cost": 106 }, { "id": "S25", "elements": [ "B", "C", "D", "E", "H" ], "cost": 30 }, { "id": "S26", "elements": [ "N", "O", "P", "R", "S" ], "cost": 465 }, { "id": "S27", "elements": [ "H", "J", "L" ], "cost": 18 }, { "id": "S28", "elements": [ "A", "B", "C", "D" ], "cost": 28 }, { "id": "S29", "elements": [ "F", "H", "I", "J", "K", "M" ], "cost": 546 }, { "id": "S30", "elements": [ "A", "B", "C" ], "cost": 204 }, { "id": "S31", "elements": [ "J", "L", "M", "N", "O" ], "cost": 5 }, { "id": "S32", "elements": [ "R", "T", "V", "W" ], "cost": 228 }, { "id": "S33", "elements": [ "J", "K", "L", "M", "N", "O", "Q" ], "cost": 427 }, { "id": "S34", "elements": [ "B", "G" ], "cost": 110 }, { "id": "S35", "elements": [ "F", "H", "L" ], "cost": 45 }, { "id": "S36", "elements": [ "R", "S", "V" ], "cost": 15 }, { "id": "S37", "elements": [ "S", "T" ], "cost": 198 }, { "id": "S38", "elements": [ "H", "I", "J", "K", "L", "N" ], "cost": 384 }, { "id": "S39", "elements": [ "Q", "R", "T", "U", "V" ], "cost": 390 }, { "id": "S40", "elements": [ "G", "H", "J", "M", "N" ], "cost": 35 }, { "id": "S41", "elements": [ "P", "Q", "R", "S", "T" ], "cost": 475 }, { "id": "S42", "elements": [ "P", "Q", "R", "U", "W" ], "cost": 290 }, { "id": "S43", "elements": [ "A", "C", "D", "E", "F", "I" ], "cost": 24 }, { "id": "S44", "elements": [ "B", "C", "E" ], "cost": 66 }, { "id": "S45", "elements": [ "J", "L", "O" ], "cost": 12 }, { "id": "S46", "elements": [ "R", "T", "U", "V", "W" ], "cost": 55 }, { "id": "S47", "elements": [ "T", "U", "V" ], "cost": 195 }, { "id": "S48", "elements": [ "P", "Q", "S", "T", "V", "W" ], "cost": 414 }, { "id": "S49", "elements": [ "S", "T", "U", "V", "W" ], "cost": 350 }, { "id": "S50", "elements": [ "F", "G", "H", "I", "J", "K", "M" ], "cost": 210 }, { "id": "S51", "elements": [ "K", "N", "O", "R", "S" ], "cost": 20 }, { "id": "S52", "elements": [ "B", "C", "E", "F", "G", "H" ], "cost": 66 }, { "id": "S53", "elements": [ "N", "P", "Q", "R", "S", "T", "U" ], "cost": 140 }, { "id": "S54", "elements": [ "L", "M", "Q", "T" ], "cost": 328 }, { "id": "S55", "elements": [ "A", "B", "C", "E", "F", "G" ], "cost": 138 }, { "id": "S56", "elements": [ "O", "P", "R" ], "cost": 249 }, { "id": "S57", "elements": [ "J", "K", "M", "N" ], "cost": 148 }, { "id": "S58", "elements": [ "H", "K", "L", "M", "N", "O" ], "cost": 240 }, { "id": "S59", "elements": [ "J", "L", "M" ], "cost": 108 }, { "id": "S60", "elements": [ "A", "C", "F", "H", "I" ], "cost": 260 }, { "id": "S61", "elements": [ "P", "Q", "R" ], "cost": 240 }, { "id": "S62", "elements": [ "B", "C", "D", "E", "F", "G" ], "cost": 276 }, { "id": "S63", "elements": [ "Q", "R", "S", "U", "V" ], "cost": 150 }, { "id": "S64", "elements": [ "M", "Q", "S" ], "cost": 18 }, { "id": "S65", "elements": [ "F", "H", "I", "J", "K", "N", "O" ], "cost": 252 }, { "id": "S66", "elements": [ "K", "M", "N", "O", "P", "Q" ], "cost": 228 }, { "id": "S67", "elements": [ "R", "S", "T", "U" ], "cost": 384 }, { "id": "S68", "elements": [ "N", "P", "R" ], "cost": 291 }, { "id": "S69", "elements": [ "B", "C", "D", "E", "F" ], "cost": 355 }, { "id": "S70", "elements": [ "F", "H", "J" ], "cost": 42 }, { "id": "S71", "elements": [ "I", "J", "K", "L", "N", "O" ], "cost": 414 }, { "id": "S72", "elements": [ "E", "G", "H", "I", "J" ], "cost": 325 }, { "id": "S73", "elements": [ "H", "J", "K", "M" ], "cost": 164 }, { "id": "S74", "elements": [ "A", "B", "C", "D", "E", "G" ], "cost": 372 }, { "id": "S75", "elements": [ "D", "E", "F", "G", "I", "J", "K" ], "cost": 56 }, { "id": "S76", "elements": [ "I", "L", "P" ], "cost": 39 }, { "id": "S77", "elements": [ "C", "D", "H", "I", "J" ], "cost": 420 }, { "id": "S78", "elements": [ "F", "H", "I", "L", "O" ], "cost": 245 }, { "id": "S79", "elements": [ "O", "R", "S", "T", "U", "V", "W" ], "cost": 392 }, { "id": "S80", "elements": [ "A", "C", "D", "E" ], "cost": 252 }, { "id": "S81", "elements": [ "F", "J", "K", "M", "O", "P" ], "cost": 150 }, { "id": "S82", "elements": [ "P", "R", "S", "U", "V", "W" ], "cost": 102 }, { "id": "S83", "elements": [ "C", "D", "E", "F", "G", "H" ], "cost": 270 }, { "id": "S84", "elements": [ "A", "C", "F" ], "cost": 72 }, { "id": "S85", "elements": [ "D", "G", "I" ], "cost": 165 }, { "id": "S86", "elements": [ "L", "M", "O", "P", "Q", "R", "S" ], "cost": 161 }, { "id": "S87", "elements": [ "B", "E", "F", "G" ], "cost": 132 }, { "id": "S88", "elements": [ "K", "P", "R", "T", "W" ], "cost": 70 }, { "id": "S89", "elements": [ "P", "Q", "R", "V" ], "cost": 104 }, { "id": "S90", "elements": [ "E", "F", "G", "H", "I", "J" ], "cost": 210 }, { "id": "S91", "elements": [ "T", "U", "V", "W" ], "cost": 40 }, { "id": "S92", "elements": [ "A", "D", "G" ], "cost": 90 }, { "id": "S93", "elements": [ "N", "O", "P", "Q", "R", "T" ], "cost": 360 }, { "id": "S94", "elements": [ "Q", "R", "S", "V", "W" ], "cost": 430 }, { "id": "S95", "elements": [ "G", "H", "J", "K", "M" ], "cost": 35 }, { "id": "S96", "elements": [ "L", "O" ], "cost": 74 }, { "id": "S97", "elements": [ "G", "H", "L", "M", "N" ], "cost": 125 }, { "id": "S98", "elements": [ "T", "V", "W" ], "cost": 294 }, { "id": "S99", "elements": [ "L", "O", "P", "Q", "R", "S" ], "cost": 588 }, { "id": "S100", "elements": [ "A", "B", "C", "D", "F" ], "cost": 305 }, { "id": "S101", "elements": [ "A", "B", "C", "E" ], "cost": 200 }, { "id": "S102", "elements": [ "A", "B", "C", "D", "E", "H", "I" ], "cost": 490 }, { "id": "S103", "elements": [ "C", "D", "E", "F", "H" ], "cost": 25 }, { "id": "S104", "elements": [ "R", "S", "T", "U", "V", "W" ], "cost": 222 }, { "id": "S105", "elements": [ "A", "B", "C", "F", "G" ], "cost": 235 }, { "id": "S106", "elements": [ "J", "N", "P", "Q" ], "cost": 356 }, { "id": "S107", "elements": [ "S", "T", "V" ], "cost": 42 }, { "id": "S108", "elements": [ "V", "W" ], "cost": 186 }, { "id": "S109", "elements": [ "E", "G", "H", "I", "M" ], "cost": 165 }, { "id": "S110", "elements": [ "I", "K", "L", "M", "N", "O" ], "cost": 588 }, { "id": "S111", "elements": [ "B", "C", "D", "E", "G" ], "cost": 70 }, { "id": "S112", "elements": [ "M", "N", "P", "R" ], "cost": 376 }, { "id": "S113", "elements": [ "P", "Q", "R", "S", "T", "U", "V" ], "cost": 532 }, { "id": "S114", "elements": [ "N", "O", "R", "V" ], "cost": 332 }, { "id": "S115", "elements": [ "L", "N", "O", "R", "S" ], "cost": 115 }, { "id": "S116", "elements": [ "N", "O", "Q", "R", "T" ], "cost": 30 }, { "id": "S117", "elements": [ "B", "C", "D", "E" ], "cost": 288 }, { "id": "S118", "elements": [ "P", "R", "S", "T", "U", "V", "W" ], "cost": 259 }, { "id": "S119", "elements": [ "F", "G", "K" ], "cost": 96 }, { "id": "S120", "elements": [ "P", "Q", "R", "S", "U", "V" ], "cost": 78 }, { "id": "S121", "elements": [ "H", "M" ], "cost": 118 }, { "id": "S122", "elements": [ "A" ], "cost": 10000 }, { "id": "S123", "elements": [ "B" ], "cost": 10000 }, { "id": "S124", "elements": [ "C" ], "cost": 10000 }, { "id": "S125", "elements": [ "D" ], "cost": 10000 }, { "id": "S126", "elements": [ "E" ], "cost": 10000 }, { "id": "S127", "elements": [ "F" ], "cost": 10000 }, { "id": "S128", "elements": [ "G" ], "cost": 10000 }, { "id": "S129", "elements": [ "H" ], "cost": 10000 }, { "id": "S130", "elements": [ "I" ], "cost": 10000 }, { "id": "S131", "elements": [ "J" ], "cost": 10000 }, { "id": "S132", "elements": [ "K" ], "cost": 10000 }, { "id": "S133", "elements": [ "L" ], "cost": 10000 }, { "id": "S134", "elements": [ "M" ], "cost": 10000 }, { "id": "S135", "elements": [ "N" ], "cost": 10000 }, { "id": "S136", "elements": [ "O" ], "cost": 10000 }, { "id": "S137", "elements": [ "P" ], "cost": 10000 }, { "id": "S138", "elements": [ "Q" ], "cost": 10000 }, { "id": "S139", "elements": [ "R" ], "cost": 10000 }, { "id": "S140", "elements": [ "S" ], "cost": 10000 }, { "id": "S141", "elements": [ "T" ], "cost": 10000 }, { "id": "S142", "elements": [ "U" ], "cost": 10000 }, { "id": "S143", "elements": [ "V" ], "cost": 10000 }, { "id": "S144", "elements": [ "W" ], "cost": 10000 } ] }, "solution_variant": [ "S27", "S34", "S43", "S66", "S104" ], "context_index": 22, "input_format": "markdown_table", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "We’ve got a pile of bundled displays to choose from and a full catalog of product models that need to be represented. The job is to decide which bundles to stock so every single model shows up in one, and only one, of the chosen bundles — no gaps, no repeats. To judge which selection is best, total up the purchase price of the bundles you pick and go with the smallest sum. See the bundle choices and prices below.\n\n{\n \"total_product_models\": 25,\n \"total_bundle_options\": 153,\n \"sets\": [\n {\n \"bundle_id\": \"S1\",\n \"bundle_models\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\"\n ],\n \"bundle_cost\": 21\n },\n {\n \"bundle_id\": \"S2\",\n \"bundle_models\": [\n \"B\",\n \"J\",\n \"K\",\n \"M\",\n \"N\",\n \"O\",\n \"P\"\n ],\n \"bundle_cost\": 406\n },\n {\n \"bundle_id\": \"S3\",\n \"bundle_models\": [\n \"B\",\n \"D\",\n \"E\",\n \"G\",\n \"H\"\n ],\n \"bundle_cost\": 30\n },\n {\n \"bundle_id\": \"S4\",\n \"bundle_models\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\"\n ],\n \"bundle_cost\": 355\n },\n {\n \"bundle_id\": \"S5\",\n \"bundle_models\": [\n \"Q\",\n \"R\",\n \"S\"\n ],\n \"bundle_cost\": 150\n },\n {\n \"bundle_id\": \"S6\",\n \"bundle_models\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\"\n ],\n \"bundle_cost\": 408\n },\n {\n \"bundle_id\": \"S7\",\n \"bundle_models\": [\n \"Q\",\n \"R\",\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"bundle_cost\": 70\n },\n {\n \"bundle_id\": \"S8\",\n \"bundle_models\": [\n \"H\",\n \"J\",\n \"K\",\n \"N\",\n \"Q\"\n ],\n \"bundle_cost\": 450\n },\n {\n \"bundle_id\": \"S9\",\n \"bundle_models\": [\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"O\",\n \"Q\"\n ],\n \"bundle_cost\": 234\n },\n {\n \"bundle_id\": \"S10\",\n \"bundle_models\": [\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\"\n ],\n \"bundle_cost\": 329\n },\n {\n \"bundle_id\": \"S11\",\n \"bundle_models\": [\n \"H\",\n \"J\",\n \"K\",\n \"P\"\n ],\n \"bundle_cost\": 48\n },\n {\n \"bundle_id\": \"S12\",\n \"bundle_models\": [\n \"A\",\n \"B\",\n \"D\",\n \"E\",\n \"G\",\n \"H\"\n ],\n \"bundle_cost\": 498\n },\n {\n \"bundle_id\": \"S13\",\n \"bundle_models\": [\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"K\"\n ],\n \"bundle_cost\": 105\n },\n {\n \"bundle_id\": \"S14\",\n \"bundle_models\": [\n \"O\",\n \"P\",\n \"Q\"\n ],\n \"bundle_cost\": 273\n },\n {\n \"bundle_id\": \"S15\",\n \"bundle_models\": [\n \"E\",\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"M\"\n ],\n \"bundle_cost\": 42\n },\n {\n \"bundle_id\": \"S16\",\n \"bundle_models\": [\n \"Q\",\n \"U\",\n \"W\",\n \"X\"\n ],\n \"bundle_cost\": 188\n },\n {\n \"bundle_id\": \"S17\",\n \"bundle_models\": [\n \"G\",\n \"H\",\n \"J\",\n \"K\",\n \"L\",\n \"N\",\n \"O\"\n ],\n \"bundle_cost\": 161\n },\n {\n \"bundle_id\": \"S18\",\n \"bundle_models\": [\n \"M\",\n \"P\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"bundle_cost\": 315\n },\n {\n \"bundle_id\": \"S19\",\n \"bundle_models\": [\n \"M\",\n \"N\",\n \"O\",\n \"R\",\n \"S\"\n ],\n \"bundle_cost\": 270\n },\n {\n \"bundle_id\": \"S20\",\n \"bundle_models\": [\n \"Q\",\n \"T\",\n \"V\"\n ],\n \"bundle_cost\": 186\n },\n {\n \"bundle_id\": \"S21\",\n \"bundle_models\": [\n \"M\",\n \"N\",\n \"P\"\n ],\n \"bundle_cost\": 24\n },\n {\n \"bundle_id\": \"S22\",\n \"bundle_models\": [\n \"K\",\n \"P\",\n \"R\"\n ],\n \"bundle_cost\": 180\n },\n {\n \"bundle_id\": \"S23\",\n \"bundle_models\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"G\",\n \"I\"\n ],\n \"bundle_cost\": 399\n },\n {\n \"bundle_id\": \"S24\",\n \"bundle_models\": [\n \"O\",\n \"P\",\n \"S\"\n ],\n \"bundle_cost\": 168\n },\n {\n \"bundle_id\": \"S25\",\n \"bundle_models\": [\n \"Q\",\n \"T\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"bundle_cost\": 60\n },\n {\n \"bundle_id\": \"S26\",\n \"bundle_models\": [\n \"O\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\"\n ],\n \"bundle_cost\": 420\n },\n {\n \"bundle_id\": \"S27\",\n \"bundle_models\": [\n \"O\",\n \"P\",\n \"Q\",\n \"R\",\n \"U\"\n ],\n \"bundle_cost\": 305\n },\n {\n \"bundle_id\": \"S28\",\n \"bundle_models\": [\n \"J\",\n \"N\",\n \"O\",\n \"P\",\n \"R\",\n \"T\"\n ],\n \"bundle_cost\": 156\n },\n {\n \"bundle_id\": \"S29\",\n \"bundle_models\": [\n \"M\",\n \"N\",\n \"O\",\n \"Q\",\n \"R\",\n \"S\",\n \"U\"\n ],\n \"bundle_cost\": 504\n },\n {\n \"bundle_id\": \"S30\",\n \"bundle_models\": [\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"P\",\n \"S\"\n ],\n \"bundle_cost\": 564\n },\n {\n \"bundle_id\": \"S31\",\n \"bundle_models\": [\n \"G\",\n \"I\",\n \"K\",\n \"L\"\n ],\n \"bundle_cost\": 80\n },\n {\n \"bundle_id\": \"S32\",\n \"bundle_models\": [\n \"P\",\n \"Q\",\n \"T\"\n ],\n \"bundle_cost\": 117\n },\n {\n \"bundle_id\": \"S33\",\n \"bundle_models\": [\n \"N\",\n \"O\",\n \"P\",\n \"Q\",\n \"T\",\n \"U\",\n \"W\"\n ],\n \"bundle_cost\": 273\n },\n {\n \"bundle_id\": \"S34\",\n \"bundle_models\": [\n \"A\",\n \"D\",\n \"E\",\n \"H\",\n \"I\"\n ],\n \"bundle_cost\": 455\n },\n {\n \"bundle_id\": \"S35\",\n \"bundle_models\": [\n \"H\",\n \"K\",\n \"L\"\n ],\n \"bundle_cost\": 153\n },\n {\n \"bundle_id\": \"S36\",\n \"bundle_models\": [\n \"Q\",\n \"S\",\n \"U\"\n ],\n \"bundle_cost\": 210\n },\n {\n \"bundle_id\": \"S37\",\n \"bundle_models\": [\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"bundle_cost\": 210\n },\n {\n \"bundle_id\": \"S38\",\n \"bundle_models\": [\n \"C\",\n \"D\",\n \"J\"\n ],\n \"bundle_cost\": 150\n },\n {\n \"bundle_id\": \"S39\",\n \"bundle_models\": [\n \"Q\",\n \"T\",\n \"U\"\n ],\n \"bundle_cost\": 171\n },\n {\n \"bundle_id\": \"S40\",\n \"bundle_models\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"F\",\n \"H\"\n ],\n \"bundle_cost\": 246\n },\n {\n \"bundle_id\": \"S41\",\n \"bundle_models\": [\n \"A\",\n \"E\",\n \"F\",\n \"G\",\n \"H\",\n \"J\",\n \"K\"\n ],\n \"bundle_cost\": 287\n },\n {\n \"bundle_id\": \"S42\",\n \"bundle_models\": [\n \"M\",\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"W\"\n ],\n \"bundle_cost\": 42\n },\n {\n \"bundle_id\": \"S43\",\n \"bundle_models\": [\n \"C\",\n \"G\",\n \"H\"\n ],\n \"bundle_cost\": 135\n },\n {\n \"bundle_id\": \"S44\",\n \"bundle_models\": [\n \"O\",\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"X\"\n ],\n \"bundle_cost\": 448\n },\n {\n \"bundle_id\": \"S45\",\n \"bundle_models\": [\n \"N\",\n \"P\",\n \"R\"\n ],\n \"bundle_cost\": 213\n },\n {\n \"bundle_id\": \"S46\",\n \"bundle_models\": [\n \"J\",\n \"L\",\n \"M\"\n ],\n \"bundle_cost\": 12\n },\n {\n \"bundle_id\": \"S47\",\n \"bundle_models\": [\n \"P\",\n \"R\",\n \"U\",\n \"V\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"bundle_cost\": 693\n },\n {\n \"bundle_id\": \"S48\",\n \"bundle_models\": [\n \"A\",\n \"E\",\n \"G\",\n \"H\",\n \"J\",\n \"K\"\n ],\n \"bundle_cost\": 330\n },\n {\n \"bundle_id\": \"S49\",\n \"bundle_models\": [\n \"P\",\n \"U\",\n \"V\"\n ],\n \"bundle_cost\": 210\n },\n {\n \"bundle_id\": \"S50\",\n \"bundle_models\": [\n \"S\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"bundle_cost\": 316\n },\n {\n \"bundle_id\": \"S51\",\n \"bundle_models\": [\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"H\",\n \"J\"\n ],\n \"bundle_cost\": 114\n },\n {\n \"bundle_id\": \"S52\",\n \"bundle_models\": [\n \"I\",\n \"J\",\n \"R\"\n ],\n \"bundle_cost\": 219\n },\n {\n \"bundle_id\": \"S53\",\n \"bundle_models\": [\n \"N\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"W\"\n ],\n \"bundle_cost\": 102\n },\n {\n \"bundle_id\": \"S54\",\n \"bundle_models\": [\n \"E\",\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"L\"\n ],\n \"bundle_cost\": 174\n },\n {\n \"bundle_id\": \"S55\",\n \"bundle_models\": [\n \"L\",\n \"M\",\n \"P\"\n ],\n \"bundle_cost\": 63\n },\n {\n \"bundle_id\": \"S56\",\n \"bundle_models\": [\n \"A\",\n \"B\",\n \"D\",\n \"E\"\n ],\n \"bundle_cost\": 112\n },\n {\n \"bundle_id\": \"S57\",\n \"bundle_models\": [\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"L\",\n \"N\"\n ],\n \"bundle_cost\": 168\n },\n {\n \"bundle_id\": \"S58\",\n \"bundle_models\": [\n \"N\",\n \"P\",\n \"R\",\n \"T\",\n \"U\",\n \"W\"\n ],\n \"bundle_cost\": 570\n },\n {\n \"bundle_id\": \"S59\",\n \"bundle_models\": [\n \"L\",\n \"O\",\n \"Q\",\n \"R\",\n \"T\",\n \"V\"\n ],\n \"bundle_cost\": 588\n },\n {\n \"bundle_id\": \"S60\",\n \"bundle_models\": [\n \"N\",\n \"O\",\n \"P\",\n \"R\",\n \"S\",\n \"U\",\n \"V\"\n ],\n \"bundle_cost\": 203\n },\n {\n \"bundle_id\": \"S61\",\n \"bundle_models\": [\n \"R\",\n \"S\",\n \"U\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"bundle_cost\": 582\n },\n {\n \"bundle_id\": \"S62\",\n \"bundle_models\": [\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"Q\"\n ],\n \"bundle_cost\": 115\n },\n {\n \"bundle_id\": \"S63\",\n \"bundle_models\": [\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"bundle_cost\": 24\n },\n {\n \"bundle_id\": \"S64\",\n \"bundle_models\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"H\"\n ],\n \"bundle_cost\": 426\n },\n {\n \"bundle_id\": \"S65\",\n \"bundle_models\": [\n \"D\",\n \"G\",\n \"K\"\n ],\n \"bundle_cost\": 234\n },\n {\n \"bundle_id\": \"S66\",\n \"bundle_models\": [\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"I\"\n ],\n \"bundle_cost\": 6\n },\n {\n \"bundle_id\": \"S67\",\n \"bundle_models\": [\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"bundle_cost\": 426\n },\n {\n \"bundle_id\": \"S68\",\n \"bundle_models\": [\n \"R\",\n \"S\",\n \"V\"\n ],\n \"bundle_cost\": 183\n },\n {\n \"bundle_id\": \"S69\",\n \"bundle_models\": [\n \"R\",\n \"T\",\n \"W\"\n ],\n \"bundle_cost\": 21\n },\n {\n \"bundle_id\": \"S70\",\n \"bundle_models\": [\n \"S\",\n \"U\",\n \"V\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"bundle_cost\": 426\n },\n {\n \"bundle_id\": \"S71\",\n \"bundle_models\": [\n \"Q\",\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"bundle_cost\": 280\n },\n {\n \"bundle_id\": \"S72\",\n \"bundle_models\": [\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"bundle_cost\": 553\n },\n {\n \"bundle_id\": \"S73\",\n \"bundle_models\": [\n \"C\",\n \"E\",\n \"F\",\n \"H\"\n ],\n \"bundle_cost\": 80\n },\n {\n \"bundle_id\": \"S74\",\n \"bundle_models\": [\n \"B\",\n \"C\",\n \"D\"\n ],\n \"bundle_cost\": 84\n },\n {\n \"bundle_id\": \"S75\",\n \"bundle_models\": [\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"W\"\n ],\n \"bundle_cost\": 42\n },\n {\n \"bundle_id\": \"S76\",\n \"bundle_models\": [\n \"T\",\n \"U\",\n \"W\",\n \"X\"\n ],\n \"bundle_cost\": 256\n },\n {\n \"bundle_id\": \"S77\",\n \"bundle_models\": [\n \"B\",\n \"C\",\n \"D\",\n \"F\"\n ],\n \"bundle_cost\": 400\n },\n {\n \"bundle_id\": \"S78\",\n \"bundle_models\": [\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"Q\",\n \"T\"\n ],\n \"bundle_cost\": 48\n },\n {\n \"bundle_id\": \"S79\",\n \"bundle_models\": [\n \"R\",\n \"S\",\n \"T\",\n \"V\"\n ],\n \"bundle_cost\": 300\n },\n {\n \"bundle_id\": \"S80\",\n \"bundle_models\": [\n \"P\",\n \"S\",\n \"T\"\n ],\n \"bundle_cost\": 252\n },\n {\n \"bundle_id\": \"S81\",\n \"bundle_models\": [\n \"Q\",\n \"U\",\n \"W\"\n ],\n \"bundle_cost\": 288\n },\n {\n \"bundle_id\": \"S82\",\n \"bundle_models\": [\n \"I\",\n \"K\",\n \"M\"\n ],\n \"bundle_cost\": 63\n },\n {\n \"bundle_id\": \"S83\",\n \"bundle_models\": [\n \"L\",\n \"O\",\n \"P\",\n \"U\",\n \"V\"\n ],\n \"bundle_cost\": 350\n },\n {\n \"bundle_id\": \"S84\",\n \"bundle_models\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"G\",\n \"H\"\n ],\n \"bundle_cost\": 14\n },\n {\n \"bundle_id\": \"S85\",\n \"bundle_models\": [\n \"K\",\n \"L\",\n \"O\",\n \"P\",\n \"Q\"\n ],\n \"bundle_cost\": 255\n },\n {\n \"bundle_id\": \"S86\",\n \"bundle_models\": [\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"Y\"\n ],\n \"bundle_cost\": 370\n },\n {\n \"bundle_id\": \"S87\",\n \"bundle_models\": [\n \"I\",\n \"L\",\n \"N\",\n \"O\",\n \"P\"\n ],\n \"bundle_cost\": 195\n },\n {\n \"bundle_id\": \"S88\",\n \"bundle_models\": [\n \"M\",\n \"N\",\n \"O\",\n \"Q\",\n \"R\",\n \"T\"\n ],\n \"bundle_cost\": 438\n },\n {\n \"bundle_id\": \"S89\",\n \"bundle_models\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"G\"\n ],\n \"bundle_cost\": 250\n },\n {\n \"bundle_id\": \"S90\",\n \"bundle_models\": [\n \"B\",\n \"C\",\n \"I\"\n ],\n \"bundle_cost\": 159\n },\n {\n \"bundle_id\": \"S91\",\n \"bundle_models\": [\n \"P\",\n \"Q\",\n \"T\",\n \"U\",\n \"V\",\n \"X\",\n \"Y\"\n ],\n \"bundle_cost\": 560\n },\n {\n \"bundle_id\": \"S92\",\n \"bundle_models\": [\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"I\"\n ],\n \"bundle_cost\": 203\n },\n {\n \"bundle_id\": \"S93\",\n \"bundle_models\": [\n \"L\",\n \"M\",\n \"N\",\n \"P\"\n ],\n \"bundle_cost\": 148\n },\n {\n \"bundle_id\": \"S94\",\n \"bundle_models\": [\n \"C\",\n \"D\",\n \"E\",\n \"G\",\n \"I\",\n \"J\",\n \"K\"\n ],\n \"bundle_cost\": 35\n },\n {\n \"bundle_id\": \"S95\",\n \"bundle_models\": [\n \"B\",\n \"D\",\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"N\"\n ],\n \"bundle_cost\": 245\n },\n {\n \"bundle_id\": \"S96\",\n \"bundle_models\": [\n \"N\",\n \"O\",\n \"P\",\n \"Q\"\n ],\n \"bundle_cost\": 256\n },\n {\n \"bundle_id\": \"S97\",\n \"bundle_models\": [\n \"N\",\n \"P\",\n \"Q\",\n \"U\"\n ],\n \"bundle_cost\": 372\n },\n {\n \"bundle_id\": \"S98\",\n \"bundle_models\": [\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"X\",\n \"Y\"\n ],\n \"bundle_cost\": 210\n },\n {\n \"bundle_id\": \"S99\",\n \"bundle_models\": [\n \"D\",\n \"F\",\n \"G\",\n \"I\"\n ],\n \"bundle_cost\": 360\n },\n {\n \"bundle_id\": \"S100\",\n \"bundle_models\": [\n \"O\",\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\"\n ],\n \"bundle_cost\": 364\n },\n {\n \"bundle_id\": \"S101\",\n \"bundle_models\": [\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"M\"\n ],\n \"bundle_cost\": 75\n },\n {\n \"bundle_id\": \"S102\",\n \"bundle_models\": [\n \"C\",\n \"D\",\n \"F\",\n \"G\",\n \"I\",\n \"J\",\n \"K\"\n ],\n \"bundle_cost\": 168\n },\n {\n \"bundle_id\": \"S103\",\n \"bundle_models\": [\n \"F\",\n \"H\",\n \"J\",\n \"K\"\n ],\n \"bundle_cost\": 104\n },\n {\n \"bundle_id\": \"S104\",\n \"bundle_models\": [\n \"D\",\n \"E\",\n \"H\",\n \"J\"\n ],\n \"bundle_cost\": 160\n },\n {\n \"bundle_id\": \"S105\",\n \"bundle_models\": [\n \"B\",\n \"C\",\n \"D\",\n \"E\"\n ],\n \"bundle_cost\": 56\n },\n {\n \"bundle_id\": \"S106\",\n \"bundle_models\": [\n \"Q\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"bundle_cost\": 154\n },\n {\n \"bundle_id\": \"S107\",\n \"bundle_models\": [\n \"A\",\n \"D\",\n \"F\"\n ],\n \"bundle_cost\": 216\n },\n {\n \"bundle_id\": \"S108\",\n \"bundle_models\": [\n \"D\",\n \"F\",\n \"G\",\n \"H\"\n ],\n \"bundle_cost\": 92\n },\n {\n \"bundle_id\": \"S109\",\n \"bundle_models\": [\n \"O\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\"\n ],\n \"bundle_cost\": 35\n },\n {\n \"bundle_id\": \"S110\",\n \"bundle_models\": [\n \"D\",\n \"H\",\n \"I\"\n ],\n \"bundle_cost\": 261\n },\n {\n \"bundle_id\": \"S111\",\n \"bundle_models\": [\n \"P\",\n \"Q\",\n \"S\"\n ],\n \"bundle_cost\": 57\n },\n {\n \"bundle_id\": \"S112\",\n \"bundle_models\": [\n \"L\",\n \"N\",\n \"O\",\n \"Q\"\n ],\n \"bundle_cost\": 212\n },\n {\n \"bundle_id\": \"S113\",\n \"bundle_models\": [\n \"H\",\n \"K\",\n \"N\"\n ],\n \"bundle_cost\": 18\n },\n {\n \"bundle_id\": \"S114\",\n \"bundle_models\": [\n \"P\",\n \"S\",\n \"U\",\n \"V\",\n \"W\",\n \"X\",\n \"Y\"\n ],\n \"bundle_cost\": 588\n },\n {\n \"bundle_id\": \"S115\",\n \"bundle_models\": [\n \"A\",\n \"C\",\n \"D\",\n \"E\",\n \"F\"\n ],\n \"bundle_cost\": 45\n },\n {\n \"bundle_id\": \"S116\",\n \"bundle_models\": [\n \"H\",\n \"J\",\n \"K\",\n \"N\"\n ],\n \"bundle_cost\": 312\n },\n {\n \"bundle_id\": \"S117\",\n \"bundle_models\": [\n \"P\",\n \"T\",\n \"U\",\n \"W\"\n ],\n \"bundle_cost\": 272\n },\n {\n \"bundle_id\": \"S118\",\n \"bundle_models\": [\n \"S\",\n \"W\",\n \"Y\"\n ],\n \"bundle_cost\": 249\n },\n {\n \"bundle_id\": \"S119\",\n \"bundle_models\": [\n \"U\",\n \"W\",\n \"X\"\n ],\n \"bundle_cost\": 51\n },\n {\n \"bundle_id\": \"S120\",\n \"bundle_models\": [\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"U\",\n \"W\"\n ],\n \"bundle_cost\": 492\n },\n {\n \"bundle_id\": \"S121\",\n \"bundle_models\": [\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"S\"\n ],\n \"bundle_cost\": 366\n },\n {\n \"bundle_id\": \"S122\",\n \"bundle_models\": [\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"N\"\n ],\n \"bundle_cost\": 366\n },\n {\n \"bundle_id\": \"S123\",\n \"bundle_models\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\"\n ],\n \"bundle_cost\": 256\n },\n {\n \"bundle_id\": \"S124\",\n \"bundle_models\": [\n \"A\",\n \"B\",\n \"C\",\n \"E\",\n \"F\",\n \"G\"\n ],\n \"bundle_cost\": 96\n },\n {\n \"bundle_id\": \"S125\",\n \"bundle_models\": [\n \"C\",\n \"F\",\n \"G\"\n ],\n \"bundle_cost\": 213\n },\n {\n \"bundle_id\": \"S126\",\n \"bundle_models\": [\n \"N\",\n \"O\",\n \"R\",\n \"S\"\n ],\n \"bundle_cost\": 256\n },\n {\n \"bundle_id\": \"S127\",\n \"bundle_models\": [\n \"U\",\n \"V\",\n \"X\",\n \"Y\"\n ],\n \"bundle_cost\": 356\n },\n {\n \"bundle_id\": \"S128\",\n \"bundle_models\": [\n \"C\",\n \"E\",\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"M\"\n ],\n \"bundle_cost\": 315\n },\n {\n \"bundle_id\": \"S129\",\n \"bundle_models\": [\n \"A\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S130\",\n \"bundle_models\": [\n \"B\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S131\",\n \"bundle_models\": [\n \"C\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S132\",\n \"bundle_models\": [\n \"D\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S133\",\n \"bundle_models\": [\n \"E\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S134\",\n \"bundle_models\": [\n \"F\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S135\",\n \"bundle_models\": [\n \"G\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S136\",\n \"bundle_models\": [\n \"H\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S137\",\n \"bundle_models\": [\n \"I\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S138\",\n \"bundle_models\": [\n \"J\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S139\",\n \"bundle_models\": [\n \"K\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S140\",\n \"bundle_models\": [\n \"L\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S141\",\n \"bundle_models\": [\n \"M\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S142\",\n \"bundle_models\": [\n \"N\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S143\",\n \"bundle_models\": [\n \"O\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S144\",\n \"bundle_models\": [\n \"P\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S145\",\n \"bundle_models\": [\n \"Q\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S146\",\n \"bundle_models\": [\n \"R\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S147\",\n \"bundle_models\": [\n \"S\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S148\",\n \"bundle_models\": [\n \"T\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S149\",\n \"bundle_models\": [\n \"U\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S150\",\n \"bundle_models\": [\n \"V\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S151\",\n \"bundle_models\": [\n \"W\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S152\",\n \"bundle_models\": [\n \"X\"\n ],\n \"bundle_cost\": 10000\n },\n {\n \"bundle_id\": \"S153\",\n \"bundle_models\": [\n \"Y\"\n ],\n \"bundle_cost\": 10000\n }\n ]\n}\n\nOh, and when you reply with your chosen bundles, please use this simple JSON layout so it's easy to parse:\n\n{\n \"solution\": [\"bundle_id\", ...]\n}\n\n\"solution\" is the list of bundle IDs you want to stock, and \"bundle_id\" is just a placeholder showing the format — replace each placeholder with the exact bundle identifier from the instance. This JSON is just a sketch of the expected shape, not your actual answer.\n\nPlease make sure to use the identifiers exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 25, "num_sets": 153, "density": 0.17281045751633986, "sets": [ { "id": 1, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 21 }, { "id": 2, "elements": [ 2, 10, 11, 13, 14, 15, 16 ], "cost": 406 }, { "id": 3, "elements": [ 2, 4, 5, 7, 8 ], "cost": 30 }, { "id": 4, "elements": [ 1, 2, 3, 4, 5 ], "cost": 355 }, { "id": 5, "elements": [ 17, 18, 19 ], "cost": 150 }, { "id": 6, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 408 }, { "id": 7, "elements": [ 17, 18, 20, 21, 22, 23, 24 ], "cost": 70 }, { "id": 8, "elements": [ 8, 10, 11, 14, 17 ], "cost": 450 }, { "id": 9, "elements": [ 10, 11, 12, 13, 15, 17 ], "cost": 234 }, { "id": 10, "elements": [ 17, 18, 19, 20, 21, 22, 23 ], "cost": 329 }, { "id": 11, "elements": [ 8, 10, 11, 16 ], "cost": 48 }, { "id": 12, "elements": [ 1, 2, 4, 5, 7, 8 ], "cost": 498 }, { "id": 13, "elements": [ 6, 7, 8, 9, 11 ], "cost": 105 }, { "id": 14, "elements": [ 15, 16, 17 ], "cost": 273 }, { "id": 15, "elements": [ 5, 7, 8, 9, 10, 11, 13 ], "cost": 42 }, { "id": 16, "elements": [ 17, 21, 23, 24 ], "cost": 188 }, { "id": 17, "elements": [ 7, 8, 10, 11, 12, 14, 15 ], "cost": 161 }, { "id": 18, "elements": [ 13, 16, 19, 20, 21 ], "cost": 315 }, { "id": 19, "elements": [ 13, 14, 15, 18, 19 ], "cost": 270 }, { "id": 20, "elements": [ 17, 20, 22 ], "cost": 186 }, { "id": 21, "elements": [ 13, 14, 16 ], "cost": 24 }, { "id": 22, "elements": [ 11, 16, 18 ], "cost": 180 }, { "id": 23, "elements": [ 1, 2, 3, 4, 5, 7, 9 ], "cost": 399 }, { "id": 24, "elements": [ 15, 16, 19 ], "cost": 168 }, { "id": 25, "elements": [ 17, 20, 23, 24, 25 ], "cost": 60 }, { "id": 26, "elements": [ 15, 17, 18, 19, 20, 21, 22 ], "cost": 420 }, { "id": 27, "elements": [ 15, 16, 17, 18, 21 ], "cost": 305 }, { "id": 28, "elements": [ 10, 14, 15, 16, 18, 20 ], "cost": 156 }, { "id": 29, "elements": [ 13, 14, 15, 17, 18, 19, 21 ], "cost": 504 }, { "id": 30, "elements": [ 11, 12, 13, 14, 16, 19 ], "cost": 564 }, { "id": 31, "elements": [ 7, 9, 11, 12 ], "cost": 80 }, { "id": 32, "elements": [ 16, 17, 20 ], "cost": 117 }, { "id": 33, "elements": [ 14, 15, 16, 17, 20, 21, 23 ], "cost": 273 }, { "id": 34, "elements": [ 1, 4, 5, 8, 9 ], "cost": 455 }, { "id": 35, "elements": [ 8, 11, 12 ], "cost": 153 }, { "id": 36, "elements": [ 17, 19, 21 ], "cost": 210 }, { "id": 37, "elements": [ 19, 20, 21, 22, 23, 24, 25 ], "cost": 210 }, { "id": 38, "elements": [ 3, 4, 10 ], "cost": 150 }, { "id": 39, "elements": [ 17, 20, 21 ], "cost": 171 }, { "id": 40, "elements": [ 1, 2, 3, 4, 6, 8 ], "cost": 246 }, { "id": 41, "elements": [ 1, 5, 6, 7, 8, 10, 11 ], "cost": 287 }, { "id": 42, "elements": [ 13, 18, 19, 20, 21, 23 ], "cost": 42 }, { "id": 43, "elements": [ 3, 7, 8 ], "cost": 135 }, { "id": 44, "elements": [ 15, 18, 19, 20, 21, 22, 24 ], "cost": 448 }, { "id": 45, "elements": [ 14, 16, 18 ], "cost": 213 }, { "id": 46, "elements": [ 10, 12, 13 ], "cost": 12 }, { "id": 47, "elements": [ 16, 18, 21, 22, 23, 24, 25 ], "cost": 693 }, { "id": 48, "elements": [ 1, 5, 7, 8, 10, 11 ], "cost": 330 }, { "id": 49, "elements": [ 16, 21, 22 ], "cost": 210 }, { "id": 50, "elements": [ 19, 23, 24, 25 ], "cost": 316 }, { "id": 51, "elements": [ 4, 5, 6, 7, 8, 10 ], "cost": 114 }, { "id": 52, "elements": [ 9, 10, 18 ], "cost": 219 }, { "id": 53, "elements": [ 14, 16, 17, 18, 19, 23 ], "cost": 102 }, { "id": 54, "elements": [ 5, 6, 7, 8, 9, 12 ], "cost": 174 }, { "id": 55, "elements": [ 12, 13, 16 ], "cost": 63 }, { "id": 56, "elements": [ 1, 2, 4, 5 ], "cost": 112 }, { "id": 57, "elements": [ 6, 7, 8, 9, 10, 12, 14 ], "cost": 168 }, { "id": 58, "elements": [ 14, 16, 18, 20, 21, 23 ], "cost": 570 }, { "id": 59, "elements": [ 12, 15, 17, 18, 20, 22 ], "cost": 588 }, { "id": 60, "elements": [ 14, 15, 16, 18, 19, 21, 22 ], "cost": 203 }, { "id": 61, "elements": [ 18, 19, 21, 22, 23, 24 ], "cost": 582 }, { "id": 62, "elements": [ 12, 13, 14, 15, 17 ], "cost": 115 }, { "id": 63, "elements": [ 19, 20, 21, 22, 23, 24 ], "cost": 24 }, { "id": 64, "elements": [ 1, 2, 3, 4, 5, 8 ], "cost": 426 }, { "id": 65, "elements": [ 4, 7, 11 ], "cost": 234 }, { "id": 66, "elements": [ 3, 4, 5, 6, 7, 9 ], "cost": 6 }, { "id": 67, "elements": [ 16, 17, 18, 19, 20, 21 ], "cost": 426 }, { "id": 68, "elements": [ 18, 19, 22 ], "cost": 183 }, { "id": 69, "elements": [ 18, 20, 23 ], "cost": 21 }, { "id": 70, "elements": [ 19, 21, 22, 23, 24, 25 ], "cost": 426 }, { "id": 71, "elements": [ 17, 20, 21, 22, 23, 24, 25 ], "cost": 280 }, { "id": 72, "elements": [ 17, 18, 19, 20, 22, 23, 24 ], "cost": 553 }, { "id": 73, "elements": [ 3, 5, 6, 8 ], "cost": 80 }, { "id": 74, "elements": [ 2, 3, 4 ], "cost": 84 }, { "id": 75, "elements": [ 17, 18, 19, 20, 21, 23 ], "cost": 42 }, { "id": 76, "elements": [ 20, 21, 23, 24 ], "cost": 256 }, { "id": 77, "elements": [ 2, 3, 4, 6 ], "cost": 400 }, { "id": 78, "elements": [ 13, 14, 15, 16, 17, 20 ], "cost": 48 }, { "id": 79, "elements": [ 18, 19, 20, 22 ], "cost": 300 }, { "id": 80, "elements": [ 16, 19, 20 ], "cost": 252 }, { "id": 81, "elements": [ 17, 21, 23 ], "cost": 288 }, { "id": 82, "elements": [ 9, 11, 13 ], "cost": 63 }, { "id": 83, "elements": [ 12, 15, 16, 21, 22 ], "cost": 350 }, { "id": 84, "elements": [ 1, 2, 3, 4, 5, 7, 8 ], "cost": 14 }, { "id": 85, "elements": [ 11, 12, 15, 16, 17 ], "cost": 255 }, { "id": 86, "elements": [ 20, 21, 22, 23, 25 ], "cost": 370 }, { "id": 87, "elements": [ 9, 12, 14, 15, 16 ], "cost": 195 }, { "id": 88, "elements": [ 13, 14, 15, 17, 18, 20 ], "cost": 438 }, { "id": 89, "elements": [ 1, 2, 3, 4, 7 ], "cost": 250 }, { "id": 90, "elements": [ 2, 3, 9 ], "cost": 159 }, { "id": 91, "elements": [ 16, 17, 20, 21, 22, 24, 25 ], "cost": 560 }, { "id": 92, "elements": [ 2, 3, 4, 5, 6, 7, 9 ], "cost": 203 }, { "id": 93, "elements": [ 12, 13, 14, 16 ], "cost": 148 }, { "id": 94, "elements": [ 3, 4, 5, 7, 9, 10, 11 ], "cost": 35 }, { "id": 95, "elements": [ 2, 4, 7, 8, 9, 10, 14 ], "cost": 245 }, { "id": 96, "elements": [ 14, 15, 16, 17 ], "cost": 256 }, { "id": 97, "elements": [ 14, 16, 17, 21 ], "cost": 372 }, { "id": 98, "elements": [ 19, 20, 21, 22, 24, 25 ], "cost": 210 }, { "id": 99, "elements": [ 4, 6, 7, 9 ], "cost": 360 }, { "id": 100, "elements": [ 15, 18, 19, 20, 21, 22, 23 ], "cost": 364 }, { "id": 101, "elements": [ 9, 10, 11, 12, 13 ], "cost": 75 }, { "id": 102, "elements": [ 3, 4, 6, 7, 9, 10, 11 ], "cost": 168 }, { "id": 103, "elements": [ 6, 8, 10, 11 ], "cost": 104 }, { "id": 104, "elements": [ 4, 5, 8, 10 ], "cost": 160 }, { "id": 105, "elements": [ 2, 3, 4, 5 ], "cost": 56 }, { "id": 106, "elements": [ 17, 19, 20, 21, 22, 23, 24 ], "cost": 154 }, { "id": 107, "elements": [ 1, 4, 6 ], "cost": 216 }, { "id": 108, "elements": [ 4, 6, 7, 8 ], "cost": 92 }, { "id": 109, "elements": [ 15, 17, 18, 19, 20 ], "cost": 35 }, { "id": 110, "elements": [ 4, 8, 9 ], "cost": 261 }, { "id": 111, "elements": [ 16, 17, 19 ], "cost": 57 }, { "id": 112, "elements": [ 12, 14, 15, 17 ], "cost": 212 }, { "id": 113, "elements": [ 8, 11, 14 ], "cost": 18 }, { "id": 114, "elements": [ 16, 19, 21, 22, 23, 24, 25 ], "cost": 588 }, { "id": 115, "elements": [ 1, 3, 4, 5, 6 ], "cost": 45 }, { "id": 116, "elements": [ 8, 10, 11, 14 ], "cost": 312 }, { "id": 117, "elements": [ 16, 20, 21, 23 ], "cost": 272 }, { "id": 118, "elements": [ 19, 23, 25 ], "cost": 249 }, { "id": 119, "elements": [ 21, 23, 24 ], "cost": 51 }, { "id": 120, "elements": [ 16, 17, 18, 19, 21, 23 ], "cost": 492 }, { "id": 121, "elements": [ 12, 13, 14, 15, 16, 19 ], "cost": 366 }, { "id": 122, "elements": [ 6, 7, 8, 9, 10, 14 ], "cost": 366 }, { "id": 123, "elements": [ 1, 2, 3, 4 ], "cost": 256 }, { "id": 124, "elements": [ 1, 2, 3, 5, 6, 7 ], "cost": 96 }, { "id": 125, "elements": [ 3, 6, 7 ], "cost": 213 }, { "id": 126, "elements": [ 14, 15, 18, 19 ], "cost": 256 }, { "id": 127, "elements": [ 21, 22, 24, 25 ], "cost": 356 }, { "id": 128, "elements": [ 3, 5, 6, 7, 8, 9, 13 ], "cost": 315 }, { "id": 129, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 130, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 131, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 132, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 133, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 134, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 135, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 136, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 137, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 138, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 139, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 140, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 141, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 142, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 143, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 144, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 145, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 146, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 147, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 148, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 149, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 150, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 151, "elements": [ 23 ], "cost": 10000, "singleton": true }, { "id": 152, "elements": [ 24 ], "cost": 10000, "singleton": true }, { "id": 153, "elements": [ 25 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0022_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0022_bag_stats.png" }, "solution": [ 4, 13, 25, 46, 60 ], "obj": 735.0, "instance_variant": { "num_elements": 25, "num_sets": 153, "sets": [ { "id": "S1", "elements": [ "A", "B", "C", "D", "E", "F", "G" ], "cost": 21 }, { "id": "S2", "elements": [ "B", "J", "K", "M", "N", "O", "P" ], "cost": 406 }, { "id": "S3", "elements": [ "B", "D", "E", "G", "H" ], "cost": 30 }, { "id": "S4", "elements": [ "A", "B", "C", "D", "E" ], "cost": 355 }, { "id": "S5", "elements": [ "Q", "R", "S" ], "cost": 150 }, { "id": "S6", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 408 }, { "id": "S7", "elements": [ "Q", "R", "T", "U", "V", "W", "X" ], "cost": 70 }, { "id": "S8", "elements": [ "H", "J", "K", "N", "Q" ], "cost": 450 }, { "id": "S9", "elements": [ "J", "K", "L", "M", "O", "Q" ], "cost": 234 }, { "id": "S10", "elements": [ "Q", "R", "S", "T", "U", "V", "W" ], "cost": 329 }, { "id": "S11", "elements": [ "H", "J", "K", "P" ], "cost": 48 }, { "id": "S12", "elements": [ "A", "B", "D", "E", "G", "H" ], "cost": 498 }, { "id": "S13", "elements": [ "F", "G", "H", "I", "K" ], "cost": 105 }, { "id": "S14", "elements": [ "O", "P", "Q" ], "cost": 273 }, { "id": "S15", "elements": [ "E", "G", "H", "I", "J", "K", "M" ], "cost": 42 }, { "id": "S16", "elements": [ "Q", "U", "W", "X" ], "cost": 188 }, { "id": "S17", "elements": [ "G", "H", "J", "K", "L", "N", "O" ], "cost": 161 }, { "id": "S18", "elements": [ "M", "P", "S", "T", "U" ], "cost": 315 }, { "id": "S19", "elements": [ "M", "N", "O", "R", "S" ], "cost": 270 }, { "id": "S20", "elements": [ "Q", "T", "V" ], "cost": 186 }, { "id": "S21", "elements": [ "M", "N", "P" ], "cost": 24 }, { "id": "S22", "elements": [ "K", "P", "R" ], "cost": 180 }, { "id": "S23", "elements": [ "A", "B", "C", "D", "E", "G", "I" ], "cost": 399 }, { "id": "S24", "elements": [ "O", "P", "S" ], "cost": 168 }, { "id": "S25", "elements": [ "Q", "T", "W", "X", "Y" ], "cost": 60 }, { "id": "S26", "elements": [ "O", "Q", "R", "S", "T", "U", "V" ], "cost": 420 }, { "id": "S27", "elements": [ "O", "P", "Q", "R", "U" ], "cost": 305 }, { "id": "S28", "elements": [ "J", "N", "O", "P", "R", "T" ], "cost": 156 }, { "id": "S29", "elements": [ "M", "N", "O", "Q", "R", "S", "U" ], "cost": 504 }, { "id": "S30", "elements": [ "K", "L", "M", "N", "P", "S" ], "cost": 564 }, { "id": "S31", "elements": [ "G", "I", "K", "L" ], "cost": 80 }, { "id": "S32", "elements": [ "P", "Q", "T" ], "cost": 117 }, { "id": "S33", "elements": [ "N", "O", "P", "Q", "T", "U", "W" ], "cost": 273 }, { "id": "S34", "elements": [ "A", "D", "E", "H", "I" ], "cost": 455 }, { "id": "S35", "elements": [ "H", "K", "L" ], "cost": 153 }, { "id": "S36", "elements": [ "Q", "S", "U" ], "cost": 210 }, { "id": "S37", "elements": [ "S", "T", "U", "V", "W", "X", "Y" ], "cost": 210 }, { "id": "S38", "elements": [ "C", "D", "J" ], "cost": 150 }, { "id": "S39", "elements": [ "Q", "T", "U" ], "cost": 171 }, { "id": "S40", "elements": [ "A", "B", "C", "D", "F", "H" ], "cost": 246 }, { "id": "S41", "elements": [ "A", "E", "F", "G", "H", "J", "K" ], "cost": 287 }, { "id": "S42", "elements": [ "M", "R", "S", "T", "U", "W" ], "cost": 42 }, { "id": "S43", "elements": [ "C", "G", "H" ], "cost": 135 }, { "id": "S44", "elements": [ "O", "R", "S", "T", "U", "V", "X" ], "cost": 448 }, { "id": "S45", "elements": [ "N", "P", "R" ], "cost": 213 }, { "id": "S46", "elements": [ "J", "L", "M" ], "cost": 12 }, { "id": "S47", "elements": [ "P", "R", "U", "V", "W", "X", "Y" ], "cost": 693 }, { "id": "S48", "elements": [ "A", "E", "G", "H", "J", "K" ], "cost": 330 }, { "id": "S49", "elements": [ "P", "U", "V" ], "cost": 210 }, { "id": "S50", "elements": [ "S", "W", "X", "Y" ], "cost": 316 }, { "id": "S51", "elements": [ "D", "E", "F", "G", "H", "J" ], "cost": 114 }, { "id": "S52", "elements": [ "I", "J", "R" ], "cost": 219 }, { "id": "S53", "elements": [ "N", "P", "Q", "R", "S", "W" ], "cost": 102 }, { "id": "S54", "elements": [ "E", "F", "G", "H", "I", "L" ], "cost": 174 }, { "id": "S55", "elements": [ "L", "M", "P" ], "cost": 63 }, { "id": "S56", "elements": [ "A", "B", "D", "E" ], "cost": 112 }, { "id": "S57", "elements": [ "F", "G", "H", "I", "J", "L", "N" ], "cost": 168 }, { "id": "S58", "elements": [ "N", "P", "R", "T", "U", "W" ], "cost": 570 }, { "id": "S59", "elements": [ "L", "O", "Q", "R", "T", "V" ], "cost": 588 }, { "id": "S60", "elements": [ "N", "O", "P", "R", "S", "U", "V" ], "cost": 203 }, { "id": "S61", "elements": [ "R", "S", "U", "V", "W", "X" ], "cost": 582 }, { "id": "S62", "elements": [ "L", "M", "N", "O", "Q" ], "cost": 115 }, { "id": "S63", "elements": [ "S", "T", "U", "V", "W", "X" ], "cost": 24 }, { "id": "S64", "elements": [ "A", "B", "C", "D", "E", "H" ], "cost": 426 }, { "id": "S65", "elements": [ "D", "G", "K" ], "cost": 234 }, { "id": "S66", "elements": [ "C", "D", "E", "F", "G", "I" ], "cost": 6 }, { "id": "S67", "elements": [ "P", "Q", "R", "S", "T", "U" ], "cost": 426 }, { "id": "S68", "elements": [ "R", "S", "V" ], "cost": 183 }, { "id": "S69", "elements": [ "R", "T", "W" ], "cost": 21 }, { "id": "S70", "elements": [ "S", "U", "V", "W", "X", "Y" ], "cost": 426 }, { "id": "S71", "elements": [ "Q", "T", "U", "V", "W", "X", "Y" ], "cost": 280 }, { "id": "S72", "elements": [ "Q", "R", "S", "T", "V", "W", "X" ], "cost": 553 }, { "id": "S73", "elements": [ "C", "E", "F", "H" ], "cost": 80 }, { "id": "S74", "elements": [ "B", "C", "D" ], "cost": 84 }, { "id": "S75", "elements": [ "Q", "R", "S", "T", "U", "W" ], "cost": 42 }, { "id": "S76", "elements": [ "T", "U", "W", "X" ], "cost": 256 }, { "id": "S77", "elements": [ "B", "C", "D", "F" ], "cost": 400 }, { "id": "S78", "elements": [ "M", "N", "O", "P", "Q", "T" ], "cost": 48 }, { "id": "S79", "elements": [ "R", "S", "T", "V" ], "cost": 300 }, { "id": "S80", "elements": [ "P", "S", "T" ], "cost": 252 }, { "id": "S81", "elements": [ "Q", "U", "W" ], "cost": 288 }, { "id": "S82", "elements": [ "I", "K", "M" ], "cost": 63 }, { "id": "S83", "elements": [ "L", "O", "P", "U", "V" ], "cost": 350 }, { "id": "S84", "elements": [ "A", "B", "C", "D", "E", "G", "H" ], "cost": 14 }, { "id": "S85", "elements": [ "K", "L", "O", "P", "Q" ], "cost": 255 }, { "id": "S86", "elements": [ "T", "U", "V", "W", "Y" ], "cost": 370 }, { "id": "S87", "elements": [ "I", "L", "N", "O", "P" ], "cost": 195 }, { "id": "S88", "elements": [ "M", "N", "O", "Q", "R", "T" ], "cost": 438 }, { "id": "S89", "elements": [ "A", "B", "C", "D", "G" ], "cost": 250 }, { "id": "S90", "elements": [ "B", "C", "I" ], "cost": 159 }, { "id": "S91", "elements": [ "P", "Q", "T", "U", "V", "X", "Y" ], "cost": 560 }, { "id": "S92", "elements": [ "B", "C", "D", "E", "F", "G", "I" ], "cost": 203 }, { "id": "S93", "elements": [ "L", "M", "N", "P" ], "cost": 148 }, { "id": "S94", "elements": [ "C", "D", "E", "G", "I", "J", "K" ], "cost": 35 }, { "id": "S95", "elements": [ "B", "D", "G", "H", "I", "J", "N" ], "cost": 245 }, { "id": "S96", "elements": [ "N", "O", "P", "Q" ], "cost": 256 }, { "id": "S97", "elements": [ "N", "P", "Q", "U" ], "cost": 372 }, { "id": "S98", "elements": [ "S", "T", "U", "V", "X", "Y" ], "cost": 210 }, { "id": "S99", "elements": [ "D", "F", "G", "I" ], "cost": 360 }, { "id": "S100", "elements": [ "O", "R", "S", "T", "U", "V", "W" ], "cost": 364 }, { "id": "S101", "elements": [ "I", "J", "K", "L", "M" ], "cost": 75 }, { "id": "S102", "elements": [ "C", "D", "F", "G", "I", "J", "K" ], "cost": 168 }, { "id": "S103", "elements": [ "F", "H", "J", "K" ], "cost": 104 }, { "id": "S104", "elements": [ "D", "E", "H", "J" ], "cost": 160 }, { "id": "S105", "elements": [ "B", "C", "D", "E" ], "cost": 56 }, { "id": "S106", "elements": [ "Q", "S", "T", "U", "V", "W", "X" ], "cost": 154 }, { "id": "S107", "elements": [ "A", "D", "F" ], "cost": 216 }, { "id": "S108", "elements": [ "D", "F", "G", "H" ], "cost": 92 }, { "id": "S109", "elements": [ "O", "Q", "R", "S", "T" ], "cost": 35 }, { "id": "S110", "elements": [ "D", "H", "I" ], "cost": 261 }, { "id": "S111", "elements": [ "P", "Q", "S" ], "cost": 57 }, { "id": "S112", "elements": [ "L", "N", "O", "Q" ], "cost": 212 }, { "id": "S113", "elements": [ "H", "K", "N" ], "cost": 18 }, { "id": "S114", "elements": [ "P", "S", "U", "V", "W", "X", "Y" ], "cost": 588 }, { "id": "S115", "elements": [ "A", "C", "D", "E", "F" ], "cost": 45 }, { "id": "S116", "elements": [ "H", "J", "K", "N" ], "cost": 312 }, { "id": "S117", "elements": [ "P", "T", "U", "W" ], "cost": 272 }, { "id": "S118", "elements": [ "S", "W", "Y" ], "cost": 249 }, { "id": "S119", "elements": [ "U", "W", "X" ], "cost": 51 }, { "id": "S120", "elements": [ "P", "Q", "R", "S", "U", "W" ], "cost": 492 }, { "id": "S121", "elements": [ "L", "M", "N", "O", "P", "S" ], "cost": 366 }, { "id": "S122", "elements": [ "F", "G", "H", "I", "J", "N" ], "cost": 366 }, { "id": "S123", "elements": [ "A", "B", "C", "D" ], "cost": 256 }, { "id": "S124", "elements": [ "A", "B", "C", "E", "F", "G" ], "cost": 96 }, { "id": "S125", "elements": [ "C", "F", "G" ], "cost": 213 }, { "id": "S126", "elements": [ "N", "O", "R", "S" ], "cost": 256 }, { "id": "S127", "elements": [ "U", "V", "X", "Y" ], "cost": 356 }, { "id": "S128", "elements": [ "C", "E", "F", "G", "H", "I", "M" ], "cost": 315 }, { "id": "S129", "elements": [ "A" ], "cost": 10000 }, { "id": "S130", "elements": [ "B" ], "cost": 10000 }, { "id": "S131", "elements": [ "C" ], "cost": 10000 }, { "id": "S132", "elements": [ "D" ], "cost": 10000 }, { "id": "S133", "elements": [ "E" ], "cost": 10000 }, { "id": "S134", "elements": [ "F" ], "cost": 10000 }, { "id": "S135", "elements": [ "G" ], "cost": 10000 }, { "id": "S136", "elements": [ "H" ], "cost": 10000 }, { "id": "S137", "elements": [ "I" ], "cost": 10000 }, { "id": "S138", "elements": [ "J" ], "cost": 10000 }, { "id": "S139", "elements": [ "K" ], "cost": 10000 }, { "id": "S140", "elements": [ "L" ], "cost": 10000 }, { "id": "S141", "elements": [ "M" ], "cost": 10000 }, { "id": "S142", "elements": [ "N" ], "cost": 10000 }, { "id": "S143", "elements": [ "O" ], "cost": 10000 }, { "id": "S144", "elements": [ "P" ], "cost": 10000 }, { "id": "S145", "elements": [ "Q" ], "cost": 10000 }, { "id": "S146", "elements": [ "R" ], "cost": 10000 }, { "id": "S147", "elements": [ "S" ], "cost": 10000 }, { "id": "S148", "elements": [ "T" ], "cost": 10000 }, { "id": "S149", "elements": [ "U" ], "cost": 10000 }, { "id": "S150", "elements": [ "V" ], "cost": 10000 }, { "id": "S151", "elements": [ "W" ], "cost": 10000 }, { "id": "S152", "elements": [ "X" ], "cost": 10000 }, { "id": "S153", "elements": [ "Y" ], "cost": 10000 } ] }, "solution_variant": [ "S4", "S13", "S25", "S46", "S60" ], "context_index": 23, "input_format": "json", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "We have a bunch of students and a bunch of ready-made project kits to choose from; the task is to decide which kits to buy so each student is covered by exactly a single kit. The way to judge any choice is by totaling the prices of the kits purchased — the lower that sum, the better the plan. It’s important that no student is skipped and no student gets more than one kit, and the concrete list of students, kits, and costs appears below.\n\n# num_students=23\n# num_kits=148\nkit_id,kit_cost,covered_students\nS1,276,15 16 18 19 20 22\nS2,192,10 11 15 18\nS3,45,10 12 13 14 15\nS4,378,0 1 2 3 4 5\nS5,10,5 7 8 9 11\nS6,180,0 1 2 3 5 6\nS7,235,13 14 15 16 17\nS8,48,14 15 16 19\nS9,220,5 7 9 11\nS10,112,15 18 19 21\nS11,90,6 11 13 14 15\nS12,4,3 5\nS13,60,17 20 21\nS14,208,5 6 8 10\nS15,112,16 17 18 19\nS16,234,9 12 13\nS17,483,1 3 5 6 7 9 10\nS18,440,18 19 20 21 22\nS19,200,17 19 20 21 22\nS20,72,16 17 18\nS21,12,1 2 3\nS22,6,12 13 14 16 17 18\nS23,130,5 9 10 12 13\nS24,12,9 10 11 17\nS25,200,9 10 11 13\nS26,511,14 15 16 17 18 19 21\nS27,90,1 3 5 6 7 8\nS28,93,4 5 6\nS29,259,7 8 9 10 11 12 13\nS30,152,11 15 16 17\nS31,468,5 7 8 9 10 11\nS32,145,17 18 19 20 21\nS33,360,6 8 9 10 11\nS34,234,3 6 7\nS35,210,5 6 7 8 9 11\nS36,222,0 2 8\nS37,168,0 1 2 3\nS38,176,14 15 16 18\nS39,156,12 14 15 20\nS40,116,9 12 14 15\nS41,470,0 1 2 3 4\nS42,486,7 9 10 11 12 13\nS43,395,14 15 17 18 19\nS44,210,6 8 9\nS45,430,0 1 2 4 5\nS46,18,0 1 2 3 4 6\nS47,340,19 20 21 22\nS48,336,13 18 19 22\nS49,175,2 3 4 5 6 7 9\nS50,147,4 6 7 8 9 10 12\nS51,588,17 18 19 20 21 22\nS52,90,3 7 12\nS53,28,0 2\nS54,180,0 1 2\nS55,576,11 12 14 15 16 17\nS56,69,4 6 9\nS57,102,15 16 17 18 19 21\nS58,225,13 16 21\nS59,280,12 15 16 18\nS60,136,2 3 4 5\nS61,171,9 12 14\nS62,27,18 20 21\nS63,267,10 11 12\nS64,380,10 11 12 14\nS65,297,6 7 8\nS66,315,2 3 4 5 6 7 10\nS67,49,14 16 17 18 19 20 21\nS68,480,11 12 13 14 15 16\nS69,70,16 19 20 21 22\nS70,96,5 7 8\nS71,60,9 12 17\nS72,102,10 13 14\nS73,350,12 19 20 21 22\nS74,402,1 5 6 7 9 11\nS75,240,14 16 17 18 19\nS76,440,8 10 12 13 14\nS77,108,7 8 9 11\nS78,174,17 18 20\nS79,490,10 11 13 14 15 16 18\nS80,63,16 19 20\nS81,297,6 8 12\nS82,208,0 2 3 4\nS83,159,4 7 9\nS84,18,19 20 22\nS85,284,1 2 3 5\nS86,24,3 4 7 8\nS87,63,18 19 20\nS88,69,17 18 22\nS89,336,8 9 10 11 13 15\nS90,340,13 16 17 18 20\nS91,348,16 17 19 22\nS92,128,10 15\nS93,200,15 16 18 19\nS94,465,7 8 10 11 13\nS95,168,6 7 10\nS96,414,1 2 3 4 5 7\nS97,219,2 3 4\nS98,75,15 16 18 20 21\nS99,42,0 1\nS100,115,1 3 4 5 6\nS101,356,9 15 16 19\nS102,219,13 14 15\nS103,195,5 8 10\nS104,480,10 11 13 14 15\nS105,291,11 13 14\nS106,60,8 10 11 12 14 17\nS107,276,18 20 22\nS108,300,1 3 4 6 7\nS109,6,13 14 15 16 17 18\nS110,85,13 15 17 19 20\nS111,184,11 15 17 18\nS112,21,10 11 12 13 14 15 17\nS113,322,1 2 3 4 5 6 7\nS114,105,1 2 3 4 5\nS115,228,1 7 8 9\nS116,330,1 3 4 5 6 7\nS117,90,6 8 9 10 11 12\nS118,208,5 6 7 11\nS119,246,14 15 16 17 18 19\nS120,102,17 20\nS121,36,5 7 9 12 14 15\nS122,140,4 9 10 12\nS123,360,5 6 7 9 10\nS124,270,13 15 17 18 19 20\nS125,8,2 6 7 10\nS126,10000,0\nS127,10000,1\nS128,10000,2\nS129,10000,3\nS130,10000,4\nS131,10000,5\nS132,10000,6\nS133,10000,7\nS134,10000,8\nS135,10000,9\nS136,10000,10\nS137,10000,11\nS138,10000,12\nS139,10000,13\nS140,10000,14\nS141,10000,15\nS142,10000,16\nS143,10000,17\nS144,10000,18\nS145,10000,19\nS146,10000,20\nS147,10000,21\nS148,10000,22\n\nAlso, when you send back your pick, tuck it into a tiny JSON snippet so it's easy to parse. Something like this will do:\n\n{\n \"solution\": [\"kit_id\", ...]\n}\n\nIn plain words: \"solution\" should list the kits you want to buy, and each item in that array is the kit's identifier (the \"kit_id\" placeholder is just there to show the shape — replace it with the real ids from the instance). This is just a sketch of the expected shape, not the actual answer.\n\nPlease make sure you use the identifiers exactly as they're given in the instance input — don't rename them or invent new labels. \nValid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.", "instance": { "problem_type": "SPP", "num_elements": 23, "num_sets": 148, "density": 0.1709753231492362, "sets": [ { "id": 1, "elements": [ 16, 17, 19, 20, 21, 23 ], "cost": 276 }, { "id": 2, "elements": [ 11, 12, 16, 19 ], "cost": 192 }, { "id": 3, "elements": [ 11, 13, 14, 15, 16 ], "cost": 45 }, { "id": 4, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 378 }, { "id": 5, "elements": [ 6, 8, 9, 10, 12 ], "cost": 10 }, { "id": 6, "elements": [ 1, 2, 3, 4, 6, 7 ], "cost": 180 }, { "id": 7, "elements": [ 14, 15, 16, 17, 18 ], "cost": 235 }, { "id": 8, "elements": [ 15, 16, 17, 20 ], "cost": 48 }, { "id": 9, "elements": [ 6, 8, 10, 12 ], "cost": 220 }, { "id": 10, "elements": [ 16, 19, 20, 22 ], "cost": 112 }, { "id": 11, "elements": [ 7, 12, 14, 15, 16 ], "cost": 90 }, { "id": 12, "elements": [ 4, 6 ], "cost": 4 }, { "id": 13, "elements": [ 18, 21, 22 ], "cost": 60 }, { "id": 14, "elements": [ 6, 7, 9, 11 ], "cost": 208 }, { "id": 15, "elements": [ 17, 18, 19, 20 ], "cost": 112 }, { "id": 16, "elements": [ 10, 13, 14 ], "cost": 234 }, { "id": 17, "elements": [ 2, 4, 6, 7, 8, 10, 11 ], "cost": 483 }, { "id": 18, "elements": [ 19, 20, 21, 22, 23 ], "cost": 440 }, { "id": 19, "elements": [ 18, 20, 21, 22, 23 ], "cost": 200 }, { "id": 20, "elements": [ 17, 18, 19 ], "cost": 72 }, { "id": 21, "elements": [ 2, 3, 4 ], "cost": 12 }, { "id": 22, "elements": [ 13, 14, 15, 17, 18, 19 ], "cost": 6 }, { "id": 23, "elements": [ 6, 10, 11, 13, 14 ], "cost": 130 }, { "id": 24, "elements": [ 10, 11, 12, 18 ], "cost": 12 }, { "id": 25, "elements": [ 10, 11, 12, 14 ], "cost": 200 }, { "id": 26, "elements": [ 15, 16, 17, 18, 19, 20, 22 ], "cost": 511 }, { "id": 27, "elements": [ 2, 4, 6, 7, 8, 9 ], "cost": 90 }, { "id": 28, "elements": [ 5, 6, 7 ], "cost": 93 }, { "id": 29, "elements": [ 8, 9, 10, 11, 12, 13, 14 ], "cost": 259 }, { "id": 30, "elements": [ 12, 16, 17, 18 ], "cost": 152 }, { "id": 31, "elements": [ 6, 8, 9, 10, 11, 12 ], "cost": 468 }, { "id": 32, "elements": [ 18, 19, 20, 21, 22 ], "cost": 145 }, { "id": 33, "elements": [ 7, 9, 10, 11, 12 ], "cost": 360 }, { "id": 34, "elements": [ 4, 7, 8 ], "cost": 234 }, { "id": 35, "elements": [ 6, 7, 8, 9, 10, 12 ], "cost": 210 }, { "id": 36, "elements": [ 1, 3, 9 ], "cost": 222 }, { "id": 37, "elements": [ 1, 2, 3, 4 ], "cost": 168 }, { "id": 38, "elements": [ 15, 16, 17, 19 ], "cost": 176 }, { "id": 39, "elements": [ 13, 15, 16, 21 ], "cost": 156 }, { "id": 40, "elements": [ 10, 13, 15, 16 ], "cost": 116 }, { "id": 41, "elements": [ 1, 2, 3, 4, 5 ], "cost": 470 }, { "id": 42, "elements": [ 8, 10, 11, 12, 13, 14 ], "cost": 486 }, { "id": 43, "elements": [ 15, 16, 18, 19, 20 ], "cost": 395 }, { "id": 44, "elements": [ 7, 9, 10 ], "cost": 210 }, { "id": 45, "elements": [ 1, 2, 3, 5, 6 ], "cost": 430 }, { "id": 46, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 18 }, { "id": 47, "elements": [ 20, 21, 22, 23 ], "cost": 340 }, { "id": 48, "elements": [ 14, 19, 20, 23 ], "cost": 336 }, { "id": 49, "elements": [ 3, 4, 5, 6, 7, 8, 10 ], "cost": 175 }, { "id": 50, "elements": [ 5, 7, 8, 9, 10, 11, 13 ], "cost": 147 }, { "id": 51, "elements": [ 18, 19, 20, 21, 22, 23 ], "cost": 588 }, { "id": 52, "elements": [ 4, 8, 13 ], "cost": 90 }, { "id": 53, "elements": [ 1, 3 ], "cost": 28 }, { "id": 54, "elements": [ 1, 2, 3 ], "cost": 180 }, { "id": 55, "elements": [ 12, 13, 15, 16, 17, 18 ], "cost": 576 }, { "id": 56, "elements": [ 5, 7, 10 ], "cost": 69 }, { "id": 57, "elements": [ 16, 17, 18, 19, 20, 22 ], "cost": 102 }, { "id": 58, "elements": [ 14, 17, 22 ], "cost": 225 }, { "id": 59, "elements": [ 13, 16, 17, 19 ], "cost": 280 }, { "id": 60, "elements": [ 3, 4, 5, 6 ], "cost": 136 }, { "id": 61, "elements": [ 10, 13, 15 ], "cost": 171 }, { "id": 62, "elements": [ 19, 21, 22 ], "cost": 27 }, { "id": 63, "elements": [ 11, 12, 13 ], "cost": 267 }, { "id": 64, "elements": [ 11, 12, 13, 15 ], "cost": 380 }, { "id": 65, "elements": [ 7, 8, 9 ], "cost": 297 }, { "id": 66, "elements": [ 3, 4, 5, 6, 7, 8, 11 ], "cost": 315 }, { "id": 67, "elements": [ 15, 17, 18, 19, 20, 21, 22 ], "cost": 49 }, { "id": 68, "elements": [ 12, 13, 14, 15, 16, 17 ], "cost": 480 }, { "id": 69, "elements": [ 17, 20, 21, 22, 23 ], "cost": 70 }, { "id": 70, "elements": [ 6, 8, 9 ], "cost": 96 }, { "id": 71, "elements": [ 10, 13, 18 ], "cost": 60 }, { "id": 72, "elements": [ 11, 14, 15 ], "cost": 102 }, { "id": 73, "elements": [ 13, 20, 21, 22, 23 ], "cost": 350 }, { "id": 74, "elements": [ 2, 6, 7, 8, 10, 12 ], "cost": 402 }, { "id": 75, "elements": [ 15, 17, 18, 19, 20 ], "cost": 240 }, { "id": 76, "elements": [ 9, 11, 13, 14, 15 ], "cost": 440 }, { "id": 77, "elements": [ 8, 9, 10, 12 ], "cost": 108 }, { "id": 78, "elements": [ 18, 19, 21 ], "cost": 174 }, { "id": 79, "elements": [ 11, 12, 14, 15, 16, 17, 19 ], "cost": 490 }, { "id": 80, "elements": [ 17, 20, 21 ], "cost": 63 }, { "id": 81, "elements": [ 7, 9, 13 ], "cost": 297 }, { "id": 82, "elements": [ 1, 3, 4, 5 ], "cost": 208 }, { "id": 83, "elements": [ 5, 8, 10 ], "cost": 159 }, { "id": 84, "elements": [ 20, 21, 23 ], "cost": 18 }, { "id": 85, "elements": [ 2, 3, 4, 6 ], "cost": 284 }, { "id": 86, "elements": [ 4, 5, 8, 9 ], "cost": 24 }, { "id": 87, "elements": [ 19, 20, 21 ], "cost": 63 }, { "id": 88, "elements": [ 18, 19, 23 ], "cost": 69 }, { "id": 89, "elements": [ 9, 10, 11, 12, 14, 16 ], "cost": 336 }, { "id": 90, "elements": [ 14, 17, 18, 19, 21 ], "cost": 340 }, { "id": 91, "elements": [ 17, 18, 20, 23 ], "cost": 348 }, { "id": 92, "elements": [ 11, 16 ], "cost": 128 }, { "id": 93, "elements": [ 16, 17, 19, 20 ], "cost": 200 }, { "id": 94, "elements": [ 8, 9, 11, 12, 14 ], "cost": 465 }, { "id": 95, "elements": [ 7, 8, 11 ], "cost": 168 }, { "id": 96, "elements": [ 2, 3, 4, 5, 6, 8 ], "cost": 414 }, { "id": 97, "elements": [ 3, 4, 5 ], "cost": 219 }, { "id": 98, "elements": [ 16, 17, 19, 21, 22 ], "cost": 75 }, { "id": 99, "elements": [ 1, 2 ], "cost": 42 }, { "id": 100, "elements": [ 2, 4, 5, 6, 7 ], "cost": 115 }, { "id": 101, "elements": [ 10, 16, 17, 20 ], "cost": 356 }, { "id": 102, "elements": [ 14, 15, 16 ], "cost": 219 }, { "id": 103, "elements": [ 6, 9, 11 ], "cost": 195 }, { "id": 104, "elements": [ 11, 12, 14, 15, 16 ], "cost": 480 }, { "id": 105, "elements": [ 12, 14, 15 ], "cost": 291 }, { "id": 106, "elements": [ 9, 11, 12, 13, 15, 18 ], "cost": 60 }, { "id": 107, "elements": [ 19, 21, 23 ], "cost": 276 }, { "id": 108, "elements": [ 2, 4, 5, 7, 8 ], "cost": 300 }, { "id": 109, "elements": [ 14, 15, 16, 17, 18, 19 ], "cost": 6 }, { "id": 110, "elements": [ 14, 16, 18, 20, 21 ], "cost": 85 }, { "id": 111, "elements": [ 12, 16, 18, 19 ], "cost": 184 }, { "id": 112, "elements": [ 11, 12, 13, 14, 15, 16, 18 ], "cost": 21 }, { "id": 113, "elements": [ 2, 3, 4, 5, 6, 7, 8 ], "cost": 322 }, { "id": 114, "elements": [ 2, 3, 4, 5, 6 ], "cost": 105 }, { "id": 115, "elements": [ 2, 8, 9, 10 ], "cost": 228 }, { "id": 116, "elements": [ 2, 4, 5, 6, 7, 8 ], "cost": 330 }, { "id": 117, "elements": [ 7, 9, 10, 11, 12, 13 ], "cost": 90 }, { "id": 118, "elements": [ 6, 7, 8, 12 ], "cost": 208 }, { "id": 119, "elements": [ 15, 16, 17, 18, 19, 20 ], "cost": 246 }, { "id": 120, "elements": [ 18, 21 ], "cost": 102 }, { "id": 121, "elements": [ 6, 8, 10, 13, 15, 16 ], "cost": 36 }, { "id": 122, "elements": [ 5, 10, 11, 13 ], "cost": 140 }, { "id": 123, "elements": [ 6, 7, 8, 10, 11 ], "cost": 360 }, { "id": 124, "elements": [ 14, 16, 18, 19, 20, 21 ], "cost": 270 }, { "id": 125, "elements": [ 3, 7, 8, 11 ], "cost": 8 }, { "id": 126, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 127, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 128, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 129, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 130, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 131, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 132, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 133, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 134, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 135, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 136, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 137, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 138, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 139, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 140, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 141, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 142, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 143, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 144, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 145, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 146, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 147, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 148, "elements": [ 23 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0023_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0023_bag_stats.png" }, "solution": [ 3, 5, 46, 62, 91 ], "obj": 448.0, "instance_variant": { "num_elements": 23, "num_sets": 148, "sets": [ { "id": "S1", "elements": [ 15, 16, 18, 19, 20, 22 ], "cost": 276 }, { "id": "S2", "elements": [ 10, 11, 15, 18 ], "cost": 192 }, { "id": "S3", "elements": [ 10, 12, 13, 14, 15 ], "cost": 45 }, { "id": "S4", "elements": [ 0, 1, 2, 3, 4, 5 ], "cost": 378 }, { "id": "S5", "elements": [ 5, 7, 8, 9, 11 ], "cost": 10 }, { "id": "S6", "elements": [ 0, 1, 2, 3, 5, 6 ], "cost": 180 }, { "id": "S7", "elements": [ 13, 14, 15, 16, 17 ], "cost": 235 }, { "id": "S8", "elements": [ 14, 15, 16, 19 ], "cost": 48 }, { "id": "S9", "elements": [ 5, 7, 9, 11 ], "cost": 220 }, { "id": "S10", "elements": [ 15, 18, 19, 21 ], "cost": 112 }, { "id": "S11", "elements": [ 6, 11, 13, 14, 15 ], "cost": 90 }, { "id": "S12", "elements": [ 3, 5 ], "cost": 4 }, { "id": "S13", "elements": [ 17, 20, 21 ], "cost": 60 }, { "id": "S14", "elements": [ 5, 6, 8, 10 ], "cost": 208 }, { "id": "S15", "elements": [ 16, 17, 18, 19 ], "cost": 112 }, { "id": "S16", "elements": [ 9, 12, 13 ], "cost": 234 }, { "id": "S17", "elements": [ 1, 3, 5, 6, 7, 9, 10 ], "cost": 483 }, { "id": "S18", "elements": [ 18, 19, 20, 21, 22 ], "cost": 440 }, { "id": "S19", "elements": [ 17, 19, 20, 21, 22 ], "cost": 200 }, { "id": "S20", "elements": [ 16, 17, 18 ], "cost": 72 }, { "id": "S21", "elements": [ 1, 2, 3 ], "cost": 12 }, { "id": "S22", "elements": [ 12, 13, 14, 16, 17, 18 ], "cost": 6 }, { "id": "S23", "elements": [ 5, 9, 10, 12, 13 ], "cost": 130 }, { "id": "S24", "elements": [ 9, 10, 11, 17 ], "cost": 12 }, { "id": "S25", "elements": [ 9, 10, 11, 13 ], "cost": 200 }, { "id": "S26", "elements": [ 14, 15, 16, 17, 18, 19, 21 ], "cost": 511 }, { "id": "S27", "elements": [ 1, 3, 5, 6, 7, 8 ], "cost": 90 }, { "id": "S28", "elements": [ 4, 5, 6 ], "cost": 93 }, { "id": "S29", "elements": [ 7, 8, 9, 10, 11, 12, 13 ], "cost": 259 }, { "id": "S30", "elements": [ 11, 15, 16, 17 ], "cost": 152 }, { "id": "S31", "elements": [ 5, 7, 8, 9, 10, 11 ], "cost": 468 }, { "id": "S32", "elements": [ 17, 18, 19, 20, 21 ], "cost": 145 }, { "id": "S33", "elements": [ 6, 8, 9, 10, 11 ], "cost": 360 }, { "id": "S34", "elements": [ 3, 6, 7 ], "cost": 234 }, { "id": "S35", "elements": [ 5, 6, 7, 8, 9, 11 ], "cost": 210 }, { "id": "S36", "elements": [ 0, 2, 8 ], "cost": 222 }, { "id": "S37", "elements": [ 0, 1, 2, 3 ], "cost": 168 }, { "id": "S38", "elements": [ 14, 15, 16, 18 ], "cost": 176 }, { "id": "S39", "elements": [ 12, 14, 15, 20 ], "cost": 156 }, { "id": "S40", "elements": [ 9, 12, 14, 15 ], "cost": 116 }, { "id": "S41", "elements": [ 0, 1, 2, 3, 4 ], "cost": 470 }, { "id": "S42", "elements": [ 7, 9, 10, 11, 12, 13 ], "cost": 486 }, { "id": "S43", "elements": [ 14, 15, 17, 18, 19 ], "cost": 395 }, { "id": "S44", "elements": [ 6, 8, 9 ], "cost": 210 }, { "id": "S45", "elements": [ 0, 1, 2, 4, 5 ], "cost": 430 }, { "id": "S46", "elements": [ 0, 1, 2, 3, 4, 6 ], "cost": 18 }, { "id": "S47", "elements": [ 19, 20, 21, 22 ], "cost": 340 }, { "id": "S48", "elements": [ 13, 18, 19, 22 ], "cost": 336 }, { "id": "S49", "elements": [ 2, 3, 4, 5, 6, 7, 9 ], "cost": 175 }, { "id": "S50", "elements": [ 4, 6, 7, 8, 9, 10, 12 ], "cost": 147 }, { "id": "S51", "elements": [ 17, 18, 19, 20, 21, 22 ], "cost": 588 }, { "id": "S52", "elements": [ 3, 7, 12 ], "cost": 90 }, { "id": "S53", "elements": [ 0, 2 ], "cost": 28 }, { "id": "S54", "elements": [ 0, 1, 2 ], "cost": 180 }, { "id": "S55", "elements": [ 11, 12, 14, 15, 16, 17 ], "cost": 576 }, { "id": "S56", "elements": [ 4, 6, 9 ], "cost": 69 }, { "id": "S57", "elements": [ 15, 16, 17, 18, 19, 21 ], "cost": 102 }, { "id": "S58", "elements": [ 13, 16, 21 ], "cost": 225 }, { "id": "S59", "elements": [ 12, 15, 16, 18 ], "cost": 280 }, { "id": "S60", "elements": [ 2, 3, 4, 5 ], "cost": 136 }, { "id": "S61", "elements": [ 9, 12, 14 ], "cost": 171 }, { "id": "S62", "elements": [ 18, 20, 21 ], "cost": 27 }, { "id": "S63", "elements": [ 10, 11, 12 ], "cost": 267 }, { "id": "S64", "elements": [ 10, 11, 12, 14 ], "cost": 380 }, { "id": "S65", "elements": [ 6, 7, 8 ], "cost": 297 }, { "id": "S66", "elements": [ 2, 3, 4, 5, 6, 7, 10 ], "cost": 315 }, { "id": "S67", "elements": [ 14, 16, 17, 18, 19, 20, 21 ], "cost": 49 }, { "id": "S68", "elements": [ 11, 12, 13, 14, 15, 16 ], "cost": 480 }, { "id": "S69", "elements": [ 16, 19, 20, 21, 22 ], "cost": 70 }, { "id": "S70", "elements": [ 5, 7, 8 ], "cost": 96 }, { "id": "S71", "elements": [ 9, 12, 17 ], "cost": 60 }, { "id": "S72", "elements": [ 10, 13, 14 ], "cost": 102 }, { "id": "S73", "elements": [ 12, 19, 20, 21, 22 ], "cost": 350 }, { "id": "S74", "elements": [ 1, 5, 6, 7, 9, 11 ], "cost": 402 }, { "id": "S75", "elements": [ 14, 16, 17, 18, 19 ], "cost": 240 }, { "id": "S76", "elements": [ 8, 10, 12, 13, 14 ], "cost": 440 }, { "id": "S77", "elements": [ 7, 8, 9, 11 ], "cost": 108 }, { "id": "S78", "elements": [ 17, 18, 20 ], "cost": 174 }, { "id": "S79", "elements": [ 10, 11, 13, 14, 15, 16, 18 ], "cost": 490 }, { "id": "S80", "elements": [ 16, 19, 20 ], "cost": 63 }, { "id": "S81", "elements": [ 6, 8, 12 ], "cost": 297 }, { "id": "S82", "elements": [ 0, 2, 3, 4 ], "cost": 208 }, { "id": "S83", "elements": [ 4, 7, 9 ], "cost": 159 }, { "id": "S84", "elements": [ 19, 20, 22 ], "cost": 18 }, { "id": "S85", "elements": [ 1, 2, 3, 5 ], "cost": 284 }, { "id": "S86", "elements": [ 3, 4, 7, 8 ], "cost": 24 }, { "id": "S87", "elements": [ 18, 19, 20 ], "cost": 63 }, { "id": "S88", "elements": [ 17, 18, 22 ], "cost": 69 }, { "id": "S89", "elements": [ 8, 9, 10, 11, 13, 15 ], "cost": 336 }, { "id": "S90", "elements": [ 13, 16, 17, 18, 20 ], "cost": 340 }, { "id": "S91", "elements": [ 16, 17, 19, 22 ], "cost": 348 }, { "id": "S92", "elements": [ 10, 15 ], "cost": 128 }, { "id": "S93", "elements": [ 15, 16, 18, 19 ], "cost": 200 }, { "id": "S94", "elements": [ 7, 8, 10, 11, 13 ], "cost": 465 }, { "id": "S95", "elements": [ 6, 7, 10 ], "cost": 168 }, { "id": "S96", "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 414 }, { "id": "S97", "elements": [ 2, 3, 4 ], "cost": 219 }, { "id": "S98", "elements": [ 15, 16, 18, 20, 21 ], "cost": 75 }, { "id": "S99", "elements": [ 0, 1 ], "cost": 42 }, { "id": "S100", "elements": [ 1, 3, 4, 5, 6 ], "cost": 115 }, { "id": "S101", "elements": [ 9, 15, 16, 19 ], "cost": 356 }, { "id": "S102", "elements": [ 13, 14, 15 ], "cost": 219 }, { "id": "S103", "elements": [ 5, 8, 10 ], "cost": 195 }, { "id": "S104", "elements": [ 10, 11, 13, 14, 15 ], "cost": 480 }, { "id": "S105", "elements": [ 11, 13, 14 ], "cost": 291 }, { "id": "S106", "elements": [ 8, 10, 11, 12, 14, 17 ], "cost": 60 }, { "id": "S107", "elements": [ 18, 20, 22 ], "cost": 276 }, { "id": "S108", "elements": [ 1, 3, 4, 6, 7 ], "cost": 300 }, { "id": "S109", "elements": [ 13, 14, 15, 16, 17, 18 ], "cost": 6 }, { "id": "S110", "elements": [ 13, 15, 17, 19, 20 ], "cost": 85 }, { "id": "S111", "elements": [ 11, 15, 17, 18 ], "cost": 184 }, { "id": "S112", "elements": [ 10, 11, 12, 13, 14, 15, 17 ], "cost": 21 }, { "id": "S113", "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 322 }, { "id": "S114", "elements": [ 1, 2, 3, 4, 5 ], "cost": 105 }, { "id": "S115", "elements": [ 1, 7, 8, 9 ], "cost": 228 }, { "id": "S116", "elements": [ 1, 3, 4, 5, 6, 7 ], "cost": 330 }, { "id": "S117", "elements": [ 6, 8, 9, 10, 11, 12 ], "cost": 90 }, { "id": "S118", "elements": [ 5, 6, 7, 11 ], "cost": 208 }, { "id": "S119", "elements": [ 14, 15, 16, 17, 18, 19 ], "cost": 246 }, { "id": "S120", "elements": [ 17, 20 ], "cost": 102 }, { "id": "S121", "elements": [ 5, 7, 9, 12, 14, 15 ], "cost": 36 }, { "id": "S122", "elements": [ 4, 9, 10, 12 ], "cost": 140 }, { "id": "S123", "elements": [ 5, 6, 7, 9, 10 ], "cost": 360 }, { "id": "S124", "elements": [ 13, 15, 17, 18, 19, 20 ], "cost": 270 }, { "id": "S125", "elements": [ 2, 6, 7, 10 ], "cost": 8 }, { "id": "S126", "elements": [ 0 ], "cost": 10000 }, { "id": "S127", "elements": [ 1 ], "cost": 10000 }, { "id": "S128", "elements": [ 2 ], "cost": 10000 }, { "id": "S129", "elements": [ 3 ], "cost": 10000 }, { "id": "S130", "elements": [ 4 ], "cost": 10000 }, { "id": "S131", "elements": [ 5 ], "cost": 10000 }, { "id": "S132", "elements": [ 6 ], "cost": 10000 }, { "id": "S133", "elements": [ 7 ], "cost": 10000 }, { "id": "S134", "elements": [ 8 ], "cost": 10000 }, { "id": "S135", "elements": [ 9 ], "cost": 10000 }, { "id": "S136", "elements": [ 10 ], "cost": 10000 }, { "id": "S137", "elements": [ 11 ], "cost": 10000 }, { "id": "S138", "elements": [ 12 ], "cost": 10000 }, { "id": "S139", "elements": [ 13 ], "cost": 10000 }, { "id": "S140", "elements": [ 14 ], "cost": 10000 }, { "id": "S141", "elements": [ 15 ], "cost": 10000 }, { "id": "S142", "elements": [ 16 ], "cost": 10000 }, { "id": "S143", "elements": [ 17 ], "cost": 10000 }, { "id": "S144", "elements": [ 18 ], "cost": 10000 }, { "id": "S145", "elements": [ 19 ], "cost": 10000 }, { "id": "S146", "elements": [ 20 ], "cost": 10000 }, { "id": "S147", "elements": [ 21 ], "cost": 10000 }, { "id": "S148", "elements": [ 22 ], "cost": 10000 } ] }, "solution_variant": [ "S3", "S5", "S46", "S62", "S91" ], "context_index": 24, "input_format": "csv", "input_index_base": 0 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "At a local bookstore they’re prepping themed bundles and need to choose a selection of boxes so that every title is covered exactly once — every book must be in one box and cannot appear in more than one. To judge a selection, add up the cost of each chosen box; the selection with the smallest total cost is preferred. The exact titles and box choices are posted below.\n\n{\n \"total_titles\": 19,\n \"num_boxes\": 87,\n \"sets\": [\n {\n \"box_id\": \"S1\",\n \"titles_in_box\": [\n 13,\n 14\n ],\n \"box_cost\": 104\n },\n {\n \"box_id\": \"S2\",\n \"titles_in_box\": [\n 8,\n 13\n ],\n \"box_cost\": 92\n },\n {\n \"box_id\": \"S3\",\n \"titles_in_box\": [\n 2,\n 4,\n 6\n ],\n \"box_cost\": 171\n },\n {\n \"box_id\": \"S4\",\n \"titles_in_box\": [\n 5,\n 8,\n 9,\n 10,\n 11,\n 13\n ],\n \"box_cost\": 96\n },\n {\n \"box_id\": \"S5\",\n \"titles_in_box\": [\n 2,\n 3,\n 6,\n 7\n ],\n \"box_cost\": 388\n },\n {\n \"box_id\": \"S6\",\n \"titles_in_box\": [\n 10,\n 11,\n 14\n ],\n \"box_cost\": 114\n },\n {\n \"box_id\": \"S7\",\n \"titles_in_box\": [\n 5,\n 7,\n 10\n ],\n \"box_cost\": 186\n },\n {\n \"box_id\": \"S8\",\n \"titles_in_box\": [\n 11,\n 13,\n 14\n ],\n \"box_cost\": 144\n },\n {\n \"box_id\": \"S9\",\n \"titles_in_box\": [\n 1,\n 2,\n 4\n ],\n \"box_cost\": 126\n },\n {\n \"box_id\": \"S10\",\n \"titles_in_box\": [\n 12,\n 13,\n 15,\n 16,\n 17\n ],\n \"box_cost\": 120\n },\n {\n \"box_id\": \"S11\",\n \"titles_in_box\": [\n 1,\n 2\n ],\n \"box_cost\": 164\n },\n {\n \"box_id\": \"S12\",\n \"titles_in_box\": [\n 6,\n 9,\n 10,\n 11\n ],\n \"box_cost\": 200\n },\n {\n \"box_id\": \"S13\",\n \"titles_in_box\": [\n 9,\n 10,\n 11\n ],\n \"box_cost\": 231\n },\n {\n \"box_id\": \"S14\",\n \"titles_in_box\": [\n 4,\n 5,\n 6,\n 8\n ],\n \"box_cost\": 4\n },\n {\n \"box_id\": \"S15\",\n \"titles_in_box\": [\n 1,\n 2,\n 4,\n 7\n ],\n \"box_cost\": 88\n },\n {\n \"box_id\": \"S16\",\n \"titles_in_box\": [\n 1,\n 2,\n 3,\n 4,\n 5\n ],\n \"box_cost\": 45\n },\n {\n \"box_id\": \"S17\",\n \"titles_in_box\": [\n 14,\n 15,\n 16,\n 18,\n 19\n ],\n \"box_cost\": 275\n },\n {\n \"box_id\": \"S18\",\n \"titles_in_box\": [\n 15,\n 16,\n 17,\n 18,\n 19\n ],\n \"box_cost\": 210\n },\n {\n \"box_id\": \"S19\",\n \"titles_in_box\": [\n 4,\n 5,\n 7,\n 9\n ],\n \"box_cost\": 72\n },\n {\n \"box_id\": \"S20\",\n \"titles_in_box\": [\n 9,\n 10,\n 11,\n 12,\n 13\n ],\n \"box_cost\": 290\n },\n {\n \"box_id\": \"S21\",\n \"titles_in_box\": [\n 11,\n 14\n ],\n \"box_cost\": 164\n },\n {\n \"box_id\": \"S22\",\n \"titles_in_box\": [\n 8,\n 10\n ],\n \"box_cost\": 118\n },\n {\n \"box_id\": \"S23\",\n \"titles_in_box\": [\n 14,\n 15,\n 16\n ],\n \"box_cost\": 297\n },\n {\n \"box_id\": \"S24\",\n \"titles_in_box\": [\n 16,\n 17\n ],\n \"box_cost\": 188\n },\n {\n \"box_id\": \"S25\",\n \"titles_in_box\": [\n 13,\n 15,\n 16\n ],\n \"box_cost\": 222\n },\n {\n \"box_id\": \"S26\",\n \"titles_in_box\": [\n 7,\n 8,\n 9,\n 12\n ],\n \"box_cost\": 252\n },\n {\n \"box_id\": \"S27\",\n \"titles_in_box\": [\n 1,\n 2,\n 3,\n 4\n ],\n \"box_cost\": 36\n },\n {\n \"box_id\": \"S28\",\n \"titles_in_box\": [\n 5,\n 8,\n 9,\n 10,\n 12\n ],\n \"box_cost\": 330\n },\n {\n \"box_id\": \"S29\",\n \"titles_in_box\": [\n 15,\n 17,\n 18,\n 19\n ],\n \"box_cost\": 92\n },\n {\n \"box_id\": \"S30\",\n \"titles_in_box\": [\n 13,\n 14,\n 15,\n 17\n ],\n \"box_cost\": 352\n },\n {\n \"box_id\": \"S31\",\n \"titles_in_box\": [\n 4,\n 7\n ],\n \"box_cost\": 106\n },\n {\n \"box_id\": \"S32\",\n \"titles_in_box\": [\n 6,\n 8,\n 10\n ],\n \"box_cost\": 198\n },\n {\n \"box_id\": \"S33\",\n \"titles_in_box\": [\n 14,\n 17\n ],\n \"box_cost\": 110\n },\n {\n \"box_id\": \"S34\",\n \"titles_in_box\": [\n 5,\n 6,\n 7,\n 8,\n 9\n ],\n \"box_cost\": 245\n },\n {\n \"box_id\": \"S35\",\n \"titles_in_box\": [\n 17,\n 18,\n 19\n ],\n \"box_cost\": 168\n },\n {\n \"box_id\": \"S36\",\n \"titles_in_box\": [\n 8,\n 11,\n 12\n ],\n \"box_cost\": 12\n },\n {\n \"box_id\": \"S37\",\n \"titles_in_box\": [\n 4,\n 5,\n 6,\n 7,\n 8\n ],\n \"box_cost\": 400\n },\n {\n \"box_id\": \"S38\",\n \"titles_in_box\": [\n 14,\n 15,\n 16,\n 17,\n 18\n ],\n \"box_cost\": 315\n },\n {\n \"box_id\": \"S39\",\n \"titles_in_box\": [\n 2,\n 3,\n 4,\n 5\n ],\n \"box_cost\": 232\n },\n {\n \"box_id\": \"S40\",\n \"titles_in_box\": [\n 11,\n 13,\n 14,\n 16,\n 19\n ],\n \"box_cost\": 215\n },\n {\n \"box_id\": \"S41\",\n \"titles_in_box\": [\n 7,\n 8,\n 9,\n 11,\n 12\n ],\n \"box_cost\": 255\n },\n {\n \"box_id\": \"S42\",\n \"titles_in_box\": [\n 2,\n 3,\n 4,\n 5,\n 6,\n 7\n ],\n \"box_cost\": 414\n },\n {\n \"box_id\": \"S43\",\n \"titles_in_box\": [\n 3,\n 4\n ],\n \"box_cost\": 126\n },\n {\n \"box_id\": \"S44\",\n \"titles_in_box\": [\n 1,\n 2,\n 4,\n 5,\n 6\n ],\n \"box_cost\": 400\n },\n {\n \"box_id\": \"S45\",\n \"titles_in_box\": [\n 16,\n 17,\n 18,\n 19\n ],\n \"box_cost\": 28\n },\n {\n \"box_id\": \"S46\",\n \"titles_in_box\": [\n 4,\n 5,\n 6\n ],\n \"box_cost\": 81\n },\n {\n \"box_id\": \"S47\",\n \"titles_in_box\": [\n 9,\n 11,\n 14\n ],\n \"box_cost\": 195\n },\n {\n \"box_id\": \"S48\",\n \"titles_in_box\": [\n 15,\n 16,\n 17,\n 19\n ],\n \"box_cost\": 296\n },\n {\n \"box_id\": \"S49\",\n \"titles_in_box\": [\n 8,\n 9,\n 10,\n 11,\n 12,\n 13\n ],\n \"box_cost\": 534\n },\n {\n \"box_id\": \"S50\",\n \"titles_in_box\": [\n 14,\n 17,\n 18,\n 19\n ],\n \"box_cost\": 168\n },\n {\n \"box_id\": \"S51\",\n \"titles_in_box\": [\n 4,\n 6\n ],\n \"box_cost\": 54\n },\n {\n \"box_id\": \"S52\",\n \"titles_in_box\": [\n 11,\n 12,\n 14\n ],\n \"box_cost\": 81\n },\n {\n \"box_id\": \"S53\",\n \"titles_in_box\": [\n 11,\n 12,\n 13\n ],\n \"box_cost\": 135\n },\n {\n \"box_id\": \"S54\",\n \"titles_in_box\": [\n 11,\n 12,\n 13,\n 15\n ],\n \"box_cost\": 364\n },\n {\n \"box_id\": \"S55\",\n \"titles_in_box\": [\n 11,\n 15\n ],\n \"box_cost\": 194\n },\n {\n \"box_id\": \"S56\",\n \"titles_in_box\": [\n 11,\n 12,\n 13,\n 15,\n 16\n ],\n \"box_cost\": 215\n },\n {\n \"box_id\": \"S57\",\n \"titles_in_box\": [\n 11,\n 12,\n 13,\n 14,\n 16\n ],\n \"box_cost\": 65\n },\n {\n \"box_id\": \"S58\",\n \"titles_in_box\": [\n 4,\n 7,\n 9\n ],\n \"box_cost\": 54\n },\n {\n \"box_id\": \"S59\",\n \"titles_in_box\": [\n 4,\n 6,\n 7,\n 8,\n 10\n ],\n \"box_cost\": 495\n },\n {\n \"box_id\": \"S60\",\n \"titles_in_box\": [\n 9,\n 13,\n 14,\n 17\n ],\n \"box_cost\": 196\n },\n {\n \"box_id\": \"S61\",\n \"titles_in_box\": [\n 5,\n 6,\n 7,\n 8,\n 10\n ],\n \"box_cost\": 450\n },\n {\n \"box_id\": \"S62\",\n \"titles_in_box\": [\n 9,\n 11,\n 12,\n 14\n ],\n \"box_cost\": 156\n },\n {\n \"box_id\": \"S63\",\n \"titles_in_box\": [\n 6,\n 7,\n 9\n ],\n \"box_cost\": 75\n },\n {\n \"box_id\": \"S64\",\n \"titles_in_box\": [\n 12,\n 16,\n 17,\n 18\n ],\n \"box_cost\": 252\n },\n {\n \"box_id\": \"S65\",\n \"titles_in_box\": [\n 12,\n 15,\n 16\n ],\n \"box_cost\": 102\n },\n {\n \"box_id\": \"S66\",\n \"titles_in_box\": [\n 8,\n 9,\n 10,\n 11\n ],\n \"box_cost\": 112\n },\n {\n \"box_id\": \"S67\",\n \"titles_in_box\": [\n 1,\n 2,\n 3,\n 5,\n 6\n ],\n \"box_cost\": 330\n },\n {\n \"box_id\": \"S68\",\n \"titles_in_box\": [\n 8,\n 10,\n 11,\n 12\n ],\n \"box_cost\": 124\n },\n {\n \"box_id\": \"S69\",\n \"titles_in_box\": [\n 1\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S70\",\n \"titles_in_box\": [\n 2\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S71\",\n \"titles_in_box\": [\n 3\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S72\",\n \"titles_in_box\": [\n 4\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S73\",\n \"titles_in_box\": [\n 5\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S74\",\n \"titles_in_box\": [\n 6\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S75\",\n \"titles_in_box\": [\n 7\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S76\",\n \"titles_in_box\": [\n 8\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S77\",\n \"titles_in_box\": [\n 9\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S78\",\n \"titles_in_box\": [\n 10\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S79\",\n \"titles_in_box\": [\n 11\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S80\",\n \"titles_in_box\": [\n 12\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S81\",\n \"titles_in_box\": [\n 13\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S82\",\n \"titles_in_box\": [\n 14\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S83\",\n \"titles_in_box\": [\n 15\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S84\",\n \"titles_in_box\": [\n 16\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S85\",\n \"titles_in_box\": [\n 17\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S86\",\n \"titles_in_box\": [\n 18\n ],\n \"box_cost\": 10000\n },\n {\n \"box_id\": \"S87\",\n \"titles_in_box\": [\n 19\n ],\n \"box_cost\": 10000\n }\n ]\n}\n\nAlso, when you're ready to tell me which boxes you picked, please send it in this little JSON shape so I can read it easily:\n\n{\n \"solution\": [\"box_id\", ...]\n}\n\nHere \"solution\" is the list where you put the identifiers of the boxes you chose — each box goes in the array. The \"box_id\" is just a placeholder showing the kind of thing to put there (it's not the real answer).\n\nThis is only a sketch of the expected shape, not the actual answer. Please make sure every identifier you include matches the instance input exactly — no renaming and no new labels. \n\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 19, "num_sets": 87, "density": 0.16575922565033274, "sets": [ { "id": 1, "elements": [ 13, 14 ], "cost": 104 }, { "id": 2, "elements": [ 8, 13 ], "cost": 92 }, { "id": 3, "elements": [ 2, 4, 6 ], "cost": 171 }, { "id": 4, "elements": [ 5, 8, 9, 10, 11, 13 ], "cost": 96 }, { "id": 5, "elements": [ 2, 3, 6, 7 ], "cost": 388 }, { "id": 6, "elements": [ 10, 11, 14 ], "cost": 114 }, { "id": 7, "elements": [ 5, 7, 10 ], "cost": 186 }, { "id": 8, "elements": [ 11, 13, 14 ], "cost": 144 }, { "id": 9, "elements": [ 1, 2, 4 ], "cost": 126 }, { "id": 10, "elements": [ 12, 13, 15, 16, 17 ], "cost": 120 }, { "id": 11, "elements": [ 1, 2 ], "cost": 164 }, { "id": 12, "elements": [ 6, 9, 10, 11 ], "cost": 200 }, { "id": 13, "elements": [ 9, 10, 11 ], "cost": 231 }, { "id": 14, "elements": [ 4, 5, 6, 8 ], "cost": 4 }, { "id": 15, "elements": [ 1, 2, 4, 7 ], "cost": 88 }, { "id": 16, "elements": [ 1, 2, 3, 4, 5 ], "cost": 45 }, { "id": 17, "elements": [ 14, 15, 16, 18, 19 ], "cost": 275 }, { "id": 18, "elements": [ 15, 16, 17, 18, 19 ], "cost": 210 }, { "id": 19, "elements": [ 4, 5, 7, 9 ], "cost": 72 }, { "id": 20, "elements": [ 9, 10, 11, 12, 13 ], "cost": 290 }, { "id": 21, "elements": [ 11, 14 ], "cost": 164 }, { "id": 22, "elements": [ 8, 10 ], "cost": 118 }, { "id": 23, "elements": [ 14, 15, 16 ], "cost": 297 }, { "id": 24, "elements": [ 16, 17 ], "cost": 188 }, { "id": 25, "elements": [ 13, 15, 16 ], "cost": 222 }, { "id": 26, "elements": [ 7, 8, 9, 12 ], "cost": 252 }, { "id": 27, "elements": [ 1, 2, 3, 4 ], "cost": 36 }, { "id": 28, "elements": [ 5, 8, 9, 10, 12 ], "cost": 330 }, { "id": 29, "elements": [ 15, 17, 18, 19 ], "cost": 92 }, { "id": 30, "elements": [ 13, 14, 15, 17 ], "cost": 352 }, { "id": 31, "elements": [ 4, 7 ], "cost": 106 }, { "id": 32, "elements": [ 6, 8, 10 ], "cost": 198 }, { "id": 33, "elements": [ 14, 17 ], "cost": 110 }, { "id": 34, "elements": [ 5, 6, 7, 8, 9 ], "cost": 245 }, { "id": 35, "elements": [ 17, 18, 19 ], "cost": 168 }, { "id": 36, "elements": [ 8, 11, 12 ], "cost": 12 }, { "id": 37, "elements": [ 4, 5, 6, 7, 8 ], "cost": 400 }, { "id": 38, "elements": [ 14, 15, 16, 17, 18 ], "cost": 315 }, { "id": 39, "elements": [ 2, 3, 4, 5 ], "cost": 232 }, { "id": 40, "elements": [ 11, 13, 14, 16, 19 ], "cost": 215 }, { "id": 41, "elements": [ 7, 8, 9, 11, 12 ], "cost": 255 }, { "id": 42, "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 414 }, { "id": 43, "elements": [ 3, 4 ], "cost": 126 }, { "id": 44, "elements": [ 1, 2, 4, 5, 6 ], "cost": 400 }, { "id": 45, "elements": [ 16, 17, 18, 19 ], "cost": 28 }, { "id": 46, "elements": [ 4, 5, 6 ], "cost": 81 }, { "id": 47, "elements": [ 9, 11, 14 ], "cost": 195 }, { "id": 48, "elements": [ 15, 16, 17, 19 ], "cost": 296 }, { "id": 49, "elements": [ 8, 9, 10, 11, 12, 13 ], "cost": 534 }, { "id": 50, "elements": [ 14, 17, 18, 19 ], "cost": 168 }, { "id": 51, "elements": [ 4, 6 ], "cost": 54 }, { "id": 52, "elements": [ 11, 12, 14 ], "cost": 81 }, { "id": 53, "elements": [ 11, 12, 13 ], "cost": 135 }, { "id": 54, "elements": [ 11, 12, 13, 15 ], "cost": 364 }, { "id": 55, "elements": [ 11, 15 ], "cost": 194 }, { "id": 56, "elements": [ 11, 12, 13, 15, 16 ], "cost": 215 }, { "id": 57, "elements": [ 11, 12, 13, 14, 16 ], "cost": 65 }, { "id": 58, "elements": [ 4, 7, 9 ], "cost": 54 }, { "id": 59, "elements": [ 4, 6, 7, 8, 10 ], "cost": 495 }, { "id": 60, "elements": [ 9, 13, 14, 17 ], "cost": 196 }, { "id": 61, "elements": [ 5, 6, 7, 8, 10 ], "cost": 450 }, { "id": 62, "elements": [ 9, 11, 12, 14 ], "cost": 156 }, { "id": 63, "elements": [ 6, 7, 9 ], "cost": 75 }, { "id": 64, "elements": [ 12, 16, 17, 18 ], "cost": 252 }, { "id": 65, "elements": [ 12, 15, 16 ], "cost": 102 }, { "id": 66, "elements": [ 8, 9, 10, 11 ], "cost": 112 }, { "id": 67, "elements": [ 1, 2, 3, 5, 6 ], "cost": 330 }, { "id": 68, "elements": [ 8, 10, 11, 12 ], "cost": 124 }, { "id": 69, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 70, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 71, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 72, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 73, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 74, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 75, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 76, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 77, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 78, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 79, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 80, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 81, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 82, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 83, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 84, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 85, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 86, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 87, "elements": [ 19 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0024_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0024_bag_stats.png" }, "solution": [ 16, 22, 29, 57, 63 ], "obj": 395.0, "instance_variant": { "num_elements": 19, "num_sets": 87, "sets": [ { "id": "S1", "elements": [ 13, 14 ], "cost": 104 }, { "id": "S2", "elements": [ 8, 13 ], "cost": 92 }, { "id": "S3", "elements": [ 2, 4, 6 ], "cost": 171 }, { "id": "S4", "elements": [ 5, 8, 9, 10, 11, 13 ], "cost": 96 }, { "id": "S5", "elements": [ 2, 3, 6, 7 ], "cost": 388 }, { "id": "S6", "elements": [ 10, 11, 14 ], "cost": 114 }, { "id": "S7", "elements": [ 5, 7, 10 ], "cost": 186 }, { "id": "S8", "elements": [ 11, 13, 14 ], "cost": 144 }, { "id": "S9", "elements": [ 1, 2, 4 ], "cost": 126 }, { "id": "S10", "elements": [ 12, 13, 15, 16, 17 ], "cost": 120 }, { "id": "S11", "elements": [ 1, 2 ], "cost": 164 }, { "id": "S12", "elements": [ 6, 9, 10, 11 ], "cost": 200 }, { "id": "S13", "elements": [ 9, 10, 11 ], "cost": 231 }, { "id": "S14", "elements": [ 4, 5, 6, 8 ], "cost": 4 }, { "id": "S15", "elements": [ 1, 2, 4, 7 ], "cost": 88 }, { "id": "S16", "elements": [ 1, 2, 3, 4, 5 ], "cost": 45 }, { "id": "S17", "elements": [ 14, 15, 16, 18, 19 ], "cost": 275 }, { "id": "S18", "elements": [ 15, 16, 17, 18, 19 ], "cost": 210 }, { "id": "S19", "elements": [ 4, 5, 7, 9 ], "cost": 72 }, { "id": "S20", "elements": [ 9, 10, 11, 12, 13 ], "cost": 290 }, { "id": "S21", "elements": [ 11, 14 ], "cost": 164 }, { "id": "S22", "elements": [ 8, 10 ], "cost": 118 }, { "id": "S23", "elements": [ 14, 15, 16 ], "cost": 297 }, { "id": "S24", "elements": [ 16, 17 ], "cost": 188 }, { "id": "S25", "elements": [ 13, 15, 16 ], "cost": 222 }, { "id": "S26", "elements": [ 7, 8, 9, 12 ], "cost": 252 }, { "id": "S27", "elements": [ 1, 2, 3, 4 ], "cost": 36 }, { "id": "S28", "elements": [ 5, 8, 9, 10, 12 ], "cost": 330 }, { "id": "S29", "elements": [ 15, 17, 18, 19 ], "cost": 92 }, { "id": "S30", "elements": [ 13, 14, 15, 17 ], "cost": 352 }, { "id": "S31", "elements": [ 4, 7 ], "cost": 106 }, { "id": "S32", "elements": [ 6, 8, 10 ], "cost": 198 }, { "id": "S33", "elements": [ 14, 17 ], "cost": 110 }, { "id": "S34", "elements": [ 5, 6, 7, 8, 9 ], "cost": 245 }, { "id": "S35", "elements": [ 17, 18, 19 ], "cost": 168 }, { "id": "S36", "elements": [ 8, 11, 12 ], "cost": 12 }, { "id": "S37", "elements": [ 4, 5, 6, 7, 8 ], "cost": 400 }, { "id": "S38", "elements": [ 14, 15, 16, 17, 18 ], "cost": 315 }, { "id": "S39", "elements": [ 2, 3, 4, 5 ], "cost": 232 }, { "id": "S40", "elements": [ 11, 13, 14, 16, 19 ], "cost": 215 }, { "id": "S41", "elements": [ 7, 8, 9, 11, 12 ], "cost": 255 }, { "id": "S42", "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 414 }, { "id": "S43", "elements": [ 3, 4 ], "cost": 126 }, { "id": "S44", "elements": [ 1, 2, 4, 5, 6 ], "cost": 400 }, { "id": "S45", "elements": [ 16, 17, 18, 19 ], "cost": 28 }, { "id": "S46", "elements": [ 4, 5, 6 ], "cost": 81 }, { "id": "S47", "elements": [ 9, 11, 14 ], "cost": 195 }, { "id": "S48", "elements": [ 15, 16, 17, 19 ], "cost": 296 }, { "id": "S49", "elements": [ 8, 9, 10, 11, 12, 13 ], "cost": 534 }, { "id": "S50", "elements": [ 14, 17, 18, 19 ], "cost": 168 }, { "id": "S51", "elements": [ 4, 6 ], "cost": 54 }, { "id": "S52", "elements": [ 11, 12, 14 ], "cost": 81 }, { "id": "S53", "elements": [ 11, 12, 13 ], "cost": 135 }, { "id": "S54", "elements": [ 11, 12, 13, 15 ], "cost": 364 }, { "id": "S55", "elements": [ 11, 15 ], "cost": 194 }, { "id": "S56", "elements": [ 11, 12, 13, 15, 16 ], "cost": 215 }, { "id": "S57", "elements": [ 11, 12, 13, 14, 16 ], "cost": 65 }, { "id": "S58", "elements": [ 4, 7, 9 ], "cost": 54 }, { "id": "S59", "elements": [ 4, 6, 7, 8, 10 ], "cost": 495 }, { "id": "S60", "elements": [ 9, 13, 14, 17 ], "cost": 196 }, { "id": "S61", "elements": [ 5, 6, 7, 8, 10 ], "cost": 450 }, { "id": "S62", "elements": [ 9, 11, 12, 14 ], "cost": 156 }, { "id": "S63", "elements": [ 6, 7, 9 ], "cost": 75 }, { "id": "S64", "elements": [ 12, 16, 17, 18 ], "cost": 252 }, { "id": "S65", "elements": [ 12, 15, 16 ], "cost": 102 }, { "id": "S66", "elements": [ 8, 9, 10, 11 ], "cost": 112 }, { "id": "S67", "elements": [ 1, 2, 3, 5, 6 ], "cost": 330 }, { "id": "S68", "elements": [ 8, 10, 11, 12 ], "cost": 124 }, { "id": "S69", "elements": [ 1 ], "cost": 10000 }, { "id": "S70", "elements": [ 2 ], "cost": 10000 }, { "id": "S71", "elements": [ 3 ], "cost": 10000 }, { "id": "S72", "elements": [ 4 ], "cost": 10000 }, { "id": "S73", "elements": [ 5 ], "cost": 10000 }, { "id": "S74", "elements": [ 6 ], "cost": 10000 }, { "id": "S75", "elements": [ 7 ], "cost": 10000 }, { "id": "S76", "elements": [ 8 ], "cost": 10000 }, { "id": "S77", "elements": [ 9 ], "cost": 10000 }, { "id": "S78", "elements": [ 10 ], "cost": 10000 }, { "id": "S79", "elements": [ 11 ], "cost": 10000 }, { "id": "S80", "elements": [ 12 ], "cost": 10000 }, { "id": "S81", "elements": [ 13 ], "cost": 10000 }, { "id": "S82", "elements": [ 14 ], "cost": 10000 }, { "id": "S83", "elements": [ 15 ], "cost": 10000 }, { "id": "S84", "elements": [ 16 ], "cost": 10000 }, { "id": "S85", "elements": [ 17 ], "cost": 10000 }, { "id": "S86", "elements": [ 18 ], "cost": 10000 }, { "id": "S87", "elements": [ 19 ], "cost": 10000 } ] }, "solution_variant": [ "S16", "S22", "S29", "S57", "S63" ], "context_index": 25, "input_format": "json", "input_index_base": 1 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "Recently the company reviewed its software needs and a stack of licensing proposals arrived, each proposal bundling different capabilities and a price. The challenge is to accept a subset of proposals so that every capability the company needs is covered by exactly one accepted proposal—no duplicates or misses—and then compute the total licensing bill by summing the prices of the accepted proposals. The cheapest total that still covers everything is the aim. The detailed proposals and numbers follow below.\n\n# num_required_capabilities=25\n# num_proposals=159\nproposal_id,license_fee,capabilities_covered\nS1,104,19 21 22 24\nS2,42,15 17 19\nS3,402,3 5 6 8 9 10\nS4,364,13 14 15 16 17 19 21\nS5,312,11 15 16 18\nS6,246,2 4 7\nS7,588,0 1 2 3 4 5 6\nS8,63,0 5 6 7 8 9 10\nS9,105,22 23 24\nS10,213,4 6 8\nS11,216,0 1 2 3 5 6\nS12,165,9 11 13 14 16\nS13,14,14 15 17 18 20 21 24\nS14,320,18 19 21 22\nS15,147,2 5 7\nS16,32,9 14 15 17\nS17,63,0 3 4 5 6 7 8\nS18,205,1 4 5 6 7\nS19,219,1 3 8\nS20,198,1 5 6\nS21,205,0 1 2 3 6\nS22,63,17 18 20 21 22 23 24\nS23,145,0 1 2 3 7\nS24,114,5 6 8 9 12 13\nS25,344,2 4 9 11\nS26,18,16 17 18 20 21 22\nS27,68,2 3 5 6\nS28,435,1 2 3 4 8\nS29,45,10 11 13\nS30,276,1 2 5 9\nS31,470,1 3 4 6 9\nS32,324,15 16 17 19 20 21\nS33,204,8 10 13\nS34,288,12 13 14 17\nS35,255,20 21 22 23 24\nS36,336,18 19 21 22 23 24\nS37,164,0 2 3 4\nS38,104,2 3 8 12\nS39,588,13 14 16 17 18 19\nS40,384,0 2 3 5\nS41,438,5 11 12 13 14 15\nS42,665,0 1 2 3 5 6 7\nS43,594,4 6 7 8 9 11\nS44,195,13 14 15 16 17\nS45,160,0 2 3 5 6\nS46,198,3 4 6\nS47,232,3 4 5 10\nS48,234,0 1 3\nS49,100,8 10 12 13 14\nS50,371,2 3 4 6 7 8 9\nS51,111,9 10 11\nS52,355,17 20 21 22 23\nS53,210,20 21 22\nS54,140,18 19 20 21 22 23 24\nS55,210,8 9 10 11 12 13 14\nS56,18,8 9 10 12 16 18\nS57,54,0 1 5\nS58,405,6 7 8 9 10\nS59,203,6 9 10 11 12 14 15\nS60,96,9 10 12\nS61,85,11 12 13 17 18\nS62,345,17 18 19 20 22\nS63,582,0 1 2 4 5 6\nS64,76,9 10 13 17\nS65,7,8 11 12 13 15 16 19\nS66,252,20 22 23 24\nS67,679,4 6 7 8 9 10 13\nS68,153,18 21 24\nS69,450,0 1 2 3 4\nS70,318,1 3 5 6 7 8\nS71,56,9 11 12 13 14 16 17\nS72,234,6 9 11 12 13 15\nS73,343,2 3 4 5 6 7 8\nS74,485,1 3 4 5 6\nS75,356,5 9 11 13\nS76,316,5 6 10 11\nS77,275,2 4 5 6 10\nS78,260,5 6 8 11\nS79,700,17 19 20 21 22 23 24\nS80,63,1 5 9\nS81,364,7 8 9 10 12 14 15\nS82,93,12 19 21\nS83,105,0 1 2 4 5\nS84,228,20 22 23\nS85,90,0 1 2 3 4 7\nS86,276,9 10 14 15 16 17\nS87,492,16 17 19 20 23 24\nS88,285,10 13 14 15 16\nS89,288,5 10 12 14\nS90,460,7 11 14 15 17\nS91,60,0 1 5 7\nS92,300,12 14 16 17 18 20\nS93,24,14 15 17 18\nS94,10,10 11 12 13 17\nS95,198,16 18 20 22 23 24\nS96,96,6 7 8 9 11 15\nS97,184,0 1 3 4\nS98,112,3 6 7 8 9 10 11\nS99,32,7 12 14 15\nS100,282,6 10 12 15 16 17\nS101,111,18 20 22\nS102,231,0 3 4\nS103,128,20 21 22 23\nS104,168,15 17 19 22\nS105,102,2 3 6\nS106,192,2 3 4 5 7 9\nS107,290,6 7 10 12 14\nS108,140,6 7 8 9 10 12 13\nS109,376,16 17 20 21\nS110,192,3 4 6 7\nS111,48,8 9 10 12 13 15\nS112,602,10 11 13 14 15 16 17\nS113,228,3 5 7 8 9 11\nS114,693,4 6 7 8 9 10 11\nS115,36,18 19 20 21 23 24\nS116,310,7 10 11 12 14\nS117,546,16 19 20 21 22 23 24\nS118,21,15 16 17\nS119,225,4 5 8 9 10\nS120,252,19 20 21 22 23 24\nS121,210,8 11 12 14 16\nS122,165,0 1 2 3 5\nS123,204,1 3 5\nS124,376,2 5 7 8\nS125,56,0 1 2 3\nS126,132,10 13 14 15 16 19\nS127,111,7 8 12\nS128,56,12 14 16 18\nS129,511,12 15 18 19 20 22 24\nS130,294,12 14 17\nS131,174,2 8 10\nS132,168,0 1 2 3 4 6 7\nS133,84,5 6 7 11\nS134,25,12 14 15 16 19\nS135,10000,0\nS136,10000,1\nS137,10000,2\nS138,10000,3\nS139,10000,4\nS140,10000,5\nS141,10000,6\nS142,10000,7\nS143,10000,8\nS144,10000,9\nS145,10000,10\nS146,10000,11\nS147,10000,12\nS148,10000,13\nS149,10000,14\nS150,10000,15\nS151,10000,16\nS152,10000,17\nS153,10000,18\nS154,10000,19\nS155,10000,20\nS156,10000,21\nS157,10000,22\nS158,10000,23\nS159,10000,24\n\nWhen you send the chosen bundle list back, please use this simple JSON layout so it's easy to read and add up the bill:\n\n{\n \"solution\": [\"proposal_id\", ...]\n}\n\n\"solution\" should be an array of the proposals you decide to accept; each entry is the id of a chosen proposal (the placeholder \"proposal_id\" just stands in for whatever id the instance uses). This is just a sketch of the expected shape — not the actual answer.\n\nPlease make sure to use the exact identifiers from the instance input — no renaming and no invented labels.\n\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 25, "num_sets": 159, "density": 0.17232704402515722, "sets": [ { "id": 1, "elements": [ 20, 22, 23, 25 ], "cost": 104 }, { "id": 2, "elements": [ 16, 18, 20 ], "cost": 42 }, { "id": 3, "elements": [ 4, 6, 7, 9, 10, 11 ], "cost": 402 }, { "id": 4, "elements": [ 14, 15, 16, 17, 18, 20, 22 ], "cost": 364 }, { "id": 5, "elements": [ 12, 16, 17, 19 ], "cost": 312 }, { "id": 6, "elements": [ 3, 5, 8 ], "cost": 246 }, { "id": 7, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 588 }, { "id": 8, "elements": [ 1, 6, 7, 8, 9, 10, 11 ], "cost": 63 }, { "id": 9, "elements": [ 23, 24, 25 ], "cost": 105 }, { "id": 10, "elements": [ 5, 7, 9 ], "cost": 213 }, { "id": 11, "elements": [ 1, 2, 3, 4, 6, 7 ], "cost": 216 }, { "id": 12, "elements": [ 10, 12, 14, 15, 17 ], "cost": 165 }, { "id": 13, "elements": [ 15, 16, 18, 19, 21, 22, 25 ], "cost": 14 }, { "id": 14, "elements": [ 19, 20, 22, 23 ], "cost": 320 }, { "id": 15, "elements": [ 3, 6, 8 ], "cost": 147 }, { "id": 16, "elements": [ 10, 15, 16, 18 ], "cost": 32 }, { "id": 17, "elements": [ 1, 4, 5, 6, 7, 8, 9 ], "cost": 63 }, { "id": 18, "elements": [ 2, 5, 6, 7, 8 ], "cost": 205 }, { "id": 19, "elements": [ 2, 4, 9 ], "cost": 219 }, { "id": 20, "elements": [ 2, 6, 7 ], "cost": 198 }, { "id": 21, "elements": [ 1, 2, 3, 4, 7 ], "cost": 205 }, { "id": 22, "elements": [ 18, 19, 21, 22, 23, 24, 25 ], "cost": 63 }, { "id": 23, "elements": [ 1, 2, 3, 4, 8 ], "cost": 145 }, { "id": 24, "elements": [ 6, 7, 9, 10, 13, 14 ], "cost": 114 }, { "id": 25, "elements": [ 3, 5, 10, 12 ], "cost": 344 }, { "id": 26, "elements": [ 17, 18, 19, 21, 22, 23 ], "cost": 18 }, { "id": 27, "elements": [ 3, 4, 6, 7 ], "cost": 68 }, { "id": 28, "elements": [ 2, 3, 4, 5, 9 ], "cost": 435 }, { "id": 29, "elements": [ 11, 12, 14 ], "cost": 45 }, { "id": 30, "elements": [ 2, 3, 6, 10 ], "cost": 276 }, { "id": 31, "elements": [ 2, 4, 5, 7, 10 ], "cost": 470 }, { "id": 32, "elements": [ 16, 17, 18, 20, 21, 22 ], "cost": 324 }, { "id": 33, "elements": [ 9, 11, 14 ], "cost": 204 }, { "id": 34, "elements": [ 13, 14, 15, 18 ], "cost": 288 }, { "id": 35, "elements": [ 21, 22, 23, 24, 25 ], "cost": 255 }, { "id": 36, "elements": [ 19, 20, 22, 23, 24, 25 ], "cost": 336 }, { "id": 37, "elements": [ 1, 3, 4, 5 ], "cost": 164 }, { "id": 38, "elements": [ 3, 4, 9, 13 ], "cost": 104 }, { "id": 39, "elements": [ 14, 15, 17, 18, 19, 20 ], "cost": 588 }, { "id": 40, "elements": [ 1, 3, 4, 6 ], "cost": 384 }, { "id": 41, "elements": [ 6, 12, 13, 14, 15, 16 ], "cost": 438 }, { "id": 42, "elements": [ 1, 2, 3, 4, 6, 7, 8 ], "cost": 665 }, { "id": 43, "elements": [ 5, 7, 8, 9, 10, 12 ], "cost": 594 }, { "id": 44, "elements": [ 14, 15, 16, 17, 18 ], "cost": 195 }, { "id": 45, "elements": [ 1, 3, 4, 6, 7 ], "cost": 160 }, { "id": 46, "elements": [ 4, 5, 7 ], "cost": 198 }, { "id": 47, "elements": [ 4, 5, 6, 11 ], "cost": 232 }, { "id": 48, "elements": [ 1, 2, 4 ], "cost": 234 }, { "id": 49, "elements": [ 9, 11, 13, 14, 15 ], "cost": 100 }, { "id": 50, "elements": [ 3, 4, 5, 7, 8, 9, 10 ], "cost": 371 }, { "id": 51, "elements": [ 10, 11, 12 ], "cost": 111 }, { "id": 52, "elements": [ 18, 21, 22, 23, 24 ], "cost": 355 }, { "id": 53, "elements": [ 21, 22, 23 ], "cost": 210 }, { "id": 54, "elements": [ 19, 20, 21, 22, 23, 24, 25 ], "cost": 140 }, { "id": 55, "elements": [ 9, 10, 11, 12, 13, 14, 15 ], "cost": 210 }, { "id": 56, "elements": [ 9, 10, 11, 13, 17, 19 ], "cost": 18 }, { "id": 57, "elements": [ 1, 2, 6 ], "cost": 54 }, { "id": 58, "elements": [ 7, 8, 9, 10, 11 ], "cost": 405 }, { "id": 59, "elements": [ 7, 10, 11, 12, 13, 15, 16 ], "cost": 203 }, { "id": 60, "elements": [ 10, 11, 13 ], "cost": 96 }, { "id": 61, "elements": [ 12, 13, 14, 18, 19 ], "cost": 85 }, { "id": 62, "elements": [ 18, 19, 20, 21, 23 ], "cost": 345 }, { "id": 63, "elements": [ 1, 2, 3, 5, 6, 7 ], "cost": 582 }, { "id": 64, "elements": [ 10, 11, 14, 18 ], "cost": 76 }, { "id": 65, "elements": [ 9, 12, 13, 14, 16, 17, 20 ], "cost": 7 }, { "id": 66, "elements": [ 21, 23, 24, 25 ], "cost": 252 }, { "id": 67, "elements": [ 5, 7, 8, 9, 10, 11, 14 ], "cost": 679 }, { "id": 68, "elements": [ 19, 22, 25 ], "cost": 153 }, { "id": 69, "elements": [ 1, 2, 3, 4, 5 ], "cost": 450 }, { "id": 70, "elements": [ 2, 4, 6, 7, 8, 9 ], "cost": 318 }, { "id": 71, "elements": [ 10, 12, 13, 14, 15, 17, 18 ], "cost": 56 }, { "id": 72, "elements": [ 7, 10, 12, 13, 14, 16 ], "cost": 234 }, { "id": 73, "elements": [ 3, 4, 5, 6, 7, 8, 9 ], "cost": 343 }, { "id": 74, "elements": [ 2, 4, 5, 6, 7 ], "cost": 485 }, { "id": 75, "elements": [ 6, 10, 12, 14 ], "cost": 356 }, { "id": 76, "elements": [ 6, 7, 11, 12 ], "cost": 316 }, { "id": 77, "elements": [ 3, 5, 6, 7, 11 ], "cost": 275 }, { "id": 78, "elements": [ 6, 7, 9, 12 ], "cost": 260 }, { "id": 79, "elements": [ 18, 20, 21, 22, 23, 24, 25 ], "cost": 700 }, { "id": 80, "elements": [ 2, 6, 10 ], "cost": 63 }, { "id": 81, "elements": [ 8, 9, 10, 11, 13, 15, 16 ], "cost": 364 }, { "id": 82, "elements": [ 13, 20, 22 ], "cost": 93 }, { "id": 83, "elements": [ 1, 2, 3, 5, 6 ], "cost": 105 }, { "id": 84, "elements": [ 21, 23, 24 ], "cost": 228 }, { "id": 85, "elements": [ 1, 2, 3, 4, 5, 8 ], "cost": 90 }, { "id": 86, "elements": [ 10, 11, 15, 16, 17, 18 ], "cost": 276 }, { "id": 87, "elements": [ 17, 18, 20, 21, 24, 25 ], "cost": 492 }, { "id": 88, "elements": [ 11, 14, 15, 16, 17 ], "cost": 285 }, { "id": 89, "elements": [ 6, 11, 13, 15 ], "cost": 288 }, { "id": 90, "elements": [ 8, 12, 15, 16, 18 ], "cost": 460 }, { "id": 91, "elements": [ 1, 2, 6, 8 ], "cost": 60 }, { "id": 92, "elements": [ 13, 15, 17, 18, 19, 21 ], "cost": 300 }, { "id": 93, "elements": [ 15, 16, 18, 19 ], "cost": 24 }, { "id": 94, "elements": [ 11, 12, 13, 14, 18 ], "cost": 10 }, { "id": 95, "elements": [ 17, 19, 21, 23, 24, 25 ], "cost": 198 }, { "id": 96, "elements": [ 7, 8, 9, 10, 12, 16 ], "cost": 96 }, { "id": 97, "elements": [ 1, 2, 4, 5 ], "cost": 184 }, { "id": 98, "elements": [ 4, 7, 8, 9, 10, 11, 12 ], "cost": 112 }, { "id": 99, "elements": [ 8, 13, 15, 16 ], "cost": 32 }, { "id": 100, "elements": [ 7, 11, 13, 16, 17, 18 ], "cost": 282 }, { "id": 101, "elements": [ 19, 21, 23 ], "cost": 111 }, { "id": 102, "elements": [ 1, 4, 5 ], "cost": 231 }, { "id": 103, "elements": [ 21, 22, 23, 24 ], "cost": 128 }, { "id": 104, "elements": [ 16, 18, 20, 23 ], "cost": 168 }, { "id": 105, "elements": [ 3, 4, 7 ], "cost": 102 }, { "id": 106, "elements": [ 3, 4, 5, 6, 8, 10 ], "cost": 192 }, { "id": 107, "elements": [ 7, 8, 11, 13, 15 ], "cost": 290 }, { "id": 108, "elements": [ 7, 8, 9, 10, 11, 13, 14 ], "cost": 140 }, { "id": 109, "elements": [ 17, 18, 21, 22 ], "cost": 376 }, { "id": 110, "elements": [ 4, 5, 7, 8 ], "cost": 192 }, { "id": 111, "elements": [ 9, 10, 11, 13, 14, 16 ], "cost": 48 }, { "id": 112, "elements": [ 11, 12, 14, 15, 16, 17, 18 ], "cost": 602 }, { "id": 113, "elements": [ 4, 6, 8, 9, 10, 12 ], "cost": 228 }, { "id": 114, "elements": [ 5, 7, 8, 9, 10, 11, 12 ], "cost": 693 }, { "id": 115, "elements": [ 19, 20, 21, 22, 24, 25 ], "cost": 36 }, { "id": 116, "elements": [ 8, 11, 12, 13, 15 ], "cost": 310 }, { "id": 117, "elements": [ 17, 20, 21, 22, 23, 24, 25 ], "cost": 546 }, { "id": 118, "elements": [ 16, 17, 18 ], "cost": 21 }, { "id": 119, "elements": [ 5, 6, 9, 10, 11 ], "cost": 225 }, { "id": 120, "elements": [ 20, 21, 22, 23, 24, 25 ], "cost": 252 }, { "id": 121, "elements": [ 9, 12, 13, 15, 17 ], "cost": 210 }, { "id": 122, "elements": [ 1, 2, 3, 4, 6 ], "cost": 165 }, { "id": 123, "elements": [ 2, 4, 6 ], "cost": 204 }, { "id": 124, "elements": [ 3, 6, 8, 9 ], "cost": 376 }, { "id": 125, "elements": [ 1, 2, 3, 4 ], "cost": 56 }, { "id": 126, "elements": [ 11, 14, 15, 16, 17, 20 ], "cost": 132 }, { "id": 127, "elements": [ 8, 9, 13 ], "cost": 111 }, { "id": 128, "elements": [ 13, 15, 17, 19 ], "cost": 56 }, { "id": 129, "elements": [ 13, 16, 19, 20, 21, 23, 25 ], "cost": 511 }, { "id": 130, "elements": [ 13, 15, 18 ], "cost": 294 }, { "id": 131, "elements": [ 3, 9, 11 ], "cost": 174 }, { "id": 132, "elements": [ 1, 2, 3, 4, 5, 7, 8 ], "cost": 168 }, { "id": 133, "elements": [ 6, 7, 8, 12 ], "cost": 84 }, { "id": 134, "elements": [ 13, 15, 16, 17, 20 ], "cost": 25 }, { "id": 135, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 136, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 137, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 138, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 139, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 140, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 141, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 142, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 143, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 144, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 145, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 146, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 147, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 148, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 149, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 150, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 151, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 152, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 153, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 154, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 155, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 156, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 157, "elements": [ 23 ], "cost": 10000, "singleton": true }, { "id": 158, "elements": [ 24 ], "cost": 10000, "singleton": true }, { "id": 159, "elements": [ 25 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0025_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0025_bag_stats.png" }, "solution": [ 22, 29, 50, 57, 134 ], "obj": 558.0, "instance_variant": { "num_elements": 25, "num_sets": 159, "sets": [ { "id": "S1", "elements": [ 19, 21, 22, 24 ], "cost": 104 }, { "id": "S2", "elements": [ 15, 17, 19 ], "cost": 42 }, { "id": "S3", "elements": [ 3, 5, 6, 8, 9, 10 ], "cost": 402 }, { "id": "S4", "elements": [ 13, 14, 15, 16, 17, 19, 21 ], "cost": 364 }, { "id": "S5", "elements": [ 11, 15, 16, 18 ], "cost": 312 }, { "id": "S6", "elements": [ 2, 4, 7 ], "cost": 246 }, { "id": "S7", "elements": [ 0, 1, 2, 3, 4, 5, 6 ], "cost": 588 }, { "id": "S8", "elements": [ 0, 5, 6, 7, 8, 9, 10 ], "cost": 63 }, { "id": "S9", "elements": [ 22, 23, 24 ], "cost": 105 }, { "id": "S10", "elements": [ 4, 6, 8 ], "cost": 213 }, { "id": "S11", "elements": [ 0, 1, 2, 3, 5, 6 ], "cost": 216 }, { "id": "S12", "elements": [ 9, 11, 13, 14, 16 ], "cost": 165 }, { "id": "S13", "elements": [ 14, 15, 17, 18, 20, 21, 24 ], "cost": 14 }, { "id": "S14", "elements": [ 18, 19, 21, 22 ], "cost": 320 }, { "id": "S15", "elements": [ 2, 5, 7 ], "cost": 147 }, { "id": "S16", "elements": [ 9, 14, 15, 17 ], "cost": 32 }, { "id": "S17", "elements": [ 0, 3, 4, 5, 6, 7, 8 ], "cost": 63 }, { "id": "S18", "elements": [ 1, 4, 5, 6, 7 ], "cost": 205 }, { "id": "S19", "elements": [ 1, 3, 8 ], "cost": 219 }, { "id": "S20", "elements": [ 1, 5, 6 ], "cost": 198 }, { "id": "S21", "elements": [ 0, 1, 2, 3, 6 ], "cost": 205 }, { "id": "S22", "elements": [ 17, 18, 20, 21, 22, 23, 24 ], "cost": 63 }, { "id": "S23", "elements": [ 0, 1, 2, 3, 7 ], "cost": 145 }, { "id": "S24", "elements": [ 5, 6, 8, 9, 12, 13 ], "cost": 114 }, { "id": "S25", "elements": [ 2, 4, 9, 11 ], "cost": 344 }, { "id": "S26", "elements": [ 16, 17, 18, 20, 21, 22 ], "cost": 18 }, { "id": "S27", "elements": [ 2, 3, 5, 6 ], "cost": 68 }, { "id": "S28", "elements": [ 1, 2, 3, 4, 8 ], "cost": 435 }, { "id": "S29", "elements": [ 10, 11, 13 ], "cost": 45 }, { "id": "S30", "elements": [ 1, 2, 5, 9 ], "cost": 276 }, { "id": "S31", "elements": [ 1, 3, 4, 6, 9 ], "cost": 470 }, { "id": "S32", "elements": [ 15, 16, 17, 19, 20, 21 ], "cost": 324 }, { "id": "S33", "elements": [ 8, 10, 13 ], "cost": 204 }, { "id": "S34", "elements": [ 12, 13, 14, 17 ], "cost": 288 }, { "id": "S35", "elements": [ 20, 21, 22, 23, 24 ], "cost": 255 }, { "id": "S36", "elements": [ 18, 19, 21, 22, 23, 24 ], "cost": 336 }, { "id": "S37", "elements": [ 0, 2, 3, 4 ], "cost": 164 }, { "id": "S38", "elements": [ 2, 3, 8, 12 ], "cost": 104 }, { "id": "S39", "elements": [ 13, 14, 16, 17, 18, 19 ], "cost": 588 }, { "id": "S40", "elements": [ 0, 2, 3, 5 ], "cost": 384 }, { "id": "S41", "elements": [ 5, 11, 12, 13, 14, 15 ], "cost": 438 }, { "id": "S42", "elements": [ 0, 1, 2, 3, 5, 6, 7 ], "cost": 665 }, { "id": "S43", "elements": [ 4, 6, 7, 8, 9, 11 ], "cost": 594 }, { "id": "S44", "elements": [ 13, 14, 15, 16, 17 ], "cost": 195 }, { "id": "S45", "elements": [ 0, 2, 3, 5, 6 ], "cost": 160 }, { "id": "S46", "elements": [ 3, 4, 6 ], "cost": 198 }, { "id": "S47", "elements": [ 3, 4, 5, 10 ], "cost": 232 }, { "id": "S48", "elements": [ 0, 1, 3 ], "cost": 234 }, { "id": "S49", "elements": [ 8, 10, 12, 13, 14 ], "cost": 100 }, { "id": "S50", "elements": [ 2, 3, 4, 6, 7, 8, 9 ], "cost": 371 }, { "id": "S51", "elements": [ 9, 10, 11 ], "cost": 111 }, { "id": "S52", "elements": [ 17, 20, 21, 22, 23 ], "cost": 355 }, { "id": "S53", "elements": [ 20, 21, 22 ], "cost": 210 }, { "id": "S54", "elements": [ 18, 19, 20, 21, 22, 23, 24 ], "cost": 140 }, { "id": "S55", "elements": [ 8, 9, 10, 11, 12, 13, 14 ], "cost": 210 }, { "id": "S56", "elements": [ 8, 9, 10, 12, 16, 18 ], "cost": 18 }, { "id": "S57", "elements": [ 0, 1, 5 ], "cost": 54 }, { "id": "S58", "elements": [ 6, 7, 8, 9, 10 ], "cost": 405 }, { "id": "S59", "elements": [ 6, 9, 10, 11, 12, 14, 15 ], "cost": 203 }, { "id": "S60", "elements": [ 9, 10, 12 ], "cost": 96 }, { "id": "S61", "elements": [ 11, 12, 13, 17, 18 ], "cost": 85 }, { "id": "S62", "elements": [ 17, 18, 19, 20, 22 ], "cost": 345 }, { "id": "S63", "elements": [ 0, 1, 2, 4, 5, 6 ], "cost": 582 }, { "id": "S64", "elements": [ 9, 10, 13, 17 ], "cost": 76 }, { "id": "S65", "elements": [ 8, 11, 12, 13, 15, 16, 19 ], "cost": 7 }, { "id": "S66", "elements": [ 20, 22, 23, 24 ], "cost": 252 }, { "id": "S67", "elements": [ 4, 6, 7, 8, 9, 10, 13 ], "cost": 679 }, { "id": "S68", "elements": [ 18, 21, 24 ], "cost": 153 }, { "id": "S69", "elements": [ 0, 1, 2, 3, 4 ], "cost": 450 }, { "id": "S70", "elements": [ 1, 3, 5, 6, 7, 8 ], "cost": 318 }, { "id": "S71", "elements": [ 9, 11, 12, 13, 14, 16, 17 ], "cost": 56 }, { "id": "S72", "elements": [ 6, 9, 11, 12, 13, 15 ], "cost": 234 }, { "id": "S73", "elements": [ 2, 3, 4, 5, 6, 7, 8 ], "cost": 343 }, { "id": "S74", "elements": [ 1, 3, 4, 5, 6 ], "cost": 485 }, { "id": "S75", "elements": [ 5, 9, 11, 13 ], "cost": 356 }, { "id": "S76", "elements": [ 5, 6, 10, 11 ], "cost": 316 }, { "id": "S77", "elements": [ 2, 4, 5, 6, 10 ], "cost": 275 }, { "id": "S78", "elements": [ 5, 6, 8, 11 ], "cost": 260 }, { "id": "S79", "elements": [ 17, 19, 20, 21, 22, 23, 24 ], "cost": 700 }, { "id": "S80", "elements": [ 1, 5, 9 ], "cost": 63 }, { "id": "S81", "elements": [ 7, 8, 9, 10, 12, 14, 15 ], "cost": 364 }, { "id": "S82", "elements": [ 12, 19, 21 ], "cost": 93 }, { "id": "S83", "elements": [ 0, 1, 2, 4, 5 ], "cost": 105 }, { "id": "S84", "elements": [ 20, 22, 23 ], "cost": 228 }, { "id": "S85", "elements": [ 0, 1, 2, 3, 4, 7 ], "cost": 90 }, { "id": "S86", "elements": [ 9, 10, 14, 15, 16, 17 ], "cost": 276 }, { "id": "S87", "elements": [ 16, 17, 19, 20, 23, 24 ], "cost": 492 }, { "id": "S88", "elements": [ 10, 13, 14, 15, 16 ], "cost": 285 }, { "id": "S89", "elements": [ 5, 10, 12, 14 ], "cost": 288 }, { "id": "S90", "elements": [ 7, 11, 14, 15, 17 ], "cost": 460 }, { "id": "S91", "elements": [ 0, 1, 5, 7 ], "cost": 60 }, { "id": "S92", "elements": [ 12, 14, 16, 17, 18, 20 ], "cost": 300 }, { "id": "S93", "elements": [ 14, 15, 17, 18 ], "cost": 24 }, { "id": "S94", "elements": [ 10, 11, 12, 13, 17 ], "cost": 10 }, { "id": "S95", "elements": [ 16, 18, 20, 22, 23, 24 ], "cost": 198 }, { "id": "S96", "elements": [ 6, 7, 8, 9, 11, 15 ], "cost": 96 }, { "id": "S97", "elements": [ 0, 1, 3, 4 ], "cost": 184 }, { "id": "S98", "elements": [ 3, 6, 7, 8, 9, 10, 11 ], "cost": 112 }, { "id": "S99", "elements": [ 7, 12, 14, 15 ], "cost": 32 }, { "id": "S100", "elements": [ 6, 10, 12, 15, 16, 17 ], "cost": 282 }, { "id": "S101", "elements": [ 18, 20, 22 ], "cost": 111 }, { "id": "S102", "elements": [ 0, 3, 4 ], "cost": 231 }, { "id": "S103", "elements": [ 20, 21, 22, 23 ], "cost": 128 }, { "id": "S104", "elements": [ 15, 17, 19, 22 ], "cost": 168 }, { "id": "S105", "elements": [ 2, 3, 6 ], "cost": 102 }, { "id": "S106", "elements": [ 2, 3, 4, 5, 7, 9 ], "cost": 192 }, { "id": "S107", "elements": [ 6, 7, 10, 12, 14 ], "cost": 290 }, { "id": "S108", "elements": [ 6, 7, 8, 9, 10, 12, 13 ], "cost": 140 }, { "id": "S109", "elements": [ 16, 17, 20, 21 ], "cost": 376 }, { "id": "S110", "elements": [ 3, 4, 6, 7 ], "cost": 192 }, { "id": "S111", "elements": [ 8, 9, 10, 12, 13, 15 ], "cost": 48 }, { "id": "S112", "elements": [ 10, 11, 13, 14, 15, 16, 17 ], "cost": 602 }, { "id": "S113", "elements": [ 3, 5, 7, 8, 9, 11 ], "cost": 228 }, { "id": "S114", "elements": [ 4, 6, 7, 8, 9, 10, 11 ], "cost": 693 }, { "id": "S115", "elements": [ 18, 19, 20, 21, 23, 24 ], "cost": 36 }, { "id": "S116", "elements": [ 7, 10, 11, 12, 14 ], "cost": 310 }, { "id": "S117", "elements": [ 16, 19, 20, 21, 22, 23, 24 ], "cost": 546 }, { "id": "S118", "elements": [ 15, 16, 17 ], "cost": 21 }, { "id": "S119", "elements": [ 4, 5, 8, 9, 10 ], "cost": 225 }, { "id": "S120", "elements": [ 19, 20, 21, 22, 23, 24 ], "cost": 252 }, { "id": "S121", "elements": [ 8, 11, 12, 14, 16 ], "cost": 210 }, { "id": "S122", "elements": [ 0, 1, 2, 3, 5 ], "cost": 165 }, { "id": "S123", "elements": [ 1, 3, 5 ], "cost": 204 }, { "id": "S124", "elements": [ 2, 5, 7, 8 ], "cost": 376 }, { "id": "S125", "elements": [ 0, 1, 2, 3 ], "cost": 56 }, { "id": "S126", "elements": [ 10, 13, 14, 15, 16, 19 ], "cost": 132 }, { "id": "S127", "elements": [ 7, 8, 12 ], "cost": 111 }, { "id": "S128", "elements": [ 12, 14, 16, 18 ], "cost": 56 }, { "id": "S129", "elements": [ 12, 15, 18, 19, 20, 22, 24 ], "cost": 511 }, { "id": "S130", "elements": [ 12, 14, 17 ], "cost": 294 }, { "id": "S131", "elements": [ 2, 8, 10 ], "cost": 174 }, { "id": "S132", "elements": [ 0, 1, 2, 3, 4, 6, 7 ], "cost": 168 }, { "id": "S133", "elements": [ 5, 6, 7, 11 ], "cost": 84 }, { "id": "S134", "elements": [ 12, 14, 15, 16, 19 ], "cost": 25 }, { "id": "S135", "elements": [ 0 ], "cost": 10000 }, { "id": "S136", "elements": [ 1 ], "cost": 10000 }, { "id": "S137", "elements": [ 2 ], "cost": 10000 }, { "id": "S138", "elements": [ 3 ], "cost": 10000 }, { "id": "S139", "elements": [ 4 ], "cost": 10000 }, { "id": "S140", "elements": [ 5 ], "cost": 10000 }, { "id": "S141", "elements": [ 6 ], "cost": 10000 }, { "id": "S142", "elements": [ 7 ], "cost": 10000 }, { "id": "S143", "elements": [ 8 ], "cost": 10000 }, { "id": "S144", "elements": [ 9 ], "cost": 10000 }, { "id": "S145", "elements": [ 10 ], "cost": 10000 }, { "id": "S146", "elements": [ 11 ], "cost": 10000 }, { "id": "S147", "elements": [ 12 ], "cost": 10000 }, { "id": "S148", "elements": [ 13 ], "cost": 10000 }, { "id": "S149", "elements": [ 14 ], "cost": 10000 }, { "id": "S150", "elements": [ 15 ], "cost": 10000 }, { "id": "S151", "elements": [ 16 ], "cost": 10000 }, { "id": "S152", "elements": [ 17 ], "cost": 10000 }, { "id": "S153", "elements": [ 18 ], "cost": 10000 }, { "id": "S154", "elements": [ 19 ], "cost": 10000 }, { "id": "S155", "elements": [ 20 ], "cost": 10000 }, { "id": "S156", "elements": [ 21 ], "cost": 10000 }, { "id": "S157", "elements": [ 22 ], "cost": 10000 }, { "id": "S158", "elements": [ 23 ], "cost": 10000 }, { "id": "S159", "elements": [ 24 ], "cost": 10000 } ] }, "solution_variant": [ "S22", "S29", "S50", "S57", "S134" ], "context_index": 26, "input_format": "csv", "input_index_base": 0 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "I’m juggling a menu and a pile of pre-packed ingredient kits: the job is to pick which kits to order so every dish on the menu is made from exactly one of those kits — nothing left without a kit, and nothing made from two kits at once. The better choices are the ones that let all dishes be covered while keeping the bill as low as possible: total cost here means adding up the prices of the kits we actually order, and the goal is to make that sum as small as possible. The concrete details are listed below.\n\n{\n \"num_dishes\": 23,\n \"num_kits_available\": 120,\n \"sets\": [\n {\n \"kit_id\": \"S1\",\n \"dishes_in_kit\": [\n 4,\n 5,\n 6,\n 7,\n 8,\n 10\n ],\n \"kit_price\": 72\n },\n {\n \"kit_id\": \"S2\",\n \"dishes_in_kit\": [\n 15,\n 17,\n 18\n ],\n \"kit_price\": 21\n },\n {\n \"kit_id\": \"S3\",\n \"dishes_in_kit\": [\n 14,\n 18,\n 19,\n 20\n ],\n \"kit_price\": 232\n },\n {\n \"kit_id\": \"S4\",\n \"dishes_in_kit\": [\n 7,\n 9,\n 11,\n 12\n ],\n \"kit_price\": 184\n },\n {\n \"kit_id\": \"S5\",\n \"dishes_in_kit\": [\n 10,\n 16,\n 17,\n 18,\n 19,\n 20,\n 22\n ],\n \"kit_price\": 392\n },\n {\n \"kit_id\": \"S6\",\n \"dishes_in_kit\": [\n 0,\n 1,\n 2,\n 3\n ],\n \"kit_price\": 12\n },\n {\n \"kit_id\": \"S7\",\n \"dishes_in_kit\": [\n 12,\n 14,\n 18\n ],\n \"kit_price\": 66\n },\n {\n \"kit_id\": \"S8\",\n \"dishes_in_kit\": [\n 2,\n 3,\n 4,\n 5,\n 8\n ],\n \"kit_price\": 155\n },\n {\n \"kit_id\": \"S9\",\n \"dishes_in_kit\": [\n 3,\n 6,\n 7,\n 9,\n 12\n ],\n \"kit_price\": 225\n },\n {\n \"kit_id\": \"S10\",\n \"dishes_in_kit\": [\n 1,\n 3,\n 5,\n 6,\n 7,\n 10,\n 11\n ],\n \"kit_price\": 301\n },\n {\n \"kit_id\": \"S11\",\n \"dishes_in_kit\": [\n 0,\n 1,\n 2,\n 4\n ],\n \"kit_price\": 172\n },\n {\n \"kit_id\": \"S12\",\n \"dishes_in_kit\": [\n 17,\n 18,\n 19,\n 20,\n 21,\n 22\n ],\n \"kit_price\": 534\n },\n {\n \"kit_id\": \"S13\",\n \"dishes_in_kit\": [\n 16,\n 17,\n 18,\n 19,\n 21,\n 22\n ],\n \"kit_price\": 126\n },\n {\n \"kit_id\": \"S14\",\n \"dishes_in_kit\": [\n 0,\n 3,\n 4,\n 5,\n 6\n ],\n \"kit_price\": 415\n },\n {\n \"kit_id\": \"S15\",\n \"dishes_in_kit\": [\n 4,\n 5,\n 6\n ],\n \"kit_price\": 36\n },\n {\n \"kit_id\": \"S16\",\n \"dishes_in_kit\": [\n 14,\n 19,\n 20\n ],\n \"kit_price\": 153\n },\n {\n \"kit_id\": \"S17\",\n \"dishes_in_kit\": [\n 15,\n 16,\n 17,\n 19\n ],\n \"kit_price\": 256\n },\n {\n \"kit_id\": \"S18\",\n \"dishes_in_kit\": [\n 8,\n 10,\n 11,\n 12,\n 13,\n 14\n ],\n \"kit_price\": 480\n },\n {\n \"kit_id\": \"S19\",\n \"dishes_in_kit\": [\n 0,\n 1,\n 2,\n 3,\n 4,\n 5,\n 7\n ],\n \"kit_price\": 119\n },\n {\n \"kit_id\": \"S20\",\n \"dishes_in_kit\": [\n 5,\n 6,\n 7,\n 8,\n 9,\n 11,\n 12\n ],\n \"kit_price\": 133\n },\n {\n \"kit_id\": \"S21\",\n \"dishes_in_kit\": [\n 14,\n 16,\n 18,\n 19,\n 20,\n 21\n ],\n \"kit_price\": 570\n },\n {\n \"kit_id\": \"S22\",\n \"dishes_in_kit\": [\n 4,\n 5,\n 6,\n 7,\n 8,\n 9\n ],\n \"kit_price\": 180\n },\n {\n \"kit_id\": \"S23\",\n \"dishes_in_kit\": [\n 6,\n 8,\n 9,\n 10,\n 12,\n 13\n ],\n \"kit_price\": 234\n },\n {\n \"kit_id\": \"S24\",\n \"dishes_in_kit\": [\n 1,\n 2,\n 5\n ],\n \"kit_price\": 180\n },\n {\n \"kit_id\": \"S25\",\n \"dishes_in_kit\": [\n 7,\n 8,\n 10,\n 11,\n 13\n ],\n \"kit_price\": 100\n },\n {\n \"kit_id\": \"S26\",\n \"dishes_in_kit\": [\n 14,\n 15,\n 16,\n 17,\n 18,\n 19,\n 20\n ],\n \"kit_price\": 532\n },\n {\n \"kit_id\": \"S27\",\n \"dishes_in_kit\": [\n 0,\n 1,\n 2\n ],\n \"kit_price\": 168\n },\n {\n \"kit_id\": \"S28\",\n \"dishes_in_kit\": [\n 9,\n 10,\n 11,\n 12,\n 14\n ],\n \"kit_price\": 380\n },\n {\n \"kit_id\": \"S29\",\n \"dishes_in_kit\": [\n 9,\n 11,\n 12,\n 13,\n 14,\n 15,\n 16\n ],\n \"kit_price\": 392\n },\n {\n \"kit_id\": \"S30\",\n \"dishes_in_kit\": [\n 15,\n 16,\n 17,\n 18\n ],\n \"kit_price\": 272\n },\n {\n \"kit_id\": \"S31\",\n \"dishes_in_kit\": [\n 15,\n 18,\n 19\n ],\n \"kit_price\": 273\n },\n {\n \"kit_id\": \"S32\",\n \"dishes_in_kit\": [\n 2,\n 3,\n 4,\n 5,\n 6,\n 7\n ],\n \"kit_price\": 600\n },\n {\n \"kit_id\": \"S33\",\n \"dishes_in_kit\": [\n 9,\n 11,\n 12\n ],\n \"kit_price\": 123\n },\n {\n \"kit_id\": \"S34\",\n \"dishes_in_kit\": [\n 0,\n 1,\n 2,\n 3,\n 4,\n 6\n ],\n \"kit_price\": 84\n },\n {\n \"kit_id\": \"S35\",\n \"dishes_in_kit\": [\n 8,\n 11,\n 12,\n 13,\n 15\n ],\n \"kit_price\": 195\n },\n {\n \"kit_id\": \"S36\",\n \"dishes_in_kit\": [\n 14,\n 15,\n 16,\n 18\n ],\n \"kit_price\": 8\n },\n {\n \"kit_id\": \"S37\",\n \"dishes_in_kit\": [\n 3,\n 4,\n 7\n ],\n \"kit_price\": 111\n },\n {\n \"kit_id\": \"S38\",\n \"dishes_in_kit\": [\n 3,\n 4,\n 5,\n 6,\n 7\n ],\n \"kit_price\": 445\n },\n {\n \"kit_id\": \"S39\",\n \"dishes_in_kit\": [\n 0,\n 4,\n 7,\n 8\n ],\n \"kit_price\": 28\n },\n {\n \"kit_id\": \"S40\",\n \"dishes_in_kit\": [\n 6,\n 9,\n 10,\n 11,\n 12,\n 13,\n 14\n ],\n \"kit_price\": 49\n },\n {\n \"kit_id\": \"S41\",\n \"dishes_in_kit\": [\n 0,\n 1,\n 2,\n 3,\n 4,\n 5\n ],\n \"kit_price\": 360\n },\n {\n \"kit_id\": \"S42\",\n \"dishes_in_kit\": [\n 16,\n 19,\n 20\n ],\n \"kit_price\": 162\n },\n {\n \"kit_id\": \"S43\",\n \"dishes_in_kit\": [\n 12,\n 14,\n 15,\n 16,\n 17,\n 19,\n 20\n ],\n \"kit_price\": 56\n },\n {\n \"kit_id\": \"S44\",\n \"dishes_in_kit\": [\n 11,\n 12,\n 14,\n 16\n ],\n \"kit_price\": 184\n },\n {\n \"kit_id\": \"S45\",\n \"dishes_in_kit\": [\n 17,\n 18\n ],\n \"kit_price\": 52\n },\n {\n \"kit_id\": \"S46\",\n \"dishes_in_kit\": [\n 3,\n 7,\n 8\n ],\n \"kit_price\": 240\n },\n {\n \"kit_id\": \"S47\",\n \"dishes_in_kit\": [\n 18,\n 19,\n 21\n ],\n \"kit_price\": 78\n },\n {\n \"kit_id\": \"S48\",\n \"dishes_in_kit\": [\n 9,\n 10,\n 11,\n 12,\n 15,\n 16\n ],\n \"kit_price\": 366\n },\n {\n \"kit_id\": \"S49\",\n \"dishes_in_kit\": [\n 7,\n 11,\n 12,\n 13,\n 16\n ],\n \"kit_price\": 445\n },\n {\n \"kit_id\": \"S50\",\n \"dishes_in_kit\": [\n 6,\n 9,\n 13\n ],\n \"kit_price\": 108\n },\n {\n \"kit_id\": \"S51\",\n \"dishes_in_kit\": [\n 16,\n 17,\n 18,\n 19,\n 20,\n 22\n ],\n \"kit_price\": 174\n },\n {\n \"kit_id\": \"S52\",\n \"dishes_in_kit\": [\n 5,\n 6,\n 8,\n 10\n ],\n \"kit_price\": 180\n },\n {\n \"kit_id\": \"S53\",\n \"dishes_in_kit\": [\n 8,\n 9,\n 10,\n 12\n ],\n \"kit_price\": 292\n },\n {\n \"kit_id\": \"S54\",\n \"dishes_in_kit\": [\n 11,\n 14,\n 15\n ],\n \"kit_price\": 3\n },\n {\n \"kit_id\": \"S55\",\n \"dishes_in_kit\": [\n 6,\n 7,\n 8,\n 10,\n 11\n ],\n \"kit_price\": 115\n },\n {\n \"kit_id\": \"S56\",\n \"dishes_in_kit\": [\n 6,\n 8,\n 10\n ],\n \"kit_price\": 15\n },\n {\n \"kit_id\": \"S57\",\n \"dishes_in_kit\": [\n 3,\n 4,\n 6,\n 7,\n 9\n ],\n \"kit_price\": 415\n },\n {\n \"kit_id\": \"S58\",\n \"dishes_in_kit\": [\n 1,\n 2,\n 3,\n 4,\n 7\n ],\n \"kit_price\": 295\n },\n {\n \"kit_id\": \"S59\",\n \"dishes_in_kit\": [\n 9,\n 14,\n 15\n ],\n \"kit_price\": 135\n },\n {\n \"kit_id\": \"S60\",\n \"dishes_in_kit\": [\n 5,\n 6,\n 8,\n 9,\n 10,\n 11\n ],\n \"kit_price\": 54\n },\n {\n \"kit_id\": \"S61\",\n \"dishes_in_kit\": [\n 6,\n 8,\n 10,\n 12\n ],\n \"kit_price\": 312\n },\n {\n \"kit_id\": \"S62\",\n \"dishes_in_kit\": [\n 0,\n 1,\n 2,\n 3,\n 4\n ],\n \"kit_price\": 500\n },\n {\n \"kit_id\": \"S63\",\n \"dishes_in_kit\": [\n 5,\n 6,\n 10,\n 11,\n 13\n ],\n \"kit_price\": 155\n },\n {\n \"kit_id\": \"S64\",\n \"dishes_in_kit\": [\n 9,\n 11,\n 12,\n 16\n ],\n \"kit_price\": 140\n },\n {\n \"kit_id\": \"S65\",\n \"dishes_in_kit\": [\n 12,\n 15,\n 16,\n 17,\n 18,\n 19,\n 20\n ],\n \"kit_price\": 630\n },\n {\n \"kit_id\": \"S66\",\n \"dishes_in_kit\": [\n 1,\n 2,\n 3,\n 4,\n 5,\n 7\n ],\n \"kit_price\": 444\n },\n {\n \"kit_id\": \"S67\",\n \"dishes_in_kit\": [\n 6,\n 11,\n 12\n ],\n \"kit_price\": 183\n },\n {\n \"kit_id\": \"S68\",\n \"dishes_in_kit\": [\n 10,\n 15,\n 16,\n 17,\n 20\n ],\n \"kit_price\": 15\n },\n {\n \"kit_id\": \"S69\",\n \"dishes_in_kit\": [\n 15,\n 17,\n 19,\n 20,\n 21,\n 22\n ],\n \"kit_price\": 432\n },\n {\n \"kit_id\": \"S70\",\n \"dishes_in_kit\": [\n 19,\n 20,\n 21,\n 22\n ],\n \"kit_price\": 16\n },\n {\n \"kit_id\": \"S71\",\n \"dishes_in_kit\": [\n 2,\n 3,\n 4,\n 8\n ],\n \"kit_price\": 296\n },\n {\n \"kit_id\": \"S72\",\n \"dishes_in_kit\": [\n 16,\n 17,\n 18,\n 19,\n 20,\n 21,\n 22\n ],\n \"kit_price\": 553\n },\n {\n \"kit_id\": \"S73\",\n \"dishes_in_kit\": [\n 17,\n 20,\n 21\n ],\n \"kit_price\": 216\n },\n {\n \"kit_id\": \"S74\",\n \"dishes_in_kit\": [\n 18,\n 20,\n 22\n ],\n \"kit_price\": 90\n },\n {\n \"kit_id\": \"S75\",\n \"dishes_in_kit\": [\n 13,\n 15,\n 16,\n 17\n ],\n \"kit_price\": 160\n },\n {\n \"kit_id\": \"S76\",\n \"dishes_in_kit\": [\n 1,\n 3,\n 4,\n 6,\n 7\n ],\n \"kit_price\": 440\n },\n {\n \"kit_id\": \"S77\",\n \"dishes_in_kit\": [\n 4,\n 6,\n 7,\n 9,\n 10,\n 11,\n 13\n ],\n \"kit_price\": 21\n },\n {\n \"kit_id\": \"S78\",\n \"dishes_in_kit\": [\n 10,\n 11,\n 13\n ],\n \"kit_price\": 12\n },\n {\n \"kit_id\": \"S79\",\n \"dishes_in_kit\": [\n 12,\n 16,\n 17,\n 18,\n 19,\n 20,\n 21\n ],\n \"kit_price\": 147\n },\n {\n \"kit_id\": \"S80\",\n \"dishes_in_kit\": [\n 6,\n 7,\n 8,\n 9,\n 10,\n 11,\n 12\n ],\n \"kit_price\": 623\n },\n {\n \"kit_id\": \"S81\",\n \"dishes_in_kit\": [\n 9,\n 10,\n 11,\n 16\n ],\n \"kit_price\": 332\n },\n {\n \"kit_id\": \"S82\",\n \"dishes_in_kit\": [\n 0,\n 1,\n 3,\n 4,\n 6\n ],\n \"kit_price\": 425\n },\n {\n \"kit_id\": \"S83\",\n \"dishes_in_kit\": [\n 17,\n 18,\n 20\n ],\n \"kit_price\": 174\n },\n {\n \"kit_id\": \"S84\",\n \"dishes_in_kit\": [\n 1,\n 2,\n 3,\n 4,\n 7,\n 8\n ],\n \"kit_price\": 42\n },\n {\n \"kit_id\": \"S85\",\n \"dishes_in_kit\": [\n 12,\n 16,\n 17\n ],\n \"kit_price\": 27\n },\n {\n \"kit_id\": \"S86\",\n \"dishes_in_kit\": [\n 10,\n 11,\n 13,\n 16\n ],\n \"kit_price\": 336\n },\n {\n \"kit_id\": \"S87\",\n \"dishes_in_kit\": [\n 3,\n 7,\n 9,\n 10,\n 11,\n 13\n ],\n \"kit_price\": 114\n },\n {\n \"kit_id\": \"S88\",\n \"dishes_in_kit\": [\n 18,\n 19,\n 20,\n 21,\n 22\n ],\n \"kit_price\": 180\n },\n {\n \"kit_id\": \"S89\",\n \"dishes_in_kit\": [\n 12,\n 13,\n 14\n ],\n \"kit_price\": 45\n },\n {\n \"kit_id\": \"S90\",\n \"dishes_in_kit\": [\n 6,\n 7,\n 10,\n 11,\n 12,\n 14\n ],\n \"kit_price\": 108\n },\n {\n \"kit_id\": \"S91\",\n \"dishes_in_kit\": [\n 5,\n 6,\n 8,\n 9,\n 10,\n 13\n ],\n \"kit_price\": 522\n },\n {\n \"kit_id\": \"S92\",\n \"dishes_in_kit\": [\n 6,\n 7,\n 8,\n 9,\n 10,\n 11\n ],\n \"kit_price\": 258\n },\n {\n \"kit_id\": \"S93\",\n \"dishes_in_kit\": [\n 15,\n 17,\n 18,\n 19,\n 20,\n 21,\n 22\n ],\n \"kit_price\": 595\n },\n {\n \"kit_id\": \"S94\",\n \"dishes_in_kit\": [\n 1,\n 6\n ],\n \"kit_price\": 58\n },\n {\n \"kit_id\": \"S95\",\n \"dishes_in_kit\": [\n 2,\n 5,\n 7,\n 8\n ],\n \"kit_price\": 364\n },\n {\n \"kit_id\": \"S96\",\n \"dishes_in_kit\": [\n 12,\n 18,\n 21\n ],\n \"kit_price\": 291\n },\n {\n \"kit_id\": \"S97\",\n \"dishes_in_kit\": [\n 10,\n 11,\n 14,\n 15\n ],\n \"kit_price\": 184\n },\n {\n \"kit_id\": \"S98\",\n \"dishes_in_kit\": [\n 0\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S99\",\n \"dishes_in_kit\": [\n 1\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S100\",\n \"dishes_in_kit\": [\n 2\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S101\",\n \"dishes_in_kit\": [\n 3\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S102\",\n \"dishes_in_kit\": [\n 4\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S103\",\n \"dishes_in_kit\": [\n 5\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S104\",\n \"dishes_in_kit\": [\n 6\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S105\",\n \"dishes_in_kit\": [\n 7\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S106\",\n \"dishes_in_kit\": [\n 8\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S107\",\n \"dishes_in_kit\": [\n 9\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S108\",\n \"dishes_in_kit\": [\n 10\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S109\",\n \"dishes_in_kit\": [\n 11\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S110\",\n \"dishes_in_kit\": [\n 12\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S111\",\n \"dishes_in_kit\": [\n 13\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S112\",\n \"dishes_in_kit\": [\n 14\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S113\",\n \"dishes_in_kit\": [\n 15\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S114\",\n \"dishes_in_kit\": [\n 16\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S115\",\n \"dishes_in_kit\": [\n 17\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S116\",\n \"dishes_in_kit\": [\n 18\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S117\",\n \"dishes_in_kit\": [\n 19\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S118\",\n \"dishes_in_kit\": [\n 20\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S119\",\n \"dishes_in_kit\": [\n 21\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S120\",\n \"dishes_in_kit\": [\n 22\n ],\n \"kit_price\": 10000\n }\n ]\n}\n\nAlso, when you send your pick, please use this little JSON shape so it's easy to read and check. It's just a sketch of what I expect back:\n\n{\n \"solution\": [\"kit_id\", ...]\n}\n\nHere \"solution\" is a list of the kits you want to order — each entry is the identifier for one pre-packed kit that will cover a dish. Think of it like ticking boxes on a form: put the exact kit IDs you choose into that array. This JSON is just the shape I want, not the actual final answer.\n\nOne more thing: use the identifiers exactly as they appear in the instance input — do not rename them or invent new labels. Valid identifiers look like plain numbers such as \"1\" or \"23\", single capital letters like \"A\" or \"B\", or a capital letter followed by digits like \"A1\" or \"X7\".", "instance": { "problem_type": "SPP", "num_elements": 23, "num_sets": 120, "density": 0.1746376811594203, "sets": [ { "id": 1, "elements": [ 5, 6, 7, 8, 9, 11 ], "cost": 72 }, { "id": 2, "elements": [ 16, 18, 19 ], "cost": 21 }, { "id": 3, "elements": [ 15, 19, 20, 21 ], "cost": 232 }, { "id": 4, "elements": [ 8, 10, 12, 13 ], "cost": 184 }, { "id": 5, "elements": [ 11, 17, 18, 19, 20, 21, 23 ], "cost": 392 }, { "id": 6, "elements": [ 1, 2, 3, 4 ], "cost": 12 }, { "id": 7, "elements": [ 13, 15, 19 ], "cost": 66 }, { "id": 8, "elements": [ 3, 4, 5, 6, 9 ], "cost": 155 }, { "id": 9, "elements": [ 4, 7, 8, 10, 13 ], "cost": 225 }, { "id": 10, "elements": [ 2, 4, 6, 7, 8, 11, 12 ], "cost": 301 }, { "id": 11, "elements": [ 1, 2, 3, 5 ], "cost": 172 }, { "id": 12, "elements": [ 18, 19, 20, 21, 22, 23 ], "cost": 534 }, { "id": 13, "elements": [ 17, 18, 19, 20, 22, 23 ], "cost": 126 }, { "id": 14, "elements": [ 1, 4, 5, 6, 7 ], "cost": 415 }, { "id": 15, "elements": [ 5, 6, 7 ], "cost": 36 }, { "id": 16, "elements": [ 15, 20, 21 ], "cost": 153 }, { "id": 17, "elements": [ 16, 17, 18, 20 ], "cost": 256 }, { "id": 18, "elements": [ 9, 11, 12, 13, 14, 15 ], "cost": 480 }, { "id": 19, "elements": [ 1, 2, 3, 4, 5, 6, 8 ], "cost": 119 }, { "id": 20, "elements": [ 6, 7, 8, 9, 10, 12, 13 ], "cost": 133 }, { "id": 21, "elements": [ 15, 17, 19, 20, 21, 22 ], "cost": 570 }, { "id": 22, "elements": [ 5, 6, 7, 8, 9, 10 ], "cost": 180 }, { "id": 23, "elements": [ 7, 9, 10, 11, 13, 14 ], "cost": 234 }, { "id": 24, "elements": [ 2, 3, 6 ], "cost": 180 }, { "id": 25, "elements": [ 8, 9, 11, 12, 14 ], "cost": 100 }, { "id": 26, "elements": [ 15, 16, 17, 18, 19, 20, 21 ], "cost": 532 }, { "id": 27, "elements": [ 1, 2, 3 ], "cost": 168 }, { "id": 28, "elements": [ 10, 11, 12, 13, 15 ], "cost": 380 }, { "id": 29, "elements": [ 10, 12, 13, 14, 15, 16, 17 ], "cost": 392 }, { "id": 30, "elements": [ 16, 17, 18, 19 ], "cost": 272 }, { "id": 31, "elements": [ 16, 19, 20 ], "cost": 273 }, { "id": 32, "elements": [ 3, 4, 5, 6, 7, 8 ], "cost": 600 }, { "id": 33, "elements": [ 10, 12, 13 ], "cost": 123 }, { "id": 34, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 84 }, { "id": 35, "elements": [ 9, 12, 13, 14, 16 ], "cost": 195 }, { "id": 36, "elements": [ 15, 16, 17, 19 ], "cost": 8 }, { "id": 37, "elements": [ 4, 5, 8 ], "cost": 111 }, { "id": 38, "elements": [ 4, 5, 6, 7, 8 ], "cost": 445 }, { "id": 39, "elements": [ 1, 5, 8, 9 ], "cost": 28 }, { "id": 40, "elements": [ 7, 10, 11, 12, 13, 14, 15 ], "cost": 49 }, { "id": 41, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 360 }, { "id": 42, "elements": [ 17, 20, 21 ], "cost": 162 }, { "id": 43, "elements": [ 13, 15, 16, 17, 18, 20, 21 ], "cost": 56 }, { "id": 44, "elements": [ 12, 13, 15, 17 ], "cost": 184 }, { "id": 45, "elements": [ 18, 19 ], "cost": 52 }, { "id": 46, "elements": [ 4, 8, 9 ], "cost": 240 }, { "id": 47, "elements": [ 19, 20, 22 ], "cost": 78 }, { "id": 48, "elements": [ 10, 11, 12, 13, 16, 17 ], "cost": 366 }, { "id": 49, "elements": [ 8, 12, 13, 14, 17 ], "cost": 445 }, { "id": 50, "elements": [ 7, 10, 14 ], "cost": 108 }, { "id": 51, "elements": [ 17, 18, 19, 20, 21, 23 ], "cost": 174 }, { "id": 52, "elements": [ 6, 7, 9, 11 ], "cost": 180 }, { "id": 53, "elements": [ 9, 10, 11, 13 ], "cost": 292 }, { "id": 54, "elements": [ 12, 15, 16 ], "cost": 3 }, { "id": 55, "elements": [ 7, 8, 9, 11, 12 ], "cost": 115 }, { "id": 56, "elements": [ 7, 9, 11 ], "cost": 15 }, { "id": 57, "elements": [ 4, 5, 7, 8, 10 ], "cost": 415 }, { "id": 58, "elements": [ 2, 3, 4, 5, 8 ], "cost": 295 }, { "id": 59, "elements": [ 10, 15, 16 ], "cost": 135 }, { "id": 60, "elements": [ 6, 7, 9, 10, 11, 12 ], "cost": 54 }, { "id": 61, "elements": [ 7, 9, 11, 13 ], "cost": 312 }, { "id": 62, "elements": [ 1, 2, 3, 4, 5 ], "cost": 500 }, { "id": 63, "elements": [ 6, 7, 11, 12, 14 ], "cost": 155 }, { "id": 64, "elements": [ 10, 12, 13, 17 ], "cost": 140 }, { "id": 65, "elements": [ 13, 16, 17, 18, 19, 20, 21 ], "cost": 630 }, { "id": 66, "elements": [ 2, 3, 4, 5, 6, 8 ], "cost": 444 }, { "id": 67, "elements": [ 7, 12, 13 ], "cost": 183 }, { "id": 68, "elements": [ 11, 16, 17, 18, 21 ], "cost": 15 }, { "id": 69, "elements": [ 16, 18, 20, 21, 22, 23 ], "cost": 432 }, { "id": 70, "elements": [ 20, 21, 22, 23 ], "cost": 16 }, { "id": 71, "elements": [ 3, 4, 5, 9 ], "cost": 296 }, { "id": 72, "elements": [ 17, 18, 19, 20, 21, 22, 23 ], "cost": 553 }, { "id": 73, "elements": [ 18, 21, 22 ], "cost": 216 }, { "id": 74, "elements": [ 19, 21, 23 ], "cost": 90 }, { "id": 75, "elements": [ 14, 16, 17, 18 ], "cost": 160 }, { "id": 76, "elements": [ 2, 4, 5, 7, 8 ], "cost": 440 }, { "id": 77, "elements": [ 5, 7, 8, 10, 11, 12, 14 ], "cost": 21 }, { "id": 78, "elements": [ 11, 12, 14 ], "cost": 12 }, { "id": 79, "elements": [ 13, 17, 18, 19, 20, 21, 22 ], "cost": 147 }, { "id": 80, "elements": [ 7, 8, 9, 10, 11, 12, 13 ], "cost": 623 }, { "id": 81, "elements": [ 10, 11, 12, 17 ], "cost": 332 }, { "id": 82, "elements": [ 1, 2, 4, 5, 7 ], "cost": 425 }, { "id": 83, "elements": [ 18, 19, 21 ], "cost": 174 }, { "id": 84, "elements": [ 2, 3, 4, 5, 8, 9 ], "cost": 42 }, { "id": 85, "elements": [ 13, 17, 18 ], "cost": 27 }, { "id": 86, "elements": [ 11, 12, 14, 17 ], "cost": 336 }, { "id": 87, "elements": [ 4, 8, 10, 11, 12, 14 ], "cost": 114 }, { "id": 88, "elements": [ 19, 20, 21, 22, 23 ], "cost": 180 }, { "id": 89, "elements": [ 13, 14, 15 ], "cost": 45 }, { "id": 90, "elements": [ 7, 8, 11, 12, 13, 15 ], "cost": 108 }, { "id": 91, "elements": [ 6, 7, 9, 10, 11, 14 ], "cost": 522 }, { "id": 92, "elements": [ 7, 8, 9, 10, 11, 12 ], "cost": 258 }, { "id": 93, "elements": [ 16, 18, 19, 20, 21, 22, 23 ], "cost": 595 }, { "id": 94, "elements": [ 2, 7 ], "cost": 58 }, { "id": 95, "elements": [ 3, 6, 8, 9 ], "cost": 364 }, { "id": 96, "elements": [ 13, 19, 22 ], "cost": 291 }, { "id": 97, "elements": [ 11, 12, 15, 16 ], "cost": 184 }, { "id": 98, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 118, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 119, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 120, "elements": [ 23 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0026_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0026_bag_stats.png" }, "solution": [ 6, 15, 25, 59, 85, 88 ], "obj": 490.0, "instance_variant": { "num_elements": 23, "num_sets": 120, "sets": [ { "id": "S1", "elements": [ 4, 5, 6, 7, 8, 10 ], "cost": 72 }, { "id": "S2", "elements": [ 15, 17, 18 ], "cost": 21 }, { "id": "S3", "elements": [ 14, 18, 19, 20 ], "cost": 232 }, { "id": "S4", "elements": [ 7, 9, 11, 12 ], "cost": 184 }, { "id": "S5", "elements": [ 10, 16, 17, 18, 19, 20, 22 ], "cost": 392 }, { "id": "S6", "elements": [ 0, 1, 2, 3 ], "cost": 12 }, { "id": "S7", "elements": [ 12, 14, 18 ], "cost": 66 }, { "id": "S8", "elements": [ 2, 3, 4, 5, 8 ], "cost": 155 }, { "id": "S9", "elements": [ 3, 6, 7, 9, 12 ], "cost": 225 }, { "id": "S10", "elements": [ 1, 3, 5, 6, 7, 10, 11 ], "cost": 301 }, { "id": "S11", "elements": [ 0, 1, 2, 4 ], "cost": 172 }, { "id": "S12", "elements": [ 17, 18, 19, 20, 21, 22 ], "cost": 534 }, { "id": "S13", "elements": [ 16, 17, 18, 19, 21, 22 ], "cost": 126 }, { "id": "S14", "elements": [ 0, 3, 4, 5, 6 ], "cost": 415 }, { "id": "S15", "elements": [ 4, 5, 6 ], "cost": 36 }, { "id": "S16", "elements": [ 14, 19, 20 ], "cost": 153 }, { "id": "S17", "elements": [ 15, 16, 17, 19 ], "cost": 256 }, { "id": "S18", "elements": [ 8, 10, 11, 12, 13, 14 ], "cost": 480 }, { "id": "S19", "elements": [ 0, 1, 2, 3, 4, 5, 7 ], "cost": 119 }, { "id": "S20", "elements": [ 5, 6, 7, 8, 9, 11, 12 ], "cost": 133 }, { "id": "S21", "elements": [ 14, 16, 18, 19, 20, 21 ], "cost": 570 }, { "id": "S22", "elements": [ 4, 5, 6, 7, 8, 9 ], "cost": 180 }, { "id": "S23", "elements": [ 6, 8, 9, 10, 12, 13 ], "cost": 234 }, { "id": "S24", "elements": [ 1, 2, 5 ], "cost": 180 }, { "id": "S25", "elements": [ 7, 8, 10, 11, 13 ], "cost": 100 }, { "id": "S26", "elements": [ 14, 15, 16, 17, 18, 19, 20 ], "cost": 532 }, { "id": "S27", "elements": [ 0, 1, 2 ], "cost": 168 }, { "id": "S28", "elements": [ 9, 10, 11, 12, 14 ], "cost": 380 }, { "id": "S29", "elements": [ 9, 11, 12, 13, 14, 15, 16 ], "cost": 392 }, { "id": "S30", "elements": [ 15, 16, 17, 18 ], "cost": 272 }, { "id": "S31", "elements": [ 15, 18, 19 ], "cost": 273 }, { "id": "S32", "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 600 }, { "id": "S33", "elements": [ 9, 11, 12 ], "cost": 123 }, { "id": "S34", "elements": [ 0, 1, 2, 3, 4, 6 ], "cost": 84 }, { "id": "S35", "elements": [ 8, 11, 12, 13, 15 ], "cost": 195 }, { "id": "S36", "elements": [ 14, 15, 16, 18 ], "cost": 8 }, { "id": "S37", "elements": [ 3, 4, 7 ], "cost": 111 }, { "id": "S38", "elements": [ 3, 4, 5, 6, 7 ], "cost": 445 }, { "id": "S39", "elements": [ 0, 4, 7, 8 ], "cost": 28 }, { "id": "S40", "elements": [ 6, 9, 10, 11, 12, 13, 14 ], "cost": 49 }, { "id": "S41", "elements": [ 0, 1, 2, 3, 4, 5 ], "cost": 360 }, { "id": "S42", "elements": [ 16, 19, 20 ], "cost": 162 }, { "id": "S43", "elements": [ 12, 14, 15, 16, 17, 19, 20 ], "cost": 56 }, { "id": "S44", "elements": [ 11, 12, 14, 16 ], "cost": 184 }, { "id": "S45", "elements": [ 17, 18 ], "cost": 52 }, { "id": "S46", "elements": [ 3, 7, 8 ], "cost": 240 }, { "id": "S47", "elements": [ 18, 19, 21 ], "cost": 78 }, { "id": "S48", "elements": [ 9, 10, 11, 12, 15, 16 ], "cost": 366 }, { "id": "S49", "elements": [ 7, 11, 12, 13, 16 ], "cost": 445 }, { "id": "S50", "elements": [ 6, 9, 13 ], "cost": 108 }, { "id": "S51", "elements": [ 16, 17, 18, 19, 20, 22 ], "cost": 174 }, { "id": "S52", "elements": [ 5, 6, 8, 10 ], "cost": 180 }, { "id": "S53", "elements": [ 8, 9, 10, 12 ], "cost": 292 }, { "id": "S54", "elements": [ 11, 14, 15 ], "cost": 3 }, { "id": "S55", "elements": [ 6, 7, 8, 10, 11 ], "cost": 115 }, { "id": "S56", "elements": [ 6, 8, 10 ], "cost": 15 }, { "id": "S57", "elements": [ 3, 4, 6, 7, 9 ], "cost": 415 }, { "id": "S58", "elements": [ 1, 2, 3, 4, 7 ], "cost": 295 }, { "id": "S59", "elements": [ 9, 14, 15 ], "cost": 135 }, { "id": "S60", "elements": [ 5, 6, 8, 9, 10, 11 ], "cost": 54 }, { "id": "S61", "elements": [ 6, 8, 10, 12 ], "cost": 312 }, { "id": "S62", "elements": [ 0, 1, 2, 3, 4 ], "cost": 500 }, { "id": "S63", "elements": [ 5, 6, 10, 11, 13 ], "cost": 155 }, { "id": "S64", "elements": [ 9, 11, 12, 16 ], "cost": 140 }, { "id": "S65", "elements": [ 12, 15, 16, 17, 18, 19, 20 ], "cost": 630 }, { "id": "S66", "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 444 }, { "id": "S67", "elements": [ 6, 11, 12 ], "cost": 183 }, { "id": "S68", "elements": [ 10, 15, 16, 17, 20 ], "cost": 15 }, { "id": "S69", "elements": [ 15, 17, 19, 20, 21, 22 ], "cost": 432 }, { "id": "S70", "elements": [ 19, 20, 21, 22 ], "cost": 16 }, { "id": "S71", "elements": [ 2, 3, 4, 8 ], "cost": 296 }, { "id": "S72", "elements": [ 16, 17, 18, 19, 20, 21, 22 ], "cost": 553 }, { "id": "S73", "elements": [ 17, 20, 21 ], "cost": 216 }, { "id": "S74", "elements": [ 18, 20, 22 ], "cost": 90 }, { "id": "S75", "elements": [ 13, 15, 16, 17 ], "cost": 160 }, { "id": "S76", "elements": [ 1, 3, 4, 6, 7 ], "cost": 440 }, { "id": "S77", "elements": [ 4, 6, 7, 9, 10, 11, 13 ], "cost": 21 }, { "id": "S78", "elements": [ 10, 11, 13 ], "cost": 12 }, { "id": "S79", "elements": [ 12, 16, 17, 18, 19, 20, 21 ], "cost": 147 }, { "id": "S80", "elements": [ 6, 7, 8, 9, 10, 11, 12 ], "cost": 623 }, { "id": "S81", "elements": [ 9, 10, 11, 16 ], "cost": 332 }, { "id": "S82", "elements": [ 0, 1, 3, 4, 6 ], "cost": 425 }, { "id": "S83", "elements": [ 17, 18, 20 ], "cost": 174 }, { "id": "S84", "elements": [ 1, 2, 3, 4, 7, 8 ], "cost": 42 }, { "id": "S85", "elements": [ 12, 16, 17 ], "cost": 27 }, { "id": "S86", "elements": [ 10, 11, 13, 16 ], "cost": 336 }, { "id": "S87", "elements": [ 3, 7, 9, 10, 11, 13 ], "cost": 114 }, { "id": "S88", "elements": [ 18, 19, 20, 21, 22 ], "cost": 180 }, { "id": "S89", "elements": [ 12, 13, 14 ], "cost": 45 }, { "id": "S90", "elements": [ 6, 7, 10, 11, 12, 14 ], "cost": 108 }, { "id": "S91", "elements": [ 5, 6, 8, 9, 10, 13 ], "cost": 522 }, { "id": "S92", "elements": [ 6, 7, 8, 9, 10, 11 ], "cost": 258 }, { "id": "S93", "elements": [ 15, 17, 18, 19, 20, 21, 22 ], "cost": 595 }, { "id": "S94", "elements": [ 1, 6 ], "cost": 58 }, { "id": "S95", "elements": [ 2, 5, 7, 8 ], "cost": 364 }, { "id": "S96", "elements": [ 12, 18, 21 ], "cost": 291 }, { "id": "S97", "elements": [ 10, 11, 14, 15 ], "cost": 184 }, { "id": "S98", "elements": [ 0 ], "cost": 10000 }, { "id": "S99", "elements": [ 1 ], "cost": 10000 }, { "id": "S100", "elements": [ 2 ], "cost": 10000 }, { "id": "S101", "elements": [ 3 ], "cost": 10000 }, { "id": "S102", "elements": [ 4 ], "cost": 10000 }, { "id": "S103", "elements": [ 5 ], "cost": 10000 }, { "id": "S104", "elements": [ 6 ], "cost": 10000 }, { "id": "S105", "elements": [ 7 ], "cost": 10000 }, { "id": "S106", "elements": [ 8 ], "cost": 10000 }, { "id": "S107", "elements": [ 9 ], "cost": 10000 }, { "id": "S108", "elements": [ 10 ], "cost": 10000 }, { "id": "S109", "elements": [ 11 ], "cost": 10000 }, { "id": "S110", "elements": [ 12 ], "cost": 10000 }, { "id": "S111", "elements": [ 13 ], "cost": 10000 }, { "id": "S112", "elements": [ 14 ], "cost": 10000 }, { "id": "S113", "elements": [ 15 ], "cost": 10000 }, { "id": "S114", "elements": [ 16 ], "cost": 10000 }, { "id": "S115", "elements": [ 17 ], "cost": 10000 }, { "id": "S116", "elements": [ 18 ], "cost": 10000 }, { "id": "S117", "elements": [ 19 ], "cost": 10000 }, { "id": "S118", "elements": [ 20 ], "cost": 10000 }, { "id": "S119", "elements": [ 21 ], "cost": 10000 }, { "id": "S120", "elements": [ 22 ], "cost": 10000 } ] }, "solution_variant": [ "S6", "S15", "S25", "S59", "S85", "S88" ], "context_index": 27, "input_format": "json", "input_index_base": 0 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "There’s an itinerary with a bunch of attractions and several overlapping tour options, so the trick is choosing tours so each attraction is handled by exactly one of the tours bought. After selecting a combination, the cost is just the sum of the prices of those tours, and the aim is to make that sum as small as possible while making sure nothing is uncovered or double-booked. The exact tour and attraction list appears below.\n\n# total_attractions_count=25\n# total_tour_packages=134\ntour_package_id,tour_price,attractions_covered\nS1,140,J K L N\nS2,644,N O P Q R T X\nS3,300,A C E I M\nS4,378,Q S T U V X\nS5,371,R S T U V W X\nS6,375,A B E F I\nS7,425,F I J K M\nS8,364,P R S T U V W\nS9,195,L N O P S\nS10,99,E I M\nS11,364,D E F G I K O\nS12,255,A B E\nS13,510,I J M N O P\nS14,64,A B D E\nS15,246,N Q S\nS16,602,L M N P Q R S\nS17,210,A B C D E F G\nS18,42,P T V\nS19,420,F I J K M N\nS20,120,A B C D E I\nS21,240,N R U V W X\nS22,400,C D E F I\nS23,88,I J K M\nS24,294,H L N\nS25,156,A B C D E F\nS26,200,C E F G H\nS27,16,N O P R\nS28,5,O P Q V X\nS29,534,K L M O P R\nS30,345,T U W X Y\nS31,360,A B C F H\nS32,348,J O P R S T\nS33,60,U V W X\nS34,644,A B C D E G H\nS35,312,E G I L\nS36,693,Q T U V W X Y\nS37,623,S T U V W X Y\nS38,294,S T U V W X\nS39,372,K L M O\nS40,325,C E F J M\nS41,42,A B C D E H J\nS42,465,B C D E G\nS43,343,Q S T V W X Y\nS44,132,J K L\nS45,252,J K N\nS46,145,D G K L P\nS47,540,N O P Q R U\nS48,282,M O P\nS49,68,I K L P\nS50,350,B D E F H J L\nS51,216,O Q S\nS52,332,K M N P\nS53,228,T V W\nS54,24,G H M\nS55,413,E F G I J K L\nS56,306,O P Q S T U\nS57,180,Q S U\nS58,324,N O Q T\nS59,215,A D F G H\nS60,308,A B E F\nS61,392,S T W X\nS62,64,J K M O\nS63,70,L N O Q S\nS64,364,B C D E F G I\nS65,141,F G M\nS66,90,H M P\nS67,444,P S T U W X\nS68,228,C D E\nS69,384,K M N O\nS70,300,C E G H\nS71,294,W X Y\nS72,450,N P Q R S T\nS73,368,H I J O\nS74,594,G H I J K N\nS75,282,I K N\nS76,219,C G J\nS77,75,A B C\nS78,266,B D E F G H J\nS79,204,E G I J\nS80,637,M N O P Q R T\nS81,3,T U V\nS82,108,Q R V\nS83,80,C G H K\nS84,112,A B C D E F I\nS85,80,C F G K\nS86,357,L M O P Q T U\nS87,35,C E J K L\nS88,72,B C D F\nS89,275,A B C D E\nS90,30,B D E G I\nS91,208,K M O S\nS92,98,G H I J K L M\nS93,350,F G H I L\nS94,246,O R T U V X\nS95,68,C E G J\nS96,456,C E G H I J\nS97,420,R S U V W X Y\nS98,348,R U V W X Y\nS99,250,N P R S W\nS100,240,K M N O S\nS101,4,U V X Y\nS102,350,J L M O P R S\nS103,265,A B D E F\nS104,91,I J K L M N O\nS105,378,A B D E F G J\nS106,152,H I J L\nS107,165,E G H\nS108,18,V W X\nS109,120,J M N O R T\nS110,10000,A\nS111,10000,B\nS112,10000,C\nS113,10000,D\nS114,10000,E\nS115,10000,F\nS116,10000,G\nS117,10000,H\nS118,10000,I\nS119,10000,J\nS120,10000,K\nS121,10000,L\nS122,10000,M\nS123,10000,N\nS124,10000,O\nS125,10000,P\nS126,10000,Q\nS127,10000,R\nS128,10000,S\nS129,10000,T\nS130,10000,U\nS131,10000,V\nS132,10000,W\nS133,10000,X\nS134,10000,Y\n\nIf you want to send the picks back, just use this simple JSON shape — nothing fancy, just the expected layout.\n\n{\n \"solution\": [\"tour_id\", ...]\n}\n\n\"solution\" is an array that should list the IDs of the tours you choose (one entry per chosen tour). The \"tour_id\" item is just a placeholder showing the kind of values to put in there. This is only a sketch of the expected shape — not the actual answer.\n\nPlease make sure to use the exact identifiers from the instance input, with no renaming or extra labels. \nValid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.", "instance": { "problem_type": "SPP", "num_elements": 25, "num_sets": 134, "density": 0.16835820895522388, "sets": [ { "id": 1, "elements": [ 10, 11, 12, 14 ], "cost": 140 }, { "id": 2, "elements": [ 14, 15, 16, 17, 18, 20, 24 ], "cost": 644 }, { "id": 3, "elements": [ 1, 3, 5, 9, 13 ], "cost": 300 }, { "id": 4, "elements": [ 17, 19, 20, 21, 22, 24 ], "cost": 378 }, { "id": 5, "elements": [ 18, 19, 20, 21, 22, 23, 24 ], "cost": 371 }, { "id": 6, "elements": [ 1, 2, 5, 6, 9 ], "cost": 375 }, { "id": 7, "elements": [ 6, 9, 10, 11, 13 ], "cost": 425 }, { "id": 8, "elements": [ 16, 18, 19, 20, 21, 22, 23 ], "cost": 364 }, { "id": 9, "elements": [ 12, 14, 15, 16, 19 ], "cost": 195 }, { "id": 10, "elements": [ 5, 9, 13 ], "cost": 99 }, { "id": 11, "elements": [ 4, 5, 6, 7, 9, 11, 15 ], "cost": 364 }, { "id": 12, "elements": [ 1, 2, 5 ], "cost": 255 }, { "id": 13, "elements": [ 9, 10, 13, 14, 15, 16 ], "cost": 510 }, { "id": 14, "elements": [ 1, 2, 4, 5 ], "cost": 64 }, { "id": 15, "elements": [ 14, 17, 19 ], "cost": 246 }, { "id": 16, "elements": [ 12, 13, 14, 16, 17, 18, 19 ], "cost": 602 }, { "id": 17, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 210 }, { "id": 18, "elements": [ 16, 20, 22 ], "cost": 42 }, { "id": 19, "elements": [ 6, 9, 10, 11, 13, 14 ], "cost": 420 }, { "id": 20, "elements": [ 1, 2, 3, 4, 5, 9 ], "cost": 120 }, { "id": 21, "elements": [ 14, 18, 21, 22, 23, 24 ], "cost": 240 }, { "id": 22, "elements": [ 3, 4, 5, 6, 9 ], "cost": 400 }, { "id": 23, "elements": [ 9, 10, 11, 13 ], "cost": 88 }, { "id": 24, "elements": [ 8, 12, 14 ], "cost": 294 }, { "id": 25, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 156 }, { "id": 26, "elements": [ 3, 5, 6, 7, 8 ], "cost": 200 }, { "id": 27, "elements": [ 14, 15, 16, 18 ], "cost": 16 }, { "id": 28, "elements": [ 15, 16, 17, 22, 24 ], "cost": 5 }, { "id": 29, "elements": [ 11, 12, 13, 15, 16, 18 ], "cost": 534 }, { "id": 30, "elements": [ 20, 21, 23, 24, 25 ], "cost": 345 }, { "id": 31, "elements": [ 1, 2, 3, 6, 8 ], "cost": 360 }, { "id": 32, "elements": [ 10, 15, 16, 18, 19, 20 ], "cost": 348 }, { "id": 33, "elements": [ 21, 22, 23, 24 ], "cost": 60 }, { "id": 34, "elements": [ 1, 2, 3, 4, 5, 7, 8 ], "cost": 644 }, { "id": 35, "elements": [ 5, 7, 9, 12 ], "cost": 312 }, { "id": 36, "elements": [ 17, 20, 21, 22, 23, 24, 25 ], "cost": 693 }, { "id": 37, "elements": [ 19, 20, 21, 22, 23, 24, 25 ], "cost": 623 }, { "id": 38, "elements": [ 19, 20, 21, 22, 23, 24 ], "cost": 294 }, { "id": 39, "elements": [ 11, 12, 13, 15 ], "cost": 372 }, { "id": 40, "elements": [ 3, 5, 6, 10, 13 ], "cost": 325 }, { "id": 41, "elements": [ 1, 2, 3, 4, 5, 8, 10 ], "cost": 42 }, { "id": 42, "elements": [ 2, 3, 4, 5, 7 ], "cost": 465 }, { "id": 43, "elements": [ 17, 19, 20, 22, 23, 24, 25 ], "cost": 343 }, { "id": 44, "elements": [ 10, 11, 12 ], "cost": 132 }, { "id": 45, "elements": [ 10, 11, 14 ], "cost": 252 }, { "id": 46, "elements": [ 4, 7, 11, 12, 16 ], "cost": 145 }, { "id": 47, "elements": [ 14, 15, 16, 17, 18, 21 ], "cost": 540 }, { "id": 48, "elements": [ 13, 15, 16 ], "cost": 282 }, { "id": 49, "elements": [ 9, 11, 12, 16 ], "cost": 68 }, { "id": 50, "elements": [ 2, 4, 5, 6, 8, 10, 12 ], "cost": 350 }, { "id": 51, "elements": [ 15, 17, 19 ], "cost": 216 }, { "id": 52, "elements": [ 11, 13, 14, 16 ], "cost": 332 }, { "id": 53, "elements": [ 20, 22, 23 ], "cost": 228 }, { "id": 54, "elements": [ 7, 8, 13 ], "cost": 24 }, { "id": 55, "elements": [ 5, 6, 7, 9, 10, 11, 12 ], "cost": 413 }, { "id": 56, "elements": [ 15, 16, 17, 19, 20, 21 ], "cost": 306 }, { "id": 57, "elements": [ 17, 19, 21 ], "cost": 180 }, { "id": 58, "elements": [ 14, 15, 17, 20 ], "cost": 324 }, { "id": 59, "elements": [ 1, 4, 6, 7, 8 ], "cost": 215 }, { "id": 60, "elements": [ 1, 2, 5, 6 ], "cost": 308 }, { "id": 61, "elements": [ 19, 20, 23, 24 ], "cost": 392 }, { "id": 62, "elements": [ 10, 11, 13, 15 ], "cost": 64 }, { "id": 63, "elements": [ 12, 14, 15, 17, 19 ], "cost": 70 }, { "id": 64, "elements": [ 2, 3, 4, 5, 6, 7, 9 ], "cost": 364 }, { "id": 65, "elements": [ 6, 7, 13 ], "cost": 141 }, { "id": 66, "elements": [ 8, 13, 16 ], "cost": 90 }, { "id": 67, "elements": [ 16, 19, 20, 21, 23, 24 ], "cost": 444 }, { "id": 68, "elements": [ 3, 4, 5 ], "cost": 228 }, { "id": 69, "elements": [ 11, 13, 14, 15 ], "cost": 384 }, { "id": 70, "elements": [ 3, 5, 7, 8 ], "cost": 300 }, { "id": 71, "elements": [ 23, 24, 25 ], "cost": 294 }, { "id": 72, "elements": [ 14, 16, 17, 18, 19, 20 ], "cost": 450 }, { "id": 73, "elements": [ 8, 9, 10, 15 ], "cost": 368 }, { "id": 74, "elements": [ 7, 8, 9, 10, 11, 14 ], "cost": 594 }, { "id": 75, "elements": [ 9, 11, 14 ], "cost": 282 }, { "id": 76, "elements": [ 3, 7, 10 ], "cost": 219 }, { "id": 77, "elements": [ 1, 2, 3 ], "cost": 75 }, { "id": 78, "elements": [ 2, 4, 5, 6, 7, 8, 10 ], "cost": 266 }, { "id": 79, "elements": [ 5, 7, 9, 10 ], "cost": 204 }, { "id": 80, "elements": [ 13, 14, 15, 16, 17, 18, 20 ], "cost": 637 }, { "id": 81, "elements": [ 20, 21, 22 ], "cost": 3 }, { "id": 82, "elements": [ 17, 18, 22 ], "cost": 108 }, { "id": 83, "elements": [ 3, 7, 8, 11 ], "cost": 80 }, { "id": 84, "elements": [ 1, 2, 3, 4, 5, 6, 9 ], "cost": 112 }, { "id": 85, "elements": [ 3, 6, 7, 11 ], "cost": 80 }, { "id": 86, "elements": [ 12, 13, 15, 16, 17, 20, 21 ], "cost": 357 }, { "id": 87, "elements": [ 3, 5, 10, 11, 12 ], "cost": 35 }, { "id": 88, "elements": [ 2, 3, 4, 6 ], "cost": 72 }, { "id": 89, "elements": [ 1, 2, 3, 4, 5 ], "cost": 275 }, { "id": 90, "elements": [ 2, 4, 5, 7, 9 ], "cost": 30 }, { "id": 91, "elements": [ 11, 13, 15, 19 ], "cost": 208 }, { "id": 92, "elements": [ 7, 8, 9, 10, 11, 12, 13 ], "cost": 98 }, { "id": 93, "elements": [ 6, 7, 8, 9, 12 ], "cost": 350 }, { "id": 94, "elements": [ 15, 18, 20, 21, 22, 24 ], "cost": 246 }, { "id": 95, "elements": [ 3, 5, 7, 10 ], "cost": 68 }, { "id": 96, "elements": [ 3, 5, 7, 8, 9, 10 ], "cost": 456 }, { "id": 97, "elements": [ 18, 19, 21, 22, 23, 24, 25 ], "cost": 420 }, { "id": 98, "elements": [ 18, 21, 22, 23, 24, 25 ], "cost": 348 }, { "id": 99, "elements": [ 14, 16, 18, 19, 23 ], "cost": 250 }, { "id": 100, "elements": [ 11, 13, 14, 15, 19 ], "cost": 240 }, { "id": 101, "elements": [ 21, 22, 24, 25 ], "cost": 4 }, { "id": 102, "elements": [ 10, 12, 13, 15, 16, 18, 19 ], "cost": 350 }, { "id": 103, "elements": [ 1, 2, 4, 5, 6 ], "cost": 265 }, { "id": 104, "elements": [ 9, 10, 11, 12, 13, 14, 15 ], "cost": 91 }, { "id": 105, "elements": [ 1, 2, 4, 5, 6, 7, 10 ], "cost": 378 }, { "id": 106, "elements": [ 8, 9, 10, 12 ], "cost": 152 }, { "id": 107, "elements": [ 5, 7, 8 ], "cost": 165 }, { "id": 108, "elements": [ 22, 23, 24 ], "cost": 18 }, { "id": 109, "elements": [ 10, 13, 14, 15, 18, 20 ], "cost": 120 }, { "id": 110, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 118, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 119, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 120, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 121, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 122, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 123, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 124, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 125, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 126, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 127, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 128, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 129, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 130, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 131, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 132, "elements": [ 23 ], "cost": 10000, "singleton": true }, { "id": 133, "elements": [ 24 ], "cost": 10000, "singleton": true }, { "id": 134, "elements": [ 25 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0027_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0027_bag_stats.png" }, "solution": [ 41, 49, 58, 65, 97 ], "obj": 995.0, "instance_variant": { "num_elements": 25, "num_sets": 134, "sets": [ { "id": "S1", "elements": [ "J", "K", "L", "N" ], "cost": 140 }, { "id": "S2", "elements": [ "N", "O", "P", "Q", "R", "T", "X" ], "cost": 644 }, { "id": "S3", "elements": [ "A", "C", "E", "I", "M" ], "cost": 300 }, { "id": "S4", "elements": [ "Q", "S", "T", "U", "V", "X" ], "cost": 378 }, { "id": "S5", "elements": [ "R", "S", "T", "U", "V", "W", "X" ], "cost": 371 }, { "id": "S6", "elements": [ "A", "B", "E", "F", "I" ], "cost": 375 }, { "id": "S7", "elements": [ "F", "I", "J", "K", "M" ], "cost": 425 }, { "id": "S8", "elements": [ "P", "R", "S", "T", "U", "V", "W" ], "cost": 364 }, { "id": "S9", "elements": [ "L", "N", "O", "P", "S" ], "cost": 195 }, { "id": "S10", "elements": [ "E", "I", "M" ], "cost": 99 }, { "id": "S11", "elements": [ "D", "E", "F", "G", "I", "K", "O" ], "cost": 364 }, { "id": "S12", "elements": [ "A", "B", "E" ], "cost": 255 }, { "id": "S13", "elements": [ "I", "J", "M", "N", "O", "P" ], "cost": 510 }, { "id": "S14", "elements": [ "A", "B", "D", "E" ], "cost": 64 }, { "id": "S15", "elements": [ "N", "Q", "S" ], "cost": 246 }, { "id": "S16", "elements": [ "L", "M", "N", "P", "Q", "R", "S" ], "cost": 602 }, { "id": "S17", "elements": [ "A", "B", "C", "D", "E", "F", "G" ], "cost": 210 }, { "id": "S18", "elements": [ "P", "T", "V" ], "cost": 42 }, { "id": "S19", "elements": [ "F", "I", "J", "K", "M", "N" ], "cost": 420 }, { "id": "S20", "elements": [ "A", "B", "C", "D", "E", "I" ], "cost": 120 }, { "id": "S21", "elements": [ "N", "R", "U", "V", "W", "X" ], "cost": 240 }, { "id": "S22", "elements": [ "C", "D", "E", "F", "I" ], "cost": 400 }, { "id": "S23", "elements": [ "I", "J", "K", "M" ], "cost": 88 }, { "id": "S24", "elements": [ "H", "L", "N" ], "cost": 294 }, { "id": "S25", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 156 }, { "id": "S26", "elements": [ "C", "E", "F", "G", "H" ], "cost": 200 }, { "id": "S27", "elements": [ "N", "O", "P", "R" ], "cost": 16 }, { "id": "S28", "elements": [ "O", "P", "Q", "V", "X" ], "cost": 5 }, { "id": "S29", "elements": [ "K", "L", "M", "O", "P", "R" ], "cost": 534 }, { "id": "S30", "elements": [ "T", "U", "W", "X", "Y" ], "cost": 345 }, { "id": "S31", "elements": [ "A", "B", "C", "F", "H" ], "cost": 360 }, { "id": "S32", "elements": [ "J", "O", "P", "R", "S", "T" ], "cost": 348 }, { "id": "S33", "elements": [ "U", "V", "W", "X" ], "cost": 60 }, { "id": "S34", "elements": [ "A", "B", "C", "D", "E", "G", "H" ], "cost": 644 }, { "id": "S35", "elements": [ "E", "G", "I", "L" ], "cost": 312 }, { "id": "S36", "elements": [ "Q", "T", "U", "V", "W", "X", "Y" ], "cost": 693 }, { "id": "S37", "elements": [ "S", "T", "U", "V", "W", "X", "Y" ], "cost": 623 }, { "id": "S38", "elements": [ "S", "T", "U", "V", "W", "X" ], "cost": 294 }, { "id": "S39", "elements": [ "K", "L", "M", "O" ], "cost": 372 }, { "id": "S40", "elements": [ "C", "E", "F", "J", "M" ], "cost": 325 }, { "id": "S41", "elements": [ "A", "B", "C", "D", "E", "H", "J" ], "cost": 42 }, { "id": "S42", "elements": [ "B", "C", "D", "E", "G" ], "cost": 465 }, { "id": "S43", "elements": [ "Q", "S", "T", "V", "W", "X", "Y" ], "cost": 343 }, { "id": "S44", "elements": [ "J", "K", "L" ], "cost": 132 }, { "id": "S45", "elements": [ "J", "K", "N" ], "cost": 252 }, { "id": "S46", "elements": [ "D", "G", "K", "L", "P" ], "cost": 145 }, { "id": "S47", "elements": [ "N", "O", "P", "Q", "R", "U" ], "cost": 540 }, { "id": "S48", "elements": [ "M", "O", "P" ], "cost": 282 }, { "id": "S49", "elements": [ "I", "K", "L", "P" ], "cost": 68 }, { "id": "S50", "elements": [ "B", "D", "E", "F", "H", "J", "L" ], "cost": 350 }, { "id": "S51", "elements": [ "O", "Q", "S" ], "cost": 216 }, { "id": "S52", "elements": [ "K", "M", "N", "P" ], "cost": 332 }, { "id": "S53", "elements": [ "T", "V", "W" ], "cost": 228 }, { "id": "S54", "elements": [ "G", "H", "M" ], "cost": 24 }, { "id": "S55", "elements": [ "E", "F", "G", "I", "J", "K", "L" ], "cost": 413 }, { "id": "S56", "elements": [ "O", "P", "Q", "S", "T", "U" ], "cost": 306 }, { "id": "S57", "elements": [ "Q", "S", "U" ], "cost": 180 }, { "id": "S58", "elements": [ "N", "O", "Q", "T" ], "cost": 324 }, { "id": "S59", "elements": [ "A", "D", "F", "G", "H" ], "cost": 215 }, { "id": "S60", "elements": [ "A", "B", "E", "F" ], "cost": 308 }, { "id": "S61", "elements": [ "S", "T", "W", "X" ], "cost": 392 }, { "id": "S62", "elements": [ "J", "K", "M", "O" ], "cost": 64 }, { "id": "S63", "elements": [ "L", "N", "O", "Q", "S" ], "cost": 70 }, { "id": "S64", "elements": [ "B", "C", "D", "E", "F", "G", "I" ], "cost": 364 }, { "id": "S65", "elements": [ "F", "G", "M" ], "cost": 141 }, { "id": "S66", "elements": [ "H", "M", "P" ], "cost": 90 }, { "id": "S67", "elements": [ "P", "S", "T", "U", "W", "X" ], "cost": 444 }, { "id": "S68", "elements": [ "C", "D", "E" ], "cost": 228 }, { "id": "S69", "elements": [ "K", "M", "N", "O" ], "cost": 384 }, { "id": "S70", "elements": [ "C", "E", "G", "H" ], "cost": 300 }, { "id": "S71", "elements": [ "W", "X", "Y" ], "cost": 294 }, { "id": "S72", "elements": [ "N", "P", "Q", "R", "S", "T" ], "cost": 450 }, { "id": "S73", "elements": [ "H", "I", "J", "O" ], "cost": 368 }, { "id": "S74", "elements": [ "G", "H", "I", "J", "K", "N" ], "cost": 594 }, { "id": "S75", "elements": [ "I", "K", "N" ], "cost": 282 }, { "id": "S76", "elements": [ "C", "G", "J" ], "cost": 219 }, { "id": "S77", "elements": [ "A", "B", "C" ], "cost": 75 }, { "id": "S78", "elements": [ "B", "D", "E", "F", "G", "H", "J" ], "cost": 266 }, { "id": "S79", "elements": [ "E", "G", "I", "J" ], "cost": 204 }, { "id": "S80", "elements": [ "M", "N", "O", "P", "Q", "R", "T" ], "cost": 637 }, { "id": "S81", "elements": [ "T", "U", "V" ], "cost": 3 }, { "id": "S82", "elements": [ "Q", "R", "V" ], "cost": 108 }, { "id": "S83", "elements": [ "C", "G", "H", "K" ], "cost": 80 }, { "id": "S84", "elements": [ "A", "B", "C", "D", "E", "F", "I" ], "cost": 112 }, { "id": "S85", "elements": [ "C", "F", "G", "K" ], "cost": 80 }, { "id": "S86", "elements": [ "L", "M", "O", "P", "Q", "T", "U" ], "cost": 357 }, { "id": "S87", "elements": [ "C", "E", "J", "K", "L" ], "cost": 35 }, { "id": "S88", "elements": [ "B", "C", "D", "F" ], "cost": 72 }, { "id": "S89", "elements": [ "A", "B", "C", "D", "E" ], "cost": 275 }, { "id": "S90", "elements": [ "B", "D", "E", "G", "I" ], "cost": 30 }, { "id": "S91", "elements": [ "K", "M", "O", "S" ], "cost": 208 }, { "id": "S92", "elements": [ "G", "H", "I", "J", "K", "L", "M" ], "cost": 98 }, { "id": "S93", "elements": [ "F", "G", "H", "I", "L" ], "cost": 350 }, { "id": "S94", "elements": [ "O", "R", "T", "U", "V", "X" ], "cost": 246 }, { "id": "S95", "elements": [ "C", "E", "G", "J" ], "cost": 68 }, { "id": "S96", "elements": [ "C", "E", "G", "H", "I", "J" ], "cost": 456 }, { "id": "S97", "elements": [ "R", "S", "U", "V", "W", "X", "Y" ], "cost": 420 }, { "id": "S98", "elements": [ "R", "U", "V", "W", "X", "Y" ], "cost": 348 }, { "id": "S99", "elements": [ "N", "P", "R", "S", "W" ], "cost": 250 }, { "id": "S100", "elements": [ "K", "M", "N", "O", "S" ], "cost": 240 }, { "id": "S101", "elements": [ "U", "V", "X", "Y" ], "cost": 4 }, { "id": "S102", "elements": [ "J", "L", "M", "O", "P", "R", "S" ], "cost": 350 }, { "id": "S103", "elements": [ "A", "B", "D", "E", "F" ], "cost": 265 }, { "id": "S104", "elements": [ "I", "J", "K", "L", "M", "N", "O" ], "cost": 91 }, { "id": "S105", "elements": [ "A", "B", "D", "E", "F", "G", "J" ], "cost": 378 }, { "id": "S106", "elements": [ "H", "I", "J", "L" ], "cost": 152 }, { "id": "S107", "elements": [ "E", "G", "H" ], "cost": 165 }, { "id": "S108", "elements": [ "V", "W", "X" ], "cost": 18 }, { "id": "S109", "elements": [ "J", "M", "N", "O", "R", "T" ], "cost": 120 }, { "id": "S110", "elements": [ "A" ], "cost": 10000 }, { "id": "S111", "elements": [ "B" ], "cost": 10000 }, { "id": "S112", "elements": [ "C" ], "cost": 10000 }, { "id": "S113", "elements": [ "D" ], "cost": 10000 }, { "id": "S114", "elements": [ "E" ], "cost": 10000 }, { "id": "S115", "elements": [ "F" ], "cost": 10000 }, { "id": "S116", "elements": [ "G" ], "cost": 10000 }, { "id": "S117", "elements": [ "H" ], "cost": 10000 }, { "id": "S118", "elements": [ "I" ], "cost": 10000 }, { "id": "S119", "elements": [ "J" ], "cost": 10000 }, { "id": "S120", "elements": [ "K" ], "cost": 10000 }, { "id": "S121", "elements": [ "L" ], "cost": 10000 }, { "id": "S122", "elements": [ "M" ], "cost": 10000 }, { "id": "S123", "elements": [ "N" ], "cost": 10000 }, { "id": "S124", "elements": [ "O" ], "cost": 10000 }, { "id": "S125", "elements": [ "P" ], "cost": 10000 }, { "id": "S126", "elements": [ "Q" ], "cost": 10000 }, { "id": "S127", "elements": [ "R" ], "cost": 10000 }, { "id": "S128", "elements": [ "S" ], "cost": 10000 }, { "id": "S129", "elements": [ "T" ], "cost": 10000 }, { "id": "S130", "elements": [ "U" ], "cost": 10000 }, { "id": "S131", "elements": [ "V" ], "cost": 10000 }, { "id": "S132", "elements": [ "W" ], "cost": 10000 }, { "id": "S133", "elements": [ "X" ], "cost": 10000 }, { "id": "S134", "elements": [ "Y" ], "cost": 10000 } ] }, "solution_variant": [ "S41", "S49", "S58", "S65", "S97" ], "context_index": 28, "input_format": "csv", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "Many people on the floor have noticed the assortment of prepacked box types and are trying to figure out which ones to use for an upcoming shipment. Each box type holds particular product units and comes with a cost, so the decision is which box types to select so that every product unit is put into one—and only one—selected box type. The better plan is the cheaper plan: add up the costs of the chosen box types to get the total packing cost, and prefer the smallest total. The concrete box and unit details are shown below.\n\nThere are 20 product units and 107 prepacked box types listed below:\nBox type S1 holds product units 3 6 7 8 and costs 380.\nBox type S2 holds product units 1 2 3 6 and costs 140.\nBox type S3 holds product units 1 2 3 4 5 and costs 370.\nBox type S4 holds product units 2 3 4 5 6 7 and costs 210.\nBox type S5 holds product units 6 8 9 12 and costs 196.\nBox type S6 holds product units 14 15 16 17 20 and costs 410.\nBox type S7 holds product units 14 15 17 18 19 20 and costs 510.\nBox type S8 holds product units 1 2 3 4 5 6 and costs 180.\nBox type S9 holds product units 10 11 12 13 14 15 and costs 18.\nBox type S10 holds product units 8 9 10 11 and costs 236.\nBox type S11 holds product units 16 17 18 20 and costs 340.\nBox type S12 holds product units 18 19 20 and costs 264.\nBox type S13 holds product units 3 4 6 8 and costs 312.\nBox type S14 holds product units 17 18 19 and costs 108.\nBox type S15 holds product units 12 15 and costs 58.\nBox type S16 holds product units 10 15 16 and costs 147.\nBox type S17 holds product units 6 7 9 10 13 and costs 285.\nBox type S18 holds product units 1 2 4 and costs 255.\nBox type S19 holds product units 10 11 12 13 and costs 292.\nBox type S20 holds product units 6 7 8 9 10 11 and costs 450.\nBox type S21 holds product units 3 7 and costs 174.\nBox type S22 holds product units 1 2 3 4 6 and costs 460.\nBox type S23 holds product units 12 14 15 17 and costs 232.\nBox type S24 holds product units 6 8 9 11 and costs 104.\nBox type S25 holds product units 3 4 5 8 and costs 24.\nBox type S26 holds product units 4 6 7 8 9 and costs 360.\nBox type S27 holds product units 10 12 13 14 and costs 208.\nBox type S28 holds product units 2 4 6 7 8 and costs 100.\nBox type S29 holds product units 13 14 15 17 18 20 and costs 516.\nBox type S30 holds product units 8 11 12 13 and costs 292.\nBox type S31 holds product units 16 17 18 19 20 and costs 90.\nBox type S32 holds product units 16 17 18 and costs 273.\nBox type S33 holds product units 10 13 14 15 and costs 368.\nBox type S34 holds product units 13 14 15 and costs 201.\nBox type S35 holds product units 6 8 9 10 13 and costs 285.\nBox type S36 holds product units 4 5 6 and costs 84.\nBox type S37 holds product units 7 10 and costs 198.\nBox type S38 holds product units 2 3 4 and costs 240.\nBox type S39 holds product units 4 6 8 9 and costs 192.\nBox type S40 holds product units 13 16 17 18 19 20 and costs 540.\nBox type S41 holds product units 8 9 and costs 84.\nBox type S42 holds product units 4 6 7 8 12 and costs 500.\nBox type S43 holds product units 2 3 4 6 and costs 320.\nBox type S44 holds product units 1 2 3 5 and costs 400.\nBox type S45 holds product units 10 11 12 13 15 and costs 20.\nBox type S46 holds product units 4 7 9 10 11 12 and costs 168.\nBox type S47 holds product units 15 16 17 18 19 20 and costs 384.\nBox type S48 holds product units 3 6 7 and costs 171.\nBox type S49 holds product units 16 18 19 and costs 273.\nBox type S50 holds product units 17 20 and costs 96.\nBox type S51 holds product units 7 8 9 11 13 15 and costs 570.\nBox type S52 holds product units 9 11 12 13 and costs 200.\nBox type S53 holds product units 7 8 9 10 11 and costs 185.\nBox type S54 holds product units 12 13 15 18 19 and costs 5.\nBox type S55 holds product units 3 7 9 and costs 156.\nBox type S56 holds product units 1 4 5 6 and costs 392.\nBox type S57 holds product units 8 10 12 and costs 273.\nBox type S58 holds product units 13 16 17 18 and costs 68.\nBox type S59 holds product units 3 4 5 7 and costs 104.\nBox type S60 holds product units 1 2 3 4 7 and costs 20.\nBox type S61 holds product units 16 18 19 20 and costs 216.\nBox type S62 holds product units 13 15 16 20 and costs 40.\nBox type S63 holds product units 12 14 and costs 72.\nBox type S64 holds product units 1 2 5 and costs 264.\nBox type S65 holds product units 4 6 7 9 11 and costs 285.\nBox type S66 holds product units 14 16 17 18 19 20 and costs 474.\nBox type S67 holds product units 4 6 and costs 104.\nBox type S68 holds product units 6 8 and costs 64.\nBox type S69 holds product units 9 11 12 13 15 and costs 90.\nBox type S70 holds product units 5 7 8 11 and costs 284.\nBox type S71 holds product units 3 5 7 and costs 279.\nBox type S72 holds product units 6 7 8 9 11 12 and costs 414.\nBox type S73 holds product units 1 4 5 and costs 27.\nBox type S74 holds product units 11 13 14 and costs 144.\nBox type S75 holds product units 7 8 9 10 12 and costs 380.\nBox type S76 holds product units 14 15 17 19 and costs 356.\nBox type S77 holds product units 1 2 and costs 114.\nBox type S78 holds product units 1 3 4 and costs 96.\nBox type S79 holds product units 6 7 and costs 18.\nBox type S80 holds product units 7 8 10 13 15 and costs 120.\nBox type S81 holds product units 5 6 9 and costs 249.\nBox type S82 holds product units 7 9 10 13 14 and costs 475.\nBox type S83 holds product units 6 12 and costs 106.\nBox type S84 holds product units 3 4 5 6 and costs 12.\nBox type S85 holds product units 11 13 and costs 190.\nBox type S86 holds product units 1 3 4 10 and costs 120.\nBox type S87 holds product units 15 16 18 19 20 and costs 65.\nBox type S88 holds product units 1 and costs 10000.\nBox type S89 holds product units 2 and costs 10000.\nBox type S90 holds product units 3 and costs 10000.\nBox type S91 holds product units 4 and costs 10000.\nBox type S92 holds product units 5 and costs 10000.\nBox type S93 holds product units 6 and costs 10000.\nBox type S94 holds product units 7 and costs 10000.\nBox type S95 holds product units 8 and costs 10000.\nBox type S96 holds product units 9 and costs 10000.\nBox type S97 holds product units 10 and costs 10000.\nBox type S98 holds product units 11 and costs 10000.\nBox type S99 holds product units 12 and costs 10000.\nBox type S100 holds product units 13 and costs 10000.\nBox type S101 holds product units 14 and costs 10000.\nBox type S102 holds product units 15 and costs 10000.\nBox type S103 holds product units 16 and costs 10000.\nBox type S104 holds product units 17 and costs 10000.\nBox type S105 holds product units 18 and costs 10000.\nBox type S106 holds product units 19 and costs 10000.\nBox type S107 holds product units 20 and costs 10000.\nThey will prefer a selection that covers all 20 units exactly once while minimizing the total cost.\n\nIf you want to tell me which box types to pick, just send it in this simple JSON shape:\n\n{\n \"solution\": [\"box_type_id\", ...]\n}\n\nThis is just a tiny sketch of the form I expect: \"solution\" holds a list of the chosen box type identifiers (one identifier per chosen box type). Think of each entry as the label of a prepacked box type you'd like to use.\n\nThe JSON above is only the expected shape, not the actual answer — fill the array with the exact identifiers from the instance when you reply. Remember: all identifiers must be used exactly as they appear in the input — no renaming and no new labels.\n\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 20, "num_sets": 107, "density": 0.1724299065420561, "sets": [ { "id": 1, "elements": [ 3, 6, 7, 8 ], "cost": 380 }, { "id": 2, "elements": [ 1, 2, 3, 6 ], "cost": 140 }, { "id": 3, "elements": [ 1, 2, 3, 4, 5 ], "cost": 370 }, { "id": 4, "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 210 }, { "id": 5, "elements": [ 6, 8, 9, 12 ], "cost": 196 }, { "id": 6, "elements": [ 14, 15, 16, 17, 20 ], "cost": 410 }, { "id": 7, "elements": [ 14, 15, 17, 18, 19, 20 ], "cost": 510 }, { "id": 8, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 180 }, { "id": 9, "elements": [ 10, 11, 12, 13, 14, 15 ], "cost": 18 }, { "id": 10, "elements": [ 8, 9, 10, 11 ], "cost": 236 }, { "id": 11, "elements": [ 16, 17, 18, 20 ], "cost": 340 }, { "id": 12, "elements": [ 18, 19, 20 ], "cost": 264 }, { "id": 13, "elements": [ 3, 4, 6, 8 ], "cost": 312 }, { "id": 14, "elements": [ 17, 18, 19 ], "cost": 108 }, { "id": 15, "elements": [ 12, 15 ], "cost": 58 }, { "id": 16, "elements": [ 10, 15, 16 ], "cost": 147 }, { "id": 17, "elements": [ 6, 7, 9, 10, 13 ], "cost": 285 }, { "id": 18, "elements": [ 1, 2, 4 ], "cost": 255 }, { "id": 19, "elements": [ 10, 11, 12, 13 ], "cost": 292 }, { "id": 20, "elements": [ 6, 7, 8, 9, 10, 11 ], "cost": 450 }, { "id": 21, "elements": [ 3, 7 ], "cost": 174 }, { "id": 22, "elements": [ 1, 2, 3, 4, 6 ], "cost": 460 }, { "id": 23, "elements": [ 12, 14, 15, 17 ], "cost": 232 }, { "id": 24, "elements": [ 6, 8, 9, 11 ], "cost": 104 }, { "id": 25, "elements": [ 3, 4, 5, 8 ], "cost": 24 }, { "id": 26, "elements": [ 4, 6, 7, 8, 9 ], "cost": 360 }, { "id": 27, "elements": [ 10, 12, 13, 14 ], "cost": 208 }, { "id": 28, "elements": [ 2, 4, 6, 7, 8 ], "cost": 100 }, { "id": 29, "elements": [ 13, 14, 15, 17, 18, 20 ], "cost": 516 }, { "id": 30, "elements": [ 8, 11, 12, 13 ], "cost": 292 }, { "id": 31, "elements": [ 16, 17, 18, 19, 20 ], "cost": 90 }, { "id": 32, "elements": [ 16, 17, 18 ], "cost": 273 }, { "id": 33, "elements": [ 10, 13, 14, 15 ], "cost": 368 }, { "id": 34, "elements": [ 13, 14, 15 ], "cost": 201 }, { "id": 35, "elements": [ 6, 8, 9, 10, 13 ], "cost": 285 }, { "id": 36, "elements": [ 4, 5, 6 ], "cost": 84 }, { "id": 37, "elements": [ 7, 10 ], "cost": 198 }, { "id": 38, "elements": [ 2, 3, 4 ], "cost": 240 }, { "id": 39, "elements": [ 4, 6, 8, 9 ], "cost": 192 }, { "id": 40, "elements": [ 13, 16, 17, 18, 19, 20 ], "cost": 540 }, { "id": 41, "elements": [ 8, 9 ], "cost": 84 }, { "id": 42, "elements": [ 4, 6, 7, 8, 12 ], "cost": 500 }, { "id": 43, "elements": [ 2, 3, 4, 6 ], "cost": 320 }, { "id": 44, "elements": [ 1, 2, 3, 5 ], "cost": 400 }, { "id": 45, "elements": [ 10, 11, 12, 13, 15 ], "cost": 20 }, { "id": 46, "elements": [ 4, 7, 9, 10, 11, 12 ], "cost": 168 }, { "id": 47, "elements": [ 15, 16, 17, 18, 19, 20 ], "cost": 384 }, { "id": 48, "elements": [ 3, 6, 7 ], "cost": 171 }, { "id": 49, "elements": [ 16, 18, 19 ], "cost": 273 }, { "id": 50, "elements": [ 17, 20 ], "cost": 96 }, { "id": 51, "elements": [ 7, 8, 9, 11, 13, 15 ], "cost": 570 }, { "id": 52, "elements": [ 9, 11, 12, 13 ], "cost": 200 }, { "id": 53, "elements": [ 7, 8, 9, 10, 11 ], "cost": 185 }, { "id": 54, "elements": [ 12, 13, 15, 18, 19 ], "cost": 5 }, { "id": 55, "elements": [ 3, 7, 9 ], "cost": 156 }, { "id": 56, "elements": [ 1, 4, 5, 6 ], "cost": 392 }, { "id": 57, "elements": [ 8, 10, 12 ], "cost": 273 }, { "id": 58, "elements": [ 13, 16, 17, 18 ], "cost": 68 }, { "id": 59, "elements": [ 3, 4, 5, 7 ], "cost": 104 }, { "id": 60, "elements": [ 1, 2, 3, 4, 7 ], "cost": 20 }, { "id": 61, "elements": [ 16, 18, 19, 20 ], "cost": 216 }, { "id": 62, "elements": [ 13, 15, 16, 20 ], "cost": 40 }, { "id": 63, "elements": [ 12, 14 ], "cost": 72 }, { "id": 64, "elements": [ 1, 2, 5 ], "cost": 264 }, { "id": 65, "elements": [ 4, 6, 7, 9, 11 ], "cost": 285 }, { "id": 66, "elements": [ 14, 16, 17, 18, 19, 20 ], "cost": 474 }, { "id": 67, "elements": [ 4, 6 ], "cost": 104 }, { "id": 68, "elements": [ 6, 8 ], "cost": 64 }, { "id": 69, "elements": [ 9, 11, 12, 13, 15 ], "cost": 90 }, { "id": 70, "elements": [ 5, 7, 8, 11 ], "cost": 284 }, { "id": 71, "elements": [ 3, 5, 7 ], "cost": 279 }, { "id": 72, "elements": [ 6, 7, 8, 9, 11, 12 ], "cost": 414 }, { "id": 73, "elements": [ 1, 4, 5 ], "cost": 27 }, { "id": 74, "elements": [ 11, 13, 14 ], "cost": 144 }, { "id": 75, "elements": [ 7, 8, 9, 10, 12 ], "cost": 380 }, { "id": 76, "elements": [ 14, 15, 17, 19 ], "cost": 356 }, { "id": 77, "elements": [ 1, 2 ], "cost": 114 }, { "id": 78, "elements": [ 1, 3, 4 ], "cost": 96 }, { "id": 79, "elements": [ 6, 7 ], "cost": 18 }, { "id": 80, "elements": [ 7, 8, 10, 13, 15 ], "cost": 120 }, { "id": 81, "elements": [ 5, 6, 9 ], "cost": 249 }, { "id": 82, "elements": [ 7, 9, 10, 13, 14 ], "cost": 475 }, { "id": 83, "elements": [ 6, 12 ], "cost": 106 }, { "id": 84, "elements": [ 3, 4, 5, 6 ], "cost": 12 }, { "id": 85, "elements": [ 11, 13 ], "cost": 190 }, { "id": 86, "elements": [ 1, 3, 4, 10 ], "cost": 120 }, { "id": 87, "elements": [ 15, 16, 18, 19, 20 ], "cost": 65 }, { "id": 88, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 89, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 90, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 91, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 92, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 93, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 94, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 95, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 96, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 97, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 98, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 20 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0028_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0028_bag_stats.png" }, "solution": [ 14, 53, 62, 63, 77, 84 ], "obj": 531.0, "instance_variant": { "num_elements": 20, "num_sets": 107, "sets": [ { "id": "S1", "elements": [ 3, 6, 7, 8 ], "cost": 380 }, { "id": "S2", "elements": [ 1, 2, 3, 6 ], "cost": 140 }, { "id": "S3", "elements": [ 1, 2, 3, 4, 5 ], "cost": 370 }, { "id": "S4", "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 210 }, { "id": "S5", "elements": [ 6, 8, 9, 12 ], "cost": 196 }, { "id": "S6", "elements": [ 14, 15, 16, 17, 20 ], "cost": 410 }, { "id": "S7", "elements": [ 14, 15, 17, 18, 19, 20 ], "cost": 510 }, { "id": "S8", "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 180 }, { "id": "S9", "elements": [ 10, 11, 12, 13, 14, 15 ], "cost": 18 }, { "id": "S10", "elements": [ 8, 9, 10, 11 ], "cost": 236 }, { "id": "S11", "elements": [ 16, 17, 18, 20 ], "cost": 340 }, { "id": "S12", "elements": [ 18, 19, 20 ], "cost": 264 }, { "id": "S13", "elements": [ 3, 4, 6, 8 ], "cost": 312 }, { "id": "S14", "elements": [ 17, 18, 19 ], "cost": 108 }, { "id": "S15", "elements": [ 12, 15 ], "cost": 58 }, { "id": "S16", "elements": [ 10, 15, 16 ], "cost": 147 }, { "id": "S17", "elements": [ 6, 7, 9, 10, 13 ], "cost": 285 }, { "id": "S18", "elements": [ 1, 2, 4 ], "cost": 255 }, { "id": "S19", "elements": [ 10, 11, 12, 13 ], "cost": 292 }, { "id": "S20", "elements": [ 6, 7, 8, 9, 10, 11 ], "cost": 450 }, { "id": "S21", "elements": [ 3, 7 ], "cost": 174 }, { "id": "S22", "elements": [ 1, 2, 3, 4, 6 ], "cost": 460 }, { "id": "S23", "elements": [ 12, 14, 15, 17 ], "cost": 232 }, { "id": "S24", "elements": [ 6, 8, 9, 11 ], "cost": 104 }, { "id": "S25", "elements": [ 3, 4, 5, 8 ], "cost": 24 }, { "id": "S26", "elements": [ 4, 6, 7, 8, 9 ], "cost": 360 }, { "id": "S27", "elements": [ 10, 12, 13, 14 ], "cost": 208 }, { "id": "S28", "elements": [ 2, 4, 6, 7, 8 ], "cost": 100 }, { "id": "S29", "elements": [ 13, 14, 15, 17, 18, 20 ], "cost": 516 }, { "id": "S30", "elements": [ 8, 11, 12, 13 ], "cost": 292 }, { "id": "S31", "elements": [ 16, 17, 18, 19, 20 ], "cost": 90 }, { "id": "S32", "elements": [ 16, 17, 18 ], "cost": 273 }, { "id": "S33", "elements": [ 10, 13, 14, 15 ], "cost": 368 }, { "id": "S34", "elements": [ 13, 14, 15 ], "cost": 201 }, { "id": "S35", "elements": [ 6, 8, 9, 10, 13 ], "cost": 285 }, { "id": "S36", "elements": [ 4, 5, 6 ], "cost": 84 }, { "id": "S37", "elements": [ 7, 10 ], "cost": 198 }, { "id": "S38", "elements": [ 2, 3, 4 ], "cost": 240 }, { "id": "S39", "elements": [ 4, 6, 8, 9 ], "cost": 192 }, { "id": "S40", "elements": [ 13, 16, 17, 18, 19, 20 ], "cost": 540 }, { "id": "S41", "elements": [ 8, 9 ], "cost": 84 }, { "id": "S42", "elements": [ 4, 6, 7, 8, 12 ], "cost": 500 }, { "id": "S43", "elements": [ 2, 3, 4, 6 ], "cost": 320 }, { "id": "S44", "elements": [ 1, 2, 3, 5 ], "cost": 400 }, { "id": "S45", "elements": [ 10, 11, 12, 13, 15 ], "cost": 20 }, { "id": "S46", "elements": [ 4, 7, 9, 10, 11, 12 ], "cost": 168 }, { "id": "S47", "elements": [ 15, 16, 17, 18, 19, 20 ], "cost": 384 }, { "id": "S48", "elements": [ 3, 6, 7 ], "cost": 171 }, { "id": "S49", "elements": [ 16, 18, 19 ], "cost": 273 }, { "id": "S50", "elements": [ 17, 20 ], "cost": 96 }, { "id": "S51", "elements": [ 7, 8, 9, 11, 13, 15 ], "cost": 570 }, { "id": "S52", "elements": [ 9, 11, 12, 13 ], "cost": 200 }, { "id": "S53", "elements": [ 7, 8, 9, 10, 11 ], "cost": 185 }, { "id": "S54", "elements": [ 12, 13, 15, 18, 19 ], "cost": 5 }, { "id": "S55", "elements": [ 3, 7, 9 ], "cost": 156 }, { "id": "S56", "elements": [ 1, 4, 5, 6 ], "cost": 392 }, { "id": "S57", "elements": [ 8, 10, 12 ], "cost": 273 }, { "id": "S58", "elements": [ 13, 16, 17, 18 ], "cost": 68 }, { "id": "S59", "elements": [ 3, 4, 5, 7 ], "cost": 104 }, { "id": "S60", "elements": [ 1, 2, 3, 4, 7 ], "cost": 20 }, { "id": "S61", "elements": [ 16, 18, 19, 20 ], "cost": 216 }, { "id": "S62", "elements": [ 13, 15, 16, 20 ], "cost": 40 }, { "id": "S63", "elements": [ 12, 14 ], "cost": 72 }, { "id": "S64", "elements": [ 1, 2, 5 ], "cost": 264 }, { "id": "S65", "elements": [ 4, 6, 7, 9, 11 ], "cost": 285 }, { "id": "S66", "elements": [ 14, 16, 17, 18, 19, 20 ], "cost": 474 }, { "id": "S67", "elements": [ 4, 6 ], "cost": 104 }, { "id": "S68", "elements": [ 6, 8 ], "cost": 64 }, { "id": "S69", "elements": [ 9, 11, 12, 13, 15 ], "cost": 90 }, { "id": "S70", "elements": [ 5, 7, 8, 11 ], "cost": 284 }, { "id": "S71", "elements": [ 3, 5, 7 ], "cost": 279 }, { "id": "S72", "elements": [ 6, 7, 8, 9, 11, 12 ], "cost": 414 }, { "id": "S73", "elements": [ 1, 4, 5 ], "cost": 27 }, { "id": "S74", "elements": [ 11, 13, 14 ], "cost": 144 }, { "id": "S75", "elements": [ 7, 8, 9, 10, 12 ], "cost": 380 }, { "id": "S76", "elements": [ 14, 15, 17, 19 ], "cost": 356 }, { "id": "S77", "elements": [ 1, 2 ], "cost": 114 }, { "id": "S78", "elements": [ 1, 3, 4 ], "cost": 96 }, { "id": "S79", "elements": [ 6, 7 ], "cost": 18 }, { "id": "S80", "elements": [ 7, 8, 10, 13, 15 ], "cost": 120 }, { "id": "S81", "elements": [ 5, 6, 9 ], "cost": 249 }, { "id": "S82", "elements": [ 7, 9, 10, 13, 14 ], "cost": 475 }, { "id": "S83", "elements": [ 6, 12 ], "cost": 106 }, { "id": "S84", "elements": [ 3, 4, 5, 6 ], "cost": 12 }, { "id": "S85", "elements": [ 11, 13 ], "cost": 190 }, { "id": "S86", "elements": [ 1, 3, 4, 10 ], "cost": 120 }, { "id": "S87", "elements": [ 15, 16, 18, 19, 20 ], "cost": 65 }, { "id": "S88", "elements": [ 1 ], "cost": 10000 }, { "id": "S89", "elements": [ 2 ], "cost": 10000 }, { "id": "S90", "elements": [ 3 ], "cost": 10000 }, { "id": "S91", "elements": [ 4 ], "cost": 10000 }, { "id": "S92", "elements": [ 5 ], "cost": 10000 }, { "id": "S93", "elements": [ 6 ], "cost": 10000 }, { "id": "S94", "elements": [ 7 ], "cost": 10000 }, { "id": "S95", "elements": [ 8 ], "cost": 10000 }, { "id": "S96", "elements": [ 9 ], "cost": 10000 }, { "id": "S97", "elements": [ 10 ], "cost": 10000 }, { "id": "S98", "elements": [ 11 ], "cost": 10000 }, { "id": "S99", "elements": [ 12 ], "cost": 10000 }, { "id": "S100", "elements": [ 13 ], "cost": 10000 }, { "id": "S101", "elements": [ 14 ], "cost": 10000 }, { "id": "S102", "elements": [ 15 ], "cost": 10000 }, { "id": "S103", "elements": [ 16 ], "cost": 10000 }, { "id": "S104", "elements": [ 17 ], "cost": 10000 }, { "id": "S105", "elements": [ 18 ], "cost": 10000 }, { "id": "S106", "elements": [ 19 ], "cost": 10000 }, { "id": "S107", "elements": [ 20 ], "cost": 10000 } ] }, "solution_variant": [ "S14", "S53", "S62", "S63", "S77", "S84" ], "context_index": 29, "input_format": "nl", "input_index_base": 1 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "Imagine needing to buy a set of compilation albums to cover a playlist where each song must be available from just one licensed compilation — that means no missing songs and no duplicates among the purchased albums. The winning plan is simply the one with the lowest total license cost, which is the sum of the fees for the selected albums. The full details on albums, which songs they include, and their prices are provided below.\n\n{\n \"num_songs\": 23,\n \"num_albums\": 111,\n \"sets\": [\n {\n \"album_id\": \"S1\",\n \"album_songs\": [\n \"E\",\n \"G\",\n \"J\"\n ],\n \"license_fee\": 45\n },\n {\n \"album_id\": \"S2\",\n \"album_songs\": [\n \"C\",\n \"D\",\n \"E\",\n \"G\",\n \"H\",\n \"I\"\n ],\n \"license_fee\": 168\n },\n {\n \"album_id\": \"S3\",\n \"album_songs\": [\n \"O\",\n \"P\",\n \"Q\",\n \"S\"\n ],\n \"license_fee\": 4\n },\n {\n \"album_id\": \"S4\",\n \"album_songs\": [\n \"Q\",\n \"R\",\n \"S\"\n ],\n \"license_fee\": 288\n },\n {\n \"album_id\": \"S5\",\n \"album_songs\": [\n \"Q\",\n \"T\",\n \"V\"\n ],\n \"license_fee\": 246\n },\n {\n \"album_id\": \"S6\",\n \"album_songs\": [\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\"\n ],\n \"license_fee\": 235\n },\n {\n \"album_id\": \"S7\",\n \"album_songs\": [\n \"O\",\n \"R\",\n \"S\"\n ],\n \"license_fee\": 36\n },\n {\n \"album_id\": \"S8\",\n \"album_songs\": [\n \"E\",\n \"H\",\n \"K\",\n \"L\"\n ],\n \"license_fee\": 292\n },\n {\n \"album_id\": \"S9\",\n \"album_songs\": [\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"L\",\n \"M\"\n ],\n \"license_fee\": 48\n },\n {\n \"album_id\": \"S10\",\n \"album_songs\": [\n \"K\",\n \"L\",\n \"N\",\n \"O\",\n \"T\"\n ],\n \"license_fee\": 260\n },\n {\n \"album_id\": \"S11\",\n \"album_songs\": [\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\"\n ],\n \"license_fee\": 336\n },\n {\n \"album_id\": \"S12\",\n \"album_songs\": [\n \"P\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\"\n ],\n \"license_fee\": 492\n },\n {\n \"album_id\": \"S13\",\n \"album_songs\": [\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\"\n ],\n \"license_fee\": 300\n },\n {\n \"album_id\": \"S14\",\n \"album_songs\": [\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\"\n ],\n \"license_fee\": 415\n },\n {\n \"album_id\": \"S15\",\n \"album_songs\": [\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"R\",\n \"S\"\n ],\n \"license_fee\": 222\n },\n {\n \"album_id\": \"S16\",\n \"album_songs\": [\n \"F\",\n \"J\",\n \"K\",\n \"M\"\n ],\n \"license_fee\": 52\n },\n {\n \"album_id\": \"S17\",\n \"album_songs\": [\n \"O\",\n \"P\",\n \"R\",\n \"S\"\n ],\n \"license_fee\": 312\n },\n {\n \"album_id\": \"S18\",\n \"album_songs\": [\n \"O\",\n \"P\",\n \"R\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"license_fee\": 78\n },\n {\n \"album_id\": \"S19\",\n \"album_songs\": [\n \"M\",\n \"O\",\n \"Q\",\n \"R\"\n ],\n \"license_fee\": 256\n },\n {\n \"album_id\": \"S20\",\n \"album_songs\": [\n \"L\",\n \"M\",\n \"P\",\n \"Q\",\n \"R\",\n \"T\"\n ],\n \"license_fee\": 216\n },\n {\n \"album_id\": \"S21\",\n \"album_songs\": [\n \"I\",\n \"O\",\n \"P\",\n \"Q\",\n \"R\"\n ],\n \"license_fee\": 250\n },\n {\n \"album_id\": \"S22\",\n \"album_songs\": [\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"U\",\n \"V\"\n ],\n \"license_fee\": 234\n },\n {\n \"album_id\": \"S23\",\n \"album_songs\": [\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"Q\",\n \"S\"\n ],\n \"license_fee\": 132\n },\n {\n \"album_id\": \"S24\",\n \"album_songs\": [\n \"S\",\n \"U\",\n \"V\",\n \"W\"\n ],\n \"license_fee\": 308\n },\n {\n \"album_id\": \"S25\",\n \"album_songs\": [\n \"E\",\n \"G\",\n \"H\"\n ],\n \"license_fee\": 243\n },\n {\n \"album_id\": \"S26\",\n \"album_songs\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\"\n ],\n \"license_fee\": 144\n },\n {\n \"album_id\": \"S27\",\n \"album_songs\": [\n \"R\",\n \"T\",\n \"U\",\n \"V\",\n \"W\"\n ],\n \"license_fee\": 435\n },\n {\n \"album_id\": \"S28\",\n \"album_songs\": [\n \"S\",\n \"W\"\n ],\n \"license_fee\": 86\n },\n {\n \"album_id\": \"S29\",\n \"album_songs\": [\n \"A\",\n \"B\",\n \"H\"\n ],\n \"license_fee\": 135\n },\n {\n \"album_id\": \"S30\",\n \"album_songs\": [\n \"O\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"license_fee\": 588\n },\n {\n \"album_id\": \"S31\",\n \"album_songs\": [\n \"E\",\n \"F\",\n \"G\",\n \"H\",\n \"J\",\n \"K\"\n ],\n \"license_fee\": 510\n },\n {\n \"album_id\": \"S32\",\n \"album_songs\": [\n \"B\",\n \"E\",\n \"F\"\n ],\n \"license_fee\": 84\n },\n {\n \"album_id\": \"S33\",\n \"album_songs\": [\n \"F\",\n \"L\"\n ],\n \"license_fee\": 104\n },\n {\n \"album_id\": \"S34\",\n \"album_songs\": [\n \"J\",\n \"L\"\n ],\n \"license_fee\": 170\n },\n {\n \"album_id\": \"S35\",\n \"album_songs\": [\n \"A\",\n \"B\",\n \"C\",\n \"E\",\n \"F\"\n ],\n \"license_fee\": 100\n },\n {\n \"album_id\": \"S36\",\n \"album_songs\": [\n \"I\",\n \"M\",\n \"N\",\n \"P\",\n \"Q\"\n ],\n \"license_fee\": 50\n },\n {\n \"album_id\": \"S37\",\n \"album_songs\": [\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"L\"\n ],\n \"license_fee\": 48\n },\n {\n \"album_id\": \"S38\",\n \"album_songs\": [\n \"P\",\n \"Q\",\n \"U\",\n \"V\",\n \"W\"\n ],\n \"license_fee\": 350\n },\n {\n \"album_id\": \"S39\",\n \"album_songs\": [\n \"C\",\n \"D\",\n \"F\",\n \"G\"\n ],\n \"license_fee\": 256\n },\n {\n \"album_id\": \"S40\",\n \"album_songs\": [\n \"B\",\n \"F\",\n \"G\"\n ],\n \"license_fee\": 9\n },\n {\n \"album_id\": \"S41\",\n \"album_songs\": [\n \"D\",\n \"F\",\n \"I\",\n \"K\",\n \"L\",\n \"M\",\n \"O\"\n ],\n \"license_fee\": 84\n },\n {\n \"album_id\": \"S42\",\n \"album_songs\": [\n \"P\",\n \"R\",\n \"S\"\n ],\n \"license_fee\": 189\n },\n {\n \"album_id\": \"S43\",\n \"album_songs\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"H\"\n ],\n \"license_fee\": 385\n },\n {\n \"album_id\": \"S44\",\n \"album_songs\": [\n \"A\",\n \"B\",\n \"E\"\n ],\n \"license_fee\": 246\n },\n {\n \"album_id\": \"S45\",\n \"album_songs\": [\n \"R\",\n \"T\",\n \"W\"\n ],\n \"license_fee\": 30\n },\n {\n \"album_id\": \"S46\",\n \"album_songs\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"F\",\n \"G\"\n ],\n \"license_fee\": 300\n },\n {\n \"album_id\": \"S47\",\n \"album_songs\": [\n \"O\",\n \"P\",\n \"R\"\n ],\n \"license_fee\": 126\n },\n {\n \"album_id\": \"S48\",\n \"album_songs\": [\n \"Q\",\n \"S\",\n \"U\"\n ],\n \"license_fee\": 54\n },\n {\n \"album_id\": \"S49\",\n \"album_songs\": [\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"N\"\n ],\n \"license_fee\": 455\n },\n {\n \"album_id\": \"S50\",\n \"album_songs\": [\n \"O\",\n \"P\",\n \"Q\",\n \"R\",\n \"T\"\n ],\n \"license_fee\": 385\n },\n {\n \"album_id\": \"S51\",\n \"album_songs\": [\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"O\"\n ],\n \"license_fee\": 228\n },\n {\n \"album_id\": \"S52\",\n \"album_songs\": [\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"H\",\n \"I\"\n ],\n \"license_fee\": 312\n },\n {\n \"album_id\": \"S53\",\n \"album_songs\": [\n \"C\",\n \"E\",\n \"F\",\n \"H\"\n ],\n \"license_fee\": 152\n },\n {\n \"album_id\": \"S54\",\n \"album_songs\": [\n \"A\",\n \"B\",\n \"E\",\n \"F\",\n \"H\",\n \"I\",\n \"K\"\n ],\n \"license_fee\": 371\n },\n {\n \"album_id\": \"S55\",\n \"album_songs\": [\n \"P\",\n \"S\",\n \"U\",\n \"W\"\n ],\n \"license_fee\": 252\n },\n {\n \"album_id\": \"S56\",\n \"album_songs\": [\n \"A\",\n \"C\",\n \"E\"\n ],\n \"license_fee\": 279\n },\n {\n \"album_id\": \"S57\",\n \"album_songs\": [\n \"I\",\n \"M\",\n \"O\"\n ],\n \"license_fee\": 57\n },\n {\n \"album_id\": \"S58\",\n \"album_songs\": [\n \"A\",\n \"D\",\n \"E\",\n \"F\",\n \"G\"\n ],\n \"license_fee\": 100\n },\n {\n \"album_id\": \"S59\",\n \"album_songs\": [\n \"E\",\n \"I\",\n \"J\",\n \"M\",\n \"U\"\n ],\n \"license_fee\": 465\n },\n {\n \"album_id\": \"S60\",\n \"album_songs\": [\n \"E\",\n \"F\",\n \"G\",\n \"I\",\n \"K\",\n \"L\"\n ],\n \"license_fee\": 192\n },\n {\n \"album_id\": \"S61\",\n \"album_songs\": [\n \"A\",\n \"B\",\n \"D\",\n \"E\"\n ],\n \"license_fee\": 400\n },\n {\n \"album_id\": \"S62\",\n \"album_songs\": [\n \"A\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"H\",\n \"I\"\n ],\n \"license_fee\": 574\n },\n {\n \"album_id\": \"S63\",\n \"album_songs\": [\n \"Q\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\"\n ],\n \"license_fee\": 102\n },\n {\n \"album_id\": \"S64\",\n \"album_songs\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\"\n ],\n \"license_fee\": 273\n },\n {\n \"album_id\": \"S65\",\n \"album_songs\": [\n \"H\",\n \"L\",\n \"N\"\n ],\n \"license_fee\": 294\n },\n {\n \"album_id\": \"S66\",\n \"album_songs\": [\n \"A\",\n \"C\",\n \"F\",\n \"G\",\n \"H\"\n ],\n \"license_fee\": 145\n },\n {\n \"album_id\": \"S67\",\n \"album_songs\": [\n \"K\",\n \"O\",\n \"P\",\n \"R\",\n \"U\",\n \"V\"\n ],\n \"license_fee\": 450\n },\n {\n \"album_id\": \"S68\",\n \"album_songs\": [\n \"E\",\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\"\n ],\n \"license_fee\": 448\n },\n {\n \"album_id\": \"S69\",\n \"album_songs\": [\n \"N\",\n \"P\",\n \"S\",\n \"T\",\n \"U\",\n \"V\"\n ],\n \"license_fee\": 6\n },\n {\n \"album_id\": \"S70\",\n \"album_songs\": [\n \"P\",\n \"R\",\n \"S\",\n \"T\",\n \"V\",\n \"W\"\n ],\n \"license_fee\": 420\n },\n {\n \"album_id\": \"S71\",\n \"album_songs\": [\n \"I\",\n \"J\",\n \"L\",\n \"N\"\n ],\n \"license_fee\": 120\n },\n {\n \"album_id\": \"S72\",\n \"album_songs\": [\n \"F\",\n \"H\",\n \"I\",\n \"L\"\n ],\n \"license_fee\": 228\n },\n {\n \"album_id\": \"S73\",\n \"album_songs\": [\n \"G\",\n \"J\",\n \"K\"\n ],\n \"license_fee\": 186\n },\n {\n \"album_id\": \"S74\",\n \"album_songs\": [\n \"Q\",\n \"R\",\n \"T\",\n \"U\",\n \"V\",\n \"W\"\n ],\n \"license_fee\": 384\n },\n {\n \"album_id\": \"S75\",\n \"album_songs\": [\n \"A\",\n \"D\",\n \"E\",\n \"F\",\n \"H\",\n \"I\"\n ],\n \"license_fee\": 198\n },\n {\n \"album_id\": \"S76\",\n \"album_songs\": [\n \"H\",\n \"J\",\n \"L\"\n ],\n \"license_fee\": 177\n },\n {\n \"album_id\": \"S77\",\n \"album_songs\": [\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"M\"\n ],\n \"license_fee\": 264\n },\n {\n \"album_id\": \"S78\",\n \"album_songs\": [\n \"E\",\n \"H\",\n \"J\",\n \"L\"\n ],\n \"license_fee\": 256\n },\n {\n \"album_id\": \"S79\",\n \"album_songs\": [\n \"H\",\n \"I\",\n \"L\",\n \"M\",\n \"T\"\n ],\n \"license_fee\": 140\n },\n {\n \"album_id\": \"S80\",\n \"album_songs\": [\n \"D\",\n \"E\",\n \"F\"\n ],\n \"license_fee\": 138\n },\n {\n \"album_id\": \"S81\",\n \"album_songs\": [\n \"G\",\n \"H\",\n \"J\"\n ],\n \"license_fee\": 54\n },\n {\n \"album_id\": \"S82\",\n \"album_songs\": [\n \"C\",\n \"D\",\n \"G\",\n \"H\",\n \"J\"\n ],\n \"license_fee\": 445\n },\n {\n \"album_id\": \"S83\",\n \"album_songs\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\"\n ],\n \"license_fee\": 85\n },\n {\n \"album_id\": \"S84\",\n \"album_songs\": [\n \"C\",\n \"D\"\n ],\n \"license_fee\": 198\n },\n {\n \"album_id\": \"S85\",\n \"album_songs\": [\n \"B\",\n \"C\",\n \"D\"\n ],\n \"license_fee\": 156\n },\n {\n \"album_id\": \"S86\",\n \"album_songs\": [\n \"A\",\n \"E\",\n \"G\",\n \"M\"\n ],\n \"license_fee\": 68\n },\n {\n \"album_id\": \"S87\",\n \"album_songs\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"G\"\n ],\n \"license_fee\": 276\n },\n {\n \"album_id\": \"S88\",\n \"album_songs\": [\n \"B\",\n \"C\"\n ],\n \"license_fee\": 150\n },\n {\n \"album_id\": \"S89\",\n \"album_songs\": [\n \"A\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S90\",\n \"album_songs\": [\n \"B\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S91\",\n \"album_songs\": [\n \"C\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S92\",\n \"album_songs\": [\n \"D\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S93\",\n \"album_songs\": [\n \"E\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S94\",\n \"album_songs\": [\n \"F\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S95\",\n \"album_songs\": [\n \"G\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S96\",\n \"album_songs\": [\n \"H\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S97\",\n \"album_songs\": [\n \"I\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S98\",\n \"album_songs\": [\n \"J\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S99\",\n \"album_songs\": [\n \"K\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S100\",\n \"album_songs\": [\n \"L\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S101\",\n \"album_songs\": [\n \"M\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S102\",\n \"album_songs\": [\n \"N\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S103\",\n \"album_songs\": [\n \"O\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S104\",\n \"album_songs\": [\n \"P\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S105\",\n \"album_songs\": [\n \"Q\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S106\",\n \"album_songs\": [\n \"R\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S107\",\n \"album_songs\": [\n \"S\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S108\",\n \"album_songs\": [\n \"T\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S109\",\n \"album_songs\": [\n \"U\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S110\",\n \"album_songs\": [\n \"V\"\n ],\n \"license_fee\": 10000\n },\n {\n \"album_id\": \"S111\",\n \"album_songs\": [\n \"W\"\n ],\n \"license_fee\": 10000\n }\n ]\n}\n\nOh, and when you send back the chosen albums, just stick to a tiny JSON snippet like this so it’s easy to parse:\n\n{\n \"solution\": [\"album_id\"]\n}\n\nThink of the \"solution\" array as the shopping list — drop in the album identifiers you want to buy so every song is covered exactly once. This JSON is just the shape I expect, not the final answer itself.\n\nPlease use the exact identifiers from the instance input — no renaming and no invented labels. \n\"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 23, "num_sets": 111, "density": 0.16842929886408148, "sets": [ { "id": 1, "elements": [ 5, 7, 10 ], "cost": 45 }, { "id": 2, "elements": [ 3, 4, 5, 7, 8, 9 ], "cost": 168 }, { "id": 3, "elements": [ 15, 16, 17, 19 ], "cost": 4 }, { "id": 4, "elements": [ 17, 18, 19 ], "cost": 288 }, { "id": 5, "elements": [ 17, 20, 22 ], "cost": 246 }, { "id": 6, "elements": [ 19, 20, 21, 22, 23 ], "cost": 235 }, { "id": 7, "elements": [ 15, 18, 19 ], "cost": 36 }, { "id": 8, "elements": [ 5, 8, 11, 12 ], "cost": 292 }, { "id": 9, "elements": [ 7, 8, 9, 10, 12, 13 ], "cost": 48 }, { "id": 10, "elements": [ 11, 12, 14, 15, 20 ], "cost": 260 }, { "id": 11, "elements": [ 18, 19, 20, 21, 22, 23 ], "cost": 336 }, { "id": 12, "elements": [ 16, 19, 20, 21, 22, 23 ], "cost": 492 }, { "id": 13, "elements": [ 17, 18, 19, 20, 21, 22 ], "cost": 300 }, { "id": 14, "elements": [ 16, 17, 18, 19, 20 ], "cost": 415 }, { "id": 15, "elements": [ 12, 13, 14, 15, 18, 19 ], "cost": 222 }, { "id": 16, "elements": [ 6, 10, 11, 13 ], "cost": 52 }, { "id": 17, "elements": [ 15, 16, 18, 19 ], "cost": 312 }, { "id": 18, "elements": [ 15, 16, 18, 19, 20, 21 ], "cost": 78 }, { "id": 19, "elements": [ 13, 15, 17, 18 ], "cost": 256 }, { "id": 20, "elements": [ 12, 13, 16, 17, 18, 20 ], "cost": 216 }, { "id": 21, "elements": [ 9, 15, 16, 17, 18 ], "cost": 250 }, { "id": 22, "elements": [ 16, 17, 18, 19, 21, 22 ], "cost": 234 }, { "id": 23, "elements": [ 12, 13, 14, 15, 17, 19 ], "cost": 132 }, { "id": 24, "elements": [ 19, 21, 22, 23 ], "cost": 308 }, { "id": 25, "elements": [ 5, 7, 8 ], "cost": 243 }, { "id": 26, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 144 }, { "id": 27, "elements": [ 18, 20, 21, 22, 23 ], "cost": 435 }, { "id": 28, "elements": [ 19, 23 ], "cost": 86 }, { "id": 29, "elements": [ 1, 2, 8 ], "cost": 135 }, { "id": 30, "elements": [ 15, 17, 18, 19, 20, 21 ], "cost": 588 }, { "id": 31, "elements": [ 5, 6, 7, 8, 10, 11 ], "cost": 510 }, { "id": 32, "elements": [ 2, 5, 6 ], "cost": 84 }, { "id": 33, "elements": [ 6, 12 ], "cost": 104 }, { "id": 34, "elements": [ 10, 12 ], "cost": 170 }, { "id": 35, "elements": [ 1, 2, 3, 5, 6 ], "cost": 100 }, { "id": 36, "elements": [ 9, 13, 14, 16, 17 ], "cost": 50 }, { "id": 37, "elements": [ 7, 8, 9, 10, 11, 12 ], "cost": 48 }, { "id": 38, "elements": [ 16, 17, 21, 22, 23 ], "cost": 350 }, { "id": 39, "elements": [ 3, 4, 6, 7 ], "cost": 256 }, { "id": 40, "elements": [ 2, 6, 7 ], "cost": 9 }, { "id": 41, "elements": [ 4, 6, 9, 11, 12, 13, 15 ], "cost": 84 }, { "id": 42, "elements": [ 16, 18, 19 ], "cost": 189 }, { "id": 43, "elements": [ 1, 2, 3, 4, 5, 6, 8 ], "cost": 385 }, { "id": 44, "elements": [ 1, 2, 5 ], "cost": 246 }, { "id": 45, "elements": [ 18, 20, 23 ], "cost": 30 }, { "id": 46, "elements": [ 1, 2, 3, 4, 6, 7 ], "cost": 300 }, { "id": 47, "elements": [ 15, 16, 18 ], "cost": 126 }, { "id": 48, "elements": [ 17, 19, 21 ], "cost": 54 }, { "id": 49, "elements": [ 9, 10, 11, 12, 14 ], "cost": 455 }, { "id": 50, "elements": [ 15, 16, 17, 18, 20 ], "cost": 385 }, { "id": 51, "elements": [ 9, 10, 11, 12, 13, 15 ], "cost": 228 }, { "id": 52, "elements": [ 3, 4, 5, 6, 8, 9 ], "cost": 312 }, { "id": 53, "elements": [ 3, 5, 6, 8 ], "cost": 152 }, { "id": 54, "elements": [ 1, 2, 5, 6, 8, 9, 11 ], "cost": 371 }, { "id": 55, "elements": [ 16, 19, 21, 23 ], "cost": 252 }, { "id": 56, "elements": [ 1, 3, 5 ], "cost": 279 }, { "id": 57, "elements": [ 9, 13, 15 ], "cost": 57 }, { "id": 58, "elements": [ 1, 4, 5, 6, 7 ], "cost": 100 }, { "id": 59, "elements": [ 5, 9, 10, 13, 21 ], "cost": 465 }, { "id": 60, "elements": [ 5, 6, 7, 9, 11, 12 ], "cost": 192 }, { "id": 61, "elements": [ 1, 2, 4, 5 ], "cost": 400 }, { "id": 62, "elements": [ 1, 3, 4, 5, 6, 8, 9 ], "cost": 574 }, { "id": 63, "elements": [ 17, 19, 20, 21, 22, 23 ], "cost": 102 }, { "id": 64, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 273 }, { "id": 65, "elements": [ 8, 12, 14 ], "cost": 294 }, { "id": 66, "elements": [ 1, 3, 6, 7, 8 ], "cost": 145 }, { "id": 67, "elements": [ 11, 15, 16, 18, 21, 22 ], "cost": 450 }, { "id": 68, "elements": [ 5, 6, 7, 8, 9, 10, 11 ], "cost": 448 }, { "id": 69, "elements": [ 14, 16, 19, 20, 21, 22 ], "cost": 6 }, { "id": 70, "elements": [ 16, 18, 19, 20, 22, 23 ], "cost": 420 }, { "id": 71, "elements": [ 9, 10, 12, 14 ], "cost": 120 }, { "id": 72, "elements": [ 6, 8, 9, 12 ], "cost": 228 }, { "id": 73, "elements": [ 7, 10, 11 ], "cost": 186 }, { "id": 74, "elements": [ 17, 18, 20, 21, 22, 23 ], "cost": 384 }, { "id": 75, "elements": [ 1, 4, 5, 6, 8, 9 ], "cost": 198 }, { "id": 76, "elements": [ 8, 10, 12 ], "cost": 177 }, { "id": 77, "elements": [ 7, 8, 9, 10, 11, 13 ], "cost": 264 }, { "id": 78, "elements": [ 5, 8, 10, 12 ], "cost": 256 }, { "id": 79, "elements": [ 8, 9, 12, 13, 20 ], "cost": 140 }, { "id": 80, "elements": [ 4, 5, 6 ], "cost": 138 }, { "id": 81, "elements": [ 7, 8, 10 ], "cost": 54 }, { "id": 82, "elements": [ 3, 4, 7, 8, 10 ], "cost": 445 }, { "id": 83, "elements": [ 1, 2, 3, 4, 5 ], "cost": 85 }, { "id": 84, "elements": [ 3, 4 ], "cost": 198 }, { "id": 85, "elements": [ 2, 3, 4 ], "cost": 156 }, { "id": 86, "elements": [ 1, 5, 7, 13 ], "cost": 68 }, { "id": 87, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 276 }, { "id": 88, "elements": [ 2, 3 ], "cost": 150 }, { "id": 89, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 90, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 91, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 92, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 93, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 94, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 95, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 96, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 97, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 98, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 23 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0029_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0029_bag_stats.png" }, "solution": [ 6, 16, 21, 65, 87 ], "obj": 1107.0, "instance_variant": { "num_elements": 23, "num_sets": 111, "sets": [ { "id": "S1", "elements": [ "E", "G", "J" ], "cost": 45 }, { "id": "S2", "elements": [ "C", "D", "E", "G", "H", "I" ], "cost": 168 }, { "id": "S3", "elements": [ "O", "P", "Q", "S" ], "cost": 4 }, { "id": "S4", "elements": [ "Q", "R", "S" ], "cost": 288 }, { "id": "S5", "elements": [ "Q", "T", "V" ], "cost": 246 }, { "id": "S6", "elements": [ "S", "T", "U", "V", "W" ], "cost": 235 }, { "id": "S7", "elements": [ "O", "R", "S" ], "cost": 36 }, { "id": "S8", "elements": [ "E", "H", "K", "L" ], "cost": 292 }, { "id": "S9", "elements": [ "G", "H", "I", "J", "L", "M" ], "cost": 48 }, { "id": "S10", "elements": [ "K", "L", "N", "O", "T" ], "cost": 260 }, { "id": "S11", "elements": [ "R", "S", "T", "U", "V", "W" ], "cost": 336 }, { "id": "S12", "elements": [ "P", "S", "T", "U", "V", "W" ], "cost": 492 }, { "id": "S13", "elements": [ "Q", "R", "S", "T", "U", "V" ], "cost": 300 }, { "id": "S14", "elements": [ "P", "Q", "R", "S", "T" ], "cost": 415 }, { "id": "S15", "elements": [ "L", "M", "N", "O", "R", "S" ], "cost": 222 }, { "id": "S16", "elements": [ "F", "J", "K", "M" ], "cost": 52 }, { "id": "S17", "elements": [ "O", "P", "R", "S" ], "cost": 312 }, { "id": "S18", "elements": [ "O", "P", "R", "S", "T", "U" ], "cost": 78 }, { "id": "S19", "elements": [ "M", "O", "Q", "R" ], "cost": 256 }, { "id": "S20", "elements": [ "L", "M", "P", "Q", "R", "T" ], "cost": 216 }, { "id": "S21", "elements": [ "I", "O", "P", "Q", "R" ], "cost": 250 }, { "id": "S22", "elements": [ "P", "Q", "R", "S", "U", "V" ], "cost": 234 }, { "id": "S23", "elements": [ "L", "M", "N", "O", "Q", "S" ], "cost": 132 }, { "id": "S24", "elements": [ "S", "U", "V", "W" ], "cost": 308 }, { "id": "S25", "elements": [ "E", "G", "H" ], "cost": 243 }, { "id": "S26", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 144 }, { "id": "S27", "elements": [ "R", "T", "U", "V", "W" ], "cost": 435 }, { "id": "S28", "elements": [ "S", "W" ], "cost": 86 }, { "id": "S29", "elements": [ "A", "B", "H" ], "cost": 135 }, { "id": "S30", "elements": [ "O", "Q", "R", "S", "T", "U" ], "cost": 588 }, { "id": "S31", "elements": [ "E", "F", "G", "H", "J", "K" ], "cost": 510 }, { "id": "S32", "elements": [ "B", "E", "F" ], "cost": 84 }, { "id": "S33", "elements": [ "F", "L" ], "cost": 104 }, { "id": "S34", "elements": [ "J", "L" ], "cost": 170 }, { "id": "S35", "elements": [ "A", "B", "C", "E", "F" ], "cost": 100 }, { "id": "S36", "elements": [ "I", "M", "N", "P", "Q" ], "cost": 50 }, { "id": "S37", "elements": [ "G", "H", "I", "J", "K", "L" ], "cost": 48 }, { "id": "S38", "elements": [ "P", "Q", "U", "V", "W" ], "cost": 350 }, { "id": "S39", "elements": [ "C", "D", "F", "G" ], "cost": 256 }, { "id": "S40", "elements": [ "B", "F", "G" ], "cost": 9 }, { "id": "S41", "elements": [ "D", "F", "I", "K", "L", "M", "O" ], "cost": 84 }, { "id": "S42", "elements": [ "P", "R", "S" ], "cost": 189 }, { "id": "S43", "elements": [ "A", "B", "C", "D", "E", "F", "H" ], "cost": 385 }, { "id": "S44", "elements": [ "A", "B", "E" ], "cost": 246 }, { "id": "S45", "elements": [ "R", "T", "W" ], "cost": 30 }, { "id": "S46", "elements": [ "A", "B", "C", "D", "F", "G" ], "cost": 300 }, { "id": "S47", "elements": [ "O", "P", "R" ], "cost": 126 }, { "id": "S48", "elements": [ "Q", "S", "U" ], "cost": 54 }, { "id": "S49", "elements": [ "I", "J", "K", "L", "N" ], "cost": 455 }, { "id": "S50", "elements": [ "O", "P", "Q", "R", "T" ], "cost": 385 }, { "id": "S51", "elements": [ "I", "J", "K", "L", "M", "O" ], "cost": 228 }, { "id": "S52", "elements": [ "C", "D", "E", "F", "H", "I" ], "cost": 312 }, { "id": "S53", "elements": [ "C", "E", "F", "H" ], "cost": 152 }, { "id": "S54", "elements": [ "A", "B", "E", "F", "H", "I", "K" ], "cost": 371 }, { "id": "S55", "elements": [ "P", "S", "U", "W" ], "cost": 252 }, { "id": "S56", "elements": [ "A", "C", "E" ], "cost": 279 }, { "id": "S57", "elements": [ "I", "M", "O" ], "cost": 57 }, { "id": "S58", "elements": [ "A", "D", "E", "F", "G" ], "cost": 100 }, { "id": "S59", "elements": [ "E", "I", "J", "M", "U" ], "cost": 465 }, { "id": "S60", "elements": [ "E", "F", "G", "I", "K", "L" ], "cost": 192 }, { "id": "S61", "elements": [ "A", "B", "D", "E" ], "cost": 400 }, { "id": "S62", "elements": [ "A", "C", "D", "E", "F", "H", "I" ], "cost": 574 }, { "id": "S63", "elements": [ "Q", "S", "T", "U", "V", "W" ], "cost": 102 }, { "id": "S64", "elements": [ "A", "B", "C", "D", "E", "F", "G" ], "cost": 273 }, { "id": "S65", "elements": [ "H", "L", "N" ], "cost": 294 }, { "id": "S66", "elements": [ "A", "C", "F", "G", "H" ], "cost": 145 }, { "id": "S67", "elements": [ "K", "O", "P", "R", "U", "V" ], "cost": 450 }, { "id": "S68", "elements": [ "E", "F", "G", "H", "I", "J", "K" ], "cost": 448 }, { "id": "S69", "elements": [ "N", "P", "S", "T", "U", "V" ], "cost": 6 }, { "id": "S70", "elements": [ "P", "R", "S", "T", "V", "W" ], "cost": 420 }, { "id": "S71", "elements": [ "I", "J", "L", "N" ], "cost": 120 }, { "id": "S72", "elements": [ "F", "H", "I", "L" ], "cost": 228 }, { "id": "S73", "elements": [ "G", "J", "K" ], "cost": 186 }, { "id": "S74", "elements": [ "Q", "R", "T", "U", "V", "W" ], "cost": 384 }, { "id": "S75", "elements": [ "A", "D", "E", "F", "H", "I" ], "cost": 198 }, { "id": "S76", "elements": [ "H", "J", "L" ], "cost": 177 }, { "id": "S77", "elements": [ "G", "H", "I", "J", "K", "M" ], "cost": 264 }, { "id": "S78", "elements": [ "E", "H", "J", "L" ], "cost": 256 }, { "id": "S79", "elements": [ "H", "I", "L", "M", "T" ], "cost": 140 }, { "id": "S80", "elements": [ "D", "E", "F" ], "cost": 138 }, { "id": "S81", "elements": [ "G", "H", "J" ], "cost": 54 }, { "id": "S82", "elements": [ "C", "D", "G", "H", "J" ], "cost": 445 }, { "id": "S83", "elements": [ "A", "B", "C", "D", "E" ], "cost": 85 }, { "id": "S84", "elements": [ "C", "D" ], "cost": 198 }, { "id": "S85", "elements": [ "B", "C", "D" ], "cost": 156 }, { "id": "S86", "elements": [ "A", "E", "G", "M" ], "cost": 68 }, { "id": "S87", "elements": [ "A", "B", "C", "D", "E", "G" ], "cost": 276 }, { "id": "S88", "elements": [ "B", "C" ], "cost": 150 }, { "id": "S89", "elements": [ "A" ], "cost": 10000 }, { "id": "S90", "elements": [ "B" ], "cost": 10000 }, { "id": "S91", "elements": [ "C" ], "cost": 10000 }, { "id": "S92", "elements": [ "D" ], "cost": 10000 }, { "id": "S93", "elements": [ "E" ], "cost": 10000 }, { "id": "S94", "elements": [ "F" ], "cost": 10000 }, { "id": "S95", "elements": [ "G" ], "cost": 10000 }, { "id": "S96", "elements": [ "H" ], "cost": 10000 }, { "id": "S97", "elements": [ "I" ], "cost": 10000 }, { "id": "S98", "elements": [ "J" ], "cost": 10000 }, { "id": "S99", "elements": [ "K" ], "cost": 10000 }, { "id": "S100", "elements": [ "L" ], "cost": 10000 }, { "id": "S101", "elements": [ "M" ], "cost": 10000 }, { "id": "S102", "elements": [ "N" ], "cost": 10000 }, { "id": "S103", "elements": [ "O" ], "cost": 10000 }, { "id": "S104", "elements": [ "P" ], "cost": 10000 }, { "id": "S105", "elements": [ "Q" ], "cost": 10000 }, { "id": "S106", "elements": [ "R" ], "cost": 10000 }, { "id": "S107", "elements": [ "S" ], "cost": 10000 }, { "id": "S108", "elements": [ "T" ], "cost": 10000 }, { "id": "S109", "elements": [ "U" ], "cost": 10000 }, { "id": "S110", "elements": [ "V" ], "cost": 10000 }, { "id": "S111", "elements": [ "W" ], "cost": 10000 } ] }, "solution_variant": [ "S6", "S16", "S21", "S65", "S87" ], "context_index": 30, "input_format": "json", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "I was helping a small club put together next season’s training schedule: there’s a list of athletes, and there’s a menu of training blocks that each address certain development needs. The job is to pick which blocks to run so that every athlete’s specific need is matched by one and only one block — nobody gets two different blocks and nobody is left uncovered — and then tally up the price tags for the blocks chosen; the plan with the lowest total cost is the one everyone prefers. The concrete list of athletes, blocks, and costs is shown below.\n\n- **num_athletes**: 20\n- **num_blocks**: 112\n\n| block_id | block_cost | athletes_covered |\n|---|---|---|\n| S1 | 332 | O P Q T |\n| S2 | 276 | B C D |\n| S3 | 5 | F H I J K |\n| S4 | 225 | L P R S T |\n| S5 | 345 | E F G H K |\n| S6 | 475 | H I J L M |\n| S7 | 174 | I M |\n| S8 | 216 | C E G H |\n| S9 | 85 | H J K L N |\n| S10 | 180 | P Q |\n| S11 | 85 | L M O P R |\n| S12 | 171 | D E F |\n| S13 | 64 | N Q |\n| S14 | 60 | D F H |\n| S15 | 186 | D F |\n| S16 | 33 | O P R |\n| S17 | 288 | B C F |\n| S18 | 219 | I L M |\n| S19 | 486 | B C D E F G |\n| S20 | 276 | K L O Q |\n| S21 | 50 | B C D E F |\n| S22 | 24 | H I K L M N |\n| S23 | 145 | J K L M O |\n| S24 | 145 | I J K L P |\n| S25 | 318 | F G H I J K |\n| S26 | 108 | L O P |\n| S27 | 205 | P Q R S T |\n| S28 | 108 | N R |\n| S29 | 261 | O S T |\n| S30 | 108 | O P |\n| S31 | 500 | B D E F G |\n| S32 | 117 | F I K |\n| S33 | 256 | J K M O |\n| S34 | 72 | A B C D |\n| S35 | 81 | P Q R |\n| S36 | 264 | Q S T |\n| S37 | 365 | C E F G H |\n| S38 | 10 | G H I J L |\n| S39 | 268 | N O P Q |\n| S40 | 16 | A C D F |\n| S41 | 250 | A B C D E |\n| S42 | 185 | A B C D F |\n| S43 | 90 | O P Q R S T |\n| S44 | 240 | D E G H I |\n| S45 | 76 | A D |\n| S46 | 332 | O Q R S |\n| S47 | 132 | A C |\n| S48 | 264 | K L O |\n| S49 | 198 | H I L |\n| S50 | 10 | M O P Q R |\n| S51 | 74 | L M |\n| S52 | 312 | M N O R |\n| S53 | 104 | O Q |\n| S54 | 402 | M O P Q R S |\n| S55 | 300 | K M N O Q R |\n| S56 | 36 | O R S T |\n| S57 | 108 | B D E G |\n| S58 | 165 | H I J |\n| S59 | 96 | C E I |\n| S60 | 340 | Q R S T |\n| S61 | 18 | N P R |\n| S62 | 110 | I K |\n| S63 | 4 | N P Q R |\n| S64 | 69 | O P S |\n| S65 | 36 | K M N O |\n| S66 | 366 | A B C D E F |\n| S67 | 276 | A B C |\n| S68 | 490 | M P Q R S |\n| S69 | 200 | D E F G |\n| S70 | 30 | P Q S |\n| S71 | 135 | J M N |\n| S72 | 460 | O P Q R S |\n| S73 | 340 | I L M N O |\n| S74 | 185 | C E F G I |\n| S75 | 72 | J L M N O Q |\n| S76 | 356 | D E H J |\n| S77 | 246 | G H J |\n| S78 | 36 | A B D F |\n| S79 | 288 | A B D E |\n| S80 | 288 | F H I J |\n| S81 | 144 | C E F G H I |\n| S82 | 66 | R S |\n| S83 | 124 | O P S T |\n| S84 | 425 | A B C D G |\n| S85 | 136 | J K L M |\n| S86 | 25 | M O Q S T |\n| S87 | 15 | A B D E G |\n| S88 | 57 | O Q T |\n| S89 | 385 | D F G H K |\n| S90 | 4 | P Q R S |\n| S91 | 150 | F I J L O |\n| S92 | 291 | F I L |\n| S93 | 10000 | A |\n| S94 | 10000 | B |\n| S95 | 10000 | C |\n| S96 | 10000 | D |\n| S97 | 10000 | E |\n| S98 | 10000 | F |\n| S99 | 10000 | G |\n| S100 | 10000 | H |\n| S101 | 10000 | I |\n| S102 | 10000 | J |\n| S103 | 10000 | K |\n| S104 | 10000 | L |\n| S105 | 10000 | M |\n| S106 | 10000 | N |\n| S107 | 10000 | O |\n| S108 | 10000 | P |\n| S109 | 10000 | Q |\n| S110 | 10000 | R |\n| S111 | 10000 | S |\n| S112 | 10000 | T |\n\nAlso, when you send the picks back, please use this simple JSON layout so it's easy to read by whatever tool is collecting the plans:\n\n{\n \"solution\": [\"block_id\", ...]\n}\n\nThink of that as a tiny form: \"solution\" holds an array of the training block IDs you choose (one string per block). The JSON above is just a sketch of the shape I expect — not the actual answer.\n\nPlease make sure to use the exact identifiers from the instance input — do not rename them or invent new ones. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 20, "num_sets": 112, "density": 0.171875, "sets": [ { "id": 1, "elements": [ 15, 16, 17, 20 ], "cost": 332 }, { "id": 2, "elements": [ 2, 3, 4 ], "cost": 276 }, { "id": 3, "elements": [ 6, 8, 9, 10, 11 ], "cost": 5 }, { "id": 4, "elements": [ 12, 16, 18, 19, 20 ], "cost": 225 }, { "id": 5, "elements": [ 5, 6, 7, 8, 11 ], "cost": 345 }, { "id": 6, "elements": [ 8, 9, 10, 12, 13 ], "cost": 475 }, { "id": 7, "elements": [ 9, 13 ], "cost": 174 }, { "id": 8, "elements": [ 3, 5, 7, 8 ], "cost": 216 }, { "id": 9, "elements": [ 8, 10, 11, 12, 14 ], "cost": 85 }, { "id": 10, "elements": [ 16, 17 ], "cost": 180 }, { "id": 11, "elements": [ 12, 13, 15, 16, 18 ], "cost": 85 }, { "id": 12, "elements": [ 4, 5, 6 ], "cost": 171 }, { "id": 13, "elements": [ 14, 17 ], "cost": 64 }, { "id": 14, "elements": [ 4, 6, 8 ], "cost": 60 }, { "id": 15, "elements": [ 4, 6 ], "cost": 186 }, { "id": 16, "elements": [ 15, 16, 18 ], "cost": 33 }, { "id": 17, "elements": [ 2, 3, 6 ], "cost": 288 }, { "id": 18, "elements": [ 9, 12, 13 ], "cost": 219 }, { "id": 19, "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 486 }, { "id": 20, "elements": [ 11, 12, 15, 17 ], "cost": 276 }, { "id": 21, "elements": [ 2, 3, 4, 5, 6 ], "cost": 50 }, { "id": 22, "elements": [ 8, 9, 11, 12, 13, 14 ], "cost": 24 }, { "id": 23, "elements": [ 10, 11, 12, 13, 15 ], "cost": 145 }, { "id": 24, "elements": [ 9, 10, 11, 12, 16 ], "cost": 145 }, { "id": 25, "elements": [ 6, 7, 8, 9, 10, 11 ], "cost": 318 }, { "id": 26, "elements": [ 12, 15, 16 ], "cost": 108 }, { "id": 27, "elements": [ 16, 17, 18, 19, 20 ], "cost": 205 }, { "id": 28, "elements": [ 14, 18 ], "cost": 108 }, { "id": 29, "elements": [ 15, 19, 20 ], "cost": 261 }, { "id": 30, "elements": [ 15, 16 ], "cost": 108 }, { "id": 31, "elements": [ 2, 4, 5, 6, 7 ], "cost": 500 }, { "id": 32, "elements": [ 6, 9, 11 ], "cost": 117 }, { "id": 33, "elements": [ 10, 11, 13, 15 ], "cost": 256 }, { "id": 34, "elements": [ 1, 2, 3, 4 ], "cost": 72 }, { "id": 35, "elements": [ 16, 17, 18 ], "cost": 81 }, { "id": 36, "elements": [ 17, 19, 20 ], "cost": 264 }, { "id": 37, "elements": [ 3, 5, 6, 7, 8 ], "cost": 365 }, { "id": 38, "elements": [ 7, 8, 9, 10, 12 ], "cost": 10 }, { "id": 39, "elements": [ 14, 15, 16, 17 ], "cost": 268 }, { "id": 40, "elements": [ 1, 3, 4, 6 ], "cost": 16 }, { "id": 41, "elements": [ 1, 2, 3, 4, 5 ], "cost": 250 }, { "id": 42, "elements": [ 1, 2, 3, 4, 6 ], "cost": 185 }, { "id": 43, "elements": [ 15, 16, 17, 18, 19, 20 ], "cost": 90 }, { "id": 44, "elements": [ 4, 5, 7, 8, 9 ], "cost": 240 }, { "id": 45, "elements": [ 1, 4 ], "cost": 76 }, { "id": 46, "elements": [ 15, 17, 18, 19 ], "cost": 332 }, { "id": 47, "elements": [ 1, 3 ], "cost": 132 }, { "id": 48, "elements": [ 11, 12, 15 ], "cost": 264 }, { "id": 49, "elements": [ 8, 9, 12 ], "cost": 198 }, { "id": 50, "elements": [ 13, 15, 16, 17, 18 ], "cost": 10 }, { "id": 51, "elements": [ 12, 13 ], "cost": 74 }, { "id": 52, "elements": [ 13, 14, 15, 18 ], "cost": 312 }, { "id": 53, "elements": [ 15, 17 ], "cost": 104 }, { "id": 54, "elements": [ 13, 15, 16, 17, 18, 19 ], "cost": 402 }, { "id": 55, "elements": [ 11, 13, 14, 15, 17, 18 ], "cost": 300 }, { "id": 56, "elements": [ 15, 18, 19, 20 ], "cost": 36 }, { "id": 57, "elements": [ 2, 4, 5, 7 ], "cost": 108 }, { "id": 58, "elements": [ 8, 9, 10 ], "cost": 165 }, { "id": 59, "elements": [ 3, 5, 9 ], "cost": 96 }, { "id": 60, "elements": [ 17, 18, 19, 20 ], "cost": 340 }, { "id": 61, "elements": [ 14, 16, 18 ], "cost": 18 }, { "id": 62, "elements": [ 9, 11 ], "cost": 110 }, { "id": 63, "elements": [ 14, 16, 17, 18 ], "cost": 4 }, { "id": 64, "elements": [ 15, 16, 19 ], "cost": 69 }, { "id": 65, "elements": [ 11, 13, 14, 15 ], "cost": 36 }, { "id": 66, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 366 }, { "id": 67, "elements": [ 1, 2, 3 ], "cost": 276 }, { "id": 68, "elements": [ 13, 16, 17, 18, 19 ], "cost": 490 }, { "id": 69, "elements": [ 4, 5, 6, 7 ], "cost": 200 }, { "id": 70, "elements": [ 16, 17, 19 ], "cost": 30 }, { "id": 71, "elements": [ 10, 13, 14 ], "cost": 135 }, { "id": 72, "elements": [ 15, 16, 17, 18, 19 ], "cost": 460 }, { "id": 73, "elements": [ 9, 12, 13, 14, 15 ], "cost": 340 }, { "id": 74, "elements": [ 3, 5, 6, 7, 9 ], "cost": 185 }, { "id": 75, "elements": [ 10, 12, 13, 14, 15, 17 ], "cost": 72 }, { "id": 76, "elements": [ 4, 5, 8, 10 ], "cost": 356 }, { "id": 77, "elements": [ 7, 8, 10 ], "cost": 246 }, { "id": 78, "elements": [ 1, 2, 4, 6 ], "cost": 36 }, { "id": 79, "elements": [ 1, 2, 4, 5 ], "cost": 288 }, { "id": 80, "elements": [ 6, 8, 9, 10 ], "cost": 288 }, { "id": 81, "elements": [ 3, 5, 6, 7, 8, 9 ], "cost": 144 }, { "id": 82, "elements": [ 18, 19 ], "cost": 66 }, { "id": 83, "elements": [ 15, 16, 19, 20 ], "cost": 124 }, { "id": 84, "elements": [ 1, 2, 3, 4, 7 ], "cost": 425 }, { "id": 85, "elements": [ 10, 11, 12, 13 ], "cost": 136 }, { "id": 86, "elements": [ 13, 15, 17, 19, 20 ], "cost": 25 }, { "id": 87, "elements": [ 1, 2, 4, 5, 7 ], "cost": 15 }, { "id": 88, "elements": [ 15, 17, 20 ], "cost": 57 }, { "id": 89, "elements": [ 4, 6, 7, 8, 11 ], "cost": 385 }, { "id": 90, "elements": [ 16, 17, 18, 19 ], "cost": 4 }, { "id": 91, "elements": [ 6, 9, 10, 12, 15 ], "cost": 150 }, { "id": 92, "elements": [ 6, 9, 12 ], "cost": 291 }, { "id": 93, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 94, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 95, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 96, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 97, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 98, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 20 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0030_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0030_bag_stats.png" }, "solution": [ 8, 24, 28, 78, 86 ], "obj": 530.0, "instance_variant": { "num_elements": 20, "num_sets": 112, "sets": [ { "id": "S1", "elements": [ "O", "P", "Q", "T" ], "cost": 332 }, { "id": "S2", "elements": [ "B", "C", "D" ], "cost": 276 }, { "id": "S3", "elements": [ "F", "H", "I", "J", "K" ], "cost": 5 }, { "id": "S4", "elements": [ "L", "P", "R", "S", "T" ], "cost": 225 }, { "id": "S5", "elements": [ "E", "F", "G", "H", "K" ], "cost": 345 }, { "id": "S6", "elements": [ "H", "I", "J", "L", "M" ], "cost": 475 }, { "id": "S7", "elements": [ "I", "M" ], "cost": 174 }, { "id": "S8", "elements": [ "C", "E", "G", "H" ], "cost": 216 }, { "id": "S9", "elements": [ "H", "J", "K", "L", "N" ], "cost": 85 }, { "id": "S10", "elements": [ "P", "Q" ], "cost": 180 }, { "id": "S11", "elements": [ "L", "M", "O", "P", "R" ], "cost": 85 }, { "id": "S12", "elements": [ "D", "E", "F" ], "cost": 171 }, { "id": "S13", "elements": [ "N", "Q" ], "cost": 64 }, { "id": "S14", "elements": [ "D", "F", "H" ], "cost": 60 }, { "id": "S15", "elements": [ "D", "F" ], "cost": 186 }, { "id": "S16", "elements": [ "O", "P", "R" ], "cost": 33 }, { "id": "S17", "elements": [ "B", "C", "F" ], "cost": 288 }, { "id": "S18", "elements": [ "I", "L", "M" ], "cost": 219 }, { "id": "S19", "elements": [ "B", "C", "D", "E", "F", "G" ], "cost": 486 }, { "id": "S20", "elements": [ "K", "L", "O", "Q" ], "cost": 276 }, { "id": "S21", "elements": [ "B", "C", "D", "E", "F" ], "cost": 50 }, { "id": "S22", "elements": [ "H", "I", "K", "L", "M", "N" ], "cost": 24 }, { "id": "S23", "elements": [ "J", "K", "L", "M", "O" ], "cost": 145 }, { "id": "S24", "elements": [ "I", "J", "K", "L", "P" ], "cost": 145 }, { "id": "S25", "elements": [ "F", "G", "H", "I", "J", "K" ], "cost": 318 }, { "id": "S26", "elements": [ "L", "O", "P" ], "cost": 108 }, { "id": "S27", "elements": [ "P", "Q", "R", "S", "T" ], "cost": 205 }, { "id": "S28", "elements": [ "N", "R" ], "cost": 108 }, { "id": "S29", "elements": [ "O", "S", "T" ], "cost": 261 }, { "id": "S30", "elements": [ "O", "P" ], "cost": 108 }, { "id": "S31", "elements": [ "B", "D", "E", "F", "G" ], "cost": 500 }, { "id": "S32", "elements": [ "F", "I", "K" ], "cost": 117 }, { "id": "S33", "elements": [ "J", "K", "M", "O" ], "cost": 256 }, { "id": "S34", "elements": [ "A", "B", "C", "D" ], "cost": 72 }, { "id": "S35", "elements": [ "P", "Q", "R" ], "cost": 81 }, { "id": "S36", "elements": [ "Q", "S", "T" ], "cost": 264 }, { "id": "S37", "elements": [ "C", "E", "F", "G", "H" ], "cost": 365 }, { "id": "S38", "elements": [ "G", "H", "I", "J", "L" ], "cost": 10 }, { "id": "S39", "elements": [ "N", "O", "P", "Q" ], "cost": 268 }, { "id": "S40", "elements": [ "A", "C", "D", "F" ], "cost": 16 }, { "id": "S41", "elements": [ "A", "B", "C", "D", "E" ], "cost": 250 }, { "id": "S42", "elements": [ "A", "B", "C", "D", "F" ], "cost": 185 }, { "id": "S43", "elements": [ "O", "P", "Q", "R", "S", "T" ], "cost": 90 }, { "id": "S44", "elements": [ "D", "E", "G", "H", "I" ], "cost": 240 }, { "id": "S45", "elements": [ "A", "D" ], "cost": 76 }, { "id": "S46", "elements": [ "O", "Q", "R", "S" ], "cost": 332 }, { "id": "S47", "elements": [ "A", "C" ], "cost": 132 }, { "id": "S48", "elements": [ "K", "L", "O" ], "cost": 264 }, { "id": "S49", "elements": [ "H", "I", "L" ], "cost": 198 }, { "id": "S50", "elements": [ "M", "O", "P", "Q", "R" ], "cost": 10 }, { "id": "S51", "elements": [ "L", "M" ], "cost": 74 }, { "id": "S52", "elements": [ "M", "N", "O", "R" ], "cost": 312 }, { "id": "S53", "elements": [ "O", "Q" ], "cost": 104 }, { "id": "S54", "elements": [ "M", "O", "P", "Q", "R", "S" ], "cost": 402 }, { "id": "S55", "elements": [ "K", "M", "N", "O", "Q", "R" ], "cost": 300 }, { "id": "S56", "elements": [ "O", "R", "S", "T" ], "cost": 36 }, { "id": "S57", "elements": [ "B", "D", "E", "G" ], "cost": 108 }, { "id": "S58", "elements": [ "H", "I", "J" ], "cost": 165 }, { "id": "S59", "elements": [ "C", "E", "I" ], "cost": 96 }, { "id": "S60", "elements": [ "Q", "R", "S", "T" ], "cost": 340 }, { "id": "S61", "elements": [ "N", "P", "R" ], "cost": 18 }, { "id": "S62", "elements": [ "I", "K" ], "cost": 110 }, { "id": "S63", "elements": [ "N", "P", "Q", "R" ], "cost": 4 }, { "id": "S64", "elements": [ "O", "P", "S" ], "cost": 69 }, { "id": "S65", "elements": [ "K", "M", "N", "O" ], "cost": 36 }, { "id": "S66", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 366 }, { "id": "S67", "elements": [ "A", "B", "C" ], "cost": 276 }, { "id": "S68", "elements": [ "M", "P", "Q", "R", "S" ], "cost": 490 }, { "id": "S69", "elements": [ "D", "E", "F", "G" ], "cost": 200 }, { "id": "S70", "elements": [ "P", "Q", "S" ], "cost": 30 }, { "id": "S71", "elements": [ "J", "M", "N" ], "cost": 135 }, { "id": "S72", "elements": [ "O", "P", "Q", "R", "S" ], "cost": 460 }, { "id": "S73", "elements": [ "I", "L", "M", "N", "O" ], "cost": 340 }, { "id": "S74", "elements": [ "C", "E", "F", "G", "I" ], "cost": 185 }, { "id": "S75", "elements": [ "J", "L", "M", "N", "O", "Q" ], "cost": 72 }, { "id": "S76", "elements": [ "D", "E", "H", "J" ], "cost": 356 }, { "id": "S77", "elements": [ "G", "H", "J" ], "cost": 246 }, { "id": "S78", "elements": [ "A", "B", "D", "F" ], "cost": 36 }, { "id": "S79", "elements": [ "A", "B", "D", "E" ], "cost": 288 }, { "id": "S80", "elements": [ "F", "H", "I", "J" ], "cost": 288 }, { "id": "S81", "elements": [ "C", "E", "F", "G", "H", "I" ], "cost": 144 }, { "id": "S82", "elements": [ "R", "S" ], "cost": 66 }, { "id": "S83", "elements": [ "O", "P", "S", "T" ], "cost": 124 }, { "id": "S84", "elements": [ "A", "B", "C", "D", "G" ], "cost": 425 }, { "id": "S85", "elements": [ "J", "K", "L", "M" ], "cost": 136 }, { "id": "S86", "elements": [ "M", "O", "Q", "S", "T" ], "cost": 25 }, { "id": "S87", "elements": [ "A", "B", "D", "E", "G" ], "cost": 15 }, { "id": "S88", "elements": [ "O", "Q", "T" ], "cost": 57 }, { "id": "S89", "elements": [ "D", "F", "G", "H", "K" ], "cost": 385 }, { "id": "S90", "elements": [ "P", "Q", "R", "S" ], "cost": 4 }, { "id": "S91", "elements": [ "F", "I", "J", "L", "O" ], "cost": 150 }, { "id": "S92", "elements": [ "F", "I", "L" ], "cost": 291 }, { "id": "S93", "elements": [ "A" ], "cost": 10000 }, { "id": "S94", "elements": [ "B" ], "cost": 10000 }, { "id": "S95", "elements": [ "C" ], "cost": 10000 }, { "id": "S96", "elements": [ "D" ], "cost": 10000 }, { "id": "S97", "elements": [ "E" ], "cost": 10000 }, { "id": "S98", "elements": [ "F" ], "cost": 10000 }, { "id": "S99", "elements": [ "G" ], "cost": 10000 }, { "id": "S100", "elements": [ "H" ], "cost": 10000 }, { "id": "S101", "elements": [ "I" ], "cost": 10000 }, { "id": "S102", "elements": [ "J" ], "cost": 10000 }, { "id": "S103", "elements": [ "K" ], "cost": 10000 }, { "id": "S104", "elements": [ "L" ], "cost": 10000 }, { "id": "S105", "elements": [ "M" ], "cost": 10000 }, { "id": "S106", "elements": [ "N" ], "cost": 10000 }, { "id": "S107", "elements": [ "O" ], "cost": 10000 }, { "id": "S108", "elements": [ "P" ], "cost": 10000 }, { "id": "S109", "elements": [ "Q" ], "cost": 10000 }, { "id": "S110", "elements": [ "R" ], "cost": 10000 }, { "id": "S111", "elements": [ "S" ], "cost": 10000 }, { "id": "S112", "elements": [ "T" ], "cost": 10000 } ] }, "solution_variant": [ "S8", "S24", "S28", "S78", "S86" ], "context_index": 31, "input_format": "markdown_table", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "We’ve got a pile of artifacts and a bunch of pre-made exhibit bundles, and the job is to decide which bundles to run so each artifact appears in one and only one display (no misses, no duplicates), and the whole setup costs the least amount; the way to see how good a plan is is simply to add up the prices of the selected bundles. The exact bundle options and artifact assignments are listed below.\n\nWe've got 20 artifacts and 111 candidate bundles listed below:\nBundle S1: if we run it, it displays 4 6 and costs 58.\nBundle S2: if we run it, it displays 10 11 13 14 17 and costs 20.\nBundle S3: if we run it, it displays 0 1 3 4 and costs 180.\nBundle S4: if we run it, it displays 15 16 17 18 19 and costs 5.\nBundle S5: if we run it, it displays 3 4 6 9 13 and costs 355.\nBundle S6: if we run it, it displays 2 5 6 7 8 9 and costs 108.\nBundle S7: if we run it, it displays 18 19 and costs 36.\nBundle S8: if we run it, it displays 13 15 17 18 19 and costs 470.\nBundle S9: if we run it, it displays 15 16 and costs 26.\nBundle S10: if we run it, it displays 14 15 16 18 and costs 112.\nBundle S11: if we run it, it displays 5 7 8 9 and costs 48.\nBundle S12: if we run it, it displays 9 10 13 14 15 and costs 105.\nBundle S13: if we run it, it displays 13 14 15 and costs 6.\nBundle S14: if we run it, it displays 5 6 10 11 and costs 168.\nBundle S15: if we run it, it displays 17 18 19 and costs 207.\nBundle S16: if we run it, it displays 0 1 2 3 4 5 and costs 264.\nBundle S17: if we run it, it displays 14 15 17 19 and costs 196.\nBundle S18: if we run it, it displays 15 18 19 and costs 285.\nBundle S19: if we run it, it displays 6 7 11 13 and costs 64.\nBundle S20: if we run it, it displays 0 2 3 4 5 and costs 45.\nBundle S21: if we run it, it displays 2 3 4 5 6 7 and costs 342.\nBundle S22: if we run it, it displays 2 5 and costs 110.\nBundle S23: if we run it, it displays 6 12 and costs 156.\nBundle S24: if we run it, it displays 7 8 9 and costs 174.\nBundle S25: if we run it, it displays 11 13 14 15 and costs 12.\nBundle S26: if we run it, it displays 15 17 and costs 128.\nBundle S27: if we run it, it displays 14 17 18 and costs 231.\nBundle S28: if we run it, it displays 3 5 7 and costs 81.\nBundle S29: if we run it, it displays 13 14 15 17 18 and costs 130.\nBundle S30: if we run it, it displays 5 6 and costs 100.\nBundle S31: if we run it, it displays 11 12 13 14 15 and costs 475.\nBundle S32: if we run it, it displays 0 1 2 3 4 and costs 490.\nBundle S33: if we run it, it displays 11 15 16 and costs 159.\nBundle S34: if we run it, it displays 12 13 14 15 16 and costs 415.\nBundle S35: if we run it, it displays 8 9 11 and costs 147.\nBundle S36: if we run it, it displays 7 8 9 10 11 and costs 230.\nBundle S37: if we run it, it displays 11 13 14 15 18 and costs 195.\nBundle S38: if we run it, it displays 5 6 8 9 and costs 272.\nBundle S39: if we run it, it displays 14 15 18 19 and costs 212.\nBundle S40: if we run it, it displays 14 15 17 18 19 and costs 200.\nBundle S41: if we run it, it displays 1 2 3 and costs 207.\nBundle S42: if we run it, it displays 11 12 16 and costs 78.\nBundle S43: if we run it, it displays 13 15 16 17 and costs 180.\nBundle S44: if we run it, it displays 7 9 12 and costs 3.\nBundle S45: if we run it, it displays 6 8 11 and costs 120.\nBundle S46: if we run it, it displays 5 7 9 and costs 102.\nBundle S47: if we run it, it displays 5 7 8 9 10 and costs 55.\nBundle S48: if we run it, it displays 0 1 2 3 9 and costs 110.\nBundle S49: if we run it, it displays 3 4 6 7 and costs 348.\nBundle S50: if we run it, it displays 4 5 6 7 and costs 256.\nBundle S51: if we run it, it displays 4 6 7 and costs 132.\nBundle S52: if we run it, it displays 4 5 7 8 9 12 and costs 180.\nBundle S53: if we run it, it displays 1 2 5 and costs 138.\nBundle S54: if we run it, it displays 7 8 10 and costs 162.\nBundle S55: if we run it, it displays 2 5 6 and costs 96.\nBundle S56: if we run it, it displays 9 12 13 14 and costs 244.\nBundle S57: if we run it, it displays 6 7 8 9 10 and costs 330.\nBundle S58: if we run it, it displays 10 11 12 13 14 15 and costs 336.\nBundle S59: if we run it, it displays 7 10 11 and costs 168.\nBundle S60: if we run it, it displays 1 5 6 and costs 90.\nBundle S61: if we run it, it displays 12 15 16 17 18 and costs 60.\nBundle S62: if we run it, it displays 1 3 4 5 6 and costs 435.\nBundle S63: if we run it, it displays 3 4 7 and costs 51.\nBundle S64: if we run it, it displays 7 10 11 12 and costs 368.\nBundle S65: if we run it, it displays 14 16 17 and costs 33.\nBundle S66: if we run it, it displays 4 5 7 9 and costs 44.\nBundle S67: if we run it, it displays 0 1 4 5 6 7 and costs 114.\nBundle S68: if we run it, it displays 2 3 4 5 6 and costs 220.\nBundle S69: if we run it, it displays 11 12 13 15 16 and costs 55.\nBundle S70: if we run it, it displays 13 14 16 17 18 and costs 255.\nBundle S71: if we run it, it displays 1 2 3 4 and costs 248.\nBundle S72: if we run it, it displays 1 2 and costs 176.\nBundle S73: if we run it, it displays 5 7 and costs 90.\nBundle S74: if we run it, it displays 2 3 4 5 6 8 and costs 210.\nBundle S75: if we run it, it displays 8 9 10 13 and costs 220.\nBundle S76: if we run it, it displays 0 3 5 6 and costs 232.\nBundle S77: if we run it, it displays 0 1 4 and costs 108.\nBundle S78: if we run it, it displays 0 1 2 3 and costs 340.\nBundle S79: if we run it, it displays 9 11 12 13 and costs 352.\nBundle S80: if we run it, it displays 10 12 13 14 15 and costs 155.\nBundle S81: if we run it, it displays 2 3 4 7 and costs 308.\nBundle S82: if we run it, it displays 11 12 13 15 and costs 360.\nBundle S83: if we run it, it displays 0 1 2 4 5 and costs 235.\nBundle S84: if we run it, it displays 15 16 17 18 and costs 44.\nBundle S85: if we run it, it displays 8 9 10 12 and costs 316.\nBundle S86: if we run it, it displays 7 8 9 11 and costs 44.\nBundle S87: if we run it, it displays 14 15 16 and costs 261.\nBundle S88: if we run it, it displays 11 12 14 15 and costs 56.\nBundle S89: if we run it, it displays 11 13 14 and costs 210.\nBundle S90: if we run it, it displays 0 1 2 and costs 78.\nBundle S91: if we run it, it displays 13 14 16 19 and costs 72.\nBundle S92: if we run it, it displays 0 and costs 10000.\nBundle S93: if we run it, it displays 1 and costs 10000.\nBundle S94: if we run it, it displays 2 and costs 10000.\nBundle S95: if we run it, it displays 3 and costs 10000.\nBundle S96: if we run it, it displays 4 and costs 10000.\nBundle S97: if we run it, it displays 5 and costs 10000.\nBundle S98: if we run it, it displays 6 and costs 10000.\nBundle S99: if we run it, it displays 7 and costs 10000.\nBundle S100: if we run it, it displays 8 and costs 10000.\nBundle S101: if we run it, it displays 9 and costs 10000.\nBundle S102: if we run it, it displays 10 and costs 10000.\nBundle S103: if we run it, it displays 11 and costs 10000.\nBundle S104: if we run it, it displays 12 and costs 10000.\nBundle S105: if we run it, it displays 13 and costs 10000.\nBundle S106: if we run it, it displays 14 and costs 10000.\nBundle S107: if we run it, it displays 15 and costs 10000.\nBundle S108: if we run it, it displays 16 and costs 10000.\nBundle S109: if we run it, it displays 17 and costs 10000.\nBundle S110: if we run it, it displays 18 and costs 10000.\nBundle S111: if we run it, it displays 19 and costs 10000.\nWe must pick bundles so every artifact appears in exactly one display and the total cost is minimized.\n\nIf you want to hand me a candidate plan, just drop it in the little JSON shape below so I can read it programmatically — nothing fancy, just a list of the bundles you'd run.\n\n{\n \"solution\": [\"bundle_id\", ...]\n}\n\n\"solution\" holds the list of bundles to run. Each \"bundle_id\" is a placeholder for one bundle label from the instance (so put the actual bundle labels there). Think of it like a short form: the file has one key, and that key points to a list of the bundle names you want to activate.\n\nThis JSON is just a sketch of the expected shape, not the final answer itself — replace the placeholder(s) with the exact bundle identifiers from the instance.\n\nPlease use the identifiers exactly as they appear in the instance input — no renaming and no new labels. \nFor example: Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.", "instance": { "problem_type": "SPP", "num_elements": 20, "num_sets": 111, "density": 0.17027027027027028, "sets": [ { "id": 1, "elements": [ 5, 7 ], "cost": 58 }, { "id": 2, "elements": [ 11, 12, 14, 15, 18 ], "cost": 20 }, { "id": 3, "elements": [ 1, 2, 4, 5 ], "cost": 180 }, { "id": 4, "elements": [ 16, 17, 18, 19, 20 ], "cost": 5 }, { "id": 5, "elements": [ 4, 5, 7, 10, 14 ], "cost": 355 }, { "id": 6, "elements": [ 3, 6, 7, 8, 9, 10 ], "cost": 108 }, { "id": 7, "elements": [ 19, 20 ], "cost": 36 }, { "id": 8, "elements": [ 14, 16, 18, 19, 20 ], "cost": 470 }, { "id": 9, "elements": [ 16, 17 ], "cost": 26 }, { "id": 10, "elements": [ 15, 16, 17, 19 ], "cost": 112 }, { "id": 11, "elements": [ 6, 8, 9, 10 ], "cost": 48 }, { "id": 12, "elements": [ 10, 11, 14, 15, 16 ], "cost": 105 }, { "id": 13, "elements": [ 14, 15, 16 ], "cost": 6 }, { "id": 14, "elements": [ 6, 7, 11, 12 ], "cost": 168 }, { "id": 15, "elements": [ 18, 19, 20 ], "cost": 207 }, { "id": 16, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 264 }, { "id": 17, "elements": [ 15, 16, 18, 20 ], "cost": 196 }, { "id": 18, "elements": [ 16, 19, 20 ], "cost": 285 }, { "id": 19, "elements": [ 7, 8, 12, 14 ], "cost": 64 }, { "id": 20, "elements": [ 1, 3, 4, 5, 6 ], "cost": 45 }, { "id": 21, "elements": [ 3, 4, 5, 6, 7, 8 ], "cost": 342 }, { "id": 22, "elements": [ 3, 6 ], "cost": 110 }, { "id": 23, "elements": [ 7, 13 ], "cost": 156 }, { "id": 24, "elements": [ 8, 9, 10 ], "cost": 174 }, { "id": 25, "elements": [ 12, 14, 15, 16 ], "cost": 12 }, { "id": 26, "elements": [ 16, 18 ], "cost": 128 }, { "id": 27, "elements": [ 15, 18, 19 ], "cost": 231 }, { "id": 28, "elements": [ 4, 6, 8 ], "cost": 81 }, { "id": 29, "elements": [ 14, 15, 16, 18, 19 ], "cost": 130 }, { "id": 30, "elements": [ 6, 7 ], "cost": 100 }, { "id": 31, "elements": [ 12, 13, 14, 15, 16 ], "cost": 475 }, { "id": 32, "elements": [ 1, 2, 3, 4, 5 ], "cost": 490 }, { "id": 33, "elements": [ 12, 16, 17 ], "cost": 159 }, { "id": 34, "elements": [ 13, 14, 15, 16, 17 ], "cost": 415 }, { "id": 35, "elements": [ 9, 10, 12 ], "cost": 147 }, { "id": 36, "elements": [ 8, 9, 10, 11, 12 ], "cost": 230 }, { "id": 37, "elements": [ 12, 14, 15, 16, 19 ], "cost": 195 }, { "id": 38, "elements": [ 6, 7, 9, 10 ], "cost": 272 }, { "id": 39, "elements": [ 15, 16, 19, 20 ], "cost": 212 }, { "id": 40, "elements": [ 15, 16, 18, 19, 20 ], "cost": 200 }, { "id": 41, "elements": [ 2, 3, 4 ], "cost": 207 }, { "id": 42, "elements": [ 12, 13, 17 ], "cost": 78 }, { "id": 43, "elements": [ 14, 16, 17, 18 ], "cost": 180 }, { "id": 44, "elements": [ 8, 10, 13 ], "cost": 3 }, { "id": 45, "elements": [ 7, 9, 12 ], "cost": 120 }, { "id": 46, "elements": [ 6, 8, 10 ], "cost": 102 }, { "id": 47, "elements": [ 6, 8, 9, 10, 11 ], "cost": 55 }, { "id": 48, "elements": [ 1, 2, 3, 4, 10 ], "cost": 110 }, { "id": 49, "elements": [ 4, 5, 7, 8 ], "cost": 348 }, { "id": 50, "elements": [ 5, 6, 7, 8 ], "cost": 256 }, { "id": 51, "elements": [ 5, 7, 8 ], "cost": 132 }, { "id": 52, "elements": [ 5, 6, 8, 9, 10, 13 ], "cost": 180 }, { "id": 53, "elements": [ 2, 3, 6 ], "cost": 138 }, { "id": 54, "elements": [ 8, 9, 11 ], "cost": 162 }, { "id": 55, "elements": [ 3, 6, 7 ], "cost": 96 }, { "id": 56, "elements": [ 10, 13, 14, 15 ], "cost": 244 }, { "id": 57, "elements": [ 7, 8, 9, 10, 11 ], "cost": 330 }, { "id": 58, "elements": [ 11, 12, 13, 14, 15, 16 ], "cost": 336 }, { "id": 59, "elements": [ 8, 11, 12 ], "cost": 168 }, { "id": 60, "elements": [ 2, 6, 7 ], "cost": 90 }, { "id": 61, "elements": [ 13, 16, 17, 18, 19 ], "cost": 60 }, { "id": 62, "elements": [ 2, 4, 5, 6, 7 ], "cost": 435 }, { "id": 63, "elements": [ 4, 5, 8 ], "cost": 51 }, { "id": 64, "elements": [ 8, 11, 12, 13 ], "cost": 368 }, { "id": 65, "elements": [ 15, 17, 18 ], "cost": 33 }, { "id": 66, "elements": [ 5, 6, 8, 10 ], "cost": 44 }, { "id": 67, "elements": [ 1, 2, 5, 6, 7, 8 ], "cost": 114 }, { "id": 68, "elements": [ 3, 4, 5, 6, 7 ], "cost": 220 }, { "id": 69, "elements": [ 12, 13, 14, 16, 17 ], "cost": 55 }, { "id": 70, "elements": [ 14, 15, 17, 18, 19 ], "cost": 255 }, { "id": 71, "elements": [ 2, 3, 4, 5 ], "cost": 248 }, { "id": 72, "elements": [ 2, 3 ], "cost": 176 }, { "id": 73, "elements": [ 6, 8 ], "cost": 90 }, { "id": 74, "elements": [ 3, 4, 5, 6, 7, 9 ], "cost": 210 }, { "id": 75, "elements": [ 9, 10, 11, 14 ], "cost": 220 }, { "id": 76, "elements": [ 1, 4, 6, 7 ], "cost": 232 }, { "id": 77, "elements": [ 1, 2, 5 ], "cost": 108 }, { "id": 78, "elements": [ 1, 2, 3, 4 ], "cost": 340 }, { "id": 79, "elements": [ 10, 12, 13, 14 ], "cost": 352 }, { "id": 80, "elements": [ 11, 13, 14, 15, 16 ], "cost": 155 }, { "id": 81, "elements": [ 3, 4, 5, 8 ], "cost": 308 }, { "id": 82, "elements": [ 12, 13, 14, 16 ], "cost": 360 }, { "id": 83, "elements": [ 1, 2, 3, 5, 6 ], "cost": 235 }, { "id": 84, "elements": [ 16, 17, 18, 19 ], "cost": 44 }, { "id": 85, "elements": [ 9, 10, 11, 13 ], "cost": 316 }, { "id": 86, "elements": [ 8, 9, 10, 12 ], "cost": 44 }, { "id": 87, "elements": [ 15, 16, 17 ], "cost": 261 }, { "id": 88, "elements": [ 12, 13, 15, 16 ], "cost": 56 }, { "id": 89, "elements": [ 12, 14, 15 ], "cost": 210 }, { "id": 90, "elements": [ 1, 2, 3 ], "cost": 78 }, { "id": 91, "elements": [ 14, 15, 17, 20 ], "cost": 72 }, { "id": 92, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 93, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 94, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 95, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 96, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 97, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 98, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 20 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0031_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0031_bag_stats.png" }, "solution": [ 2, 7, 9, 16, 23, 24 ], "obj": 676.0, "instance_variant": { "num_elements": 20, "num_sets": 111, "sets": [ { "id": "S1", "elements": [ 4, 6 ], "cost": 58 }, { "id": "S2", "elements": [ 10, 11, 13, 14, 17 ], "cost": 20 }, { "id": "S3", "elements": [ 0, 1, 3, 4 ], "cost": 180 }, { "id": "S4", "elements": [ 15, 16, 17, 18, 19 ], "cost": 5 }, { "id": "S5", "elements": [ 3, 4, 6, 9, 13 ], "cost": 355 }, { "id": "S6", "elements": [ 2, 5, 6, 7, 8, 9 ], "cost": 108 }, { "id": "S7", "elements": [ 18, 19 ], "cost": 36 }, { "id": "S8", "elements": [ 13, 15, 17, 18, 19 ], "cost": 470 }, { "id": "S9", "elements": [ 15, 16 ], "cost": 26 }, { "id": "S10", "elements": [ 14, 15, 16, 18 ], "cost": 112 }, { "id": "S11", "elements": [ 5, 7, 8, 9 ], "cost": 48 }, { "id": "S12", "elements": [ 9, 10, 13, 14, 15 ], "cost": 105 }, { "id": "S13", "elements": [ 13, 14, 15 ], "cost": 6 }, { "id": "S14", "elements": [ 5, 6, 10, 11 ], "cost": 168 }, { "id": "S15", "elements": [ 17, 18, 19 ], "cost": 207 }, { "id": "S16", "elements": [ 0, 1, 2, 3, 4, 5 ], "cost": 264 }, { "id": "S17", "elements": [ 14, 15, 17, 19 ], "cost": 196 }, { "id": "S18", "elements": [ 15, 18, 19 ], "cost": 285 }, { "id": "S19", "elements": [ 6, 7, 11, 13 ], "cost": 64 }, { "id": "S20", "elements": [ 0, 2, 3, 4, 5 ], "cost": 45 }, { "id": "S21", "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 342 }, { "id": "S22", "elements": [ 2, 5 ], "cost": 110 }, { "id": "S23", "elements": [ 6, 12 ], "cost": 156 }, { "id": "S24", "elements": [ 7, 8, 9 ], "cost": 174 }, { "id": "S25", "elements": [ 11, 13, 14, 15 ], "cost": 12 }, { "id": "S26", "elements": [ 15, 17 ], "cost": 128 }, { "id": "S27", "elements": [ 14, 17, 18 ], "cost": 231 }, { "id": "S28", "elements": [ 3, 5, 7 ], "cost": 81 }, { "id": "S29", "elements": [ 13, 14, 15, 17, 18 ], "cost": 130 }, { "id": "S30", "elements": [ 5, 6 ], "cost": 100 }, { "id": "S31", "elements": [ 11, 12, 13, 14, 15 ], "cost": 475 }, { "id": "S32", "elements": [ 0, 1, 2, 3, 4 ], "cost": 490 }, { "id": "S33", "elements": [ 11, 15, 16 ], "cost": 159 }, { "id": "S34", "elements": [ 12, 13, 14, 15, 16 ], "cost": 415 }, { "id": "S35", "elements": [ 8, 9, 11 ], "cost": 147 }, { "id": "S36", "elements": [ 7, 8, 9, 10, 11 ], "cost": 230 }, { "id": "S37", "elements": [ 11, 13, 14, 15, 18 ], "cost": 195 }, { "id": "S38", "elements": [ 5, 6, 8, 9 ], "cost": 272 }, { "id": "S39", "elements": [ 14, 15, 18, 19 ], "cost": 212 }, { "id": "S40", "elements": [ 14, 15, 17, 18, 19 ], "cost": 200 }, { "id": "S41", "elements": [ 1, 2, 3 ], "cost": 207 }, { "id": "S42", "elements": [ 11, 12, 16 ], "cost": 78 }, { "id": "S43", "elements": [ 13, 15, 16, 17 ], "cost": 180 }, { "id": "S44", "elements": [ 7, 9, 12 ], "cost": 3 }, { "id": "S45", "elements": [ 6, 8, 11 ], "cost": 120 }, { "id": "S46", "elements": [ 5, 7, 9 ], "cost": 102 }, { "id": "S47", "elements": [ 5, 7, 8, 9, 10 ], "cost": 55 }, { "id": "S48", "elements": [ 0, 1, 2, 3, 9 ], "cost": 110 }, { "id": "S49", "elements": [ 3, 4, 6, 7 ], "cost": 348 }, { "id": "S50", "elements": [ 4, 5, 6, 7 ], "cost": 256 }, { "id": "S51", "elements": [ 4, 6, 7 ], "cost": 132 }, { "id": "S52", "elements": [ 4, 5, 7, 8, 9, 12 ], "cost": 180 }, { "id": "S53", "elements": [ 1, 2, 5 ], "cost": 138 }, { "id": "S54", "elements": [ 7, 8, 10 ], "cost": 162 }, { "id": "S55", "elements": [ 2, 5, 6 ], "cost": 96 }, { "id": "S56", "elements": [ 9, 12, 13, 14 ], "cost": 244 }, { "id": "S57", "elements": [ 6, 7, 8, 9, 10 ], "cost": 330 }, { "id": "S58", "elements": [ 10, 11, 12, 13, 14, 15 ], "cost": 336 }, { "id": "S59", "elements": [ 7, 10, 11 ], "cost": 168 }, { "id": "S60", "elements": [ 1, 5, 6 ], "cost": 90 }, { "id": "S61", "elements": [ 12, 15, 16, 17, 18 ], "cost": 60 }, { "id": "S62", "elements": [ 1, 3, 4, 5, 6 ], "cost": 435 }, { "id": "S63", "elements": [ 3, 4, 7 ], "cost": 51 }, { "id": "S64", "elements": [ 7, 10, 11, 12 ], "cost": 368 }, { "id": "S65", "elements": [ 14, 16, 17 ], "cost": 33 }, { "id": "S66", "elements": [ 4, 5, 7, 9 ], "cost": 44 }, { "id": "S67", "elements": [ 0, 1, 4, 5, 6, 7 ], "cost": 114 }, { "id": "S68", "elements": [ 2, 3, 4, 5, 6 ], "cost": 220 }, { "id": "S69", "elements": [ 11, 12, 13, 15, 16 ], "cost": 55 }, { "id": "S70", "elements": [ 13, 14, 16, 17, 18 ], "cost": 255 }, { "id": "S71", "elements": [ 1, 2, 3, 4 ], "cost": 248 }, { "id": "S72", "elements": [ 1, 2 ], "cost": 176 }, { "id": "S73", "elements": [ 5, 7 ], "cost": 90 }, { "id": "S74", "elements": [ 2, 3, 4, 5, 6, 8 ], "cost": 210 }, { "id": "S75", "elements": [ 8, 9, 10, 13 ], "cost": 220 }, { "id": "S76", "elements": [ 0, 3, 5, 6 ], "cost": 232 }, { "id": "S77", "elements": [ 0, 1, 4 ], "cost": 108 }, { "id": "S78", "elements": [ 0, 1, 2, 3 ], "cost": 340 }, { "id": "S79", "elements": [ 9, 11, 12, 13 ], "cost": 352 }, { "id": "S80", "elements": [ 10, 12, 13, 14, 15 ], "cost": 155 }, { "id": "S81", "elements": [ 2, 3, 4, 7 ], "cost": 308 }, { "id": "S82", "elements": [ 11, 12, 13, 15 ], "cost": 360 }, { "id": "S83", "elements": [ 0, 1, 2, 4, 5 ], "cost": 235 }, { "id": "S84", "elements": [ 15, 16, 17, 18 ], "cost": 44 }, { "id": "S85", "elements": [ 8, 9, 10, 12 ], "cost": 316 }, { "id": "S86", "elements": [ 7, 8, 9, 11 ], "cost": 44 }, { "id": "S87", "elements": [ 14, 15, 16 ], "cost": 261 }, { "id": "S88", "elements": [ 11, 12, 14, 15 ], "cost": 56 }, { "id": "S89", "elements": [ 11, 13, 14 ], "cost": 210 }, { "id": "S90", "elements": [ 0, 1, 2 ], "cost": 78 }, { "id": "S91", "elements": [ 13, 14, 16, 19 ], "cost": 72 }, { "id": "S92", "elements": [ 0 ], "cost": 10000 }, { "id": "S93", "elements": [ 1 ], "cost": 10000 }, { "id": "S94", "elements": [ 2 ], "cost": 10000 }, { "id": "S95", "elements": [ 3 ], "cost": 10000 }, { "id": "S96", "elements": [ 4 ], "cost": 10000 }, { "id": "S97", "elements": [ 5 ], "cost": 10000 }, { "id": "S98", "elements": [ 6 ], "cost": 10000 }, { "id": "S99", "elements": [ 7 ], "cost": 10000 }, { "id": "S100", "elements": [ 8 ], "cost": 10000 }, { "id": "S101", "elements": [ 9 ], "cost": 10000 }, { "id": "S102", "elements": [ 10 ], "cost": 10000 }, { "id": "S103", "elements": [ 11 ], "cost": 10000 }, { "id": "S104", "elements": [ 12 ], "cost": 10000 }, { "id": "S105", "elements": [ 13 ], "cost": 10000 }, { "id": "S106", "elements": [ 14 ], "cost": 10000 }, { "id": "S107", "elements": [ 15 ], "cost": 10000 }, { "id": "S108", "elements": [ 16 ], "cost": 10000 }, { "id": "S109", "elements": [ 17 ], "cost": 10000 }, { "id": "S110", "elements": [ 18 ], "cost": 10000 }, { "id": "S111", "elements": [ 19 ], "cost": 10000 } ] }, "solution_variant": [ "S2", "S7", "S9", "S16", "S23", "S24" ], "context_index": 32, "input_format": "nl", "input_index_base": 0 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "Recently a couple looked over a bunch of vendor packages, and each package takes care of different pieces of the day’s logistics. They must pick some packages so every single task is covered exactly once (no missing services and no two packages doing the same job), and they want the cheapest overall bill — the cost is simply the sum of the chosen package prices. The concrete list of packages and tasks follows below.\n\nThere are 20 distinct wedding tasks they must cover exactly once, and 115 vendor bundles available to choose from.\nVendor bundle S1 handles tasks 6 8 10 11 12 13 and costs 330.\nVendor bundle S2 handles tasks 0 4 9 and costs 177.\nVendor bundle S3 handles tasks 1 2 4 5 and costs 152.\nVendor bundle S4 handles tasks 1 4 5 8 and costs 216.\nVendor bundle S5 handles tasks 0 1 3 and costs 219.\nVendor bundle S6 handles tasks 14 16 18 19 and costs 400.\nVendor bundle S7 handles tasks 0 1 2 3 4 5 and costs 180.\nVendor bundle S8 handles tasks 0 2 3 4 6 and costs 455.\nVendor bundle S9 handles tasks 14 15 16 17 18 19 and costs 54.\nVendor bundle S10 handles tasks 5 9 and costs 176.\nVendor bundle S11 handles tasks 1 2 3 and costs 63.\nVendor bundle S12 handles tasks 0 3 4 and costs 174.\nVendor bundle S13 handles tasks 10 11 12 13 14 and costs 470.\nVendor bundle S14 handles tasks 14 15 16 17 18 and costs 400.\nVendor bundle S15 handles tasks 7 8 9 12 and costs 144.\nVendor bundle S16 handles tasks 12 14 15 18 and costs 284.\nVendor bundle S17 handles tasks 0 1 5 8 and costs 112.\nVendor bundle S18 handles tasks 7 8 9 10 15 and costs 385.\nVendor bundle S19 handles tasks 9 10 13 14 and costs 216.\nVendor bundle S20 handles tasks 17 18 and costs 126.\nVendor bundle S21 handles tasks 13 14 15 16 17 18 and costs 450.\nVendor bundle S22 handles tasks 2 3 and costs 50.\nVendor bundle S23 handles tasks 3 4 5 8 10 and costs 155.\nVendor bundle S24 handles tasks 17 19 and costs 150.\nVendor bundle S25 handles tasks 1 2 and costs 16.\nVendor bundle S26 handles tasks 4 6 7 and costs 300.\nVendor bundle S27 handles tasks 6 7 11 and costs 93.\nVendor bundle S28 handles tasks 15 16 17 18 19 and costs 140.\nVendor bundle S29 handles tasks 2 3 4 8 and costs 228.\nVendor bundle S30 handles tasks 5 8 9 10 11 and costs 455.\nVendor bundle S31 handles tasks 2 4 5 6 7 8 and costs 204.\nVendor bundle S32 handles tasks 6 7 and costs 82.\nVendor bundle S33 handles tasks 9 11 12 and costs 6.\nVendor bundle S34 handles tasks 6 8 9 11 and costs 144.\nVendor bundle S35 handles tasks 4 5 6 8 9 and costs 45.\nVendor bundle S36 handles tasks 3 5 6 7 9 10 and costs 372.\nVendor bundle S37 handles tasks 6 7 8 10 13 and costs 110.\nVendor bundle S38 handles tasks 5 6 9 10 11 and costs 40.\nVendor bundle S39 handles tasks 5 6 7 10 and costs 52.\nVendor bundle S40 handles tasks 1 2 3 4 6 8 and costs 210.\nVendor bundle S41 handles tasks 0 1 2 4 5 and costs 80.\nVendor bundle S42 handles tasks 11 13 15 16 19 and costs 370.\nVendor bundle S43 handles tasks 7 9 10 and costs 273.\nVendor bundle S44 handles tasks 5 6 and costs 164.\nVendor bundle S45 handles tasks 10 11 15 and costs 198.\nVendor bundle S46 handles tasks 1 2 3 4 7 8 and costs 300.\nVendor bundle S47 handles tasks 6 9 13 and costs 15.\nVendor bundle S48 handles tasks 4 5 6 7 9 and costs 275.\nVendor bundle S49 handles tasks 2 3 5 6 and costs 132.\nVendor bundle S50 handles tasks 3 4 6 and costs 198.\nVendor bundle S51 handles tasks 6 7 8 9 10 and costs 155.\nVendor bundle S52 handles tasks 14 16 18 and costs 147.\nVendor bundle S53 handles tasks 15 17 18 19 and costs 148.\nVendor bundle S54 handles tasks 5 8 10 and costs 54.\nVendor bundle S55 handles tasks 6 9 10 13 and costs 296.\nVendor bundle S56 handles tasks 5 6 7 8 9 and costs 240.\nVendor bundle S57 handles tasks 13 14 16 17 and costs 312.\nVendor bundle S58 handles tasks 11 12 13 14 16 and costs 215.\nVendor bundle S59 handles tasks 12 15 and costs 52.\nVendor bundle S60 handles tasks 16 19 and costs 36.\nVendor bundle S61 handles tasks 12 13 14 16 and costs 332.\nVendor bundle S62 handles tasks 11 12 13 14 and costs 300.\nVendor bundle S63 handles tasks 0 1 2 3 5 and costs 400.\nVendor bundle S64 handles tasks 0 1 3 6 and costs 368.\nVendor bundle S65 handles tasks 6 11 13 14 15 16 and costs 264.\nVendor bundle S66 handles tasks 12 16 and costs 26.\nVendor bundle S67 handles tasks 7 8 10 12 and costs 356.\nVendor bundle S68 handles tasks 10 11 12 and costs 201.\nVendor bundle S69 handles tasks 7 9 11 and costs 159.\nVendor bundle S70 handles tasks 14 16 17 19 and costs 368.\nVendor bundle S71 handles tasks 7 8 9 10 11 and costs 380.\nVendor bundle S72 handles tasks 8 12 14 and costs 51.\nVendor bundle S73 handles tasks 7 10 13 and costs 165.\nVendor bundle S74 handles tasks 9 11 12 14 17 and costs 485.\nVendor bundle S75 handles tasks 8 9 11 and costs 216.\nVendor bundle S76 handles tasks 8 10 11 12 14 and costs 495.\nVendor bundle S77 handles tasks 4 5 and costs 154.\nVendor bundle S78 handles tasks 13 17 18 and costs 195.\nVendor bundle S79 handles tasks 2 3 4 6 7 and costs 305.\nVendor bundle S80 handles tasks 4 7 and costs 78.\nVendor bundle S81 handles tasks 4 6 7 8 9 and costs 460.\nVendor bundle S82 handles tasks 1 3 4 and costs 87.\nVendor bundle S83 handles tasks 10 14 15 and costs 21.\nVendor bundle S84 handles tasks 6 8 9 10 and costs 240.\nVendor bundle S85 handles tasks 10 11 12 13 and costs 204.\nVendor bundle S86 handles tasks 9 10 11 13 and costs 312.\nVendor bundle S87 handles tasks 13 14 15 17 and costs 140.\nVendor bundle S88 handles tasks 4 6 7 9 10 and costs 125.\nVendor bundle S89 handles tasks 15 16 17 18 and costs 124.\nVendor bundle S90 handles tasks 10 13 14 17 and costs 12.\nVendor bundle S91 handles tasks 11 12 13 14 15 16 and costs 246.\nVendor bundle S92 handles tasks 5 6 7 8 and costs 172.\nVendor bundle S93 handles tasks 10 11 12 13 15 and costs 265.\nVendor bundle S94 handles tasks 0 1 2 4 and costs 20.\nVendor bundle S95 handles tasks 7 9 and costs 194.\nVendor bundle S96 handles tasks 0 and costs 10000.\nVendor bundle S97 handles tasks 1 and costs 10000.\nVendor bundle S98 handles tasks 2 and costs 10000.\nVendor bundle S99 handles tasks 3 and costs 10000.\nVendor bundle S100 handles tasks 4 and costs 10000.\nVendor bundle S101 handles tasks 5 and costs 10000.\nVendor bundle S102 handles tasks 6 and costs 10000.\nVendor bundle S103 handles tasks 7 and costs 10000.\nVendor bundle S104 handles tasks 8 and costs 10000.\nVendor bundle S105 handles tasks 9 and costs 10000.\nVendor bundle S106 handles tasks 10 and costs 10000.\nVendor bundle S107 handles tasks 11 and costs 10000.\nVendor bundle S108 handles tasks 12 and costs 10000.\nVendor bundle S109 handles tasks 13 and costs 10000.\nVendor bundle S110 handles tasks 14 and costs 10000.\nVendor bundle S111 handles tasks 15 and costs 10000.\nVendor bundle S112 handles tasks 16 and costs 10000.\nVendor bundle S113 handles tasks 17 and costs 10000.\nVendor bundle S114 handles tasks 18 and costs 10000.\nVendor bundle S115 handles tasks 19 and costs 10000.\nThey will select some of these bundles so every one of the 20 tasks is covered exactly once at the lowest total cost.\n\nOh, and when you send your pick, please stick to this simple JSON shape so it's easy to read and check:\n\n{\n \"solution\": [\"package_id\", ...]\n}\n\nHere \"solution\" is the list of packages you choose for the day — each entry should be the exact package identifier from the instance (one per chosen package). This JSON is just a sketch of the expected shape, not the final answer itself.\n\nPlease use the identifiers exactly as they appear in the instance input — do not rename them or invent new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 20, "num_sets": 115, "density": 0.1726086956521739, "sets": [ { "id": 1, "elements": [ 7, 9, 11, 12, 13, 14 ], "cost": 330 }, { "id": 2, "elements": [ 1, 5, 10 ], "cost": 177 }, { "id": 3, "elements": [ 2, 3, 5, 6 ], "cost": 152 }, { "id": 4, "elements": [ 2, 5, 6, 9 ], "cost": 216 }, { "id": 5, "elements": [ 1, 2, 4 ], "cost": 219 }, { "id": 6, "elements": [ 15, 17, 19, 20 ], "cost": 400 }, { "id": 7, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 180 }, { "id": 8, "elements": [ 1, 3, 4, 5, 7 ], "cost": 455 }, { "id": 9, "elements": [ 15, 16, 17, 18, 19, 20 ], "cost": 54 }, { "id": 10, "elements": [ 6, 10 ], "cost": 176 }, { "id": 11, "elements": [ 2, 3, 4 ], "cost": 63 }, { "id": 12, "elements": [ 1, 4, 5 ], "cost": 174 }, { "id": 13, "elements": [ 11, 12, 13, 14, 15 ], "cost": 470 }, { "id": 14, "elements": [ 15, 16, 17, 18, 19 ], "cost": 400 }, { "id": 15, "elements": [ 8, 9, 10, 13 ], "cost": 144 }, { "id": 16, "elements": [ 13, 15, 16, 19 ], "cost": 284 }, { "id": 17, "elements": [ 1, 2, 6, 9 ], "cost": 112 }, { "id": 18, "elements": [ 8, 9, 10, 11, 16 ], "cost": 385 }, { "id": 19, "elements": [ 10, 11, 14, 15 ], "cost": 216 }, { "id": 20, "elements": [ 18, 19 ], "cost": 126 }, { "id": 21, "elements": [ 14, 15, 16, 17, 18, 19 ], "cost": 450 }, { "id": 22, "elements": [ 3, 4 ], "cost": 50 }, { "id": 23, "elements": [ 4, 5, 6, 9, 11 ], "cost": 155 }, { "id": 24, "elements": [ 18, 20 ], "cost": 150 }, { "id": 25, "elements": [ 2, 3 ], "cost": 16 }, { "id": 26, "elements": [ 5, 7, 8 ], "cost": 300 }, { "id": 27, "elements": [ 7, 8, 12 ], "cost": 93 }, { "id": 28, "elements": [ 16, 17, 18, 19, 20 ], "cost": 140 }, { "id": 29, "elements": [ 3, 4, 5, 9 ], "cost": 228 }, { "id": 30, "elements": [ 6, 9, 10, 11, 12 ], "cost": 455 }, { "id": 31, "elements": [ 3, 5, 6, 7, 8, 9 ], "cost": 204 }, { "id": 32, "elements": [ 7, 8 ], "cost": 82 }, { "id": 33, "elements": [ 10, 12, 13 ], "cost": 6 }, { "id": 34, "elements": [ 7, 9, 10, 12 ], "cost": 144 }, { "id": 35, "elements": [ 5, 6, 7, 9, 10 ], "cost": 45 }, { "id": 36, "elements": [ 4, 6, 7, 8, 10, 11 ], "cost": 372 }, { "id": 37, "elements": [ 7, 8, 9, 11, 14 ], "cost": 110 }, { "id": 38, "elements": [ 6, 7, 10, 11, 12 ], "cost": 40 }, { "id": 39, "elements": [ 6, 7, 8, 11 ], "cost": 52 }, { "id": 40, "elements": [ 2, 3, 4, 5, 7, 9 ], "cost": 210 }, { "id": 41, "elements": [ 1, 2, 3, 5, 6 ], "cost": 80 }, { "id": 42, "elements": [ 12, 14, 16, 17, 20 ], "cost": 370 }, { "id": 43, "elements": [ 8, 10, 11 ], "cost": 273 }, { "id": 44, "elements": [ 6, 7 ], "cost": 164 }, { "id": 45, "elements": [ 11, 12, 16 ], "cost": 198 }, { "id": 46, "elements": [ 2, 3, 4, 5, 8, 9 ], "cost": 300 }, { "id": 47, "elements": [ 7, 10, 14 ], "cost": 15 }, { "id": 48, "elements": [ 5, 6, 7, 8, 10 ], "cost": 275 }, { "id": 49, "elements": [ 3, 4, 6, 7 ], "cost": 132 }, { "id": 50, "elements": [ 4, 5, 7 ], "cost": 198 }, { "id": 51, "elements": [ 7, 8, 9, 10, 11 ], "cost": 155 }, { "id": 52, "elements": [ 15, 17, 19 ], "cost": 147 }, { "id": 53, "elements": [ 16, 18, 19, 20 ], "cost": 148 }, { "id": 54, "elements": [ 6, 9, 11 ], "cost": 54 }, { "id": 55, "elements": [ 7, 10, 11, 14 ], "cost": 296 }, { "id": 56, "elements": [ 6, 7, 8, 9, 10 ], "cost": 240 }, { "id": 57, "elements": [ 14, 15, 17, 18 ], "cost": 312 }, { "id": 58, "elements": [ 12, 13, 14, 15, 17 ], "cost": 215 }, { "id": 59, "elements": [ 13, 16 ], "cost": 52 }, { "id": 60, "elements": [ 17, 20 ], "cost": 36 }, { "id": 61, "elements": [ 13, 14, 15, 17 ], "cost": 332 }, { "id": 62, "elements": [ 12, 13, 14, 15 ], "cost": 300 }, { "id": 63, "elements": [ 1, 2, 3, 4, 6 ], "cost": 400 }, { "id": 64, "elements": [ 1, 2, 4, 7 ], "cost": 368 }, { "id": 65, "elements": [ 7, 12, 14, 15, 16, 17 ], "cost": 264 }, { "id": 66, "elements": [ 13, 17 ], "cost": 26 }, { "id": 67, "elements": [ 8, 9, 11, 13 ], "cost": 356 }, { "id": 68, "elements": [ 11, 12, 13 ], "cost": 201 }, { "id": 69, "elements": [ 8, 10, 12 ], "cost": 159 }, { "id": 70, "elements": [ 15, 17, 18, 20 ], "cost": 368 }, { "id": 71, "elements": [ 8, 9, 10, 11, 12 ], "cost": 380 }, { "id": 72, "elements": [ 9, 13, 15 ], "cost": 51 }, { "id": 73, "elements": [ 8, 11, 14 ], "cost": 165 }, { "id": 74, "elements": [ 10, 12, 13, 15, 18 ], "cost": 485 }, { "id": 75, "elements": [ 9, 10, 12 ], "cost": 216 }, { "id": 76, "elements": [ 9, 11, 12, 13, 15 ], "cost": 495 }, { "id": 77, "elements": [ 5, 6 ], "cost": 154 }, { "id": 78, "elements": [ 14, 18, 19 ], "cost": 195 }, { "id": 79, "elements": [ 3, 4, 5, 7, 8 ], "cost": 305 }, { "id": 80, "elements": [ 5, 8 ], "cost": 78 }, { "id": 81, "elements": [ 5, 7, 8, 9, 10 ], "cost": 460 }, { "id": 82, "elements": [ 2, 4, 5 ], "cost": 87 }, { "id": 83, "elements": [ 11, 15, 16 ], "cost": 21 }, { "id": 84, "elements": [ 7, 9, 10, 11 ], "cost": 240 }, { "id": 85, "elements": [ 11, 12, 13, 14 ], "cost": 204 }, { "id": 86, "elements": [ 10, 11, 12, 14 ], "cost": 312 }, { "id": 87, "elements": [ 14, 15, 16, 18 ], "cost": 140 }, { "id": 88, "elements": [ 5, 7, 8, 10, 11 ], "cost": 125 }, { "id": 89, "elements": [ 16, 17, 18, 19 ], "cost": 124 }, { "id": 90, "elements": [ 11, 14, 15, 18 ], "cost": 12 }, { "id": 91, "elements": [ 12, 13, 14, 15, 16, 17 ], "cost": 246 }, { "id": 92, "elements": [ 6, 7, 8, 9 ], "cost": 172 }, { "id": 93, "elements": [ 11, 12, 13, 14, 16 ], "cost": 265 }, { "id": 94, "elements": [ 1, 2, 3, 5 ], "cost": 20 }, { "id": 95, "elements": [ 8, 10 ], "cost": 194 }, { "id": 96, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 97, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 98, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 20 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0032_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0032_bag_stats.png" }, "solution": [ 7, 9, 33, 37 ], "obj": 350.0, "instance_variant": { "num_elements": 20, "num_sets": 115, "sets": [ { "id": "S1", "elements": [ 6, 8, 10, 11, 12, 13 ], "cost": 330 }, { "id": "S2", "elements": [ 0, 4, 9 ], "cost": 177 }, { "id": "S3", "elements": [ 1, 2, 4, 5 ], "cost": 152 }, { "id": "S4", "elements": [ 1, 4, 5, 8 ], "cost": 216 }, { "id": "S5", "elements": [ 0, 1, 3 ], "cost": 219 }, { "id": "S6", "elements": [ 14, 16, 18, 19 ], "cost": 400 }, { "id": "S7", "elements": [ 0, 1, 2, 3, 4, 5 ], "cost": 180 }, { "id": "S8", "elements": [ 0, 2, 3, 4, 6 ], "cost": 455 }, { "id": "S9", "elements": [ 14, 15, 16, 17, 18, 19 ], "cost": 54 }, { "id": "S10", "elements": [ 5, 9 ], "cost": 176 }, { "id": "S11", "elements": [ 1, 2, 3 ], "cost": 63 }, { "id": "S12", "elements": [ 0, 3, 4 ], "cost": 174 }, { "id": "S13", "elements": [ 10, 11, 12, 13, 14 ], "cost": 470 }, { "id": "S14", "elements": [ 14, 15, 16, 17, 18 ], "cost": 400 }, { "id": "S15", "elements": [ 7, 8, 9, 12 ], "cost": 144 }, { "id": "S16", "elements": [ 12, 14, 15, 18 ], "cost": 284 }, { "id": "S17", "elements": [ 0, 1, 5, 8 ], "cost": 112 }, { "id": "S18", "elements": [ 7, 8, 9, 10, 15 ], "cost": 385 }, { "id": "S19", "elements": [ 9, 10, 13, 14 ], "cost": 216 }, { "id": "S20", "elements": [ 17, 18 ], "cost": 126 }, { "id": "S21", "elements": [ 13, 14, 15, 16, 17, 18 ], "cost": 450 }, { "id": "S22", "elements": [ 2, 3 ], "cost": 50 }, { "id": "S23", "elements": [ 3, 4, 5, 8, 10 ], "cost": 155 }, { "id": "S24", "elements": [ 17, 19 ], "cost": 150 }, { "id": "S25", "elements": [ 1, 2 ], "cost": 16 }, { "id": "S26", "elements": [ 4, 6, 7 ], "cost": 300 }, { "id": "S27", "elements": [ 6, 7, 11 ], "cost": 93 }, { "id": "S28", "elements": [ 15, 16, 17, 18, 19 ], "cost": 140 }, { "id": "S29", "elements": [ 2, 3, 4, 8 ], "cost": 228 }, { "id": "S30", "elements": [ 5, 8, 9, 10, 11 ], "cost": 455 }, { "id": "S31", "elements": [ 2, 4, 5, 6, 7, 8 ], "cost": 204 }, { "id": "S32", "elements": [ 6, 7 ], "cost": 82 }, { "id": "S33", "elements": [ 9, 11, 12 ], "cost": 6 }, { "id": "S34", "elements": [ 6, 8, 9, 11 ], "cost": 144 }, { "id": "S35", "elements": [ 4, 5, 6, 8, 9 ], "cost": 45 }, { "id": "S36", "elements": [ 3, 5, 6, 7, 9, 10 ], "cost": 372 }, { "id": "S37", "elements": [ 6, 7, 8, 10, 13 ], "cost": 110 }, { "id": "S38", "elements": [ 5, 6, 9, 10, 11 ], "cost": 40 }, { "id": "S39", "elements": [ 5, 6, 7, 10 ], "cost": 52 }, { "id": "S40", "elements": [ 1, 2, 3, 4, 6, 8 ], "cost": 210 }, { "id": "S41", "elements": [ 0, 1, 2, 4, 5 ], "cost": 80 }, { "id": "S42", "elements": [ 11, 13, 15, 16, 19 ], "cost": 370 }, { "id": "S43", "elements": [ 7, 9, 10 ], "cost": 273 }, { "id": "S44", "elements": [ 5, 6 ], "cost": 164 }, { "id": "S45", "elements": [ 10, 11, 15 ], "cost": 198 }, { "id": "S46", "elements": [ 1, 2, 3, 4, 7, 8 ], "cost": 300 }, { "id": "S47", "elements": [ 6, 9, 13 ], "cost": 15 }, { "id": "S48", "elements": [ 4, 5, 6, 7, 9 ], "cost": 275 }, { "id": "S49", "elements": [ 2, 3, 5, 6 ], "cost": 132 }, { "id": "S50", "elements": [ 3, 4, 6 ], "cost": 198 }, { "id": "S51", "elements": [ 6, 7, 8, 9, 10 ], "cost": 155 }, { "id": "S52", "elements": [ 14, 16, 18 ], "cost": 147 }, { "id": "S53", "elements": [ 15, 17, 18, 19 ], "cost": 148 }, { "id": "S54", "elements": [ 5, 8, 10 ], "cost": 54 }, { "id": "S55", "elements": [ 6, 9, 10, 13 ], "cost": 296 }, { "id": "S56", "elements": [ 5, 6, 7, 8, 9 ], "cost": 240 }, { "id": "S57", "elements": [ 13, 14, 16, 17 ], "cost": 312 }, { "id": "S58", "elements": [ 11, 12, 13, 14, 16 ], "cost": 215 }, { "id": "S59", "elements": [ 12, 15 ], "cost": 52 }, { "id": "S60", "elements": [ 16, 19 ], "cost": 36 }, { "id": "S61", "elements": [ 12, 13, 14, 16 ], "cost": 332 }, { "id": "S62", "elements": [ 11, 12, 13, 14 ], "cost": 300 }, { "id": "S63", "elements": [ 0, 1, 2, 3, 5 ], "cost": 400 }, { "id": "S64", "elements": [ 0, 1, 3, 6 ], "cost": 368 }, { "id": "S65", "elements": [ 6, 11, 13, 14, 15, 16 ], "cost": 264 }, { "id": "S66", "elements": [ 12, 16 ], "cost": 26 }, { "id": "S67", "elements": [ 7, 8, 10, 12 ], "cost": 356 }, { "id": "S68", "elements": [ 10, 11, 12 ], "cost": 201 }, { "id": "S69", "elements": [ 7, 9, 11 ], "cost": 159 }, { "id": "S70", "elements": [ 14, 16, 17, 19 ], "cost": 368 }, { "id": "S71", "elements": [ 7, 8, 9, 10, 11 ], "cost": 380 }, { "id": "S72", "elements": [ 8, 12, 14 ], "cost": 51 }, { "id": "S73", "elements": [ 7, 10, 13 ], "cost": 165 }, { "id": "S74", "elements": [ 9, 11, 12, 14, 17 ], "cost": 485 }, { "id": "S75", "elements": [ 8, 9, 11 ], "cost": 216 }, { "id": "S76", "elements": [ 8, 10, 11, 12, 14 ], "cost": 495 }, { "id": "S77", "elements": [ 4, 5 ], "cost": 154 }, { "id": "S78", "elements": [ 13, 17, 18 ], "cost": 195 }, { "id": "S79", "elements": [ 2, 3, 4, 6, 7 ], "cost": 305 }, { "id": "S80", "elements": [ 4, 7 ], "cost": 78 }, { "id": "S81", "elements": [ 4, 6, 7, 8, 9 ], "cost": 460 }, { "id": "S82", "elements": [ 1, 3, 4 ], "cost": 87 }, { "id": "S83", "elements": [ 10, 14, 15 ], "cost": 21 }, { "id": "S84", "elements": [ 6, 8, 9, 10 ], "cost": 240 }, { "id": "S85", "elements": [ 10, 11, 12, 13 ], "cost": 204 }, { "id": "S86", "elements": [ 9, 10, 11, 13 ], "cost": 312 }, { "id": "S87", "elements": [ 13, 14, 15, 17 ], "cost": 140 }, { "id": "S88", "elements": [ 4, 6, 7, 9, 10 ], "cost": 125 }, { "id": "S89", "elements": [ 15, 16, 17, 18 ], "cost": 124 }, { "id": "S90", "elements": [ 10, 13, 14, 17 ], "cost": 12 }, { "id": "S91", "elements": [ 11, 12, 13, 14, 15, 16 ], "cost": 246 }, { "id": "S92", "elements": [ 5, 6, 7, 8 ], "cost": 172 }, { "id": "S93", "elements": [ 10, 11, 12, 13, 15 ], "cost": 265 }, { "id": "S94", "elements": [ 0, 1, 2, 4 ], "cost": 20 }, { "id": "S95", "elements": [ 7, 9 ], "cost": 194 }, { "id": "S96", "elements": [ 0 ], "cost": 10000 }, { "id": "S97", "elements": [ 1 ], "cost": 10000 }, { "id": "S98", "elements": [ 2 ], "cost": 10000 }, { "id": "S99", "elements": [ 3 ], "cost": 10000 }, { "id": "S100", "elements": [ 4 ], "cost": 10000 }, { "id": "S101", "elements": [ 5 ], "cost": 10000 }, { "id": "S102", "elements": [ 6 ], "cost": 10000 }, { "id": "S103", "elements": [ 7 ], "cost": 10000 }, { "id": "S104", "elements": [ 8 ], "cost": 10000 }, { "id": "S105", "elements": [ 9 ], "cost": 10000 }, { "id": "S106", "elements": [ 10 ], "cost": 10000 }, { "id": "S107", "elements": [ 11 ], "cost": 10000 }, { "id": "S108", "elements": [ 12 ], "cost": 10000 }, { "id": "S109", "elements": [ 13 ], "cost": 10000 }, { "id": "S110", "elements": [ 14 ], "cost": 10000 }, { "id": "S111", "elements": [ 15 ], "cost": 10000 }, { "id": "S112", "elements": [ 16 ], "cost": 10000 }, { "id": "S113", "elements": [ 17 ], "cost": 10000 }, { "id": "S114", "elements": [ 18 ], "cost": 10000 }, { "id": "S115", "elements": [ 19 ], "cost": 10000 } ] }, "solution_variant": [ "S7", "S9", "S33", "S37" ], "context_index": 33, "input_format": "nl", "input_index_base": 0 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "Imagine the HR team must pick which benefit packages to use and slot every person into just one package — nobody can be left uncovered or put into two packages. Each package has a price tag for the employer, and the goal is to make the company’s total payout as small as possible by adding up the employer contributions of the packages that are selected. The concrete options and numbers are shown below.\n\n{\n \"total_employees\": 24,\n \"num_bundles\": 131,\n \"sets\": [\n {\n \"bundle_id\": \"S1\",\n \"covered_employee_ids\": [\n \"B\",\n \"C\",\n \"D\",\n \"E\"\n ],\n \"employer_contribution\": 208\n },\n {\n \"bundle_id\": \"S2\",\n \"covered_employee_ids\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"F\",\n \"G\"\n ],\n \"employer_contribution\": 216\n },\n {\n \"bundle_id\": \"S3\",\n \"covered_employee_ids\": [\n \"B\",\n \"C\",\n \"D\",\n \"H\"\n ],\n \"employer_contribution\": 264\n },\n {\n \"bundle_id\": \"S4\",\n \"covered_employee_ids\": [\n \"L\",\n \"T\",\n \"U\",\n \"V\",\n \"W\"\n ],\n \"employer_contribution\": 265\n },\n {\n \"bundle_id\": \"S5\",\n \"covered_employee_ids\": [\n \"R\",\n \"T\",\n \"V\",\n \"W\"\n ],\n \"employer_contribution\": 316\n },\n {\n \"bundle_id\": \"S6\",\n \"covered_employee_ids\": [\n \"K\",\n \"M\"\n ],\n \"employer_contribution\": 88\n },\n {\n \"bundle_id\": \"S7\",\n \"covered_employee_ids\": [\n \"O\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"employer_contribution\": 658\n },\n {\n \"bundle_id\": \"S8\",\n \"covered_employee_ids\": [\n \"T\",\n \"U\",\n \"V\"\n ],\n \"employer_contribution\": 9\n },\n {\n \"bundle_id\": \"S9\",\n \"covered_employee_ids\": [\n \"O\",\n \"P\",\n \"Q\",\n \"T\"\n ],\n \"employer_contribution\": 36\n },\n {\n \"bundle_id\": \"S10\",\n \"covered_employee_ids\": [\n \"H\",\n \"K\",\n \"L\",\n \"O\"\n ],\n \"employer_contribution\": 160\n },\n {\n \"bundle_id\": \"S11\",\n \"covered_employee_ids\": [\n \"K\",\n \"M\",\n \"O\",\n \"P\",\n \"R\"\n ],\n \"employer_contribution\": 475\n },\n {\n \"bundle_id\": \"S12\",\n \"covered_employee_ids\": [\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\"\n ],\n \"employer_contribution\": 390\n },\n {\n \"bundle_id\": \"S13\",\n \"covered_employee_ids\": [\n \"A\",\n \"B\",\n \"F\",\n \"G\"\n ],\n \"employer_contribution\": 372\n },\n {\n \"bundle_id\": \"S14\",\n \"covered_employee_ids\": [\n \"O\",\n \"P\",\n \"Q\"\n ],\n \"employer_contribution\": 33\n },\n {\n \"bundle_id\": \"S15\",\n \"covered_employee_ids\": [\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"employer_contribution\": 320\n },\n {\n \"bundle_id\": \"S16\",\n \"covered_employee_ids\": [\n \"A\",\n \"B\",\n \"E\",\n \"G\",\n \"I\"\n ],\n \"employer_contribution\": 10\n },\n {\n \"bundle_id\": \"S17\",\n \"covered_employee_ids\": [\n \"O\",\n \"P\",\n \"U\"\n ],\n \"employer_contribution\": 162\n },\n {\n \"bundle_id\": \"S18\",\n \"covered_employee_ids\": [\n \"P\",\n \"Q\",\n \"S\",\n \"T\",\n \"U\",\n \"V\"\n ],\n \"employer_contribution\": 132\n },\n {\n \"bundle_id\": \"S19\",\n \"covered_employee_ids\": [\n \"K\",\n \"O\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\"\n ],\n \"employer_contribution\": 234\n },\n {\n \"bundle_id\": \"S20\",\n \"covered_employee_ids\": [\n \"R\",\n \"S\",\n \"T\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"employer_contribution\": 402\n },\n {\n \"bundle_id\": \"S21\",\n \"covered_employee_ids\": [\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"H\"\n ],\n \"employer_contribution\": 456\n },\n {\n \"bundle_id\": \"S22\",\n \"covered_employee_ids\": [\n \"Q\",\n \"R\",\n \"T\"\n ],\n \"employer_contribution\": 276\n },\n {\n \"bundle_id\": \"S23\",\n \"covered_employee_ids\": [\n \"R\",\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"employer_contribution\": 102\n },\n {\n \"bundle_id\": \"S24\",\n \"covered_employee_ids\": [\n \"C\",\n \"E\",\n \"F\",\n \"G\",\n \"H\",\n \"I\"\n ],\n \"employer_contribution\": 186\n },\n {\n \"bundle_id\": \"S25\",\n \"covered_employee_ids\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\"\n ],\n \"employer_contribution\": 84\n },\n {\n \"bundle_id\": \"S26\",\n \"covered_employee_ids\": [\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"N\"\n ],\n \"employer_contribution\": 63\n },\n {\n \"bundle_id\": \"S27\",\n \"covered_employee_ids\": [\n \"A\",\n \"B\",\n \"D\",\n \"E\"\n ],\n \"employer_contribution\": 276\n },\n {\n \"bundle_id\": \"S28\",\n \"covered_employee_ids\": [\n \"S\",\n \"T\",\n \"X\"\n ],\n \"employer_contribution\": 285\n },\n {\n \"bundle_id\": \"S29\",\n \"covered_employee_ids\": [\n \"R\",\n \"S\",\n \"T\",\n \"U\"\n ],\n \"employer_contribution\": 380\n },\n {\n \"bundle_id\": \"S30\",\n \"covered_employee_ids\": [\n \"J\",\n \"L\",\n \"M\",\n \"O\",\n \"P\",\n \"R\"\n ],\n \"employer_contribution\": 144\n },\n {\n \"bundle_id\": \"S31\",\n \"covered_employee_ids\": [\n \"B\",\n \"C\",\n \"D\",\n \"F\",\n \"H\"\n ],\n \"employer_contribution\": 40\n },\n {\n \"bundle_id\": \"S32\",\n \"covered_employee_ids\": [\n \"R\",\n \"S\",\n \"V\",\n \"W\"\n ],\n \"employer_contribution\": 216\n },\n {\n \"bundle_id\": \"S33\",\n \"covered_employee_ids\": [\n \"P\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"employer_contribution\": 630\n },\n {\n \"bundle_id\": \"S34\",\n \"covered_employee_ids\": [\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"V\",\n \"W\"\n ],\n \"employer_contribution\": 390\n },\n {\n \"bundle_id\": \"S35\",\n \"covered_employee_ids\": [\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"I\"\n ],\n \"employer_contribution\": 560\n },\n {\n \"bundle_id\": \"S36\",\n \"covered_employee_ids\": [\n \"H\",\n \"K\",\n \"L\",\n \"N\",\n \"Q\"\n ],\n \"employer_contribution\": 455\n },\n {\n \"bundle_id\": \"S37\",\n \"covered_employee_ids\": [\n \"A\",\n \"B\",\n \"C\"\n ],\n \"employer_contribution\": 246\n },\n {\n \"bundle_id\": \"S38\",\n \"covered_employee_ids\": [\n \"P\",\n \"R\",\n \"S\",\n \"T\",\n \"V\"\n ],\n \"employer_contribution\": 130\n },\n {\n \"bundle_id\": \"S39\",\n \"covered_employee_ids\": [\n \"O\",\n \"Q\",\n \"R\",\n \"S\",\n \"U\",\n \"W\"\n ],\n \"employer_contribution\": 114\n },\n {\n \"bundle_id\": \"S40\",\n \"covered_employee_ids\": [\n \"A\",\n \"B\",\n \"C\",\n \"E\",\n \"F\"\n ],\n \"employer_contribution\": 330\n },\n {\n \"bundle_id\": \"S41\",\n \"covered_employee_ids\": [\n \"P\",\n \"Q\",\n \"S\",\n \"U\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"employer_contribution\": 497\n },\n {\n \"bundle_id\": \"S42\",\n \"covered_employee_ids\": [\n \"D\",\n \"E\",\n \"F\"\n ],\n \"employer_contribution\": 117\n },\n {\n \"bundle_id\": \"S43\",\n \"covered_employee_ids\": [\n \"C\",\n \"F\",\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"L\"\n ],\n \"employer_contribution\": 77\n },\n {\n \"bundle_id\": \"S44\",\n \"covered_employee_ids\": [\n \"S\",\n \"U\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"employer_contribution\": 65\n },\n {\n \"bundle_id\": \"S45\",\n \"covered_employee_ids\": [\n \"H\",\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"P\"\n ],\n \"employer_contribution\": 399\n },\n {\n \"bundle_id\": \"S46\",\n \"covered_employee_ids\": [\n \"I\",\n \"L\",\n \"M\",\n \"O\",\n \"P\",\n \"Q\"\n ],\n \"employer_contribution\": 300\n },\n {\n \"bundle_id\": \"S47\",\n \"covered_employee_ids\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\"\n ],\n \"employer_contribution\": 78\n },\n {\n \"bundle_id\": \"S48\",\n \"covered_employee_ids\": [\n \"B\",\n \"E\",\n \"F\",\n \"G\",\n \"I\",\n \"J\"\n ],\n \"employer_contribution\": 66\n },\n {\n \"bundle_id\": \"S49\",\n \"covered_employee_ids\": [\n \"O\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"V\"\n ],\n \"employer_contribution\": 402\n },\n {\n \"bundle_id\": \"S50\",\n \"covered_employee_ids\": [\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"O\"\n ],\n \"employer_contribution\": 380\n },\n {\n \"bundle_id\": \"S51\",\n \"covered_employee_ids\": [\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"I\"\n ],\n \"employer_contribution\": 432\n },\n {\n \"bundle_id\": \"S52\",\n \"covered_employee_ids\": [\n \"J\",\n \"K\",\n \"O\",\n \"P\"\n ],\n \"employer_contribution\": 32\n },\n {\n \"bundle_id\": \"S53\",\n \"covered_employee_ids\": [\n \"H\",\n \"I\",\n \"J\",\n \"L\",\n \"M\",\n \"N\"\n ],\n \"employer_contribution\": 396\n },\n {\n \"bundle_id\": \"S54\",\n \"covered_employee_ids\": [\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"P\"\n ],\n \"employer_contribution\": 120\n },\n {\n \"bundle_id\": \"S55\",\n \"covered_employee_ids\": [\n \"E\",\n \"F\",\n \"G\",\n \"H\",\n \"J\"\n ],\n \"employer_contribution\": 105\n },\n {\n \"bundle_id\": \"S56\",\n \"covered_employee_ids\": [\n \"C\",\n \"D\",\n \"G\",\n \"H\",\n \"I\"\n ],\n \"employer_contribution\": 160\n },\n {\n \"bundle_id\": \"S57\",\n \"covered_employee_ids\": [\n \"A\",\n \"B\",\n \"C\",\n \"E\"\n ],\n \"employer_contribution\": 156\n },\n {\n \"bundle_id\": \"S58\",\n \"covered_employee_ids\": [\n \"H\",\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"S\"\n ],\n \"employer_contribution\": 426\n },\n {\n \"bundle_id\": \"S59\",\n \"covered_employee_ids\": [\n \"I\",\n \"J\",\n \"L\"\n ],\n \"employer_contribution\": 183\n },\n {\n \"bundle_id\": \"S60\",\n \"covered_employee_ids\": [\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"employer_contribution\": 413\n },\n {\n \"bundle_id\": \"S61\",\n \"covered_employee_ids\": [\n \"A\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"H\"\n ],\n \"employer_contribution\": 385\n },\n {\n \"bundle_id\": \"S62\",\n \"covered_employee_ids\": [\n \"B\",\n \"D\",\n \"E\",\n \"F\",\n \"G\"\n ],\n \"employer_contribution\": 215\n },\n {\n \"bundle_id\": \"S63\",\n \"covered_employee_ids\": [\n \"P\",\n \"S\",\n \"V\"\n ],\n \"employer_contribution\": 138\n },\n {\n \"bundle_id\": \"S64\",\n \"covered_employee_ids\": [\n \"K\",\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"S\"\n ],\n \"employer_contribution\": 12\n },\n {\n \"bundle_id\": \"S65\",\n \"covered_employee_ids\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"H\"\n ],\n \"employer_contribution\": 35\n },\n {\n \"bundle_id\": \"S66\",\n \"covered_employee_ids\": [\n \"V\",\n \"W\",\n \"X\"\n ],\n \"employer_contribution\": 69\n },\n {\n \"bundle_id\": \"S67\",\n \"covered_employee_ids\": [\n \"N\",\n \"Q\",\n \"S\",\n \"T\"\n ],\n \"employer_contribution\": 356\n },\n {\n \"bundle_id\": \"S68\",\n \"covered_employee_ids\": [\n \"E\",\n \"F\",\n \"H\"\n ],\n \"employer_contribution\": 108\n },\n {\n \"bundle_id\": \"S69\",\n \"covered_employee_ids\": [\n \"E\",\n \"F\"\n ],\n \"employer_contribution\": 2\n },\n {\n \"bundle_id\": \"S70\",\n \"covered_employee_ids\": [\n \"I\",\n \"J\",\n \"N\"\n ],\n \"employer_contribution\": 204\n },\n {\n \"bundle_id\": \"S71\",\n \"covered_employee_ids\": [\n \"T\",\n \"U\",\n \"W\",\n \"X\"\n ],\n \"employer_contribution\": 12\n },\n {\n \"bundle_id\": \"S72\",\n \"covered_employee_ids\": [\n \"Q\",\n \"R\",\n \"S\",\n \"T\"\n ],\n \"employer_contribution\": 140\n },\n {\n \"bundle_id\": \"S73\",\n \"covered_employee_ids\": [\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"P\"\n ],\n \"employer_contribution\": 273\n },\n {\n \"bundle_id\": \"S74\",\n \"covered_employee_ids\": [\n \"A\",\n \"B\",\n \"D\",\n \"E\",\n \"F\"\n ],\n \"employer_contribution\": 285\n },\n {\n \"bundle_id\": \"S75\",\n \"covered_employee_ids\": [\n \"Q\",\n \"S\",\n \"T\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"employer_contribution\": 276\n },\n {\n \"bundle_id\": \"S76\",\n \"covered_employee_ids\": [\n \"Q\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"employer_contribution\": 308\n },\n {\n \"bundle_id\": \"S77\",\n \"covered_employee_ids\": [\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"M\"\n ],\n \"employer_contribution\": 310\n },\n {\n \"bundle_id\": \"S78\",\n \"covered_employee_ids\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"F\",\n \"H\"\n ],\n \"employer_contribution\": 372\n },\n {\n \"bundle_id\": \"S79\",\n \"covered_employee_ids\": [\n \"D\",\n \"E\",\n \"F\",\n \"G\"\n ],\n \"employer_contribution\": 280\n },\n {\n \"bundle_id\": \"S80\",\n \"covered_employee_ids\": [\n \"U\",\n \"V\",\n \"W\"\n ],\n \"employer_contribution\": 255\n },\n {\n \"bundle_id\": \"S81\",\n \"covered_employee_ids\": [\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"M\"\n ],\n \"employer_contribution\": 310\n },\n {\n \"bundle_id\": \"S82\",\n \"covered_employee_ids\": [\n \"J\",\n \"L\"\n ],\n \"employer_contribution\": 168\n },\n {\n \"bundle_id\": \"S83\",\n \"covered_employee_ids\": [\n \"E\",\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"L\"\n ],\n \"employer_contribution\": 264\n },\n {\n \"bundle_id\": \"S84\",\n \"covered_employee_ids\": [\n \"L\",\n \"N\",\n \"P\",\n \"Q\",\n \"T\",\n \"U\",\n \"V\"\n ],\n \"employer_contribution\": 364\n },\n {\n \"bundle_id\": \"S85\",\n \"covered_employee_ids\": [\n \"Q\",\n \"R\",\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"employer_contribution\": 49\n },\n {\n \"bundle_id\": \"S86\",\n \"covered_employee_ids\": [\n \"J\",\n \"L\",\n \"N\"\n ],\n \"employer_contribution\": 297\n },\n {\n \"bundle_id\": \"S87\",\n \"covered_employee_ids\": [\n \"E\",\n \"G\",\n \"H\",\n \"L\"\n ],\n \"employer_contribution\": 276\n },\n {\n \"bundle_id\": \"S88\",\n \"covered_employee_ids\": [\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"K\"\n ],\n \"employer_contribution\": 145\n },\n {\n \"bundle_id\": \"S89\",\n \"covered_employee_ids\": [\n \"S\",\n \"T\",\n \"U\",\n \"X\"\n ],\n \"employer_contribution\": 172\n },\n {\n \"bundle_id\": \"S90\",\n \"covered_employee_ids\": [\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\"\n ],\n \"employer_contribution\": 497\n },\n {\n \"bundle_id\": \"S91\",\n \"covered_employee_ids\": [\n \"V\",\n \"W\"\n ],\n \"employer_contribution\": 90\n },\n {\n \"bundle_id\": \"S92\",\n \"covered_employee_ids\": [\n \"P\",\n \"R\",\n \"S\",\n \"T\",\n \"W\"\n ],\n \"employer_contribution\": 30\n },\n {\n \"bundle_id\": \"S93\",\n \"covered_employee_ids\": [\n \"O\",\n \"R\",\n \"S\"\n ],\n \"employer_contribution\": 300\n },\n {\n \"bundle_id\": \"S94\",\n \"covered_employee_ids\": [\n \"D\",\n \"G\",\n \"H\"\n ],\n \"employer_contribution\": 246\n },\n {\n \"bundle_id\": \"S95\",\n \"covered_employee_ids\": [\n \"A\",\n \"B\",\n \"C\",\n \"F\"\n ],\n \"employer_contribution\": 228\n },\n {\n \"bundle_id\": \"S96\",\n \"covered_employee_ids\": [\n \"C\",\n \"D\",\n \"E\",\n \"G\",\n \"I\"\n ],\n \"employer_contribution\": 445\n },\n {\n \"bundle_id\": \"S97\",\n \"covered_employee_ids\": [\n \"G\",\n \"I\",\n \"J\",\n \"K\"\n ],\n \"employer_contribution\": 24\n },\n {\n \"bundle_id\": \"S98\",\n \"covered_employee_ids\": [\n \"C\",\n \"E\",\n \"F\",\n \"G\",\n \"I\"\n ],\n \"employer_contribution\": 20\n },\n {\n \"bundle_id\": \"S99\",\n \"covered_employee_ids\": [\n \"J\",\n \"K\",\n \"M\"\n ],\n \"employer_contribution\": 261\n },\n {\n \"bundle_id\": \"S100\",\n \"covered_employee_ids\": [\n \"A\",\n \"B\",\n \"D\"\n ],\n \"employer_contribution\": 150\n },\n {\n \"bundle_id\": \"S101\",\n \"covered_employee_ids\": [\n \"P\",\n \"R\",\n \"S\",\n \"U\",\n \"W\"\n ],\n \"employer_contribution\": 340\n },\n {\n \"bundle_id\": \"S102\",\n \"covered_employee_ids\": [\n \"S\",\n \"T\",\n \"W\",\n \"X\"\n ],\n \"employer_contribution\": 188\n },\n {\n \"bundle_id\": \"S103\",\n \"covered_employee_ids\": [\n \"F\",\n \"G\",\n \"I\",\n \"J\",\n \"L\"\n ],\n \"employer_contribution\": 160\n },\n {\n \"bundle_id\": \"S104\",\n \"covered_employee_ids\": [\n \"E\",\n \"G\",\n \"I\",\n \"J\"\n ],\n \"employer_contribution\": 184\n },\n {\n \"bundle_id\": \"S105\",\n \"covered_employee_ids\": [\n \"D\",\n \"E\",\n \"G\"\n ],\n \"employer_contribution\": 75\n },\n {\n \"bundle_id\": \"S106\",\n \"covered_employee_ids\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"G\"\n ],\n \"employer_contribution\": 410\n },\n {\n \"bundle_id\": \"S107\",\n \"covered_employee_ids\": [\n \"Q\",\n \"R\",\n \"T\",\n \"V\"\n ],\n \"employer_contribution\": 248\n },\n {\n \"bundle_id\": \"S108\",\n \"covered_employee_ids\": [\n \"A\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S109\",\n \"covered_employee_ids\": [\n \"B\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S110\",\n \"covered_employee_ids\": [\n \"C\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S111\",\n \"covered_employee_ids\": [\n \"D\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S112\",\n \"covered_employee_ids\": [\n \"E\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S113\",\n \"covered_employee_ids\": [\n \"F\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S114\",\n \"covered_employee_ids\": [\n \"G\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S115\",\n \"covered_employee_ids\": [\n \"H\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S116\",\n \"covered_employee_ids\": [\n \"I\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S117\",\n \"covered_employee_ids\": [\n \"J\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S118\",\n \"covered_employee_ids\": [\n \"K\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S119\",\n \"covered_employee_ids\": [\n \"L\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S120\",\n \"covered_employee_ids\": [\n \"M\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S121\",\n \"covered_employee_ids\": [\n \"N\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S122\",\n \"covered_employee_ids\": [\n \"O\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S123\",\n \"covered_employee_ids\": [\n \"P\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S124\",\n \"covered_employee_ids\": [\n \"Q\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S125\",\n \"covered_employee_ids\": [\n \"R\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S126\",\n \"covered_employee_ids\": [\n \"S\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S127\",\n \"covered_employee_ids\": [\n \"T\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S128\",\n \"covered_employee_ids\": [\n \"U\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S129\",\n \"covered_employee_ids\": [\n \"V\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S130\",\n \"covered_employee_ids\": [\n \"W\"\n ],\n \"employer_contribution\": 10000\n },\n {\n \"bundle_id\": \"S131\",\n \"covered_employee_ids\": [\n \"X\"\n ],\n \"employer_contribution\": 10000\n }\n ]\n}\n\nYou can reply using this simple JSON shape so it's easy to parse — just put the package IDs you pick into the array under \"solution\".\n\n{\n \"solution\": [\"package_id\", ...]\n}\n\nThis is just a sketch: \"solution\" should hold the IDs of the benefit packages you want to use (one ID per selected package). Think of it like a tiny form: drop the exact package identifiers from the instance into that array and you're done.\n\nPlease use the identifiers exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 24, "num_sets": 131, "density": 0.1720737913486005, "sets": [ { "id": 1, "elements": [ 2, 3, 4, 5 ], "cost": 208 }, { "id": 2, "elements": [ 1, 2, 3, 4, 6, 7 ], "cost": 216 }, { "id": 3, "elements": [ 2, 3, 4, 8 ], "cost": 264 }, { "id": 4, "elements": [ 12, 20, 21, 22, 23 ], "cost": 265 }, { "id": 5, "elements": [ 18, 20, 22, 23 ], "cost": 316 }, { "id": 6, "elements": [ 11, 13 ], "cost": 88 }, { "id": 7, "elements": [ 15, 16, 17, 18, 19, 20, 21 ], "cost": 658 }, { "id": 8, "elements": [ 20, 21, 22 ], "cost": 9 }, { "id": 9, "elements": [ 15, 16, 17, 20 ], "cost": 36 }, { "id": 10, "elements": [ 8, 11, 12, 15 ], "cost": 160 }, { "id": 11, "elements": [ 11, 13, 15, 16, 18 ], "cost": 475 }, { "id": 12, "elements": [ 3, 4, 5, 6, 7 ], "cost": 390 }, { "id": 13, "elements": [ 1, 2, 6, 7 ], "cost": 372 }, { "id": 14, "elements": [ 15, 16, 17 ], "cost": 33 }, { "id": 15, "elements": [ 20, 21, 22, 23, 24 ], "cost": 320 }, { "id": 16, "elements": [ 1, 2, 5, 7, 9 ], "cost": 10 }, { "id": 17, "elements": [ 15, 16, 21 ], "cost": 162 }, { "id": 18, "elements": [ 16, 17, 19, 20, 21, 22 ], "cost": 132 }, { "id": 19, "elements": [ 11, 15, 16, 17, 18, 19 ], "cost": 234 }, { "id": 20, "elements": [ 18, 19, 20, 22, 23, 24 ], "cost": 402 }, { "id": 21, "elements": [ 2, 3, 4, 5, 6, 8 ], "cost": 456 }, { "id": 22, "elements": [ 17, 18, 20 ], "cost": 276 }, { "id": 23, "elements": [ 18, 20, 21, 22, 23, 24 ], "cost": 102 }, { "id": 24, "elements": [ 3, 5, 6, 7, 8, 9 ], "cost": 186 }, { "id": 25, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 84 }, { "id": 26, "elements": [ 7, 8, 9, 10, 11, 12, 14 ], "cost": 63 }, { "id": 27, "elements": [ 1, 2, 4, 5 ], "cost": 276 }, { "id": 28, "elements": [ 19, 20, 24 ], "cost": 285 }, { "id": 29, "elements": [ 18, 19, 20, 21 ], "cost": 380 }, { "id": 30, "elements": [ 10, 12, 13, 15, 16, 18 ], "cost": 144 }, { "id": 31, "elements": [ 2, 3, 4, 6, 8 ], "cost": 40 }, { "id": 32, "elements": [ 18, 19, 22, 23 ], "cost": 216 }, { "id": 33, "elements": [ 16, 19, 20, 21, 22, 23, 24 ], "cost": 630 }, { "id": 34, "elements": [ 17, 18, 19, 20, 22, 23 ], "cost": 390 }, { "id": 35, "elements": [ 2, 3, 4, 5, 6, 7, 9 ], "cost": 560 }, { "id": 36, "elements": [ 8, 11, 12, 14, 17 ], "cost": 455 }, { "id": 37, "elements": [ 1, 2, 3 ], "cost": 246 }, { "id": 38, "elements": [ 16, 18, 19, 20, 22 ], "cost": 130 }, { "id": 39, "elements": [ 15, 17, 18, 19, 21, 23 ], "cost": 114 }, { "id": 40, "elements": [ 1, 2, 3, 5, 6 ], "cost": 330 }, { "id": 41, "elements": [ 16, 17, 19, 21, 22, 23, 24 ], "cost": 497 }, { "id": 42, "elements": [ 4, 5, 6 ], "cost": 117 }, { "id": 43, "elements": [ 3, 6, 8, 9, 10, 11, 12 ], "cost": 77 }, { "id": 44, "elements": [ 19, 21, 22, 23, 24 ], "cost": 65 }, { "id": 45, "elements": [ 8, 10, 11, 12, 13, 14, 16 ], "cost": 399 }, { "id": 46, "elements": [ 9, 12, 13, 15, 16, 17 ], "cost": 300 }, { "id": 47, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 78 }, { "id": 48, "elements": [ 2, 5, 6, 7, 9, 10 ], "cost": 66 }, { "id": 49, "elements": [ 15, 17, 18, 19, 20, 22 ], "cost": 402 }, { "id": 50, "elements": [ 9, 10, 11, 12, 15 ], "cost": 380 }, { "id": 51, "elements": [ 2, 3, 4, 5, 6, 9 ], "cost": 432 }, { "id": 52, "elements": [ 10, 11, 15, 16 ], "cost": 32 }, { "id": 53, "elements": [ 8, 9, 10, 12, 13, 14 ], "cost": 396 }, { "id": 54, "elements": [ 11, 12, 13, 14, 15, 16 ], "cost": 120 }, { "id": 55, "elements": [ 5, 6, 7, 8, 10 ], "cost": 105 }, { "id": 56, "elements": [ 3, 4, 7, 8, 9 ], "cost": 160 }, { "id": 57, "elements": [ 1, 2, 3, 5 ], "cost": 156 }, { "id": 58, "elements": [ 8, 12, 13, 14, 15, 19 ], "cost": 426 }, { "id": 59, "elements": [ 9, 10, 12 ], "cost": 183 }, { "id": 60, "elements": [ 18, 19, 20, 21, 22, 23, 24 ], "cost": 413 }, { "id": 61, "elements": [ 1, 3, 4, 5, 6, 7, 8 ], "cost": 385 }, { "id": 62, "elements": [ 2, 4, 5, 6, 7 ], "cost": 215 }, { "id": 63, "elements": [ 16, 19, 22 ], "cost": 138 }, { "id": 64, "elements": [ 11, 13, 14, 15, 16, 19 ], "cost": 12 }, { "id": 65, "elements": [ 1, 2, 3, 4, 5, 6, 8 ], "cost": 35 }, { "id": 66, "elements": [ 22, 23, 24 ], "cost": 69 }, { "id": 67, "elements": [ 14, 17, 19, 20 ], "cost": 356 }, { "id": 68, "elements": [ 5, 6, 8 ], "cost": 108 }, { "id": 69, "elements": [ 5, 6 ], "cost": 2 }, { "id": 70, "elements": [ 9, 10, 14 ], "cost": 204 }, { "id": 71, "elements": [ 20, 21, 23, 24 ], "cost": 12 }, { "id": 72, "elements": [ 17, 18, 19, 20 ], "cost": 140 }, { "id": 73, "elements": [ 10, 11, 12, 13, 14, 15, 16 ], "cost": 273 }, { "id": 74, "elements": [ 1, 2, 4, 5, 6 ], "cost": 285 }, { "id": 75, "elements": [ 17, 19, 20, 22, 23, 24 ], "cost": 276 }, { "id": 76, "elements": [ 17, 19, 20, 21, 22, 23, 24 ], "cost": 308 }, { "id": 77, "elements": [ 8, 9, 10, 11, 13 ], "cost": 310 }, { "id": 78, "elements": [ 1, 2, 3, 4, 6, 8 ], "cost": 372 }, { "id": 79, "elements": [ 4, 5, 6, 7 ], "cost": 280 }, { "id": 80, "elements": [ 21, 22, 23 ], "cost": 255 }, { "id": 81, "elements": [ 7, 8, 9, 10, 13 ], "cost": 310 }, { "id": 82, "elements": [ 10, 12 ], "cost": 168 }, { "id": 83, "elements": [ 5, 6, 7, 8, 9, 12 ], "cost": 264 }, { "id": 84, "elements": [ 12, 14, 16, 17, 20, 21, 22 ], "cost": 364 }, { "id": 85, "elements": [ 17, 18, 20, 21, 22, 23, 24 ], "cost": 49 }, { "id": 86, "elements": [ 10, 12, 14 ], "cost": 297 }, { "id": 87, "elements": [ 5, 7, 8, 12 ], "cost": 276 }, { "id": 88, "elements": [ 3, 4, 5, 6, 11 ], "cost": 145 }, { "id": 89, "elements": [ 19, 20, 21, 24 ], "cost": 172 }, { "id": 90, "elements": [ 8, 9, 10, 11, 12, 13, 14 ], "cost": 497 }, { "id": 91, "elements": [ 22, 23 ], "cost": 90 }, { "id": 92, "elements": [ 16, 18, 19, 20, 23 ], "cost": 30 }, { "id": 93, "elements": [ 15, 18, 19 ], "cost": 300 }, { "id": 94, "elements": [ 4, 7, 8 ], "cost": 246 }, { "id": 95, "elements": [ 1, 2, 3, 6 ], "cost": 228 }, { "id": 96, "elements": [ 3, 4, 5, 7, 9 ], "cost": 445 }, { "id": 97, "elements": [ 7, 9, 10, 11 ], "cost": 24 }, { "id": 98, "elements": [ 3, 5, 6, 7, 9 ], "cost": 20 }, { "id": 99, "elements": [ 10, 11, 13 ], "cost": 261 }, { "id": 100, "elements": [ 1, 2, 4 ], "cost": 150 }, { "id": 101, "elements": [ 16, 18, 19, 21, 23 ], "cost": 340 }, { "id": 102, "elements": [ 19, 20, 23, 24 ], "cost": 188 }, { "id": 103, "elements": [ 6, 7, 9, 10, 12 ], "cost": 160 }, { "id": 104, "elements": [ 5, 7, 9, 10 ], "cost": 184 }, { "id": 105, "elements": [ 4, 5, 7 ], "cost": 75 }, { "id": 106, "elements": [ 1, 2, 3, 4, 7 ], "cost": 410 }, { "id": 107, "elements": [ 17, 18, 20, 22 ], "cost": 248 }, { "id": 108, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 118, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 119, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 120, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 121, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 122, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 123, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 124, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 125, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 126, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 127, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 128, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 129, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 130, "elements": [ 23 ], "cost": 10000, "singleton": true }, { "id": 131, "elements": [ 24 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0033_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0033_bag_stats.png" }, "solution": [ 24, 64, 82, 85, 100 ], "obj": 565.0, "instance_variant": { "num_elements": 24, "num_sets": 131, "sets": [ { "id": "S1", "elements": [ "B", "C", "D", "E" ], "cost": 208 }, { "id": "S2", "elements": [ "A", "B", "C", "D", "F", "G" ], "cost": 216 }, { "id": "S3", "elements": [ "B", "C", "D", "H" ], "cost": 264 }, { "id": "S4", "elements": [ "L", "T", "U", "V", "W" ], "cost": 265 }, { "id": "S5", "elements": [ "R", "T", "V", "W" ], "cost": 316 }, { "id": "S6", "elements": [ "K", "M" ], "cost": 88 }, { "id": "S7", "elements": [ "O", "P", "Q", "R", "S", "T", "U" ], "cost": 658 }, { "id": "S8", "elements": [ "T", "U", "V" ], "cost": 9 }, { "id": "S9", "elements": [ "O", "P", "Q", "T" ], "cost": 36 }, { "id": "S10", "elements": [ "H", "K", "L", "O" ], "cost": 160 }, { "id": "S11", "elements": [ "K", "M", "O", "P", "R" ], "cost": 475 }, { "id": "S12", "elements": [ "C", "D", "E", "F", "G" ], "cost": 390 }, { "id": "S13", "elements": [ "A", "B", "F", "G" ], "cost": 372 }, { "id": "S14", "elements": [ "O", "P", "Q" ], "cost": 33 }, { "id": "S15", "elements": [ "T", "U", "V", "W", "X" ], "cost": 320 }, { "id": "S16", "elements": [ "A", "B", "E", "G", "I" ], "cost": 10 }, { "id": "S17", "elements": [ "O", "P", "U" ], "cost": 162 }, { "id": "S18", "elements": [ "P", "Q", "S", "T", "U", "V" ], "cost": 132 }, { "id": "S19", "elements": [ "K", "O", "P", "Q", "R", "S" ], "cost": 234 }, { "id": "S20", "elements": [ "R", "S", "T", "V", "W", "X" ], "cost": 402 }, { "id": "S21", "elements": [ "B", "C", "D", "E", "F", "H" ], "cost": 456 }, { "id": "S22", "elements": [ "Q", "R", "T" ], "cost": 276 }, { "id": "S23", "elements": [ "R", "T", "U", "V", "W", "X" ], "cost": 102 }, { "id": "S24", "elements": [ "C", "E", "F", "G", "H", "I" ], "cost": 186 }, { "id": "S25", "elements": [ "A", "B", "C", "D", "E", "F", "G" ], "cost": 84 }, { "id": "S26", "elements": [ "G", "H", "I", "J", "K", "L", "N" ], "cost": 63 }, { "id": "S27", "elements": [ "A", "B", "D", "E" ], "cost": 276 }, { "id": "S28", "elements": [ "S", "T", "X" ], "cost": 285 }, { "id": "S29", "elements": [ "R", "S", "T", "U" ], "cost": 380 }, { "id": "S30", "elements": [ "J", "L", "M", "O", "P", "R" ], "cost": 144 }, { "id": "S31", "elements": [ "B", "C", "D", "F", "H" ], "cost": 40 }, { "id": "S32", "elements": [ "R", "S", "V", "W" ], "cost": 216 }, { "id": "S33", "elements": [ "P", "S", "T", "U", "V", "W", "X" ], "cost": 630 }, { "id": "S34", "elements": [ "Q", "R", "S", "T", "V", "W" ], "cost": 390 }, { "id": "S35", "elements": [ "B", "C", "D", "E", "F", "G", "I" ], "cost": 560 }, { "id": "S36", "elements": [ "H", "K", "L", "N", "Q" ], "cost": 455 }, { "id": "S37", "elements": [ "A", "B", "C" ], "cost": 246 }, { "id": "S38", "elements": [ "P", "R", "S", "T", "V" ], "cost": 130 }, { "id": "S39", "elements": [ "O", "Q", "R", "S", "U", "W" ], "cost": 114 }, { "id": "S40", "elements": [ "A", "B", "C", "E", "F" ], "cost": 330 }, { "id": "S41", "elements": [ "P", "Q", "S", "U", "V", "W", "X" ], "cost": 497 }, { "id": "S42", "elements": [ "D", "E", "F" ], "cost": 117 }, { "id": "S43", "elements": [ "C", "F", "H", "I", "J", "K", "L" ], "cost": 77 }, { "id": "S44", "elements": [ "S", "U", "V", "W", "X" ], "cost": 65 }, { "id": "S45", "elements": [ "H", "J", "K", "L", "M", "N", "P" ], "cost": 399 }, { "id": "S46", "elements": [ "I", "L", "M", "O", "P", "Q" ], "cost": 300 }, { "id": "S47", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 78 }, { "id": "S48", "elements": [ "B", "E", "F", "G", "I", "J" ], "cost": 66 }, { "id": "S49", "elements": [ "O", "Q", "R", "S", "T", "V" ], "cost": 402 }, { "id": "S50", "elements": [ "I", "J", "K", "L", "O" ], "cost": 380 }, { "id": "S51", "elements": [ "B", "C", "D", "E", "F", "I" ], "cost": 432 }, { "id": "S52", "elements": [ "J", "K", "O", "P" ], "cost": 32 }, { "id": "S53", "elements": [ "H", "I", "J", "L", "M", "N" ], "cost": 396 }, { "id": "S54", "elements": [ "K", "L", "M", "N", "O", "P" ], "cost": 120 }, { "id": "S55", "elements": [ "E", "F", "G", "H", "J" ], "cost": 105 }, { "id": "S56", "elements": [ "C", "D", "G", "H", "I" ], "cost": 160 }, { "id": "S57", "elements": [ "A", "B", "C", "E" ], "cost": 156 }, { "id": "S58", "elements": [ "H", "L", "M", "N", "O", "S" ], "cost": 426 }, { "id": "S59", "elements": [ "I", "J", "L" ], "cost": 183 }, { "id": "S60", "elements": [ "R", "S", "T", "U", "V", "W", "X" ], "cost": 413 }, { "id": "S61", "elements": [ "A", "C", "D", "E", "F", "G", "H" ], "cost": 385 }, { "id": "S62", "elements": [ "B", "D", "E", "F", "G" ], "cost": 215 }, { "id": "S63", "elements": [ "P", "S", "V" ], "cost": 138 }, { "id": "S64", "elements": [ "K", "M", "N", "O", "P", "S" ], "cost": 12 }, { "id": "S65", "elements": [ "A", "B", "C", "D", "E", "F", "H" ], "cost": 35 }, { "id": "S66", "elements": [ "V", "W", "X" ], "cost": 69 }, { "id": "S67", "elements": [ "N", "Q", "S", "T" ], "cost": 356 }, { "id": "S68", "elements": [ "E", "F", "H" ], "cost": 108 }, { "id": "S69", "elements": [ "E", "F" ], "cost": 2 }, { "id": "S70", "elements": [ "I", "J", "N" ], "cost": 204 }, { "id": "S71", "elements": [ "T", "U", "W", "X" ], "cost": 12 }, { "id": "S72", "elements": [ "Q", "R", "S", "T" ], "cost": 140 }, { "id": "S73", "elements": [ "J", "K", "L", "M", "N", "O", "P" ], "cost": 273 }, { "id": "S74", "elements": [ "A", "B", "D", "E", "F" ], "cost": 285 }, { "id": "S75", "elements": [ "Q", "S", "T", "V", "W", "X" ], "cost": 276 }, { "id": "S76", "elements": [ "Q", "S", "T", "U", "V", "W", "X" ], "cost": 308 }, { "id": "S77", "elements": [ "H", "I", "J", "K", "M" ], "cost": 310 }, { "id": "S78", "elements": [ "A", "B", "C", "D", "F", "H" ], "cost": 372 }, { "id": "S79", "elements": [ "D", "E", "F", "G" ], "cost": 280 }, { "id": "S80", "elements": [ "U", "V", "W" ], "cost": 255 }, { "id": "S81", "elements": [ "G", "H", "I", "J", "M" ], "cost": 310 }, { "id": "S82", "elements": [ "J", "L" ], "cost": 168 }, { "id": "S83", "elements": [ "E", "F", "G", "H", "I", "L" ], "cost": 264 }, { "id": "S84", "elements": [ "L", "N", "P", "Q", "T", "U", "V" ], "cost": 364 }, { "id": "S85", "elements": [ "Q", "R", "T", "U", "V", "W", "X" ], "cost": 49 }, { "id": "S86", "elements": [ "J", "L", "N" ], "cost": 297 }, { "id": "S87", "elements": [ "E", "G", "H", "L" ], "cost": 276 }, { "id": "S88", "elements": [ "C", "D", "E", "F", "K" ], "cost": 145 }, { "id": "S89", "elements": [ "S", "T", "U", "X" ], "cost": 172 }, { "id": "S90", "elements": [ "H", "I", "J", "K", "L", "M", "N" ], "cost": 497 }, { "id": "S91", "elements": [ "V", "W" ], "cost": 90 }, { "id": "S92", "elements": [ "P", "R", "S", "T", "W" ], "cost": 30 }, { "id": "S93", "elements": [ "O", "R", "S" ], "cost": 300 }, { "id": "S94", "elements": [ "D", "G", "H" ], "cost": 246 }, { "id": "S95", "elements": [ "A", "B", "C", "F" ], "cost": 228 }, { "id": "S96", "elements": [ "C", "D", "E", "G", "I" ], "cost": 445 }, { "id": "S97", "elements": [ "G", "I", "J", "K" ], "cost": 24 }, { "id": "S98", "elements": [ "C", "E", "F", "G", "I" ], "cost": 20 }, { "id": "S99", "elements": [ "J", "K", "M" ], "cost": 261 }, { "id": "S100", "elements": [ "A", "B", "D" ], "cost": 150 }, { "id": "S101", "elements": [ "P", "R", "S", "U", "W" ], "cost": 340 }, { "id": "S102", "elements": [ "S", "T", "W", "X" ], "cost": 188 }, { "id": "S103", "elements": [ "F", "G", "I", "J", "L" ], "cost": 160 }, { "id": "S104", "elements": [ "E", "G", "I", "J" ], "cost": 184 }, { "id": "S105", "elements": [ "D", "E", "G" ], "cost": 75 }, { "id": "S106", "elements": [ "A", "B", "C", "D", "G" ], "cost": 410 }, { "id": "S107", "elements": [ "Q", "R", "T", "V" ], "cost": 248 }, { "id": "S108", "elements": [ "A" ], "cost": 10000 }, { "id": "S109", "elements": [ "B" ], "cost": 10000 }, { "id": "S110", "elements": [ "C" ], "cost": 10000 }, { "id": "S111", "elements": [ "D" ], "cost": 10000 }, { "id": "S112", "elements": [ "E" ], "cost": 10000 }, { "id": "S113", "elements": [ "F" ], "cost": 10000 }, { "id": "S114", "elements": [ "G" ], "cost": 10000 }, { "id": "S115", "elements": [ "H" ], "cost": 10000 }, { "id": "S116", "elements": [ "I" ], "cost": 10000 }, { "id": "S117", "elements": [ "J" ], "cost": 10000 }, { "id": "S118", "elements": [ "K" ], "cost": 10000 }, { "id": "S119", "elements": [ "L" ], "cost": 10000 }, { "id": "S120", "elements": [ "M" ], "cost": 10000 }, { "id": "S121", "elements": [ "N" ], "cost": 10000 }, { "id": "S122", "elements": [ "O" ], "cost": 10000 }, { "id": "S123", "elements": [ "P" ], "cost": 10000 }, { "id": "S124", "elements": [ "Q" ], "cost": 10000 }, { "id": "S125", "elements": [ "R" ], "cost": 10000 }, { "id": "S126", "elements": [ "S" ], "cost": 10000 }, { "id": "S127", "elements": [ "T" ], "cost": 10000 }, { "id": "S128", "elements": [ "U" ], "cost": 10000 }, { "id": "S129", "elements": [ "V" ], "cost": 10000 }, { "id": "S130", "elements": [ "W" ], "cost": 10000 }, { "id": "S131", "elements": [ "X" ], "cost": 10000 } ] }, "solution_variant": [ "S24", "S64", "S82", "S85", "S100" ], "context_index": 34, "input_format": "json", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "At my job we’re choosing from several route contracts, and the rule is simple: every delivery stop must belong to exactly one chosen contract — nothing left out and nothing covered twice. Each contract charges a fee, so better choices are those with the lowest combined bill; you get that combined bill by adding together the fees of all picked contracts. The concrete instance details appear below.\n\n- **num_stops**: 24\n- **num_contracts_available**: 146\n\n| contract_id | contract_fee | stops_covered |\n|---|---|---|\n| S1 | 130 | 13 17 18 19 20 |\n| S2 | 336 | 16 18 21 22 23 24 |\n| S3 | 516 | 13 15 16 17 20 21 |\n| S4 | 36 | 21 22 23 24 |\n| S5 | 225 | 16 17 20 |\n| S6 | 66 | 6 7 9 |\n| S7 | 147 | 18 19 22 |\n| S8 | 57 | 4 5 6 |\n| S9 | 210 | 18 19 20 21 22 24 |\n| S10 | 148 | 10 11 13 14 |\n| S11 | 343 | 1 2 3 4 5 6 7 |\n| S12 | 623 | 2 3 5 6 7 8 9 |\n| S13 | 180 | 14 15 16 17 18 |\n| S14 | 294 | 1 2 3 4 5 6 8 |\n| S15 | 200 | 1 3 4 5 |\n| S16 | 102 | 2 5 |\n| S17 | 490 | 11 12 14 15 16 18 21 |\n| S18 | 336 | 14 15 16 17 |\n| S19 | 340 | 1 2 3 4 5 |\n| S20 | 516 | 8 9 12 13 14 15 |\n| S21 | 165 | 8 11 12 |\n| S22 | 150 | 1 2 5 6 7 |\n| S23 | 465 | 2 3 4 5 7 |\n| S24 | 462 | 13 14 17 18 19 23 |\n| S25 | 255 | 1 2 3 4 6 |\n| S26 | 5 | 17 19 21 22 23 |\n| S27 | 390 | 14 16 17 19 20 |\n| S28 | 320 | 9 12 14 15 |\n| S29 | 16 | 7 8 9 11 |\n| S30 | 301 | 17 19 20 21 22 23 24 |\n| S31 | 176 | 20 21 22 24 |\n| S32 | 270 | 16 19 20 21 22 23 |\n| S33 | 360 | 12 13 17 18 |\n| S34 | 328 | 8 9 11 12 |\n| S35 | 295 | 16 17 18 19 22 |\n| S36 | 252 | 9 10 11 12 13 16 |\n| S37 | 574 | 18 19 20 21 22 23 24 |\n| S38 | 348 | 4 6 7 9 10 11 |\n| S39 | 4 | 13 14 18 19 |\n| S40 | 228 | 9 10 11 12 |\n| S41 | 126 | 6 7 8 9 10 13 |\n| S42 | 195 | 13 15 16 17 18 |\n| S43 | 348 | 10 11 13 14 16 18 |\n| S44 | 236 | 20 21 22 23 |\n| S45 | 105 | 11 13 14 |\n| S46 | 665 | 16 17 20 21 22 23 24 |\n| S47 | 336 | 13 14 15 18 |\n| S48 | 342 | 10 13 14 15 16 18 |\n| S49 | 343 | 10 13 14 15 16 17 19 |\n| S50 | 171 | 16 17 18 |\n| S51 | 138 | 1 2 4 |\n| S52 | 15 | 15 19 20 |\n| S53 | 400 | 18 19 20 23 |\n| S54 | 12 | 3 4 5 6 |\n| S55 | 180 | 18 20 21 22 23 24 |\n| S56 | 600 | 1 2 3 4 5 6 |\n| S57 | 276 | 7 8 10 |\n| S58 | 10 | 8 9 12 13 14 |\n| S59 | 168 | 9 10 11 15 |\n| S60 | 16 | 14 15 19 21 |\n| S61 | 30 | 12 13 14 15 17 |\n| S62 | 30 | 17 18 23 |\n| S63 | 532 | 1 3 4 5 6 7 8 |\n| S64 | 384 | 18 19 21 22 23 24 |\n| S65 | 219 | 19 23 24 |\n| S66 | 204 | 12 14 15 16 |\n| S67 | 260 | 2 4 5 6 |\n| S68 | 12 | 1 2 |\n| S69 | 9 | 13 15 17 |\n| S70 | 279 | 1 6 7 |\n| S71 | 63 | 8 9 10 11 13 15 16 |\n| S72 | 252 | 6 8 10 |\n| S73 | 180 | 2 3 4 5 6 8 |\n| S74 | 14 | 9 10 11 13 14 15 18 |\n| S75 | 345 | 9 11 12 14 16 |\n| S76 | 108 | 1 2 3 4 5 7 |\n| S77 | 320 | 2 3 5 6 |\n| S78 | 432 | 11 13 14 15 16 17 |\n| S79 | 60 | 20 21 24 |\n| S80 | 532 | 12 13 14 15 16 17 18 |\n| S81 | 546 | 7 10 11 12 13 14 15 |\n| S82 | 234 | 1 2 3 4 6 8 |\n| S83 | 450 | 3 5 6 7 8 10 |\n| S84 | 356 | 10 12 13 14 |\n| S85 | 183 | 11 14 15 |\n| S86 | 234 | 5 6 8 9 10 11 |\n| S87 | 240 | 8 12 13 |\n| S88 | 280 | 13 15 17 18 19 20 21 |\n| S89 | 237 | 18 19 20 |\n| S90 | 90 | 3 4 5 7 8 |\n| S91 | 231 | 7 10 13 |\n| S92 | 32 | 18 21 22 23 |\n| S93 | 280 | 17 18 19 21 22 23 24 |\n| S94 | 245 | 3 4 5 8 9 12 14 |\n| S95 | 490 | 20 21 22 23 24 |\n| S96 | 276 | 9 10 11 12 15 16 |\n| S97 | 380 | 15 18 20 23 |\n| S98 | 76 | 17 22 23 24 |\n| S99 | 380 | 18 19 20 21 |\n| S100 | 480 | 7 9 10 12 13 15 |\n| S101 | 175 | 3 4 5 6 7 |\n| S102 | 665 | 5 6 7 8 9 10 12 |\n| S103 | 258 | 16 18 19 20 21 22 |\n| S104 | 310 | 3 5 6 9 10 |\n| S105 | 120 | 7 9 12 14 |\n| S106 | 12 | 2 6 7 8 |\n| S107 | 30 | 14 16 17 18 20 |\n| S108 | 16 | 1 3 4 6 |\n| S109 | 308 | 6 7 8 9 10 11 12 |\n| S110 | 189 | 6 9 13 |\n| S111 | 270 | 3 4 5 6 7 8 |\n| S112 | 366 | 6 8 9 11 12 13 |\n| S113 | 48 | 21 22 24 |\n| S114 | 392 | 9 10 11 12 13 14 17 |\n| S115 | 207 | 11 15 17 |\n| S116 | 246 | 17 20 21 22 23 24 |\n| S117 | 195 | 2 6 7 |\n| S118 | 330 | 18 19 20 21 22 23 |\n| S119 | 18 | 13 14 19 |\n| S120 | 21 | 6 9 12 |\n| S121 | 216 | 16 17 19 20 21 22 |\n| S122 | 546 | 11 12 13 14 15 16 |\n| S123 | 10000 | 1 |\n| S124 | 10000 | 2 |\n| S125 | 10000 | 3 |\n| S126 | 10000 | 4 |\n| S127 | 10000 | 5 |\n| S128 | 10000 | 6 |\n| S129 | 10000 | 7 |\n| S130 | 10000 | 8 |\n| S131 | 10000 | 9 |\n| S132 | 10000 | 10 |\n| S133 | 10000 | 11 |\n| S134 | 10000 | 12 |\n| S135 | 10000 | 13 |\n| S136 | 10000 | 14 |\n| S137 | 10000 | 15 |\n| S138 | 10000 | 16 |\n| S139 | 10000 | 17 |\n| S140 | 10000 | 18 |\n| S141 | 10000 | 19 |\n| S142 | 10000 | 20 |\n| S143 | 10000 | 21 |\n| S144 | 10000 | 22 |\n| S145 | 10000 | 23 |\n| S146 | 10000 | 24 |\n\nPretty simple — when you send back the selected contracts, just stick to this little JSON shape so it's easy to parse:\n\n{\n \"solution\": [\"contract_id\", ...]\n}\n\nHere \"solution\" is the list of the route contracts you pick (one entry per chosen contract). The placeholder \"contract_id\" is just a stand-in — in your actual reply put the real contract identifiers from the instance. This block is just a sketch of the expected shape, not the final answer.\n\nPlease be sure to use the identifiers exactly as they appear in the instance input — don't rename them or invent new labels. Valid identifiers look like plain numbers such as \"1\" or \"23\", single capital letters like \"A\" or \"B\", or a capital letter followed by digits like \"A1\" or \"X7\".", "instance": { "problem_type": "SPP", "num_elements": 24, "num_sets": 146, "density": 0.175513698630137, "sets": [ { "id": 1, "elements": [ 13, 17, 18, 19, 20 ], "cost": 130 }, { "id": 2, "elements": [ 16, 18, 21, 22, 23, 24 ], "cost": 336 }, { "id": 3, "elements": [ 13, 15, 16, 17, 20, 21 ], "cost": 516 }, { "id": 4, "elements": [ 21, 22, 23, 24 ], "cost": 36 }, { "id": 5, "elements": [ 16, 17, 20 ], "cost": 225 }, { "id": 6, "elements": [ 6, 7, 9 ], "cost": 66 }, { "id": 7, "elements": [ 18, 19, 22 ], "cost": 147 }, { "id": 8, "elements": [ 4, 5, 6 ], "cost": 57 }, { "id": 9, "elements": [ 18, 19, 20, 21, 22, 24 ], "cost": 210 }, { "id": 10, "elements": [ 10, 11, 13, 14 ], "cost": 148 }, { "id": 11, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 343 }, { "id": 12, "elements": [ 2, 3, 5, 6, 7, 8, 9 ], "cost": 623 }, { "id": 13, "elements": [ 14, 15, 16, 17, 18 ], "cost": 180 }, { "id": 14, "elements": [ 1, 2, 3, 4, 5, 6, 8 ], "cost": 294 }, { "id": 15, "elements": [ 1, 3, 4, 5 ], "cost": 200 }, { "id": 16, "elements": [ 2, 5 ], "cost": 102 }, { "id": 17, "elements": [ 11, 12, 14, 15, 16, 18, 21 ], "cost": 490 }, { "id": 18, "elements": [ 14, 15, 16, 17 ], "cost": 336 }, { "id": 19, "elements": [ 1, 2, 3, 4, 5 ], "cost": 340 }, { "id": 20, "elements": [ 8, 9, 12, 13, 14, 15 ], "cost": 516 }, { "id": 21, "elements": [ 8, 11, 12 ], "cost": 165 }, { "id": 22, "elements": [ 1, 2, 5, 6, 7 ], "cost": 150 }, { "id": 23, "elements": [ 2, 3, 4, 5, 7 ], "cost": 465 }, { "id": 24, "elements": [ 13, 14, 17, 18, 19, 23 ], "cost": 462 }, { "id": 25, "elements": [ 1, 2, 3, 4, 6 ], "cost": 255 }, { "id": 26, "elements": [ 17, 19, 21, 22, 23 ], "cost": 5 }, { "id": 27, "elements": [ 14, 16, 17, 19, 20 ], "cost": 390 }, { "id": 28, "elements": [ 9, 12, 14, 15 ], "cost": 320 }, { "id": 29, "elements": [ 7, 8, 9, 11 ], "cost": 16 }, { "id": 30, "elements": [ 17, 19, 20, 21, 22, 23, 24 ], "cost": 301 }, { "id": 31, "elements": [ 20, 21, 22, 24 ], "cost": 176 }, { "id": 32, "elements": [ 16, 19, 20, 21, 22, 23 ], "cost": 270 }, { "id": 33, "elements": [ 12, 13, 17, 18 ], "cost": 360 }, { "id": 34, "elements": [ 8, 9, 11, 12 ], "cost": 328 }, { "id": 35, "elements": [ 16, 17, 18, 19, 22 ], "cost": 295 }, { "id": 36, "elements": [ 9, 10, 11, 12, 13, 16 ], "cost": 252 }, { "id": 37, "elements": [ 18, 19, 20, 21, 22, 23, 24 ], "cost": 574 }, { "id": 38, "elements": [ 4, 6, 7, 9, 10, 11 ], "cost": 348 }, { "id": 39, "elements": [ 13, 14, 18, 19 ], "cost": 4 }, { "id": 40, "elements": [ 9, 10, 11, 12 ], "cost": 228 }, { "id": 41, "elements": [ 6, 7, 8, 9, 10, 13 ], "cost": 126 }, { "id": 42, "elements": [ 13, 15, 16, 17, 18 ], "cost": 195 }, { "id": 43, "elements": [ 10, 11, 13, 14, 16, 18 ], "cost": 348 }, { "id": 44, "elements": [ 20, 21, 22, 23 ], "cost": 236 }, { "id": 45, "elements": [ 11, 13, 14 ], "cost": 105 }, { "id": 46, "elements": [ 16, 17, 20, 21, 22, 23, 24 ], "cost": 665 }, { "id": 47, "elements": [ 13, 14, 15, 18 ], "cost": 336 }, { "id": 48, "elements": [ 10, 13, 14, 15, 16, 18 ], "cost": 342 }, { "id": 49, "elements": [ 10, 13, 14, 15, 16, 17, 19 ], "cost": 343 }, { "id": 50, "elements": [ 16, 17, 18 ], "cost": 171 }, { "id": 51, "elements": [ 1, 2, 4 ], "cost": 138 }, { "id": 52, "elements": [ 15, 19, 20 ], "cost": 15 }, { "id": 53, "elements": [ 18, 19, 20, 23 ], "cost": 400 }, { "id": 54, "elements": [ 3, 4, 5, 6 ], "cost": 12 }, { "id": 55, "elements": [ 18, 20, 21, 22, 23, 24 ], "cost": 180 }, { "id": 56, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 600 }, { "id": 57, "elements": [ 7, 8, 10 ], "cost": 276 }, { "id": 58, "elements": [ 8, 9, 12, 13, 14 ], "cost": 10 }, { "id": 59, "elements": [ 9, 10, 11, 15 ], "cost": 168 }, { "id": 60, "elements": [ 14, 15, 19, 21 ], "cost": 16 }, { "id": 61, "elements": [ 12, 13, 14, 15, 17 ], "cost": 30 }, { "id": 62, "elements": [ 17, 18, 23 ], "cost": 30 }, { "id": 63, "elements": [ 1, 3, 4, 5, 6, 7, 8 ], "cost": 532 }, { "id": 64, "elements": [ 18, 19, 21, 22, 23, 24 ], "cost": 384 }, { "id": 65, "elements": [ 19, 23, 24 ], "cost": 219 }, { "id": 66, "elements": [ 12, 14, 15, 16 ], "cost": 204 }, { "id": 67, "elements": [ 2, 4, 5, 6 ], "cost": 260 }, { "id": 68, "elements": [ 1, 2 ], "cost": 12 }, { "id": 69, "elements": [ 13, 15, 17 ], "cost": 9 }, { "id": 70, "elements": [ 1, 6, 7 ], "cost": 279 }, { "id": 71, "elements": [ 8, 9, 10, 11, 13, 15, 16 ], "cost": 63 }, { "id": 72, "elements": [ 6, 8, 10 ], "cost": 252 }, { "id": 73, "elements": [ 2, 3, 4, 5, 6, 8 ], "cost": 180 }, { "id": 74, "elements": [ 9, 10, 11, 13, 14, 15, 18 ], "cost": 14 }, { "id": 75, "elements": [ 9, 11, 12, 14, 16 ], "cost": 345 }, { "id": 76, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 108 }, { "id": 77, "elements": [ 2, 3, 5, 6 ], "cost": 320 }, { "id": 78, "elements": [ 11, 13, 14, 15, 16, 17 ], "cost": 432 }, { "id": 79, "elements": [ 20, 21, 24 ], "cost": 60 }, { "id": 80, "elements": [ 12, 13, 14, 15, 16, 17, 18 ], "cost": 532 }, { "id": 81, "elements": [ 7, 10, 11, 12, 13, 14, 15 ], "cost": 546 }, { "id": 82, "elements": [ 1, 2, 3, 4, 6, 8 ], "cost": 234 }, { "id": 83, "elements": [ 3, 5, 6, 7, 8, 10 ], "cost": 450 }, { "id": 84, "elements": [ 10, 12, 13, 14 ], "cost": 356 }, { "id": 85, "elements": [ 11, 14, 15 ], "cost": 183 }, { "id": 86, "elements": [ 5, 6, 8, 9, 10, 11 ], "cost": 234 }, { "id": 87, "elements": [ 8, 12, 13 ], "cost": 240 }, { "id": 88, "elements": [ 13, 15, 17, 18, 19, 20, 21 ], "cost": 280 }, { "id": 89, "elements": [ 18, 19, 20 ], "cost": 237 }, { "id": 90, "elements": [ 3, 4, 5, 7, 8 ], "cost": 90 }, { "id": 91, "elements": [ 7, 10, 13 ], "cost": 231 }, { "id": 92, "elements": [ 18, 21, 22, 23 ], "cost": 32 }, { "id": 93, "elements": [ 17, 18, 19, 21, 22, 23, 24 ], "cost": 280 }, { "id": 94, "elements": [ 3, 4, 5, 8, 9, 12, 14 ], "cost": 245 }, { "id": 95, "elements": [ 20, 21, 22, 23, 24 ], "cost": 490 }, { "id": 96, "elements": [ 9, 10, 11, 12, 15, 16 ], "cost": 276 }, { "id": 97, "elements": [ 15, 18, 20, 23 ], "cost": 380 }, { "id": 98, "elements": [ 17, 22, 23, 24 ], "cost": 76 }, { "id": 99, "elements": [ 18, 19, 20, 21 ], "cost": 380 }, { "id": 100, "elements": [ 7, 9, 10, 12, 13, 15 ], "cost": 480 }, { "id": 101, "elements": [ 3, 4, 5, 6, 7 ], "cost": 175 }, { "id": 102, "elements": [ 5, 6, 7, 8, 9, 10, 12 ], "cost": 665 }, { "id": 103, "elements": [ 16, 18, 19, 20, 21, 22 ], "cost": 258 }, { "id": 104, "elements": [ 3, 5, 6, 9, 10 ], "cost": 310 }, { "id": 105, "elements": [ 7, 9, 12, 14 ], "cost": 120 }, { "id": 106, "elements": [ 2, 6, 7, 8 ], "cost": 12 }, { "id": 107, "elements": [ 14, 16, 17, 18, 20 ], "cost": 30 }, { "id": 108, "elements": [ 1, 3, 4, 6 ], "cost": 16 }, { "id": 109, "elements": [ 6, 7, 8, 9, 10, 11, 12 ], "cost": 308 }, { "id": 110, "elements": [ 6, 9, 13 ], "cost": 189 }, { "id": 111, "elements": [ 3, 4, 5, 6, 7, 8 ], "cost": 270 }, { "id": 112, "elements": [ 6, 8, 9, 11, 12, 13 ], "cost": 366 }, { "id": 113, "elements": [ 21, 22, 24 ], "cost": 48 }, { "id": 114, "elements": [ 9, 10, 11, 12, 13, 14, 17 ], "cost": 392 }, { "id": 115, "elements": [ 11, 15, 17 ], "cost": 207 }, { "id": 116, "elements": [ 17, 20, 21, 22, 23, 24 ], "cost": 246 }, { "id": 117, "elements": [ 2, 6, 7 ], "cost": 195 }, { "id": 118, "elements": [ 18, 19, 20, 21, 22, 23 ], "cost": 330 }, { "id": 119, "elements": [ 13, 14, 19 ], "cost": 18 }, { "id": 120, "elements": [ 6, 9, 12 ], "cost": 21 }, { "id": 121, "elements": [ 16, 17, 19, 20, 21, 22 ], "cost": 216 }, { "id": 122, "elements": [ 11, 12, 13, 14, 15, 16 ], "cost": 546 }, { "id": 123, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 124, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 125, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 126, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 127, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 128, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 129, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 130, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 131, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 132, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 133, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 134, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 135, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 136, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 137, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 138, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 139, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 140, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 141, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 142, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 143, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 144, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 145, "elements": [ 23 ], "cost": 10000, "singleton": true }, { "id": 146, "elements": [ 24 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0034_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0034_bag_stats.png" }, "solution": [ 4, 10, 50, 52, 68, 90, 120 ], "obj": 493.0, "instance_variant": { "num_elements": 24, "num_sets": 146, "sets": [ { "id": "S1", "elements": [ 13, 17, 18, 19, 20 ], "cost": 130 }, { "id": "S2", "elements": [ 16, 18, 21, 22, 23, 24 ], "cost": 336 }, { "id": "S3", "elements": [ 13, 15, 16, 17, 20, 21 ], "cost": 516 }, { "id": "S4", "elements": [ 21, 22, 23, 24 ], "cost": 36 }, { "id": "S5", "elements": [ 16, 17, 20 ], "cost": 225 }, { "id": "S6", "elements": [ 6, 7, 9 ], "cost": 66 }, { "id": "S7", "elements": [ 18, 19, 22 ], "cost": 147 }, { "id": "S8", "elements": [ 4, 5, 6 ], "cost": 57 }, { "id": "S9", "elements": [ 18, 19, 20, 21, 22, 24 ], "cost": 210 }, { "id": "S10", "elements": [ 10, 11, 13, 14 ], "cost": 148 }, { "id": "S11", "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 343 }, { "id": "S12", "elements": [ 2, 3, 5, 6, 7, 8, 9 ], "cost": 623 }, { "id": "S13", "elements": [ 14, 15, 16, 17, 18 ], "cost": 180 }, { "id": "S14", "elements": [ 1, 2, 3, 4, 5, 6, 8 ], "cost": 294 }, { "id": "S15", "elements": [ 1, 3, 4, 5 ], "cost": 200 }, { "id": "S16", "elements": [ 2, 5 ], "cost": 102 }, { "id": "S17", "elements": [ 11, 12, 14, 15, 16, 18, 21 ], "cost": 490 }, { "id": "S18", "elements": [ 14, 15, 16, 17 ], "cost": 336 }, { "id": "S19", "elements": [ 1, 2, 3, 4, 5 ], "cost": 340 }, { "id": "S20", "elements": [ 8, 9, 12, 13, 14, 15 ], "cost": 516 }, { "id": "S21", "elements": [ 8, 11, 12 ], "cost": 165 }, { "id": "S22", "elements": [ 1, 2, 5, 6, 7 ], "cost": 150 }, { "id": "S23", "elements": [ 2, 3, 4, 5, 7 ], "cost": 465 }, { "id": "S24", "elements": [ 13, 14, 17, 18, 19, 23 ], "cost": 462 }, { "id": "S25", "elements": [ 1, 2, 3, 4, 6 ], "cost": 255 }, { "id": "S26", "elements": [ 17, 19, 21, 22, 23 ], "cost": 5 }, { "id": "S27", "elements": [ 14, 16, 17, 19, 20 ], "cost": 390 }, { "id": "S28", "elements": [ 9, 12, 14, 15 ], "cost": 320 }, { "id": "S29", "elements": [ 7, 8, 9, 11 ], "cost": 16 }, { "id": "S30", "elements": [ 17, 19, 20, 21, 22, 23, 24 ], "cost": 301 }, { "id": "S31", "elements": [ 20, 21, 22, 24 ], "cost": 176 }, { "id": "S32", "elements": [ 16, 19, 20, 21, 22, 23 ], "cost": 270 }, { "id": "S33", "elements": [ 12, 13, 17, 18 ], "cost": 360 }, { "id": "S34", "elements": [ 8, 9, 11, 12 ], "cost": 328 }, { "id": "S35", "elements": [ 16, 17, 18, 19, 22 ], "cost": 295 }, { "id": "S36", "elements": [ 9, 10, 11, 12, 13, 16 ], "cost": 252 }, { "id": "S37", "elements": [ 18, 19, 20, 21, 22, 23, 24 ], "cost": 574 }, { "id": "S38", "elements": [ 4, 6, 7, 9, 10, 11 ], "cost": 348 }, { "id": "S39", "elements": [ 13, 14, 18, 19 ], "cost": 4 }, { "id": "S40", "elements": [ 9, 10, 11, 12 ], "cost": 228 }, { "id": "S41", "elements": [ 6, 7, 8, 9, 10, 13 ], "cost": 126 }, { "id": "S42", "elements": [ 13, 15, 16, 17, 18 ], "cost": 195 }, { "id": "S43", "elements": [ 10, 11, 13, 14, 16, 18 ], "cost": 348 }, { "id": "S44", "elements": [ 20, 21, 22, 23 ], "cost": 236 }, { "id": "S45", "elements": [ 11, 13, 14 ], "cost": 105 }, { "id": "S46", "elements": [ 16, 17, 20, 21, 22, 23, 24 ], "cost": 665 }, { "id": "S47", "elements": [ 13, 14, 15, 18 ], "cost": 336 }, { "id": "S48", "elements": [ 10, 13, 14, 15, 16, 18 ], "cost": 342 }, { "id": "S49", "elements": [ 10, 13, 14, 15, 16, 17, 19 ], "cost": 343 }, { "id": "S50", "elements": [ 16, 17, 18 ], "cost": 171 }, { "id": "S51", "elements": [ 1, 2, 4 ], "cost": 138 }, { "id": "S52", "elements": [ 15, 19, 20 ], "cost": 15 }, { "id": "S53", "elements": [ 18, 19, 20, 23 ], "cost": 400 }, { "id": "S54", "elements": [ 3, 4, 5, 6 ], "cost": 12 }, { "id": "S55", "elements": [ 18, 20, 21, 22, 23, 24 ], "cost": 180 }, { "id": "S56", "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 600 }, { "id": "S57", "elements": [ 7, 8, 10 ], "cost": 276 }, { "id": "S58", "elements": [ 8, 9, 12, 13, 14 ], "cost": 10 }, { "id": "S59", "elements": [ 9, 10, 11, 15 ], "cost": 168 }, { "id": "S60", "elements": [ 14, 15, 19, 21 ], "cost": 16 }, { "id": "S61", "elements": [ 12, 13, 14, 15, 17 ], "cost": 30 }, { "id": "S62", "elements": [ 17, 18, 23 ], "cost": 30 }, { "id": "S63", "elements": [ 1, 3, 4, 5, 6, 7, 8 ], "cost": 532 }, { "id": "S64", "elements": [ 18, 19, 21, 22, 23, 24 ], "cost": 384 }, { "id": "S65", "elements": [ 19, 23, 24 ], "cost": 219 }, { "id": "S66", "elements": [ 12, 14, 15, 16 ], "cost": 204 }, { "id": "S67", "elements": [ 2, 4, 5, 6 ], "cost": 260 }, { "id": "S68", "elements": [ 1, 2 ], "cost": 12 }, { "id": "S69", "elements": [ 13, 15, 17 ], "cost": 9 }, { "id": "S70", "elements": [ 1, 6, 7 ], "cost": 279 }, { "id": "S71", "elements": [ 8, 9, 10, 11, 13, 15, 16 ], "cost": 63 }, { "id": "S72", "elements": [ 6, 8, 10 ], "cost": 252 }, { "id": "S73", "elements": [ 2, 3, 4, 5, 6, 8 ], "cost": 180 }, { "id": "S74", "elements": [ 9, 10, 11, 13, 14, 15, 18 ], "cost": 14 }, { "id": "S75", "elements": [ 9, 11, 12, 14, 16 ], "cost": 345 }, { "id": "S76", "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 108 }, { "id": "S77", "elements": [ 2, 3, 5, 6 ], "cost": 320 }, { "id": "S78", "elements": [ 11, 13, 14, 15, 16, 17 ], "cost": 432 }, { "id": "S79", "elements": [ 20, 21, 24 ], "cost": 60 }, { "id": "S80", "elements": [ 12, 13, 14, 15, 16, 17, 18 ], "cost": 532 }, { "id": "S81", "elements": [ 7, 10, 11, 12, 13, 14, 15 ], "cost": 546 }, { "id": "S82", "elements": [ 1, 2, 3, 4, 6, 8 ], "cost": 234 }, { "id": "S83", "elements": [ 3, 5, 6, 7, 8, 10 ], "cost": 450 }, { "id": "S84", "elements": [ 10, 12, 13, 14 ], "cost": 356 }, { "id": "S85", "elements": [ 11, 14, 15 ], "cost": 183 }, { "id": "S86", "elements": [ 5, 6, 8, 9, 10, 11 ], "cost": 234 }, { "id": "S87", "elements": [ 8, 12, 13 ], "cost": 240 }, { "id": "S88", "elements": [ 13, 15, 17, 18, 19, 20, 21 ], "cost": 280 }, { "id": "S89", "elements": [ 18, 19, 20 ], "cost": 237 }, { "id": "S90", "elements": [ 3, 4, 5, 7, 8 ], "cost": 90 }, { "id": "S91", "elements": [ 7, 10, 13 ], "cost": 231 }, { "id": "S92", "elements": [ 18, 21, 22, 23 ], "cost": 32 }, { "id": "S93", "elements": [ 17, 18, 19, 21, 22, 23, 24 ], "cost": 280 }, { "id": "S94", "elements": [ 3, 4, 5, 8, 9, 12, 14 ], "cost": 245 }, { "id": "S95", "elements": [ 20, 21, 22, 23, 24 ], "cost": 490 }, { "id": "S96", "elements": [ 9, 10, 11, 12, 15, 16 ], "cost": 276 }, { "id": "S97", "elements": [ 15, 18, 20, 23 ], "cost": 380 }, { "id": "S98", "elements": [ 17, 22, 23, 24 ], "cost": 76 }, { "id": "S99", "elements": [ 18, 19, 20, 21 ], "cost": 380 }, { "id": "S100", "elements": [ 7, 9, 10, 12, 13, 15 ], "cost": 480 }, { "id": "S101", "elements": [ 3, 4, 5, 6, 7 ], "cost": 175 }, { "id": "S102", "elements": [ 5, 6, 7, 8, 9, 10, 12 ], "cost": 665 }, { "id": "S103", "elements": [ 16, 18, 19, 20, 21, 22 ], "cost": 258 }, { "id": "S104", "elements": [ 3, 5, 6, 9, 10 ], "cost": 310 }, { "id": "S105", "elements": [ 7, 9, 12, 14 ], "cost": 120 }, { "id": "S106", "elements": [ 2, 6, 7, 8 ], "cost": 12 }, { "id": "S107", "elements": [ 14, 16, 17, 18, 20 ], "cost": 30 }, { "id": "S108", "elements": [ 1, 3, 4, 6 ], "cost": 16 }, { "id": "S109", "elements": [ 6, 7, 8, 9, 10, 11, 12 ], "cost": 308 }, { "id": "S110", "elements": [ 6, 9, 13 ], "cost": 189 }, { "id": "S111", "elements": [ 3, 4, 5, 6, 7, 8 ], "cost": 270 }, { "id": "S112", "elements": [ 6, 8, 9, 11, 12, 13 ], "cost": 366 }, { "id": "S113", "elements": [ 21, 22, 24 ], "cost": 48 }, { "id": "S114", "elements": [ 9, 10, 11, 12, 13, 14, 17 ], "cost": 392 }, { "id": "S115", "elements": [ 11, 15, 17 ], "cost": 207 }, { "id": "S116", "elements": [ 17, 20, 21, 22, 23, 24 ], "cost": 246 }, { "id": "S117", "elements": [ 2, 6, 7 ], "cost": 195 }, { "id": "S118", "elements": [ 18, 19, 20, 21, 22, 23 ], "cost": 330 }, { "id": "S119", "elements": [ 13, 14, 19 ], "cost": 18 }, { "id": "S120", "elements": [ 6, 9, 12 ], "cost": 21 }, { "id": "S121", "elements": [ 16, 17, 19, 20, 21, 22 ], "cost": 216 }, { "id": "S122", "elements": [ 11, 12, 13, 14, 15, 16 ], "cost": 546 }, { "id": "S123", "elements": [ 1 ], "cost": 10000 }, { "id": "S124", "elements": [ 2 ], "cost": 10000 }, { "id": "S125", "elements": [ 3 ], "cost": 10000 }, { "id": "S126", "elements": [ 4 ], "cost": 10000 }, { "id": "S127", "elements": [ 5 ], "cost": 10000 }, { "id": "S128", "elements": [ 6 ], "cost": 10000 }, { "id": "S129", "elements": [ 7 ], "cost": 10000 }, { "id": "S130", "elements": [ 8 ], "cost": 10000 }, { "id": "S131", "elements": [ 9 ], "cost": 10000 }, { "id": "S132", "elements": [ 10 ], "cost": 10000 }, { "id": "S133", "elements": [ 11 ], "cost": 10000 }, { "id": "S134", "elements": [ 12 ], "cost": 10000 }, { "id": "S135", "elements": [ 13 ], "cost": 10000 }, { "id": "S136", "elements": [ 14 ], "cost": 10000 }, { "id": "S137", "elements": [ 15 ], "cost": 10000 }, { "id": "S138", "elements": [ 16 ], "cost": 10000 }, { "id": "S139", "elements": [ 17 ], "cost": 10000 }, { "id": "S140", "elements": [ 18 ], "cost": 10000 }, { "id": "S141", "elements": [ 19 ], "cost": 10000 }, { "id": "S142", "elements": [ 20 ], "cost": 10000 }, { "id": "S143", "elements": [ 21 ], "cost": 10000 }, { "id": "S144", "elements": [ 22 ], "cost": 10000 }, { "id": "S145", "elements": [ 23 ], "cost": 10000 }, { "id": "S146", "elements": [ 24 ], "cost": 10000 } ] }, "solution_variant": [ "S4", "S10", "S50", "S52", "S68", "S90", "S120" ], "context_index": 35, "input_format": "markdown_table", "input_index_base": 1 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "We had a checklist of customer wants and a pile of possible feature modules, and the task was to choose some modules so every want is handled by a single module — no gaps and no overlaps. The better choice is the cheaper one overall: total development cost is simply the sum of the costs of the modules that get picked, and the goal is to make that sum as small as possible. The exact list of requirements and module costs follows below.\n\n# num_requirements=23\n# num_modules=140\nmodule_id,development_cost,covered_requirements\nS1,135,T U V\nS2,245,S T U V W\nS3,465,M O Q R T\nS4,295,O Q R S U\nS5,279,L M N\nS6,462,A B C D E F\nS7,265,Q R S T U\nS8,196,B D E H\nS9,55,J N O P R\nS10,208,J K L M\nS11,243,H J L\nS12,225,D E G H I\nS13,12,E G H J\nS14,270,A C D E F\nS15,42,E J K\nS16,114,R V W\nS17,260,A C G H\nS18,469,A B C D E F G\nS19,185,B C D E H\nS20,33,D E J\nS21,252,I J O P\nS22,147,K L M N O P R\nS23,3,D E F\nS24,15,D F L\nS25,44,L O R S\nS26,110,M O Q T U\nS27,90,L M N Q R T\nS28,450,E F G H I J\nS29,185,F G H I J\nS30,80,A C F J\nS31,180,N P T\nS32,45,A B G\nS33,390,P R T U V W\nS34,245,C D E F G\nS35,42,P R S T U V W\nS36,42,A B C\nS37,132,Q S V\nS38,35,N Q S T U V W\nS39,295,G N O P R\nS40,9,S U W\nS41,105,H K L\nS42,308,C D I J\nS43,232,A B C F\nS44,312,R T U W\nS45,364,D G H I J K M\nS46,630,A B C D E G I\nS47,55,A B C E F\nS48,76,R S\nS49,12,A B C H\nS50,117,N O P\nS51,360,K L M N O P\nS52,8,S V\nS53,287,I L O P Q R S\nS54,70,I K L M N\nS55,21,C D F\nS56,532,E F G H I J N\nS57,292,G H I K\nS58,24,R S T U V W\nS59,450,F G H I J K\nS60,475,I K L N R\nS61,195,R S U V W\nS62,576,Q S T U V W\nS63,372,H J K M N O\nS64,155,I K M N Q\nS65,228,F G I J\nS66,112,H K L M N O Q\nS67,288,A C D E F G\nS68,228,I K L M N O\nS69,155,O S T U W\nS70,210,B E H\nS71,28,H J L M N P Q\nS72,75,B C D E J\nS73,125,A B C D F\nS74,150,E I J L M\nS75,168,U V\nS76,546,L P Q R S T\nS77,96,I K L M N Q\nS78,205,M Q R S U\nS79,18,A B D E G I\nS80,200,A D E H\nS81,60,U V W\nS82,93,H J K\nS83,240,E F G H\nS84,368,A B C D\nS85,410,E F I K M\nS86,480,M N O P Q R\nS87,195,C E F H I\nS88,470,G I J K M\nS89,145,N O Q U V\nS90,445,M O P Q R\nS91,186,R U W\nS92,470,N P T U W\nS93,372,J K M Q\nS94,287,I J K L M N P\nS95,236,Q S T W\nS96,581,P Q R S T U W\nS97,354,E F G J K O\nS98,92,T V\nS99,288,Q S T U\nS100,88,D H I L\nS101,75,R T U V W\nS102,52,P Q\nS103,294,T V W\nS104,126,L P Q R S T U\nS105,280,E G I J K\nS106,528,D E F G H I\nS107,21,C D H\nS108,34,I N\nS109,273,A B D\nS110,216,B F I\nS111,102,Q T\nS112,198,M N O P Q S\nS113,16,A B C E\nS114,159,R S T\nS115,504,A C D F G I\nS116,80,M N O P S\nS117,75,Q T U\nS118,10000,A\nS119,10000,B\nS120,10000,C\nS121,10000,D\nS122,10000,E\nS123,10000,F\nS124,10000,G\nS125,10000,H\nS126,10000,I\nS127,10000,J\nS128,10000,K\nS129,10000,L\nS130,10000,M\nS131,10000,N\nS132,10000,O\nS133,10000,P\nS134,10000,Q\nS135,10000,R\nS136,10000,S\nS137,10000,T\nS138,10000,U\nS139,10000,V\nS140,10000,W\n\nAlso, just to keep things tidy when you send the answer back, please use this simple JSON layout so I can read it automatically.\n\n{\n \"solution\": [\"module_id\", ...]\n}\n\n\"solution\" is the list of chosen modules (one entry per module). The array items are placeholders showing where the actual module identifiers go — replace \"module_id\" with the real IDs from the instance when you submit the final selection. This is just a sketch of the shape I expect, not the final answer.\n\nOne more thing: all identifiers must be used exactly as they appear in the instance input — do not rename them or invent new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 23, "num_sets": 140, "density": 0.17360248447204968, "sets": [ { "id": 1, "elements": [ 20, 21, 22 ], "cost": 135 }, { "id": 2, "elements": [ 19, 20, 21, 22, 23 ], "cost": 245 }, { "id": 3, "elements": [ 13, 15, 17, 18, 20 ], "cost": 465 }, { "id": 4, "elements": [ 15, 17, 18, 19, 21 ], "cost": 295 }, { "id": 5, "elements": [ 12, 13, 14 ], "cost": 279 }, { "id": 6, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 462 }, { "id": 7, "elements": [ 17, 18, 19, 20, 21 ], "cost": 265 }, { "id": 8, "elements": [ 2, 4, 5, 8 ], "cost": 196 }, { "id": 9, "elements": [ 10, 14, 15, 16, 18 ], "cost": 55 }, { "id": 10, "elements": [ 10, 11, 12, 13 ], "cost": 208 }, { "id": 11, "elements": [ 8, 10, 12 ], "cost": 243 }, { "id": 12, "elements": [ 4, 5, 7, 8, 9 ], "cost": 225 }, { "id": 13, "elements": [ 5, 7, 8, 10 ], "cost": 12 }, { "id": 14, "elements": [ 1, 3, 4, 5, 6 ], "cost": 270 }, { "id": 15, "elements": [ 5, 10, 11 ], "cost": 42 }, { "id": 16, "elements": [ 18, 22, 23 ], "cost": 114 }, { "id": 17, "elements": [ 1, 3, 7, 8 ], "cost": 260 }, { "id": 18, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 469 }, { "id": 19, "elements": [ 2, 3, 4, 5, 8 ], "cost": 185 }, { "id": 20, "elements": [ 4, 5, 10 ], "cost": 33 }, { "id": 21, "elements": [ 9, 10, 15, 16 ], "cost": 252 }, { "id": 22, "elements": [ 11, 12, 13, 14, 15, 16, 18 ], "cost": 147 }, { "id": 23, "elements": [ 4, 5, 6 ], "cost": 3 }, { "id": 24, "elements": [ 4, 6, 12 ], "cost": 15 }, { "id": 25, "elements": [ 12, 15, 18, 19 ], "cost": 44 }, { "id": 26, "elements": [ 13, 15, 17, 20, 21 ], "cost": 110 }, { "id": 27, "elements": [ 12, 13, 14, 17, 18, 20 ], "cost": 90 }, { "id": 28, "elements": [ 5, 6, 7, 8, 9, 10 ], "cost": 450 }, { "id": 29, "elements": [ 6, 7, 8, 9, 10 ], "cost": 185 }, { "id": 30, "elements": [ 1, 3, 6, 10 ], "cost": 80 }, { "id": 31, "elements": [ 14, 16, 20 ], "cost": 180 }, { "id": 32, "elements": [ 1, 2, 7 ], "cost": 45 }, { "id": 33, "elements": [ 16, 18, 20, 21, 22, 23 ], "cost": 390 }, { "id": 34, "elements": [ 3, 4, 5, 6, 7 ], "cost": 245 }, { "id": 35, "elements": [ 16, 18, 19, 20, 21, 22, 23 ], "cost": 42 }, { "id": 36, "elements": [ 1, 2, 3 ], "cost": 42 }, { "id": 37, "elements": [ 17, 19, 22 ], "cost": 132 }, { "id": 38, "elements": [ 14, 17, 19, 20, 21, 22, 23 ], "cost": 35 }, { "id": 39, "elements": [ 7, 14, 15, 16, 18 ], "cost": 295 }, { "id": 40, "elements": [ 19, 21, 23 ], "cost": 9 }, { "id": 41, "elements": [ 8, 11, 12 ], "cost": 105 }, { "id": 42, "elements": [ 3, 4, 9, 10 ], "cost": 308 }, { "id": 43, "elements": [ 1, 2, 3, 6 ], "cost": 232 }, { "id": 44, "elements": [ 18, 20, 21, 23 ], "cost": 312 }, { "id": 45, "elements": [ 4, 7, 8, 9, 10, 11, 13 ], "cost": 364 }, { "id": 46, "elements": [ 1, 2, 3, 4, 5, 7, 9 ], "cost": 630 }, { "id": 47, "elements": [ 1, 2, 3, 5, 6 ], "cost": 55 }, { "id": 48, "elements": [ 18, 19 ], "cost": 76 }, { "id": 49, "elements": [ 1, 2, 3, 8 ], "cost": 12 }, { "id": 50, "elements": [ 14, 15, 16 ], "cost": 117 }, { "id": 51, "elements": [ 11, 12, 13, 14, 15, 16 ], "cost": 360 }, { "id": 52, "elements": [ 19, 22 ], "cost": 8 }, { "id": 53, "elements": [ 9, 12, 15, 16, 17, 18, 19 ], "cost": 287 }, { "id": 54, "elements": [ 9, 11, 12, 13, 14 ], "cost": 70 }, { "id": 55, "elements": [ 3, 4, 6 ], "cost": 21 }, { "id": 56, "elements": [ 5, 6, 7, 8, 9, 10, 14 ], "cost": 532 }, { "id": 57, "elements": [ 7, 8, 9, 11 ], "cost": 292 }, { "id": 58, "elements": [ 18, 19, 20, 21, 22, 23 ], "cost": 24 }, { "id": 59, "elements": [ 6, 7, 8, 9, 10, 11 ], "cost": 450 }, { "id": 60, "elements": [ 9, 11, 12, 14, 18 ], "cost": 475 }, { "id": 61, "elements": [ 18, 19, 21, 22, 23 ], "cost": 195 }, { "id": 62, "elements": [ 17, 19, 20, 21, 22, 23 ], "cost": 576 }, { "id": 63, "elements": [ 8, 10, 11, 13, 14, 15 ], "cost": 372 }, { "id": 64, "elements": [ 9, 11, 13, 14, 17 ], "cost": 155 }, { "id": 65, "elements": [ 6, 7, 9, 10 ], "cost": 228 }, { "id": 66, "elements": [ 8, 11, 12, 13, 14, 15, 17 ], "cost": 112 }, { "id": 67, "elements": [ 1, 3, 4, 5, 6, 7 ], "cost": 288 }, { "id": 68, "elements": [ 9, 11, 12, 13, 14, 15 ], "cost": 228 }, { "id": 69, "elements": [ 15, 19, 20, 21, 23 ], "cost": 155 }, { "id": 70, "elements": [ 2, 5, 8 ], "cost": 210 }, { "id": 71, "elements": [ 8, 10, 12, 13, 14, 16, 17 ], "cost": 28 }, { "id": 72, "elements": [ 2, 3, 4, 5, 10 ], "cost": 75 }, { "id": 73, "elements": [ 1, 2, 3, 4, 6 ], "cost": 125 }, { "id": 74, "elements": [ 5, 9, 10, 12, 13 ], "cost": 150 }, { "id": 75, "elements": [ 21, 22 ], "cost": 168 }, { "id": 76, "elements": [ 12, 16, 17, 18, 19, 20 ], "cost": 546 }, { "id": 77, "elements": [ 9, 11, 12, 13, 14, 17 ], "cost": 96 }, { "id": 78, "elements": [ 13, 17, 18, 19, 21 ], "cost": 205 }, { "id": 79, "elements": [ 1, 2, 4, 5, 7, 9 ], "cost": 18 }, { "id": 80, "elements": [ 1, 4, 5, 8 ], "cost": 200 }, { "id": 81, "elements": [ 21, 22, 23 ], "cost": 60 }, { "id": 82, "elements": [ 8, 10, 11 ], "cost": 93 }, { "id": 83, "elements": [ 5, 6, 7, 8 ], "cost": 240 }, { "id": 84, "elements": [ 1, 2, 3, 4 ], "cost": 368 }, { "id": 85, "elements": [ 5, 6, 9, 11, 13 ], "cost": 410 }, { "id": 86, "elements": [ 13, 14, 15, 16, 17, 18 ], "cost": 480 }, { "id": 87, "elements": [ 3, 5, 6, 8, 9 ], "cost": 195 }, { "id": 88, "elements": [ 7, 9, 10, 11, 13 ], "cost": 470 }, { "id": 89, "elements": [ 14, 15, 17, 21, 22 ], "cost": 145 }, { "id": 90, "elements": [ 13, 15, 16, 17, 18 ], "cost": 445 }, { "id": 91, "elements": [ 18, 21, 23 ], "cost": 186 }, { "id": 92, "elements": [ 14, 16, 20, 21, 23 ], "cost": 470 }, { "id": 93, "elements": [ 10, 11, 13, 17 ], "cost": 372 }, { "id": 94, "elements": [ 9, 10, 11, 12, 13, 14, 16 ], "cost": 287 }, { "id": 95, "elements": [ 17, 19, 20, 23 ], "cost": 236 }, { "id": 96, "elements": [ 16, 17, 18, 19, 20, 21, 23 ], "cost": 581 }, { "id": 97, "elements": [ 5, 6, 7, 10, 11, 15 ], "cost": 354 }, { "id": 98, "elements": [ 20, 22 ], "cost": 92 }, { "id": 99, "elements": [ 17, 19, 20, 21 ], "cost": 288 }, { "id": 100, "elements": [ 4, 8, 9, 12 ], "cost": 88 }, { "id": 101, "elements": [ 18, 20, 21, 22, 23 ], "cost": 75 }, { "id": 102, "elements": [ 16, 17 ], "cost": 52 }, { "id": 103, "elements": [ 20, 22, 23 ], "cost": 294 }, { "id": 104, "elements": [ 12, 16, 17, 18, 19, 20, 21 ], "cost": 126 }, { "id": 105, "elements": [ 5, 7, 9, 10, 11 ], "cost": 280 }, { "id": 106, "elements": [ 4, 5, 6, 7, 8, 9 ], "cost": 528 }, { "id": 107, "elements": [ 3, 4, 8 ], "cost": 21 }, { "id": 108, "elements": [ 9, 14 ], "cost": 34 }, { "id": 109, "elements": [ 1, 2, 4 ], "cost": 273 }, { "id": 110, "elements": [ 2, 6, 9 ], "cost": 216 }, { "id": 111, "elements": [ 17, 20 ], "cost": 102 }, { "id": 112, "elements": [ 13, 14, 15, 16, 17, 19 ], "cost": 198 }, { "id": 113, "elements": [ 1, 2, 3, 5 ], "cost": 16 }, { "id": 114, "elements": [ 18, 19, 20 ], "cost": 159 }, { "id": 115, "elements": [ 1, 3, 4, 6, 7, 9 ], "cost": 504 }, { "id": 116, "elements": [ 13, 14, 15, 16, 19 ], "cost": 80 }, { "id": 117, "elements": [ 17, 20, 21 ], "cost": 75 }, { "id": 118, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 119, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 120, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 121, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 122, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 123, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 124, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 125, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 126, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 127, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 128, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 129, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 130, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 131, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 132, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 133, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 134, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 135, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 136, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 137, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 138, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 139, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 140, "elements": [ 23 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0035_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0035_bag_stats.png" }, "solution": [ 13, 58, 68, 73, 102 ], "obj": 441.0, "instance_variant": { "num_elements": 23, "num_sets": 140, "sets": [ { "id": "S1", "elements": [ "T", "U", "V" ], "cost": 135 }, { "id": "S2", "elements": [ "S", "T", "U", "V", "W" ], "cost": 245 }, { "id": "S3", "elements": [ "M", "O", "Q", "R", "T" ], "cost": 465 }, { "id": "S4", "elements": [ "O", "Q", "R", "S", "U" ], "cost": 295 }, { "id": "S5", "elements": [ "L", "M", "N" ], "cost": 279 }, { "id": "S6", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 462 }, { "id": "S7", "elements": [ "Q", "R", "S", "T", "U" ], "cost": 265 }, { "id": "S8", "elements": [ "B", "D", "E", "H" ], "cost": 196 }, { "id": "S9", "elements": [ "J", "N", "O", "P", "R" ], "cost": 55 }, { "id": "S10", "elements": [ "J", "K", "L", "M" ], "cost": 208 }, { "id": "S11", "elements": [ "H", "J", "L" ], "cost": 243 }, { "id": "S12", "elements": [ "D", "E", "G", "H", "I" ], "cost": 225 }, { "id": "S13", "elements": [ "E", "G", "H", "J" ], "cost": 12 }, { "id": "S14", "elements": [ "A", "C", "D", "E", "F" ], "cost": 270 }, { "id": "S15", "elements": [ "E", "J", "K" ], "cost": 42 }, { "id": "S16", "elements": [ "R", "V", "W" ], "cost": 114 }, { "id": "S17", "elements": [ "A", "C", "G", "H" ], "cost": 260 }, { "id": "S18", "elements": [ "A", "B", "C", "D", "E", "F", "G" ], "cost": 469 }, { "id": "S19", "elements": [ "B", "C", "D", "E", "H" ], "cost": 185 }, { "id": "S20", "elements": [ "D", "E", "J" ], "cost": 33 }, { "id": "S21", "elements": [ "I", "J", "O", "P" ], "cost": 252 }, { "id": "S22", "elements": [ "K", "L", "M", "N", "O", "P", "R" ], "cost": 147 }, { "id": "S23", "elements": [ "D", "E", "F" ], "cost": 3 }, { "id": "S24", "elements": [ "D", "F", "L" ], "cost": 15 }, { "id": "S25", "elements": [ "L", "O", "R", "S" ], "cost": 44 }, { "id": "S26", "elements": [ "M", "O", "Q", "T", "U" ], "cost": 110 }, { "id": "S27", "elements": [ "L", "M", "N", "Q", "R", "T" ], "cost": 90 }, { "id": "S28", "elements": [ "E", "F", "G", "H", "I", "J" ], "cost": 450 }, { "id": "S29", "elements": [ "F", "G", "H", "I", "J" ], "cost": 185 }, { "id": "S30", "elements": [ "A", "C", "F", "J" ], "cost": 80 }, { "id": "S31", "elements": [ "N", "P", "T" ], "cost": 180 }, { "id": "S32", "elements": [ "A", "B", "G" ], "cost": 45 }, { "id": "S33", "elements": [ "P", "R", "T", "U", "V", "W" ], "cost": 390 }, { "id": "S34", "elements": [ "C", "D", "E", "F", "G" ], "cost": 245 }, { "id": "S35", "elements": [ "P", "R", "S", "T", "U", "V", "W" ], "cost": 42 }, { "id": "S36", "elements": [ "A", "B", "C" ], "cost": 42 }, { "id": "S37", "elements": [ "Q", "S", "V" ], "cost": 132 }, { "id": "S38", "elements": [ "N", "Q", "S", "T", "U", "V", "W" ], "cost": 35 }, { "id": "S39", "elements": [ "G", "N", "O", "P", "R" ], "cost": 295 }, { "id": "S40", "elements": [ "S", "U", "W" ], "cost": 9 }, { "id": "S41", "elements": [ "H", "K", "L" ], "cost": 105 }, { "id": "S42", "elements": [ "C", "D", "I", "J" ], "cost": 308 }, { "id": "S43", "elements": [ "A", "B", "C", "F" ], "cost": 232 }, { "id": "S44", "elements": [ "R", "T", "U", "W" ], "cost": 312 }, { "id": "S45", "elements": [ "D", "G", "H", "I", "J", "K", "M" ], "cost": 364 }, { "id": "S46", "elements": [ "A", "B", "C", "D", "E", "G", "I" ], "cost": 630 }, { "id": "S47", "elements": [ "A", "B", "C", "E", "F" ], "cost": 55 }, { "id": "S48", "elements": [ "R", "S" ], "cost": 76 }, { "id": "S49", "elements": [ "A", "B", "C", "H" ], "cost": 12 }, { "id": "S50", "elements": [ "N", "O", "P" ], "cost": 117 }, { "id": "S51", "elements": [ "K", "L", "M", "N", "O", "P" ], "cost": 360 }, { "id": "S52", "elements": [ "S", "V" ], "cost": 8 }, { "id": "S53", "elements": [ "I", "L", "O", "P", "Q", "R", "S" ], "cost": 287 }, { "id": "S54", "elements": [ "I", "K", "L", "M", "N" ], "cost": 70 }, { "id": "S55", "elements": [ "C", "D", "F" ], "cost": 21 }, { "id": "S56", "elements": [ "E", "F", "G", "H", "I", "J", "N" ], "cost": 532 }, { "id": "S57", "elements": [ "G", "H", "I", "K" ], "cost": 292 }, { "id": "S58", "elements": [ "R", "S", "T", "U", "V", "W" ], "cost": 24 }, { "id": "S59", "elements": [ "F", "G", "H", "I", "J", "K" ], "cost": 450 }, { "id": "S60", "elements": [ "I", "K", "L", "N", "R" ], "cost": 475 }, { "id": "S61", "elements": [ "R", "S", "U", "V", "W" ], "cost": 195 }, { "id": "S62", "elements": [ "Q", "S", "T", "U", "V", "W" ], "cost": 576 }, { "id": "S63", "elements": [ "H", "J", "K", "M", "N", "O" ], "cost": 372 }, { "id": "S64", "elements": [ "I", "K", "M", "N", "Q" ], "cost": 155 }, { "id": "S65", "elements": [ "F", "G", "I", "J" ], "cost": 228 }, { "id": "S66", "elements": [ "H", "K", "L", "M", "N", "O", "Q" ], "cost": 112 }, { "id": "S67", "elements": [ "A", "C", "D", "E", "F", "G" ], "cost": 288 }, { "id": "S68", "elements": [ "I", "K", "L", "M", "N", "O" ], "cost": 228 }, { "id": "S69", "elements": [ "O", "S", "T", "U", "W" ], "cost": 155 }, { "id": "S70", "elements": [ "B", "E", "H" ], "cost": 210 }, { "id": "S71", "elements": [ "H", "J", "L", "M", "N", "P", "Q" ], "cost": 28 }, { "id": "S72", "elements": [ "B", "C", "D", "E", "J" ], "cost": 75 }, { "id": "S73", "elements": [ "A", "B", "C", "D", "F" ], "cost": 125 }, { "id": "S74", "elements": [ "E", "I", "J", "L", "M" ], "cost": 150 }, { "id": "S75", "elements": [ "U", "V" ], "cost": 168 }, { "id": "S76", "elements": [ "L", "P", "Q", "R", "S", "T" ], "cost": 546 }, { "id": "S77", "elements": [ "I", "K", "L", "M", "N", "Q" ], "cost": 96 }, { "id": "S78", "elements": [ "M", "Q", "R", "S", "U" ], "cost": 205 }, { "id": "S79", "elements": [ "A", "B", "D", "E", "G", "I" ], "cost": 18 }, { "id": "S80", "elements": [ "A", "D", "E", "H" ], "cost": 200 }, { "id": "S81", "elements": [ "U", "V", "W" ], "cost": 60 }, { "id": "S82", "elements": [ "H", "J", "K" ], "cost": 93 }, { "id": "S83", "elements": [ "E", "F", "G", "H" ], "cost": 240 }, { "id": "S84", "elements": [ "A", "B", "C", "D" ], "cost": 368 }, { "id": "S85", "elements": [ "E", "F", "I", "K", "M" ], "cost": 410 }, { "id": "S86", "elements": [ "M", "N", "O", "P", "Q", "R" ], "cost": 480 }, { "id": "S87", "elements": [ "C", "E", "F", "H", "I" ], "cost": 195 }, { "id": "S88", "elements": [ "G", "I", "J", "K", "M" ], "cost": 470 }, { "id": "S89", "elements": [ "N", "O", "Q", "U", "V" ], "cost": 145 }, { "id": "S90", "elements": [ "M", "O", "P", "Q", "R" ], "cost": 445 }, { "id": "S91", "elements": [ "R", "U", "W" ], "cost": 186 }, { "id": "S92", "elements": [ "N", "P", "T", "U", "W" ], "cost": 470 }, { "id": "S93", "elements": [ "J", "K", "M", "Q" ], "cost": 372 }, { "id": "S94", "elements": [ "I", "J", "K", "L", "M", "N", "P" ], "cost": 287 }, { "id": "S95", "elements": [ "Q", "S", "T", "W" ], "cost": 236 }, { "id": "S96", "elements": [ "P", "Q", "R", "S", "T", "U", "W" ], "cost": 581 }, { "id": "S97", "elements": [ "E", "F", "G", "J", "K", "O" ], "cost": 354 }, { "id": "S98", "elements": [ "T", "V" ], "cost": 92 }, { "id": "S99", "elements": [ "Q", "S", "T", "U" ], "cost": 288 }, { "id": "S100", "elements": [ "D", "H", "I", "L" ], "cost": 88 }, { "id": "S101", "elements": [ "R", "T", "U", "V", "W" ], "cost": 75 }, { "id": "S102", "elements": [ "P", "Q" ], "cost": 52 }, { "id": "S103", "elements": [ "T", "V", "W" ], "cost": 294 }, { "id": "S104", "elements": [ "L", "P", "Q", "R", "S", "T", "U" ], "cost": 126 }, { "id": "S105", "elements": [ "E", "G", "I", "J", "K" ], "cost": 280 }, { "id": "S106", "elements": [ "D", "E", "F", "G", "H", "I" ], "cost": 528 }, { "id": "S107", "elements": [ "C", "D", "H" ], "cost": 21 }, { "id": "S108", "elements": [ "I", "N" ], "cost": 34 }, { "id": "S109", "elements": [ "A", "B", "D" ], "cost": 273 }, { "id": "S110", "elements": [ "B", "F", "I" ], "cost": 216 }, { "id": "S111", "elements": [ "Q", "T" ], "cost": 102 }, { "id": "S112", "elements": [ "M", "N", "O", "P", "Q", "S" ], "cost": 198 }, { "id": "S113", "elements": [ "A", "B", "C", "E" ], "cost": 16 }, { "id": "S114", "elements": [ "R", "S", "T" ], "cost": 159 }, { "id": "S115", "elements": [ "A", "C", "D", "F", "G", "I" ], "cost": 504 }, { "id": "S116", "elements": [ "M", "N", "O", "P", "S" ], "cost": 80 }, { "id": "S117", "elements": [ "Q", "T", "U" ], "cost": 75 }, { "id": "S118", "elements": [ "A" ], "cost": 10000 }, { "id": "S119", "elements": [ "B" ], "cost": 10000 }, { "id": "S120", "elements": [ "C" ], "cost": 10000 }, { "id": "S121", "elements": [ "D" ], "cost": 10000 }, { "id": "S122", "elements": [ "E" ], "cost": 10000 }, { "id": "S123", "elements": [ "F" ], "cost": 10000 }, { "id": "S124", "elements": [ "G" ], "cost": 10000 }, { "id": "S125", "elements": [ "H" ], "cost": 10000 }, { "id": "S126", "elements": [ "I" ], "cost": 10000 }, { "id": "S127", "elements": [ "J" ], "cost": 10000 }, { "id": "S128", "elements": [ "K" ], "cost": 10000 }, { "id": "S129", "elements": [ "L" ], "cost": 10000 }, { "id": "S130", "elements": [ "M" ], "cost": 10000 }, { "id": "S131", "elements": [ "N" ], "cost": 10000 }, { "id": "S132", "elements": [ "O" ], "cost": 10000 }, { "id": "S133", "elements": [ "P" ], "cost": 10000 }, { "id": "S134", "elements": [ "Q" ], "cost": 10000 }, { "id": "S135", "elements": [ "R" ], "cost": 10000 }, { "id": "S136", "elements": [ "S" ], "cost": 10000 }, { "id": "S137", "elements": [ "T" ], "cost": 10000 }, { "id": "S138", "elements": [ "U" ], "cost": 10000 }, { "id": "S139", "elements": [ "V" ], "cost": 10000 }, { "id": "S140", "elements": [ "W" ], "cost": 10000 } ] }, "solution_variant": [ "S13", "S58", "S68", "S73", "S102" ], "context_index": 36, "input_format": "csv", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "There’s a renovation on the table where each tile box can cover particular patches of floor; the job is to choose boxes so every patch ends up covered by one box and no two chosen boxes cover the same patch. To compare options, just sum the prices of the boxes you pick and aim for the combination with the smallest total cost. The detailed floor map and box price list follow below.\n\nThere are 22 floor patches and 126 tile boxes available.\nBox S1 covers patches 3 5 6 8 9 and costs 320.\nBox S2 covers patches 1 2 3 4 5 6 8 and costs 301.\nBox S3 covers patches 8 9 10 11 12 13 and costs 138.\nBox S4 covers patches 2 4 7 8 and costs 124.\nBox S5 covers patches 11 13 and costs 130.\nBox S6 covers patches 18 20 and costs 112.\nBox S7 covers patches 6 11 and costs 156.\nBox S8 covers patches 4 6 7 8 9 10 11 and costs 42.\nBox S9 covers patches 10 11 12 13 and costs 48.\nBox S10 covers patches 7 8 11 and costs 87.\nBox S11 covers patches 9 11 and costs 38.\nBox S12 covers patches 5 7 11 14 and costs 4.\nBox S13 covers patches 7 11 12 14 15 and costs 225.\nBox S14 covers patches 3 4 5 6 7 and costs 245.\nBox S15 covers patches 14 16 and costs 176.\nBox S16 covers patches 14 18 and costs 190.\nBox S17 covers patches 7 8 9 and costs 186.\nBox S18 covers patches 16 17 18 19 21 and costs 35.\nBox S19 covers patches 5 7 8 9 10 11 and costs 114.\nBox S20 covers patches 17 19 20 21 and costs 336.\nBox S21 covers patches 7 10 12 and costs 270.\nBox S22 covers patches 12 13 16 and costs 228.\nBox S23 covers patches 19 20 21 and costs 252.\nBox S24 covers patches 5 6 7 9 10 11 and costs 36.\nBox S25 covers patches 16 17 18 20 and costs 396.\nBox S26 covers patches 7 8 9 10 11 12 and costs 24.\nBox S27 covers patches 6 7 8 9 10 12 and costs 486.\nBox S28 covers patches 17 18 19 20 21 and costs 235.\nBox S29 covers patches 9 10 12 13 and costs 216.\nBox S30 covers patches 14 16 17 20 21 and costs 435.\nBox S31 covers patches 3 6 and costs 172.\nBox S32 covers patches 18 19 21 and costs 264.\nBox S33 covers patches 2 5 6 7 8 and costs 250.\nBox S34 covers patches 1 2 3 4 6 and costs 125.\nBox S35 covers patches 15 18 19 20 and costs 24.\nBox S36 covers patches 4 5 8 9 14 and costs 205.\nBox S37 covers patches 12 15 16 and costs 48.\nBox S38 covers patches 10 12 15 and costs 54.\nBox S39 covers patches 2 5 6 7 8 10 and costs 192.\nBox S40 covers patches 6 9 11 14 15 and costs 25.\nBox S41 covers patches 0 3 and costs 108.\nBox S42 covers patches 4 7 8 9 10 11 and costs 126.\nBox S43 covers patches 15 16 19 and costs 246.\nBox S44 covers patches 11 12 15 17 and costs 280.\nBox S45 covers patches 15 18 19 20 21 and costs 230.\nBox S46 covers patches 0 1 4 and costs 132.\nBox S47 covers patches 0 1 2 3 4 and costs 315.\nBox S48 covers patches 15 16 17 18 19 20 and costs 150.\nBox S49 covers patches 14 15 17 18 20 and costs 475.\nBox S50 covers patches 10 12 13 14 16 17 and costs 438.\nBox S51 covers patches 17 18 and costs 114.\nBox S52 covers patches 9 10 11 12 and costs 224.\nBox S53 covers patches 10 12 14 15 18 and costs 40.\nBox S54 covers patches 5 6 7 and costs 231.\nBox S55 covers patches 4 5 6 8 and costs 100.\nBox S56 covers patches 9 12 14 and costs 132.\nBox S57 covers patches 12 13 15 and costs 24.\nBox S58 covers patches 2 3 4 5 6 7 and costs 42.\nBox S59 covers patches 18 19 20 21 and costs 88.\nBox S60 covers patches 1 2 3 5 6 7 and costs 24.\nBox S61 covers patches 1 3 4 5 6 7 and costs 390.\nBox S62 covers patches 2 3 4 5 7 and costs 445.\nBox S63 covers patches 16 17 18 19 20 21 and costs 264.\nBox S64 covers patches 15 17 19 20 and costs 360.\nBox S65 covers patches 9 12 13 14 15 and costs 95.\nBox S66 covers patches 13 15 16 17 18 19 and costs 462.\nBox S67 covers patches 1 5 7 and costs 249.\nBox S68 covers patches 4 5 6 8 9 11 and costs 330.\nBox S69 covers patches 3 5 6 8 and costs 244.\nBox S70 covers patches 14 15 16 17 18 20 and costs 138.\nBox S71 covers patches 4 6 7 8 9 11 and costs 312.\nBox S72 covers patches 0 1 2 3 4 5 and costs 6.\nBox S73 covers patches 1 2 3 4 5 7 and costs 282.\nBox S74 covers patches 12 15 19 and costs 183.\nBox S75 covers patches 0 1 2 3 4 6 and costs 348.\nBox S76 covers patches 15 17 18 20 21 and costs 25.\nBox S77 covers patches 14 17 18 19 20 21 and costs 24.\nBox S78 covers patches 13 14 15 16 17 18 and costs 510.\nBox S79 covers patches 4 5 6 7 and costs 380.\nBox S80 covers patches 17 19 20 and costs 45.\nBox S81 covers patches 2 6 7 8 and costs 188.\nBox S82 covers patches 4 9 11 and costs 297.\nBox S83 covers patches 13 15 16 and costs 222.\nBox S84 covers patches 5 8 9 10 11 12 and costs 282.\nBox S85 covers patches 15 17 18 20 and costs 316.\nBox S86 covers patches 0 4 and costs 88.\nBox S87 covers patches 1 3 4 5 and costs 172.\nBox S88 covers patches 18 19 and costs 48.\nBox S89 covers patches 6 8 11 and costs 42.\nBox S90 covers patches 1 6 7 and costs 201.\nBox S91 covers patches 1 3 5 6 and costs 316.\nBox S92 covers patches 7 9 10 11 12 and costs 80.\nBox S93 covers patches 12 18 20 and costs 57.\nBox S94 covers patches 12 13 14 16 17 and costs 10.\nBox S95 covers patches 12 15 16 17 18 20 and costs 222.\nBox S96 covers patches 5 9 and costs 28.\nBox S97 covers patches 10 11 13 and costs 21.\nBox S98 covers patches 10 11 12 13 14 15 and costs 480.\nBox S99 covers patches 14 15 16 17 20 21 and costs 216.\nBox S100 covers patches 6 9 11 12 and costs 4.\nBox S101 covers patches 16 20 21 and costs 267.\nBox S102 covers patches 11 12 13 14 16 17 19 and costs 63.\nBox S103 covers patches 14 16 17 18 19 20 21 and costs 476.\nBox S104 covers patches 7 9 11 and costs 258.\nBox S105 covers patches 0 and costs 10000.\nBox S106 covers patches 1 and costs 10000.\nBox S107 covers patches 2 and costs 10000.\nBox S108 covers patches 3 and costs 10000.\nBox S109 covers patches 4 and costs 10000.\nBox S110 covers patches 5 and costs 10000.\nBox S111 covers patches 6 and costs 10000.\nBox S112 covers patches 7 and costs 10000.\nBox S113 covers patches 8 and costs 10000.\nBox S114 covers patches 9 and costs 10000.\nBox S115 covers patches 10 and costs 10000.\nBox S116 covers patches 11 and costs 10000.\nBox S117 covers patches 12 and costs 10000.\nBox S118 covers patches 13 and costs 10000.\nBox S119 covers patches 14 and costs 10000.\nBox S120 covers patches 15 and costs 10000.\nBox S121 covers patches 16 and costs 10000.\nBox S122 covers patches 17 and costs 10000.\nBox S123 covers patches 18 and costs 10000.\nBox S124 covers patches 19 and costs 10000.\nBox S125 covers patches 20 and costs 10000.\nBox S126 covers patches 21 and costs 10000.\nChoose boxes so all 22 patches are covered exactly once while minimizing the total cost.\n\nWhenever you’re ready to give the chosen boxes, just drop them into a tiny JSON snippet like this so it’s easy to read and machine-friendly:\n\n{\n \"solution\": [\"box_id\", ...]\n}\n\nHere \"solution\" is the list of the box identifiers you picked to cover every patch exactly once. The \"box_id\" entries are placeholders — replace each one with the exact box label from the instance input when you submit your final choice. This JSON is just a sketch of the shape I’m expecting, not the final answer itself.\n\nPlease make sure you use the identifiers exactly as they appear in the instance input — no renaming, no new labels. Valid identifiers look like plain numbers such as \"1\" or \"23\", single capital letters like \"A\" or \"B\", or a capital letter followed by digits like \"A1\" or \"X7\".", "instance": { "problem_type": "SPP", "num_elements": 22, "num_sets": 126, "density": 0.17063492063492064, "sets": [ { "id": 1, "elements": [ 4, 6, 7, 9, 10 ], "cost": 320 }, { "id": 2, "elements": [ 2, 3, 4, 5, 6, 7, 9 ], "cost": 301 }, { "id": 3, "elements": [ 9, 10, 11, 12, 13, 14 ], "cost": 138 }, { "id": 4, "elements": [ 3, 5, 8, 9 ], "cost": 124 }, { "id": 5, "elements": [ 12, 14 ], "cost": 130 }, { "id": 6, "elements": [ 19, 21 ], "cost": 112 }, { "id": 7, "elements": [ 7, 12 ], "cost": 156 }, { "id": 8, "elements": [ 5, 7, 8, 9, 10, 11, 12 ], "cost": 42 }, { "id": 9, "elements": [ 11, 12, 13, 14 ], "cost": 48 }, { "id": 10, "elements": [ 8, 9, 12 ], "cost": 87 }, { "id": 11, "elements": [ 10, 12 ], "cost": 38 }, { "id": 12, "elements": [ 6, 8, 12, 15 ], "cost": 4 }, { "id": 13, "elements": [ 8, 12, 13, 15, 16 ], "cost": 225 }, { "id": 14, "elements": [ 4, 5, 6, 7, 8 ], "cost": 245 }, { "id": 15, "elements": [ 15, 17 ], "cost": 176 }, { "id": 16, "elements": [ 15, 19 ], "cost": 190 }, { "id": 17, "elements": [ 8, 9, 10 ], "cost": 186 }, { "id": 18, "elements": [ 17, 18, 19, 20, 22 ], "cost": 35 }, { "id": 19, "elements": [ 6, 8, 9, 10, 11, 12 ], "cost": 114 }, { "id": 20, "elements": [ 18, 20, 21, 22 ], "cost": 336 }, { "id": 21, "elements": [ 8, 11, 13 ], "cost": 270 }, { "id": 22, "elements": [ 13, 14, 17 ], "cost": 228 }, { "id": 23, "elements": [ 20, 21, 22 ], "cost": 252 }, { "id": 24, "elements": [ 6, 7, 8, 10, 11, 12 ], "cost": 36 }, { "id": 25, "elements": [ 17, 18, 19, 21 ], "cost": 396 }, { "id": 26, "elements": [ 8, 9, 10, 11, 12, 13 ], "cost": 24 }, { "id": 27, "elements": [ 7, 8, 9, 10, 11, 13 ], "cost": 486 }, { "id": 28, "elements": [ 18, 19, 20, 21, 22 ], "cost": 235 }, { "id": 29, "elements": [ 10, 11, 13, 14 ], "cost": 216 }, { "id": 30, "elements": [ 15, 17, 18, 21, 22 ], "cost": 435 }, { "id": 31, "elements": [ 4, 7 ], "cost": 172 }, { "id": 32, "elements": [ 19, 20, 22 ], "cost": 264 }, { "id": 33, "elements": [ 3, 6, 7, 8, 9 ], "cost": 250 }, { "id": 34, "elements": [ 2, 3, 4, 5, 7 ], "cost": 125 }, { "id": 35, "elements": [ 16, 19, 20, 21 ], "cost": 24 }, { "id": 36, "elements": [ 5, 6, 9, 10, 15 ], "cost": 205 }, { "id": 37, "elements": [ 13, 16, 17 ], "cost": 48 }, { "id": 38, "elements": [ 11, 13, 16 ], "cost": 54 }, { "id": 39, "elements": [ 3, 6, 7, 8, 9, 11 ], "cost": 192 }, { "id": 40, "elements": [ 7, 10, 12, 15, 16 ], "cost": 25 }, { "id": 41, "elements": [ 1, 4 ], "cost": 108 }, { "id": 42, "elements": [ 5, 8, 9, 10, 11, 12 ], "cost": 126 }, { "id": 43, "elements": [ 16, 17, 20 ], "cost": 246 }, { "id": 44, "elements": [ 12, 13, 16, 18 ], "cost": 280 }, { "id": 45, "elements": [ 16, 19, 20, 21, 22 ], "cost": 230 }, { "id": 46, "elements": [ 1, 2, 5 ], "cost": 132 }, { "id": 47, "elements": [ 1, 2, 3, 4, 5 ], "cost": 315 }, { "id": 48, "elements": [ 16, 17, 18, 19, 20, 21 ], "cost": 150 }, { "id": 49, "elements": [ 15, 16, 18, 19, 21 ], "cost": 475 }, { "id": 50, "elements": [ 11, 13, 14, 15, 17, 18 ], "cost": 438 }, { "id": 51, "elements": [ 18, 19 ], "cost": 114 }, { "id": 52, "elements": [ 10, 11, 12, 13 ], "cost": 224 }, { "id": 53, "elements": [ 11, 13, 15, 16, 19 ], "cost": 40 }, { "id": 54, "elements": [ 6, 7, 8 ], "cost": 231 }, { "id": 55, "elements": [ 5, 6, 7, 9 ], "cost": 100 }, { "id": 56, "elements": [ 10, 13, 15 ], "cost": 132 }, { "id": 57, "elements": [ 13, 14, 16 ], "cost": 24 }, { "id": 58, "elements": [ 3, 4, 5, 6, 7, 8 ], "cost": 42 }, { "id": 59, "elements": [ 19, 20, 21, 22 ], "cost": 88 }, { "id": 60, "elements": [ 2, 3, 4, 6, 7, 8 ], "cost": 24 }, { "id": 61, "elements": [ 2, 4, 5, 6, 7, 8 ], "cost": 390 }, { "id": 62, "elements": [ 3, 4, 5, 6, 8 ], "cost": 445 }, { "id": 63, "elements": [ 17, 18, 19, 20, 21, 22 ], "cost": 264 }, { "id": 64, "elements": [ 16, 18, 20, 21 ], "cost": 360 }, { "id": 65, "elements": [ 10, 13, 14, 15, 16 ], "cost": 95 }, { "id": 66, "elements": [ 14, 16, 17, 18, 19, 20 ], "cost": 462 }, { "id": 67, "elements": [ 2, 6, 8 ], "cost": 249 }, { "id": 68, "elements": [ 5, 6, 7, 9, 10, 12 ], "cost": 330 }, { "id": 69, "elements": [ 4, 6, 7, 9 ], "cost": 244 }, { "id": 70, "elements": [ 15, 16, 17, 18, 19, 21 ], "cost": 138 }, { "id": 71, "elements": [ 5, 7, 8, 9, 10, 12 ], "cost": 312 }, { "id": 72, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 6 }, { "id": 73, "elements": [ 2, 3, 4, 5, 6, 8 ], "cost": 282 }, { "id": 74, "elements": [ 13, 16, 20 ], "cost": 183 }, { "id": 75, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 348 }, { "id": 76, "elements": [ 16, 18, 19, 21, 22 ], "cost": 25 }, { "id": 77, "elements": [ 15, 18, 19, 20, 21, 22 ], "cost": 24 }, { "id": 78, "elements": [ 14, 15, 16, 17, 18, 19 ], "cost": 510 }, { "id": 79, "elements": [ 5, 6, 7, 8 ], "cost": 380 }, { "id": 80, "elements": [ 18, 20, 21 ], "cost": 45 }, { "id": 81, "elements": [ 3, 7, 8, 9 ], "cost": 188 }, { "id": 82, "elements": [ 5, 10, 12 ], "cost": 297 }, { "id": 83, "elements": [ 14, 16, 17 ], "cost": 222 }, { "id": 84, "elements": [ 6, 9, 10, 11, 12, 13 ], "cost": 282 }, { "id": 85, "elements": [ 16, 18, 19, 21 ], "cost": 316 }, { "id": 86, "elements": [ 1, 5 ], "cost": 88 }, { "id": 87, "elements": [ 2, 4, 5, 6 ], "cost": 172 }, { "id": 88, "elements": [ 19, 20 ], "cost": 48 }, { "id": 89, "elements": [ 7, 9, 12 ], "cost": 42 }, { "id": 90, "elements": [ 2, 7, 8 ], "cost": 201 }, { "id": 91, "elements": [ 2, 4, 6, 7 ], "cost": 316 }, { "id": 92, "elements": [ 8, 10, 11, 12, 13 ], "cost": 80 }, { "id": 93, "elements": [ 13, 19, 21 ], "cost": 57 }, { "id": 94, "elements": [ 13, 14, 15, 17, 18 ], "cost": 10 }, { "id": 95, "elements": [ 13, 16, 17, 18, 19, 21 ], "cost": 222 }, { "id": 96, "elements": [ 6, 10 ], "cost": 28 }, { "id": 97, "elements": [ 11, 12, 14 ], "cost": 21 }, { "id": 98, "elements": [ 11, 12, 13, 14, 15, 16 ], "cost": 480 }, { "id": 99, "elements": [ 15, 16, 17, 18, 21, 22 ], "cost": 216 }, { "id": 100, "elements": [ 7, 10, 12, 13 ], "cost": 4 }, { "id": 101, "elements": [ 17, 21, 22 ], "cost": 267 }, { "id": 102, "elements": [ 12, 13, 14, 15, 17, 18, 20 ], "cost": 63 }, { "id": 103, "elements": [ 15, 17, 18, 19, 20, 21, 22 ], "cost": 476 }, { "id": 104, "elements": [ 8, 10, 12 ], "cost": 258 }, { "id": 105, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 118, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 119, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 120, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 121, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 122, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 123, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 124, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 125, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 126, "elements": [ 22 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0036_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0036_bag_stats.png" }, "solution": [ 3, 60, 86, 88, 99 ], "obj": 514.0, "instance_variant": { "num_elements": 22, "num_sets": 126, "sets": [ { "id": "S1", "elements": [ 3, 5, 6, 8, 9 ], "cost": 320 }, { "id": "S2", "elements": [ 1, 2, 3, 4, 5, 6, 8 ], "cost": 301 }, { "id": "S3", "elements": [ 8, 9, 10, 11, 12, 13 ], "cost": 138 }, { "id": "S4", "elements": [ 2, 4, 7, 8 ], "cost": 124 }, { "id": "S5", "elements": [ 11, 13 ], "cost": 130 }, { "id": "S6", "elements": [ 18, 20 ], "cost": 112 }, { "id": "S7", "elements": [ 6, 11 ], "cost": 156 }, { "id": "S8", "elements": [ 4, 6, 7, 8, 9, 10, 11 ], "cost": 42 }, { "id": "S9", "elements": [ 10, 11, 12, 13 ], "cost": 48 }, { "id": "S10", "elements": [ 7, 8, 11 ], "cost": 87 }, { "id": "S11", "elements": [ 9, 11 ], "cost": 38 }, { "id": "S12", "elements": [ 5, 7, 11, 14 ], "cost": 4 }, { "id": "S13", "elements": [ 7, 11, 12, 14, 15 ], "cost": 225 }, { "id": "S14", "elements": [ 3, 4, 5, 6, 7 ], "cost": 245 }, { "id": "S15", "elements": [ 14, 16 ], "cost": 176 }, { "id": "S16", "elements": [ 14, 18 ], "cost": 190 }, { "id": "S17", "elements": [ 7, 8, 9 ], "cost": 186 }, { "id": "S18", "elements": [ 16, 17, 18, 19, 21 ], "cost": 35 }, { "id": "S19", "elements": [ 5, 7, 8, 9, 10, 11 ], "cost": 114 }, { "id": "S20", "elements": [ 17, 19, 20, 21 ], "cost": 336 }, { "id": "S21", "elements": [ 7, 10, 12 ], "cost": 270 }, { "id": "S22", "elements": [ 12, 13, 16 ], "cost": 228 }, { "id": "S23", "elements": [ 19, 20, 21 ], "cost": 252 }, { "id": "S24", "elements": [ 5, 6, 7, 9, 10, 11 ], "cost": 36 }, { "id": "S25", "elements": [ 16, 17, 18, 20 ], "cost": 396 }, { "id": "S26", "elements": [ 7, 8, 9, 10, 11, 12 ], "cost": 24 }, { "id": "S27", "elements": [ 6, 7, 8, 9, 10, 12 ], "cost": 486 }, { "id": "S28", "elements": [ 17, 18, 19, 20, 21 ], "cost": 235 }, { "id": "S29", "elements": [ 9, 10, 12, 13 ], "cost": 216 }, { "id": "S30", "elements": [ 14, 16, 17, 20, 21 ], "cost": 435 }, { "id": "S31", "elements": [ 3, 6 ], "cost": 172 }, { "id": "S32", "elements": [ 18, 19, 21 ], "cost": 264 }, { "id": "S33", "elements": [ 2, 5, 6, 7, 8 ], "cost": 250 }, { "id": "S34", "elements": [ 1, 2, 3, 4, 6 ], "cost": 125 }, { "id": "S35", "elements": [ 15, 18, 19, 20 ], "cost": 24 }, { "id": "S36", "elements": [ 4, 5, 8, 9, 14 ], "cost": 205 }, { "id": "S37", "elements": [ 12, 15, 16 ], "cost": 48 }, { "id": "S38", "elements": [ 10, 12, 15 ], "cost": 54 }, { "id": "S39", "elements": [ 2, 5, 6, 7, 8, 10 ], "cost": 192 }, { "id": "S40", "elements": [ 6, 9, 11, 14, 15 ], "cost": 25 }, { "id": "S41", "elements": [ 0, 3 ], "cost": 108 }, { "id": "S42", "elements": [ 4, 7, 8, 9, 10, 11 ], "cost": 126 }, { "id": "S43", "elements": [ 15, 16, 19 ], "cost": 246 }, { "id": "S44", "elements": [ 11, 12, 15, 17 ], "cost": 280 }, { "id": "S45", "elements": [ 15, 18, 19, 20, 21 ], "cost": 230 }, { "id": "S46", "elements": [ 0, 1, 4 ], "cost": 132 }, { "id": "S47", "elements": [ 0, 1, 2, 3, 4 ], "cost": 315 }, { "id": "S48", "elements": [ 15, 16, 17, 18, 19, 20 ], "cost": 150 }, { "id": "S49", "elements": [ 14, 15, 17, 18, 20 ], "cost": 475 }, { "id": "S50", "elements": [ 10, 12, 13, 14, 16, 17 ], "cost": 438 }, { "id": "S51", "elements": [ 17, 18 ], "cost": 114 }, { "id": "S52", "elements": [ 9, 10, 11, 12 ], "cost": 224 }, { "id": "S53", "elements": [ 10, 12, 14, 15, 18 ], "cost": 40 }, { "id": "S54", "elements": [ 5, 6, 7 ], "cost": 231 }, { "id": "S55", "elements": [ 4, 5, 6, 8 ], "cost": 100 }, { "id": "S56", "elements": [ 9, 12, 14 ], "cost": 132 }, { "id": "S57", "elements": [ 12, 13, 15 ], "cost": 24 }, { "id": "S58", "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 42 }, { "id": "S59", "elements": [ 18, 19, 20, 21 ], "cost": 88 }, { "id": "S60", "elements": [ 1, 2, 3, 5, 6, 7 ], "cost": 24 }, { "id": "S61", "elements": [ 1, 3, 4, 5, 6, 7 ], "cost": 390 }, { "id": "S62", "elements": [ 2, 3, 4, 5, 7 ], "cost": 445 }, { "id": "S63", "elements": [ 16, 17, 18, 19, 20, 21 ], "cost": 264 }, { "id": "S64", "elements": [ 15, 17, 19, 20 ], "cost": 360 }, { "id": "S65", "elements": [ 9, 12, 13, 14, 15 ], "cost": 95 }, { "id": "S66", "elements": [ 13, 15, 16, 17, 18, 19 ], "cost": 462 }, { "id": "S67", "elements": [ 1, 5, 7 ], "cost": 249 }, { "id": "S68", "elements": [ 4, 5, 6, 8, 9, 11 ], "cost": 330 }, { "id": "S69", "elements": [ 3, 5, 6, 8 ], "cost": 244 }, { "id": "S70", "elements": [ 14, 15, 16, 17, 18, 20 ], "cost": 138 }, { "id": "S71", "elements": [ 4, 6, 7, 8, 9, 11 ], "cost": 312 }, { "id": "S72", "elements": [ 0, 1, 2, 3, 4, 5 ], "cost": 6 }, { "id": "S73", "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 282 }, { "id": "S74", "elements": [ 12, 15, 19 ], "cost": 183 }, { "id": "S75", "elements": [ 0, 1, 2, 3, 4, 6 ], "cost": 348 }, { "id": "S76", "elements": [ 15, 17, 18, 20, 21 ], "cost": 25 }, { "id": "S77", "elements": [ 14, 17, 18, 19, 20, 21 ], "cost": 24 }, { "id": "S78", "elements": [ 13, 14, 15, 16, 17, 18 ], "cost": 510 }, { "id": "S79", "elements": [ 4, 5, 6, 7 ], "cost": 380 }, { "id": "S80", "elements": [ 17, 19, 20 ], "cost": 45 }, { "id": "S81", "elements": [ 2, 6, 7, 8 ], "cost": 188 }, { "id": "S82", "elements": [ 4, 9, 11 ], "cost": 297 }, { "id": "S83", "elements": [ 13, 15, 16 ], "cost": 222 }, { "id": "S84", "elements": [ 5, 8, 9, 10, 11, 12 ], "cost": 282 }, { "id": "S85", "elements": [ 15, 17, 18, 20 ], "cost": 316 }, { "id": "S86", "elements": [ 0, 4 ], "cost": 88 }, { "id": "S87", "elements": [ 1, 3, 4, 5 ], "cost": 172 }, { "id": "S88", "elements": [ 18, 19 ], "cost": 48 }, { "id": "S89", "elements": [ 6, 8, 11 ], "cost": 42 }, { "id": "S90", "elements": [ 1, 6, 7 ], "cost": 201 }, { "id": "S91", "elements": [ 1, 3, 5, 6 ], "cost": 316 }, { "id": "S92", "elements": [ 7, 9, 10, 11, 12 ], "cost": 80 }, { "id": "S93", "elements": [ 12, 18, 20 ], "cost": 57 }, { "id": "S94", "elements": [ 12, 13, 14, 16, 17 ], "cost": 10 }, { "id": "S95", "elements": [ 12, 15, 16, 17, 18, 20 ], "cost": 222 }, { "id": "S96", "elements": [ 5, 9 ], "cost": 28 }, { "id": "S97", "elements": [ 10, 11, 13 ], "cost": 21 }, { "id": "S98", "elements": [ 10, 11, 12, 13, 14, 15 ], "cost": 480 }, { "id": "S99", "elements": [ 14, 15, 16, 17, 20, 21 ], "cost": 216 }, { "id": "S100", "elements": [ 6, 9, 11, 12 ], "cost": 4 }, { "id": "S101", "elements": [ 16, 20, 21 ], "cost": 267 }, { "id": "S102", "elements": [ 11, 12, 13, 14, 16, 17, 19 ], "cost": 63 }, { "id": "S103", "elements": [ 14, 16, 17, 18, 19, 20, 21 ], "cost": 476 }, { "id": "S104", "elements": [ 7, 9, 11 ], "cost": 258 }, { "id": "S105", "elements": [ 0 ], "cost": 10000 }, { "id": "S106", "elements": [ 1 ], "cost": 10000 }, { "id": "S107", "elements": [ 2 ], "cost": 10000 }, { "id": "S108", "elements": [ 3 ], "cost": 10000 }, { "id": "S109", "elements": [ 4 ], "cost": 10000 }, { "id": "S110", "elements": [ 5 ], "cost": 10000 }, { "id": "S111", "elements": [ 6 ], "cost": 10000 }, { "id": "S112", "elements": [ 7 ], "cost": 10000 }, { "id": "S113", "elements": [ 8 ], "cost": 10000 }, { "id": "S114", "elements": [ 9 ], "cost": 10000 }, { "id": "S115", "elements": [ 10 ], "cost": 10000 }, { "id": "S116", "elements": [ 11 ], "cost": 10000 }, { "id": "S117", "elements": [ 12 ], "cost": 10000 }, { "id": "S118", "elements": [ 13 ], "cost": 10000 }, { "id": "S119", "elements": [ 14 ], "cost": 10000 }, { "id": "S120", "elements": [ 15 ], "cost": 10000 }, { "id": "S121", "elements": [ 16 ], "cost": 10000 }, { "id": "S122", "elements": [ 17 ], "cost": 10000 }, { "id": "S123", "elements": [ 18 ], "cost": 10000 }, { "id": "S124", "elements": [ 19 ], "cost": 10000 }, { "id": "S125", "elements": [ 20 ], "cost": 10000 }, { "id": "S126", "elements": [ 21 ], "cost": 10000 } ] }, "solution_variant": [ "S3", "S60", "S86", "S88", "S99" ], "context_index": 37, "input_format": "nl", "input_index_base": 0 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "There’s a scheduling puzzle at the depot: choose a handful of preplanned routes so that every neighborhood gets a single delivery and no neighborhood appears on more than one picked route. Each route carries a fuel price tag — total fuel is just the sum of those tags for the routes we pick, and the aim is to keep that sum as low as possible. The concrete route and cost details are listed below.\n\n- **num_neighborhoods**: 21\n- **num_routes**: 112\n\n| route_id | fuel_cost | route_neighborhoods |\n|---|---|---|\n| S1 | 48 | 1 2 3 5 6 7 |\n| S2 | 84 | 11 14 15 16 18 20 |\n| S3 | 28 | 3 4 6 8 |\n| S4 | 252 | 13 17 19 |\n| S5 | 230 | 13 14 15 16 17 |\n| S6 | 360 | 1 2 4 5 |\n| S7 | 126 | 18 20 |\n| S8 | 180 | 1 2 5 6 |\n| S9 | 190 | 15 16 17 18 20 |\n| S10 | 176 | 1 4 |\n| S11 | 280 | 9 11 13 14 |\n| S12 | 200 | 15 18 20 21 |\n| S13 | 345 | 17 18 19 20 21 |\n| S14 | 474 | 11 12 13 14 16 17 |\n| S15 | 84 | 12 13 16 18 |\n| S16 | 576 | 4 6 7 9 10 12 |\n| S17 | 96 | 11 14 15 16 |\n| S18 | 4 | 13 15 16 17 |\n| S19 | 384 | 14 15 16 20 |\n| S20 | 58 | 4 7 |\n| S21 | 246 | 19 20 21 |\n| S22 | 290 | 13 14 15 19 20 |\n| S23 | 300 | 1 2 3 4 |\n| S24 | 68 | 1 3 4 6 |\n| S25 | 9 | 13 14 15 |\n| S26 | 60 | 2 7 8 |\n| S27 | 270 | 10 11 12 |\n| S28 | 70 | 16 18 19 20 21 |\n| S29 | 328 | 17 18 19 21 |\n| S30 | 220 | 8 9 10 11 13 |\n| S31 | 136 | 1 3 5 6 |\n| S32 | 445 | 11 12 13 14 17 |\n| S33 | 267 | 17 18 19 |\n| S34 | 136 | 1 2 3 5 |\n| S35 | 24 | 16 17 18 19 20 21 |\n| S36 | 160 | 16 17 18 19 21 |\n| S37 | 245 | 1 2 3 5 6 |\n| S38 | 126 | 7 8 9 |\n| S39 | 162 | 13 15 16 18 19 20 |\n| S40 | 252 | 4 6 7 9 |\n| S41 | 234 | 12 14 15 |\n| S42 | 184 | 18 19 20 21 |\n| S43 | 504 | 8 9 10 11 12 15 |\n| S44 | 40 | 8 10 13 14 |\n| S45 | 415 | 10 12 13 14 15 |\n| S46 | 144 | 2 3 4 6 7 8 |\n| S47 | 82 | 12 14 |\n| S48 | 243 | 3 5 9 |\n| S49 | 284 | 15 19 20 21 |\n| S50 | 93 | 1 3 4 |\n| S51 | 111 | 2 4 5 |\n| S52 | 112 | 6 12 |\n| S53 | 94 | 5 6 |\n| S54 | 310 | 2 3 5 6 7 |\n| S55 | 265 | 6 8 10 11 12 |\n| S56 | 30 | 15 20 |\n| S57 | 195 | 8 11 12 |\n| S58 | 300 | 13 15 16 17 18 |\n| S59 | 396 | 11 12 13 14 |\n| S60 | 273 | 2 3 5 |\n| S61 | 120 | 1 3 8 |\n| S62 | 272 | 2 6 7 10 |\n| S63 | 370 | 15 16 18 19 21 |\n| S64 | 258 | 15 17 21 |\n| S65 | 81 | 10 12 13 |\n| S66 | 162 | 4 5 |\n| S67 | 36 | 3 4 6 7 |\n| S68 | 390 | 1 2 3 4 5 6 |\n| S69 | 105 | 13 15 16 18 19 |\n| S70 | 152 | 7 10 |\n| S71 | 249 | 9 11 13 |\n| S72 | 32 | 1 4 7 8 |\n| S73 | 154 | 10 12 |\n| S74 | 360 | 10 11 12 14 15 16 |\n| S75 | 134 | 13 17 |\n| S76 | 132 | 6 8 12 |\n| S77 | 132 | 13 15 16 17 19 20 |\n| S78 | 72 | 6 11 12 |\n| S79 | 144 | 8 13 14 |\n| S80 | 320 | 8 10 12 18 |\n| S81 | 213 | 8 9 11 |\n| S82 | 15 | 4 5 6 7 8 |\n| S83 | 134 | 7 8 |\n| S84 | 186 | 13 14 17 |\n| S85 | 258 | 5 6 7 8 9 11 |\n| S86 | 264 | 2 4 5 6 |\n| S87 | 33 | 14 16 18 |\n| S88 | 36 | 5 6 7 8 |\n| S89 | 100 | 15 17 |\n| S90 | 258 | 2 4 5 7 8 9 |\n| S91 | 204 | 17 19 20 21 |\n| S92 | 10000 | 1 |\n| S93 | 10000 | 2 |\n| S94 | 10000 | 3 |\n| S95 | 10000 | 4 |\n| S96 | 10000 | 5 |\n| S97 | 10000 | 6 |\n| S98 | 10000 | 7 |\n| S99 | 10000 | 8 |\n| S100 | 10000 | 9 |\n| S101 | 10000 | 10 |\n| S102 | 10000 | 11 |\n| S103 | 10000 | 12 |\n| S104 | 10000 | 13 |\n| S105 | 10000 | 14 |\n| S106 | 10000 | 15 |\n| S107 | 10000 | 16 |\n| S108 | 10000 | 17 |\n| S109 | 10000 | 18 |\n| S110 | 10000 | 19 |\n| S111 | 10000 | 20 |\n| S112 | 10000 | 21 |\n\nIf you want to hand the chosen routes back in a tidy, machine-friendly way, just reply with a little JSON that looks like this:\n\n{\n \"solution\": [\"route_id\", ...]\n}\n\nKeep in mind this is just a sketch of the shape I expect: \"solution\" is the list of picked routes, and each \"route_id\" is a placeholder for the exact route identifier from the instance (use the identifier exactly as given). This isn't the actual answer — just the format to use.\n\nPlease don’t rename any identifiers or invent new labels; use the IDs exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 21, "num_sets": 112, "density": 0.16071428571428573, "sets": [ { "id": 1, "elements": [ 1, 2, 3, 5, 6, 7 ], "cost": 48 }, { "id": 2, "elements": [ 11, 14, 15, 16, 18, 20 ], "cost": 84 }, { "id": 3, "elements": [ 3, 4, 6, 8 ], "cost": 28 }, { "id": 4, "elements": [ 13, 17, 19 ], "cost": 252 }, { "id": 5, "elements": [ 13, 14, 15, 16, 17 ], "cost": 230 }, { "id": 6, "elements": [ 1, 2, 4, 5 ], "cost": 360 }, { "id": 7, "elements": [ 18, 20 ], "cost": 126 }, { "id": 8, "elements": [ 1, 2, 5, 6 ], "cost": 180 }, { "id": 9, "elements": [ 15, 16, 17, 18, 20 ], "cost": 190 }, { "id": 10, "elements": [ 1, 4 ], "cost": 176 }, { "id": 11, "elements": [ 9, 11, 13, 14 ], "cost": 280 }, { "id": 12, "elements": [ 15, 18, 20, 21 ], "cost": 200 }, { "id": 13, "elements": [ 17, 18, 19, 20, 21 ], "cost": 345 }, { "id": 14, "elements": [ 11, 12, 13, 14, 16, 17 ], "cost": 474 }, { "id": 15, "elements": [ 12, 13, 16, 18 ], "cost": 84 }, { "id": 16, "elements": [ 4, 6, 7, 9, 10, 12 ], "cost": 576 }, { "id": 17, "elements": [ 11, 14, 15, 16 ], "cost": 96 }, { "id": 18, "elements": [ 13, 15, 16, 17 ], "cost": 4 }, { "id": 19, "elements": [ 14, 15, 16, 20 ], "cost": 384 }, { "id": 20, "elements": [ 4, 7 ], "cost": 58 }, { "id": 21, "elements": [ 19, 20, 21 ], "cost": 246 }, { "id": 22, "elements": [ 13, 14, 15, 19, 20 ], "cost": 290 }, { "id": 23, "elements": [ 1, 2, 3, 4 ], "cost": 300 }, { "id": 24, "elements": [ 1, 3, 4, 6 ], "cost": 68 }, { "id": 25, "elements": [ 13, 14, 15 ], "cost": 9 }, { "id": 26, "elements": [ 2, 7, 8 ], "cost": 60 }, { "id": 27, "elements": [ 10, 11, 12 ], "cost": 270 }, { "id": 28, "elements": [ 16, 18, 19, 20, 21 ], "cost": 70 }, { "id": 29, "elements": [ 17, 18, 19, 21 ], "cost": 328 }, { "id": 30, "elements": [ 8, 9, 10, 11, 13 ], "cost": 220 }, { "id": 31, "elements": [ 1, 3, 5, 6 ], "cost": 136 }, { "id": 32, "elements": [ 11, 12, 13, 14, 17 ], "cost": 445 }, { "id": 33, "elements": [ 17, 18, 19 ], "cost": 267 }, { "id": 34, "elements": [ 1, 2, 3, 5 ], "cost": 136 }, { "id": 35, "elements": [ 16, 17, 18, 19, 20, 21 ], "cost": 24 }, { "id": 36, "elements": [ 16, 17, 18, 19, 21 ], "cost": 160 }, { "id": 37, "elements": [ 1, 2, 3, 5, 6 ], "cost": 245 }, { "id": 38, "elements": [ 7, 8, 9 ], "cost": 126 }, { "id": 39, "elements": [ 13, 15, 16, 18, 19, 20 ], "cost": 162 }, { "id": 40, "elements": [ 4, 6, 7, 9 ], "cost": 252 }, { "id": 41, "elements": [ 12, 14, 15 ], "cost": 234 }, { "id": 42, "elements": [ 18, 19, 20, 21 ], "cost": 184 }, { "id": 43, "elements": [ 8, 9, 10, 11, 12, 15 ], "cost": 504 }, { "id": 44, "elements": [ 8, 10, 13, 14 ], "cost": 40 }, { "id": 45, "elements": [ 10, 12, 13, 14, 15 ], "cost": 415 }, { "id": 46, "elements": [ 2, 3, 4, 6, 7, 8 ], "cost": 144 }, { "id": 47, "elements": [ 12, 14 ], "cost": 82 }, { "id": 48, "elements": [ 3, 5, 9 ], "cost": 243 }, { "id": 49, "elements": [ 15, 19, 20, 21 ], "cost": 284 }, { "id": 50, "elements": [ 1, 3, 4 ], "cost": 93 }, { "id": 51, "elements": [ 2, 4, 5 ], "cost": 111 }, { "id": 52, "elements": [ 6, 12 ], "cost": 112 }, { "id": 53, "elements": [ 5, 6 ], "cost": 94 }, { "id": 54, "elements": [ 2, 3, 5, 6, 7 ], "cost": 310 }, { "id": 55, "elements": [ 6, 8, 10, 11, 12 ], "cost": 265 }, { "id": 56, "elements": [ 15, 20 ], "cost": 30 }, { "id": 57, "elements": [ 8, 11, 12 ], "cost": 195 }, { "id": 58, "elements": [ 13, 15, 16, 17, 18 ], "cost": 300 }, { "id": 59, "elements": [ 11, 12, 13, 14 ], "cost": 396 }, { "id": 60, "elements": [ 2, 3, 5 ], "cost": 273 }, { "id": 61, "elements": [ 1, 3, 8 ], "cost": 120 }, { "id": 62, "elements": [ 2, 6, 7, 10 ], "cost": 272 }, { "id": 63, "elements": [ 15, 16, 18, 19, 21 ], "cost": 370 }, { "id": 64, "elements": [ 15, 17, 21 ], "cost": 258 }, { "id": 65, "elements": [ 10, 12, 13 ], "cost": 81 }, { "id": 66, "elements": [ 4, 5 ], "cost": 162 }, { "id": 67, "elements": [ 3, 4, 6, 7 ], "cost": 36 }, { "id": 68, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 390 }, { "id": 69, "elements": [ 13, 15, 16, 18, 19 ], "cost": 105 }, { "id": 70, "elements": [ 7, 10 ], "cost": 152 }, { "id": 71, "elements": [ 9, 11, 13 ], "cost": 249 }, { "id": 72, "elements": [ 1, 4, 7, 8 ], "cost": 32 }, { "id": 73, "elements": [ 10, 12 ], "cost": 154 }, { "id": 74, "elements": [ 10, 11, 12, 14, 15, 16 ], "cost": 360 }, { "id": 75, "elements": [ 13, 17 ], "cost": 134 }, { "id": 76, "elements": [ 6, 8, 12 ], "cost": 132 }, { "id": 77, "elements": [ 13, 15, 16, 17, 19, 20 ], "cost": 132 }, { "id": 78, "elements": [ 6, 11, 12 ], "cost": 72 }, { "id": 79, "elements": [ 8, 13, 14 ], "cost": 144 }, { "id": 80, "elements": [ 8, 10, 12, 18 ], "cost": 320 }, { "id": 81, "elements": [ 8, 9, 11 ], "cost": 213 }, { "id": 82, "elements": [ 4, 5, 6, 7, 8 ], "cost": 15 }, { "id": 83, "elements": [ 7, 8 ], "cost": 134 }, { "id": 84, "elements": [ 13, 14, 17 ], "cost": 186 }, { "id": 85, "elements": [ 5, 6, 7, 8, 9, 11 ], "cost": 258 }, { "id": 86, "elements": [ 2, 4, 5, 6 ], "cost": 264 }, { "id": 87, "elements": [ 14, 16, 18 ], "cost": 33 }, { "id": 88, "elements": [ 5, 6, 7, 8 ], "cost": 36 }, { "id": 89, "elements": [ 15, 17 ], "cost": 100 }, { "id": 90, "elements": [ 2, 4, 5, 7, 8, 9 ], "cost": 258 }, { "id": 91, "elements": [ 17, 19, 20, 21 ], "cost": 204 }, { "id": 92, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 93, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 94, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 95, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 96, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 97, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 98, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 21 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0037_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0037_bag_stats.png" }, "solution": [ 20, 25, 35, 37, 73, 81 ], "obj": 703.0, "instance_variant": { "num_elements": 21, "num_sets": 112, "sets": [ { "id": "S1", "elements": [ 1, 2, 3, 5, 6, 7 ], "cost": 48 }, { "id": "S2", "elements": [ 11, 14, 15, 16, 18, 20 ], "cost": 84 }, { "id": "S3", "elements": [ 3, 4, 6, 8 ], "cost": 28 }, { "id": "S4", "elements": [ 13, 17, 19 ], "cost": 252 }, { "id": "S5", "elements": [ 13, 14, 15, 16, 17 ], "cost": 230 }, { "id": "S6", "elements": [ 1, 2, 4, 5 ], "cost": 360 }, { "id": "S7", "elements": [ 18, 20 ], "cost": 126 }, { "id": "S8", "elements": [ 1, 2, 5, 6 ], "cost": 180 }, { "id": "S9", "elements": [ 15, 16, 17, 18, 20 ], "cost": 190 }, { "id": "S10", "elements": [ 1, 4 ], "cost": 176 }, { "id": "S11", "elements": [ 9, 11, 13, 14 ], "cost": 280 }, { "id": "S12", "elements": [ 15, 18, 20, 21 ], "cost": 200 }, { "id": "S13", "elements": [ 17, 18, 19, 20, 21 ], "cost": 345 }, { "id": "S14", "elements": [ 11, 12, 13, 14, 16, 17 ], "cost": 474 }, { "id": "S15", "elements": [ 12, 13, 16, 18 ], "cost": 84 }, { "id": "S16", "elements": [ 4, 6, 7, 9, 10, 12 ], "cost": 576 }, { "id": "S17", "elements": [ 11, 14, 15, 16 ], "cost": 96 }, { "id": "S18", "elements": [ 13, 15, 16, 17 ], "cost": 4 }, { "id": "S19", "elements": [ 14, 15, 16, 20 ], "cost": 384 }, { "id": "S20", "elements": [ 4, 7 ], "cost": 58 }, { "id": "S21", "elements": [ 19, 20, 21 ], "cost": 246 }, { "id": "S22", "elements": [ 13, 14, 15, 19, 20 ], "cost": 290 }, { "id": "S23", "elements": [ 1, 2, 3, 4 ], "cost": 300 }, { "id": "S24", "elements": [ 1, 3, 4, 6 ], "cost": 68 }, { "id": "S25", "elements": [ 13, 14, 15 ], "cost": 9 }, { "id": "S26", "elements": [ 2, 7, 8 ], "cost": 60 }, { "id": "S27", "elements": [ 10, 11, 12 ], "cost": 270 }, { "id": "S28", "elements": [ 16, 18, 19, 20, 21 ], "cost": 70 }, { "id": "S29", "elements": [ 17, 18, 19, 21 ], "cost": 328 }, { "id": "S30", "elements": [ 8, 9, 10, 11, 13 ], "cost": 220 }, { "id": "S31", "elements": [ 1, 3, 5, 6 ], "cost": 136 }, { "id": "S32", "elements": [ 11, 12, 13, 14, 17 ], "cost": 445 }, { "id": "S33", "elements": [ 17, 18, 19 ], "cost": 267 }, { "id": "S34", "elements": [ 1, 2, 3, 5 ], "cost": 136 }, { "id": "S35", "elements": [ 16, 17, 18, 19, 20, 21 ], "cost": 24 }, { "id": "S36", "elements": [ 16, 17, 18, 19, 21 ], "cost": 160 }, { "id": "S37", "elements": [ 1, 2, 3, 5, 6 ], "cost": 245 }, { "id": "S38", "elements": [ 7, 8, 9 ], "cost": 126 }, { "id": "S39", "elements": [ 13, 15, 16, 18, 19, 20 ], "cost": 162 }, { "id": "S40", "elements": [ 4, 6, 7, 9 ], "cost": 252 }, { "id": "S41", "elements": [ 12, 14, 15 ], "cost": 234 }, { "id": "S42", "elements": [ 18, 19, 20, 21 ], "cost": 184 }, { "id": "S43", "elements": [ 8, 9, 10, 11, 12, 15 ], "cost": 504 }, { "id": "S44", "elements": [ 8, 10, 13, 14 ], "cost": 40 }, { "id": "S45", "elements": [ 10, 12, 13, 14, 15 ], "cost": 415 }, { "id": "S46", "elements": [ 2, 3, 4, 6, 7, 8 ], "cost": 144 }, { "id": "S47", "elements": [ 12, 14 ], "cost": 82 }, { "id": "S48", "elements": [ 3, 5, 9 ], "cost": 243 }, { "id": "S49", "elements": [ 15, 19, 20, 21 ], "cost": 284 }, { "id": "S50", "elements": [ 1, 3, 4 ], "cost": 93 }, { "id": "S51", "elements": [ 2, 4, 5 ], "cost": 111 }, { "id": "S52", "elements": [ 6, 12 ], "cost": 112 }, { "id": "S53", "elements": [ 5, 6 ], "cost": 94 }, { "id": "S54", "elements": [ 2, 3, 5, 6, 7 ], "cost": 310 }, { "id": "S55", "elements": [ 6, 8, 10, 11, 12 ], "cost": 265 }, { "id": "S56", "elements": [ 15, 20 ], "cost": 30 }, { "id": "S57", "elements": [ 8, 11, 12 ], "cost": 195 }, { "id": "S58", "elements": [ 13, 15, 16, 17, 18 ], "cost": 300 }, { "id": "S59", "elements": [ 11, 12, 13, 14 ], "cost": 396 }, { "id": "S60", "elements": [ 2, 3, 5 ], "cost": 273 }, { "id": "S61", "elements": [ 1, 3, 8 ], "cost": 120 }, { "id": "S62", "elements": [ 2, 6, 7, 10 ], "cost": 272 }, { "id": "S63", "elements": [ 15, 16, 18, 19, 21 ], "cost": 370 }, { "id": "S64", "elements": [ 15, 17, 21 ], "cost": 258 }, { "id": "S65", "elements": [ 10, 12, 13 ], "cost": 81 }, { "id": "S66", "elements": [ 4, 5 ], "cost": 162 }, { "id": "S67", "elements": [ 3, 4, 6, 7 ], "cost": 36 }, { "id": "S68", "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 390 }, { "id": "S69", "elements": [ 13, 15, 16, 18, 19 ], "cost": 105 }, { "id": "S70", "elements": [ 7, 10 ], "cost": 152 }, { "id": "S71", "elements": [ 9, 11, 13 ], "cost": 249 }, { "id": "S72", "elements": [ 1, 4, 7, 8 ], "cost": 32 }, { "id": "S73", "elements": [ 10, 12 ], "cost": 154 }, { "id": "S74", "elements": [ 10, 11, 12, 14, 15, 16 ], "cost": 360 }, { "id": "S75", "elements": [ 13, 17 ], "cost": 134 }, { "id": "S76", "elements": [ 6, 8, 12 ], "cost": 132 }, { "id": "S77", "elements": [ 13, 15, 16, 17, 19, 20 ], "cost": 132 }, { "id": "S78", "elements": [ 6, 11, 12 ], "cost": 72 }, { "id": "S79", "elements": [ 8, 13, 14 ], "cost": 144 }, { "id": "S80", "elements": [ 8, 10, 12, 18 ], "cost": 320 }, { "id": "S81", "elements": [ 8, 9, 11 ], "cost": 213 }, { "id": "S82", "elements": [ 4, 5, 6, 7, 8 ], "cost": 15 }, { "id": "S83", "elements": [ 7, 8 ], "cost": 134 }, { "id": "S84", "elements": [ 13, 14, 17 ], "cost": 186 }, { "id": "S85", "elements": [ 5, 6, 7, 8, 9, 11 ], "cost": 258 }, { "id": "S86", "elements": [ 2, 4, 5, 6 ], "cost": 264 }, { "id": "S87", "elements": [ 14, 16, 18 ], "cost": 33 }, { "id": "S88", "elements": [ 5, 6, 7, 8 ], "cost": 36 }, { "id": "S89", "elements": [ 15, 17 ], "cost": 100 }, { "id": "S90", "elements": [ 2, 4, 5, 7, 8, 9 ], "cost": 258 }, { "id": "S91", "elements": [ 17, 19, 20, 21 ], "cost": 204 }, { "id": "S92", "elements": [ 1 ], "cost": 10000 }, { "id": "S93", "elements": [ 2 ], "cost": 10000 }, { "id": "S94", "elements": [ 3 ], "cost": 10000 }, { "id": "S95", "elements": [ 4 ], "cost": 10000 }, { "id": "S96", "elements": [ 5 ], "cost": 10000 }, { "id": "S97", "elements": [ 6 ], "cost": 10000 }, { "id": "S98", "elements": [ 7 ], "cost": 10000 }, { "id": "S99", "elements": [ 8 ], "cost": 10000 }, { "id": "S100", "elements": [ 9 ], "cost": 10000 }, { "id": "S101", "elements": [ 10 ], "cost": 10000 }, { "id": "S102", "elements": [ 11 ], "cost": 10000 }, { "id": "S103", "elements": [ 12 ], "cost": 10000 }, { "id": "S104", "elements": [ 13 ], "cost": 10000 }, { "id": "S105", "elements": [ 14 ], "cost": 10000 }, { "id": "S106", "elements": [ 15 ], "cost": 10000 }, { "id": "S107", "elements": [ 16 ], "cost": 10000 }, { "id": "S108", "elements": [ 17 ], "cost": 10000 }, { "id": "S109", "elements": [ 18 ], "cost": 10000 }, { "id": "S110", "elements": [ 19 ], "cost": 10000 }, { "id": "S111", "elements": [ 20 ], "cost": 10000 }, { "id": "S112", "elements": [ 21 ], "cost": 10000 } ] }, "solution_variant": [ "S20", "S25", "S35", "S37", "S73", "S81" ], "context_index": 38, "input_format": "markdown_table", "input_index_base": 1 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "Recently the library received a bunch of curated book bundles and the job is to choose which ones to shelve so every genre is represented, with each genre coming from exactly one bundle. Because some bundles share genres, the choices must be made so those overlaps don’t end up in the final set, and the practical goal is to minimize how much is spent by summing the prices of the chosen bundles. The complete list of bundles and prices is shown below.\n\n- **total_genres**: 23\n- **total_bundles**: 142\n\n| bundle_id | bundle_price | bundle_genres |\n|---|---|---|\n| S1 | 256 | 10 11 12 14 |\n| S2 | 324 | 6 8 10 11 |\n| S3 | 10 | 6 7 8 10 13 |\n| S4 | 249 | 0 3 5 |\n| S5 | 119 | 1 2 3 4 5 6 7 |\n| S6 | 340 | 16 19 20 21 22 |\n| S7 | 126 | 12 14 15 |\n| S8 | 66 | 17 18 19 20 21 22 |\n| S9 | 136 | 7 9 10 14 |\n| S10 | 366 | 0 1 2 3 4 6 |\n| S11 | 258 | 0 1 2 3 4 5 |\n| S12 | 444 | 1 2 3 4 5 7 |\n| S13 | 360 | 3 5 6 9 |\n| S14 | 576 | 7 8 10 11 13 16 |\n| S15 | 368 | 8 10 12 13 |\n| S16 | 198 | 6 9 10 |\n| S17 | 300 | 17 18 20 21 22 |\n| S18 | 50 | 1 2 3 4 5 |\n| S19 | 72 | 3 6 7 8 11 14 |\n| S20 | 600 | 6 7 9 10 11 13 |\n| S21 | 207 | 14 15 16 |\n| S22 | 108 | 16 17 19 20 21 22 |\n| S23 | 44 | 9 10 11 14 |\n| S24 | 240 | 14 18 19 22 |\n| S25 | 105 | 16 17 18 19 20 21 22 |\n| S26 | 455 | 0 2 3 5 7 8 11 |\n| S27 | 297 | 15 18 20 |\n| S28 | 280 | 13 15 16 17 |\n| S29 | 522 | 7 8 9 10 12 14 |\n| S30 | 138 | 11 13 14 15 16 18 |\n| S31 | 16 | 1 3 4 5 |\n| S32 | 430 | 3 4 5 6 7 |\n| S33 | 284 | 0 2 3 4 |\n| S34 | 98 | 14 15 16 17 18 19 20 |\n| S35 | 115 | 18 19 20 21 22 |\n| S36 | 255 | 7 10 11 13 14 |\n| S37 | 308 | 1 3 4 6 |\n| S38 | 30 | 13 15 16 17 22 |\n| S39 | 168 | 6 11 12 |\n| S40 | 315 | 8 10 12 13 14 16 17 |\n| S41 | 276 | 6 9 15 |\n| S42 | 336 | 9 12 13 14 15 16 |\n| S43 | 32 | 15 17 19 22 |\n| S44 | 150 | 9 10 12 13 14 15 |\n| S45 | 207 | 10 11 19 |\n| S46 | 42 | 18 20 21 |\n| S47 | 188 | 8 9 |\n| S48 | 162 | 0 1 2 3 4 7 |\n| S49 | 378 | 12 14 15 16 17 19 |\n| S50 | 486 | 16 18 19 20 21 22 |\n| S51 | 280 | 19 20 21 22 |\n| S52 | 182 | 18 20 |\n| S53 | 360 | 6 8 9 10 |\n| S54 | 95 | 2 4 6 7 8 |\n| S55 | 395 | 12 14 15 17 18 |\n| S56 | 222 | 4 5 6 |\n| S57 | 120 | 19 22 |\n| S58 | 440 | 14 16 17 18 20 |\n| S59 | 516 | 2 3 4 5 6 8 |\n| S60 | 438 | 8 10 11 12 13 14 |\n| S61 | 470 | 13 14 17 18 19 |\n| S62 | 122 | 15 16 |\n| S63 | 348 | 1 2 4 6 |\n| S64 | 316 | 4 6 7 11 |\n| S65 | 384 | 17 18 19 20 |\n| S66 | 32 | 7 9 11 12 |\n| S67 | 300 | 1 2 4 5 9 |\n| S68 | 651 | 9 10 11 12 13 14 15 |\n| S69 | 54 | 2 3 4 5 6 7 |\n| S70 | 156 | 6 9 10 13 |\n| S71 | 480 | 8 12 13 17 18 |\n| S72 | 216 | 8 10 11 |\n| S73 | 65 | 8 10 13 14 15 |\n| S74 | 679 | 5 7 8 9 10 12 14 |\n| S75 | 320 | 4 6 8 9 11 |\n| S76 | 490 | 15 16 17 18 19 20 22 |\n| S77 | 205 | 12 13 14 15 16 |\n| S78 | 370 | 11 12 14 15 16 |\n| S79 | 145 | 17 19 20 21 22 |\n| S80 | 60 | 4 6 7 8 |\n| S81 | 195 | 2 4 6 |\n| S82 | 320 | 0 1 2 3 |\n| S83 | 208 | 6 11 12 15 |\n| S84 | 255 | 7 8 10 |\n| S85 | 444 | 0 1 2 3 4 8 |\n| S86 | 54 | 10 11 12 13 16 17 |\n| S87 | 140 | 16 18 19 20 21 |\n| S88 | 156 | 3 5 6 |\n| S89 | 2 | 6 7 |\n| S90 | 450 | 13 14 16 17 18 19 |\n| S91 | 138 | 8 13 14 |\n| S92 | 296 | 15 16 19 20 |\n| S93 | 4 | 10 11 14 15 |\n| S94 | 360 | 16 17 19 20 22 |\n| S95 | 174 | 6 8 11 12 14 16 |\n| S96 | 63 | 1 3 5 |\n| S97 | 434 | 0 1 2 3 4 5 6 |\n| S98 | 198 | 1 2 3 4 5 6 |\n| S99 | 270 | 8 10 11 15 16 |\n| S100 | 60 | 3 4 5 7 8 10 |\n| S101 | 332 | 12 13 15 16 |\n| S102 | 192 | 9 10 13 |\n| S103 | 220 | 1 3 5 6 7 |\n| S104 | 15 | 13 16 17 18 20 |\n| S105 | 14 | 2 3 |\n| S106 | 594 | 7 10 14 15 16 17 |\n| S107 | 12 | 9 10 |\n| S108 | 138 | 11 13 14 |\n| S109 | 190 | 7 13 |\n| S110 | 176 | 12 15 17 18 |\n| S111 | 180 | 18 19 20 22 |\n| S112 | 400 | 0 1 2 3 7 |\n| S113 | 115 | 8 9 10 12 13 |\n| S114 | 200 | 4 5 6 7 |\n| S115 | 320 | 16 17 20 21 22 |\n| S116 | 276 | 18 19 20 |\n| S117 | 352 | 13 15 18 19 |\n| S118 | 200 | 16 19 21 22 |\n| S119 | 231 | 11 15 16 |\n| S120 | 10000 | 0 |\n| S121 | 10000 | 1 |\n| S122 | 10000 | 2 |\n| S123 | 10000 | 3 |\n| S124 | 10000 | 4 |\n| S125 | 10000 | 5 |\n| S126 | 10000 | 6 |\n| S127 | 10000 | 7 |\n| S128 | 10000 | 8 |\n| S129 | 10000 | 9 |\n| S130 | 10000 | 10 |\n| S131 | 10000 | 11 |\n| S132 | 10000 | 12 |\n| S133 | 10000 | 13 |\n| S134 | 10000 | 14 |\n| S135 | 10000 | 15 |\n| S136 | 10000 | 16 |\n| S137 | 10000 | 17 |\n| S138 | 10000 | 18 |\n| S139 | 10000 | 19 |\n| S140 | 10000 | 20 |\n| S141 | 10000 | 21 |\n| S142 | 10000 | 22 |\n\nIf you want to tell me which bundles to shelve, a tiny JSON snippet like this works great — keeps things neat and machine-friendly but still simple to read:\n\n{\n \"solution\": [\"bundle_id\", ...]\n}\n\n\"solution\" is a list of the bundle identifiers you choose to shelve so every genre is represented exactly once. Each item in that array should be the identifier for a bundle from the list above. Think of it like filling out a short form: list the bundle IDs you pick, and that's it.\n\nThis JSON is just a sketch of the shape I expect, not the actual answer — put the real bundle IDs from the instance into the array when you're ready.\n\nAlso, please make sure all identifiers are used exactly as they appear in the instance input — no renaming and no new labels.\n\nfor example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 23, "num_sets": 142, "density": 0.1736068585425597, "sets": [ { "id": 1, "elements": [ 11, 12, 13, 15 ], "cost": 256 }, { "id": 2, "elements": [ 7, 9, 11, 12 ], "cost": 324 }, { "id": 3, "elements": [ 7, 8, 9, 11, 14 ], "cost": 10 }, { "id": 4, "elements": [ 1, 4, 6 ], "cost": 249 }, { "id": 5, "elements": [ 2, 3, 4, 5, 6, 7, 8 ], "cost": 119 }, { "id": 6, "elements": [ 17, 20, 21, 22, 23 ], "cost": 340 }, { "id": 7, "elements": [ 13, 15, 16 ], "cost": 126 }, { "id": 8, "elements": [ 18, 19, 20, 21, 22, 23 ], "cost": 66 }, { "id": 9, "elements": [ 8, 10, 11, 15 ], "cost": 136 }, { "id": 10, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 366 }, { "id": 11, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 258 }, { "id": 12, "elements": [ 2, 3, 4, 5, 6, 8 ], "cost": 444 }, { "id": 13, "elements": [ 4, 6, 7, 10 ], "cost": 360 }, { "id": 14, "elements": [ 8, 9, 11, 12, 14, 17 ], "cost": 576 }, { "id": 15, "elements": [ 9, 11, 13, 14 ], "cost": 368 }, { "id": 16, "elements": [ 7, 10, 11 ], "cost": 198 }, { "id": 17, "elements": [ 18, 19, 21, 22, 23 ], "cost": 300 }, { "id": 18, "elements": [ 2, 3, 4, 5, 6 ], "cost": 50 }, { "id": 19, "elements": [ 4, 7, 8, 9, 12, 15 ], "cost": 72 }, { "id": 20, "elements": [ 7, 8, 10, 11, 12, 14 ], "cost": 600 }, { "id": 21, "elements": [ 15, 16, 17 ], "cost": 207 }, { "id": 22, "elements": [ 17, 18, 20, 21, 22, 23 ], "cost": 108 }, { "id": 23, "elements": [ 10, 11, 12, 15 ], "cost": 44 }, { "id": 24, "elements": [ 15, 19, 20, 23 ], "cost": 240 }, { "id": 25, "elements": [ 17, 18, 19, 20, 21, 22, 23 ], "cost": 105 }, { "id": 26, "elements": [ 1, 3, 4, 6, 8, 9, 12 ], "cost": 455 }, { "id": 27, "elements": [ 16, 19, 21 ], "cost": 297 }, { "id": 28, "elements": [ 14, 16, 17, 18 ], "cost": 280 }, { "id": 29, "elements": [ 8, 9, 10, 11, 13, 15 ], "cost": 522 }, { "id": 30, "elements": [ 12, 14, 15, 16, 17, 19 ], "cost": 138 }, { "id": 31, "elements": [ 2, 4, 5, 6 ], "cost": 16 }, { "id": 32, "elements": [ 4, 5, 6, 7, 8 ], "cost": 430 }, { "id": 33, "elements": [ 1, 3, 4, 5 ], "cost": 284 }, { "id": 34, "elements": [ 15, 16, 17, 18, 19, 20, 21 ], "cost": 98 }, { "id": 35, "elements": [ 19, 20, 21, 22, 23 ], "cost": 115 }, { "id": 36, "elements": [ 8, 11, 12, 14, 15 ], "cost": 255 }, { "id": 37, "elements": [ 2, 4, 5, 7 ], "cost": 308 }, { "id": 38, "elements": [ 14, 16, 17, 18, 23 ], "cost": 30 }, { "id": 39, "elements": [ 7, 12, 13 ], "cost": 168 }, { "id": 40, "elements": [ 9, 11, 13, 14, 15, 17, 18 ], "cost": 315 }, { "id": 41, "elements": [ 7, 10, 16 ], "cost": 276 }, { "id": 42, "elements": [ 10, 13, 14, 15, 16, 17 ], "cost": 336 }, { "id": 43, "elements": [ 16, 18, 20, 23 ], "cost": 32 }, { "id": 44, "elements": [ 10, 11, 13, 14, 15, 16 ], "cost": 150 }, { "id": 45, "elements": [ 11, 12, 20 ], "cost": 207 }, { "id": 46, "elements": [ 19, 21, 22 ], "cost": 42 }, { "id": 47, "elements": [ 9, 10 ], "cost": 188 }, { "id": 48, "elements": [ 1, 2, 3, 4, 5, 8 ], "cost": 162 }, { "id": 49, "elements": [ 13, 15, 16, 17, 18, 20 ], "cost": 378 }, { "id": 50, "elements": [ 17, 19, 20, 21, 22, 23 ], "cost": 486 }, { "id": 51, "elements": [ 20, 21, 22, 23 ], "cost": 280 }, { "id": 52, "elements": [ 19, 21 ], "cost": 182 }, { "id": 53, "elements": [ 7, 9, 10, 11 ], "cost": 360 }, { "id": 54, "elements": [ 3, 5, 7, 8, 9 ], "cost": 95 }, { "id": 55, "elements": [ 13, 15, 16, 18, 19 ], "cost": 395 }, { "id": 56, "elements": [ 5, 6, 7 ], "cost": 222 }, { "id": 57, "elements": [ 20, 23 ], "cost": 120 }, { "id": 58, "elements": [ 15, 17, 18, 19, 21 ], "cost": 440 }, { "id": 59, "elements": [ 3, 4, 5, 6, 7, 9 ], "cost": 516 }, { "id": 60, "elements": [ 9, 11, 12, 13, 14, 15 ], "cost": 438 }, { "id": 61, "elements": [ 14, 15, 18, 19, 20 ], "cost": 470 }, { "id": 62, "elements": [ 16, 17 ], "cost": 122 }, { "id": 63, "elements": [ 2, 3, 5, 7 ], "cost": 348 }, { "id": 64, "elements": [ 5, 7, 8, 12 ], "cost": 316 }, { "id": 65, "elements": [ 18, 19, 20, 21 ], "cost": 384 }, { "id": 66, "elements": [ 8, 10, 12, 13 ], "cost": 32 }, { "id": 67, "elements": [ 2, 3, 5, 6, 10 ], "cost": 300 }, { "id": 68, "elements": [ 10, 11, 12, 13, 14, 15, 16 ], "cost": 651 }, { "id": 69, "elements": [ 3, 4, 5, 6, 7, 8 ], "cost": 54 }, { "id": 70, "elements": [ 7, 10, 11, 14 ], "cost": 156 }, { "id": 71, "elements": [ 9, 13, 14, 18, 19 ], "cost": 480 }, { "id": 72, "elements": [ 9, 11, 12 ], "cost": 216 }, { "id": 73, "elements": [ 9, 11, 14, 15, 16 ], "cost": 65 }, { "id": 74, "elements": [ 6, 8, 9, 10, 11, 13, 15 ], "cost": 679 }, { "id": 75, "elements": [ 5, 7, 9, 10, 12 ], "cost": 320 }, { "id": 76, "elements": [ 16, 17, 18, 19, 20, 21, 23 ], "cost": 490 }, { "id": 77, "elements": [ 13, 14, 15, 16, 17 ], "cost": 205 }, { "id": 78, "elements": [ 12, 13, 15, 16, 17 ], "cost": 370 }, { "id": 79, "elements": [ 18, 20, 21, 22, 23 ], "cost": 145 }, { "id": 80, "elements": [ 5, 7, 8, 9 ], "cost": 60 }, { "id": 81, "elements": [ 3, 5, 7 ], "cost": 195 }, { "id": 82, "elements": [ 1, 2, 3, 4 ], "cost": 320 }, { "id": 83, "elements": [ 7, 12, 13, 16 ], "cost": 208 }, { "id": 84, "elements": [ 8, 9, 11 ], "cost": 255 }, { "id": 85, "elements": [ 1, 2, 3, 4, 5, 9 ], "cost": 444 }, { "id": 86, "elements": [ 11, 12, 13, 14, 17, 18 ], "cost": 54 }, { "id": 87, "elements": [ 17, 19, 20, 21, 22 ], "cost": 140 }, { "id": 88, "elements": [ 4, 6, 7 ], "cost": 156 }, { "id": 89, "elements": [ 7, 8 ], "cost": 2 }, { "id": 90, "elements": [ 14, 15, 17, 18, 19, 20 ], "cost": 450 }, { "id": 91, "elements": [ 9, 14, 15 ], "cost": 138 }, { "id": 92, "elements": [ 16, 17, 20, 21 ], "cost": 296 }, { "id": 93, "elements": [ 11, 12, 15, 16 ], "cost": 4 }, { "id": 94, "elements": [ 17, 18, 20, 21, 23 ], "cost": 360 }, { "id": 95, "elements": [ 7, 9, 12, 13, 15, 17 ], "cost": 174 }, { "id": 96, "elements": [ 2, 4, 6 ], "cost": 63 }, { "id": 97, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 434 }, { "id": 98, "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 198 }, { "id": 99, "elements": [ 9, 11, 12, 16, 17 ], "cost": 270 }, { "id": 100, "elements": [ 4, 5, 6, 8, 9, 11 ], "cost": 60 }, { "id": 101, "elements": [ 13, 14, 16, 17 ], "cost": 332 }, { "id": 102, "elements": [ 10, 11, 14 ], "cost": 192 }, { "id": 103, "elements": [ 2, 4, 6, 7, 8 ], "cost": 220 }, { "id": 104, "elements": [ 14, 17, 18, 19, 21 ], "cost": 15 }, { "id": 105, "elements": [ 3, 4 ], "cost": 14 }, { "id": 106, "elements": [ 8, 11, 15, 16, 17, 18 ], "cost": 594 }, { "id": 107, "elements": [ 10, 11 ], "cost": 12 }, { "id": 108, "elements": [ 12, 14, 15 ], "cost": 138 }, { "id": 109, "elements": [ 8, 14 ], "cost": 190 }, { "id": 110, "elements": [ 13, 16, 18, 19 ], "cost": 176 }, { "id": 111, "elements": [ 19, 20, 21, 23 ], "cost": 180 }, { "id": 112, "elements": [ 1, 2, 3, 4, 8 ], "cost": 400 }, { "id": 113, "elements": [ 9, 10, 11, 13, 14 ], "cost": 115 }, { "id": 114, "elements": [ 5, 6, 7, 8 ], "cost": 200 }, { "id": 115, "elements": [ 17, 18, 21, 22, 23 ], "cost": 320 }, { "id": 116, "elements": [ 19, 20, 21 ], "cost": 276 }, { "id": 117, "elements": [ 14, 16, 19, 20 ], "cost": 352 }, { "id": 118, "elements": [ 17, 20, 22, 23 ], "cost": 200 }, { "id": 119, "elements": [ 12, 16, 17 ], "cost": 231 }, { "id": 120, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 121, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 122, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 123, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 124, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 125, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 126, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 127, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 128, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 129, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 130, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 131, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 132, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 133, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 134, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 135, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 136, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 137, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 138, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 139, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 140, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 141, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 142, "elements": [ 23 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0038_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0038_bag_stats.png" }, "solution": [ 25, 66, 73, 97 ], "obj": 636.0, "instance_variant": { "num_elements": 23, "num_sets": 142, "sets": [ { "id": "S1", "elements": [ 10, 11, 12, 14 ], "cost": 256 }, { "id": "S2", "elements": [ 6, 8, 10, 11 ], "cost": 324 }, { "id": "S3", "elements": [ 6, 7, 8, 10, 13 ], "cost": 10 }, { "id": "S4", "elements": [ 0, 3, 5 ], "cost": 249 }, { "id": "S5", "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 119 }, { "id": "S6", "elements": [ 16, 19, 20, 21, 22 ], "cost": 340 }, { "id": "S7", "elements": [ 12, 14, 15 ], "cost": 126 }, { "id": "S8", "elements": [ 17, 18, 19, 20, 21, 22 ], "cost": 66 }, { "id": "S9", "elements": [ 7, 9, 10, 14 ], "cost": 136 }, { "id": "S10", "elements": [ 0, 1, 2, 3, 4, 6 ], "cost": 366 }, { "id": "S11", "elements": [ 0, 1, 2, 3, 4, 5 ], "cost": 258 }, { "id": "S12", "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 444 }, { "id": "S13", "elements": [ 3, 5, 6, 9 ], "cost": 360 }, { "id": "S14", "elements": [ 7, 8, 10, 11, 13, 16 ], "cost": 576 }, { "id": "S15", "elements": [ 8, 10, 12, 13 ], "cost": 368 }, { "id": "S16", "elements": [ 6, 9, 10 ], "cost": 198 }, { "id": "S17", "elements": [ 17, 18, 20, 21, 22 ], "cost": 300 }, { "id": "S18", "elements": [ 1, 2, 3, 4, 5 ], "cost": 50 }, { "id": "S19", "elements": [ 3, 6, 7, 8, 11, 14 ], "cost": 72 }, { "id": "S20", "elements": [ 6, 7, 9, 10, 11, 13 ], "cost": 600 }, { "id": "S21", "elements": [ 14, 15, 16 ], "cost": 207 }, { "id": "S22", "elements": [ 16, 17, 19, 20, 21, 22 ], "cost": 108 }, { "id": "S23", "elements": [ 9, 10, 11, 14 ], "cost": 44 }, { "id": "S24", "elements": [ 14, 18, 19, 22 ], "cost": 240 }, { "id": "S25", "elements": [ 16, 17, 18, 19, 20, 21, 22 ], "cost": 105 }, { "id": "S26", "elements": [ 0, 2, 3, 5, 7, 8, 11 ], "cost": 455 }, { "id": "S27", "elements": [ 15, 18, 20 ], "cost": 297 }, { "id": "S28", "elements": [ 13, 15, 16, 17 ], "cost": 280 }, { "id": "S29", "elements": [ 7, 8, 9, 10, 12, 14 ], "cost": 522 }, { "id": "S30", "elements": [ 11, 13, 14, 15, 16, 18 ], "cost": 138 }, { "id": "S31", "elements": [ 1, 3, 4, 5 ], "cost": 16 }, { "id": "S32", "elements": [ 3, 4, 5, 6, 7 ], "cost": 430 }, { "id": "S33", "elements": [ 0, 2, 3, 4 ], "cost": 284 }, { "id": "S34", "elements": [ 14, 15, 16, 17, 18, 19, 20 ], "cost": 98 }, { "id": "S35", "elements": [ 18, 19, 20, 21, 22 ], "cost": 115 }, { "id": "S36", "elements": [ 7, 10, 11, 13, 14 ], "cost": 255 }, { "id": "S37", "elements": [ 1, 3, 4, 6 ], "cost": 308 }, { "id": "S38", "elements": [ 13, 15, 16, 17, 22 ], "cost": 30 }, { "id": "S39", "elements": [ 6, 11, 12 ], "cost": 168 }, { "id": "S40", "elements": [ 8, 10, 12, 13, 14, 16, 17 ], "cost": 315 }, { "id": "S41", "elements": [ 6, 9, 15 ], "cost": 276 }, { "id": "S42", "elements": [ 9, 12, 13, 14, 15, 16 ], "cost": 336 }, { "id": "S43", "elements": [ 15, 17, 19, 22 ], "cost": 32 }, { "id": "S44", "elements": [ 9, 10, 12, 13, 14, 15 ], "cost": 150 }, { "id": "S45", "elements": [ 10, 11, 19 ], "cost": 207 }, { "id": "S46", "elements": [ 18, 20, 21 ], "cost": 42 }, { "id": "S47", "elements": [ 8, 9 ], "cost": 188 }, { "id": "S48", "elements": [ 0, 1, 2, 3, 4, 7 ], "cost": 162 }, { "id": "S49", "elements": [ 12, 14, 15, 16, 17, 19 ], "cost": 378 }, { "id": "S50", "elements": [ 16, 18, 19, 20, 21, 22 ], "cost": 486 }, { "id": "S51", "elements": [ 19, 20, 21, 22 ], "cost": 280 }, { "id": "S52", "elements": [ 18, 20 ], "cost": 182 }, { "id": "S53", "elements": [ 6, 8, 9, 10 ], "cost": 360 }, { "id": "S54", "elements": [ 2, 4, 6, 7, 8 ], "cost": 95 }, { "id": "S55", "elements": [ 12, 14, 15, 17, 18 ], "cost": 395 }, { "id": "S56", "elements": [ 4, 5, 6 ], "cost": 222 }, { "id": "S57", "elements": [ 19, 22 ], "cost": 120 }, { "id": "S58", "elements": [ 14, 16, 17, 18, 20 ], "cost": 440 }, { "id": "S59", "elements": [ 2, 3, 4, 5, 6, 8 ], "cost": 516 }, { "id": "S60", "elements": [ 8, 10, 11, 12, 13, 14 ], "cost": 438 }, { "id": "S61", "elements": [ 13, 14, 17, 18, 19 ], "cost": 470 }, { "id": "S62", "elements": [ 15, 16 ], "cost": 122 }, { "id": "S63", "elements": [ 1, 2, 4, 6 ], "cost": 348 }, { "id": "S64", "elements": [ 4, 6, 7, 11 ], "cost": 316 }, { "id": "S65", "elements": [ 17, 18, 19, 20 ], "cost": 384 }, { "id": "S66", "elements": [ 7, 9, 11, 12 ], "cost": 32 }, { "id": "S67", "elements": [ 1, 2, 4, 5, 9 ], "cost": 300 }, { "id": "S68", "elements": [ 9, 10, 11, 12, 13, 14, 15 ], "cost": 651 }, { "id": "S69", "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 54 }, { "id": "S70", "elements": [ 6, 9, 10, 13 ], "cost": 156 }, { "id": "S71", "elements": [ 8, 12, 13, 17, 18 ], "cost": 480 }, { "id": "S72", "elements": [ 8, 10, 11 ], "cost": 216 }, { "id": "S73", "elements": [ 8, 10, 13, 14, 15 ], "cost": 65 }, { "id": "S74", "elements": [ 5, 7, 8, 9, 10, 12, 14 ], "cost": 679 }, { "id": "S75", "elements": [ 4, 6, 8, 9, 11 ], "cost": 320 }, { "id": "S76", "elements": [ 15, 16, 17, 18, 19, 20, 22 ], "cost": 490 }, { "id": "S77", "elements": [ 12, 13, 14, 15, 16 ], "cost": 205 }, { "id": "S78", "elements": [ 11, 12, 14, 15, 16 ], "cost": 370 }, { "id": "S79", "elements": [ 17, 19, 20, 21, 22 ], "cost": 145 }, { "id": "S80", "elements": [ 4, 6, 7, 8 ], "cost": 60 }, { "id": "S81", "elements": [ 2, 4, 6 ], "cost": 195 }, { "id": "S82", "elements": [ 0, 1, 2, 3 ], "cost": 320 }, { "id": "S83", "elements": [ 6, 11, 12, 15 ], "cost": 208 }, { "id": "S84", "elements": [ 7, 8, 10 ], "cost": 255 }, { "id": "S85", "elements": [ 0, 1, 2, 3, 4, 8 ], "cost": 444 }, { "id": "S86", "elements": [ 10, 11, 12, 13, 16, 17 ], "cost": 54 }, { "id": "S87", "elements": [ 16, 18, 19, 20, 21 ], "cost": 140 }, { "id": "S88", "elements": [ 3, 5, 6 ], "cost": 156 }, { "id": "S89", "elements": [ 6, 7 ], "cost": 2 }, { "id": "S90", "elements": [ 13, 14, 16, 17, 18, 19 ], "cost": 450 }, { "id": "S91", "elements": [ 8, 13, 14 ], "cost": 138 }, { "id": "S92", "elements": [ 15, 16, 19, 20 ], "cost": 296 }, { "id": "S93", "elements": [ 10, 11, 14, 15 ], "cost": 4 }, { "id": "S94", "elements": [ 16, 17, 19, 20, 22 ], "cost": 360 }, { "id": "S95", "elements": [ 6, 8, 11, 12, 14, 16 ], "cost": 174 }, { "id": "S96", "elements": [ 1, 3, 5 ], "cost": 63 }, { "id": "S97", "elements": [ 0, 1, 2, 3, 4, 5, 6 ], "cost": 434 }, { "id": "S98", "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 198 }, { "id": "S99", "elements": [ 8, 10, 11, 15, 16 ], "cost": 270 }, { "id": "S100", "elements": [ 3, 4, 5, 7, 8, 10 ], "cost": 60 }, { "id": "S101", "elements": [ 12, 13, 15, 16 ], "cost": 332 }, { "id": "S102", "elements": [ 9, 10, 13 ], "cost": 192 }, { "id": "S103", "elements": [ 1, 3, 5, 6, 7 ], "cost": 220 }, { "id": "S104", "elements": [ 13, 16, 17, 18, 20 ], "cost": 15 }, { "id": "S105", "elements": [ 2, 3 ], "cost": 14 }, { "id": "S106", "elements": [ 7, 10, 14, 15, 16, 17 ], "cost": 594 }, { "id": "S107", "elements": [ 9, 10 ], "cost": 12 }, { "id": "S108", "elements": [ 11, 13, 14 ], "cost": 138 }, { "id": "S109", "elements": [ 7, 13 ], "cost": 190 }, { "id": "S110", "elements": [ 12, 15, 17, 18 ], "cost": 176 }, { "id": "S111", "elements": [ 18, 19, 20, 22 ], "cost": 180 }, { "id": "S112", "elements": [ 0, 1, 2, 3, 7 ], "cost": 400 }, { "id": "S113", "elements": [ 8, 9, 10, 12, 13 ], "cost": 115 }, { "id": "S114", "elements": [ 4, 5, 6, 7 ], "cost": 200 }, { "id": "S115", "elements": [ 16, 17, 20, 21, 22 ], "cost": 320 }, { "id": "S116", "elements": [ 18, 19, 20 ], "cost": 276 }, { "id": "S117", "elements": [ 13, 15, 18, 19 ], "cost": 352 }, { "id": "S118", "elements": [ 16, 19, 21, 22 ], "cost": 200 }, { "id": "S119", "elements": [ 11, 15, 16 ], "cost": 231 }, { "id": "S120", "elements": [ 0 ], "cost": 10000 }, { "id": "S121", "elements": [ 1 ], "cost": 10000 }, { "id": "S122", "elements": [ 2 ], "cost": 10000 }, { "id": "S123", "elements": [ 3 ], "cost": 10000 }, { "id": "S124", "elements": [ 4 ], "cost": 10000 }, { "id": "S125", "elements": [ 5 ], "cost": 10000 }, { "id": "S126", "elements": [ 6 ], "cost": 10000 }, { "id": "S127", "elements": [ 7 ], "cost": 10000 }, { "id": "S128", "elements": [ 8 ], "cost": 10000 }, { "id": "S129", "elements": [ 9 ], "cost": 10000 }, { "id": "S130", "elements": [ 10 ], "cost": 10000 }, { "id": "S131", "elements": [ 11 ], "cost": 10000 }, { "id": "S132", "elements": [ 12 ], "cost": 10000 }, { "id": "S133", "elements": [ 13 ], "cost": 10000 }, { "id": "S134", "elements": [ 14 ], "cost": 10000 }, { "id": "S135", "elements": [ 15 ], "cost": 10000 }, { "id": "S136", "elements": [ 16 ], "cost": 10000 }, { "id": "S137", "elements": [ 17 ], "cost": 10000 }, { "id": "S138", "elements": [ 18 ], "cost": 10000 }, { "id": "S139", "elements": [ 19 ], "cost": 10000 }, { "id": "S140", "elements": [ 20 ], "cost": 10000 }, { "id": "S141", "elements": [ 21 ], "cost": 10000 }, { "id": "S142", "elements": [ 22 ], "cost": 10000 } ] }, "solution_variant": [ "S25", "S66", "S73", "S97" ], "context_index": 39, "input_format": "markdown_table", "input_index_base": 0 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "I manage a mid‑size building and need to pick which cleaning packages to hire so every room gets cleaned exactly once. The idea is to choose a set of packages that together cover all rooms without any room being in two different packages or being skipped, and then add up the labor fees of the chosen packages — the plan with the lowest total fee is the one to go with. The specific rooms, package options, and fees are shown below.\n\n- **total_rooms**: 23\n- **num_packages**: 118\n\n| package_id | labor_fee | package_rooms |\n|---|---|---|\n| S1 | 96 | K L M N O P |\n| S2 | 24 | K M N O |\n| S3 | 36 | I L N |\n| S4 | 30 | C D E F H I |\n| S5 | 160 | K M |\n| S6 | 32 | J M P Q |\n| S7 | 104 | A C D G |\n| S8 | 480 | D E F G H L |\n| S9 | 276 | S T W |\n| S10 | 105 | Q R V |\n| S11 | 483 | C E F H I J K |\n| S12 | 434 | N O P Q R S T |\n| S13 | 275 | N P Q U V |\n| S14 | 98 | A C D E F G H |\n| S15 | 225 | A B C D H |\n| S16 | 268 | J K N O |\n| S17 | 574 | A B C D E F G |\n| S18 | 388 | H L N O |\n| S19 | 141 | A B G |\n| S20 | 102 | N O U |\n| S21 | 280 | O P Q S |\n| S22 | 85 | O P Q R S |\n| S23 | 84 | L O S |\n| S24 | 78 | J K M N O P |\n| S25 | 170 | A B C E F |\n| S26 | 108 | P T V |\n| S27 | 189 | I J K L M N P |\n| S28 | 48 | Q U V |\n| S29 | 110 | J M N O P |\n| S30 | 195 | Q S T U V |\n| S31 | 210 | J K L |\n| S32 | 396 | N Q R S T V |\n| S33 | 304 | S T V W |\n| S34 | 87 | R T W |\n| S35 | 105 | L N P S T |\n| S36 | 171 | G H K |\n| S37 | 80 | L M |\n| S38 | 6 | O P Q R S T |\n| S39 | 208 | G H J O |\n| S40 | 184 | T W |\n| S41 | 264 | O T V |\n| S42 | 345 | C D E G H |\n| S43 | 102 | O S T |\n| S44 | 336 | B E F H I J |\n| S45 | 175 | P T U V W |\n| S46 | 335 | E F G I O |\n| S47 | 336 | F I J M N Q |\n| S48 | 124 | A C E I |\n| S49 | 372 | Q S U V |\n| S50 | 665 | J K M N O Q R |\n| S51 | 182 | N P Q R S T U |\n| S52 | 385 | C E F G H I J |\n| S53 | 500 | L N P Q R |\n| S54 | 288 | M O P Q |\n| S55 | 46 | S T |\n| S56 | 252 | J N P R |\n| S57 | 408 | A B C D E F |\n| S58 | 525 | F H I J K L M |\n| S59 | 500 | A B E F G |\n| S60 | 444 | D I J K L N |\n| S61 | 64 | D F G J |\n| S62 | 87 | E F H |\n| S63 | 175 | A B C D E |\n| S64 | 480 | Q R T U V W |\n| S65 | 120 | M P R |\n| S66 | 265 | A C D E F |\n| S67 | 132 | R S |\n| S68 | 340 | N Q S T V |\n| S69 | 180 | P Q R U V |\n| S70 | 256 | L N O P |\n| S71 | 420 | J K L M N |\n| S72 | 486 | J L M N O R |\n| S73 | 198 | C F G H I K |\n| S74 | 154 | J L |\n| S75 | 150 | O Q R S T |\n| S76 | 25 | D G H I L |\n| S77 | 450 | M O P Q R T |\n| S78 | 285 | Q S T |\n| S79 | 180 | Q S T V W |\n| S80 | 273 | A B C |\n| S81 | 57 | A B D |\n| S82 | 104 | O P Q R |\n| S83 | 336 | K M N P Q R |\n| S84 | 366 | E F H I J K |\n| S85 | 148 | K L M N |\n| S86 | 90 | A B C D E G |\n| S87 | 305 | R S T V W |\n| S88 | 170 | N O P Q S |\n| S89 | 183 | I K L |\n| S90 | 415 | B F H I J |\n| S91 | 215 | J K N O P |\n| S92 | 352 | L M Q R |\n| S93 | 441 | B D E G H I J |\n| S94 | 56 | L M P Q |\n| S95 | 390 | A E G H J K |\n| S96 | 10000 | A |\n| S97 | 10000 | B |\n| S98 | 10000 | C |\n| S99 | 10000 | D |\n| S100 | 10000 | E |\n| S101 | 10000 | F |\n| S102 | 10000 | G |\n| S103 | 10000 | H |\n| S104 | 10000 | I |\n| S105 | 10000 | J |\n| S106 | 10000 | K |\n| S107 | 10000 | L |\n| S108 | 10000 | M |\n| S109 | 10000 | N |\n| S110 | 10000 | O |\n| S111 | 10000 | P |\n| S112 | 10000 | Q |\n| S113 | 10000 | R |\n| S114 | 10000 | S |\n| S115 | 10000 | T |\n| S116 | 10000 | U |\n| S117 | 10000 | V |\n| S118 | 10000 | W |\n\nIf you want to return the chosen cleaning packages, just use this simple JSON layout when you reply:\n\n{\n \"solution\": [\"package_id\", ...]\n}\n\n\"solution\" should hold a list of the package identifiers you picked — each entry is the exact id of a cleaning package to hire. This JSON is just a sketch of the shape I expect, not the final answer; when you give the real plan, replace the placeholder package_id items with the actual ids from the instance. Please don’t rename or invent ids.\n\nFor example: Valid identifiers look like plain numbers such as \"1\" or \"23\", single capital letters like \"A\" or \"B\", or a capital letter followed by digits like \"A1\" or \"X7\".", "instance": { "problem_type": "SPP", "num_elements": 23, "num_sets": 118, "density": 0.17022844509948415, "sets": [ { "id": 1, "elements": [ 11, 12, 13, 14, 15, 16 ], "cost": 96 }, { "id": 2, "elements": [ 11, 13, 14, 15 ], "cost": 24 }, { "id": 3, "elements": [ 9, 12, 14 ], "cost": 36 }, { "id": 4, "elements": [ 3, 4, 5, 6, 8, 9 ], "cost": 30 }, { "id": 5, "elements": [ 11, 13 ], "cost": 160 }, { "id": 6, "elements": [ 10, 13, 16, 17 ], "cost": 32 }, { "id": 7, "elements": [ 1, 3, 4, 7 ], "cost": 104 }, { "id": 8, "elements": [ 4, 5, 6, 7, 8, 12 ], "cost": 480 }, { "id": 9, "elements": [ 19, 20, 23 ], "cost": 276 }, { "id": 10, "elements": [ 17, 18, 22 ], "cost": 105 }, { "id": 11, "elements": [ 3, 5, 6, 8, 9, 10, 11 ], "cost": 483 }, { "id": 12, "elements": [ 14, 15, 16, 17, 18, 19, 20 ], "cost": 434 }, { "id": 13, "elements": [ 14, 16, 17, 21, 22 ], "cost": 275 }, { "id": 14, "elements": [ 1, 3, 4, 5, 6, 7, 8 ], "cost": 98 }, { "id": 15, "elements": [ 1, 2, 3, 4, 8 ], "cost": 225 }, { "id": 16, "elements": [ 10, 11, 14, 15 ], "cost": 268 }, { "id": 17, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 574 }, { "id": 18, "elements": [ 8, 12, 14, 15 ], "cost": 388 }, { "id": 19, "elements": [ 1, 2, 7 ], "cost": 141 }, { "id": 20, "elements": [ 14, 15, 21 ], "cost": 102 }, { "id": 21, "elements": [ 15, 16, 17, 19 ], "cost": 280 }, { "id": 22, "elements": [ 15, 16, 17, 18, 19 ], "cost": 85 }, { "id": 23, "elements": [ 12, 15, 19 ], "cost": 84 }, { "id": 24, "elements": [ 10, 11, 13, 14, 15, 16 ], "cost": 78 }, { "id": 25, "elements": [ 1, 2, 3, 5, 6 ], "cost": 170 }, { "id": 26, "elements": [ 16, 20, 22 ], "cost": 108 }, { "id": 27, "elements": [ 9, 10, 11, 12, 13, 14, 16 ], "cost": 189 }, { "id": 28, "elements": [ 17, 21, 22 ], "cost": 48 }, { "id": 29, "elements": [ 10, 13, 14, 15, 16 ], "cost": 110 }, { "id": 30, "elements": [ 17, 19, 20, 21, 22 ], "cost": 195 }, { "id": 31, "elements": [ 10, 11, 12 ], "cost": 210 }, { "id": 32, "elements": [ 14, 17, 18, 19, 20, 22 ], "cost": 396 }, { "id": 33, "elements": [ 19, 20, 22, 23 ], "cost": 304 }, { "id": 34, "elements": [ 18, 20, 23 ], "cost": 87 }, { "id": 35, "elements": [ 12, 14, 16, 19, 20 ], "cost": 105 }, { "id": 36, "elements": [ 7, 8, 11 ], "cost": 171 }, { "id": 37, "elements": [ 12, 13 ], "cost": 80 }, { "id": 38, "elements": [ 15, 16, 17, 18, 19, 20 ], "cost": 6 }, { "id": 39, "elements": [ 7, 8, 10, 15 ], "cost": 208 }, { "id": 40, "elements": [ 20, 23 ], "cost": 184 }, { "id": 41, "elements": [ 15, 20, 22 ], "cost": 264 }, { "id": 42, "elements": [ 3, 4, 5, 7, 8 ], "cost": 345 }, { "id": 43, "elements": [ 15, 19, 20 ], "cost": 102 }, { "id": 44, "elements": [ 2, 5, 6, 8, 9, 10 ], "cost": 336 }, { "id": 45, "elements": [ 16, 20, 21, 22, 23 ], "cost": 175 }, { "id": 46, "elements": [ 5, 6, 7, 9, 15 ], "cost": 335 }, { "id": 47, "elements": [ 6, 9, 10, 13, 14, 17 ], "cost": 336 }, { "id": 48, "elements": [ 1, 3, 5, 9 ], "cost": 124 }, { "id": 49, "elements": [ 17, 19, 21, 22 ], "cost": 372 }, { "id": 50, "elements": [ 10, 11, 13, 14, 15, 17, 18 ], "cost": 665 }, { "id": 51, "elements": [ 14, 16, 17, 18, 19, 20, 21 ], "cost": 182 }, { "id": 52, "elements": [ 3, 5, 6, 7, 8, 9, 10 ], "cost": 385 }, { "id": 53, "elements": [ 12, 14, 16, 17, 18 ], "cost": 500 }, { "id": 54, "elements": [ 13, 15, 16, 17 ], "cost": 288 }, { "id": 55, "elements": [ 19, 20 ], "cost": 46 }, { "id": 56, "elements": [ 10, 14, 16, 18 ], "cost": 252 }, { "id": 57, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 408 }, { "id": 58, "elements": [ 6, 8, 9, 10, 11, 12, 13 ], "cost": 525 }, { "id": 59, "elements": [ 1, 2, 5, 6, 7 ], "cost": 500 }, { "id": 60, "elements": [ 4, 9, 10, 11, 12, 14 ], "cost": 444 }, { "id": 61, "elements": [ 4, 6, 7, 10 ], "cost": 64 }, { "id": 62, "elements": [ 5, 6, 8 ], "cost": 87 }, { "id": 63, "elements": [ 1, 2, 3, 4, 5 ], "cost": 175 }, { "id": 64, "elements": [ 17, 18, 20, 21, 22, 23 ], "cost": 480 }, { "id": 65, "elements": [ 13, 16, 18 ], "cost": 120 }, { "id": 66, "elements": [ 1, 3, 4, 5, 6 ], "cost": 265 }, { "id": 67, "elements": [ 18, 19 ], "cost": 132 }, { "id": 68, "elements": [ 14, 17, 19, 20, 22 ], "cost": 340 }, { "id": 69, "elements": [ 16, 17, 18, 21, 22 ], "cost": 180 }, { "id": 70, "elements": [ 12, 14, 15, 16 ], "cost": 256 }, { "id": 71, "elements": [ 10, 11, 12, 13, 14 ], "cost": 420 }, { "id": 72, "elements": [ 10, 12, 13, 14, 15, 18 ], "cost": 486 }, { "id": 73, "elements": [ 3, 6, 7, 8, 9, 11 ], "cost": 198 }, { "id": 74, "elements": [ 10, 12 ], "cost": 154 }, { "id": 75, "elements": [ 15, 17, 18, 19, 20 ], "cost": 150 }, { "id": 76, "elements": [ 4, 7, 8, 9, 12 ], "cost": 25 }, { "id": 77, "elements": [ 13, 15, 16, 17, 18, 20 ], "cost": 450 }, { "id": 78, "elements": [ 17, 19, 20 ], "cost": 285 }, { "id": 79, "elements": [ 17, 19, 20, 22, 23 ], "cost": 180 }, { "id": 80, "elements": [ 1, 2, 3 ], "cost": 273 }, { "id": 81, "elements": [ 1, 2, 4 ], "cost": 57 }, { "id": 82, "elements": [ 15, 16, 17, 18 ], "cost": 104 }, { "id": 83, "elements": [ 11, 13, 14, 16, 17, 18 ], "cost": 336 }, { "id": 84, "elements": [ 5, 6, 8, 9, 10, 11 ], "cost": 366 }, { "id": 85, "elements": [ 11, 12, 13, 14 ], "cost": 148 }, { "id": 86, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 90 }, { "id": 87, "elements": [ 18, 19, 20, 22, 23 ], "cost": 305 }, { "id": 88, "elements": [ 14, 15, 16, 17, 19 ], "cost": 170 }, { "id": 89, "elements": [ 9, 11, 12 ], "cost": 183 }, { "id": 90, "elements": [ 2, 6, 8, 9, 10 ], "cost": 415 }, { "id": 91, "elements": [ 10, 11, 14, 15, 16 ], "cost": 215 }, { "id": 92, "elements": [ 12, 13, 17, 18 ], "cost": 352 }, { "id": 93, "elements": [ 2, 4, 5, 7, 8, 9, 10 ], "cost": 441 }, { "id": 94, "elements": [ 12, 13, 16, 17 ], "cost": 56 }, { "id": 95, "elements": [ 1, 5, 7, 8, 10, 11 ], "cost": 390 }, { "id": 96, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 97, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 98, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 118, "elements": [ 23 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0039_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0039_bag_stats.png" }, "solution": [ 24, 25, 28, 40, 67, 76 ], "obj": 637.0, "instance_variant": { "num_elements": 23, "num_sets": 118, "sets": [ { "id": "S1", "elements": [ "K", "L", "M", "N", "O", "P" ], "cost": 96 }, { "id": "S2", "elements": [ "K", "M", "N", "O" ], "cost": 24 }, { "id": "S3", "elements": [ "I", "L", "N" ], "cost": 36 }, { "id": "S4", "elements": [ "C", "D", "E", "F", "H", "I" ], "cost": 30 }, { "id": "S5", "elements": [ "K", "M" ], "cost": 160 }, { "id": "S6", "elements": [ "J", "M", "P", "Q" ], "cost": 32 }, { "id": "S7", "elements": [ "A", "C", "D", "G" ], "cost": 104 }, { "id": "S8", "elements": [ "D", "E", "F", "G", "H", "L" ], "cost": 480 }, { "id": "S9", "elements": [ "S", "T", "W" ], "cost": 276 }, { "id": "S10", "elements": [ "Q", "R", "V" ], "cost": 105 }, { "id": "S11", "elements": [ "C", "E", "F", "H", "I", "J", "K" ], "cost": 483 }, { "id": "S12", "elements": [ "N", "O", "P", "Q", "R", "S", "T" ], "cost": 434 }, { "id": "S13", "elements": [ "N", "P", "Q", "U", "V" ], "cost": 275 }, { "id": "S14", "elements": [ "A", "C", "D", "E", "F", "G", "H" ], "cost": 98 }, { "id": "S15", "elements": [ "A", "B", "C", "D", "H" ], "cost": 225 }, { "id": "S16", "elements": [ "J", "K", "N", "O" ], "cost": 268 }, { "id": "S17", "elements": [ "A", "B", "C", "D", "E", "F", "G" ], "cost": 574 }, { "id": "S18", "elements": [ "H", "L", "N", "O" ], "cost": 388 }, { "id": "S19", "elements": [ "A", "B", "G" ], "cost": 141 }, { "id": "S20", "elements": [ "N", "O", "U" ], "cost": 102 }, { "id": "S21", "elements": [ "O", "P", "Q", "S" ], "cost": 280 }, { "id": "S22", "elements": [ "O", "P", "Q", "R", "S" ], "cost": 85 }, { "id": "S23", "elements": [ "L", "O", "S" ], "cost": 84 }, { "id": "S24", "elements": [ "J", "K", "M", "N", "O", "P" ], "cost": 78 }, { "id": "S25", "elements": [ "A", "B", "C", "E", "F" ], "cost": 170 }, { "id": "S26", "elements": [ "P", "T", "V" ], "cost": 108 }, { "id": "S27", "elements": [ "I", "J", "K", "L", "M", "N", "P" ], "cost": 189 }, { "id": "S28", "elements": [ "Q", "U", "V" ], "cost": 48 }, { "id": "S29", "elements": [ "J", "M", "N", "O", "P" ], "cost": 110 }, { "id": "S30", "elements": [ "Q", "S", "T", "U", "V" ], "cost": 195 }, { "id": "S31", "elements": [ "J", "K", "L" ], "cost": 210 }, { "id": "S32", "elements": [ "N", "Q", "R", "S", "T", "V" ], "cost": 396 }, { "id": "S33", "elements": [ "S", "T", "V", "W" ], "cost": 304 }, { "id": "S34", "elements": [ "R", "T", "W" ], "cost": 87 }, { "id": "S35", "elements": [ "L", "N", "P", "S", "T" ], "cost": 105 }, { "id": "S36", "elements": [ "G", "H", "K" ], "cost": 171 }, { "id": "S37", "elements": [ "L", "M" ], "cost": 80 }, { "id": "S38", "elements": [ "O", "P", "Q", "R", "S", "T" ], "cost": 6 }, { "id": "S39", "elements": [ "G", "H", "J", "O" ], "cost": 208 }, { "id": "S40", "elements": [ "T", "W" ], "cost": 184 }, { "id": "S41", "elements": [ "O", "T", "V" ], "cost": 264 }, { "id": "S42", "elements": [ "C", "D", "E", "G", "H" ], "cost": 345 }, { "id": "S43", "elements": [ "O", "S", "T" ], "cost": 102 }, { "id": "S44", "elements": [ "B", "E", "F", "H", "I", "J" ], "cost": 336 }, { "id": "S45", "elements": [ "P", "T", "U", "V", "W" ], "cost": 175 }, { "id": "S46", "elements": [ "E", "F", "G", "I", "O" ], "cost": 335 }, { "id": "S47", "elements": [ "F", "I", "J", "M", "N", "Q" ], "cost": 336 }, { "id": "S48", "elements": [ "A", "C", "E", "I" ], "cost": 124 }, { "id": "S49", "elements": [ "Q", "S", "U", "V" ], "cost": 372 }, { "id": "S50", "elements": [ "J", "K", "M", "N", "O", "Q", "R" ], "cost": 665 }, { "id": "S51", "elements": [ "N", "P", "Q", "R", "S", "T", "U" ], "cost": 182 }, { "id": "S52", "elements": [ "C", "E", "F", "G", "H", "I", "J" ], "cost": 385 }, { "id": "S53", "elements": [ "L", "N", "P", "Q", "R" ], "cost": 500 }, { "id": "S54", "elements": [ "M", "O", "P", "Q" ], "cost": 288 }, { "id": "S55", "elements": [ "S", "T" ], "cost": 46 }, { "id": "S56", "elements": [ "J", "N", "P", "R" ], "cost": 252 }, { "id": "S57", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 408 }, { "id": "S58", "elements": [ "F", "H", "I", "J", "K", "L", "M" ], "cost": 525 }, { "id": "S59", "elements": [ "A", "B", "E", "F", "G" ], "cost": 500 }, { "id": "S60", "elements": [ "D", "I", "J", "K", "L", "N" ], "cost": 444 }, { "id": "S61", "elements": [ "D", "F", "G", "J" ], "cost": 64 }, { "id": "S62", "elements": [ "E", "F", "H" ], "cost": 87 }, { "id": "S63", "elements": [ "A", "B", "C", "D", "E" ], "cost": 175 }, { "id": "S64", "elements": [ "Q", "R", "T", "U", "V", "W" ], "cost": 480 }, { "id": "S65", "elements": [ "M", "P", "R" ], "cost": 120 }, { "id": "S66", "elements": [ "A", "C", "D", "E", "F" ], "cost": 265 }, { "id": "S67", "elements": [ "R", "S" ], "cost": 132 }, { "id": "S68", "elements": [ "N", "Q", "S", "T", "V" ], "cost": 340 }, { "id": "S69", "elements": [ "P", "Q", "R", "U", "V" ], "cost": 180 }, { "id": "S70", "elements": [ "L", "N", "O", "P" ], "cost": 256 }, { "id": "S71", "elements": [ "J", "K", "L", "M", "N" ], "cost": 420 }, { "id": "S72", "elements": [ "J", "L", "M", "N", "O", "R" ], "cost": 486 }, { "id": "S73", "elements": [ "C", "F", "G", "H", "I", "K" ], "cost": 198 }, { "id": "S74", "elements": [ "J", "L" ], "cost": 154 }, { "id": "S75", "elements": [ "O", "Q", "R", "S", "T" ], "cost": 150 }, { "id": "S76", "elements": [ "D", "G", "H", "I", "L" ], "cost": 25 }, { "id": "S77", "elements": [ "M", "O", "P", "Q", "R", "T" ], "cost": 450 }, { "id": "S78", "elements": [ "Q", "S", "T" ], "cost": 285 }, { "id": "S79", "elements": [ "Q", "S", "T", "V", "W" ], "cost": 180 }, { "id": "S80", "elements": [ "A", "B", "C" ], "cost": 273 }, { "id": "S81", "elements": [ "A", "B", "D" ], "cost": 57 }, { "id": "S82", "elements": [ "O", "P", "Q", "R" ], "cost": 104 }, { "id": "S83", "elements": [ "K", "M", "N", "P", "Q", "R" ], "cost": 336 }, { "id": "S84", "elements": [ "E", "F", "H", "I", "J", "K" ], "cost": 366 }, { "id": "S85", "elements": [ "K", "L", "M", "N" ], "cost": 148 }, { "id": "S86", "elements": [ "A", "B", "C", "D", "E", "G" ], "cost": 90 }, { "id": "S87", "elements": [ "R", "S", "T", "V", "W" ], "cost": 305 }, { "id": "S88", "elements": [ "N", "O", "P", "Q", "S" ], "cost": 170 }, { "id": "S89", "elements": [ "I", "K", "L" ], "cost": 183 }, { "id": "S90", "elements": [ "B", "F", "H", "I", "J" ], "cost": 415 }, { "id": "S91", "elements": [ "J", "K", "N", "O", "P" ], "cost": 215 }, { "id": "S92", "elements": [ "L", "M", "Q", "R" ], "cost": 352 }, { "id": "S93", "elements": [ "B", "D", "E", "G", "H", "I", "J" ], "cost": 441 }, { "id": "S94", "elements": [ "L", "M", "P", "Q" ], "cost": 56 }, { "id": "S95", "elements": [ "A", "E", "G", "H", "J", "K" ], "cost": 390 }, { "id": "S96", "elements": [ "A" ], "cost": 10000 }, { "id": "S97", "elements": [ "B" ], "cost": 10000 }, { "id": "S98", "elements": [ "C" ], "cost": 10000 }, { "id": "S99", "elements": [ "D" ], "cost": 10000 }, { "id": "S100", "elements": [ "E" ], "cost": 10000 }, { "id": "S101", "elements": [ "F" ], "cost": 10000 }, { "id": "S102", "elements": [ "G" ], "cost": 10000 }, { "id": "S103", "elements": [ "H" ], "cost": 10000 }, { "id": "S104", "elements": [ "I" ], "cost": 10000 }, { "id": "S105", "elements": [ "J" ], "cost": 10000 }, { "id": "S106", "elements": [ "K" ], "cost": 10000 }, { "id": "S107", "elements": [ "L" ], "cost": 10000 }, { "id": "S108", "elements": [ "M" ], "cost": 10000 }, { "id": "S109", "elements": [ "N" ], "cost": 10000 }, { "id": "S110", "elements": [ "O" ], "cost": 10000 }, { "id": "S111", "elements": [ "P" ], "cost": 10000 }, { "id": "S112", "elements": [ "Q" ], "cost": 10000 }, { "id": "S113", "elements": [ "R" ], "cost": 10000 }, { "id": "S114", "elements": [ "S" ], "cost": 10000 }, { "id": "S115", "elements": [ "T" ], "cost": 10000 }, { "id": "S116", "elements": [ "U" ], "cost": 10000 }, { "id": "S117", "elements": [ "V" ], "cost": 10000 }, { "id": "S118", "elements": [ "W" ], "cost": 10000 } ] }, "solution_variant": [ "S24", "S25", "S28", "S40", "S67", "S76" ], "context_index": 40, "input_format": "markdown_table", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "We put together a shortlist of vendor modules and needed to decide which ones to bring into the product. The plan was to make sure each product feature ends up covered by a single chosen module, with no overlap between selected vendors and nothing left unhandled. To compare options, simply total the license costs of the modules you pick; the smaller that sum, the more attractive the combination. The concrete options and feature list are shown below.\n\nWe are considering 163 candidate modules to cover 25 distinct product features.\nModule S1 covers features 1 3 4 5 6 and has a license cost of 20.\nModule S2 covers features 3 4 6 7 8 9 10 and has a license cost of 658.\nModule S3 covers features 4 6 8 9 and has a license cost of 240.\nModule S4 covers features 4 8 11 12 and has a license cost of 136.\nModule S5 covers features 9 11 12 13 16 19 and has a license cost of 390.\nModule S6 covers features 19 21 22 23 and has a license cost of 44.\nModule S7 covers features 19 20 21 22 24 and has a license cost of 175.\nModule S8 covers features 13 14 15 16 17 18 20 and has a license cost of 567.\nModule S9 covers features 19 22 23 and has a license cost of 132.\nModule S10 covers features 15 20 21 and has a license cost of 63.\nModule S11 covers features 16 18 21 24 and has a license cost of 208.\nModule S12 covers features 2 8 10 11 and has a license cost of 184.\nModule S13 covers features 7 11 12 13 15 16 17 and has a license cost of 252.\nModule S14 covers features 0 1 2 3 4 5 7 and has a license cost of 322.\nModule S15 covers features 12 14 16 22 and has a license cost of 44.\nModule S16 covers features 13 14 15 20 21 24 and has a license cost of 48.\nModule S17 covers features 0 2 5 8 and has a license cost of 180.\nModule S18 covers features 7 8 9 10 and has a license cost of 252.\nModule S19 covers features 4 5 6 7 8 9 and has a license cost of 252.\nModule S20 covers features 0 1 2 3 and has a license cost of 348.\nModule S21 covers features 14 15 19 and has a license cost of 51.\nModule S22 covers features 12 14 16 17 20 21 and has a license cost of 96.\nModule S23 covers features 20 21 22 23 24 and has a license cost of 295.\nModule S24 covers features 15 19 21 23 and has a license cost of 284.\nModule S25 covers features 17 20 21 23 and has a license cost of 296.\nModule S26 covers features 3 7 10 and has a license cost of 129.\nModule S27 covers features 12 14 16 and has a license cost of 231.\nModule S28 covers features 19 20 21 22 23 24 and has a license cost of 552.\nModule S29 covers features 2 4 6 and has a license cost of 297.\nModule S30 covers features 15 18 19 20 21 23 24 and has a license cost of 196.\nModule S31 covers features 2 4 5 6 7 and has a license cost of 395.\nModule S32 covers features 11 14 15 and has a license cost of 141.\nModule S33 covers features 7 8 9 11 13 and has a license cost of 465.\nModule S34 covers features 16 19 21 22 and has a license cost of 224.\nModule S35 covers features 14 17 20 21 23 and has a license cost of 100.\nModule S36 covers features 16 18 20 21 22 23 24 and has a license cost of 329.\nModule S37 covers features 5 6 8 9 and has a license cost of 188.\nModule S38 covers features 2 4 5 6 9 10 11 and has a license cost of 595.\nModule S39 covers features 19 21 22 23 24 and has a license cost of 5.\nModule S40 covers features 18 21 23 24 and has a license cost of 152.\nModule S41 covers features 12 15 16 18 and has a license cost of 396.\nModule S42 covers features 17 19 21 and has a license cost of 153.\nModule S43 covers features 1 2 3 5 6 and has a license cost of 270.\nModule S44 covers features 20 22 23 and has a license cost of 210.\nModule S45 covers features 16 18 20 23 and has a license cost of 284.\nModule S46 covers features 0 1 2 and has a license cost of 18.\nModule S47 covers features 5 7 10 and has a license cost of 261.\nModule S48 covers features 14 18 19 20 22 23 and has a license cost of 516.\nModule S49 covers features 14 16 17 18 19 20 22 and has a license cost of 476.\nModule S50 covers features 12 13 14 16 and has a license cost of 208.\nModule S51 covers features 7 9 10 11 and has a license cost of 156.\nModule S52 covers features 1 2 3 4 5 and has a license cost of 100.\nModule S53 covers features 10 11 12 14 15 16 and has a license cost of 132.\nModule S54 covers features 14 17 18 19 21 22 and has a license cost of 534.\nModule S55 covers features 11 12 13 15 16 and has a license cost of 290.\nModule S56 covers features 7 8 9 10 11 13 and has a license cost of 258.\nModule S57 covers features 0 1 2 3 4 5 and has a license cost of 438.\nModule S58 covers features 3 7 8 9 10 11 13 and has a license cost of 595.\nModule S59 covers features 3 4 5 6 7 8 10 and has a license cost of 126.\nModule S60 covers features 13 14 17 18 and has a license cost of 120.\nModule S61 covers features 13 14 16 18 19 and has a license cost of 265.\nModule S62 covers features 7 8 10 13 15 and has a license cost of 170.\nModule S63 covers features 8 9 10 11 12 14 and has a license cost of 504.\nModule S64 covers features 6 7 9 10 11 12 13 and has a license cost of 665.\nModule S65 covers features 11 12 13 and has a license cost of 246.\nModule S66 covers features 16 17 22 23 and has a license cost of 20.\nModule S67 covers features 1 2 3 4 and has a license cost of 132.\nModule S68 covers features 0 1 4 and has a license cost of 168.\nModule S69 covers features 10 11 12 15 17 and has a license cost of 370.\nModule S70 covers features 11 15 20 and has a license cost of 132.\nModule S71 covers features 15 17 19 21 23 and has a license cost of 60.\nModule S72 covers features 5 6 7 8 9 10 and has a license cost of 318.\nModule S73 covers features 2 4 6 7 8 9 and has a license cost of 162.\nModule S74 covers features 11 12 13 14 15 17 and has a license cost of 504.\nModule S75 covers features 5 6 7 9 10 11 and has a license cost of 18.\nModule S76 covers features 8 10 12 13 15 16 and has a license cost of 390.\nModule S77 covers features 0 1 2 3 4 5 6 and has a license cost of 42.\nModule S78 covers features 0 1 2 3 5 and has a license cost of 380.\nModule S79 covers features 8 9 10 11 12 and has a license cost of 100.\nModule S80 covers features 4 5 7 and has a license cost of 36.\nModule S81 covers features 8 12 13 and has a license cost of 87.\nModule S82 covers features 0 1 5 7 and has a license cost of 392.\nModule S83 covers features 13 14 15 and has a license cost of 54.\nModule S84 covers features 14 16 18 19 20 21 22 and has a license cost of 322.\nModule S85 covers features 16 19 20 21 22 23 24 and has a license cost of 161.\nModule S86 covers features 13 16 19 and has a license cost of 231.\nModule S87 covers features 4 5 7 10 and has a license cost of 120.\nModule S88 covers features 2 5 6 7 8 10 and has a license cost of 66.\nModule S89 covers features 10 12 13 14 15 16 17 and has a license cost of 210.\nModule S90 covers features 6 7 9 and has a license cost of 204.\nModule S91 covers features 15 17 18 19 21 23 and has a license cost of 600.\nModule S92 covers features 10 13 14 and has a license cost of 162.\nModule S93 covers features 1 7 8 9 and has a license cost of 100.\nModule S94 covers features 0 1 3 4 5 and has a license cost of 170.\nModule S95 covers features 9 11 14 15 and has a license cost of 384.\nModule S96 covers features 0 1 4 5 and has a license cost of 88.\nModule S97 covers features 18 19 22 23 24 and has a license cost of 480.\nModule S98 covers features 12 13 14 15 17 and has a license cost of 390.\nModule S99 covers features 0 1 2 4 6 and has a license cost of 365.\nModule S100 covers features 21 22 23 24 and has a license cost of 304.\nModule S101 covers features 10 12 13 and has a license cost of 252.\nModule S102 covers features 16 17 21 and has a license cost of 192.\nModule S103 covers features 17 18 20 and has a license cost of 150.\nModule S104 covers features 20 21 23 24 and has a license cost of 80.\nModule S105 covers features 5 8 9 10 11 14 15 and has a license cost of 336.\nModule S106 covers features 7 8 11 and has a license cost of 249.\nModule S107 covers features 3 4 5 6 and has a license cost of 84.\nModule S108 covers features 0 2 3 4 5 6 9 and has a license cost of 203.\nModule S109 covers features 0 1 2 4 5 and has a license cost of 45.\nModule S110 covers features 19 20 21 22 and has a license cost of 256.\nModule S111 covers features 0 1 2 3 4 8 and has a license cost of 84.\nModule S112 covers features 11 12 13 14 15 and has a license cost of 315.\nModule S113 covers features 18 19 21 22 23 and has a license cost of 380.\nModule S114 covers features 0 2 3 4 6 7 8 and has a license cost of 658.\nModule S115 covers features 11 13 14 16 and has a license cost of 328.\nModule S116 covers features 9 10 11 12 13 and has a license cost of 330.\nModule S117 covers features 7 9 10 13 15 and has a license cost of 430.\nModule S118 covers features 8 9 10 11 12 13 and has a license cost of 582.\nModule S119 covers features 6 8 9 and has a license cost of 78.\nModule S120 covers features 11 13 15 and has a license cost of 60.\nModule S121 covers features 12 13 15 17 and has a license cost of 100.\nModule S122 covers features 3 5 8 9 and has a license cost of 248.\nModule S123 covers features 2 4 5 8 10 and has a license cost of 385.\nModule S124 covers features 19 20 21 22 23 and has a license cost of 110.\nModule S125 covers features 1 2 4 5 6 7 9 and has a license cost of 511.\nModule S126 covers features 17 19 20 21 22 23 and has a license cost of 372.\nModule S127 covers features 12 13 14 16 17 20 and has a license cost of 516.\nModule S128 covers features 7 8 10 13 and has a license cost of 40.\nModule S129 covers features 5 10 11 and has a license cost of 177.\nModule S130 covers features 14 17 21 and has a license cost of 147.\nModule S131 covers features 8 10 11 12 15 and has a license cost of 130.\nModule S132 covers features 13 14 15 16 18 and has a license cost of 345.\nModule S133 covers features 0 1 3 5 and has a license cost of 332.\nModule S134 covers features 2 4 5 8 9 10 11 and has a license cost of 427.\nModule S135 covers features 3 5 9 10 and has a license cost of 4.\nModule S136 covers features 16 18 19 and has a license cost of 237.\nModule S137 covers features 16 17 19 20 22 and has a license cost of 160.\nModule S138 covers features 8 9 11 12 and has a license cost of 52.\nModule S139 covers features 0 and has a license cost of 10000.\nModule S140 covers features 1 and has a license cost of 10000.\nModule S141 covers features 2 and has a license cost of 10000.\nModule S142 covers features 3 and has a license cost of 10000.\nModule S143 covers features 4 and has a license cost of 10000.\nModule S144 covers features 5 and has a license cost of 10000.\nModule S145 covers features 6 and has a license cost of 10000.\nModule S146 covers features 7 and has a license cost of 10000.\nModule S147 covers features 8 and has a license cost of 10000.\nModule S148 covers features 9 and has a license cost of 10000.\nModule S149 covers features 10 and has a license cost of 10000.\nModule S150 covers features 11 and has a license cost of 10000.\nModule S151 covers features 12 and has a license cost of 10000.\nModule S152 covers features 13 and has a license cost of 10000.\nModule S153 covers features 14 and has a license cost of 10000.\nModule S154 covers features 15 and has a license cost of 10000.\nModule S155 covers features 16 and has a license cost of 10000.\nModule S156 covers features 17 and has a license cost of 10000.\nModule S157 covers features 18 and has a license cost of 10000.\nModule S158 covers features 19 and has a license cost of 10000.\nModule S159 covers features 20 and has a license cost of 10000.\nModule S160 covers features 21 and has a license cost of 10000.\nModule S161 covers features 22 and has a license cost of 10000.\nModule S162 covers features 23 and has a license cost of 10000.\nModule S163 covers features 24 and has a license cost of 10000.\nWe must pick modules that together assign each of the 25 features exactly once, with no overlap, and minimize the total license cost across the chosen modules.\n\nWhen you send your pick, just return a tiny JSON object in this shape — super simple, like filling out a short form:\n\n{\n \"solution\": [\"module_id\", ...]\n}\n\n\"solution\" is the list of vendor modules you choose (one entry per feature, no overlaps). The placeholder module_id shows where each chosen module's identifier goes; the JSON above is just a sketch of the shape to follow, not the actual answer — replace the placeholder with the real IDs from the instance.\n\nQuick reminder: all identifiers must be used exactly as they appear in the instance input — no renaming and no new labels.\n\nValid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.", "instance": { "problem_type": "SPP", "num_elements": 25, "num_sets": 163, "density": 0.16736196319018404, "sets": [ { "id": 1, "elements": [ 2, 4, 5, 6, 7 ], "cost": 20 }, { "id": 2, "elements": [ 4, 5, 7, 8, 9, 10, 11 ], "cost": 658 }, { "id": 3, "elements": [ 5, 7, 9, 10 ], "cost": 240 }, { "id": 4, "elements": [ 5, 9, 12, 13 ], "cost": 136 }, { "id": 5, "elements": [ 10, 12, 13, 14, 17, 20 ], "cost": 390 }, { "id": 6, "elements": [ 20, 22, 23, 24 ], "cost": 44 }, { "id": 7, "elements": [ 20, 21, 22, 23, 25 ], "cost": 175 }, { "id": 8, "elements": [ 14, 15, 16, 17, 18, 19, 21 ], "cost": 567 }, { "id": 9, "elements": [ 20, 23, 24 ], "cost": 132 }, { "id": 10, "elements": [ 16, 21, 22 ], "cost": 63 }, { "id": 11, "elements": [ 17, 19, 22, 25 ], "cost": 208 }, { "id": 12, "elements": [ 3, 9, 11, 12 ], "cost": 184 }, { "id": 13, "elements": [ 8, 12, 13, 14, 16, 17, 18 ], "cost": 252 }, { "id": 14, "elements": [ 1, 2, 3, 4, 5, 6, 8 ], "cost": 322 }, { "id": 15, "elements": [ 13, 15, 17, 23 ], "cost": 44 }, { "id": 16, "elements": [ 14, 15, 16, 21, 22, 25 ], "cost": 48 }, { "id": 17, "elements": [ 1, 3, 6, 9 ], "cost": 180 }, { "id": 18, "elements": [ 8, 9, 10, 11 ], "cost": 252 }, { "id": 19, "elements": [ 5, 6, 7, 8, 9, 10 ], "cost": 252 }, { "id": 20, "elements": [ 1, 2, 3, 4 ], "cost": 348 }, { "id": 21, "elements": [ 15, 16, 20 ], "cost": 51 }, { "id": 22, "elements": [ 13, 15, 17, 18, 21, 22 ], "cost": 96 }, { "id": 23, "elements": [ 21, 22, 23, 24, 25 ], "cost": 295 }, { "id": 24, "elements": [ 16, 20, 22, 24 ], "cost": 284 }, { "id": 25, "elements": [ 18, 21, 22, 24 ], "cost": 296 }, { "id": 26, "elements": [ 4, 8, 11 ], "cost": 129 }, { "id": 27, "elements": [ 13, 15, 17 ], "cost": 231 }, { "id": 28, "elements": [ 20, 21, 22, 23, 24, 25 ], "cost": 552 }, { "id": 29, "elements": [ 3, 5, 7 ], "cost": 297 }, { "id": 30, "elements": [ 16, 19, 20, 21, 22, 24, 25 ], "cost": 196 }, { "id": 31, "elements": [ 3, 5, 6, 7, 8 ], "cost": 395 }, { "id": 32, "elements": [ 12, 15, 16 ], "cost": 141 }, { "id": 33, "elements": [ 8, 9, 10, 12, 14 ], "cost": 465 }, { "id": 34, "elements": [ 17, 20, 22, 23 ], "cost": 224 }, { "id": 35, "elements": [ 15, 18, 21, 22, 24 ], "cost": 100 }, { "id": 36, "elements": [ 17, 19, 21, 22, 23, 24, 25 ], "cost": 329 }, { "id": 37, "elements": [ 6, 7, 9, 10 ], "cost": 188 }, { "id": 38, "elements": [ 3, 5, 6, 7, 10, 11, 12 ], "cost": 595 }, { "id": 39, "elements": [ 20, 22, 23, 24, 25 ], "cost": 5 }, { "id": 40, "elements": [ 19, 22, 24, 25 ], "cost": 152 }, { "id": 41, "elements": [ 13, 16, 17, 19 ], "cost": 396 }, { "id": 42, "elements": [ 18, 20, 22 ], "cost": 153 }, { "id": 43, "elements": [ 2, 3, 4, 6, 7 ], "cost": 270 }, { "id": 44, "elements": [ 21, 23, 24 ], "cost": 210 }, { "id": 45, "elements": [ 17, 19, 21, 24 ], "cost": 284 }, { "id": 46, "elements": [ 1, 2, 3 ], "cost": 18 }, { "id": 47, "elements": [ 6, 8, 11 ], "cost": 261 }, { "id": 48, "elements": [ 15, 19, 20, 21, 23, 24 ], "cost": 516 }, { "id": 49, "elements": [ 15, 17, 18, 19, 20, 21, 23 ], "cost": 476 }, { "id": 50, "elements": [ 13, 14, 15, 17 ], "cost": 208 }, { "id": 51, "elements": [ 8, 10, 11, 12 ], "cost": 156 }, { "id": 52, "elements": [ 2, 3, 4, 5, 6 ], "cost": 100 }, { "id": 53, "elements": [ 11, 12, 13, 15, 16, 17 ], "cost": 132 }, { "id": 54, "elements": [ 15, 18, 19, 20, 22, 23 ], "cost": 534 }, { "id": 55, "elements": [ 12, 13, 14, 16, 17 ], "cost": 290 }, { "id": 56, "elements": [ 8, 9, 10, 11, 12, 14 ], "cost": 258 }, { "id": 57, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 438 }, { "id": 58, "elements": [ 4, 8, 9, 10, 11, 12, 14 ], "cost": 595 }, { "id": 59, "elements": [ 4, 5, 6, 7, 8, 9, 11 ], "cost": 126 }, { "id": 60, "elements": [ 14, 15, 18, 19 ], "cost": 120 }, { "id": 61, "elements": [ 14, 15, 17, 19, 20 ], "cost": 265 }, { "id": 62, "elements": [ 8, 9, 11, 14, 16 ], "cost": 170 }, { "id": 63, "elements": [ 9, 10, 11, 12, 13, 15 ], "cost": 504 }, { "id": 64, "elements": [ 7, 8, 10, 11, 12, 13, 14 ], "cost": 665 }, { "id": 65, "elements": [ 12, 13, 14 ], "cost": 246 }, { "id": 66, "elements": [ 17, 18, 23, 24 ], "cost": 20 }, { "id": 67, "elements": [ 2, 3, 4, 5 ], "cost": 132 }, { "id": 68, "elements": [ 1, 2, 5 ], "cost": 168 }, { "id": 69, "elements": [ 11, 12, 13, 16, 18 ], "cost": 370 }, { "id": 70, "elements": [ 12, 16, 21 ], "cost": 132 }, { "id": 71, "elements": [ 16, 18, 20, 22, 24 ], "cost": 60 }, { "id": 72, "elements": [ 6, 7, 8, 9, 10, 11 ], "cost": 318 }, { "id": 73, "elements": [ 3, 5, 7, 8, 9, 10 ], "cost": 162 }, { "id": 74, "elements": [ 12, 13, 14, 15, 16, 18 ], "cost": 504 }, { "id": 75, "elements": [ 6, 7, 8, 10, 11, 12 ], "cost": 18 }, { "id": 76, "elements": [ 9, 11, 13, 14, 16, 17 ], "cost": 390 }, { "id": 77, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 42 }, { "id": 78, "elements": [ 1, 2, 3, 4, 6 ], "cost": 380 }, { "id": 79, "elements": [ 9, 10, 11, 12, 13 ], "cost": 100 }, { "id": 80, "elements": [ 5, 6, 8 ], "cost": 36 }, { "id": 81, "elements": [ 9, 13, 14 ], "cost": 87 }, { "id": 82, "elements": [ 1, 2, 6, 8 ], "cost": 392 }, { "id": 83, "elements": [ 14, 15, 16 ], "cost": 54 }, { "id": 84, "elements": [ 15, 17, 19, 20, 21, 22, 23 ], "cost": 322 }, { "id": 85, "elements": [ 17, 20, 21, 22, 23, 24, 25 ], "cost": 161 }, { "id": 86, "elements": [ 14, 17, 20 ], "cost": 231 }, { "id": 87, "elements": [ 5, 6, 8, 11 ], "cost": 120 }, { "id": 88, "elements": [ 3, 6, 7, 8, 9, 11 ], "cost": 66 }, { "id": 89, "elements": [ 11, 13, 14, 15, 16, 17, 18 ], "cost": 210 }, { "id": 90, "elements": [ 7, 8, 10 ], "cost": 204 }, { "id": 91, "elements": [ 16, 18, 19, 20, 22, 24 ], "cost": 600 }, { "id": 92, "elements": [ 11, 14, 15 ], "cost": 162 }, { "id": 93, "elements": [ 2, 8, 9, 10 ], "cost": 100 }, { "id": 94, "elements": [ 1, 2, 4, 5, 6 ], "cost": 170 }, { "id": 95, "elements": [ 10, 12, 15, 16 ], "cost": 384 }, { "id": 96, "elements": [ 1, 2, 5, 6 ], "cost": 88 }, { "id": 97, "elements": [ 19, 20, 23, 24, 25 ], "cost": 480 }, { "id": 98, "elements": [ 13, 14, 15, 16, 18 ], "cost": 390 }, { "id": 99, "elements": [ 1, 2, 3, 5, 7 ], "cost": 365 }, { "id": 100, "elements": [ 22, 23, 24, 25 ], "cost": 304 }, { "id": 101, "elements": [ 11, 13, 14 ], "cost": 252 }, { "id": 102, "elements": [ 17, 18, 22 ], "cost": 192 }, { "id": 103, "elements": [ 18, 19, 21 ], "cost": 150 }, { "id": 104, "elements": [ 21, 22, 24, 25 ], "cost": 80 }, { "id": 105, "elements": [ 6, 9, 10, 11, 12, 15, 16 ], "cost": 336 }, { "id": 106, "elements": [ 8, 9, 12 ], "cost": 249 }, { "id": 107, "elements": [ 4, 5, 6, 7 ], "cost": 84 }, { "id": 108, "elements": [ 1, 3, 4, 5, 6, 7, 10 ], "cost": 203 }, { "id": 109, "elements": [ 1, 2, 3, 5, 6 ], "cost": 45 }, { "id": 110, "elements": [ 20, 21, 22, 23 ], "cost": 256 }, { "id": 111, "elements": [ 1, 2, 3, 4, 5, 9 ], "cost": 84 }, { "id": 112, "elements": [ 12, 13, 14, 15, 16 ], "cost": 315 }, { "id": 113, "elements": [ 19, 20, 22, 23, 24 ], "cost": 380 }, { "id": 114, "elements": [ 1, 3, 4, 5, 7, 8, 9 ], "cost": 658 }, { "id": 115, "elements": [ 12, 14, 15, 17 ], "cost": 328 }, { "id": 116, "elements": [ 10, 11, 12, 13, 14 ], "cost": 330 }, { "id": 117, "elements": [ 8, 10, 11, 14, 16 ], "cost": 430 }, { "id": 118, "elements": [ 9, 10, 11, 12, 13, 14 ], "cost": 582 }, { "id": 119, "elements": [ 7, 9, 10 ], "cost": 78 }, { "id": 120, "elements": [ 12, 14, 16 ], "cost": 60 }, { "id": 121, "elements": [ 13, 14, 16, 18 ], "cost": 100 }, { "id": 122, "elements": [ 4, 6, 9, 10 ], "cost": 248 }, { "id": 123, "elements": [ 3, 5, 6, 9, 11 ], "cost": 385 }, { "id": 124, "elements": [ 20, 21, 22, 23, 24 ], "cost": 110 }, { "id": 125, "elements": [ 2, 3, 5, 6, 7, 8, 10 ], "cost": 511 }, { "id": 126, "elements": [ 18, 20, 21, 22, 23, 24 ], "cost": 372 }, { "id": 127, "elements": [ 13, 14, 15, 17, 18, 21 ], "cost": 516 }, { "id": 128, "elements": [ 8, 9, 11, 14 ], "cost": 40 }, { "id": 129, "elements": [ 6, 11, 12 ], "cost": 177 }, { "id": 130, "elements": [ 15, 18, 22 ], "cost": 147 }, { "id": 131, "elements": [ 9, 11, 12, 13, 16 ], "cost": 130 }, { "id": 132, "elements": [ 14, 15, 16, 17, 19 ], "cost": 345 }, { "id": 133, "elements": [ 1, 2, 4, 6 ], "cost": 332 }, { "id": 134, "elements": [ 3, 5, 6, 9, 10, 11, 12 ], "cost": 427 }, { "id": 135, "elements": [ 4, 6, 10, 11 ], "cost": 4 }, { "id": 136, "elements": [ 17, 19, 20 ], "cost": 237 }, { "id": 137, "elements": [ 17, 18, 20, 21, 23 ], "cost": 160 }, { "id": 138, "elements": [ 9, 10, 12, 13 ], "cost": 52 }, { "id": 139, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 140, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 141, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 142, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 143, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 144, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 145, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 146, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 147, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 148, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 149, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 150, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 151, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 152, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 153, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 154, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 155, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 156, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 157, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 158, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 159, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 160, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 161, "elements": [ 23 ], "cost": 10000, "singleton": true }, { "id": 162, "elements": [ 24 ], "cost": 10000, "singleton": true }, { "id": 163, "elements": [ 25 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0040_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0040_bag_stats.png" }, "solution": [ 26, 27, 39, 103, 109, 119, 120 ], "obj": 698.0, "instance_variant": { "num_elements": 25, "num_sets": 163, "sets": [ { "id": "S1", "elements": [ 1, 3, 4, 5, 6 ], "cost": 20 }, { "id": "S2", "elements": [ 3, 4, 6, 7, 8, 9, 10 ], "cost": 658 }, { "id": "S3", "elements": [ 4, 6, 8, 9 ], "cost": 240 }, { "id": "S4", "elements": [ 4, 8, 11, 12 ], "cost": 136 }, { "id": "S5", "elements": [ 9, 11, 12, 13, 16, 19 ], "cost": 390 }, { "id": "S6", "elements": [ 19, 21, 22, 23 ], "cost": 44 }, { "id": "S7", "elements": [ 19, 20, 21, 22, 24 ], "cost": 175 }, { "id": "S8", "elements": [ 13, 14, 15, 16, 17, 18, 20 ], "cost": 567 }, { "id": "S9", "elements": [ 19, 22, 23 ], "cost": 132 }, { "id": "S10", "elements": [ 15, 20, 21 ], "cost": 63 }, { "id": "S11", "elements": [ 16, 18, 21, 24 ], "cost": 208 }, { "id": "S12", "elements": [ 2, 8, 10, 11 ], "cost": 184 }, { "id": "S13", "elements": [ 7, 11, 12, 13, 15, 16, 17 ], "cost": 252 }, { "id": "S14", "elements": [ 0, 1, 2, 3, 4, 5, 7 ], "cost": 322 }, { "id": "S15", "elements": [ 12, 14, 16, 22 ], "cost": 44 }, { "id": "S16", "elements": [ 13, 14, 15, 20, 21, 24 ], "cost": 48 }, { "id": "S17", "elements": [ 0, 2, 5, 8 ], "cost": 180 }, { "id": "S18", "elements": [ 7, 8, 9, 10 ], "cost": 252 }, { "id": "S19", "elements": [ 4, 5, 6, 7, 8, 9 ], "cost": 252 }, { "id": "S20", "elements": [ 0, 1, 2, 3 ], "cost": 348 }, { "id": "S21", "elements": [ 14, 15, 19 ], "cost": 51 }, { "id": "S22", "elements": [ 12, 14, 16, 17, 20, 21 ], "cost": 96 }, { "id": "S23", "elements": [ 20, 21, 22, 23, 24 ], "cost": 295 }, { "id": "S24", "elements": [ 15, 19, 21, 23 ], "cost": 284 }, { "id": "S25", "elements": [ 17, 20, 21, 23 ], "cost": 296 }, { "id": "S26", "elements": [ 3, 7, 10 ], "cost": 129 }, { "id": "S27", "elements": [ 12, 14, 16 ], "cost": 231 }, { "id": "S28", "elements": [ 19, 20, 21, 22, 23, 24 ], "cost": 552 }, { "id": "S29", "elements": [ 2, 4, 6 ], "cost": 297 }, { "id": "S30", "elements": [ 15, 18, 19, 20, 21, 23, 24 ], "cost": 196 }, { "id": "S31", "elements": [ 2, 4, 5, 6, 7 ], "cost": 395 }, { "id": "S32", "elements": [ 11, 14, 15 ], "cost": 141 }, { "id": "S33", "elements": [ 7, 8, 9, 11, 13 ], "cost": 465 }, { "id": "S34", "elements": [ 16, 19, 21, 22 ], "cost": 224 }, { "id": "S35", "elements": [ 14, 17, 20, 21, 23 ], "cost": 100 }, { "id": "S36", "elements": [ 16, 18, 20, 21, 22, 23, 24 ], "cost": 329 }, { "id": "S37", "elements": [ 5, 6, 8, 9 ], "cost": 188 }, { "id": "S38", "elements": [ 2, 4, 5, 6, 9, 10, 11 ], "cost": 595 }, { "id": "S39", "elements": [ 19, 21, 22, 23, 24 ], "cost": 5 }, { "id": "S40", "elements": [ 18, 21, 23, 24 ], "cost": 152 }, { "id": "S41", "elements": [ 12, 15, 16, 18 ], "cost": 396 }, { "id": "S42", "elements": [ 17, 19, 21 ], "cost": 153 }, { "id": "S43", "elements": [ 1, 2, 3, 5, 6 ], "cost": 270 }, { "id": "S44", "elements": [ 20, 22, 23 ], "cost": 210 }, { "id": "S45", "elements": [ 16, 18, 20, 23 ], "cost": 284 }, { "id": "S46", "elements": [ 0, 1, 2 ], "cost": 18 }, { "id": "S47", "elements": [ 5, 7, 10 ], "cost": 261 }, { "id": "S48", "elements": [ 14, 18, 19, 20, 22, 23 ], "cost": 516 }, { "id": "S49", "elements": [ 14, 16, 17, 18, 19, 20, 22 ], "cost": 476 }, { "id": "S50", "elements": [ 12, 13, 14, 16 ], "cost": 208 }, { "id": "S51", "elements": [ 7, 9, 10, 11 ], "cost": 156 }, { "id": "S52", "elements": [ 1, 2, 3, 4, 5 ], "cost": 100 }, { "id": "S53", "elements": [ 10, 11, 12, 14, 15, 16 ], "cost": 132 }, { "id": "S54", "elements": [ 14, 17, 18, 19, 21, 22 ], "cost": 534 }, { "id": "S55", "elements": [ 11, 12, 13, 15, 16 ], "cost": 290 }, { "id": "S56", "elements": [ 7, 8, 9, 10, 11, 13 ], "cost": 258 }, { "id": "S57", "elements": [ 0, 1, 2, 3, 4, 5 ], "cost": 438 }, { "id": "S58", "elements": [ 3, 7, 8, 9, 10, 11, 13 ], "cost": 595 }, { "id": "S59", "elements": [ 3, 4, 5, 6, 7, 8, 10 ], "cost": 126 }, { "id": "S60", "elements": [ 13, 14, 17, 18 ], "cost": 120 }, { "id": "S61", "elements": [ 13, 14, 16, 18, 19 ], "cost": 265 }, { "id": "S62", "elements": [ 7, 8, 10, 13, 15 ], "cost": 170 }, { "id": "S63", "elements": [ 8, 9, 10, 11, 12, 14 ], "cost": 504 }, { "id": "S64", "elements": [ 6, 7, 9, 10, 11, 12, 13 ], "cost": 665 }, { "id": "S65", "elements": [ 11, 12, 13 ], "cost": 246 }, { "id": "S66", "elements": [ 16, 17, 22, 23 ], "cost": 20 }, { "id": "S67", "elements": [ 1, 2, 3, 4 ], "cost": 132 }, { "id": "S68", "elements": [ 0, 1, 4 ], "cost": 168 }, { "id": "S69", "elements": [ 10, 11, 12, 15, 17 ], "cost": 370 }, { "id": "S70", "elements": [ 11, 15, 20 ], "cost": 132 }, { "id": "S71", "elements": [ 15, 17, 19, 21, 23 ], "cost": 60 }, { "id": "S72", "elements": [ 5, 6, 7, 8, 9, 10 ], "cost": 318 }, { "id": "S73", "elements": [ 2, 4, 6, 7, 8, 9 ], "cost": 162 }, { "id": "S74", "elements": [ 11, 12, 13, 14, 15, 17 ], "cost": 504 }, { "id": "S75", "elements": [ 5, 6, 7, 9, 10, 11 ], "cost": 18 }, { "id": "S76", "elements": [ 8, 10, 12, 13, 15, 16 ], "cost": 390 }, { "id": "S77", "elements": [ 0, 1, 2, 3, 4, 5, 6 ], "cost": 42 }, { "id": "S78", "elements": [ 0, 1, 2, 3, 5 ], "cost": 380 }, { "id": "S79", "elements": [ 8, 9, 10, 11, 12 ], "cost": 100 }, { "id": "S80", "elements": [ 4, 5, 7 ], "cost": 36 }, { "id": "S81", "elements": [ 8, 12, 13 ], "cost": 87 }, { "id": "S82", "elements": [ 0, 1, 5, 7 ], "cost": 392 }, { "id": "S83", "elements": [ 13, 14, 15 ], "cost": 54 }, { "id": "S84", "elements": [ 14, 16, 18, 19, 20, 21, 22 ], "cost": 322 }, { "id": "S85", "elements": [ 16, 19, 20, 21, 22, 23, 24 ], "cost": 161 }, { "id": "S86", "elements": [ 13, 16, 19 ], "cost": 231 }, { "id": "S87", "elements": [ 4, 5, 7, 10 ], "cost": 120 }, { "id": "S88", "elements": [ 2, 5, 6, 7, 8, 10 ], "cost": 66 }, { "id": "S89", "elements": [ 10, 12, 13, 14, 15, 16, 17 ], "cost": 210 }, { "id": "S90", "elements": [ 6, 7, 9 ], "cost": 204 }, { "id": "S91", "elements": [ 15, 17, 18, 19, 21, 23 ], "cost": 600 }, { "id": "S92", "elements": [ 10, 13, 14 ], "cost": 162 }, { "id": "S93", "elements": [ 1, 7, 8, 9 ], "cost": 100 }, { "id": "S94", "elements": [ 0, 1, 3, 4, 5 ], "cost": 170 }, { "id": "S95", "elements": [ 9, 11, 14, 15 ], "cost": 384 }, { "id": "S96", "elements": [ 0, 1, 4, 5 ], "cost": 88 }, { "id": "S97", "elements": [ 18, 19, 22, 23, 24 ], "cost": 480 }, { "id": "S98", "elements": [ 12, 13, 14, 15, 17 ], "cost": 390 }, { "id": "S99", "elements": [ 0, 1, 2, 4, 6 ], "cost": 365 }, { "id": "S100", "elements": [ 21, 22, 23, 24 ], "cost": 304 }, { "id": "S101", "elements": [ 10, 12, 13 ], "cost": 252 }, { "id": "S102", "elements": [ 16, 17, 21 ], "cost": 192 }, { "id": "S103", "elements": [ 17, 18, 20 ], "cost": 150 }, { "id": "S104", "elements": [ 20, 21, 23, 24 ], "cost": 80 }, { "id": "S105", "elements": [ 5, 8, 9, 10, 11, 14, 15 ], "cost": 336 }, { "id": "S106", "elements": [ 7, 8, 11 ], "cost": 249 }, { "id": "S107", "elements": [ 3, 4, 5, 6 ], "cost": 84 }, { "id": "S108", "elements": [ 0, 2, 3, 4, 5, 6, 9 ], "cost": 203 }, { "id": "S109", "elements": [ 0, 1, 2, 4, 5 ], "cost": 45 }, { "id": "S110", "elements": [ 19, 20, 21, 22 ], "cost": 256 }, { "id": "S111", "elements": [ 0, 1, 2, 3, 4, 8 ], "cost": 84 }, { "id": "S112", "elements": [ 11, 12, 13, 14, 15 ], "cost": 315 }, { "id": "S113", "elements": [ 18, 19, 21, 22, 23 ], "cost": 380 }, { "id": "S114", "elements": [ 0, 2, 3, 4, 6, 7, 8 ], "cost": 658 }, { "id": "S115", "elements": [ 11, 13, 14, 16 ], "cost": 328 }, { "id": "S116", "elements": [ 9, 10, 11, 12, 13 ], "cost": 330 }, { "id": "S117", "elements": [ 7, 9, 10, 13, 15 ], "cost": 430 }, { "id": "S118", "elements": [ 8, 9, 10, 11, 12, 13 ], "cost": 582 }, { "id": "S119", "elements": [ 6, 8, 9 ], "cost": 78 }, { "id": "S120", "elements": [ 11, 13, 15 ], "cost": 60 }, { "id": "S121", "elements": [ 12, 13, 15, 17 ], "cost": 100 }, { "id": "S122", "elements": [ 3, 5, 8, 9 ], "cost": 248 }, { "id": "S123", "elements": [ 2, 4, 5, 8, 10 ], "cost": 385 }, { "id": "S124", "elements": [ 19, 20, 21, 22, 23 ], "cost": 110 }, { "id": "S125", "elements": [ 1, 2, 4, 5, 6, 7, 9 ], "cost": 511 }, { "id": "S126", "elements": [ 17, 19, 20, 21, 22, 23 ], "cost": 372 }, { "id": "S127", "elements": [ 12, 13, 14, 16, 17, 20 ], "cost": 516 }, { "id": "S128", "elements": [ 7, 8, 10, 13 ], "cost": 40 }, { "id": "S129", "elements": [ 5, 10, 11 ], "cost": 177 }, { "id": "S130", "elements": [ 14, 17, 21 ], "cost": 147 }, { "id": "S131", "elements": [ 8, 10, 11, 12, 15 ], "cost": 130 }, { "id": "S132", "elements": [ 13, 14, 15, 16, 18 ], "cost": 345 }, { "id": "S133", "elements": [ 0, 1, 3, 5 ], "cost": 332 }, { "id": "S134", "elements": [ 2, 4, 5, 8, 9, 10, 11 ], "cost": 427 }, { "id": "S135", "elements": [ 3, 5, 9, 10 ], "cost": 4 }, { "id": "S136", "elements": [ 16, 18, 19 ], "cost": 237 }, { "id": "S137", "elements": [ 16, 17, 19, 20, 22 ], "cost": 160 }, { "id": "S138", "elements": [ 8, 9, 11, 12 ], "cost": 52 }, { "id": "S139", "elements": [ 0 ], "cost": 10000 }, { "id": "S140", "elements": [ 1 ], "cost": 10000 }, { "id": "S141", "elements": [ 2 ], "cost": 10000 }, { "id": "S142", "elements": [ 3 ], "cost": 10000 }, { "id": "S143", "elements": [ 4 ], "cost": 10000 }, { "id": "S144", "elements": [ 5 ], "cost": 10000 }, { "id": "S145", "elements": [ 6 ], "cost": 10000 }, { "id": "S146", "elements": [ 7 ], "cost": 10000 }, { "id": "S147", "elements": [ 8 ], "cost": 10000 }, { "id": "S148", "elements": [ 9 ], "cost": 10000 }, { "id": "S149", "elements": [ 10 ], "cost": 10000 }, { "id": "S150", "elements": [ 11 ], "cost": 10000 }, { "id": "S151", "elements": [ 12 ], "cost": 10000 }, { "id": "S152", "elements": [ 13 ], "cost": 10000 }, { "id": "S153", "elements": [ 14 ], "cost": 10000 }, { "id": "S154", "elements": [ 15 ], "cost": 10000 }, { "id": "S155", "elements": [ 16 ], "cost": 10000 }, { "id": "S156", "elements": [ 17 ], "cost": 10000 }, { "id": "S157", "elements": [ 18 ], "cost": 10000 }, { "id": "S158", "elements": [ 19 ], "cost": 10000 }, { "id": "S159", "elements": [ 20 ], "cost": 10000 }, { "id": "S160", "elements": [ 21 ], "cost": 10000 }, { "id": "S161", "elements": [ 22 ], "cost": 10000 }, { "id": "S162", "elements": [ 23 ], "cost": 10000 }, { "id": "S163", "elements": [ 24 ], "cost": 10000 } ] }, "solution_variant": [ "S26", "S27", "S39", "S103", "S109", "S119", "S120" ], "context_index": 41, "input_format": "nl", "input_index_base": 0 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "There’s a local relay coming up and the organizer’s juggling which squads to hire: each squad covers certain parts of the course and charges a stipend. The aim is to assign squads so every section is covered by exactly one squad, squads don’t overlap on any section, and the overall payout — add together each hired squad’s stipend — is minimized. The concrete squads, segments, and amounts are given below.\n\nThey need to cover 19 segments exactly once, choosing from 99 candidate squads.\nSquad S1 covers segments 4 5 6 7 and charges a stipend of 212.\nSquad S2 covers segments 5 6 9 and charges a stipend of 126.\nSquad S3 covers segments 1 4 5 and charges a stipend of 90.\nSquad S4 covers segments 9 10 11 and charges a stipend of 57.\nSquad S5 covers segments 16 17 18 19 and charges a stipend of 128.\nSquad S6 covers segments 5 7 8 12 and charges a stipend of 320.\nSquad S7 covers segments 6 8 9 10 12 and charges a stipend of 160.\nSquad S8 covers segments 14 15 16 18 19 and charges a stipend of 255.\nSquad S9 covers segments 2 6 7 and charges a stipend of 285.\nSquad S10 covers segments 4 6 7 and charges a stipend of 156.\nSquad S11 covers segments 14 17 and charges a stipend of 112.\nSquad S12 covers segments 2 3 6 and charges a stipend of 48.\nSquad S13 covers segments 6 7 9 and charges a stipend of 54.\nSquad S14 covers segments 2 3 4 5 and charges a stipend of 60.\nSquad S15 covers segments 3 4 5 6 and charges a stipend of 364.\nSquad S16 covers segments 8 10 12 and charges a stipend of 45.\nSquad S17 covers segments 2 4 5 6 and charges a stipend of 260.\nSquad S18 covers segments 3 6 and charges a stipend of 56.\nSquad S19 covers segments 9 10 11 12 13 and charges a stipend of 110.\nSquad S20 covers segments 1 3 4 and charges a stipend of 273.\nSquad S21 covers segments 12 14 and charges a stipend of 14.\nSquad S22 covers segments 13 15 16 17 and charges a stipend of 280.\nSquad S23 covers segments 6 9 and charges a stipend of 64.\nSquad S24 covers segments 12 14 15 17 and charges a stipend of 216.\nSquad S25 covers segments 9 10 14 and charges a stipend of 123.\nSquad S26 covers segments 1 5 6 and charges a stipend of 39.\nSquad S27 covers segments 3 5 and charges a stipend of 56.\nSquad S28 covers segments 13 15 and charges a stipend of 44.\nSquad S29 covers segments 7 8 9 11 12 and charges a stipend of 95.\nSquad S30 covers segments 6 7 8 9 12 and charges a stipend of 500.\nSquad S31 covers segments 4 8 9 and charges a stipend of 99.\nSquad S32 covers segments 17 18 19 and charges a stipend of 183.\nSquad S33 covers segments 10 11 12 and charges a stipend of 252.\nSquad S34 covers segments 14 15 16 17 18 19 and charges a stipend of 234.\nSquad S35 covers segments 18 19 and charges a stipend of 128.\nSquad S36 covers segments 2 4 5 7 and charges a stipend of 356.\nSquad S37 covers segments 9 10 and charges a stipend of 96.\nSquad S38 covers segments 1 2 3 4 5 and charges a stipend of 95.\nSquad S39 covers segments 13 14 17 and charges a stipend of 168.\nSquad S40 covers segments 3 5 8 and charges a stipend of 228.\nSquad S41 covers segments 5 6 7 8 9 and charges a stipend of 355.\nSquad S42 covers segments 16 17 19 and charges a stipend of 258.\nSquad S43 covers segments 11 14 15 18 and charges a stipend of 380.\nSquad S44 covers segments 7 8 10 11 12 and charges a stipend of 150.\nSquad S45 covers segments 9 12 and charges a stipend of 22.\nSquad S46 covers segments 15 16 17 18 19 and charges a stipend of 300.\nSquad S47 covers segments 7 11 12 13 14 and charges a stipend of 240.\nSquad S48 covers segments 12 13 14 and charges a stipend of 162.\nSquad S49 covers segments 2 6 and charges a stipend of 84.\nSquad S50 covers segments 9 11 12 13 14 and charges a stipend of 480.\nSquad S51 covers segments 4 5 7 8 9 and charges a stipend of 265.\nSquad S52 covers segments 10 13 15 and charges a stipend of 39.\nSquad S53 covers segments 13 14 and charges a stipend of 174.\nSquad S54 covers segments 13 15 16 and charges a stipend of 6.\nSquad S55 covers segments 9 10 11 12 15 and charges a stipend of 95.\nSquad S56 covers segments 7 11 and charges a stipend of 126.\nSquad S57 covers segments 13 16 17 18 19 and charges a stipend of 435.\nSquad S58 covers segments 15 17 18 and charges a stipend of 204.\nSquad S59 covers segments 9 10 12 13 14 and charges a stipend of 205.\nSquad S60 covers segments 11 12 13 15 16 17 and charges a stipend of 162.\nSquad S61 covers segments 4 6 9 and charges a stipend of 63.\nSquad S62 covers segments 1 2 3 4 5 6 and charges a stipend of 24.\nSquad S63 covers segments 15 19 and charges a stipend of 64.\nSquad S64 covers segments 6 7 8 9 10 and charges a stipend of 45.\nSquad S65 covers segments 7 9 10 11 and charges a stipend of 72.\nSquad S66 covers segments 11 13 14 15 and charges a stipend of 324.\nSquad S67 covers segments 2 3 4 and charges a stipend of 6.\nSquad S68 covers segments 4 6 8 and charges a stipend of 258.\nSquad S69 covers segments 3 4 6 8 and charges a stipend of 188.\nSquad S70 covers segments 2 4 and charges a stipend of 102.\nSquad S71 covers segments 1 2 and charges a stipend of 32.\nSquad S72 covers segments 11 12 14 and charges a stipend of 54.\nSquad S73 covers segments 17 19 and charges a stipend of 176.\nSquad S74 covers segments 7 8 10 13 and charges a stipend of 240.\nSquad S75 covers segments 17 18 and charges a stipend of 196.\nSquad S76 covers segments 14 16 17 18 and charges a stipend of 248.\nSquad S77 covers segments 1 2 4 5 and charges a stipend of 296.\nSquad S78 covers segments 11 16 17 and charges a stipend of 195.\nSquad S79 covers segments 1 2 3 4 and charges a stipend of 260.\nSquad S80 covers segments 14 17 18 and charges a stipend of 177.\nSquad S81 covers segments 1 and charges a stipend of 10000.\nSquad S82 covers segments 2 and charges a stipend of 10000.\nSquad S83 covers segments 3 and charges a stipend of 10000.\nSquad S84 covers segments 4 and charges a stipend of 10000.\nSquad S85 covers segments 5 and charges a stipend of 10000.\nSquad S86 covers segments 6 and charges a stipend of 10000.\nSquad S87 covers segments 7 and charges a stipend of 10000.\nSquad S88 covers segments 8 and charges a stipend of 10000.\nSquad S89 covers segments 9 and charges a stipend of 10000.\nSquad S90 covers segments 10 and charges a stipend of 10000.\nSquad S91 covers segments 11 and charges a stipend of 10000.\nSquad S92 covers segments 12 and charges a stipend of 10000.\nSquad S93 covers segments 13 and charges a stipend of 10000.\nSquad S94 covers segments 14 and charges a stipend of 10000.\nSquad S95 covers segments 15 and charges a stipend of 10000.\nSquad S96 covers segments 16 and charges a stipend of 10000.\nSquad S97 covers segments 17 and charges a stipend of 10000.\nSquad S98 covers segments 18 and charges a stipend of 10000.\nSquad S99 covers segments 19 and charges a stipend of 10000.\nThe organizer will choose squads to cover all 19 segments while minimizing the total stipend from the 99 candidates.\n\nAlso, when you show which squads you picked, please use this simple JSON layout so it's easy to read:\n\n{\n \"solution\": [\"squad_id\", ...]\n}\n\nHere \"solution\" is just a list of the squad IDs you want to hire (one ID per entry). This JSON is only a sketch of the shape I expect — not the final answer itself — so fill that array with the actual squad identifiers from the instance.\n\nPlease use the identifiers exactly as they appear in the instance input — no renaming and no new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 19, "num_sets": 99, "density": 0.1594896331738437, "sets": [ { "id": 1, "elements": [ 4, 5, 6, 7 ], "cost": 212 }, { "id": 2, "elements": [ 5, 6, 9 ], "cost": 126 }, { "id": 3, "elements": [ 1, 4, 5 ], "cost": 90 }, { "id": 4, "elements": [ 9, 10, 11 ], "cost": 57 }, { "id": 5, "elements": [ 16, 17, 18, 19 ], "cost": 128 }, { "id": 6, "elements": [ 5, 7, 8, 12 ], "cost": 320 }, { "id": 7, "elements": [ 6, 8, 9, 10, 12 ], "cost": 160 }, { "id": 8, "elements": [ 14, 15, 16, 18, 19 ], "cost": 255 }, { "id": 9, "elements": [ 2, 6, 7 ], "cost": 285 }, { "id": 10, "elements": [ 4, 6, 7 ], "cost": 156 }, { "id": 11, "elements": [ 14, 17 ], "cost": 112 }, { "id": 12, "elements": [ 2, 3, 6 ], "cost": 48 }, { "id": 13, "elements": [ 6, 7, 9 ], "cost": 54 }, { "id": 14, "elements": [ 2, 3, 4, 5 ], "cost": 60 }, { "id": 15, "elements": [ 3, 4, 5, 6 ], "cost": 364 }, { "id": 16, "elements": [ 8, 10, 12 ], "cost": 45 }, { "id": 17, "elements": [ 2, 4, 5, 6 ], "cost": 260 }, { "id": 18, "elements": [ 3, 6 ], "cost": 56 }, { "id": 19, "elements": [ 9, 10, 11, 12, 13 ], "cost": 110 }, { "id": 20, "elements": [ 1, 3, 4 ], "cost": 273 }, { "id": 21, "elements": [ 12, 14 ], "cost": 14 }, { "id": 22, "elements": [ 13, 15, 16, 17 ], "cost": 280 }, { "id": 23, "elements": [ 6, 9 ], "cost": 64 }, { "id": 24, "elements": [ 12, 14, 15, 17 ], "cost": 216 }, { "id": 25, "elements": [ 9, 10, 14 ], "cost": 123 }, { "id": 26, "elements": [ 1, 5, 6 ], "cost": 39 }, { "id": 27, "elements": [ 3, 5 ], "cost": 56 }, { "id": 28, "elements": [ 13, 15 ], "cost": 44 }, { "id": 29, "elements": [ 7, 8, 9, 11, 12 ], "cost": 95 }, { "id": 30, "elements": [ 6, 7, 8, 9, 12 ], "cost": 500 }, { "id": 31, "elements": [ 4, 8, 9 ], "cost": 99 }, { "id": 32, "elements": [ 17, 18, 19 ], "cost": 183 }, { "id": 33, "elements": [ 10, 11, 12 ], "cost": 252 }, { "id": 34, "elements": [ 14, 15, 16, 17, 18, 19 ], "cost": 234 }, { "id": 35, "elements": [ 18, 19 ], "cost": 128 }, { "id": 36, "elements": [ 2, 4, 5, 7 ], "cost": 356 }, { "id": 37, "elements": [ 9, 10 ], "cost": 96 }, { "id": 38, "elements": [ 1, 2, 3, 4, 5 ], "cost": 95 }, { "id": 39, "elements": [ 13, 14, 17 ], "cost": 168 }, { "id": 40, "elements": [ 3, 5, 8 ], "cost": 228 }, { "id": 41, "elements": [ 5, 6, 7, 8, 9 ], "cost": 355 }, { "id": 42, "elements": [ 16, 17, 19 ], "cost": 258 }, { "id": 43, "elements": [ 11, 14, 15, 18 ], "cost": 380 }, { "id": 44, "elements": [ 7, 8, 10, 11, 12 ], "cost": 150 }, { "id": 45, "elements": [ 9, 12 ], "cost": 22 }, { "id": 46, "elements": [ 15, 16, 17, 18, 19 ], "cost": 300 }, { "id": 47, "elements": [ 7, 11, 12, 13, 14 ], "cost": 240 }, { "id": 48, "elements": [ 12, 13, 14 ], "cost": 162 }, { "id": 49, "elements": [ 2, 6 ], "cost": 84 }, { "id": 50, "elements": [ 9, 11, 12, 13, 14 ], "cost": 480 }, { "id": 51, "elements": [ 4, 5, 7, 8, 9 ], "cost": 265 }, { "id": 52, "elements": [ 10, 13, 15 ], "cost": 39 }, { "id": 53, "elements": [ 13, 14 ], "cost": 174 }, { "id": 54, "elements": [ 13, 15, 16 ], "cost": 6 }, { "id": 55, "elements": [ 9, 10, 11, 12, 15 ], "cost": 95 }, { "id": 56, "elements": [ 7, 11 ], "cost": 126 }, { "id": 57, "elements": [ 13, 16, 17, 18, 19 ], "cost": 435 }, { "id": 58, "elements": [ 15, 17, 18 ], "cost": 204 }, { "id": 59, "elements": [ 9, 10, 12, 13, 14 ], "cost": 205 }, { "id": 60, "elements": [ 11, 12, 13, 15, 16, 17 ], "cost": 162 }, { "id": 61, "elements": [ 4, 6, 9 ], "cost": 63 }, { "id": 62, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 24 }, { "id": 63, "elements": [ 15, 19 ], "cost": 64 }, { "id": 64, "elements": [ 6, 7, 8, 9, 10 ], "cost": 45 }, { "id": 65, "elements": [ 7, 9, 10, 11 ], "cost": 72 }, { "id": 66, "elements": [ 11, 13, 14, 15 ], "cost": 324 }, { "id": 67, "elements": [ 2, 3, 4 ], "cost": 6 }, { "id": 68, "elements": [ 4, 6, 8 ], "cost": 258 }, { "id": 69, "elements": [ 3, 4, 6, 8 ], "cost": 188 }, { "id": 70, "elements": [ 2, 4 ], "cost": 102 }, { "id": 71, "elements": [ 1, 2 ], "cost": 32 }, { "id": 72, "elements": [ 11, 12, 14 ], "cost": 54 }, { "id": 73, "elements": [ 17, 19 ], "cost": 176 }, { "id": 74, "elements": [ 7, 8, 10, 13 ], "cost": 240 }, { "id": 75, "elements": [ 17, 18 ], "cost": 196 }, { "id": 76, "elements": [ 14, 16, 17, 18 ], "cost": 248 }, { "id": 77, "elements": [ 1, 2, 4, 5 ], "cost": 296 }, { "id": 78, "elements": [ 11, 16, 17 ], "cost": 195 }, { "id": 79, "elements": [ 1, 2, 3, 4 ], "cost": 260 }, { "id": 80, "elements": [ 14, 17, 18 ], "cost": 177 }, { "id": 81, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 82, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 83, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 84, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 85, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 86, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 87, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 88, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 89, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 90, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 91, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 92, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 93, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 94, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 95, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 96, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 97, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 98, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 19 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0041_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0041_bag_stats.png" }, "solution": [ 5, 28, 38, 64, 72 ], "obj": 366.0, "instance_variant": { "num_elements": 19, "num_sets": 99, "sets": [ { "id": "S1", "elements": [ 4, 5, 6, 7 ], "cost": 212 }, { "id": "S2", "elements": [ 5, 6, 9 ], "cost": 126 }, { "id": "S3", "elements": [ 1, 4, 5 ], "cost": 90 }, { "id": "S4", "elements": [ 9, 10, 11 ], "cost": 57 }, { "id": "S5", "elements": [ 16, 17, 18, 19 ], "cost": 128 }, { "id": "S6", "elements": [ 5, 7, 8, 12 ], "cost": 320 }, { "id": "S7", "elements": [ 6, 8, 9, 10, 12 ], "cost": 160 }, { "id": "S8", "elements": [ 14, 15, 16, 18, 19 ], "cost": 255 }, { "id": "S9", "elements": [ 2, 6, 7 ], "cost": 285 }, { "id": "S10", "elements": [ 4, 6, 7 ], "cost": 156 }, { "id": "S11", "elements": [ 14, 17 ], "cost": 112 }, { "id": "S12", "elements": [ 2, 3, 6 ], "cost": 48 }, { "id": "S13", "elements": [ 6, 7, 9 ], "cost": 54 }, { "id": "S14", "elements": [ 2, 3, 4, 5 ], "cost": 60 }, { "id": "S15", "elements": [ 3, 4, 5, 6 ], "cost": 364 }, { "id": "S16", "elements": [ 8, 10, 12 ], "cost": 45 }, { "id": "S17", "elements": [ 2, 4, 5, 6 ], "cost": 260 }, { "id": "S18", "elements": [ 3, 6 ], "cost": 56 }, { "id": "S19", "elements": [ 9, 10, 11, 12, 13 ], "cost": 110 }, { "id": "S20", "elements": [ 1, 3, 4 ], "cost": 273 }, { "id": "S21", "elements": [ 12, 14 ], "cost": 14 }, { "id": "S22", "elements": [ 13, 15, 16, 17 ], "cost": 280 }, { "id": "S23", "elements": [ 6, 9 ], "cost": 64 }, { "id": "S24", "elements": [ 12, 14, 15, 17 ], "cost": 216 }, { "id": "S25", "elements": [ 9, 10, 14 ], "cost": 123 }, { "id": "S26", "elements": [ 1, 5, 6 ], "cost": 39 }, { "id": "S27", "elements": [ 3, 5 ], "cost": 56 }, { "id": "S28", "elements": [ 13, 15 ], "cost": 44 }, { "id": "S29", "elements": [ 7, 8, 9, 11, 12 ], "cost": 95 }, { "id": "S30", "elements": [ 6, 7, 8, 9, 12 ], "cost": 500 }, { "id": "S31", "elements": [ 4, 8, 9 ], "cost": 99 }, { "id": "S32", "elements": [ 17, 18, 19 ], "cost": 183 }, { "id": "S33", "elements": [ 10, 11, 12 ], "cost": 252 }, { "id": "S34", "elements": [ 14, 15, 16, 17, 18, 19 ], "cost": 234 }, { "id": "S35", "elements": [ 18, 19 ], "cost": 128 }, { "id": "S36", "elements": [ 2, 4, 5, 7 ], "cost": 356 }, { "id": "S37", "elements": [ 9, 10 ], "cost": 96 }, { "id": "S38", "elements": [ 1, 2, 3, 4, 5 ], "cost": 95 }, { "id": "S39", "elements": [ 13, 14, 17 ], "cost": 168 }, { "id": "S40", "elements": [ 3, 5, 8 ], "cost": 228 }, { "id": "S41", "elements": [ 5, 6, 7, 8, 9 ], "cost": 355 }, { "id": "S42", "elements": [ 16, 17, 19 ], "cost": 258 }, { "id": "S43", "elements": [ 11, 14, 15, 18 ], "cost": 380 }, { "id": "S44", "elements": [ 7, 8, 10, 11, 12 ], "cost": 150 }, { "id": "S45", "elements": [ 9, 12 ], "cost": 22 }, { "id": "S46", "elements": [ 15, 16, 17, 18, 19 ], "cost": 300 }, { "id": "S47", "elements": [ 7, 11, 12, 13, 14 ], "cost": 240 }, { "id": "S48", "elements": [ 12, 13, 14 ], "cost": 162 }, { "id": "S49", "elements": [ 2, 6 ], "cost": 84 }, { "id": "S50", "elements": [ 9, 11, 12, 13, 14 ], "cost": 480 }, { "id": "S51", "elements": [ 4, 5, 7, 8, 9 ], "cost": 265 }, { "id": "S52", "elements": [ 10, 13, 15 ], "cost": 39 }, { "id": "S53", "elements": [ 13, 14 ], "cost": 174 }, { "id": "S54", "elements": [ 13, 15, 16 ], "cost": 6 }, { "id": "S55", "elements": [ 9, 10, 11, 12, 15 ], "cost": 95 }, { "id": "S56", "elements": [ 7, 11 ], "cost": 126 }, { "id": "S57", "elements": [ 13, 16, 17, 18, 19 ], "cost": 435 }, { "id": "S58", "elements": [ 15, 17, 18 ], "cost": 204 }, { "id": "S59", "elements": [ 9, 10, 12, 13, 14 ], "cost": 205 }, { "id": "S60", "elements": [ 11, 12, 13, 15, 16, 17 ], "cost": 162 }, { "id": "S61", "elements": [ 4, 6, 9 ], "cost": 63 }, { "id": "S62", "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 24 }, { "id": "S63", "elements": [ 15, 19 ], "cost": 64 }, { "id": "S64", "elements": [ 6, 7, 8, 9, 10 ], "cost": 45 }, { "id": "S65", "elements": [ 7, 9, 10, 11 ], "cost": 72 }, { "id": "S66", "elements": [ 11, 13, 14, 15 ], "cost": 324 }, { "id": "S67", "elements": [ 2, 3, 4 ], "cost": 6 }, { "id": "S68", "elements": [ 4, 6, 8 ], "cost": 258 }, { "id": "S69", "elements": [ 3, 4, 6, 8 ], "cost": 188 }, { "id": "S70", "elements": [ 2, 4 ], "cost": 102 }, { "id": "S71", "elements": [ 1, 2 ], "cost": 32 }, { "id": "S72", "elements": [ 11, 12, 14 ], "cost": 54 }, { "id": "S73", "elements": [ 17, 19 ], "cost": 176 }, { "id": "S74", "elements": [ 7, 8, 10, 13 ], "cost": 240 }, { "id": "S75", "elements": [ 17, 18 ], "cost": 196 }, { "id": "S76", "elements": [ 14, 16, 17, 18 ], "cost": 248 }, { "id": "S77", "elements": [ 1, 2, 4, 5 ], "cost": 296 }, { "id": "S78", "elements": [ 11, 16, 17 ], "cost": 195 }, { "id": "S79", "elements": [ 1, 2, 3, 4 ], "cost": 260 }, { "id": "S80", "elements": [ 14, 17, 18 ], "cost": 177 }, { "id": "S81", "elements": [ 1 ], "cost": 10000 }, { "id": "S82", "elements": [ 2 ], "cost": 10000 }, { "id": "S83", "elements": [ 3 ], "cost": 10000 }, { "id": "S84", "elements": [ 4 ], "cost": 10000 }, { "id": "S85", "elements": [ 5 ], "cost": 10000 }, { "id": "S86", "elements": [ 6 ], "cost": 10000 }, { "id": "S87", "elements": [ 7 ], "cost": 10000 }, { "id": "S88", "elements": [ 8 ], "cost": 10000 }, { "id": "S89", "elements": [ 9 ], "cost": 10000 }, { "id": "S90", "elements": [ 10 ], "cost": 10000 }, { "id": "S91", "elements": [ 11 ], "cost": 10000 }, { "id": "S92", "elements": [ 12 ], "cost": 10000 }, { "id": "S93", "elements": [ 13 ], "cost": 10000 }, { "id": "S94", "elements": [ 14 ], "cost": 10000 }, { "id": "S95", "elements": [ 15 ], "cost": 10000 }, { "id": "S96", "elements": [ 16 ], "cost": 10000 }, { "id": "S97", "elements": [ 17 ], "cost": 10000 }, { "id": "S98", "elements": [ 18 ], "cost": 10000 }, { "id": "S99", "elements": [ 19 ], "cost": 10000 } ] }, "solution_variant": [ "S5", "S28", "S38", "S64", "S72" ], "context_index": 42, "input_format": "nl", "input_index_base": 1 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "A friend is tossing around a few outfit kits before a vacation; each kit covers certain days and contributes a bit of weight to the luggage. The goal is to pick a set of kits so each day is assigned to one—and only one—of the chosen kits, and no chosen kits clash on the same day. The practical criterion is the lightest possible bag: just add up the weights of the chosen kits and aim for the lowest total. The exact options and schedule appear below.\n\nThere are 25 travel days and 150 available kits to consider.\nKit S1 covers days Q R T W and adds 340 weight to the luggage.\nKit S2 covers days F H J L M and adds 315 weight to the luggage.\nKit S3 covers days N O P Q R S T and adds 651 weight to the luggage.\nKit S4 covers days L N O and adds 297 weight to the luggage.\nKit S5 covers days A D F H and adds 212 weight to the luggage.\nKit S6 covers days E M N P and adds 380 weight to the luggage.\nKit S7 covers days Q R T U W and adds 440 weight to the luggage.\nKit S8 covers days I J K L M N P and adds 35 weight to the luggage.\nKit S9 covers days O P Q R S and adds 340 weight to the luggage.\nKit S10 covers days P T U W X and adds 70 weight to the luggage.\nKit S11 covers days A B C D E G and adds 582 weight to the luggage.\nKit S12 covers days L M N P R S and adds 282 weight to the luggage.\nKit S13 covers days M P R S U V and adds 240 weight to the luggage.\nKit S14 covers days L M N O S and adds 85 weight to the luggage.\nKit S15 covers days F H I J L O and adds 90 weight to the luggage.\nKit S16 covers days A C D E F G H and adds 595 weight to the luggage.\nKit S17 covers days T V W X and adds 220 weight to the luggage.\nKit S18 covers days Q S T U V Y and adds 582 weight to the luggage.\nKit S19 covers days T V X Y and adds 240 weight to the luggage.\nKit S20 covers days Q S T U W and adds 370 weight to the luggage.\nKit S21 covers days J L R and adds 264 weight to the luggage.\nKit S22 covers days B D E and adds 54 weight to the luggage.\nKit S23 covers days E J K M and adds 64 weight to the luggage.\nKit S24 covers days K M O and adds 9 weight to the luggage.\nKit S25 covers days E H J K and adds 184 weight to the luggage.\nKit S26 covers days Q R S U V X Y and adds 84 weight to the luggage.\nKit S27 covers days R S T U W X Y and adds 28 weight to the luggage.\nKit S28 covers days A B C D E F and adds 30 weight to the luggage.\nKit S29 covers days B C D J and adds 220 weight to the luggage.\nKit S30 covers days L M N P Q S and adds 108 weight to the luggage.\nKit S31 covers days K L O Q and adds 344 weight to the luggage.\nKit S32 covers days Q R S T U V and adds 258 weight to the luggage.\nKit S33 covers days N Q T V and adds 208 weight to the luggage.\nKit S34 covers days U V W X Y and adds 425 weight to the luggage.\nKit S35 covers days A B C E and adds 304 weight to the luggage.\nKit S36 covers days L M N P and adds 324 weight to the luggage.\nKit S37 covers days S T U Y and adds 256 weight to the luggage.\nKit S38 covers days O P Q T and adds 324 weight to the luggage.\nKit S39 covers days D E F H I J K and adds 448 weight to the luggage.\nKit S40 covers days G H J K M and adds 25 weight to the luggage.\nKit S41 covers days D F J L and adds 264 weight to the luggage.\nKit S42 covers days I K L M N O R and adds 420 weight to the luggage.\nKit S43 covers days L P Q S U and adds 65 weight to the luggage.\nKit S44 covers days A C E and adds 141 weight to the luggage.\nKit S45 covers days G L M N and adds 32 weight to the luggage.\nKit S46 covers days Q R T V W and adds 20 weight to the luggage.\nKit S47 covers days B C D G I and adds 310 weight to the luggage.\nKit S48 covers days B C E F and adds 336 weight to the luggage.\nKit S49 covers days S T U V W X Y and adds 245 weight to the luggage.\nKit S50 covers days K L N O P Q S and adds 238 weight to the luggage.\nKit S51 covers days I K L M N O and adds 384 weight to the luggage.\nKit S52 covers days F G I K L M and adds 114 weight to the luggage.\nKit S53 covers days F G H J K L M and adds 98 weight to the luggage.\nKit S54 covers days D G I J K and adds 250 weight to the luggage.\nKit S55 covers days I J L M and adds 64 weight to the luggage.\nKit S56 covers days F I J K L M O and adds 588 weight to the luggage.\nKit S57 covers days T U W Y and adds 368 weight to the luggage.\nKit S58 covers days L M N O P Q R and adds 238 weight to the luggage.\nKit S59 covers days U V X Y and adds 72 weight to the luggage.\nKit S60 covers days A B D and adds 66 weight to the luggage.\nKit S61 covers days G J K and adds 270 weight to the luggage.\nKit S62 covers days P R T U V and adds 320 weight to the luggage.\nKit S63 covers days G I L and adds 72 weight to the luggage.\nKit S64 covers days O R S and adds 12 weight to the luggage.\nKit S65 covers days O R T U V X and adds 180 weight to the luggage.\nKit S66 covers days F H K and adds 108 weight to the luggage.\nKit S67 covers days S T V W and adds 368 weight to the luggage.\nKit S68 covers days O P R and adds 252 weight to the luggage.\nKit S69 covers days A B D E F G and adds 144 weight to the luggage.\nKit S70 covers days Q U V W and adds 108 weight to the luggage.\nKit S71 covers days L M N and adds 201 weight to the luggage.\nKit S72 covers days F G I J K N and adds 426 weight to the luggage.\nKit S73 covers days A B D E F and adds 220 weight to the luggage.\nKit S74 covers days V X Y and adds 21 weight to the luggage.\nKit S75 covers days S T U V W and adds 220 weight to the luggage.\nKit S76 covers days E F I J M O and adds 480 weight to the luggage.\nKit S77 covers days A B C D E G J and adds 497 weight to the luggage.\nKit S78 covers days R T U Y and adds 208 weight to the luggage.\nKit S79 covers days M N S and adds 114 weight to the luggage.\nKit S80 covers days D G H I J K and adds 294 weight to the luggage.\nKit S81 covers days A B C E F and adds 330 weight to the luggage.\nKit S82 covers days T U V W X Y and adds 72 weight to the luggage.\nKit S83 covers days T V W and adds 165 weight to the luggage.\nKit S84 covers days B C D E and adds 316 weight to the luggage.\nKit S85 covers days C E F G H J L and adds 245 weight to the luggage.\nKit S86 covers days S V X Y and adds 128 weight to the luggage.\nKit S87 covers days H J L M N O Q and adds 462 weight to the luggage.\nKit S88 covers days G I K M O P R and adds 700 weight to the luggage.\nKit S89 covers days E H I K L M and adds 96 weight to the luggage.\nKit S90 covers days P R S and adds 60 weight to the luggage.\nKit S91 covers days A B C D E F G and adds 210 weight to the luggage.\nKit S92 covers days H I L and adds 3 weight to the luggage.\nKit S93 covers days O P S W X and adds 160 weight to the luggage.\nKit S94 covers days J K L M N O P and adds 609 weight to the luggage.\nKit S95 covers days J K N P Q R and adds 36 weight to the luggage.\nKit S96 covers days F H M and adds 93 weight to the luggage.\nKit S97 covers days J K N P and adds 376 weight to the luggage.\nKit S98 covers days N O P Q R S V and adds 343 weight to the luggage.\nKit S99 covers days N O Q R S and adds 325 weight to the luggage.\nKit S100 covers days A C D E F and adds 110 weight to the luggage.\nKit S101 covers days J L M N O and adds 325 weight to the luggage.\nKit S102 covers days J K L M O and adds 135 weight to the luggage.\nKit S103 covers days O S T U V W and adds 438 weight to the luggage.\nKit S104 covers days N O P Q S U and adds 516 weight to the luggage.\nKit S105 covers days Q V W Y and adds 164 weight to the luggage.\nKit S106 covers days Q S T U V and adds 200 weight to the luggage.\nKit S107 covers days T W X Y and adds 388 weight to the luggage.\nKit S108 covers days I J K L M N O and adds 119 weight to the luggage.\nKit S109 covers days E G I J K and adds 370 weight to the luggage.\nKit S110 covers days M N P Q R S and adds 570 weight to the luggage.\nKit S111 covers days A C E H and adds 24 weight to the luggage.\nKit S112 covers days R S T U V and adds 280 weight to the luggage.\nKit S113 covers days E G H I J M and adds 450 weight to the luggage.\nKit S114 covers days C E H and adds 75 weight to the luggage.\nKit S115 covers days U W Y and adds 93 weight to the luggage.\nKit S116 covers days O P Q S and adds 292 weight to the luggage.\nKit S117 covers days G I L O and adds 180 weight to the luggage.\nKit S118 covers days M O Q R S and adds 420 weight to the luggage.\nKit S119 covers days L M N P Q S T and adds 672 weight to the luggage.\nKit S120 covers days R T V W Y and adds 250 weight to the luggage.\nKit S121 covers days R S T V W X Y and adds 385 weight to the luggage.\nKit S122 covers days H I J K M N O and adds 259 weight to the luggage.\nKit S123 covers days P S T and adds 228 weight to the luggage.\nKit S124 covers days J N O P T and adds 5 weight to the luggage.\nKit S125 covers days L P R and adds 240 weight to the luggage.\nKit S126 covers days A and adds 10000 weight to the luggage.\nKit S127 covers days B and adds 10000 weight to the luggage.\nKit S128 covers days C and adds 10000 weight to the luggage.\nKit S129 covers days D and adds 10000 weight to the luggage.\nKit S130 covers days E and adds 10000 weight to the luggage.\nKit S131 covers days F and adds 10000 weight to the luggage.\nKit S132 covers days G and adds 10000 weight to the luggage.\nKit S133 covers days H and adds 10000 weight to the luggage.\nKit S134 covers days I and adds 10000 weight to the luggage.\nKit S135 covers days J and adds 10000 weight to the luggage.\nKit S136 covers days K and adds 10000 weight to the luggage.\nKit S137 covers days L and adds 10000 weight to the luggage.\nKit S138 covers days M and adds 10000 weight to the luggage.\nKit S139 covers days N and adds 10000 weight to the luggage.\nKit S140 covers days O and adds 10000 weight to the luggage.\nKit S141 covers days P and adds 10000 weight to the luggage.\nKit S142 covers days Q and adds 10000 weight to the luggage.\nKit S143 covers days R and adds 10000 weight to the luggage.\nKit S144 covers days S and adds 10000 weight to the luggage.\nKit S145 covers days T and adds 10000 weight to the luggage.\nKit S146 covers days U and adds 10000 weight to the luggage.\nKit S147 covers days V and adds 10000 weight to the luggage.\nKit S148 covers days W and adds 10000 weight to the luggage.\nKit S149 covers days X and adds 10000 weight to the luggage.\nKit S150 covers days Y and adds 10000 weight to the luggage.\nPick kits so each of the 25 days is covered exactly once without overlap, and aim for the lowest total weight.\n\nIf you want to hand me the chosen kits in a simple, predictable way, just use this little JSON shape when you reply:\n\n{\n \"solution\": [\"kit_id\", ...]\n}\n\n\"solution\" is the list of the outfit kit identifiers you picked — one entry per chosen kit. Think of it like filling in a short form: drop in the exact kit IDs from the instance so I can read them straight away. This is just a sketch of the shape I expect, not the actual answer.\n\nPlease make sure every identifier is used exactly as it appears in the instance input — no renaming and no new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 25, "num_sets": 150, "density": 0.17146666666666666, "sets": [ { "id": 1, "elements": [ 17, 18, 20, 23 ], "cost": 340 }, { "id": 2, "elements": [ 6, 8, 10, 12, 13 ], "cost": 315 }, { "id": 3, "elements": [ 14, 15, 16, 17, 18, 19, 20 ], "cost": 651 }, { "id": 4, "elements": [ 12, 14, 15 ], "cost": 297 }, { "id": 5, "elements": [ 1, 4, 6, 8 ], "cost": 212 }, { "id": 6, "elements": [ 5, 13, 14, 16 ], "cost": 380 }, { "id": 7, "elements": [ 17, 18, 20, 21, 23 ], "cost": 440 }, { "id": 8, "elements": [ 9, 10, 11, 12, 13, 14, 16 ], "cost": 35 }, { "id": 9, "elements": [ 15, 16, 17, 18, 19 ], "cost": 340 }, { "id": 10, "elements": [ 16, 20, 21, 23, 24 ], "cost": 70 }, { "id": 11, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 582 }, { "id": 12, "elements": [ 12, 13, 14, 16, 18, 19 ], "cost": 282 }, { "id": 13, "elements": [ 13, 16, 18, 19, 21, 22 ], "cost": 240 }, { "id": 14, "elements": [ 12, 13, 14, 15, 19 ], "cost": 85 }, { "id": 15, "elements": [ 6, 8, 9, 10, 12, 15 ], "cost": 90 }, { "id": 16, "elements": [ 1, 3, 4, 5, 6, 7, 8 ], "cost": 595 }, { "id": 17, "elements": [ 20, 22, 23, 24 ], "cost": 220 }, { "id": 18, "elements": [ 17, 19, 20, 21, 22, 25 ], "cost": 582 }, { "id": 19, "elements": [ 20, 22, 24, 25 ], "cost": 240 }, { "id": 20, "elements": [ 17, 19, 20, 21, 23 ], "cost": 370 }, { "id": 21, "elements": [ 10, 12, 18 ], "cost": 264 }, { "id": 22, "elements": [ 2, 4, 5 ], "cost": 54 }, { "id": 23, "elements": [ 5, 10, 11, 13 ], "cost": 64 }, { "id": 24, "elements": [ 11, 13, 15 ], "cost": 9 }, { "id": 25, "elements": [ 5, 8, 10, 11 ], "cost": 184 }, { "id": 26, "elements": [ 17, 18, 19, 21, 22, 24, 25 ], "cost": 84 }, { "id": 27, "elements": [ 18, 19, 20, 21, 23, 24, 25 ], "cost": 28 }, { "id": 28, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 30 }, { "id": 29, "elements": [ 2, 3, 4, 10 ], "cost": 220 }, { "id": 30, "elements": [ 12, 13, 14, 16, 17, 19 ], "cost": 108 }, { "id": 31, "elements": [ 11, 12, 15, 17 ], "cost": 344 }, { "id": 32, "elements": [ 17, 18, 19, 20, 21, 22 ], "cost": 258 }, { "id": 33, "elements": [ 14, 17, 20, 22 ], "cost": 208 }, { "id": 34, "elements": [ 21, 22, 23, 24, 25 ], "cost": 425 }, { "id": 35, "elements": [ 1, 2, 3, 5 ], "cost": 304 }, { "id": 36, "elements": [ 12, 13, 14, 16 ], "cost": 324 }, { "id": 37, "elements": [ 19, 20, 21, 25 ], "cost": 256 }, { "id": 38, "elements": [ 15, 16, 17, 20 ], "cost": 324 }, { "id": 39, "elements": [ 4, 5, 6, 8, 9, 10, 11 ], "cost": 448 }, { "id": 40, "elements": [ 7, 8, 10, 11, 13 ], "cost": 25 }, { "id": 41, "elements": [ 4, 6, 10, 12 ], "cost": 264 }, { "id": 42, "elements": [ 9, 11, 12, 13, 14, 15, 18 ], "cost": 420 }, { "id": 43, "elements": [ 12, 16, 17, 19, 21 ], "cost": 65 }, { "id": 44, "elements": [ 1, 3, 5 ], "cost": 141 }, { "id": 45, "elements": [ 7, 12, 13, 14 ], "cost": 32 }, { "id": 46, "elements": [ 17, 18, 20, 22, 23 ], "cost": 20 }, { "id": 47, "elements": [ 2, 3, 4, 7, 9 ], "cost": 310 }, { "id": 48, "elements": [ 2, 3, 5, 6 ], "cost": 336 }, { "id": 49, "elements": [ 19, 20, 21, 22, 23, 24, 25 ], "cost": 245 }, { "id": 50, "elements": [ 11, 12, 14, 15, 16, 17, 19 ], "cost": 238 }, { "id": 51, "elements": [ 9, 11, 12, 13, 14, 15 ], "cost": 384 }, { "id": 52, "elements": [ 6, 7, 9, 11, 12, 13 ], "cost": 114 }, { "id": 53, "elements": [ 6, 7, 8, 10, 11, 12, 13 ], "cost": 98 }, { "id": 54, "elements": [ 4, 7, 9, 10, 11 ], "cost": 250 }, { "id": 55, "elements": [ 9, 10, 12, 13 ], "cost": 64 }, { "id": 56, "elements": [ 6, 9, 10, 11, 12, 13, 15 ], "cost": 588 }, { "id": 57, "elements": [ 20, 21, 23, 25 ], "cost": 368 }, { "id": 58, "elements": [ 12, 13, 14, 15, 16, 17, 18 ], "cost": 238 }, { "id": 59, "elements": [ 21, 22, 24, 25 ], "cost": 72 }, { "id": 60, "elements": [ 1, 2, 4 ], "cost": 66 }, { "id": 61, "elements": [ 7, 10, 11 ], "cost": 270 }, { "id": 62, "elements": [ 16, 18, 20, 21, 22 ], "cost": 320 }, { "id": 63, "elements": [ 7, 9, 12 ], "cost": 72 }, { "id": 64, "elements": [ 15, 18, 19 ], "cost": 12 }, { "id": 65, "elements": [ 15, 18, 20, 21, 22, 24 ], "cost": 180 }, { "id": 66, "elements": [ 6, 8, 11 ], "cost": 108 }, { "id": 67, "elements": [ 19, 20, 22, 23 ], "cost": 368 }, { "id": 68, "elements": [ 15, 16, 18 ], "cost": 252 }, { "id": 69, "elements": [ 1, 2, 4, 5, 6, 7 ], "cost": 144 }, { "id": 70, "elements": [ 17, 21, 22, 23 ], "cost": 108 }, { "id": 71, "elements": [ 12, 13, 14 ], "cost": 201 }, { "id": 72, "elements": [ 6, 7, 9, 10, 11, 14 ], "cost": 426 }, { "id": 73, "elements": [ 1, 2, 4, 5, 6 ], "cost": 220 }, { "id": 74, "elements": [ 22, 24, 25 ], "cost": 21 }, { "id": 75, "elements": [ 19, 20, 21, 22, 23 ], "cost": 220 }, { "id": 76, "elements": [ 5, 6, 9, 10, 13, 15 ], "cost": 480 }, { "id": 77, "elements": [ 1, 2, 3, 4, 5, 7, 10 ], "cost": 497 }, { "id": 78, "elements": [ 18, 20, 21, 25 ], "cost": 208 }, { "id": 79, "elements": [ 13, 14, 19 ], "cost": 114 }, { "id": 80, "elements": [ 4, 7, 8, 9, 10, 11 ], "cost": 294 }, { "id": 81, "elements": [ 1, 2, 3, 5, 6 ], "cost": 330 }, { "id": 82, "elements": [ 20, 21, 22, 23, 24, 25 ], "cost": 72 }, { "id": 83, "elements": [ 20, 22, 23 ], "cost": 165 }, { "id": 84, "elements": [ 2, 3, 4, 5 ], "cost": 316 }, { "id": 85, "elements": [ 3, 5, 6, 7, 8, 10, 12 ], "cost": 245 }, { "id": 86, "elements": [ 19, 22, 24, 25 ], "cost": 128 }, { "id": 87, "elements": [ 8, 10, 12, 13, 14, 15, 17 ], "cost": 462 }, { "id": 88, "elements": [ 7, 9, 11, 13, 15, 16, 18 ], "cost": 700 }, { "id": 89, "elements": [ 5, 8, 9, 11, 12, 13 ], "cost": 96 }, { "id": 90, "elements": [ 16, 18, 19 ], "cost": 60 }, { "id": 91, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 210 }, { "id": 92, "elements": [ 8, 9, 12 ], "cost": 3 }, { "id": 93, "elements": [ 15, 16, 19, 23, 24 ], "cost": 160 }, { "id": 94, "elements": [ 10, 11, 12, 13, 14, 15, 16 ], "cost": 609 }, { "id": 95, "elements": [ 10, 11, 14, 16, 17, 18 ], "cost": 36 }, { "id": 96, "elements": [ 6, 8, 13 ], "cost": 93 }, { "id": 97, "elements": [ 10, 11, 14, 16 ], "cost": 376 }, { "id": 98, "elements": [ 14, 15, 16, 17, 18, 19, 22 ], "cost": 343 }, { "id": 99, "elements": [ 14, 15, 17, 18, 19 ], "cost": 325 }, { "id": 100, "elements": [ 1, 3, 4, 5, 6 ], "cost": 110 }, { "id": 101, "elements": [ 10, 12, 13, 14, 15 ], "cost": 325 }, { "id": 102, "elements": [ 10, 11, 12, 13, 15 ], "cost": 135 }, { "id": 103, "elements": [ 15, 19, 20, 21, 22, 23 ], "cost": 438 }, { "id": 104, "elements": [ 14, 15, 16, 17, 19, 21 ], "cost": 516 }, { "id": 105, "elements": [ 17, 22, 23, 25 ], "cost": 164 }, { "id": 106, "elements": [ 17, 19, 20, 21, 22 ], "cost": 200 }, { "id": 107, "elements": [ 20, 23, 24, 25 ], "cost": 388 }, { "id": 108, "elements": [ 9, 10, 11, 12, 13, 14, 15 ], "cost": 119 }, { "id": 109, "elements": [ 5, 7, 9, 10, 11 ], "cost": 370 }, { "id": 110, "elements": [ 13, 14, 16, 17, 18, 19 ], "cost": 570 }, { "id": 111, "elements": [ 1, 3, 5, 8 ], "cost": 24 }, { "id": 112, "elements": [ 18, 19, 20, 21, 22 ], "cost": 280 }, { "id": 113, "elements": [ 5, 7, 8, 9, 10, 13 ], "cost": 450 }, { "id": 114, "elements": [ 3, 5, 8 ], "cost": 75 }, { "id": 115, "elements": [ 21, 23, 25 ], "cost": 93 }, { "id": 116, "elements": [ 15, 16, 17, 19 ], "cost": 292 }, { "id": 117, "elements": [ 7, 9, 12, 15 ], "cost": 180 }, { "id": 118, "elements": [ 13, 15, 17, 18, 19 ], "cost": 420 }, { "id": 119, "elements": [ 12, 13, 14, 16, 17, 19, 20 ], "cost": 672 }, { "id": 120, "elements": [ 18, 20, 22, 23, 25 ], "cost": 250 }, { "id": 121, "elements": [ 18, 19, 20, 22, 23, 24, 25 ], "cost": 385 }, { "id": 122, "elements": [ 8, 9, 10, 11, 13, 14, 15 ], "cost": 259 }, { "id": 123, "elements": [ 16, 19, 20 ], "cost": 228 }, { "id": 124, "elements": [ 10, 14, 15, 16, 20 ], "cost": 5 }, { "id": 125, "elements": [ 12, 16, 18 ], "cost": 240 }, { "id": 126, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 127, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 128, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 129, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 130, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 131, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 132, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 133, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 134, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 135, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 136, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 137, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 138, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 139, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 140, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 141, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 142, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 143, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 144, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 145, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 146, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 147, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 148, "elements": [ 23 ], "cost": 10000, "singleton": true }, { "id": 149, "elements": [ 24 ], "cost": 10000, "singleton": true }, { "id": 150, "elements": [ 25 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0042_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0042_bag_stats.png" }, "solution": [ 82, 91, 92, 97, 118 ], "obj": 1081.0, "instance_variant": { "num_elements": 25, "num_sets": 150, "sets": [ { "id": "S1", "elements": [ "Q", "R", "T", "W" ], "cost": 340 }, { "id": "S2", "elements": [ "F", "H", "J", "L", "M" ], "cost": 315 }, { "id": "S3", "elements": [ "N", "O", "P", "Q", "R", "S", "T" ], "cost": 651 }, { "id": "S4", "elements": [ "L", "N", "O" ], "cost": 297 }, { "id": "S5", "elements": [ "A", "D", "F", "H" ], "cost": 212 }, { "id": "S6", "elements": [ "E", "M", "N", "P" ], "cost": 380 }, { "id": "S7", "elements": [ "Q", "R", "T", "U", "W" ], "cost": 440 }, { "id": "S8", "elements": [ "I", "J", "K", "L", "M", "N", "P" ], "cost": 35 }, { "id": "S9", "elements": [ "O", "P", "Q", "R", "S" ], "cost": 340 }, { "id": "S10", "elements": [ "P", "T", "U", "W", "X" ], "cost": 70 }, { "id": "S11", "elements": [ "A", "B", "C", "D", "E", "G" ], "cost": 582 }, { "id": "S12", "elements": [ "L", "M", "N", "P", "R", "S" ], "cost": 282 }, { "id": "S13", "elements": [ "M", "P", "R", "S", "U", "V" ], "cost": 240 }, { "id": "S14", "elements": [ "L", "M", "N", "O", "S" ], "cost": 85 }, { "id": "S15", "elements": [ "F", "H", "I", "J", "L", "O" ], "cost": 90 }, { "id": "S16", "elements": [ "A", "C", "D", "E", "F", "G", "H" ], "cost": 595 }, { "id": "S17", "elements": [ "T", "V", "W", "X" ], "cost": 220 }, { "id": "S18", "elements": [ "Q", "S", "T", "U", "V", "Y" ], "cost": 582 }, { "id": "S19", "elements": [ "T", "V", "X", "Y" ], "cost": 240 }, { "id": "S20", "elements": [ "Q", "S", "T", "U", "W" ], "cost": 370 }, { "id": "S21", "elements": [ "J", "L", "R" ], "cost": 264 }, { "id": "S22", "elements": [ "B", "D", "E" ], "cost": 54 }, { "id": "S23", "elements": [ "E", "J", "K", "M" ], "cost": 64 }, { "id": "S24", "elements": [ "K", "M", "O" ], "cost": 9 }, { "id": "S25", "elements": [ "E", "H", "J", "K" ], "cost": 184 }, { "id": "S26", "elements": [ "Q", "R", "S", "U", "V", "X", "Y" ], "cost": 84 }, { "id": "S27", "elements": [ "R", "S", "T", "U", "W", "X", "Y" ], "cost": 28 }, { "id": "S28", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 30 }, { "id": "S29", "elements": [ "B", "C", "D", "J" ], "cost": 220 }, { "id": "S30", "elements": [ "L", "M", "N", "P", "Q", "S" ], "cost": 108 }, { "id": "S31", "elements": [ "K", "L", "O", "Q" ], "cost": 344 }, { "id": "S32", "elements": [ "Q", "R", "S", "T", "U", "V" ], "cost": 258 }, { "id": "S33", "elements": [ "N", "Q", "T", "V" ], "cost": 208 }, { "id": "S34", "elements": [ "U", "V", "W", "X", "Y" ], "cost": 425 }, { "id": "S35", "elements": [ "A", "B", "C", "E" ], "cost": 304 }, { "id": "S36", "elements": [ "L", "M", "N", "P" ], "cost": 324 }, { "id": "S37", "elements": [ "S", "T", "U", "Y" ], "cost": 256 }, { "id": "S38", "elements": [ "O", "P", "Q", "T" ], "cost": 324 }, { "id": "S39", "elements": [ "D", "E", "F", "H", "I", "J", "K" ], "cost": 448 }, { "id": "S40", "elements": [ "G", "H", "J", "K", "M" ], "cost": 25 }, { "id": "S41", "elements": [ "D", "F", "J", "L" ], "cost": 264 }, { "id": "S42", "elements": [ "I", "K", "L", "M", "N", "O", "R" ], "cost": 420 }, { "id": "S43", "elements": [ "L", "P", "Q", "S", "U" ], "cost": 65 }, { "id": "S44", "elements": [ "A", "C", "E" ], "cost": 141 }, { "id": "S45", "elements": [ "G", "L", "M", "N" ], "cost": 32 }, { "id": "S46", "elements": [ "Q", "R", "T", "V", "W" ], "cost": 20 }, { "id": "S47", "elements": [ "B", "C", "D", "G", "I" ], "cost": 310 }, { "id": "S48", "elements": [ "B", "C", "E", "F" ], "cost": 336 }, { "id": "S49", "elements": [ "S", "T", "U", "V", "W", "X", "Y" ], "cost": 245 }, { "id": "S50", "elements": [ "K", "L", "N", "O", "P", "Q", "S" ], "cost": 238 }, { "id": "S51", "elements": [ "I", "K", "L", "M", "N", "O" ], "cost": 384 }, { "id": "S52", "elements": [ "F", "G", "I", "K", "L", "M" ], "cost": 114 }, { "id": "S53", "elements": [ "F", "G", "H", "J", "K", "L", "M" ], "cost": 98 }, { "id": "S54", "elements": [ "D", "G", "I", "J", "K" ], "cost": 250 }, { "id": "S55", "elements": [ "I", "J", "L", "M" ], "cost": 64 }, { "id": "S56", "elements": [ "F", "I", "J", "K", "L", "M", "O" ], "cost": 588 }, { "id": "S57", "elements": [ "T", "U", "W", "Y" ], "cost": 368 }, { "id": "S58", "elements": [ "L", "M", "N", "O", "P", "Q", "R" ], "cost": 238 }, { "id": "S59", "elements": [ "U", "V", "X", "Y" ], "cost": 72 }, { "id": "S60", "elements": [ "A", "B", "D" ], "cost": 66 }, { "id": "S61", "elements": [ "G", "J", "K" ], "cost": 270 }, { "id": "S62", "elements": [ "P", "R", "T", "U", "V" ], "cost": 320 }, { "id": "S63", "elements": [ "G", "I", "L" ], "cost": 72 }, { "id": "S64", "elements": [ "O", "R", "S" ], "cost": 12 }, { "id": "S65", "elements": [ "O", "R", "T", "U", "V", "X" ], "cost": 180 }, { "id": "S66", "elements": [ "F", "H", "K" ], "cost": 108 }, { "id": "S67", "elements": [ "S", "T", "V", "W" ], "cost": 368 }, { "id": "S68", "elements": [ "O", "P", "R" ], "cost": 252 }, { "id": "S69", "elements": [ "A", "B", "D", "E", "F", "G" ], "cost": 144 }, { "id": "S70", "elements": [ "Q", "U", "V", "W" ], "cost": 108 }, { "id": "S71", "elements": [ "L", "M", "N" ], "cost": 201 }, { "id": "S72", "elements": [ "F", "G", "I", "J", "K", "N" ], "cost": 426 }, { "id": "S73", "elements": [ "A", "B", "D", "E", "F" ], "cost": 220 }, { "id": "S74", "elements": [ "V", "X", "Y" ], "cost": 21 }, { "id": "S75", "elements": [ "S", "T", "U", "V", "W" ], "cost": 220 }, { "id": "S76", "elements": [ "E", "F", "I", "J", "M", "O" ], "cost": 480 }, { "id": "S77", "elements": [ "A", "B", "C", "D", "E", "G", "J" ], "cost": 497 }, { "id": "S78", "elements": [ "R", "T", "U", "Y" ], "cost": 208 }, { "id": "S79", "elements": [ "M", "N", "S" ], "cost": 114 }, { "id": "S80", "elements": [ "D", "G", "H", "I", "J", "K" ], "cost": 294 }, { "id": "S81", "elements": [ "A", "B", "C", "E", "F" ], "cost": 330 }, { "id": "S82", "elements": [ "T", "U", "V", "W", "X", "Y" ], "cost": 72 }, { "id": "S83", "elements": [ "T", "V", "W" ], "cost": 165 }, { "id": "S84", "elements": [ "B", "C", "D", "E" ], "cost": 316 }, { "id": "S85", "elements": [ "C", "E", "F", "G", "H", "J", "L" ], "cost": 245 }, { "id": "S86", "elements": [ "S", "V", "X", "Y" ], "cost": 128 }, { "id": "S87", "elements": [ "H", "J", "L", "M", "N", "O", "Q" ], "cost": 462 }, { "id": "S88", "elements": [ "G", "I", "K", "M", "O", "P", "R" ], "cost": 700 }, { "id": "S89", "elements": [ "E", "H", "I", "K", "L", "M" ], "cost": 96 }, { "id": "S90", "elements": [ "P", "R", "S" ], "cost": 60 }, { "id": "S91", "elements": [ "A", "B", "C", "D", "E", "F", "G" ], "cost": 210 }, { "id": "S92", "elements": [ "H", "I", "L" ], "cost": 3 }, { "id": "S93", "elements": [ "O", "P", "S", "W", "X" ], "cost": 160 }, { "id": "S94", "elements": [ "J", "K", "L", "M", "N", "O", "P" ], "cost": 609 }, { "id": "S95", "elements": [ "J", "K", "N", "P", "Q", "R" ], "cost": 36 }, { "id": "S96", "elements": [ "F", "H", "M" ], "cost": 93 }, { "id": "S97", "elements": [ "J", "K", "N", "P" ], "cost": 376 }, { "id": "S98", "elements": [ "N", "O", "P", "Q", "R", "S", "V" ], "cost": 343 }, { "id": "S99", "elements": [ "N", "O", "Q", "R", "S" ], "cost": 325 }, { "id": "S100", "elements": [ "A", "C", "D", "E", "F" ], "cost": 110 }, { "id": "S101", "elements": [ "J", "L", "M", "N", "O" ], "cost": 325 }, { "id": "S102", "elements": [ "J", "K", "L", "M", "O" ], "cost": 135 }, { "id": "S103", "elements": [ "O", "S", "T", "U", "V", "W" ], "cost": 438 }, { "id": "S104", "elements": [ "N", "O", "P", "Q", "S", "U" ], "cost": 516 }, { "id": "S105", "elements": [ "Q", "V", "W", "Y" ], "cost": 164 }, { "id": "S106", "elements": [ "Q", "S", "T", "U", "V" ], "cost": 200 }, { "id": "S107", "elements": [ "T", "W", "X", "Y" ], "cost": 388 }, { "id": "S108", "elements": [ "I", "J", "K", "L", "M", "N", "O" ], "cost": 119 }, { "id": "S109", "elements": [ "E", "G", "I", "J", "K" ], "cost": 370 }, { "id": "S110", "elements": [ "M", "N", "P", "Q", "R", "S" ], "cost": 570 }, { "id": "S111", "elements": [ "A", "C", "E", "H" ], "cost": 24 }, { "id": "S112", "elements": [ "R", "S", "T", "U", "V" ], "cost": 280 }, { "id": "S113", "elements": [ "E", "G", "H", "I", "J", "M" ], "cost": 450 }, { "id": "S114", "elements": [ "C", "E", "H" ], "cost": 75 }, { "id": "S115", "elements": [ "U", "W", "Y" ], "cost": 93 }, { "id": "S116", "elements": [ "O", "P", "Q", "S" ], "cost": 292 }, { "id": "S117", "elements": [ "G", "I", "L", "O" ], "cost": 180 }, { "id": "S118", "elements": [ "M", "O", "Q", "R", "S" ], "cost": 420 }, { "id": "S119", "elements": [ "L", "M", "N", "P", "Q", "S", "T" ], "cost": 672 }, { "id": "S120", "elements": [ "R", "T", "V", "W", "Y" ], "cost": 250 }, { "id": "S121", "elements": [ "R", "S", "T", "V", "W", "X", "Y" ], "cost": 385 }, { "id": "S122", "elements": [ "H", "I", "J", "K", "M", "N", "O" ], "cost": 259 }, { "id": "S123", "elements": [ "P", "S", "T" ], "cost": 228 }, { "id": "S124", "elements": [ "J", "N", "O", "P", "T" ], "cost": 5 }, { "id": "S125", "elements": [ "L", "P", "R" ], "cost": 240 }, { "id": "S126", "elements": [ "A" ], "cost": 10000 }, { "id": "S127", "elements": [ "B" ], "cost": 10000 }, { "id": "S128", "elements": [ "C" ], "cost": 10000 }, { "id": "S129", "elements": [ "D" ], "cost": 10000 }, { "id": "S130", "elements": [ "E" ], "cost": 10000 }, { "id": "S131", "elements": [ "F" ], "cost": 10000 }, { "id": "S132", "elements": [ "G" ], "cost": 10000 }, { "id": "S133", "elements": [ "H" ], "cost": 10000 }, { "id": "S134", "elements": [ "I" ], "cost": 10000 }, { "id": "S135", "elements": [ "J" ], "cost": 10000 }, { "id": "S136", "elements": [ "K" ], "cost": 10000 }, { "id": "S137", "elements": [ "L" ], "cost": 10000 }, { "id": "S138", "elements": [ "M" ], "cost": 10000 }, { "id": "S139", "elements": [ "N" ], "cost": 10000 }, { "id": "S140", "elements": [ "O" ], "cost": 10000 }, { "id": "S141", "elements": [ "P" ], "cost": 10000 }, { "id": "S142", "elements": [ "Q" ], "cost": 10000 }, { "id": "S143", "elements": [ "R" ], "cost": 10000 }, { "id": "S144", "elements": [ "S" ], "cost": 10000 }, { "id": "S145", "elements": [ "T" ], "cost": 10000 }, { "id": "S146", "elements": [ "U" ], "cost": 10000 }, { "id": "S147", "elements": [ "V" ], "cost": 10000 }, { "id": "S148", "elements": [ "W" ], "cost": 10000 }, { "id": "S149", "elements": [ "X" ], "cost": 10000 }, { "id": "S150", "elements": [ "Y" ], "cost": 10000 } ] }, "solution_variant": [ "S82", "S91", "S92", "S97", "S118" ], "context_index": 43, "input_format": "nl", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "I’m running the relief side of things and need to decide which pre-packed medical boxes to send out so that every kind of treatment needed in the field is included once and only once — no two boxes should cover the same treatment, and nothing can be left out. The aim is to keep the total shipping bill as small as possible, which is just the sum of the shipping fees for whatever boxes get sent. Concrete details about the available packs and their costs are listed below.\n\nI have 24 treatment categories to cover exactly once and 135 pre-packed medical boxes available; details follow:\nPack S1: covers treatments C E H; shipping fee 30.\nPack S2: covers treatments R S T U W; shipping fee 465.\nPack S3: covers treatments R T V W; shipping fee 112.\nPack S4: covers treatments F G H I M; shipping fee 470.\nPack S5: covers treatments B D F; shipping fee 108.\nPack S6: covers treatments B D; shipping fee 154.\nPack S7: covers treatments F G H I J K; shipping fee 582.\nPack S8: covers treatments A B C D E G H; shipping fee 462.\nPack S9: covers treatments S T V; shipping fee 180.\nPack S10: covers treatments G H K; shipping fee 36.\nPack S11: covers treatments O T U; shipping fee 69.\nPack S12: covers treatments I L M O; shipping fee 252.\nPack S13: covers treatments P S T U V; shipping fee 85.\nPack S14: covers treatments N O P S T; shipping fee 65.\nPack S15: covers treatments J L P; shipping fee 96.\nPack S16: covers treatments N P R S U; shipping fee 90.\nPack S17: covers treatments H J K L M N O; shipping fee 392.\nPack S18: covers treatments T U V; shipping fee 129.\nPack S19: covers treatments H J K L N; shipping fee 85.\nPack S20: covers treatments P R T V W; shipping fee 285.\nPack S21: covers treatments F H I J; shipping fee 48.\nPack S22: covers treatments A C F G H J L; shipping fee 357.\nPack S23: covers treatments R S T U V W X; shipping fee 196.\nPack S24: covers treatments H J K L; shipping fee 216.\nPack S25: covers treatments G I J K N; shipping fee 300.\nPack S26: covers treatments D E F I J; shipping fee 300.\nPack S27: covers treatments S T U V W; shipping fee 70.\nPack S28: covers treatments E F I; shipping fee 6.\nPack S29: covers treatments A C E; shipping fee 54.\nPack S30: covers treatments U V W X; shipping fee 172.\nPack S31: covers treatments I J L N; shipping fee 240.\nPack S32: covers treatments E G H I J K; shipping fee 276.\nPack S33: covers treatments Q S T U W X; shipping fee 318.\nPack S34: covers treatments H I J M N P; shipping fee 396.\nPack S35: covers treatments T V W X; shipping fee 364.\nPack S36: covers treatments F K L M N O; shipping fee 594.\nPack S37: covers treatments E F G H I J K; shipping fee 574.\nPack S38: covers treatments D E F G H I; shipping fee 330.\nPack S39: covers treatments A B C E; shipping fee 252.\nPack S40: covers treatments P S T; shipping fee 78.\nPack S41: covers treatments B C D J; shipping fee 312.\nPack S42: covers treatments A C D F H; shipping fee 90.\nPack S43: covers treatments A B C D E; shipping fee 390.\nPack S44: covers treatments R S V; shipping fee 255.\nPack S45: covers treatments K O Q T; shipping fee 228.\nPack S46: covers treatments I J K L M; shipping fee 145.\nPack S47: covers treatments S U V W; shipping fee 268.\nPack S48: covers treatments A G H I; shipping fee 168.\nPack S49: covers treatments C E F G; shipping fee 96.\nPack S50: covers treatments P S T U; shipping fee 168.\nPack S51: covers treatments D E F G I; shipping fee 350.\nPack S52: covers treatments P Q S T U V; shipping fee 522.\nPack S53: covers treatments E G H J L N; shipping fee 234.\nPack S54: covers treatments S U V W X; shipping fee 325.\nPack S55: covers treatments L M N O Q R; shipping fee 132.\nPack S56: covers treatments Q S T U V X; shipping fee 72.\nPack S57: covers treatments H I M O; shipping fee 336.\nPack S58: covers treatments A D E F; shipping fee 100.\nPack S59: covers treatments N O P Q; shipping fee 108.\nPack S60: covers treatments O P Q R S; shipping fee 60.\nPack S61: covers treatments B D E F G H; shipping fee 24.\nPack S62: covers treatments B C; shipping fee 40.\nPack S63: covers treatments E H I; shipping fee 264.\nPack S64: covers treatments A B E; shipping fee 33.\nPack S65: covers treatments A B D; shipping fee 165.\nPack S66: covers treatments A B C E G; shipping fee 160.\nPack S67: covers treatments N O S; shipping fee 270.\nPack S68: covers treatments A B C D E F; shipping fee 360.\nPack S69: covers treatments H L M N; shipping fee 152.\nPack S70: covers treatments M O Q S T W; shipping fee 60.\nPack S71: covers treatments S T U W X; shipping fee 25.\nPack S72: covers treatments A B C E F H; shipping fee 564.\nPack S73: covers treatments R U V; shipping fee 81.\nPack S74: covers treatments H L N; shipping fee 231.\nPack S75: covers treatments D E F G H I J; shipping fee 350.\nPack S76: covers treatments A C F; shipping fee 87.\nPack S77: covers treatments L M N; shipping fee 204.\nPack S78: covers treatments N P S; shipping fee 207.\nPack S79: covers treatments A D E I; shipping fee 260.\nPack S80: covers treatments C G H K; shipping fee 88.\nPack S81: covers treatments B C D G H I K; shipping fee 700.\nPack S82: covers treatments U V X; shipping fee 3.\nPack S83: covers treatments L O P Q R; shipping fee 320.\nPack S84: covers treatments M N O P Q; shipping fee 20.\nPack S85: covers treatments H K L M N Q; shipping fee 6.\nPack S86: covers treatments E F G; shipping fee 171.\nPack S87: covers treatments R U V X; shipping fee 284.\nPack S88: covers treatments A B C D E G I; shipping fee 350.\nPack S89: covers treatments A B C D F; shipping fee 205.\nPack S90: covers treatments T U W; shipping fee 183.\nPack S91: covers treatments S T U V W X; shipping fee 438.\nPack S92: covers treatments M Q R U; shipping fee 132.\nPack S93: covers treatments M N O Q R T; shipping fee 420.\nPack S94: covers treatments M N O Q T U W; shipping fee 630.\nPack S95: covers treatments D E I K; shipping fee 80.\nPack S96: covers treatments N O P Q S T; shipping fee 36.\nPack S97: covers treatments C E F; shipping fee 135.\nPack S98: covers treatments F G H I J K M; shipping fee 560.\nPack S99: covers treatments Q S T U V W X; shipping fee 455.\nPack S100: covers treatments L O P R S; shipping fee 220.\nPack S101: covers treatments Q S T V W X; shipping fee 390.\nPack S102: covers treatments L N O Q; shipping fee 148.\nPack S103: covers treatments F G I J L; shipping fee 255.\nPack S104: covers treatments H I K L M N; shipping fee 132.\nPack S105: covers treatments B C D E F J; shipping fee 6.\nPack S106: covers treatments P R S T U V W; shipping fee 126.\nPack S107: covers treatments K N O; shipping fee 132.\nPack S108: covers treatments L M Q; shipping fee 216.\nPack S109: covers treatments E F G H; shipping fee 188.\nPack S110: covers treatments B E F G H I; shipping fee 66.\nPack S111: covers treatments M N R; shipping fee 219.\nPack S112: covers treatments A; shipping fee 10000.\nPack S113: covers treatments B; shipping fee 10000.\nPack S114: covers treatments C; shipping fee 10000.\nPack S115: covers treatments D; shipping fee 10000.\nPack S116: covers treatments E; shipping fee 10000.\nPack S117: covers treatments F; shipping fee 10000.\nPack S118: covers treatments G; shipping fee 10000.\nPack S119: covers treatments H; shipping fee 10000.\nPack S120: covers treatments I; shipping fee 10000.\nPack S121: covers treatments J; shipping fee 10000.\nPack S122: covers treatments K; shipping fee 10000.\nPack S123: covers treatments L; shipping fee 10000.\nPack S124: covers treatments M; shipping fee 10000.\nPack S125: covers treatments N; shipping fee 10000.\nPack S126: covers treatments O; shipping fee 10000.\nPack S127: covers treatments P; shipping fee 10000.\nPack S128: covers treatments Q; shipping fee 10000.\nPack S129: covers treatments R; shipping fee 10000.\nPack S130: covers treatments S; shipping fee 10000.\nPack S131: covers treatments T; shipping fee 10000.\nPack S132: covers treatments U; shipping fee 10000.\nPack S133: covers treatments V; shipping fee 10000.\nPack S134: covers treatments W; shipping fee 10000.\nPack S135: covers treatments X; shipping fee 10000.\nI'll pick packs so each of the 24 treatments is covered exactly once while minimizing total shipping fees.\n\nAlso, when you send your pick, please use this simple JSON layout so it's easy to read and process — just drop the chosen box IDs in the array like this:\n\n{\n \"solution\": [\"box_id\", ...]\n}\n\nThis just shows the shape I expect: \"solution\" is the list of pre-packed medical boxes you want to send. The \"box_id\" placeholder stands in for whatever actual box IDs you pick from the instance; replace it with the real IDs when you answer. The JSON above is only a sketch of the expected format, not the final answer.\n\nAll identifiers must be used exactly as they appear in the instance input — no renaming and no new labels.\nValid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.", "instance": { "problem_type": "SPP", "num_elements": 24, "num_sets": 135, "density": 0.16666666666666666, "sets": [ { "id": 1, "elements": [ 3, 5, 8 ], "cost": 30 }, { "id": 2, "elements": [ 18, 19, 20, 21, 23 ], "cost": 465 }, { "id": 3, "elements": [ 18, 20, 22, 23 ], "cost": 112 }, { "id": 4, "elements": [ 6, 7, 8, 9, 13 ], "cost": 470 }, { "id": 5, "elements": [ 2, 4, 6 ], "cost": 108 }, { "id": 6, "elements": [ 2, 4 ], "cost": 154 }, { "id": 7, "elements": [ 6, 7, 8, 9, 10, 11 ], "cost": 582 }, { "id": 8, "elements": [ 1, 2, 3, 4, 5, 7, 8 ], "cost": 462 }, { "id": 9, "elements": [ 19, 20, 22 ], "cost": 180 }, { "id": 10, "elements": [ 7, 8, 11 ], "cost": 36 }, { "id": 11, "elements": [ 15, 20, 21 ], "cost": 69 }, { "id": 12, "elements": [ 9, 12, 13, 15 ], "cost": 252 }, { "id": 13, "elements": [ 16, 19, 20, 21, 22 ], "cost": 85 }, { "id": 14, "elements": [ 14, 15, 16, 19, 20 ], "cost": 65 }, { "id": 15, "elements": [ 10, 12, 16 ], "cost": 96 }, { "id": 16, "elements": [ 14, 16, 18, 19, 21 ], "cost": 90 }, { "id": 17, "elements": [ 8, 10, 11, 12, 13, 14, 15 ], "cost": 392 }, { "id": 18, "elements": [ 20, 21, 22 ], "cost": 129 }, { "id": 19, "elements": [ 8, 10, 11, 12, 14 ], "cost": 85 }, { "id": 20, "elements": [ 16, 18, 20, 22, 23 ], "cost": 285 }, { "id": 21, "elements": [ 6, 8, 9, 10 ], "cost": 48 }, { "id": 22, "elements": [ 1, 3, 6, 7, 8, 10, 12 ], "cost": 357 }, { "id": 23, "elements": [ 18, 19, 20, 21, 22, 23, 24 ], "cost": 196 }, { "id": 24, "elements": [ 8, 10, 11, 12 ], "cost": 216 }, { "id": 25, "elements": [ 7, 9, 10, 11, 14 ], "cost": 300 }, { "id": 26, "elements": [ 4, 5, 6, 9, 10 ], "cost": 300 }, { "id": 27, "elements": [ 19, 20, 21, 22, 23 ], "cost": 70 }, { "id": 28, "elements": [ 5, 6, 9 ], "cost": 6 }, { "id": 29, "elements": [ 1, 3, 5 ], "cost": 54 }, { "id": 30, "elements": [ 21, 22, 23, 24 ], "cost": 172 }, { "id": 31, "elements": [ 9, 10, 12, 14 ], "cost": 240 }, { "id": 32, "elements": [ 5, 7, 8, 9, 10, 11 ], "cost": 276 }, { "id": 33, "elements": [ 17, 19, 20, 21, 23, 24 ], "cost": 318 }, { "id": 34, "elements": [ 8, 9, 10, 13, 14, 16 ], "cost": 396 }, { "id": 35, "elements": [ 20, 22, 23, 24 ], "cost": 364 }, { "id": 36, "elements": [ 6, 11, 12, 13, 14, 15 ], "cost": 594 }, { "id": 37, "elements": [ 5, 6, 7, 8, 9, 10, 11 ], "cost": 574 }, { "id": 38, "elements": [ 4, 5, 6, 7, 8, 9 ], "cost": 330 }, { "id": 39, "elements": [ 1, 2, 3, 5 ], "cost": 252 }, { "id": 40, "elements": [ 16, 19, 20 ], "cost": 78 }, { "id": 41, "elements": [ 2, 3, 4, 10 ], "cost": 312 }, { "id": 42, "elements": [ 1, 3, 4, 6, 8 ], "cost": 90 }, { "id": 43, "elements": [ 1, 2, 3, 4, 5 ], "cost": 390 }, { "id": 44, "elements": [ 18, 19, 22 ], "cost": 255 }, { "id": 45, "elements": [ 11, 15, 17, 20 ], "cost": 228 }, { "id": 46, "elements": [ 9, 10, 11, 12, 13 ], "cost": 145 }, { "id": 47, "elements": [ 19, 21, 22, 23 ], "cost": 268 }, { "id": 48, "elements": [ 1, 7, 8, 9 ], "cost": 168 }, { "id": 49, "elements": [ 3, 5, 6, 7 ], "cost": 96 }, { "id": 50, "elements": [ 16, 19, 20, 21 ], "cost": 168 }, { "id": 51, "elements": [ 4, 5, 6, 7, 9 ], "cost": 350 }, { "id": 52, "elements": [ 16, 17, 19, 20, 21, 22 ], "cost": 522 }, { "id": 53, "elements": [ 5, 7, 8, 10, 12, 14 ], "cost": 234 }, { "id": 54, "elements": [ 19, 21, 22, 23, 24 ], "cost": 325 }, { "id": 55, "elements": [ 12, 13, 14, 15, 17, 18 ], "cost": 132 }, { "id": 56, "elements": [ 17, 19, 20, 21, 22, 24 ], "cost": 72 }, { "id": 57, "elements": [ 8, 9, 13, 15 ], "cost": 336 }, { "id": 58, "elements": [ 1, 4, 5, 6 ], "cost": 100 }, { "id": 59, "elements": [ 14, 15, 16, 17 ], "cost": 108 }, { "id": 60, "elements": [ 15, 16, 17, 18, 19 ], "cost": 60 }, { "id": 61, "elements": [ 2, 4, 5, 6, 7, 8 ], "cost": 24 }, { "id": 62, "elements": [ 2, 3 ], "cost": 40 }, { "id": 63, "elements": [ 5, 8, 9 ], "cost": 264 }, { "id": 64, "elements": [ 1, 2, 5 ], "cost": 33 }, { "id": 65, "elements": [ 1, 2, 4 ], "cost": 165 }, { "id": 66, "elements": [ 1, 2, 3, 5, 7 ], "cost": 160 }, { "id": 67, "elements": [ 14, 15, 19 ], "cost": 270 }, { "id": 68, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 360 }, { "id": 69, "elements": [ 8, 12, 13, 14 ], "cost": 152 }, { "id": 70, "elements": [ 13, 15, 17, 19, 20, 23 ], "cost": 60 }, { "id": 71, "elements": [ 19, 20, 21, 23, 24 ], "cost": 25 }, { "id": 72, "elements": [ 1, 2, 3, 5, 6, 8 ], "cost": 564 }, { "id": 73, "elements": [ 18, 21, 22 ], "cost": 81 }, { "id": 74, "elements": [ 8, 12, 14 ], "cost": 231 }, { "id": 75, "elements": [ 4, 5, 6, 7, 8, 9, 10 ], "cost": 350 }, { "id": 76, "elements": [ 1, 3, 6 ], "cost": 87 }, { "id": 77, "elements": [ 12, 13, 14 ], "cost": 204 }, { "id": 78, "elements": [ 14, 16, 19 ], "cost": 207 }, { "id": 79, "elements": [ 1, 4, 5, 9 ], "cost": 260 }, { "id": 80, "elements": [ 3, 7, 8, 11 ], "cost": 88 }, { "id": 81, "elements": [ 2, 3, 4, 7, 8, 9, 11 ], "cost": 700 }, { "id": 82, "elements": [ 21, 22, 24 ], "cost": 3 }, { "id": 83, "elements": [ 12, 15, 16, 17, 18 ], "cost": 320 }, { "id": 84, "elements": [ 13, 14, 15, 16, 17 ], "cost": 20 }, { "id": 85, "elements": [ 8, 11, 12, 13, 14, 17 ], "cost": 6 }, { "id": 86, "elements": [ 5, 6, 7 ], "cost": 171 }, { "id": 87, "elements": [ 18, 21, 22, 24 ], "cost": 284 }, { "id": 88, "elements": [ 1, 2, 3, 4, 5, 7, 9 ], "cost": 350 }, { "id": 89, "elements": [ 1, 2, 3, 4, 6 ], "cost": 205 }, { "id": 90, "elements": [ 20, 21, 23 ], "cost": 183 }, { "id": 91, "elements": [ 19, 20, 21, 22, 23, 24 ], "cost": 438 }, { "id": 92, "elements": [ 13, 17, 18, 21 ], "cost": 132 }, { "id": 93, "elements": [ 13, 14, 15, 17, 18, 20 ], "cost": 420 }, { "id": 94, "elements": [ 13, 14, 15, 17, 20, 21, 23 ], "cost": 630 }, { "id": 95, "elements": [ 4, 5, 9, 11 ], "cost": 80 }, { "id": 96, "elements": [ 14, 15, 16, 17, 19, 20 ], "cost": 36 }, { "id": 97, "elements": [ 3, 5, 6 ], "cost": 135 }, { "id": 98, "elements": [ 6, 7, 8, 9, 10, 11, 13 ], "cost": 560 }, { "id": 99, "elements": [ 17, 19, 20, 21, 22, 23, 24 ], "cost": 455 }, { "id": 100, "elements": [ 12, 15, 16, 18, 19 ], "cost": 220 }, { "id": 101, "elements": [ 17, 19, 20, 22, 23, 24 ], "cost": 390 }, { "id": 102, "elements": [ 12, 14, 15, 17 ], "cost": 148 }, { "id": 103, "elements": [ 6, 7, 9, 10, 12 ], "cost": 255 }, { "id": 104, "elements": [ 8, 9, 11, 12, 13, 14 ], "cost": 132 }, { "id": 105, "elements": [ 2, 3, 4, 5, 6, 10 ], "cost": 6 }, { "id": 106, "elements": [ 16, 18, 19, 20, 21, 22, 23 ], "cost": 126 }, { "id": 107, "elements": [ 11, 14, 15 ], "cost": 132 }, { "id": 108, "elements": [ 12, 13, 17 ], "cost": 216 }, { "id": 109, "elements": [ 5, 6, 7, 8 ], "cost": 188 }, { "id": 110, "elements": [ 2, 5, 6, 7, 8, 9 ], "cost": 66 }, { "id": 111, "elements": [ 13, 14, 18 ], "cost": 219 }, { "id": 112, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 118, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 119, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 120, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 121, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 122, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 123, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 124, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 125, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 126, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 127, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 128, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 129, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 130, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 131, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 132, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 133, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 134, "elements": [ 23 ], "cost": 10000, "singleton": true }, { "id": 135, "elements": [ 24 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0043_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0043_bag_stats.png" }, "solution": [ 30, 32, 40, 55, 89 ], "obj": 863.0, "instance_variant": { "num_elements": 24, "num_sets": 135, "sets": [ { "id": "S1", "elements": [ "C", "E", "H" ], "cost": 30 }, { "id": "S2", "elements": [ "R", "S", "T", "U", "W" ], "cost": 465 }, { "id": "S3", "elements": [ "R", "T", "V", "W" ], "cost": 112 }, { "id": "S4", "elements": [ "F", "G", "H", "I", "M" ], "cost": 470 }, { "id": "S5", "elements": [ "B", "D", "F" ], "cost": 108 }, { "id": "S6", "elements": [ "B", "D" ], "cost": 154 }, { "id": "S7", "elements": [ "F", "G", "H", "I", "J", "K" ], "cost": 582 }, { "id": "S8", "elements": [ "A", "B", "C", "D", "E", "G", "H" ], "cost": 462 }, { "id": "S9", "elements": [ "S", "T", "V" ], "cost": 180 }, { "id": "S10", "elements": [ "G", "H", "K" ], "cost": 36 }, { "id": "S11", "elements": [ "O", "T", "U" ], "cost": 69 }, { "id": "S12", "elements": [ "I", "L", "M", "O" ], "cost": 252 }, { "id": "S13", "elements": [ "P", "S", "T", "U", "V" ], "cost": 85 }, { "id": "S14", "elements": [ "N", "O", "P", "S", "T" ], "cost": 65 }, { "id": "S15", "elements": [ "J", "L", "P" ], "cost": 96 }, { "id": "S16", "elements": [ "N", "P", "R", "S", "U" ], "cost": 90 }, { "id": "S17", "elements": [ "H", "J", "K", "L", "M", "N", "O" ], "cost": 392 }, { "id": "S18", "elements": [ "T", "U", "V" ], "cost": 129 }, { "id": "S19", "elements": [ "H", "J", "K", "L", "N" ], "cost": 85 }, { "id": "S20", "elements": [ "P", "R", "T", "V", "W" ], "cost": 285 }, { "id": "S21", "elements": [ "F", "H", "I", "J" ], "cost": 48 }, { "id": "S22", "elements": [ "A", "C", "F", "G", "H", "J", "L" ], "cost": 357 }, { "id": "S23", "elements": [ "R", "S", "T", "U", "V", "W", "X" ], "cost": 196 }, { "id": "S24", "elements": [ "H", "J", "K", "L" ], "cost": 216 }, { "id": "S25", "elements": [ "G", "I", "J", "K", "N" ], "cost": 300 }, { "id": "S26", "elements": [ "D", "E", "F", "I", "J" ], "cost": 300 }, { "id": "S27", "elements": [ "S", "T", "U", "V", "W" ], "cost": 70 }, { "id": "S28", "elements": [ "E", "F", "I" ], "cost": 6 }, { "id": "S29", "elements": [ "A", "C", "E" ], "cost": 54 }, { "id": "S30", "elements": [ "U", "V", "W", "X" ], "cost": 172 }, { "id": "S31", "elements": [ "I", "J", "L", "N" ], "cost": 240 }, { "id": "S32", "elements": [ "E", "G", "H", "I", "J", "K" ], "cost": 276 }, { "id": "S33", "elements": [ "Q", "S", "T", "U", "W", "X" ], "cost": 318 }, { "id": "S34", "elements": [ "H", "I", "J", "M", "N", "P" ], "cost": 396 }, { "id": "S35", "elements": [ "T", "V", "W", "X" ], "cost": 364 }, { "id": "S36", "elements": [ "F", "K", "L", "M", "N", "O" ], "cost": 594 }, { "id": "S37", "elements": [ "E", "F", "G", "H", "I", "J", "K" ], "cost": 574 }, { "id": "S38", "elements": [ "D", "E", "F", "G", "H", "I" ], "cost": 330 }, { "id": "S39", "elements": [ "A", "B", "C", "E" ], "cost": 252 }, { "id": "S40", "elements": [ "P", "S", "T" ], "cost": 78 }, { "id": "S41", "elements": [ "B", "C", "D", "J" ], "cost": 312 }, { "id": "S42", "elements": [ "A", "C", "D", "F", "H" ], "cost": 90 }, { "id": "S43", "elements": [ "A", "B", "C", "D", "E" ], "cost": 390 }, { "id": "S44", "elements": [ "R", "S", "V" ], "cost": 255 }, { "id": "S45", "elements": [ "K", "O", "Q", "T" ], "cost": 228 }, { "id": "S46", "elements": [ "I", "J", "K", "L", "M" ], "cost": 145 }, { "id": "S47", "elements": [ "S", "U", "V", "W" ], "cost": 268 }, { "id": "S48", "elements": [ "A", "G", "H", "I" ], "cost": 168 }, { "id": "S49", "elements": [ "C", "E", "F", "G" ], "cost": 96 }, { "id": "S50", "elements": [ "P", "S", "T", "U" ], "cost": 168 }, { "id": "S51", "elements": [ "D", "E", "F", "G", "I" ], "cost": 350 }, { "id": "S52", "elements": [ "P", "Q", "S", "T", "U", "V" ], "cost": 522 }, { "id": "S53", "elements": [ "E", "G", "H", "J", "L", "N" ], "cost": 234 }, { "id": "S54", "elements": [ "S", "U", "V", "W", "X" ], "cost": 325 }, { "id": "S55", "elements": [ "L", "M", "N", "O", "Q", "R" ], "cost": 132 }, { "id": "S56", "elements": [ "Q", "S", "T", "U", "V", "X" ], "cost": 72 }, { "id": "S57", "elements": [ "H", "I", "M", "O" ], "cost": 336 }, { "id": "S58", "elements": [ "A", "D", "E", "F" ], "cost": 100 }, { "id": "S59", "elements": [ "N", "O", "P", "Q" ], "cost": 108 }, { "id": "S60", "elements": [ "O", "P", "Q", "R", "S" ], "cost": 60 }, { "id": "S61", "elements": [ "B", "D", "E", "F", "G", "H" ], "cost": 24 }, { "id": "S62", "elements": [ "B", "C" ], "cost": 40 }, { "id": "S63", "elements": [ "E", "H", "I" ], "cost": 264 }, { "id": "S64", "elements": [ "A", "B", "E" ], "cost": 33 }, { "id": "S65", "elements": [ "A", "B", "D" ], "cost": 165 }, { "id": "S66", "elements": [ "A", "B", "C", "E", "G" ], "cost": 160 }, { "id": "S67", "elements": [ "N", "O", "S" ], "cost": 270 }, { "id": "S68", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 360 }, { "id": "S69", "elements": [ "H", "L", "M", "N" ], "cost": 152 }, { "id": "S70", "elements": [ "M", "O", "Q", "S", "T", "W" ], "cost": 60 }, { "id": "S71", "elements": [ "S", "T", "U", "W", "X" ], "cost": 25 }, { "id": "S72", "elements": [ "A", "B", "C", "E", "F", "H" ], "cost": 564 }, { "id": "S73", "elements": [ "R", "U", "V" ], "cost": 81 }, { "id": "S74", "elements": [ "H", "L", "N" ], "cost": 231 }, { "id": "S75", "elements": [ "D", "E", "F", "G", "H", "I", "J" ], "cost": 350 }, { "id": "S76", "elements": [ "A", "C", "F" ], "cost": 87 }, { "id": "S77", "elements": [ "L", "M", "N" ], "cost": 204 }, { "id": "S78", "elements": [ "N", "P", "S" ], "cost": 207 }, { "id": "S79", "elements": [ "A", "D", "E", "I" ], "cost": 260 }, { "id": "S80", "elements": [ "C", "G", "H", "K" ], "cost": 88 }, { "id": "S81", "elements": [ "B", "C", "D", "G", "H", "I", "K" ], "cost": 700 }, { "id": "S82", "elements": [ "U", "V", "X" ], "cost": 3 }, { "id": "S83", "elements": [ "L", "O", "P", "Q", "R" ], "cost": 320 }, { "id": "S84", "elements": [ "M", "N", "O", "P", "Q" ], "cost": 20 }, { "id": "S85", "elements": [ "H", "K", "L", "M", "N", "Q" ], "cost": 6 }, { "id": "S86", "elements": [ "E", "F", "G" ], "cost": 171 }, { "id": "S87", "elements": [ "R", "U", "V", "X" ], "cost": 284 }, { "id": "S88", "elements": [ "A", "B", "C", "D", "E", "G", "I" ], "cost": 350 }, { "id": "S89", "elements": [ "A", "B", "C", "D", "F" ], "cost": 205 }, { "id": "S90", "elements": [ "T", "U", "W" ], "cost": 183 }, { "id": "S91", "elements": [ "S", "T", "U", "V", "W", "X" ], "cost": 438 }, { "id": "S92", "elements": [ "M", "Q", "R", "U" ], "cost": 132 }, { "id": "S93", "elements": [ "M", "N", "O", "Q", "R", "T" ], "cost": 420 }, { "id": "S94", "elements": [ "M", "N", "O", "Q", "T", "U", "W" ], "cost": 630 }, { "id": "S95", "elements": [ "D", "E", "I", "K" ], "cost": 80 }, { "id": "S96", "elements": [ "N", "O", "P", "Q", "S", "T" ], "cost": 36 }, { "id": "S97", "elements": [ "C", "E", "F" ], "cost": 135 }, { "id": "S98", "elements": [ "F", "G", "H", "I", "J", "K", "M" ], "cost": 560 }, { "id": "S99", "elements": [ "Q", "S", "T", "U", "V", "W", "X" ], "cost": 455 }, { "id": "S100", "elements": [ "L", "O", "P", "R", "S" ], "cost": 220 }, { "id": "S101", "elements": [ "Q", "S", "T", "V", "W", "X" ], "cost": 390 }, { "id": "S102", "elements": [ "L", "N", "O", "Q" ], "cost": 148 }, { "id": "S103", "elements": [ "F", "G", "I", "J", "L" ], "cost": 255 }, { "id": "S104", "elements": [ "H", "I", "K", "L", "M", "N" ], "cost": 132 }, { "id": "S105", "elements": [ "B", "C", "D", "E", "F", "J" ], "cost": 6 }, { "id": "S106", "elements": [ "P", "R", "S", "T", "U", "V", "W" ], "cost": 126 }, { "id": "S107", "elements": [ "K", "N", "O" ], "cost": 132 }, { "id": "S108", "elements": [ "L", "M", "Q" ], "cost": 216 }, { "id": "S109", "elements": [ "E", "F", "G", "H" ], "cost": 188 }, { "id": "S110", "elements": [ "B", "E", "F", "G", "H", "I" ], "cost": 66 }, { "id": "S111", "elements": [ "M", "N", "R" ], "cost": 219 }, { "id": "S112", "elements": [ "A" ], "cost": 10000 }, { "id": "S113", "elements": [ "B" ], "cost": 10000 }, { "id": "S114", "elements": [ "C" ], "cost": 10000 }, { "id": "S115", "elements": [ "D" ], "cost": 10000 }, { "id": "S116", "elements": [ "E" ], "cost": 10000 }, { "id": "S117", "elements": [ "F" ], "cost": 10000 }, { "id": "S118", "elements": [ "G" ], "cost": 10000 }, { "id": "S119", "elements": [ "H" ], "cost": 10000 }, { "id": "S120", "elements": [ "I" ], "cost": 10000 }, { "id": "S121", "elements": [ "J" ], "cost": 10000 }, { "id": "S122", "elements": [ "K" ], "cost": 10000 }, { "id": "S123", "elements": [ "L" ], "cost": 10000 }, { "id": "S124", "elements": [ "M" ], "cost": 10000 }, { "id": "S125", "elements": [ "N" ], "cost": 10000 }, { "id": "S126", "elements": [ "O" ], "cost": 10000 }, { "id": "S127", "elements": [ "P" ], "cost": 10000 }, { "id": "S128", "elements": [ "Q" ], "cost": 10000 }, { "id": "S129", "elements": [ "R" ], "cost": 10000 }, { "id": "S130", "elements": [ "S" ], "cost": 10000 }, { "id": "S131", "elements": [ "T" ], "cost": 10000 }, { "id": "S132", "elements": [ "U" ], "cost": 10000 }, { "id": "S133", "elements": [ "V" ], "cost": 10000 }, { "id": "S134", "elements": [ "W" ], "cost": 10000 }, { "id": "S135", "elements": [ "X" ], "cost": 10000 } ] }, "solution_variant": [ "S30", "S32", "S40", "S55", "S89" ], "context_index": 44, "input_format": "nl", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "There’s a campaign puzzle at hand — pick a set of advertising bundles so that all the target groups are covered, each group gets served by a single chosen bundle, and no two chosen bundles try to reach the same group. The simple measure of success is the total bill: sum the prices of the selected bundles and keep that sum as low as possible. The specific bundles and which segments they touch are listed below.\n\n# total_customer_segments=21\n# total_ad_bundles=117\nbundle_id,bundle_price,target_segments\nS1,243,B C D\nS2,265,J K L M N\nS3,150,N O Q\nS4,99,Q R S\nS5,256,F G H I\nS6,240,N R S T\nS7,204,C E I\nS8,93,I J L\nS9,68,O Q\nS10,54,C G M\nS11,246,H J M\nS12,175,L M N P R\nS13,385,A B C D E\nS14,45,E H I J K\nS15,220,L O P Q\nS16,4,H I\nS17,33,D E K\nS18,88,A C D G\nS19,120,R S T U\nS20,395,A B D F G\nS21,320,D E F G\nS22,415,Q R S T U\nS23,66,B C\nS24,276,Q S T U\nS25,462,A B C D E G\nS26,485,P R S T U\nS27,462,O Q R S T U\nS28,240,C F G H I\nS29,150,N Q R S T\nS30,272,G I J K\nS31,8,P R S U\nS32,66,I L M N O R\nS33,51,P R S\nS34,288,I J K L M N\nS35,177,E F H\nS36,384,I J K N\nS37,480,I J M N P\nS38,340,F J L M\nS39,328,J L M N\nS40,40,L O P S T\nS41,552,J M O P R S\nS42,190,D E\nS43,395,E F G H I\nS44,9,B D E\nS45,81,M P S\nS46,192,K N O\nS47,66,C D E G H K\nS48,40,J L N O R\nS49,336,I J K M\nS50,145,I K L M O\nS51,300,M N O Q R\nS52,156,L M N R\nS53,237,A C E\nS54,42,G J K\nS55,564,C D E F G H\nS56,480,D E G H K\nS57,198,G H I J L N\nS58,100,D G\nS59,260,C D F K\nS60,288,C H I J\nS61,450,E G H J K L\nS62,135,N P Q\nS63,141,A D E\nS64,276,D G H I\nS65,276,O R S\nS66,375,G H I K L\nS67,102,R U\nS68,5,G J L M P\nS69,40,B G\nS70,456,K M O P Q R\nS71,156,L O Q S\nS72,76,A B C D\nS73,445,G K L M N\nS74,246,A C F\nS75,52,F H J K\nS76,264,E J K\nS77,415,J K L M O\nS78,51,O Q R\nS79,35,J L M N P\nS80,60,K M N O\nS81,375,I K L M N\nS82,438,A B C D E F\nS83,150,G I J K L N\nS84,144,I K L\nS85,220,J K N Q\nS86,426,M N O P Q R\nS87,159,P Q R\nS88,170,A B D E F\nS89,470,K M N O P\nS90,32,C G\nS91,45,J K M\nS92,220,D H I K\nS93,87,P T U\nS94,20,A B C H\nS95,120,B F H\nS96,210,D E G H I\nS97,10000,A\nS98,10000,B\nS99,10000,C\nS100,10000,D\nS101,10000,E\nS102,10000,F\nS103,10000,G\nS104,10000,H\nS105,10000,I\nS106,10000,J\nS107,10000,K\nS108,10000,L\nS109,10000,M\nS110,10000,N\nS111,10000,O\nS112,10000,P\nS113,10000,Q\nS114,10000,R\nS115,10000,S\nS116,10000,T\nS117,10000,U\n\nWhen you send your pick, just drop it into a tiny JSON snippet like this:\n\n{\n \"solution\": [\"bundle_id\", ...]\n}\n\n\"solution\" is just the list of advertising bundles you're choosing — one identifier per chosen bundle. \"bundle_id\" is a placeholder showing the shape: replace each placeholder with an actual bundle identifier from the instance. Think of this as filling out a short form, not a full report.\n\nThis is only a sketch of the expected shape, not the actual answer. Please use the exact identifiers from the instance input — no renaming and no new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 21, "num_sets": 117, "density": 0.1684981684981685, "sets": [ { "id": 1, "elements": [ 2, 3, 4 ], "cost": 243 }, { "id": 2, "elements": [ 10, 11, 12, 13, 14 ], "cost": 265 }, { "id": 3, "elements": [ 14, 15, 17 ], "cost": 150 }, { "id": 4, "elements": [ 17, 18, 19 ], "cost": 99 }, { "id": 5, "elements": [ 6, 7, 8, 9 ], "cost": 256 }, { "id": 6, "elements": [ 14, 18, 19, 20 ], "cost": 240 }, { "id": 7, "elements": [ 3, 5, 9 ], "cost": 204 }, { "id": 8, "elements": [ 9, 10, 12 ], "cost": 93 }, { "id": 9, "elements": [ 15, 17 ], "cost": 68 }, { "id": 10, "elements": [ 3, 7, 13 ], "cost": 54 }, { "id": 11, "elements": [ 8, 10, 13 ], "cost": 246 }, { "id": 12, "elements": [ 12, 13, 14, 16, 18 ], "cost": 175 }, { "id": 13, "elements": [ 1, 2, 3, 4, 5 ], "cost": 385 }, { "id": 14, "elements": [ 5, 8, 9, 10, 11 ], "cost": 45 }, { "id": 15, "elements": [ 12, 15, 16, 17 ], "cost": 220 }, { "id": 16, "elements": [ 8, 9 ], "cost": 4 }, { "id": 17, "elements": [ 4, 5, 11 ], "cost": 33 }, { "id": 18, "elements": [ 1, 3, 4, 7 ], "cost": 88 }, { "id": 19, "elements": [ 18, 19, 20, 21 ], "cost": 120 }, { "id": 20, "elements": [ 1, 2, 4, 6, 7 ], "cost": 395 }, { "id": 21, "elements": [ 4, 5, 6, 7 ], "cost": 320 }, { "id": 22, "elements": [ 17, 18, 19, 20, 21 ], "cost": 415 }, { "id": 23, "elements": [ 2, 3 ], "cost": 66 }, { "id": 24, "elements": [ 17, 19, 20, 21 ], "cost": 276 }, { "id": 25, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 462 }, { "id": 26, "elements": [ 16, 18, 19, 20, 21 ], "cost": 485 }, { "id": 27, "elements": [ 15, 17, 18, 19, 20, 21 ], "cost": 462 }, { "id": 28, "elements": [ 3, 6, 7, 8, 9 ], "cost": 240 }, { "id": 29, "elements": [ 14, 17, 18, 19, 20 ], "cost": 150 }, { "id": 30, "elements": [ 7, 9, 10, 11 ], "cost": 272 }, { "id": 31, "elements": [ 16, 18, 19, 21 ], "cost": 8 }, { "id": 32, "elements": [ 9, 12, 13, 14, 15, 18 ], "cost": 66 }, { "id": 33, "elements": [ 16, 18, 19 ], "cost": 51 }, { "id": 34, "elements": [ 9, 10, 11, 12, 13, 14 ], "cost": 288 }, { "id": 35, "elements": [ 5, 6, 8 ], "cost": 177 }, { "id": 36, "elements": [ 9, 10, 11, 14 ], "cost": 384 }, { "id": 37, "elements": [ 9, 10, 13, 14, 16 ], "cost": 480 }, { "id": 38, "elements": [ 6, 10, 12, 13 ], "cost": 340 }, { "id": 39, "elements": [ 10, 12, 13, 14 ], "cost": 328 }, { "id": 40, "elements": [ 12, 15, 16, 19, 20 ], "cost": 40 }, { "id": 41, "elements": [ 10, 13, 15, 16, 18, 19 ], "cost": 552 }, { "id": 42, "elements": [ 4, 5 ], "cost": 190 }, { "id": 43, "elements": [ 5, 6, 7, 8, 9 ], "cost": 395 }, { "id": 44, "elements": [ 2, 4, 5 ], "cost": 9 }, { "id": 45, "elements": [ 13, 16, 19 ], "cost": 81 }, { "id": 46, "elements": [ 11, 14, 15 ], "cost": 192 }, { "id": 47, "elements": [ 3, 4, 5, 7, 8, 11 ], "cost": 66 }, { "id": 48, "elements": [ 10, 12, 14, 15, 18 ], "cost": 40 }, { "id": 49, "elements": [ 9, 10, 11, 13 ], "cost": 336 }, { "id": 50, "elements": [ 9, 11, 12, 13, 15 ], "cost": 145 }, { "id": 51, "elements": [ 13, 14, 15, 17, 18 ], "cost": 300 }, { "id": 52, "elements": [ 12, 13, 14, 18 ], "cost": 156 }, { "id": 53, "elements": [ 1, 3, 5 ], "cost": 237 }, { "id": 54, "elements": [ 7, 10, 11 ], "cost": 42 }, { "id": 55, "elements": [ 3, 4, 5, 6, 7, 8 ], "cost": 564 }, { "id": 56, "elements": [ 4, 5, 7, 8, 11 ], "cost": 480 }, { "id": 57, "elements": [ 7, 8, 9, 10, 12, 14 ], "cost": 198 }, { "id": 58, "elements": [ 4, 7 ], "cost": 100 }, { "id": 59, "elements": [ 3, 4, 6, 11 ], "cost": 260 }, { "id": 60, "elements": [ 3, 8, 9, 10 ], "cost": 288 }, { "id": 61, "elements": [ 5, 7, 8, 10, 11, 12 ], "cost": 450 }, { "id": 62, "elements": [ 14, 16, 17 ], "cost": 135 }, { "id": 63, "elements": [ 1, 4, 5 ], "cost": 141 }, { "id": 64, "elements": [ 4, 7, 8, 9 ], "cost": 276 }, { "id": 65, "elements": [ 15, 18, 19 ], "cost": 276 }, { "id": 66, "elements": [ 7, 8, 9, 11, 12 ], "cost": 375 }, { "id": 67, "elements": [ 18, 21 ], "cost": 102 }, { "id": 68, "elements": [ 7, 10, 12, 13, 16 ], "cost": 5 }, { "id": 69, "elements": [ 2, 7 ], "cost": 40 }, { "id": 70, "elements": [ 11, 13, 15, 16, 17, 18 ], "cost": 456 }, { "id": 71, "elements": [ 12, 15, 17, 19 ], "cost": 156 }, { "id": 72, "elements": [ 1, 2, 3, 4 ], "cost": 76 }, { "id": 73, "elements": [ 7, 11, 12, 13, 14 ], "cost": 445 }, { "id": 74, "elements": [ 1, 3, 6 ], "cost": 246 }, { "id": 75, "elements": [ 6, 8, 10, 11 ], "cost": 52 }, { "id": 76, "elements": [ 5, 10, 11 ], "cost": 264 }, { "id": 77, "elements": [ 10, 11, 12, 13, 15 ], "cost": 415 }, { "id": 78, "elements": [ 15, 17, 18 ], "cost": 51 }, { "id": 79, "elements": [ 10, 12, 13, 14, 16 ], "cost": 35 }, { "id": 80, "elements": [ 11, 13, 14, 15 ], "cost": 60 }, { "id": 81, "elements": [ 9, 11, 12, 13, 14 ], "cost": 375 }, { "id": 82, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 438 }, { "id": 83, "elements": [ 7, 9, 10, 11, 12, 14 ], "cost": 150 }, { "id": 84, "elements": [ 9, 11, 12 ], "cost": 144 }, { "id": 85, "elements": [ 10, 11, 14, 17 ], "cost": 220 }, { "id": 86, "elements": [ 13, 14, 15, 16, 17, 18 ], "cost": 426 }, { "id": 87, "elements": [ 16, 17, 18 ], "cost": 159 }, { "id": 88, "elements": [ 1, 2, 4, 5, 6 ], "cost": 170 }, { "id": 89, "elements": [ 11, 13, 14, 15, 16 ], "cost": 470 }, { "id": 90, "elements": [ 3, 7 ], "cost": 32 }, { "id": 91, "elements": [ 10, 11, 13 ], "cost": 45 }, { "id": 92, "elements": [ 4, 8, 9, 11 ], "cost": 220 }, { "id": 93, "elements": [ 16, 20, 21 ], "cost": 87 }, { "id": 94, "elements": [ 1, 2, 3, 8 ], "cost": 20 }, { "id": 95, "elements": [ 2, 6, 8 ], "cost": 120 }, { "id": 96, "elements": [ 4, 5, 7, 8, 9 ], "cost": 210 }, { "id": 97, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 98, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 21 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0044_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0044_bag_stats.png" }, "solution": [ 9, 16, 17, 19, 69, 74, 79 ], "obj": 546.0, "instance_variant": { "num_elements": 21, "num_sets": 117, "sets": [ { "id": "S1", "elements": [ "B", "C", "D" ], "cost": 243 }, { "id": "S2", "elements": [ "J", "K", "L", "M", "N" ], "cost": 265 }, { "id": "S3", "elements": [ "N", "O", "Q" ], "cost": 150 }, { "id": "S4", "elements": [ "Q", "R", "S" ], "cost": 99 }, { "id": "S5", "elements": [ "F", "G", "H", "I" ], "cost": 256 }, { "id": "S6", "elements": [ "N", "R", "S", "T" ], "cost": 240 }, { "id": "S7", "elements": [ "C", "E", "I" ], "cost": 204 }, { "id": "S8", "elements": [ "I", "J", "L" ], "cost": 93 }, { "id": "S9", "elements": [ "O", "Q" ], "cost": 68 }, { "id": "S10", "elements": [ "C", "G", "M" ], "cost": 54 }, { "id": "S11", "elements": [ "H", "J", "M" ], "cost": 246 }, { "id": "S12", "elements": [ "L", "M", "N", "P", "R" ], "cost": 175 }, { "id": "S13", "elements": [ "A", "B", "C", "D", "E" ], "cost": 385 }, { "id": "S14", "elements": [ "E", "H", "I", "J", "K" ], "cost": 45 }, { "id": "S15", "elements": [ "L", "O", "P", "Q" ], "cost": 220 }, { "id": "S16", "elements": [ "H", "I" ], "cost": 4 }, { "id": "S17", "elements": [ "D", "E", "K" ], "cost": 33 }, { "id": "S18", "elements": [ "A", "C", "D", "G" ], "cost": 88 }, { "id": "S19", "elements": [ "R", "S", "T", "U" ], "cost": 120 }, { "id": "S20", "elements": [ "A", "B", "D", "F", "G" ], "cost": 395 }, { "id": "S21", "elements": [ "D", "E", "F", "G" ], "cost": 320 }, { "id": "S22", "elements": [ "Q", "R", "S", "T", "U" ], "cost": 415 }, { "id": "S23", "elements": [ "B", "C" ], "cost": 66 }, { "id": "S24", "elements": [ "Q", "S", "T", "U" ], "cost": 276 }, { "id": "S25", "elements": [ "A", "B", "C", "D", "E", "G" ], "cost": 462 }, { "id": "S26", "elements": [ "P", "R", "S", "T", "U" ], "cost": 485 }, { "id": "S27", "elements": [ "O", "Q", "R", "S", "T", "U" ], "cost": 462 }, { "id": "S28", "elements": [ "C", "F", "G", "H", "I" ], "cost": 240 }, { "id": "S29", "elements": [ "N", "Q", "R", "S", "T" ], "cost": 150 }, { "id": "S30", "elements": [ "G", "I", "J", "K" ], "cost": 272 }, { "id": "S31", "elements": [ "P", "R", "S", "U" ], "cost": 8 }, { "id": "S32", "elements": [ "I", "L", "M", "N", "O", "R" ], "cost": 66 }, { "id": "S33", "elements": [ "P", "R", "S" ], "cost": 51 }, { "id": "S34", "elements": [ "I", "J", "K", "L", "M", "N" ], "cost": 288 }, { "id": "S35", "elements": [ "E", "F", "H" ], "cost": 177 }, { "id": "S36", "elements": [ "I", "J", "K", "N" ], "cost": 384 }, { "id": "S37", "elements": [ "I", "J", "M", "N", "P" ], "cost": 480 }, { "id": "S38", "elements": [ "F", "J", "L", "M" ], "cost": 340 }, { "id": "S39", "elements": [ "J", "L", "M", "N" ], "cost": 328 }, { "id": "S40", "elements": [ "L", "O", "P", "S", "T" ], "cost": 40 }, { "id": "S41", "elements": [ "J", "M", "O", "P", "R", "S" ], "cost": 552 }, { "id": "S42", "elements": [ "D", "E" ], "cost": 190 }, { "id": "S43", "elements": [ "E", "F", "G", "H", "I" ], "cost": 395 }, { "id": "S44", "elements": [ "B", "D", "E" ], "cost": 9 }, { "id": "S45", "elements": [ "M", "P", "S" ], "cost": 81 }, { "id": "S46", "elements": [ "K", "N", "O" ], "cost": 192 }, { "id": "S47", "elements": [ "C", "D", "E", "G", "H", "K" ], "cost": 66 }, { "id": "S48", "elements": [ "J", "L", "N", "O", "R" ], "cost": 40 }, { "id": "S49", "elements": [ "I", "J", "K", "M" ], "cost": 336 }, { "id": "S50", "elements": [ "I", "K", "L", "M", "O" ], "cost": 145 }, { "id": "S51", "elements": [ "M", "N", "O", "Q", "R" ], "cost": 300 }, { "id": "S52", "elements": [ "L", "M", "N", "R" ], "cost": 156 }, { "id": "S53", "elements": [ "A", "C", "E" ], "cost": 237 }, { "id": "S54", "elements": [ "G", "J", "K" ], "cost": 42 }, { "id": "S55", "elements": [ "C", "D", "E", "F", "G", "H" ], "cost": 564 }, { "id": "S56", "elements": [ "D", "E", "G", "H", "K" ], "cost": 480 }, { "id": "S57", "elements": [ "G", "H", "I", "J", "L", "N" ], "cost": 198 }, { "id": "S58", "elements": [ "D", "G" ], "cost": 100 }, { "id": "S59", "elements": [ "C", "D", "F", "K" ], "cost": 260 }, { "id": "S60", "elements": [ "C", "H", "I", "J" ], "cost": 288 }, { "id": "S61", "elements": [ "E", "G", "H", "J", "K", "L" ], "cost": 450 }, { "id": "S62", "elements": [ "N", "P", "Q" ], "cost": 135 }, { "id": "S63", "elements": [ "A", "D", "E" ], "cost": 141 }, { "id": "S64", "elements": [ "D", "G", "H", "I" ], "cost": 276 }, { "id": "S65", "elements": [ "O", "R", "S" ], "cost": 276 }, { "id": "S66", "elements": [ "G", "H", "I", "K", "L" ], "cost": 375 }, { "id": "S67", "elements": [ "R", "U" ], "cost": 102 }, { "id": "S68", "elements": [ "G", "J", "L", "M", "P" ], "cost": 5 }, { "id": "S69", "elements": [ "B", "G" ], "cost": 40 }, { "id": "S70", "elements": [ "K", "M", "O", "P", "Q", "R" ], "cost": 456 }, { "id": "S71", "elements": [ "L", "O", "Q", "S" ], "cost": 156 }, { "id": "S72", "elements": [ "A", "B", "C", "D" ], "cost": 76 }, { "id": "S73", "elements": [ "G", "K", "L", "M", "N" ], "cost": 445 }, { "id": "S74", "elements": [ "A", "C", "F" ], "cost": 246 }, { "id": "S75", "elements": [ "F", "H", "J", "K" ], "cost": 52 }, { "id": "S76", "elements": [ "E", "J", "K" ], "cost": 264 }, { "id": "S77", "elements": [ "J", "K", "L", "M", "O" ], "cost": 415 }, { "id": "S78", "elements": [ "O", "Q", "R" ], "cost": 51 }, { "id": "S79", "elements": [ "J", "L", "M", "N", "P" ], "cost": 35 }, { "id": "S80", "elements": [ "K", "M", "N", "O" ], "cost": 60 }, { "id": "S81", "elements": [ "I", "K", "L", "M", "N" ], "cost": 375 }, { "id": "S82", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 438 }, { "id": "S83", "elements": [ "G", "I", "J", "K", "L", "N" ], "cost": 150 }, { "id": "S84", "elements": [ "I", "K", "L" ], "cost": 144 }, { "id": "S85", "elements": [ "J", "K", "N", "Q" ], "cost": 220 }, { "id": "S86", "elements": [ "M", "N", "O", "P", "Q", "R" ], "cost": 426 }, { "id": "S87", "elements": [ "P", "Q", "R" ], "cost": 159 }, { "id": "S88", "elements": [ "A", "B", "D", "E", "F" ], "cost": 170 }, { "id": "S89", "elements": [ "K", "M", "N", "O", "P" ], "cost": 470 }, { "id": "S90", "elements": [ "C", "G" ], "cost": 32 }, { "id": "S91", "elements": [ "J", "K", "M" ], "cost": 45 }, { "id": "S92", "elements": [ "D", "H", "I", "K" ], "cost": 220 }, { "id": "S93", "elements": [ "P", "T", "U" ], "cost": 87 }, { "id": "S94", "elements": [ "A", "B", "C", "H" ], "cost": 20 }, { "id": "S95", "elements": [ "B", "F", "H" ], "cost": 120 }, { "id": "S96", "elements": [ "D", "E", "G", "H", "I" ], "cost": 210 }, { "id": "S97", "elements": [ "A" ], "cost": 10000 }, { "id": "S98", "elements": [ "B" ], "cost": 10000 }, { "id": "S99", "elements": [ "C" ], "cost": 10000 }, { "id": "S100", "elements": [ "D" ], "cost": 10000 }, { "id": "S101", "elements": [ "E" ], "cost": 10000 }, { "id": "S102", "elements": [ "F" ], "cost": 10000 }, { "id": "S103", "elements": [ "G" ], "cost": 10000 }, { "id": "S104", "elements": [ "H" ], "cost": 10000 }, { "id": "S105", "elements": [ "I" ], "cost": 10000 }, { "id": "S106", "elements": [ "J" ], "cost": 10000 }, { "id": "S107", "elements": [ "K" ], "cost": 10000 }, { "id": "S108", "elements": [ "L" ], "cost": 10000 }, { "id": "S109", "elements": [ "M" ], "cost": 10000 }, { "id": "S110", "elements": [ "N" ], "cost": 10000 }, { "id": "S111", "elements": [ "O" ], "cost": 10000 }, { "id": "S112", "elements": [ "P" ], "cost": 10000 }, { "id": "S113", "elements": [ "Q" ], "cost": 10000 }, { "id": "S114", "elements": [ "R" ], "cost": 10000 }, { "id": "S115", "elements": [ "S" ], "cost": 10000 }, { "id": "S116", "elements": [ "T" ], "cost": 10000 }, { "id": "S117", "elements": [ "U" ], "cost": 10000 } ] }, "solution_variant": [ "S9", "S16", "S17", "S19", "S69", "S74", "S79" ], "context_index": 45, "input_format": "csv", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "We’ve got a stack of vendor kits and the office manager needs to decide which kits to order so every department’s supply list is fulfilled by one kit alone. No department can be skipped and no department should receive items from more than one ordered kit — it’s one box per department, no overlaps. The way to judge options is simple: add up the costs of the selected kits and pick the combination with the smallest total bill. The concrete options and their prices are shown below.\n\n{\n \"total_departments\": 20,\n \"total_kits_available\": 112,\n \"sets\": [\n {\n \"kit_id\": \"S1\",\n \"covered_departments\": [\n 7,\n 10,\n 12\n ],\n \"kit_price\": 267\n },\n {\n \"kit_id\": \"S2\",\n \"covered_departments\": [\n 0,\n 1,\n 2,\n 3,\n 4\n ],\n \"kit_price\": 435\n },\n {\n \"kit_id\": \"S3\",\n \"covered_departments\": [\n 14,\n 15,\n 16,\n 17,\n 18\n ],\n \"kit_price\": 215\n },\n {\n \"kit_id\": \"S4\",\n \"covered_departments\": [\n 0,\n 3,\n 5\n ],\n \"kit_price\": 195\n },\n {\n \"kit_id\": \"S5\",\n \"covered_departments\": [\n 12,\n 13,\n 14,\n 15,\n 16\n ],\n \"kit_price\": 250\n },\n {\n \"kit_id\": \"S6\",\n \"covered_departments\": [\n 9,\n 11,\n 13,\n 15\n ],\n \"kit_price\": 40\n },\n {\n \"kit_id\": \"S7\",\n \"covered_departments\": [\n 0,\n 3,\n 4,\n 5,\n 7\n ],\n \"kit_price\": 130\n },\n {\n \"kit_id\": \"S8\",\n \"covered_departments\": [\n 10,\n 14,\n 15,\n 16,\n 17,\n 19\n ],\n \"kit_price\": 174\n },\n {\n \"kit_id\": \"S9\",\n \"covered_departments\": [\n 3,\n 5,\n 7,\n 8\n ],\n \"kit_price\": 104\n },\n {\n \"kit_id\": \"S10\",\n \"covered_departments\": [\n 6,\n 9,\n 10\n ],\n \"kit_price\": 267\n },\n {\n \"kit_id\": \"S11\",\n \"covered_departments\": [\n 13,\n 14,\n 16\n ],\n \"kit_price\": 282\n },\n {\n \"kit_id\": \"S12\",\n \"covered_departments\": [\n 7,\n 10,\n 11\n ],\n \"kit_price\": 27\n },\n {\n \"kit_id\": \"S13\",\n \"covered_departments\": [\n 11,\n 12,\n 15\n ],\n \"kit_price\": 114\n },\n {\n \"kit_id\": \"S14\",\n \"covered_departments\": [\n 15,\n 16,\n 17,\n 18,\n 19\n ],\n \"kit_price\": 325\n },\n {\n \"kit_id\": \"S15\",\n \"covered_departments\": [\n 2,\n 3,\n 7,\n 8\n ],\n \"kit_price\": 304\n },\n {\n \"kit_id\": \"S16\",\n \"covered_departments\": [\n 13,\n 15,\n 16,\n 17\n ],\n \"kit_price\": 212\n },\n {\n \"kit_id\": \"S17\",\n \"covered_departments\": [\n 13,\n 14,\n 15,\n 18,\n 19\n ],\n \"kit_price\": 455\n },\n {\n \"kit_id\": \"S18\",\n \"covered_departments\": [\n 10,\n 11,\n 12,\n 13,\n 14,\n 16\n ],\n \"kit_price\": 24\n },\n {\n \"kit_id\": \"S19\",\n \"covered_departments\": [\n 1,\n 2,\n 4,\n 5,\n 6,\n 7\n ],\n \"kit_price\": 198\n },\n {\n \"kit_id\": \"S20\",\n \"covered_departments\": [\n 7,\n 10,\n 12,\n 13,\n 15\n ],\n \"kit_price\": 155\n },\n {\n \"kit_id\": \"S21\",\n \"covered_departments\": [\n 14,\n 15,\n 16,\n 17,\n 18,\n 19\n ],\n \"kit_price\": 546\n },\n {\n \"kit_id\": \"S22\",\n \"covered_departments\": [\n 0,\n 1,\n 2,\n 3,\n 4,\n 5\n ],\n \"kit_price\": 162\n },\n {\n \"kit_id\": \"S23\",\n \"covered_departments\": [\n 9,\n 11,\n 12,\n 14\n ],\n \"kit_price\": 8\n },\n {\n \"kit_id\": \"S24\",\n \"covered_departments\": [\n 13,\n 15,\n 16,\n 17,\n 18,\n 19\n ],\n \"kit_price\": 252\n },\n {\n \"kit_id\": \"S25\",\n \"covered_departments\": [\n 9,\n 12,\n 13,\n 14,\n 18\n ],\n \"kit_price\": 5\n },\n {\n \"kit_id\": \"S26\",\n \"covered_departments\": [\n 1,\n 2,\n 3,\n 4\n ],\n \"kit_price\": 248\n },\n {\n \"kit_id\": \"S27\",\n \"covered_departments\": [\n 5,\n 6,\n 7,\n 10\n ],\n \"kit_price\": 152\n },\n {\n \"kit_id\": \"S28\",\n \"covered_departments\": [\n 1,\n 2\n ],\n \"kit_price\": 84\n },\n {\n \"kit_id\": \"S29\",\n \"covered_departments\": [\n 8,\n 9,\n 10,\n 11\n ],\n \"kit_price\": 20\n },\n {\n \"kit_id\": \"S30\",\n \"covered_departments\": [\n 10,\n 11,\n 14\n ],\n \"kit_price\": 153\n },\n {\n \"kit_id\": \"S31\",\n \"covered_departments\": [\n 14,\n 18,\n 19\n ],\n \"kit_price\": 105\n },\n {\n \"kit_id\": \"S32\",\n \"covered_departments\": [\n 12,\n 14,\n 16,\n 17,\n 18,\n 19\n ],\n \"kit_price\": 390\n },\n {\n \"kit_id\": \"S33\",\n \"covered_departments\": [\n 0,\n 1,\n 2,\n 3\n ],\n \"kit_price\": 304\n },\n {\n \"kit_id\": \"S34\",\n \"covered_departments\": [\n 11,\n 12,\n 14,\n 15,\n 18\n ],\n \"kit_price\": 35\n },\n {\n \"kit_id\": \"S35\",\n \"covered_departments\": [\n 13,\n 14,\n 15,\n 17,\n 18\n ],\n \"kit_price\": 80\n },\n {\n \"kit_id\": \"S36\",\n \"covered_departments\": [\n 14,\n 15,\n 17,\n 18,\n 19\n ],\n \"kit_price\": 430\n },\n {\n \"kit_id\": \"S37\",\n \"covered_departments\": [\n 5,\n 7,\n 8,\n 9,\n 10,\n 11\n ],\n \"kit_price\": 312\n },\n {\n \"kit_id\": \"S38\",\n \"covered_departments\": [\n 3,\n 4,\n 7,\n 8,\n 9\n ],\n \"kit_price\": 70\n },\n {\n \"kit_id\": \"S39\",\n \"covered_departments\": [\n 7,\n 8,\n 9,\n 13\n ],\n \"kit_price\": 56\n },\n {\n \"kit_id\": \"S40\",\n \"covered_departments\": [\n 0,\n 1,\n 3\n ],\n \"kit_price\": 246\n },\n {\n \"kit_id\": \"S41\",\n \"covered_departments\": [\n 2,\n 3,\n 4\n ],\n \"kit_price\": 225\n },\n {\n \"kit_id\": \"S42\",\n \"covered_departments\": [\n 1,\n 4,\n 6,\n 7,\n 9\n ],\n \"kit_price\": 425\n },\n {\n \"kit_id\": \"S43\",\n \"covered_departments\": [\n 12,\n 13,\n 17,\n 18\n ],\n \"kit_price\": 200\n },\n {\n \"kit_id\": \"S44\",\n \"covered_departments\": [\n 12,\n 14,\n 15,\n 16,\n 19\n ],\n \"kit_price\": 200\n },\n {\n \"kit_id\": \"S45\",\n \"covered_departments\": [\n 6,\n 8\n ],\n \"kit_price\": 122\n },\n {\n \"kit_id\": \"S46\",\n \"covered_departments\": [\n 14,\n 15,\n 17,\n 18\n ],\n \"kit_price\": 272\n },\n {\n \"kit_id\": \"S47\",\n \"covered_departments\": [\n 4,\n 6\n ],\n \"kit_price\": 194\n },\n {\n \"kit_id\": \"S48\",\n \"covered_departments\": [\n 5,\n 6,\n 7,\n 8\n ],\n \"kit_price\": 96\n },\n {\n \"kit_id\": \"S49\",\n \"covered_departments\": [\n 7,\n 9,\n 10,\n 11\n ],\n \"kit_price\": 160\n },\n {\n \"kit_id\": \"S50\",\n \"covered_departments\": [\n 14,\n 15,\n 16,\n 19\n ],\n \"kit_price\": 244\n },\n {\n \"kit_id\": \"S51\",\n \"covered_departments\": [\n 0,\n 1,\n 2,\n 4\n ],\n \"kit_price\": 40\n },\n {\n \"kit_id\": \"S52\",\n \"covered_departments\": [\n 13,\n 14,\n 16,\n 17\n ],\n \"kit_price\": 80\n },\n {\n \"kit_id\": \"S53\",\n \"covered_departments\": [\n 10,\n 13,\n 14,\n 15\n ],\n \"kit_price\": 388\n },\n {\n \"kit_id\": \"S54\",\n \"covered_departments\": [\n 2,\n 5,\n 6,\n 7,\n 8,\n 9\n ],\n \"kit_price\": 462\n },\n {\n \"kit_id\": \"S55\",\n \"covered_departments\": [\n 2,\n 3,\n 4,\n 6,\n 7\n ],\n \"kit_price\": 485\n },\n {\n \"kit_id\": \"S56\",\n \"covered_departments\": [\n 1,\n 2,\n 3,\n 4,\n 5,\n 6\n ],\n \"kit_price\": 78\n },\n {\n \"kit_id\": \"S57\",\n \"covered_departments\": [\n 7,\n 8,\n 9,\n 10,\n 11\n ],\n \"kit_price\": 195\n },\n {\n \"kit_id\": \"S58\",\n \"covered_departments\": [\n 8,\n 9,\n 10,\n 11,\n 13\n ],\n \"kit_price\": 50\n },\n {\n \"kit_id\": \"S59\",\n \"covered_departments\": [\n 12,\n 13,\n 18\n ],\n \"kit_price\": 252\n },\n {\n \"kit_id\": \"S60\",\n \"covered_departments\": [\n 12,\n 14,\n 15\n ],\n \"kit_price\": 234\n },\n {\n \"kit_id\": \"S61\",\n \"covered_departments\": [\n 14,\n 17,\n 18,\n 19\n ],\n \"kit_price\": 224\n },\n {\n \"kit_id\": \"S62\",\n \"covered_departments\": [\n 0,\n 2,\n 3\n ],\n \"kit_price\": 165\n },\n {\n \"kit_id\": \"S63\",\n \"covered_departments\": [\n 10,\n 12\n ],\n \"kit_price\": 98\n },\n {\n \"kit_id\": \"S64\",\n \"covered_departments\": [\n 7,\n 8,\n 9\n ],\n \"kit_price\": 96\n },\n {\n \"kit_id\": \"S65\",\n \"covered_departments\": [\n 9,\n 11,\n 13\n ],\n \"kit_price\": 132\n },\n {\n \"kit_id\": \"S66\",\n \"covered_departments\": [\n 11,\n 12\n ],\n \"kit_price\": 164\n },\n {\n \"kit_id\": \"S67\",\n \"covered_departments\": [\n 1,\n 6,\n 7\n ],\n \"kit_price\": 66\n },\n {\n \"kit_id\": \"S68\",\n \"covered_departments\": [\n 2,\n 5,\n 6,\n 7,\n 12\n ],\n \"kit_price\": 305\n },\n {\n \"kit_id\": \"S69\",\n \"covered_departments\": [\n 11,\n 13,\n 14,\n 15\n ],\n \"kit_price\": 100\n },\n {\n \"kit_id\": \"S70\",\n \"covered_departments\": [\n 1,\n 5,\n 6,\n 9,\n 10,\n 13\n ],\n \"kit_price\": 270\n },\n {\n \"kit_id\": \"S71\",\n \"covered_departments\": [\n 16,\n 18,\n 19\n ],\n \"kit_price\": 141\n },\n {\n \"kit_id\": \"S72\",\n \"covered_departments\": [\n 10,\n 12,\n 13\n ],\n \"kit_price\": 123\n },\n {\n \"kit_id\": \"S73\",\n \"covered_departments\": [\n 13,\n 16,\n 17,\n 18\n ],\n \"kit_price\": 28\n },\n {\n \"kit_id\": \"S74\",\n \"covered_departments\": [\n 7,\n 9,\n 10,\n 11,\n 12,\n 14\n ],\n \"kit_price\": 252\n },\n {\n \"kit_id\": \"S75\",\n \"covered_departments\": [\n 14,\n 16,\n 17,\n 18,\n 19\n ],\n \"kit_price\": 235\n },\n {\n \"kit_id\": \"S76\",\n \"covered_departments\": [\n 12,\n 13,\n 14,\n 15\n ],\n \"kit_price\": 48\n },\n {\n \"kit_id\": \"S77\",\n \"covered_departments\": [\n 8,\n 11,\n 13\n ],\n \"kit_price\": 159\n },\n {\n \"kit_id\": \"S78\",\n \"covered_departments\": [\n 6,\n 8,\n 9,\n 10,\n 12,\n 14\n ],\n \"kit_price\": 414\n },\n {\n \"kit_id\": \"S79\",\n \"covered_departments\": [\n 1,\n 3,\n 4,\n 5,\n 6,\n 7\n ],\n \"kit_price\": 84\n },\n {\n \"kit_id\": \"S80\",\n \"covered_departments\": [\n 10,\n 11,\n 14,\n 16\n ],\n \"kit_price\": 108\n },\n {\n \"kit_id\": \"S81\",\n \"covered_departments\": [\n 9,\n 11,\n 12,\n 14,\n 15\n ],\n \"kit_price\": 320\n },\n {\n \"kit_id\": \"S82\",\n \"covered_departments\": [\n 2,\n 3,\n 8\n ],\n \"kit_price\": 99\n },\n {\n \"kit_id\": \"S83\",\n \"covered_departments\": [\n 4,\n 5,\n 7,\n 8,\n 9\n ],\n \"kit_price\": 445\n },\n {\n \"kit_id\": \"S84\",\n \"covered_departments\": [\n 12,\n 13\n ],\n \"kit_price\": 46\n },\n {\n \"kit_id\": \"S85\",\n \"covered_departments\": [\n 15,\n 17,\n 19\n ],\n \"kit_price\": 3\n },\n {\n \"kit_id\": \"S86\",\n \"covered_departments\": [\n 12,\n 16,\n 17,\n 18\n ],\n \"kit_price\": 384\n },\n {\n \"kit_id\": \"S87\",\n \"covered_departments\": [\n 0,\n 2\n ],\n \"kit_price\": 2\n },\n {\n \"kit_id\": \"S88\",\n \"covered_departments\": [\n 14,\n 16,\n 17\n ],\n \"kit_price\": 255\n },\n {\n \"kit_id\": \"S89\",\n \"covered_departments\": [\n 3,\n 4,\n 5,\n 7,\n 8\n ],\n \"kit_price\": 350\n },\n {\n \"kit_id\": \"S90\",\n \"covered_departments\": [\n 8,\n 11,\n 12,\n 14\n ],\n \"kit_price\": 40\n },\n {\n \"kit_id\": \"S91\",\n \"covered_departments\": [\n 0,\n 2,\n 3,\n 4\n ],\n \"kit_price\": 400\n },\n {\n \"kit_id\": \"S92\",\n \"covered_departments\": [\n 1,\n 5,\n 7,\n 8\n ],\n \"kit_price\": 40\n },\n {\n \"kit_id\": \"S93\",\n \"covered_departments\": [\n 0\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S94\",\n \"covered_departments\": [\n 1\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S95\",\n \"covered_departments\": [\n 2\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S96\",\n \"covered_departments\": [\n 3\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S97\",\n \"covered_departments\": [\n 4\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S98\",\n \"covered_departments\": [\n 5\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S99\",\n \"covered_departments\": [\n 6\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S100\",\n \"covered_departments\": [\n 7\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S101\",\n \"covered_departments\": [\n 8\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S102\",\n \"covered_departments\": [\n 9\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S103\",\n \"covered_departments\": [\n 10\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S104\",\n \"covered_departments\": [\n 11\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S105\",\n \"covered_departments\": [\n 12\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S106\",\n \"covered_departments\": [\n 13\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S107\",\n \"covered_departments\": [\n 14\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S108\",\n \"covered_departments\": [\n 15\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S109\",\n \"covered_departments\": [\n 16\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S110\",\n \"covered_departments\": [\n 17\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S111\",\n \"covered_departments\": [\n 18\n ],\n \"kit_price\": 10000\n },\n {\n \"kit_id\": \"S112\",\n \"covered_departments\": [\n 19\n ],\n \"kit_price\": 10000\n }\n ]\n}\n\nAlso, when you send back the chosen kits, please use this simple JSON shape so it's easy to read and process:\n\n{\n \"solution\": [\"kit_id\", ...]\n}\n\nHere \"solution\" is just the list of kit IDs you want to order — one kit per department — and each string in the array is the identifier for a kit. Think of it like filling out a short form: list the exact box labels you picked. This JSON is just a sketch of the shape I want, not the final answer itself.\n\nPlease be sure to use the identifiers exactly as they appear in the instance input — don't rename them or invent new labels.\n\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 20, "num_sets": 112, "density": 0.17946428571428572, "sets": [ { "id": 1, "elements": [ 8, 11, 13 ], "cost": 267 }, { "id": 2, "elements": [ 1, 2, 3, 4, 5 ], "cost": 435 }, { "id": 3, "elements": [ 15, 16, 17, 18, 19 ], "cost": 215 }, { "id": 4, "elements": [ 1, 4, 6 ], "cost": 195 }, { "id": 5, "elements": [ 13, 14, 15, 16, 17 ], "cost": 250 }, { "id": 6, "elements": [ 10, 12, 14, 16 ], "cost": 40 }, { "id": 7, "elements": [ 1, 4, 5, 6, 8 ], "cost": 130 }, { "id": 8, "elements": [ 11, 15, 16, 17, 18, 20 ], "cost": 174 }, { "id": 9, "elements": [ 4, 6, 8, 9 ], "cost": 104 }, { "id": 10, "elements": [ 7, 10, 11 ], "cost": 267 }, { "id": 11, "elements": [ 14, 15, 17 ], "cost": 282 }, { "id": 12, "elements": [ 8, 11, 12 ], "cost": 27 }, { "id": 13, "elements": [ 12, 13, 16 ], "cost": 114 }, { "id": 14, "elements": [ 16, 17, 18, 19, 20 ], "cost": 325 }, { "id": 15, "elements": [ 3, 4, 8, 9 ], "cost": 304 }, { "id": 16, "elements": [ 14, 16, 17, 18 ], "cost": 212 }, { "id": 17, "elements": [ 14, 15, 16, 19, 20 ], "cost": 455 }, { "id": 18, "elements": [ 11, 12, 13, 14, 15, 17 ], "cost": 24 }, { "id": 19, "elements": [ 2, 3, 5, 6, 7, 8 ], "cost": 198 }, { "id": 20, "elements": [ 8, 11, 13, 14, 16 ], "cost": 155 }, { "id": 21, "elements": [ 15, 16, 17, 18, 19, 20 ], "cost": 546 }, { "id": 22, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 162 }, { "id": 23, "elements": [ 10, 12, 13, 15 ], "cost": 8 }, { "id": 24, "elements": [ 14, 16, 17, 18, 19, 20 ], "cost": 252 }, { "id": 25, "elements": [ 10, 13, 14, 15, 19 ], "cost": 5 }, { "id": 26, "elements": [ 2, 3, 4, 5 ], "cost": 248 }, { "id": 27, "elements": [ 6, 7, 8, 11 ], "cost": 152 }, { "id": 28, "elements": [ 2, 3 ], "cost": 84 }, { "id": 29, "elements": [ 9, 10, 11, 12 ], "cost": 20 }, { "id": 30, "elements": [ 11, 12, 15 ], "cost": 153 }, { "id": 31, "elements": [ 15, 19, 20 ], "cost": 105 }, { "id": 32, "elements": [ 13, 15, 17, 18, 19, 20 ], "cost": 390 }, { "id": 33, "elements": [ 1, 2, 3, 4 ], "cost": 304 }, { "id": 34, "elements": [ 12, 13, 15, 16, 19 ], "cost": 35 }, { "id": 35, "elements": [ 14, 15, 16, 18, 19 ], "cost": 80 }, { "id": 36, "elements": [ 15, 16, 18, 19, 20 ], "cost": 430 }, { "id": 37, "elements": [ 6, 8, 9, 10, 11, 12 ], "cost": 312 }, { "id": 38, "elements": [ 4, 5, 8, 9, 10 ], "cost": 70 }, { "id": 39, "elements": [ 8, 9, 10, 14 ], "cost": 56 }, { "id": 40, "elements": [ 1, 2, 4 ], "cost": 246 }, { "id": 41, "elements": [ 3, 4, 5 ], "cost": 225 }, { "id": 42, "elements": [ 2, 5, 7, 8, 10 ], "cost": 425 }, { "id": 43, "elements": [ 13, 14, 18, 19 ], "cost": 200 }, { "id": 44, "elements": [ 13, 15, 16, 17, 20 ], "cost": 200 }, { "id": 45, "elements": [ 7, 9 ], "cost": 122 }, { "id": 46, "elements": [ 15, 16, 18, 19 ], "cost": 272 }, { "id": 47, "elements": [ 5, 7 ], "cost": 194 }, { "id": 48, "elements": [ 6, 7, 8, 9 ], "cost": 96 }, { "id": 49, "elements": [ 8, 10, 11, 12 ], "cost": 160 }, { "id": 50, "elements": [ 15, 16, 17, 20 ], "cost": 244 }, { "id": 51, "elements": [ 1, 2, 3, 5 ], "cost": 40 }, { "id": 52, "elements": [ 14, 15, 17, 18 ], "cost": 80 }, { "id": 53, "elements": [ 11, 14, 15, 16 ], "cost": 388 }, { "id": 54, "elements": [ 3, 6, 7, 8, 9, 10 ], "cost": 462 }, { "id": 55, "elements": [ 3, 4, 5, 7, 8 ], "cost": 485 }, { "id": 56, "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 78 }, { "id": 57, "elements": [ 8, 9, 10, 11, 12 ], "cost": 195 }, { "id": 58, "elements": [ 9, 10, 11, 12, 14 ], "cost": 50 }, { "id": 59, "elements": [ 13, 14, 19 ], "cost": 252 }, { "id": 60, "elements": [ 13, 15, 16 ], "cost": 234 }, { "id": 61, "elements": [ 15, 18, 19, 20 ], "cost": 224 }, { "id": 62, "elements": [ 1, 3, 4 ], "cost": 165 }, { "id": 63, "elements": [ 11, 13 ], "cost": 98 }, { "id": 64, "elements": [ 8, 9, 10 ], "cost": 96 }, { "id": 65, "elements": [ 10, 12, 14 ], "cost": 132 }, { "id": 66, "elements": [ 12, 13 ], "cost": 164 }, { "id": 67, "elements": [ 2, 7, 8 ], "cost": 66 }, { "id": 68, "elements": [ 3, 6, 7, 8, 13 ], "cost": 305 }, { "id": 69, "elements": [ 12, 14, 15, 16 ], "cost": 100 }, { "id": 70, "elements": [ 2, 6, 7, 10, 11, 14 ], "cost": 270 }, { "id": 71, "elements": [ 17, 19, 20 ], "cost": 141 }, { "id": 72, "elements": [ 11, 13, 14 ], "cost": 123 }, { "id": 73, "elements": [ 14, 17, 18, 19 ], "cost": 28 }, { "id": 74, "elements": [ 8, 10, 11, 12, 13, 15 ], "cost": 252 }, { "id": 75, "elements": [ 15, 17, 18, 19, 20 ], "cost": 235 }, { "id": 76, "elements": [ 13, 14, 15, 16 ], "cost": 48 }, { "id": 77, "elements": [ 9, 12, 14 ], "cost": 159 }, { "id": 78, "elements": [ 7, 9, 10, 11, 13, 15 ], "cost": 414 }, { "id": 79, "elements": [ 2, 4, 5, 6, 7, 8 ], "cost": 84 }, { "id": 80, "elements": [ 11, 12, 15, 17 ], "cost": 108 }, { "id": 81, "elements": [ 10, 12, 13, 15, 16 ], "cost": 320 }, { "id": 82, "elements": [ 3, 4, 9 ], "cost": 99 }, { "id": 83, "elements": [ 5, 6, 8, 9, 10 ], "cost": 445 }, { "id": 84, "elements": [ 13, 14 ], "cost": 46 }, { "id": 85, "elements": [ 16, 18, 20 ], "cost": 3 }, { "id": 86, "elements": [ 13, 17, 18, 19 ], "cost": 384 }, { "id": 87, "elements": [ 1, 3 ], "cost": 2 }, { "id": 88, "elements": [ 15, 17, 18 ], "cost": 255 }, { "id": 89, "elements": [ 4, 5, 6, 8, 9 ], "cost": 350 }, { "id": 90, "elements": [ 9, 12, 13, 15 ], "cost": 40 }, { "id": 91, "elements": [ 1, 3, 4, 5 ], "cost": 400 }, { "id": 92, "elements": [ 2, 6, 8, 9 ], "cost": 40 }, { "id": 93, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 94, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 95, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 96, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 97, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 98, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 20 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0045_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0045_bag_stats.png" }, "solution": [ 29, 43, 50, 79, 87 ], "obj": 550.0, "instance_variant": { "num_elements": 20, "num_sets": 112, "sets": [ { "id": "S1", "elements": [ 7, 10, 12 ], "cost": 267 }, { "id": "S2", "elements": [ 0, 1, 2, 3, 4 ], "cost": 435 }, { "id": "S3", "elements": [ 14, 15, 16, 17, 18 ], "cost": 215 }, { "id": "S4", "elements": [ 0, 3, 5 ], "cost": 195 }, { "id": "S5", "elements": [ 12, 13, 14, 15, 16 ], "cost": 250 }, { "id": "S6", "elements": [ 9, 11, 13, 15 ], "cost": 40 }, { "id": "S7", "elements": [ 0, 3, 4, 5, 7 ], "cost": 130 }, { "id": "S8", "elements": [ 10, 14, 15, 16, 17, 19 ], "cost": 174 }, { "id": "S9", "elements": [ 3, 5, 7, 8 ], "cost": 104 }, { "id": "S10", "elements": [ 6, 9, 10 ], "cost": 267 }, { "id": "S11", "elements": [ 13, 14, 16 ], "cost": 282 }, { "id": "S12", "elements": [ 7, 10, 11 ], "cost": 27 }, { "id": "S13", "elements": [ 11, 12, 15 ], "cost": 114 }, { "id": "S14", "elements": [ 15, 16, 17, 18, 19 ], "cost": 325 }, { "id": "S15", "elements": [ 2, 3, 7, 8 ], "cost": 304 }, { "id": "S16", "elements": [ 13, 15, 16, 17 ], "cost": 212 }, { "id": "S17", "elements": [ 13, 14, 15, 18, 19 ], "cost": 455 }, { "id": "S18", "elements": [ 10, 11, 12, 13, 14, 16 ], "cost": 24 }, { "id": "S19", "elements": [ 1, 2, 4, 5, 6, 7 ], "cost": 198 }, { "id": "S20", "elements": [ 7, 10, 12, 13, 15 ], "cost": 155 }, { "id": "S21", "elements": [ 14, 15, 16, 17, 18, 19 ], "cost": 546 }, { "id": "S22", "elements": [ 0, 1, 2, 3, 4, 5 ], "cost": 162 }, { "id": "S23", "elements": [ 9, 11, 12, 14 ], "cost": 8 }, { "id": "S24", "elements": [ 13, 15, 16, 17, 18, 19 ], "cost": 252 }, { "id": "S25", "elements": [ 9, 12, 13, 14, 18 ], "cost": 5 }, { "id": "S26", "elements": [ 1, 2, 3, 4 ], "cost": 248 }, { "id": "S27", "elements": [ 5, 6, 7, 10 ], "cost": 152 }, { "id": "S28", "elements": [ 1, 2 ], "cost": 84 }, { "id": "S29", "elements": [ 8, 9, 10, 11 ], "cost": 20 }, { "id": "S30", "elements": [ 10, 11, 14 ], "cost": 153 }, { "id": "S31", "elements": [ 14, 18, 19 ], "cost": 105 }, { "id": "S32", "elements": [ 12, 14, 16, 17, 18, 19 ], "cost": 390 }, { "id": "S33", "elements": [ 0, 1, 2, 3 ], "cost": 304 }, { "id": "S34", "elements": [ 11, 12, 14, 15, 18 ], "cost": 35 }, { "id": "S35", "elements": [ 13, 14, 15, 17, 18 ], "cost": 80 }, { "id": "S36", "elements": [ 14, 15, 17, 18, 19 ], "cost": 430 }, { "id": "S37", "elements": [ 5, 7, 8, 9, 10, 11 ], "cost": 312 }, { "id": "S38", "elements": [ 3, 4, 7, 8, 9 ], "cost": 70 }, { "id": "S39", "elements": [ 7, 8, 9, 13 ], "cost": 56 }, { "id": "S40", "elements": [ 0, 1, 3 ], "cost": 246 }, { "id": "S41", "elements": [ 2, 3, 4 ], "cost": 225 }, { "id": "S42", "elements": [ 1, 4, 6, 7, 9 ], "cost": 425 }, { "id": "S43", "elements": [ 12, 13, 17, 18 ], "cost": 200 }, { "id": "S44", "elements": [ 12, 14, 15, 16, 19 ], "cost": 200 }, { "id": "S45", "elements": [ 6, 8 ], "cost": 122 }, { "id": "S46", "elements": [ 14, 15, 17, 18 ], "cost": 272 }, { "id": "S47", "elements": [ 4, 6 ], "cost": 194 }, { "id": "S48", "elements": [ 5, 6, 7, 8 ], "cost": 96 }, { "id": "S49", "elements": [ 7, 9, 10, 11 ], "cost": 160 }, { "id": "S50", "elements": [ 14, 15, 16, 19 ], "cost": 244 }, { "id": "S51", "elements": [ 0, 1, 2, 4 ], "cost": 40 }, { "id": "S52", "elements": [ 13, 14, 16, 17 ], "cost": 80 }, { "id": "S53", "elements": [ 10, 13, 14, 15 ], "cost": 388 }, { "id": "S54", "elements": [ 2, 5, 6, 7, 8, 9 ], "cost": 462 }, { "id": "S55", "elements": [ 2, 3, 4, 6, 7 ], "cost": 485 }, { "id": "S56", "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 78 }, { "id": "S57", "elements": [ 7, 8, 9, 10, 11 ], "cost": 195 }, { "id": "S58", "elements": [ 8, 9, 10, 11, 13 ], "cost": 50 }, { "id": "S59", "elements": [ 12, 13, 18 ], "cost": 252 }, { "id": "S60", "elements": [ 12, 14, 15 ], "cost": 234 }, { "id": "S61", "elements": [ 14, 17, 18, 19 ], "cost": 224 }, { "id": "S62", "elements": [ 0, 2, 3 ], "cost": 165 }, { "id": "S63", "elements": [ 10, 12 ], "cost": 98 }, { "id": "S64", "elements": [ 7, 8, 9 ], "cost": 96 }, { "id": "S65", "elements": [ 9, 11, 13 ], "cost": 132 }, { "id": "S66", "elements": [ 11, 12 ], "cost": 164 }, { "id": "S67", "elements": [ 1, 6, 7 ], "cost": 66 }, { "id": "S68", "elements": [ 2, 5, 6, 7, 12 ], "cost": 305 }, { "id": "S69", "elements": [ 11, 13, 14, 15 ], "cost": 100 }, { "id": "S70", "elements": [ 1, 5, 6, 9, 10, 13 ], "cost": 270 }, { "id": "S71", "elements": [ 16, 18, 19 ], "cost": 141 }, { "id": "S72", "elements": [ 10, 12, 13 ], "cost": 123 }, { "id": "S73", "elements": [ 13, 16, 17, 18 ], "cost": 28 }, { "id": "S74", "elements": [ 7, 9, 10, 11, 12, 14 ], "cost": 252 }, { "id": "S75", "elements": [ 14, 16, 17, 18, 19 ], "cost": 235 }, { "id": "S76", "elements": [ 12, 13, 14, 15 ], "cost": 48 }, { "id": "S77", "elements": [ 8, 11, 13 ], "cost": 159 }, { "id": "S78", "elements": [ 6, 8, 9, 10, 12, 14 ], "cost": 414 }, { "id": "S79", "elements": [ 1, 3, 4, 5, 6, 7 ], "cost": 84 }, { "id": "S80", "elements": [ 10, 11, 14, 16 ], "cost": 108 }, { "id": "S81", "elements": [ 9, 11, 12, 14, 15 ], "cost": 320 }, { "id": "S82", "elements": [ 2, 3, 8 ], "cost": 99 }, { "id": "S83", "elements": [ 4, 5, 7, 8, 9 ], "cost": 445 }, { "id": "S84", "elements": [ 12, 13 ], "cost": 46 }, { "id": "S85", "elements": [ 15, 17, 19 ], "cost": 3 }, { "id": "S86", "elements": [ 12, 16, 17, 18 ], "cost": 384 }, { "id": "S87", "elements": [ 0, 2 ], "cost": 2 }, { "id": "S88", "elements": [ 14, 16, 17 ], "cost": 255 }, { "id": "S89", "elements": [ 3, 4, 5, 7, 8 ], "cost": 350 }, { "id": "S90", "elements": [ 8, 11, 12, 14 ], "cost": 40 }, { "id": "S91", "elements": [ 0, 2, 3, 4 ], "cost": 400 }, { "id": "S92", "elements": [ 1, 5, 7, 8 ], "cost": 40 }, { "id": "S93", "elements": [ 0 ], "cost": 10000 }, { "id": "S94", "elements": [ 1 ], "cost": 10000 }, { "id": "S95", "elements": [ 2 ], "cost": 10000 }, { "id": "S96", "elements": [ 3 ], "cost": 10000 }, { "id": "S97", "elements": [ 4 ], "cost": 10000 }, { "id": "S98", "elements": [ 5 ], "cost": 10000 }, { "id": "S99", "elements": [ 6 ], "cost": 10000 }, { "id": "S100", "elements": [ 7 ], "cost": 10000 }, { "id": "S101", "elements": [ 8 ], "cost": 10000 }, { "id": "S102", "elements": [ 9 ], "cost": 10000 }, { "id": "S103", "elements": [ 10 ], "cost": 10000 }, { "id": "S104", "elements": [ 11 ], "cost": 10000 }, { "id": "S105", "elements": [ 12 ], "cost": 10000 }, { "id": "S106", "elements": [ 13 ], "cost": 10000 }, { "id": "S107", "elements": [ 14 ], "cost": 10000 }, { "id": "S108", "elements": [ 15 ], "cost": 10000 }, { "id": "S109", "elements": [ 16 ], "cost": 10000 }, { "id": "S110", "elements": [ 17 ], "cost": 10000 }, { "id": "S111", "elements": [ 18 ], "cost": 10000 }, { "id": "S112", "elements": [ 19 ], "cost": 10000 } ] }, "solution_variant": [ "S29", "S43", "S50", "S79", "S87" ], "context_index": 46, "input_format": "json", "input_index_base": 0 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "We’ve got a pile of screening blocks to choose from for the festival, and the job is to pick a combination so each film is scheduled in one—and only one—of the blocks, with no overlap between chosen blocks, while keeping the venue expenses down. Practically speaking, that means selecting blocks that together include every title exactly once, checking that none of the selected blocks repeat a film, and then adding up the rental costs of those blocks to get the total price to minimize. The exact lineup and prices are listed below.\n\n# total_films=25\n# total_blocks_available=122\nblock_id,block_rental_cost,films_in_block\nS1,574,15 18 19 21 22 23 24\nS2,112,0 1 2 3 4 5 6\nS3,110,0 1 2 3 6\nS4,120,10 14 15 18 19\nS5,440,17 18 20 22 23\nS6,84,0 3 5 6\nS7,150,11 12 14 16 18\nS8,623,0 1 3 4 6 7 8\nS9,186,7 10 11 12 13 14\nS10,435,15 18 19 20 21\nS11,322,12 13 14 15 16 17 19\nS12,318,0 1 2 3 4 6\nS13,210,0 1 3 5 7 9\nS14,420,11 14 15 16 17 19 21\nS15,150,0 1 2 3 5\nS16,130,16 18 19 23 24\nS17,522,8 9 10 11 12 16\nS18,56,19 21 22 24\nS19,231,10 14 17\nS20,300,17 18 20 22\nS21,600,0 1 2 3 4 5\nS22,42,14 15 16 17 19 20\nS23,164,18 19 21 23\nS24,150,18 19 20\nS25,322,2 3 4 5 6 7 8\nS26,336,6 8 12 14\nS27,455,8 9 10 11 12\nS28,441,0 1 2 3 4 6 8\nS29,316,18 19 20 22\nS30,150,9 12 13\nS31,616,4 7 8 9 10 12 14\nS32,376,9 10 13 14\nS33,205,0 1 2 4 5\nS34,70,16 18 19 21 22\nS35,532,13 15 16 17 18 19 21\nS36,120,7 9 10\nS37,90,0 2 3 4 7 10\nS38,112,8 9 11 12 13 14 15\nS39,198,15 17 18\nS40,208,16 18 21 22\nS41,216,14 15 18\nS42,162,7 9 10 13 14 15\nS43,192,9 10 11\nS44,7,10 11 12 13 14 15 16\nS45,18,21 22 24\nS46,120,20 23 24\nS47,455,0 1 2 3 4 6 7\nS48,552,7 8 9 10 12 17\nS49,385,20 21 22 23 24\nS50,470,3 4 5 8 9\nS51,434,15 16 17 18 19 20 21\nS52,392,0 1 2 3 4 5 7\nS53,228,11 12 14 15 16 18\nS54,182,17 18 19 20 21 23 24\nS55,140,4 9 10 12 14 15 16\nS56,686,11 15 17 18 19 23 24\nS57,152,16 21 22 23\nS58,72,15 16 18 19\nS59,210,1 2 3 4 5 7 8\nS60,35,8 9 10 11 17\nS61,357,7 10 12 13 14 17 18\nS62,186,0 2 3 4 5 8\nS63,80,19 22 23 24\nS64,504,17 18 19 20 21 23\nS65,252,4 5 9\nS66,270,11 12 18\nS67,258,15 18 20\nS68,15,15 20 22\nS69,504,11 12 14 15 16 17 19\nS70,16,9 10 11 12\nS71,150,7 8 9 11 12 14\nS72,224,20 21 23 24\nS73,498,5 7 9 10 11 13\nS74,294,20 21 22\nS75,196,18 19 20 21 22 23 24\nS76,99,3 5 7\nS77,245,5 6 8 9 10 11 16\nS78,105,14 21 24\nS79,516,14 15 17 18 20 22\nS80,582,16 17 18 19 21 22\nS81,301,1 2 6 7 8 10 11\nS82,497,11 12 13 14 15 18 23\nS83,651,1 2 3 4 5 6 9\nS84,371,8 9 10 11 12 13 15\nS85,228,13 15 16 17 19 23\nS86,476,10 11 12 14 15 16 17\nS87,525,8 10 12 13 14 15 16\nS88,651,14 16 19 20 21 22 23\nS89,322,17 19 20 21 22 23 24\nS90,40,14 15 16 17 18\nS91,140,12 14 16 18\nS92,168,15 16 17 19\nS93,57,0 6 7\nS94,390,15 16 17 18 20 21\nS95,288,19 20 21 22 23 24\nS96,292,7 8 9 10\nS97,130,13 14 15 16 17\nS98,10000,0\nS99,10000,1\nS100,10000,2\nS101,10000,3\nS102,10000,4\nS103,10000,5\nS104,10000,6\nS105,10000,7\nS106,10000,8\nS107,10000,9\nS108,10000,10\nS109,10000,11\nS110,10000,12\nS111,10000,13\nS112,10000,14\nS113,10000,15\nS114,10000,16\nS115,10000,17\nS116,10000,18\nS117,10000,19\nS118,10000,20\nS119,10000,21\nS120,10000,22\nS121,10000,23\nS122,10000,24\n\nOh, and when you send me the chosen blocks, please use this simple JSON layout so I can read it easily.\n\n{\n \"solution\": [\"block_id\", ...]\n}\n\nThink of \"solution\" as the list of screening blocks you pick for the lineup — each entry in the array is the identifier for one chosen block (the \"block_id\" in the sketch above). That JSON is just a template to show the shape I expect, not the actual answer itself — replace the placeholders with the real IDs from the instance.\n\nPlease be sure to use the exact identifiers from the instance input — do not rename them or invent new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.", "instance": { "problem_type": "SPP", "num_elements": 25, "num_sets": 122, "density": 0.17672131147540984, "sets": [ { "id": 1, "elements": [ 16, 19, 20, 22, 23, 24, 25 ], "cost": 574 }, { "id": 2, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 112 }, { "id": 3, "elements": [ 1, 2, 3, 4, 7 ], "cost": 110 }, { "id": 4, "elements": [ 11, 15, 16, 19, 20 ], "cost": 120 }, { "id": 5, "elements": [ 18, 19, 21, 23, 24 ], "cost": 440 }, { "id": 6, "elements": [ 1, 4, 6, 7 ], "cost": 84 }, { "id": 7, "elements": [ 12, 13, 15, 17, 19 ], "cost": 150 }, { "id": 8, "elements": [ 1, 2, 4, 5, 7, 8, 9 ], "cost": 623 }, { "id": 9, "elements": [ 8, 11, 12, 13, 14, 15 ], "cost": 186 }, { "id": 10, "elements": [ 16, 19, 20, 21, 22 ], "cost": 435 }, { "id": 11, "elements": [ 13, 14, 15, 16, 17, 18, 20 ], "cost": 322 }, { "id": 12, "elements": [ 1, 2, 3, 4, 5, 7 ], "cost": 318 }, { "id": 13, "elements": [ 1, 2, 4, 6, 8, 10 ], "cost": 210 }, { "id": 14, "elements": [ 12, 15, 16, 17, 18, 20, 22 ], "cost": 420 }, { "id": 15, "elements": [ 1, 2, 3, 4, 6 ], "cost": 150 }, { "id": 16, "elements": [ 17, 19, 20, 24, 25 ], "cost": 130 }, { "id": 17, "elements": [ 9, 10, 11, 12, 13, 17 ], "cost": 522 }, { "id": 18, "elements": [ 20, 22, 23, 25 ], "cost": 56 }, { "id": 19, "elements": [ 11, 15, 18 ], "cost": 231 }, { "id": 20, "elements": [ 18, 19, 21, 23 ], "cost": 300 }, { "id": 21, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 600 }, { "id": 22, "elements": [ 15, 16, 17, 18, 20, 21 ], "cost": 42 }, { "id": 23, "elements": [ 19, 20, 22, 24 ], "cost": 164 }, { "id": 24, "elements": [ 19, 20, 21 ], "cost": 150 }, { "id": 25, "elements": [ 3, 4, 5, 6, 7, 8, 9 ], "cost": 322 }, { "id": 26, "elements": [ 7, 9, 13, 15 ], "cost": 336 }, { "id": 27, "elements": [ 9, 10, 11, 12, 13 ], "cost": 455 }, { "id": 28, "elements": [ 1, 2, 3, 4, 5, 7, 9 ], "cost": 441 }, { "id": 29, "elements": [ 19, 20, 21, 23 ], "cost": 316 }, { "id": 30, "elements": [ 10, 13, 14 ], "cost": 150 }, { "id": 31, "elements": [ 5, 8, 9, 10, 11, 13, 15 ], "cost": 616 }, { "id": 32, "elements": [ 10, 11, 14, 15 ], "cost": 376 }, { "id": 33, "elements": [ 1, 2, 3, 5, 6 ], "cost": 205 }, { "id": 34, "elements": [ 17, 19, 20, 22, 23 ], "cost": 70 }, { "id": 35, "elements": [ 14, 16, 17, 18, 19, 20, 22 ], "cost": 532 }, { "id": 36, "elements": [ 8, 10, 11 ], "cost": 120 }, { "id": 37, "elements": [ 1, 3, 4, 5, 8, 11 ], "cost": 90 }, { "id": 38, "elements": [ 9, 10, 12, 13, 14, 15, 16 ], "cost": 112 }, { "id": 39, "elements": [ 16, 18, 19 ], "cost": 198 }, { "id": 40, "elements": [ 17, 19, 22, 23 ], "cost": 208 }, { "id": 41, "elements": [ 15, 16, 19 ], "cost": 216 }, { "id": 42, "elements": [ 8, 10, 11, 14, 15, 16 ], "cost": 162 }, { "id": 43, "elements": [ 10, 11, 12 ], "cost": 192 }, { "id": 44, "elements": [ 11, 12, 13, 14, 15, 16, 17 ], "cost": 7 }, { "id": 45, "elements": [ 22, 23, 25 ], "cost": 18 }, { "id": 46, "elements": [ 21, 24, 25 ], "cost": 120 }, { "id": 47, "elements": [ 1, 2, 3, 4, 5, 7, 8 ], "cost": 455 }, { "id": 48, "elements": [ 8, 9, 10, 11, 13, 18 ], "cost": 552 }, { "id": 49, "elements": [ 21, 22, 23, 24, 25 ], "cost": 385 }, { "id": 50, "elements": [ 4, 5, 6, 9, 10 ], "cost": 470 }, { "id": 51, "elements": [ 16, 17, 18, 19, 20, 21, 22 ], "cost": 434 }, { "id": 52, "elements": [ 1, 2, 3, 4, 5, 6, 8 ], "cost": 392 }, { "id": 53, "elements": [ 12, 13, 15, 16, 17, 19 ], "cost": 228 }, { "id": 54, "elements": [ 18, 19, 20, 21, 22, 24, 25 ], "cost": 182 }, { "id": 55, "elements": [ 5, 10, 11, 13, 15, 16, 17 ], "cost": 140 }, { "id": 56, "elements": [ 12, 16, 18, 19, 20, 24, 25 ], "cost": 686 }, { "id": 57, "elements": [ 17, 22, 23, 24 ], "cost": 152 }, { "id": 58, "elements": [ 16, 17, 19, 20 ], "cost": 72 }, { "id": 59, "elements": [ 2, 3, 4, 5, 6, 8, 9 ], "cost": 210 }, { "id": 60, "elements": [ 9, 10, 11, 12, 18 ], "cost": 35 }, { "id": 61, "elements": [ 8, 11, 13, 14, 15, 18, 19 ], "cost": 357 }, { "id": 62, "elements": [ 1, 3, 4, 5, 6, 9 ], "cost": 186 }, { "id": 63, "elements": [ 20, 23, 24, 25 ], "cost": 80 }, { "id": 64, "elements": [ 18, 19, 20, 21, 22, 24 ], "cost": 504 }, { "id": 65, "elements": [ 5, 6, 10 ], "cost": 252 }, { "id": 66, "elements": [ 12, 13, 19 ], "cost": 270 }, { "id": 67, "elements": [ 16, 19, 21 ], "cost": 258 }, { "id": 68, "elements": [ 16, 21, 23 ], "cost": 15 }, { "id": 69, "elements": [ 12, 13, 15, 16, 17, 18, 20 ], "cost": 504 }, { "id": 70, "elements": [ 10, 11, 12, 13 ], "cost": 16 }, { "id": 71, "elements": [ 8, 9, 10, 12, 13, 15 ], "cost": 150 }, { "id": 72, "elements": [ 21, 22, 24, 25 ], "cost": 224 }, { "id": 73, "elements": [ 6, 8, 10, 11, 12, 14 ], "cost": 498 }, { "id": 74, "elements": [ 21, 22, 23 ], "cost": 294 }, { "id": 75, "elements": [ 19, 20, 21, 22, 23, 24, 25 ], "cost": 196 }, { "id": 76, "elements": [ 4, 6, 8 ], "cost": 99 }, { "id": 77, "elements": [ 6, 7, 9, 10, 11, 12, 17 ], "cost": 245 }, { "id": 78, "elements": [ 15, 22, 25 ], "cost": 105 }, { "id": 79, "elements": [ 15, 16, 18, 19, 21, 23 ], "cost": 516 }, { "id": 80, "elements": [ 17, 18, 19, 20, 22, 23 ], "cost": 582 }, { "id": 81, "elements": [ 2, 3, 7, 8, 9, 11, 12 ], "cost": 301 }, { "id": 82, "elements": [ 12, 13, 14, 15, 16, 19, 24 ], "cost": 497 }, { "id": 83, "elements": [ 2, 3, 4, 5, 6, 7, 10 ], "cost": 651 }, { "id": 84, "elements": [ 9, 10, 11, 12, 13, 14, 16 ], "cost": 371 }, { "id": 85, "elements": [ 14, 16, 17, 18, 20, 24 ], "cost": 228 }, { "id": 86, "elements": [ 11, 12, 13, 15, 16, 17, 18 ], "cost": 476 }, { "id": 87, "elements": [ 9, 11, 13, 14, 15, 16, 17 ], "cost": 525 }, { "id": 88, "elements": [ 15, 17, 20, 21, 22, 23, 24 ], "cost": 651 }, { "id": 89, "elements": [ 18, 20, 21, 22, 23, 24, 25 ], "cost": 322 }, { "id": 90, "elements": [ 15, 16, 17, 18, 19 ], "cost": 40 }, { "id": 91, "elements": [ 13, 15, 17, 19 ], "cost": 140 }, { "id": 92, "elements": [ 16, 17, 18, 20 ], "cost": 168 }, { "id": 93, "elements": [ 1, 7, 8 ], "cost": 57 }, { "id": 94, "elements": [ 16, 17, 18, 19, 21, 22 ], "cost": 390 }, { "id": 95, "elements": [ 20, 21, 22, 23, 24, 25 ], "cost": 288 }, { "id": 96, "elements": [ 8, 9, 10, 11 ], "cost": 292 }, { "id": 97, "elements": [ 14, 15, 16, 17, 18 ], "cost": 130 }, { "id": 98, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 118, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 119, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 120, "elements": [ 23 ], "cost": 10000, "singleton": true }, { "id": 121, "elements": [ 24 ], "cost": 10000, "singleton": true }, { "id": 122, "elements": [ 25 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0046_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0046_bag_stats.png" }, "solution": [ 2, 66, 95, 96, 97 ], "obj": 1092.0, "instance_variant": { "num_elements": 25, "num_sets": 122, "sets": [ { "id": "S1", "elements": [ 15, 18, 19, 21, 22, 23, 24 ], "cost": 574 }, { "id": "S2", "elements": [ 0, 1, 2, 3, 4, 5, 6 ], "cost": 112 }, { "id": "S3", "elements": [ 0, 1, 2, 3, 6 ], "cost": 110 }, { "id": "S4", "elements": [ 10, 14, 15, 18, 19 ], "cost": 120 }, { "id": "S5", "elements": [ 17, 18, 20, 22, 23 ], "cost": 440 }, { "id": "S6", "elements": [ 0, 3, 5, 6 ], "cost": 84 }, { "id": "S7", "elements": [ 11, 12, 14, 16, 18 ], "cost": 150 }, { "id": "S8", "elements": [ 0, 1, 3, 4, 6, 7, 8 ], "cost": 623 }, { "id": "S9", "elements": [ 7, 10, 11, 12, 13, 14 ], "cost": 186 }, { "id": "S10", "elements": [ 15, 18, 19, 20, 21 ], "cost": 435 }, { "id": "S11", "elements": [ 12, 13, 14, 15, 16, 17, 19 ], "cost": 322 }, { "id": "S12", "elements": [ 0, 1, 2, 3, 4, 6 ], "cost": 318 }, { "id": "S13", "elements": [ 0, 1, 3, 5, 7, 9 ], "cost": 210 }, { "id": "S14", "elements": [ 11, 14, 15, 16, 17, 19, 21 ], "cost": 420 }, { "id": "S15", "elements": [ 0, 1, 2, 3, 5 ], "cost": 150 }, { "id": "S16", "elements": [ 16, 18, 19, 23, 24 ], "cost": 130 }, { "id": "S17", "elements": [ 8, 9, 10, 11, 12, 16 ], "cost": 522 }, { "id": "S18", "elements": [ 19, 21, 22, 24 ], "cost": 56 }, { "id": "S19", "elements": [ 10, 14, 17 ], "cost": 231 }, { "id": "S20", "elements": [ 17, 18, 20, 22 ], "cost": 300 }, { "id": "S21", "elements": [ 0, 1, 2, 3, 4, 5 ], "cost": 600 }, { "id": "S22", "elements": [ 14, 15, 16, 17, 19, 20 ], "cost": 42 }, { "id": "S23", "elements": [ 18, 19, 21, 23 ], "cost": 164 }, { "id": "S24", "elements": [ 18, 19, 20 ], "cost": 150 }, { "id": "S25", "elements": [ 2, 3, 4, 5, 6, 7, 8 ], "cost": 322 }, { "id": "S26", "elements": [ 6, 8, 12, 14 ], "cost": 336 }, { "id": "S27", "elements": [ 8, 9, 10, 11, 12 ], "cost": 455 }, { "id": "S28", "elements": [ 0, 1, 2, 3, 4, 6, 8 ], "cost": 441 }, { "id": "S29", "elements": [ 18, 19, 20, 22 ], "cost": 316 }, { "id": "S30", "elements": [ 9, 12, 13 ], "cost": 150 }, { "id": "S31", "elements": [ 4, 7, 8, 9, 10, 12, 14 ], "cost": 616 }, { "id": "S32", "elements": [ 9, 10, 13, 14 ], "cost": 376 }, { "id": "S33", "elements": [ 0, 1, 2, 4, 5 ], "cost": 205 }, { "id": "S34", "elements": [ 16, 18, 19, 21, 22 ], "cost": 70 }, { "id": "S35", "elements": [ 13, 15, 16, 17, 18, 19, 21 ], "cost": 532 }, { "id": "S36", "elements": [ 7, 9, 10 ], "cost": 120 }, { "id": "S37", "elements": [ 0, 2, 3, 4, 7, 10 ], "cost": 90 }, { "id": "S38", "elements": [ 8, 9, 11, 12, 13, 14, 15 ], "cost": 112 }, { "id": "S39", "elements": [ 15, 17, 18 ], "cost": 198 }, { "id": "S40", "elements": [ 16, 18, 21, 22 ], "cost": 208 }, { "id": "S41", "elements": [ 14, 15, 18 ], "cost": 216 }, { "id": "S42", "elements": [ 7, 9, 10, 13, 14, 15 ], "cost": 162 }, { "id": "S43", "elements": [ 9, 10, 11 ], "cost": 192 }, { "id": "S44", "elements": [ 10, 11, 12, 13, 14, 15, 16 ], "cost": 7 }, { "id": "S45", "elements": [ 21, 22, 24 ], "cost": 18 }, { "id": "S46", "elements": [ 20, 23, 24 ], "cost": 120 }, { "id": "S47", "elements": [ 0, 1, 2, 3, 4, 6, 7 ], "cost": 455 }, { "id": "S48", "elements": [ 7, 8, 9, 10, 12, 17 ], "cost": 552 }, { "id": "S49", "elements": [ 20, 21, 22, 23, 24 ], "cost": 385 }, { "id": "S50", "elements": [ 3, 4, 5, 8, 9 ], "cost": 470 }, { "id": "S51", "elements": [ 15, 16, 17, 18, 19, 20, 21 ], "cost": 434 }, { "id": "S52", "elements": [ 0, 1, 2, 3, 4, 5, 7 ], "cost": 392 }, { "id": "S53", "elements": [ 11, 12, 14, 15, 16, 18 ], "cost": 228 }, { "id": "S54", "elements": [ 17, 18, 19, 20, 21, 23, 24 ], "cost": 182 }, { "id": "S55", "elements": [ 4, 9, 10, 12, 14, 15, 16 ], "cost": 140 }, { "id": "S56", "elements": [ 11, 15, 17, 18, 19, 23, 24 ], "cost": 686 }, { "id": "S57", "elements": [ 16, 21, 22, 23 ], "cost": 152 }, { "id": "S58", "elements": [ 15, 16, 18, 19 ], "cost": 72 }, { "id": "S59", "elements": [ 1, 2, 3, 4, 5, 7, 8 ], "cost": 210 }, { "id": "S60", "elements": [ 8, 9, 10, 11, 17 ], "cost": 35 }, { "id": "S61", "elements": [ 7, 10, 12, 13, 14, 17, 18 ], "cost": 357 }, { "id": "S62", "elements": [ 0, 2, 3, 4, 5, 8 ], "cost": 186 }, { "id": "S63", "elements": [ 19, 22, 23, 24 ], "cost": 80 }, { "id": "S64", "elements": [ 17, 18, 19, 20, 21, 23 ], "cost": 504 }, { "id": "S65", "elements": [ 4, 5, 9 ], "cost": 252 }, { "id": "S66", "elements": [ 11, 12, 18 ], "cost": 270 }, { "id": "S67", "elements": [ 15, 18, 20 ], "cost": 258 }, { "id": "S68", "elements": [ 15, 20, 22 ], "cost": 15 }, { "id": "S69", "elements": [ 11, 12, 14, 15, 16, 17, 19 ], "cost": 504 }, { "id": "S70", "elements": [ 9, 10, 11, 12 ], "cost": 16 }, { "id": "S71", "elements": [ 7, 8, 9, 11, 12, 14 ], "cost": 150 }, { "id": "S72", "elements": [ 20, 21, 23, 24 ], "cost": 224 }, { "id": "S73", "elements": [ 5, 7, 9, 10, 11, 13 ], "cost": 498 }, { "id": "S74", "elements": [ 20, 21, 22 ], "cost": 294 }, { "id": "S75", "elements": [ 18, 19, 20, 21, 22, 23, 24 ], "cost": 196 }, { "id": "S76", "elements": [ 3, 5, 7 ], "cost": 99 }, { "id": "S77", "elements": [ 5, 6, 8, 9, 10, 11, 16 ], "cost": 245 }, { "id": "S78", "elements": [ 14, 21, 24 ], "cost": 105 }, { "id": "S79", "elements": [ 14, 15, 17, 18, 20, 22 ], "cost": 516 }, { "id": "S80", "elements": [ 16, 17, 18, 19, 21, 22 ], "cost": 582 }, { "id": "S81", "elements": [ 1, 2, 6, 7, 8, 10, 11 ], "cost": 301 }, { "id": "S82", "elements": [ 11, 12, 13, 14, 15, 18, 23 ], "cost": 497 }, { "id": "S83", "elements": [ 1, 2, 3, 4, 5, 6, 9 ], "cost": 651 }, { "id": "S84", "elements": [ 8, 9, 10, 11, 12, 13, 15 ], "cost": 371 }, { "id": "S85", "elements": [ 13, 15, 16, 17, 19, 23 ], "cost": 228 }, { "id": "S86", "elements": [ 10, 11, 12, 14, 15, 16, 17 ], "cost": 476 }, { "id": "S87", "elements": [ 8, 10, 12, 13, 14, 15, 16 ], "cost": 525 }, { "id": "S88", "elements": [ 14, 16, 19, 20, 21, 22, 23 ], "cost": 651 }, { "id": "S89", "elements": [ 17, 19, 20, 21, 22, 23, 24 ], "cost": 322 }, { "id": "S90", "elements": [ 14, 15, 16, 17, 18 ], "cost": 40 }, { "id": "S91", "elements": [ 12, 14, 16, 18 ], "cost": 140 }, { "id": "S92", "elements": [ 15, 16, 17, 19 ], "cost": 168 }, { "id": "S93", "elements": [ 0, 6, 7 ], "cost": 57 }, { "id": "S94", "elements": [ 15, 16, 17, 18, 20, 21 ], "cost": 390 }, { "id": "S95", "elements": [ 19, 20, 21, 22, 23, 24 ], "cost": 288 }, { "id": "S96", "elements": [ 7, 8, 9, 10 ], "cost": 292 }, { "id": "S97", "elements": [ 13, 14, 15, 16, 17 ], "cost": 130 }, { "id": "S98", "elements": [ 0 ], "cost": 10000 }, { "id": "S99", "elements": [ 1 ], "cost": 10000 }, { "id": "S100", "elements": [ 2 ], "cost": 10000 }, { "id": "S101", "elements": [ 3 ], "cost": 10000 }, { "id": "S102", "elements": [ 4 ], "cost": 10000 }, { "id": "S103", "elements": [ 5 ], "cost": 10000 }, { "id": "S104", "elements": [ 6 ], "cost": 10000 }, { "id": "S105", "elements": [ 7 ], "cost": 10000 }, { "id": "S106", "elements": [ 8 ], "cost": 10000 }, { "id": "S107", "elements": [ 9 ], "cost": 10000 }, { "id": "S108", "elements": [ 10 ], "cost": 10000 }, { "id": "S109", "elements": [ 11 ], "cost": 10000 }, { "id": "S110", "elements": [ 12 ], "cost": 10000 }, { "id": "S111", "elements": [ 13 ], "cost": 10000 }, { "id": "S112", "elements": [ 14 ], "cost": 10000 }, { "id": "S113", "elements": [ 15 ], "cost": 10000 }, { "id": "S114", "elements": [ 16 ], "cost": 10000 }, { "id": "S115", "elements": [ 17 ], "cost": 10000 }, { "id": "S116", "elements": [ 18 ], "cost": 10000 }, { "id": "S117", "elements": [ 19 ], "cost": 10000 }, { "id": "S118", "elements": [ 20 ], "cost": 10000 }, { "id": "S119", "elements": [ 21 ], "cost": 10000 }, { "id": "S120", "elements": [ 22 ], "cost": 10000 }, { "id": "S121", "elements": [ 23 ], "cost": 10000 }, { "id": "S122", "elements": [ 24 ], "cost": 10000 } ] }, "solution_variant": [ "S2", "S66", "S95", "S96", "S97" ], "context_index": 47, "input_format": "csv", "input_index_base": 0 }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "That Saturday the planting rota required a pick: select seed packs so each garden bed gets seeds from a single chosen pack and the chosen packs don’t overlap on any bed. What makes one selection better than another is the total money spent — just add up the costs of the packs in the selection and aim for the lowest total. The exact beds, pack coverage, and prices are shown below.\n\n# num_beds=20\n# num_packs=109\npack_id,pack_price,beds_covered\nS1,66,L M N O P Q\nS2,455,M N O R S\nS3,25,K L M N P\nS4,192,A F G\nS5,94,B C\nS6,500,A B C D E\nS7,111,F I L\nS8,340,P Q R S T\nS9,220,N P Q R\nS10,55,E F I J L\nS11,244,I K M N\nS12,210,I J M N O\nS13,141,F G K\nS14,132,M N O P Q R\nS15,30,N O P Q R T\nS16,276,K L O\nS17,480,K N O P R\nS18,160,I J K L M\nS19,24,P R S\nS20,498,N P Q R S T\nS21,430,B D E H J\nS22,430,N O P Q R\nS23,55,O P Q R T\nS24,69,A D E\nS25,177,H L N\nS26,195,N P Q\nS27,522,O P Q R S T\nS28,72,C D E F\nS29,8,O S\nS30,172,G K\nS31,176,A B C D\nS32,146,S T\nS33,10,L N O P Q\nS34,178,H J\nS35,332,E F G H\nS36,51,R S T\nS37,64,M O P T\nS38,132,B E F H\nS39,261,G H K\nS40,68,N P R S\nS41,8,G J K M\nS42,186,M P R\nS43,106,B H\nS44,68,R S\nS45,576,K L M N O Q\nS46,198,B E\nS47,285,K M N\nS48,148,C D\nS49,285,D E G\nS50,200,O P Q R\nS51,264,K O R\nS52,344,H I J K\nS53,198,B C D E F G\nS54,132,A C D\nS55,76,I K L N\nS56,330,A B C E F\nS57,380,A B D E\nS58,160,E F G H I\nS59,308,I L M O\nS60,129,Q R S\nS61,36,M Q R\nS62,84,K P Q\nS63,270,B C D E G\nS64,45,G H I J L\nS65,212,C E F G\nS66,145,M O P Q T\nS67,280,G K L M N\nS68,69,E F H\nS69,272,E G J K\nS70,336,C D F G\nS71,256,O Q R S\nS72,60,H I J\nS73,42,C E I K L N\nS74,180,A B C E\nS75,415,E F I K L\nS76,224,L M N R\nS77,188,G I\nS78,392,L M O P\nS79,288,G H J\nS80,356,E F H K\nS81,112,A C F G\nS82,28,E G H L\nS83,246,D E F\nS84,160,N P\nS85,252,Q R S T\nS86,300,M O P R\nS87,272,G H I J\nS88,264,O P Q\nS89,237,J L N\nS90,10000,A\nS91,10000,B\nS92,10000,C\nS93,10000,D\nS94,10000,E\nS95,10000,F\nS96,10000,G\nS97,10000,H\nS98,10000,I\nS99,10000,J\nS100,10000,K\nS101,10000,L\nS102,10000,M\nS103,10000,N\nS104,10000,O\nS105,10000,P\nS106,10000,Q\nS107,10000,R\nS108,10000,S\nS109,10000,T\n\nWhen you're ready to give the final pick, just send it in this little JSON shape so it's easy to read and check:\n\n{\n \"solution\": [\"pack_id\", ...]\n}\n\nThis just says \"solution\" with a list of chosen seed pack identifiers (the \"pack_id\" items are placeholders for the real pack IDs from the instance). It's just a sketch of the shape I expect — replace those placeholders with the actual pack IDs when you answer.\n\nPlease make sure all identifiers are used exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"1\" or \"23\", \"A\" or \"B\", \"A1\" or \"X7\"", "instance": { "problem_type": "SPP", "num_elements": 20, "num_sets": 109, "density": 0.16743119266055045, "sets": [ { "id": 1, "elements": [ 12, 13, 14, 15, 16, 17 ], "cost": 66 }, { "id": 2, "elements": [ 13, 14, 15, 18, 19 ], "cost": 455 }, { "id": 3, "elements": [ 11, 12, 13, 14, 16 ], "cost": 25 }, { "id": 4, "elements": [ 1, 6, 7 ], "cost": 192 }, { "id": 5, "elements": [ 2, 3 ], "cost": 94 }, { "id": 6, "elements": [ 1, 2, 3, 4, 5 ], "cost": 500 }, { "id": 7, "elements": [ 6, 9, 12 ], "cost": 111 }, { "id": 8, "elements": [ 16, 17, 18, 19, 20 ], "cost": 340 }, { "id": 9, "elements": [ 14, 16, 17, 18 ], "cost": 220 }, { "id": 10, "elements": [ 5, 6, 9, 10, 12 ], "cost": 55 }, { "id": 11, "elements": [ 9, 11, 13, 14 ], "cost": 244 }, { "id": 12, "elements": [ 9, 10, 13, 14, 15 ], "cost": 210 }, { "id": 13, "elements": [ 6, 7, 11 ], "cost": 141 }, { "id": 14, "elements": [ 13, 14, 15, 16, 17, 18 ], "cost": 132 }, { "id": 15, "elements": [ 14, 15, 16, 17, 18, 20 ], "cost": 30 }, { "id": 16, "elements": [ 11, 12, 15 ], "cost": 276 }, { "id": 17, "elements": [ 11, 14, 15, 16, 18 ], "cost": 480 }, { "id": 18, "elements": [ 9, 10, 11, 12, 13 ], "cost": 160 }, { "id": 19, "elements": [ 16, 18, 19 ], "cost": 24 }, { "id": 20, "elements": [ 14, 16, 17, 18, 19, 20 ], "cost": 498 }, { "id": 21, "elements": [ 2, 4, 5, 8, 10 ], "cost": 430 }, { "id": 22, "elements": [ 14, 15, 16, 17, 18 ], "cost": 430 }, { "id": 23, "elements": [ 15, 16, 17, 18, 20 ], "cost": 55 }, { "id": 24, "elements": [ 1, 4, 5 ], "cost": 69 }, { "id": 25, "elements": [ 8, 12, 14 ], "cost": 177 }, { "id": 26, "elements": [ 14, 16, 17 ], "cost": 195 }, { "id": 27, "elements": [ 15, 16, 17, 18, 19, 20 ], "cost": 522 }, { "id": 28, "elements": [ 3, 4, 5, 6 ], "cost": 72 }, { "id": 29, "elements": [ 15, 19 ], "cost": 8 }, { "id": 30, "elements": [ 7, 11 ], "cost": 172 }, { "id": 31, "elements": [ 1, 2, 3, 4 ], "cost": 176 }, { "id": 32, "elements": [ 19, 20 ], "cost": 146 }, { "id": 33, "elements": [ 12, 14, 15, 16, 17 ], "cost": 10 }, { "id": 34, "elements": [ 8, 10 ], "cost": 178 }, { "id": 35, "elements": [ 5, 6, 7, 8 ], "cost": 332 }, { "id": 36, "elements": [ 18, 19, 20 ], "cost": 51 }, { "id": 37, "elements": [ 13, 15, 16, 20 ], "cost": 64 }, { "id": 38, "elements": [ 2, 5, 6, 8 ], "cost": 132 }, { "id": 39, "elements": [ 7, 8, 11 ], "cost": 261 }, { "id": 40, "elements": [ 14, 16, 18, 19 ], "cost": 68 }, { "id": 41, "elements": [ 7, 10, 11, 13 ], "cost": 8 }, { "id": 42, "elements": [ 13, 16, 18 ], "cost": 186 }, { "id": 43, "elements": [ 2, 8 ], "cost": 106 }, { "id": 44, "elements": [ 18, 19 ], "cost": 68 }, { "id": 45, "elements": [ 11, 12, 13, 14, 15, 17 ], "cost": 576 }, { "id": 46, "elements": [ 2, 5 ], "cost": 198 }, { "id": 47, "elements": [ 11, 13, 14 ], "cost": 285 }, { "id": 48, "elements": [ 3, 4 ], "cost": 148 }, { "id": 49, "elements": [ 4, 5, 7 ], "cost": 285 }, { "id": 50, "elements": [ 15, 16, 17, 18 ], "cost": 200 }, { "id": 51, "elements": [ 11, 15, 18 ], "cost": 264 }, { "id": 52, "elements": [ 8, 9, 10, 11 ], "cost": 344 }, { "id": 53, "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 198 }, { "id": 54, "elements": [ 1, 3, 4 ], "cost": 132 }, { "id": 55, "elements": [ 9, 11, 12, 14 ], "cost": 76 }, { "id": 56, "elements": [ 1, 2, 3, 5, 6 ], "cost": 330 }, { "id": 57, "elements": [ 1, 2, 4, 5 ], "cost": 380 }, { "id": 58, "elements": [ 5, 6, 7, 8, 9 ], "cost": 160 }, { "id": 59, "elements": [ 9, 12, 13, 15 ], "cost": 308 }, { "id": 60, "elements": [ 17, 18, 19 ], "cost": 129 }, { "id": 61, "elements": [ 13, 17, 18 ], "cost": 36 }, { "id": 62, "elements": [ 11, 16, 17 ], "cost": 84 }, { "id": 63, "elements": [ 2, 3, 4, 5, 7 ], "cost": 270 }, { "id": 64, "elements": [ 7, 8, 9, 10, 12 ], "cost": 45 }, { "id": 65, "elements": [ 3, 5, 6, 7 ], "cost": 212 }, { "id": 66, "elements": [ 13, 15, 16, 17, 20 ], "cost": 145 }, { "id": 67, "elements": [ 7, 11, 12, 13, 14 ], "cost": 280 }, { "id": 68, "elements": [ 5, 6, 8 ], "cost": 69 }, { "id": 69, "elements": [ 5, 7, 10, 11 ], "cost": 272 }, { "id": 70, "elements": [ 3, 4, 6, 7 ], "cost": 336 }, { "id": 71, "elements": [ 15, 17, 18, 19 ], "cost": 256 }, { "id": 72, "elements": [ 8, 9, 10 ], "cost": 60 }, { "id": 73, "elements": [ 3, 5, 9, 11, 12, 14 ], "cost": 42 }, { "id": 74, "elements": [ 1, 2, 3, 5 ], "cost": 180 }, { "id": 75, "elements": [ 5, 6, 9, 11, 12 ], "cost": 415 }, { "id": 76, "elements": [ 12, 13, 14, 18 ], "cost": 224 }, { "id": 77, "elements": [ 7, 9 ], "cost": 188 }, { "id": 78, "elements": [ 12, 13, 15, 16 ], "cost": 392 }, { "id": 79, "elements": [ 7, 8, 10 ], "cost": 288 }, { "id": 80, "elements": [ 5, 6, 8, 11 ], "cost": 356 }, { "id": 81, "elements": [ 1, 3, 6, 7 ], "cost": 112 }, { "id": 82, "elements": [ 5, 7, 8, 12 ], "cost": 28 }, { "id": 83, "elements": [ 4, 5, 6 ], "cost": 246 }, { "id": 84, "elements": [ 14, 16 ], "cost": 160 }, { "id": 85, "elements": [ 17, 18, 19, 20 ], "cost": 252 }, { "id": 86, "elements": [ 13, 15, 16, 18 ], "cost": 300 }, { "id": 87, "elements": [ 7, 8, 9, 10 ], "cost": 272 }, { "id": 88, "elements": [ 15, 16, 17 ], "cost": 264 }, { "id": 89, "elements": [ 10, 12, 14 ], "cost": 237 }, { "id": 90, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 91, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 92, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 93, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 94, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 95, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 96, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 97, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 98, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 20 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0047_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0047_bag_stats.png" }, "solution": [ 1, 5, 13, 24, 36, 72 ], "obj": 481.0, "instance_variant": { "num_elements": 20, "num_sets": 109, "sets": [ { "id": "S1", "elements": [ "L", "M", "N", "O", "P", "Q" ], "cost": 66 }, { "id": "S2", "elements": [ "M", "N", "O", "R", "S" ], "cost": 455 }, { "id": "S3", "elements": [ "K", "L", "M", "N", "P" ], "cost": 25 }, { "id": "S4", "elements": [ "A", "F", "G" ], "cost": 192 }, { "id": "S5", "elements": [ "B", "C" ], "cost": 94 }, { "id": "S6", "elements": [ "A", "B", "C", "D", "E" ], "cost": 500 }, { "id": "S7", "elements": [ "F", "I", "L" ], "cost": 111 }, { "id": "S8", "elements": [ "P", "Q", "R", "S", "T" ], "cost": 340 }, { "id": "S9", "elements": [ "N", "P", "Q", "R" ], "cost": 220 }, { "id": "S10", "elements": [ "E", "F", "I", "J", "L" ], "cost": 55 }, { "id": "S11", "elements": [ "I", "K", "M", "N" ], "cost": 244 }, { "id": "S12", "elements": [ "I", "J", "M", "N", "O" ], "cost": 210 }, { "id": "S13", "elements": [ "F", "G", "K" ], "cost": 141 }, { "id": "S14", "elements": [ "M", "N", "O", "P", "Q", "R" ], "cost": 132 }, { "id": "S15", "elements": [ "N", "O", "P", "Q", "R", "T" ], "cost": 30 }, { "id": "S16", "elements": [ "K", "L", "O" ], "cost": 276 }, { "id": "S17", "elements": [ "K", "N", "O", "P", "R" ], "cost": 480 }, { "id": "S18", "elements": [ "I", "J", "K", "L", "M" ], "cost": 160 }, { "id": "S19", "elements": [ "P", "R", "S" ], "cost": 24 }, { "id": "S20", "elements": [ "N", "P", "Q", "R", "S", "T" ], "cost": 498 }, { "id": "S21", "elements": [ "B", "D", "E", "H", "J" ], "cost": 430 }, { "id": "S22", "elements": [ "N", "O", "P", "Q", "R" ], "cost": 430 }, { "id": "S23", "elements": [ "O", "P", "Q", "R", "T" ], "cost": 55 }, { "id": "S24", "elements": [ "A", "D", "E" ], "cost": 69 }, { "id": "S25", "elements": [ "H", "L", "N" ], "cost": 177 }, { "id": "S26", "elements": [ "N", "P", "Q" ], "cost": 195 }, { "id": "S27", "elements": [ "O", "P", "Q", "R", "S", "T" ], "cost": 522 }, { "id": "S28", "elements": [ "C", "D", "E", "F" ], "cost": 72 }, { "id": "S29", "elements": [ "O", "S" ], "cost": 8 }, { "id": "S30", "elements": [ "G", "K" ], "cost": 172 }, { "id": "S31", "elements": [ "A", "B", "C", "D" ], "cost": 176 }, { "id": "S32", "elements": [ "S", "T" ], "cost": 146 }, { "id": "S33", "elements": [ "L", "N", "O", "P", "Q" ], "cost": 10 }, { "id": "S34", "elements": [ "H", "J" ], "cost": 178 }, { "id": "S35", "elements": [ "E", "F", "G", "H" ], "cost": 332 }, { "id": "S36", "elements": [ "R", "S", "T" ], "cost": 51 }, { "id": "S37", "elements": [ "M", "O", "P", "T" ], "cost": 64 }, { "id": "S38", "elements": [ "B", "E", "F", "H" ], "cost": 132 }, { "id": "S39", "elements": [ "G", "H", "K" ], "cost": 261 }, { "id": "S40", "elements": [ "N", "P", "R", "S" ], "cost": 68 }, { "id": "S41", "elements": [ "G", "J", "K", "M" ], "cost": 8 }, { "id": "S42", "elements": [ "M", "P", "R" ], "cost": 186 }, { "id": "S43", "elements": [ "B", "H" ], "cost": 106 }, { "id": "S44", "elements": [ "R", "S" ], "cost": 68 }, { "id": "S45", "elements": [ "K", "L", "M", "N", "O", "Q" ], "cost": 576 }, { "id": "S46", "elements": [ "B", "E" ], "cost": 198 }, { "id": "S47", "elements": [ "K", "M", "N" ], "cost": 285 }, { "id": "S48", "elements": [ "C", "D" ], "cost": 148 }, { "id": "S49", "elements": [ "D", "E", "G" ], "cost": 285 }, { "id": "S50", "elements": [ "O", "P", "Q", "R" ], "cost": 200 }, { "id": "S51", "elements": [ "K", "O", "R" ], "cost": 264 }, { "id": "S52", "elements": [ "H", "I", "J", "K" ], "cost": 344 }, { "id": "S53", "elements": [ "B", "C", "D", "E", "F", "G" ], "cost": 198 }, { "id": "S54", "elements": [ "A", "C", "D" ], "cost": 132 }, { "id": "S55", "elements": [ "I", "K", "L", "N" ], "cost": 76 }, { "id": "S56", "elements": [ "A", "B", "C", "E", "F" ], "cost": 330 }, { "id": "S57", "elements": [ "A", "B", "D", "E" ], "cost": 380 }, { "id": "S58", "elements": [ "E", "F", "G", "H", "I" ], "cost": 160 }, { "id": "S59", "elements": [ "I", "L", "M", "O" ], "cost": 308 }, { "id": "S60", "elements": [ "Q", "R", "S" ], "cost": 129 }, { "id": "S61", "elements": [ "M", "Q", "R" ], "cost": 36 }, { "id": "S62", "elements": [ "K", "P", "Q" ], "cost": 84 }, { "id": "S63", "elements": [ "B", "C", "D", "E", "G" ], "cost": 270 }, { "id": "S64", "elements": [ "G", "H", "I", "J", "L" ], "cost": 45 }, { "id": "S65", "elements": [ "C", "E", "F", "G" ], "cost": 212 }, { "id": "S66", "elements": [ "M", "O", "P", "Q", "T" ], "cost": 145 }, { "id": "S67", "elements": [ "G", "K", "L", "M", "N" ], "cost": 280 }, { "id": "S68", "elements": [ "E", "F", "H" ], "cost": 69 }, { "id": "S69", "elements": [ "E", "G", "J", "K" ], "cost": 272 }, { "id": "S70", "elements": [ "C", "D", "F", "G" ], "cost": 336 }, { "id": "S71", "elements": [ "O", "Q", "R", "S" ], "cost": 256 }, { "id": "S72", "elements": [ "H", "I", "J" ], "cost": 60 }, { "id": "S73", "elements": [ "C", "E", "I", "K", "L", "N" ], "cost": 42 }, { "id": "S74", "elements": [ "A", "B", "C", "E" ], "cost": 180 }, { "id": "S75", "elements": [ "E", "F", "I", "K", "L" ], "cost": 415 }, { "id": "S76", "elements": [ "L", "M", "N", "R" ], "cost": 224 }, { "id": "S77", "elements": [ "G", "I" ], "cost": 188 }, { "id": "S78", "elements": [ "L", "M", "O", "P" ], "cost": 392 }, { "id": "S79", "elements": [ "G", "H", "J" ], "cost": 288 }, { "id": "S80", "elements": [ "E", "F", "H", "K" ], "cost": 356 }, { "id": "S81", "elements": [ "A", "C", "F", "G" ], "cost": 112 }, { "id": "S82", "elements": [ "E", "G", "H", "L" ], "cost": 28 }, { "id": "S83", "elements": [ "D", "E", "F" ], "cost": 246 }, { "id": "S84", "elements": [ "N", "P" ], "cost": 160 }, { "id": "S85", "elements": [ "Q", "R", "S", "T" ], "cost": 252 }, { "id": "S86", "elements": [ "M", "O", "P", "R" ], "cost": 300 }, { "id": "S87", "elements": [ "G", "H", "I", "J" ], "cost": 272 }, { "id": "S88", "elements": [ "O", "P", "Q" ], "cost": 264 }, { "id": "S89", "elements": [ "J", "L", "N" ], "cost": 237 }, { "id": "S90", "elements": [ "A" ], "cost": 10000 }, { "id": "S91", "elements": [ "B" ], "cost": 10000 }, { "id": "S92", "elements": [ "C" ], "cost": 10000 }, { "id": "S93", "elements": [ "D" ], "cost": 10000 }, { "id": "S94", "elements": [ "E" ], "cost": 10000 }, { "id": "S95", "elements": [ "F" ], "cost": 10000 }, { "id": "S96", "elements": [ "G" ], "cost": 10000 }, { "id": "S97", "elements": [ "H" ], "cost": 10000 }, { "id": "S98", "elements": [ "I" ], "cost": 10000 }, { "id": "S99", "elements": [ "J" ], "cost": 10000 }, { "id": "S100", "elements": [ "K" ], "cost": 10000 }, { "id": "S101", "elements": [ "L" ], "cost": 10000 }, { "id": "S102", "elements": [ "M" ], "cost": 10000 }, { "id": "S103", "elements": [ "N" ], "cost": 10000 }, { "id": "S104", "elements": [ "O" ], "cost": 10000 }, { "id": "S105", "elements": [ "P" ], "cost": 10000 }, { "id": "S106", "elements": [ "Q" ], "cost": 10000 }, { "id": "S107", "elements": [ "R" ], "cost": 10000 }, { "id": "S108", "elements": [ "S" ], "cost": 10000 }, { "id": "S109", "elements": [ "T" ], "cost": 10000 } ] }, "solution_variant": [ "S1", "S5", "S13", "S24", "S36", "S72" ], "context_index": 48, "input_format": "csv", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "Many people on the editorial side are weighing which bundles to publish so each interest area in the subscriber base is covered once and only once, and no two selected bundles include the same interest. What counts as a better selection is a lower editing bill: just total the editing charges for the chosen bundles to see which option is cheapest. The specific bundle choices, their topics, and associated costs are shown below.\n\n- **total_interest_areas**: 23\n- **total_bundle_options**: 120\n\n| bundle_id | bundle_cost | bundle_interest_areas |\n|---|---|---|\n| S1 | 164 | P S V W |\n| S2 | 171 | N O P |\n| S3 | 50 | A B C D E |\n| S4 | 516 | Q R S T U V |\n| S5 | 40 | C E F I J |\n| S6 | 168 | A B C D |\n| S7 | 120 | F G I J K |\n| S8 | 100 | D F G J |\n| S9 | 130 | F H I K L |\n| S10 | 300 | E F J K |\n| S11 | 385 | A B C E H |\n| S12 | 100 | R T U V |\n| S13 | 332 | K L N O |\n| S14 | 390 | G H I J K O |\n| S15 | 510 | I K L M P R |\n| S16 | 282 | T V W |\n| S17 | 133 | A B C D E F H |\n| S18 | 344 | A E F H |\n| S19 | 123 | I L M |\n| S20 | 171 | T U V |\n| S21 | 213 | R T W |\n| S22 | 368 | B C D G |\n| S23 | 176 | F G H I |\n| S24 | 295 | K N P S T |\n| S25 | 600 | A B C D E F |\n| S26 | 12 | C D E |\n| S27 | 594 | D F G H I J |\n| S28 | 33 | H K O |\n| S29 | 267 | A D G |\n| S30 | 15 | B C E |\n| S31 | 392 | H I J N |\n| S32 | 30 | B C D E G |\n| S33 | 294 | N O P Q R S V |\n| S34 | 248 | B C D E |\n| S35 | 504 | A B D E G H |\n| S36 | 290 | L O P Q R |\n| S37 | 183 | I J K |\n| S38 | 240 | A B C D F |\n| S39 | 96 | M P S |\n| S40 | 264 | H L N |\n| S41 | 30 | K M N O P R |\n| S42 | 588 | M O Q R S T V |\n| S43 | 168 | Q R T U V W |\n| S44 | 384 | N O P S |\n| S45 | 294 | A B C |\n| S46 | 318 | R S T U V W |\n| S47 | 184 | D E F I |\n| S48 | 234 | O P Q R S T |\n| S49 | 14 | R U |\n| S50 | 75 | L O Q R U |\n| S51 | 228 | K L N P Q R |\n| S52 | 132 | A C D E |\n| S53 | 180 | K L M N |\n| S54 | 153 | B D E |\n| S55 | 192 | F H J |\n| S56 | 390 | P Q S T U V |\n| S57 | 320 | K L N P R |\n| S58 | 33 | J M N |\n| S59 | 136 | G I K L |\n| S60 | 462 | J K L M N Q |\n| S61 | 90 | L M O P Q S |\n| S62 | 426 | L M O P Q R |\n| S63 | 516 | L M N O R S |\n| S64 | 136 | S T V W |\n| S65 | 276 | C F G H I J |\n| S66 | 48 | A D E G |\n| S67 | 30 | H K M |\n| S68 | 372 | O P Q R T W |\n| S69 | 80 | P Q R T |\n| S70 | 252 | D F H I |\n| S71 | 100 | B C D E F |\n| S72 | 390 | C D E F H I |\n| S73 | 582 | B C D E F G |\n| S74 | 312 | L N O P Q R |\n| S75 | 460 | L M N O P |\n| S76 | 426 | E F G H J K |\n| S77 | 470 | E F H L M |\n| S78 | 6 | A C D |\n| S79 | 231 | J K L |\n| S80 | 141 | Q R S |\n| S81 | 441 | A B C D E F G |\n| S82 | 261 | A D F |\n| S83 | 184 | I L N P |\n| S84 | 252 | I K P |\n| S85 | 234 | D F I |\n| S86 | 57 | H M Q |\n| S87 | 54 | J K M N P R |\n| S88 | 237 | F I J |\n| S89 | 474 | K L N P Q S |\n| S90 | 192 | J K L P |\n| S91 | 216 | P S T |\n| S92 | 413 | Q R S T U V W |\n| S93 | 6 | G I K |\n| S94 | 255 | A B D F H |\n| S95 | 504 | J N O P Q S |\n| S96 | 88 | M Q R T |\n| S97 | 108 | O P Q S |\n| S98 | 10000 | A |\n| S99 | 10000 | B |\n| S100 | 10000 | C |\n| S101 | 10000 | D |\n| S102 | 10000 | E |\n| S103 | 10000 | F |\n| S104 | 10000 | G |\n| S105 | 10000 | H |\n| S106 | 10000 | I |\n| S107 | 10000 | J |\n| S108 | 10000 | K |\n| S109 | 10000 | L |\n| S110 | 10000 | M |\n| S111 | 10000 | N |\n| S112 | 10000 | O |\n| S113 | 10000 | P |\n| S114 | 10000 | Q |\n| S115 | 10000 | R |\n| S116 | 10000 | S |\n| S117 | 10000 | T |\n| S118 | 10000 | U |\n| S119 | 10000 | V |\n| S120 | 10000 | W |\n\nIf you want, reply using this simple JSON shape so it’s easy to read and machine-friendly:\n\n{\n \"solution\": [\"bundle_id\"]\n}\n\nHere \"solution\" is just the list of bundle ids you’d choose to publish — put each bundle’s exact id string in that array. Think of it like filling out a tiny form: the array lists the chosen bundles, and that’s all it needs to show. This is only a sketch of the expected shape, not the actual answer.\n\nPlease use the identifiers exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 23, "num_sets": 120, "density": 0.1673913043478261, "sets": [ { "id": 1, "elements": [ 16, 19, 22, 23 ], "cost": 164 }, { "id": 2, "elements": [ 14, 15, 16 ], "cost": 171 }, { "id": 3, "elements": [ 1, 2, 3, 4, 5 ], "cost": 50 }, { "id": 4, "elements": [ 17, 18, 19, 20, 21, 22 ], "cost": 516 }, { "id": 5, "elements": [ 3, 5, 6, 9, 10 ], "cost": 40 }, { "id": 6, "elements": [ 1, 2, 3, 4 ], "cost": 168 }, { "id": 7, "elements": [ 6, 7, 9, 10, 11 ], "cost": 120 }, { "id": 8, "elements": [ 4, 6, 7, 10 ], "cost": 100 }, { "id": 9, "elements": [ 6, 8, 9, 11, 12 ], "cost": 130 }, { "id": 10, "elements": [ 5, 6, 10, 11 ], "cost": 300 }, { "id": 11, "elements": [ 1, 2, 3, 5, 8 ], "cost": 385 }, { "id": 12, "elements": [ 18, 20, 21, 22 ], "cost": 100 }, { "id": 13, "elements": [ 11, 12, 14, 15 ], "cost": 332 }, { "id": 14, "elements": [ 7, 8, 9, 10, 11, 15 ], "cost": 390 }, { "id": 15, "elements": [ 9, 11, 12, 13, 16, 18 ], "cost": 510 }, { "id": 16, "elements": [ 20, 22, 23 ], "cost": 282 }, { "id": 17, "elements": [ 1, 2, 3, 4, 5, 6, 8 ], "cost": 133 }, { "id": 18, "elements": [ 1, 5, 6, 8 ], "cost": 344 }, { "id": 19, "elements": [ 9, 12, 13 ], "cost": 123 }, { "id": 20, "elements": [ 20, 21, 22 ], "cost": 171 }, { "id": 21, "elements": [ 18, 20, 23 ], "cost": 213 }, { "id": 22, "elements": [ 2, 3, 4, 7 ], "cost": 368 }, { "id": 23, "elements": [ 6, 7, 8, 9 ], "cost": 176 }, { "id": 24, "elements": [ 11, 14, 16, 19, 20 ], "cost": 295 }, { "id": 25, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 600 }, { "id": 26, "elements": [ 3, 4, 5 ], "cost": 12 }, { "id": 27, "elements": [ 4, 6, 7, 8, 9, 10 ], "cost": 594 }, { "id": 28, "elements": [ 8, 11, 15 ], "cost": 33 }, { "id": 29, "elements": [ 1, 4, 7 ], "cost": 267 }, { "id": 30, "elements": [ 2, 3, 5 ], "cost": 15 }, { "id": 31, "elements": [ 8, 9, 10, 14 ], "cost": 392 }, { "id": 32, "elements": [ 2, 3, 4, 5, 7 ], "cost": 30 }, { "id": 33, "elements": [ 14, 15, 16, 17, 18, 19, 22 ], "cost": 294 }, { "id": 34, "elements": [ 2, 3, 4, 5 ], "cost": 248 }, { "id": 35, "elements": [ 1, 2, 4, 5, 7, 8 ], "cost": 504 }, { "id": 36, "elements": [ 12, 15, 16, 17, 18 ], "cost": 290 }, { "id": 37, "elements": [ 9, 10, 11 ], "cost": 183 }, { "id": 38, "elements": [ 1, 2, 3, 4, 6 ], "cost": 240 }, { "id": 39, "elements": [ 13, 16, 19 ], "cost": 96 }, { "id": 40, "elements": [ 8, 12, 14 ], "cost": 264 }, { "id": 41, "elements": [ 11, 13, 14, 15, 16, 18 ], "cost": 30 }, { "id": 42, "elements": [ 13, 15, 17, 18, 19, 20, 22 ], "cost": 588 }, { "id": 43, "elements": [ 17, 18, 20, 21, 22, 23 ], "cost": 168 }, { "id": 44, "elements": [ 14, 15, 16, 19 ], "cost": 384 }, { "id": 45, "elements": [ 1, 2, 3 ], "cost": 294 }, { "id": 46, "elements": [ 18, 19, 20, 21, 22, 23 ], "cost": 318 }, { "id": 47, "elements": [ 4, 5, 6, 9 ], "cost": 184 }, { "id": 48, "elements": [ 15, 16, 17, 18, 19, 20 ], "cost": 234 }, { "id": 49, "elements": [ 18, 21 ], "cost": 14 }, { "id": 50, "elements": [ 12, 15, 17, 18, 21 ], "cost": 75 }, { "id": 51, "elements": [ 11, 12, 14, 16, 17, 18 ], "cost": 228 }, { "id": 52, "elements": [ 1, 3, 4, 5 ], "cost": 132 }, { "id": 53, "elements": [ 11, 12, 13, 14 ], "cost": 180 }, { "id": 54, "elements": [ 2, 4, 5 ], "cost": 153 }, { "id": 55, "elements": [ 6, 8, 10 ], "cost": 192 }, { "id": 56, "elements": [ 16, 17, 19, 20, 21, 22 ], "cost": 390 }, { "id": 57, "elements": [ 11, 12, 14, 16, 18 ], "cost": 320 }, { "id": 58, "elements": [ 10, 13, 14 ], "cost": 33 }, { "id": 59, "elements": [ 7, 9, 11, 12 ], "cost": 136 }, { "id": 60, "elements": [ 10, 11, 12, 13, 14, 17 ], "cost": 462 }, { "id": 61, "elements": [ 12, 13, 15, 16, 17, 19 ], "cost": 90 }, { "id": 62, "elements": [ 12, 13, 15, 16, 17, 18 ], "cost": 426 }, { "id": 63, "elements": [ 12, 13, 14, 15, 18, 19 ], "cost": 516 }, { "id": 64, "elements": [ 19, 20, 22, 23 ], "cost": 136 }, { "id": 65, "elements": [ 3, 6, 7, 8, 9, 10 ], "cost": 276 }, { "id": 66, "elements": [ 1, 4, 5, 7 ], "cost": 48 }, { "id": 67, "elements": [ 8, 11, 13 ], "cost": 30 }, { "id": 68, "elements": [ 15, 16, 17, 18, 20, 23 ], "cost": 372 }, { "id": 69, "elements": [ 16, 17, 18, 20 ], "cost": 80 }, { "id": 70, "elements": [ 4, 6, 8, 9 ], "cost": 252 }, { "id": 71, "elements": [ 2, 3, 4, 5, 6 ], "cost": 100 }, { "id": 72, "elements": [ 3, 4, 5, 6, 8, 9 ], "cost": 390 }, { "id": 73, "elements": [ 2, 3, 4, 5, 6, 7 ], "cost": 582 }, { "id": 74, "elements": [ 12, 14, 15, 16, 17, 18 ], "cost": 312 }, { "id": 75, "elements": [ 12, 13, 14, 15, 16 ], "cost": 460 }, { "id": 76, "elements": [ 5, 6, 7, 8, 10, 11 ], "cost": 426 }, { "id": 77, "elements": [ 5, 6, 8, 12, 13 ], "cost": 470 }, { "id": 78, "elements": [ 1, 3, 4 ], "cost": 6 }, { "id": 79, "elements": [ 10, 11, 12 ], "cost": 231 }, { "id": 80, "elements": [ 17, 18, 19 ], "cost": 141 }, { "id": 81, "elements": [ 1, 2, 3, 4, 5, 6, 7 ], "cost": 441 }, { "id": 82, "elements": [ 1, 4, 6 ], "cost": 261 }, { "id": 83, "elements": [ 9, 12, 14, 16 ], "cost": 184 }, { "id": 84, "elements": [ 9, 11, 16 ], "cost": 252 }, { "id": 85, "elements": [ 4, 6, 9 ], "cost": 234 }, { "id": 86, "elements": [ 8, 13, 17 ], "cost": 57 }, { "id": 87, "elements": [ 10, 11, 13, 14, 16, 18 ], "cost": 54 }, { "id": 88, "elements": [ 6, 9, 10 ], "cost": 237 }, { "id": 89, "elements": [ 11, 12, 14, 16, 17, 19 ], "cost": 474 }, { "id": 90, "elements": [ 10, 11, 12, 16 ], "cost": 192 }, { "id": 91, "elements": [ 16, 19, 20 ], "cost": 216 }, { "id": 92, "elements": [ 17, 18, 19, 20, 21, 22, 23 ], "cost": 413 }, { "id": 93, "elements": [ 7, 9, 11 ], "cost": 6 }, { "id": 94, "elements": [ 1, 2, 4, 6, 8 ], "cost": 255 }, { "id": 95, "elements": [ 10, 14, 15, 16, 17, 19 ], "cost": 504 }, { "id": 96, "elements": [ 13, 17, 18, 20 ], "cost": 88 }, { "id": 97, "elements": [ 15, 16, 17, 19 ], "cost": 108 }, { "id": 98, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 99, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 100, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 101, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 102, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 103, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 104, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 105, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 106, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 107, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 108, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 109, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 110, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 111, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 112, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 113, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 114, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 115, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 116, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 117, "elements": [ 20 ], "cost": 10000, "singleton": true }, { "id": 118, "elements": [ 21 ], "cost": 10000, "singleton": true }, { "id": 119, "elements": [ 22 ], "cost": 10000, "singleton": true }, { "id": 120, "elements": [ 23 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0048_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0048_bag_stats.png" }, "solution": [ 16, 17, 49, 58, 59, 97 ], "obj": 706.0, "instance_variant": { "num_elements": 23, "num_sets": 120, "sets": [ { "id": "S1", "elements": [ "P", "S", "V", "W" ], "cost": 164 }, { "id": "S2", "elements": [ "N", "O", "P" ], "cost": 171 }, { "id": "S3", "elements": [ "A", "B", "C", "D", "E" ], "cost": 50 }, { "id": "S4", "elements": [ "Q", "R", "S", "T", "U", "V" ], "cost": 516 }, { "id": "S5", "elements": [ "C", "E", "F", "I", "J" ], "cost": 40 }, { "id": "S6", "elements": [ "A", "B", "C", "D" ], "cost": 168 }, { "id": "S7", "elements": [ "F", "G", "I", "J", "K" ], "cost": 120 }, { "id": "S8", "elements": [ "D", "F", "G", "J" ], "cost": 100 }, { "id": "S9", "elements": [ "F", "H", "I", "K", "L" ], "cost": 130 }, { "id": "S10", "elements": [ "E", "F", "J", "K" ], "cost": 300 }, { "id": "S11", "elements": [ "A", "B", "C", "E", "H" ], "cost": 385 }, { "id": "S12", "elements": [ "R", "T", "U", "V" ], "cost": 100 }, { "id": "S13", "elements": [ "K", "L", "N", "O" ], "cost": 332 }, { "id": "S14", "elements": [ "G", "H", "I", "J", "K", "O" ], "cost": 390 }, { "id": "S15", "elements": [ "I", "K", "L", "M", "P", "R" ], "cost": 510 }, { "id": "S16", "elements": [ "T", "V", "W" ], "cost": 282 }, { "id": "S17", "elements": [ "A", "B", "C", "D", "E", "F", "H" ], "cost": 133 }, { "id": "S18", "elements": [ "A", "E", "F", "H" ], "cost": 344 }, { "id": "S19", "elements": [ "I", "L", "M" ], "cost": 123 }, { "id": "S20", "elements": [ "T", "U", "V" ], "cost": 171 }, { "id": "S21", "elements": [ "R", "T", "W" ], "cost": 213 }, { "id": "S22", "elements": [ "B", "C", "D", "G" ], "cost": 368 }, { "id": "S23", "elements": [ "F", "G", "H", "I" ], "cost": 176 }, { "id": "S24", "elements": [ "K", "N", "P", "S", "T" ], "cost": 295 }, { "id": "S25", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 600 }, { "id": "S26", "elements": [ "C", "D", "E" ], "cost": 12 }, { "id": "S27", "elements": [ "D", "F", "G", "H", "I", "J" ], "cost": 594 }, { "id": "S28", "elements": [ "H", "K", "O" ], "cost": 33 }, { "id": "S29", "elements": [ "A", "D", "G" ], "cost": 267 }, { "id": "S30", "elements": [ "B", "C", "E" ], "cost": 15 }, { "id": "S31", "elements": [ "H", "I", "J", "N" ], "cost": 392 }, { "id": "S32", "elements": [ "B", "C", "D", "E", "G" ], "cost": 30 }, { "id": "S33", "elements": [ "N", "O", "P", "Q", "R", "S", "V" ], "cost": 294 }, { "id": "S34", "elements": [ "B", "C", "D", "E" ], "cost": 248 }, { "id": "S35", "elements": [ "A", "B", "D", "E", "G", "H" ], "cost": 504 }, { "id": "S36", "elements": [ "L", "O", "P", "Q", "R" ], "cost": 290 }, { "id": "S37", "elements": [ "I", "J", "K" ], "cost": 183 }, { "id": "S38", "elements": [ "A", "B", "C", "D", "F" ], "cost": 240 }, { "id": "S39", "elements": [ "M", "P", "S" ], "cost": 96 }, { "id": "S40", "elements": [ "H", "L", "N" ], "cost": 264 }, { "id": "S41", "elements": [ "K", "M", "N", "O", "P", "R" ], "cost": 30 }, { "id": "S42", "elements": [ "M", "O", "Q", "R", "S", "T", "V" ], "cost": 588 }, { "id": "S43", "elements": [ "Q", "R", "T", "U", "V", "W" ], "cost": 168 }, { "id": "S44", "elements": [ "N", "O", "P", "S" ], "cost": 384 }, { "id": "S45", "elements": [ "A", "B", "C" ], "cost": 294 }, { "id": "S46", "elements": [ "R", "S", "T", "U", "V", "W" ], "cost": 318 }, { "id": "S47", "elements": [ "D", "E", "F", "I" ], "cost": 184 }, { "id": "S48", "elements": [ "O", "P", "Q", "R", "S", "T" ], "cost": 234 }, { "id": "S49", "elements": [ "R", "U" ], "cost": 14 }, { "id": "S50", "elements": [ "L", "O", "Q", "R", "U" ], "cost": 75 }, { "id": "S51", "elements": [ "K", "L", "N", "P", "Q", "R" ], "cost": 228 }, { "id": "S52", "elements": [ "A", "C", "D", "E" ], "cost": 132 }, { "id": "S53", "elements": [ "K", "L", "M", "N" ], "cost": 180 }, { "id": "S54", "elements": [ "B", "D", "E" ], "cost": 153 }, { "id": "S55", "elements": [ "F", "H", "J" ], "cost": 192 }, { "id": "S56", "elements": [ "P", "Q", "S", "T", "U", "V" ], "cost": 390 }, { "id": "S57", "elements": [ "K", "L", "N", "P", "R" ], "cost": 320 }, { "id": "S58", "elements": [ "J", "M", "N" ], "cost": 33 }, { "id": "S59", "elements": [ "G", "I", "K", "L" ], "cost": 136 }, { "id": "S60", "elements": [ "J", "K", "L", "M", "N", "Q" ], "cost": 462 }, { "id": "S61", "elements": [ "L", "M", "O", "P", "Q", "S" ], "cost": 90 }, { "id": "S62", "elements": [ "L", "M", "O", "P", "Q", "R" ], "cost": 426 }, { "id": "S63", "elements": [ "L", "M", "N", "O", "R", "S" ], "cost": 516 }, { "id": "S64", "elements": [ "S", "T", "V", "W" ], "cost": 136 }, { "id": "S65", "elements": [ "C", "F", "G", "H", "I", "J" ], "cost": 276 }, { "id": "S66", "elements": [ "A", "D", "E", "G" ], "cost": 48 }, { "id": "S67", "elements": [ "H", "K", "M" ], "cost": 30 }, { "id": "S68", "elements": [ "O", "P", "Q", "R", "T", "W" ], "cost": 372 }, { "id": "S69", "elements": [ "P", "Q", "R", "T" ], "cost": 80 }, { "id": "S70", "elements": [ "D", "F", "H", "I" ], "cost": 252 }, { "id": "S71", "elements": [ "B", "C", "D", "E", "F" ], "cost": 100 }, { "id": "S72", "elements": [ "C", "D", "E", "F", "H", "I" ], "cost": 390 }, { "id": "S73", "elements": [ "B", "C", "D", "E", "F", "G" ], "cost": 582 }, { "id": "S74", "elements": [ "L", "N", "O", "P", "Q", "R" ], "cost": 312 }, { "id": "S75", "elements": [ "L", "M", "N", "O", "P" ], "cost": 460 }, { "id": "S76", "elements": [ "E", "F", "G", "H", "J", "K" ], "cost": 426 }, { "id": "S77", "elements": [ "E", "F", "H", "L", "M" ], "cost": 470 }, { "id": "S78", "elements": [ "A", "C", "D" ], "cost": 6 }, { "id": "S79", "elements": [ "J", "K", "L" ], "cost": 231 }, { "id": "S80", "elements": [ "Q", "R", "S" ], "cost": 141 }, { "id": "S81", "elements": [ "A", "B", "C", "D", "E", "F", "G" ], "cost": 441 }, { "id": "S82", "elements": [ "A", "D", "F" ], "cost": 261 }, { "id": "S83", "elements": [ "I", "L", "N", "P" ], "cost": 184 }, { "id": "S84", "elements": [ "I", "K", "P" ], "cost": 252 }, { "id": "S85", "elements": [ "D", "F", "I" ], "cost": 234 }, { "id": "S86", "elements": [ "H", "M", "Q" ], "cost": 57 }, { "id": "S87", "elements": [ "J", "K", "M", "N", "P", "R" ], "cost": 54 }, { "id": "S88", "elements": [ "F", "I", "J" ], "cost": 237 }, { "id": "S89", "elements": [ "K", "L", "N", "P", "Q", "S" ], "cost": 474 }, { "id": "S90", "elements": [ "J", "K", "L", "P" ], "cost": 192 }, { "id": "S91", "elements": [ "P", "S", "T" ], "cost": 216 }, { "id": "S92", "elements": [ "Q", "R", "S", "T", "U", "V", "W" ], "cost": 413 }, { "id": "S93", "elements": [ "G", "I", "K" ], "cost": 6 }, { "id": "S94", "elements": [ "A", "B", "D", "F", "H" ], "cost": 255 }, { "id": "S95", "elements": [ "J", "N", "O", "P", "Q", "S" ], "cost": 504 }, { "id": "S96", "elements": [ "M", "Q", "R", "T" ], "cost": 88 }, { "id": "S97", "elements": [ "O", "P", "Q", "S" ], "cost": 108 }, { "id": "S98", "elements": [ "A" ], "cost": 10000 }, { "id": "S99", "elements": [ "B" ], "cost": 10000 }, { "id": "S100", "elements": [ "C" ], "cost": 10000 }, { "id": "S101", "elements": [ "D" ], "cost": 10000 }, { "id": "S102", "elements": [ "E" ], "cost": 10000 }, { "id": "S103", "elements": [ "F" ], "cost": 10000 }, { "id": "S104", "elements": [ "G" ], "cost": 10000 }, { "id": "S105", "elements": [ "H" ], "cost": 10000 }, { "id": "S106", "elements": [ "I" ], "cost": 10000 }, { "id": "S107", "elements": [ "J" ], "cost": 10000 }, { "id": "S108", "elements": [ "K" ], "cost": 10000 }, { "id": "S109", "elements": [ "L" ], "cost": 10000 }, { "id": "S110", "elements": [ "M" ], "cost": 10000 }, { "id": "S111", "elements": [ "N" ], "cost": 10000 }, { "id": "S112", "elements": [ "O" ], "cost": 10000 }, { "id": "S113", "elements": [ "P" ], "cost": 10000 }, { "id": "S114", "elements": [ "Q" ], "cost": 10000 }, { "id": "S115", "elements": [ "R" ], "cost": 10000 }, { "id": "S116", "elements": [ "S" ], "cost": 10000 }, { "id": "S117", "elements": [ "T" ], "cost": 10000 }, { "id": "S118", "elements": [ "U" ], "cost": 10000 }, { "id": "S119", "elements": [ "V" ], "cost": 10000 }, { "id": "S120", "elements": [ "W" ], "cost": 10000 } ] }, "solution_variant": [ "S16", "S17", "S49", "S58", "S59", "S97" ], "context_index": 49, "input_format": "markdown_table", "input_index_base": "names" }, { "task_name": "SPP", "problem_type": "SPP", "instruction": "Many sysadmins face the same tidy-up: choose which backup archives to keep so every file is stored in one chosen archive and nowhere else, and don’t let any file fall through the cracks. The practical test is monetary — total storage cost equals the sum of costs for the kept archives, and the aim is to make that sum as small as possible. The concrete backup list, file coverage, and costs appear below.\n\nThere are 94 candidate archives covering 20 distinct files:\nArchive S1: files B E F J; cost 344.\nArchive S2: files G H I J K; cost 190.\nArchive S3: files B C D; cost 231.\nArchive S4: files O R S; cost 222.\nArchive S5: files G I J; cost 231.\nArchive S6: files G H I J L; cost 180.\nArchive S7: files F J K L M; cost 480.\nArchive S8: files A B C D; cost 76.\nArchive S9: files D E; cost 152.\nArchive S10: files A B C D E; cost 375.\nArchive S11: files B F J L; cost 224.\nArchive S12: files E G H; cost 81.\nArchive S13: files K N; cost 100.\nArchive S14: files F G H K; cost 80.\nArchive S15: files A B C; cost 234.\nArchive S16: files G H I J M; cost 430.\nArchive S17: files G H I N O; cost 485.\nArchive S18: files L M O P Q; cost 145.\nArchive S19: files I K N; cost 72.\nArchive S20: files D H I; cost 204.\nArchive S21: files J L M N O; cost 110.\nArchive S22: files E F J; cost 171.\nArchive S23: files C E G; cost 240.\nArchive S24: files M N O P Q; cost 340.\nArchive S25: files P R S T; cost 368.\nArchive S26: files O S; cost 28.\nArchive S27: files B C F; cost 45.\nArchive S28: files O P Q; cost 162.\nArchive S29: files C D E F; cost 244.\nArchive S30: files B F G I J L; cost 246.\nArchive S31: files L P Q S T; cost 420.\nArchive S32: files A B C D E F; cost 30.\nArchive S33: files E F; cost 24.\nArchive S34: files M O P Q R; cost 205.\nArchive S35: files I K L O; cost 292.\nArchive S36: files I J; cost 104.\nArchive S37: files L N O Q S; cost 180.\nArchive S38: files E G K; cost 27.\nArchive S39: files I J K L M; cost 95.\nArchive S40: files O P S; cost 249.\nArchive S41: files G H I M; cost 136.\nArchive S42: files F H I K; cost 324.\nArchive S43: files O Q R S T; cost 485.\nArchive S44: files K L M Q; cost 224.\nArchive S45: files H I J K L; cost 185.\nArchive S46: files G N; cost 168.\nArchive S47: files G I; cost 110.\nArchive S48: files C G H I J; cost 440.\nArchive S49: files P Q; cost 176.\nArchive S50: files C E; cost 72.\nArchive S51: files N O P Q R; cost 420.\nArchive S52: files I J N O; cost 360.\nArchive S53: files H J M O; cost 12.\nArchive S54: files K M N P; cost 12.\nArchive S55: files H K O; cost 60.\nArchive S56: files O P Q R; cost 204.\nArchive S57: files N P Q R S T; cost 270.\nArchive S58: files B C D E F; cost 245.\nArchive S59: files P Q S; cost 198.\nArchive S60: files A B C D F G; cost 354.\nArchive S61: files H I J L M N; cost 66.\nArchive S62: files G K M; cost 267.\nArchive S63: files E F G H I; cost 445.\nArchive S64: files B C F G; cost 256.\nArchive S65: files B C E; cost 234.\nArchive S66: files A B D; cost 45.\nArchive S67: files P Q R; cost 57.\nArchive S68: files I N P; cost 129.\nArchive S69: files C D E F H; cost 405.\nArchive S70: files A B; cost 62.\nArchive S71: files B D E; cost 81.\nArchive S72: files F H I; cost 204.\nArchive S73: files M N P Q R; cost 70.\nArchive S74: files B E F; cost 171.\nArchive S75: files A; cost 10000.\nArchive S76: files B; cost 10000.\nArchive S77: files C; cost 10000.\nArchive S78: files D; cost 10000.\nArchive S79: files E; cost 10000.\nArchive S80: files F; cost 10000.\nArchive S81: files G; cost 10000.\nArchive S82: files H; cost 10000.\nArchive S83: files I; cost 10000.\nArchive S84: files J; cost 10000.\nArchive S85: files K; cost 10000.\nArchive S86: files L; cost 10000.\nArchive S87: files M; cost 10000.\nArchive S88: files N; cost 10000.\nArchive S89: files O; cost 10000.\nArchive S90: files P; cost 10000.\nArchive S91: files Q; cost 10000.\nArchive S92: files R; cost 10000.\nArchive S93: files S; cost 10000.\nArchive S94: files T; cost 10000.\nChoose archives so every one of the 20 files is retained exactly once while minimizing the total cost across the 94 candidates.\n\nOh, and when you reply, just drop the chosen archive IDs into a tiny JSON outline like this (super simple):\n\n{\n \"solution\": [\"archive_id\", ...]\n}\n\nThink of \"solution\" as the list of archives you decide to keep — each string in that array is an archive identifier placeholder. This is just a sketch of the shape I expect, not the actual answer.\n\nPlease make sure you use the exact identifiers from the instance input — no renaming and no new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "problem_type": "SPP", "num_elements": 20, "num_sets": 94, "density": 0.16170212765957448, "sets": [ { "id": 1, "elements": [ 2, 5, 6, 10 ], "cost": 344 }, { "id": 2, "elements": [ 7, 8, 9, 10, 11 ], "cost": 190 }, { "id": 3, "elements": [ 2, 3, 4 ], "cost": 231 }, { "id": 4, "elements": [ 15, 18, 19 ], "cost": 222 }, { "id": 5, "elements": [ 7, 9, 10 ], "cost": 231 }, { "id": 6, "elements": [ 7, 8, 9, 10, 12 ], "cost": 180 }, { "id": 7, "elements": [ 6, 10, 11, 12, 13 ], "cost": 480 }, { "id": 8, "elements": [ 1, 2, 3, 4 ], "cost": 76 }, { "id": 9, "elements": [ 4, 5 ], "cost": 152 }, { "id": 10, "elements": [ 1, 2, 3, 4, 5 ], "cost": 375 }, { "id": 11, "elements": [ 2, 6, 10, 12 ], "cost": 224 }, { "id": 12, "elements": [ 5, 7, 8 ], "cost": 81 }, { "id": 13, "elements": [ 11, 14 ], "cost": 100 }, { "id": 14, "elements": [ 6, 7, 8, 11 ], "cost": 80 }, { "id": 15, "elements": [ 1, 2, 3 ], "cost": 234 }, { "id": 16, "elements": [ 7, 8, 9, 10, 13 ], "cost": 430 }, { "id": 17, "elements": [ 7, 8, 9, 14, 15 ], "cost": 485 }, { "id": 18, "elements": [ 12, 13, 15, 16, 17 ], "cost": 145 }, { "id": 19, "elements": [ 9, 11, 14 ], "cost": 72 }, { "id": 20, "elements": [ 4, 8, 9 ], "cost": 204 }, { "id": 21, "elements": [ 10, 12, 13, 14, 15 ], "cost": 110 }, { "id": 22, "elements": [ 5, 6, 10 ], "cost": 171 }, { "id": 23, "elements": [ 3, 5, 7 ], "cost": 240 }, { "id": 24, "elements": [ 13, 14, 15, 16, 17 ], "cost": 340 }, { "id": 25, "elements": [ 16, 18, 19, 20 ], "cost": 368 }, { "id": 26, "elements": [ 15, 19 ], "cost": 28 }, { "id": 27, "elements": [ 2, 3, 6 ], "cost": 45 }, { "id": 28, "elements": [ 15, 16, 17 ], "cost": 162 }, { "id": 29, "elements": [ 3, 4, 5, 6 ], "cost": 244 }, { "id": 30, "elements": [ 2, 6, 7, 9, 10, 12 ], "cost": 246 }, { "id": 31, "elements": [ 12, 16, 17, 19, 20 ], "cost": 420 }, { "id": 32, "elements": [ 1, 2, 3, 4, 5, 6 ], "cost": 30 }, { "id": 33, "elements": [ 5, 6 ], "cost": 24 }, { "id": 34, "elements": [ 13, 15, 16, 17, 18 ], "cost": 205 }, { "id": 35, "elements": [ 9, 11, 12, 15 ], "cost": 292 }, { "id": 36, "elements": [ 9, 10 ], "cost": 104 }, { "id": 37, "elements": [ 12, 14, 15, 17, 19 ], "cost": 180 }, { "id": 38, "elements": [ 5, 7, 11 ], "cost": 27 }, { "id": 39, "elements": [ 9, 10, 11, 12, 13 ], "cost": 95 }, { "id": 40, "elements": [ 15, 16, 19 ], "cost": 249 }, { "id": 41, "elements": [ 7, 8, 9, 13 ], "cost": 136 }, { "id": 42, "elements": [ 6, 8, 9, 11 ], "cost": 324 }, { "id": 43, "elements": [ 15, 17, 18, 19, 20 ], "cost": 485 }, { "id": 44, "elements": [ 11, 12, 13, 17 ], "cost": 224 }, { "id": 45, "elements": [ 8, 9, 10, 11, 12 ], "cost": 185 }, { "id": 46, "elements": [ 7, 14 ], "cost": 168 }, { "id": 47, "elements": [ 7, 9 ], "cost": 110 }, { "id": 48, "elements": [ 3, 7, 8, 9, 10 ], "cost": 440 }, { "id": 49, "elements": [ 16, 17 ], "cost": 176 }, { "id": 50, "elements": [ 3, 5 ], "cost": 72 }, { "id": 51, "elements": [ 14, 15, 16, 17, 18 ], "cost": 420 }, { "id": 52, "elements": [ 9, 10, 14, 15 ], "cost": 360 }, { "id": 53, "elements": [ 8, 10, 13, 15 ], "cost": 12 }, { "id": 54, "elements": [ 11, 13, 14, 16 ], "cost": 12 }, { "id": 55, "elements": [ 8, 11, 15 ], "cost": 60 }, { "id": 56, "elements": [ 15, 16, 17, 18 ], "cost": 204 }, { "id": 57, "elements": [ 14, 16, 17, 18, 19, 20 ], "cost": 270 }, { "id": 58, "elements": [ 2, 3, 4, 5, 6 ], "cost": 245 }, { "id": 59, "elements": [ 16, 17, 19 ], "cost": 198 }, { "id": 60, "elements": [ 1, 2, 3, 4, 6, 7 ], "cost": 354 }, { "id": 61, "elements": [ 8, 9, 10, 12, 13, 14 ], "cost": 66 }, { "id": 62, "elements": [ 7, 11, 13 ], "cost": 267 }, { "id": 63, "elements": [ 5, 6, 7, 8, 9 ], "cost": 445 }, { "id": 64, "elements": [ 2, 3, 6, 7 ], "cost": 256 }, { "id": 65, "elements": [ 2, 3, 5 ], "cost": 234 }, { "id": 66, "elements": [ 1, 2, 4 ], "cost": 45 }, { "id": 67, "elements": [ 16, 17, 18 ], "cost": 57 }, { "id": 68, "elements": [ 9, 14, 16 ], "cost": 129 }, { "id": 69, "elements": [ 3, 4, 5, 6, 8 ], "cost": 405 }, { "id": 70, "elements": [ 1, 2 ], "cost": 62 }, { "id": 71, "elements": [ 2, 4, 5 ], "cost": 81 }, { "id": 72, "elements": [ 6, 8, 9 ], "cost": 204 }, { "id": 73, "elements": [ 13, 14, 16, 17, 18 ], "cost": 70 }, { "id": 74, "elements": [ 2, 5, 6 ], "cost": 171 }, { "id": 75, "elements": [ 1 ], "cost": 10000, "singleton": true }, { "id": 76, "elements": [ 2 ], "cost": 10000, "singleton": true }, { "id": 77, "elements": [ 3 ], "cost": 10000, "singleton": true }, { "id": 78, "elements": [ 4 ], "cost": 10000, "singleton": true }, { "id": 79, "elements": [ 5 ], "cost": 10000, "singleton": true }, { "id": 80, "elements": [ 6 ], "cost": 10000, "singleton": true }, { "id": 81, "elements": [ 7 ], "cost": 10000, "singleton": true }, { "id": 82, "elements": [ 8 ], "cost": 10000, "singleton": true }, { "id": 83, "elements": [ 9 ], "cost": 10000, "singleton": true }, { "id": 84, "elements": [ 10 ], "cost": 10000, "singleton": true }, { "id": 85, "elements": [ 11 ], "cost": 10000, "singleton": true }, { "id": 86, "elements": [ 12 ], "cost": 10000, "singleton": true }, { "id": 87, "elements": [ 13 ], "cost": 10000, "singleton": true }, { "id": 88, "elements": [ 14 ], "cost": 10000, "singleton": true }, { "id": 89, "elements": [ 15 ], "cost": 10000, "singleton": true }, { "id": 90, "elements": [ 16 ], "cost": 10000, "singleton": true }, { "id": 91, "elements": [ 17 ], "cost": 10000, "singleton": true }, { "id": 92, "elements": [ 18 ], "cost": 10000, "singleton": true }, { "id": 93, "elements": [ 19 ], "cost": 10000, "singleton": true }, { "id": 94, "elements": [ 20 ], "cost": 10000, "singleton": true } ], "generator_config": { "items_min": 8, "items_max": 25, "bag_factor_min": 4.0, "bag_factor_max": 6.0, "high_cost": 10000, "cluster_sigma_ratio": 0.1 }, "viz_matrix": "generated_data/SPP/viz/SPP_L/instance_0049_matrix.png", "viz_bag_stats": "generated_data/SPP/viz/SPP_L/instance_0049_bag_stats.png" }, "solution": [ 6, 32, 43, 54 ], "obj": 707.0, "instance_variant": { "num_elements": 20, "num_sets": 94, "sets": [ { "id": "S1", "elements": [ "B", "E", "F", "J" ], "cost": 344 }, { "id": "S2", "elements": [ "G", "H", "I", "J", "K" ], "cost": 190 }, { "id": "S3", "elements": [ "B", "C", "D" ], "cost": 231 }, { "id": "S4", "elements": [ "O", "R", "S" ], "cost": 222 }, { "id": "S5", "elements": [ "G", "I", "J" ], "cost": 231 }, { "id": "S6", "elements": [ "G", "H", "I", "J", "L" ], "cost": 180 }, { "id": "S7", "elements": [ "F", "J", "K", "L", "M" ], "cost": 480 }, { "id": "S8", "elements": [ "A", "B", "C", "D" ], "cost": 76 }, { "id": "S9", "elements": [ "D", "E" ], "cost": 152 }, { "id": "S10", "elements": [ "A", "B", "C", "D", "E" ], "cost": 375 }, { "id": "S11", "elements": [ "B", "F", "J", "L" ], "cost": 224 }, { "id": "S12", "elements": [ "E", "G", "H" ], "cost": 81 }, { "id": "S13", "elements": [ "K", "N" ], "cost": 100 }, { "id": "S14", "elements": [ "F", "G", "H", "K" ], "cost": 80 }, { "id": "S15", "elements": [ "A", "B", "C" ], "cost": 234 }, { "id": "S16", "elements": [ "G", "H", "I", "J", "M" ], "cost": 430 }, { "id": "S17", "elements": [ "G", "H", "I", "N", "O" ], "cost": 485 }, { "id": "S18", "elements": [ "L", "M", "O", "P", "Q" ], "cost": 145 }, { "id": "S19", "elements": [ "I", "K", "N" ], "cost": 72 }, { "id": "S20", "elements": [ "D", "H", "I" ], "cost": 204 }, { "id": "S21", "elements": [ "J", "L", "M", "N", "O" ], "cost": 110 }, { "id": "S22", "elements": [ "E", "F", "J" ], "cost": 171 }, { "id": "S23", "elements": [ "C", "E", "G" ], "cost": 240 }, { "id": "S24", "elements": [ "M", "N", "O", "P", "Q" ], "cost": 340 }, { "id": "S25", "elements": [ "P", "R", "S", "T" ], "cost": 368 }, { "id": "S26", "elements": [ "O", "S" ], "cost": 28 }, { "id": "S27", "elements": [ "B", "C", "F" ], "cost": 45 }, { "id": "S28", "elements": [ "O", "P", "Q" ], "cost": 162 }, { "id": "S29", "elements": [ "C", "D", "E", "F" ], "cost": 244 }, { "id": "S30", "elements": [ "B", "F", "G", "I", "J", "L" ], "cost": 246 }, { "id": "S31", "elements": [ "L", "P", "Q", "S", "T" ], "cost": 420 }, { "id": "S32", "elements": [ "A", "B", "C", "D", "E", "F" ], "cost": 30 }, { "id": "S33", "elements": [ "E", "F" ], "cost": 24 }, { "id": "S34", "elements": [ "M", "O", "P", "Q", "R" ], "cost": 205 }, { "id": "S35", "elements": [ "I", "K", "L", "O" ], "cost": 292 }, { "id": "S36", "elements": [ "I", "J" ], "cost": 104 }, { "id": "S37", "elements": [ "L", "N", "O", "Q", "S" ], "cost": 180 }, { "id": "S38", "elements": [ "E", "G", "K" ], "cost": 27 }, { "id": "S39", "elements": [ "I", "J", "K", "L", "M" ], "cost": 95 }, { "id": "S40", "elements": [ "O", "P", "S" ], "cost": 249 }, { "id": "S41", "elements": [ "G", "H", "I", "M" ], "cost": 136 }, { "id": "S42", "elements": [ "F", "H", "I", "K" ], "cost": 324 }, { "id": "S43", "elements": [ "O", "Q", "R", "S", "T" ], "cost": 485 }, { "id": "S44", "elements": [ "K", "L", "M", "Q" ], "cost": 224 }, { "id": "S45", "elements": [ "H", "I", "J", "K", "L" ], "cost": 185 }, { "id": "S46", "elements": [ "G", "N" ], "cost": 168 }, { "id": "S47", "elements": [ "G", "I" ], "cost": 110 }, { "id": "S48", "elements": [ "C", "G", "H", "I", "J" ], "cost": 440 }, { "id": "S49", "elements": [ "P", "Q" ], "cost": 176 }, { "id": "S50", "elements": [ "C", "E" ], "cost": 72 }, { "id": "S51", "elements": [ "N", "O", "P", "Q", "R" ], "cost": 420 }, { "id": "S52", "elements": [ "I", "J", "N", "O" ], "cost": 360 }, { "id": "S53", "elements": [ "H", "J", "M", "O" ], "cost": 12 }, { "id": "S54", "elements": [ "K", "M", "N", "P" ], "cost": 12 }, { "id": "S55", "elements": [ "H", "K", "O" ], "cost": 60 }, { "id": "S56", "elements": [ "O", "P", "Q", "R" ], "cost": 204 }, { "id": "S57", "elements": [ "N", "P", "Q", "R", "S", "T" ], "cost": 270 }, { "id": "S58", "elements": [ "B", "C", "D", "E", "F" ], "cost": 245 }, { "id": "S59", "elements": [ "P", "Q", "S" ], "cost": 198 }, { "id": "S60", "elements": [ "A", "B", "C", "D", "F", "G" ], "cost": 354 }, { "id": "S61", "elements": [ "H", "I", "J", "L", "M", "N" ], "cost": 66 }, { "id": "S62", "elements": [ "G", "K", "M" ], "cost": 267 }, { "id": "S63", "elements": [ "E", "F", "G", "H", "I" ], "cost": 445 }, { "id": "S64", "elements": [ "B", "C", "F", "G" ], "cost": 256 }, { "id": "S65", "elements": [ "B", "C", "E" ], "cost": 234 }, { "id": "S66", "elements": [ "A", "B", "D" ], "cost": 45 }, { "id": "S67", "elements": [ "P", "Q", "R" ], "cost": 57 }, { "id": "S68", "elements": [ "I", "N", "P" ], "cost": 129 }, { "id": "S69", "elements": [ "C", "D", "E", "F", "H" ], "cost": 405 }, { "id": "S70", "elements": [ "A", "B" ], "cost": 62 }, { "id": "S71", "elements": [ "B", "D", "E" ], "cost": 81 }, { "id": "S72", "elements": [ "F", "H", "I" ], "cost": 204 }, { "id": "S73", "elements": [ "M", "N", "P", "Q", "R" ], "cost": 70 }, { "id": "S74", "elements": [ "B", "E", "F" ], "cost": 171 }, { "id": "S75", "elements": [ "A" ], "cost": 10000 }, { "id": "S76", "elements": [ "B" ], "cost": 10000 }, { "id": "S77", "elements": [ "C" ], "cost": 10000 }, { "id": "S78", "elements": [ "D" ], "cost": 10000 }, { "id": "S79", "elements": [ "E" ], "cost": 10000 }, { "id": "S80", "elements": [ "F" ], "cost": 10000 }, { "id": "S81", "elements": [ "G" ], "cost": 10000 }, { "id": "S82", "elements": [ "H" ], "cost": 10000 }, { "id": "S83", "elements": [ "I" ], "cost": 10000 }, { "id": "S84", "elements": [ "J" ], "cost": 10000 }, { "id": "S85", "elements": [ "K" ], "cost": 10000 }, { "id": "S86", "elements": [ "L" ], "cost": 10000 }, { "id": "S87", "elements": [ "M" ], "cost": 10000 }, { "id": "S88", "elements": [ "N" ], "cost": 10000 }, { "id": "S89", "elements": [ "O" ], "cost": 10000 }, { "id": "S90", "elements": [ "P" ], "cost": 10000 }, { "id": "S91", "elements": [ "Q" ], "cost": 10000 }, { "id": "S92", "elements": [ "R" ], "cost": 10000 }, { "id": "S93", "elements": [ "S" ], "cost": 10000 }, { "id": "S94", "elements": [ "T" ], "cost": 10000 } ] }, "solution_variant": [ "S6", "S32", "S43", "S54" ], "context_index": 50, "input_format": "nl", "input_index_base": "names" } ]