task_name,problem_type,instruction,instance,solution,obj,instance_variant,solution_variant,context_index,input_format,input_index_base
LOP,LOP,"Someone in the back office measured how each dish plays with every other: for any two different plates, there’s a number saying how much better it is to serve A before B. The head chef’s job is to create a single tasting order that uses every plate exactly once and then add up those pairwise numbers according to that order. The better the sum, the smoother the tasting — so the plan is to find the order with the biggest combined harmony. The detailed menu and the pairwise numbers follow below.

Here are the 20 dishes to order: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T.

*Meaning: this is a directed pairwise_harmony matrix over items in dish_ids (20 total). Entry at row preceding_dish_id=i and column following_dish_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).*

**pairwise_harmony)**

| preceding_dish_id\following_dish_id | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 3.0 | 71241.0 | 0.0 | 1141.0 | 1539.0 | 6.0 | 19.0 | 93.0 | 45.0 | 10.0 | 0.0 | 1221.0 | 6589.0 | 0.0 | 78.0 | 0.0 | 0.0 | 3347.0 | 298.0 | 394.0 |
| B | 2.0 | 1118.0 | 0.0 | 54816.0 | 10345.0 | 61.0 | 163.0 | 1043.0 | 138.0 | 35.0 | 309.0 | 1406.0 | 19.0 | 0.0 | 155.0 | 0.0 | 0.0 | 5305.0 | 327.0 | 880.0 |
| C | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| D | 162.0 | 0.0 | 0.0 | 1085632.0 | 28351.0 | 179824.0 | 63050.0 | 6145.0 | 0.0 | 3354.0 | 24687.0 | 0.0 | 1473.0 | 0.0 | 0.0 | 0.0 | 0.0 | 9948.0 | 54.0 | 0.0 |
| E | 368.0 | 319.0 | 0.0 | 38463.0 | 930910.0 | 15868.0 | 4220.0 | 52434.0 | 194088.0 | 34977.0 | 9549.0 | 28291.0 | 1134.0 | 3998.0 | 1214.0 | 2621.0 | 269.0 | 15201.0 | 1577.0 | 3459.0 |
| F | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 190817.0 | 0.0 | 0.0 | 0.0 | 0.0 | 70788.0 | 0.0 | 3278.0 | 0.0 | 0.0 | 0.0 | 0.0 | 40241.0 | 0.0 | 0.0 |
| G | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 30888.0 | 0.0 | 0.0 | 0.0 | 4956.0 | 0.0 | 50379.0 | 0.0 | 0.0 | 0.0 | 0.0 | 85689.0 | 78.0 | 0.0 |
| H | 0.0 | 0.0 | 0.0 | 1.0 | 58723.0 | 0.0 | 0.0 | 797261.0 | 1418.0 | 1861.0 | 0.0 | 6522.0 | 49.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| I | 0.0 | 0.0 | 0.0 | 0.0 | 3908.0 | 6672.0 | 132.0 | 4463.0 | 1212279.0 | 31530.0 | 455.0 | 20594.0 | 4773.0 | 1529.0 | 553.0 | 0.0 | 199.0 | 0.0 | 0.0 | 269.0 |
| J | 0.0 | 0.0 | 0.0 | 26.0 | 5388.0 | 3042.0 | 10680.0 | 3508.0 | 5634.0 | 244621.0 | 21.0 | 49747.0 | 1132.0 | 1321.0 | 1041.0 | 28789.0 | 0.0 | 10941.0 | 1569.0 | 1232.0 |
| K | 24.0 | 729.0 | 0.0 | 194271.0 | 15928.0 | 2981.0 | 1202.0 | 19683.0 | 33839.0 | 2689.0 | 1891.0 | 92246.0 | 54017.0 | 10629.0 | 3295.0 | 39744.0 | 329.0 | 17681.0 | 2594.0 | 3033.0 |
| L | 132.0 | 129.0 | 0.0 | 4680.0 | 36979.0 | 20090.0 | 7645.0 | 57312.0 | 113543.0 | 13579.0 | 33554.0 | 90317.0 | 33773.0 | 4776.0 | 5104.0 | 3231.0 | 375.0 | 28083.0 | 3120.0 | 2610.0 |
| M | 72.0 | 3357.0 | 0.0 | 58419.0 | 40729.0 | 29078.0 | 8622.0 | 46974.0 | 31722.0 | 10566.0 | 10012.0 | 57169.0 | 18412.0 | 8337.0 | 3485.0 | 886.0 | 1161.0 | 23293.0 | 3813.0 | 2019.0 |
| N | 49.0 | 50.0 | 0.0 | 2850.0 | 7790.0 | 2500.0 | 270.0 | 1034.0 | 2477.0 | 233.0 | 0.0 | 30789.0 | 86651.0 | 1401.0 | 2551.0 | 0.0 | 0.0 | 3912.0 | 115.0 | 218.0 |
| O | 87.0 | 666.0 | 0.0 | 15626.0 | 18900.0 | 10670.0 | 4353.0 | 9675.0 | 20109.0 | 7411.0 | 6891.0 | 58496.0 | 25186.0 | 690.0 | 981443.0 | 14176.0 | 997.0 | 135991.0 | 19869.0 | 9730.0 |
| P | 8.0 | 41.0 | 0.0 | 4410.0 | 7102.0 | 2112.0 | 6335.0 | 13161.0 | 30827.0 | 17153.0 | 25341.0 | 234931.0 | 9318.0 | 21069.0 | 10152.0 | 0.0 | 3126.0 | 26944.0 | 8811.0 | 4045.0 |
| Q | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| R | 0.0 | 0.0 | 0.0 | 500.0 | 393.0 | 185.0 | 0.0 | 280.0 | 600.0 | 0.0 | 0.0 | 5420.0 | 3.0 | 0.0 | 0.0 | 0.0 | 0.0 | 52063.0 | 889.0 | 1765.0 |
| S | 0.0 | 0.0 | 0.0 | 0.0 | 215.0 | 215.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 646.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 658.0 | 0.0 |
| T | 0.0 | 0.0 | 0.0 | 0.0 | 92.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

The task is to pick one tasting sequence that uses all 20 dishes exactly once and maximizes the combined harmony.

Also, when you send back the final tasting order, you can just use a simple JSON snippet like this so it's easy to read and parse:

{
  ""solution"": [<first_dish>, <second_dish>, ..., <last_dish>]
}

This just means ""solution"" is an ordered list of the plates from the one served first to the one served last — swap each placeholder with the actual plate identifier from the instance. It's just the expected shape, not the actual answer itself.

Please make sure to use the identifiers exactly as they appear in the instance input — no renaming and no new labels. 
To be clear: 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"".","[[3.0, 71241.0, 0.0, 1141.0, 1539.0, 6.0, 19.0, 93.0, 45.0, 10.0, 0.0, 1221.0, 6589.0, 0.0, 78.0, 0.0, 0.0, 3347.0, 298.0, 394.0], [2.0, 1118.0, 0.0, 54816.0, 10345.0, 61.0, 163.0, 1043.0, 138.0, 35.0, 309.0, 1406.0, 19.0, 0.0, 155.0, 0.0, 0.0, 5305.0, 327.0, 880.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [162.0, 0.0, 0.0, 1085632.0, 28351.0, 179824.0, 63050.0, 6145.0, 0.0, 3354.0, 24687.0, 0.0, 1473.0, 0.0, 0.0, 0.0, 0.0, 9948.0, 54.0, 0.0], [368.0, 319.0, 0.0, 38463.0, 930910.0, 15868.0, 4220.0, 52434.0, 194088.0, 34977.0, 9549.0, 28291.0, 1134.0, 3998.0, 1214.0, 2621.0, 269.0, 15201.0, 1577.0, 3459.0], [0.0, 0.0, 0.0, 0.0, 0.0, 190817.0, 0.0, 0.0, 0.0, 0.0, 70788.0, 0.0, 3278.0, 0.0, 0.0, 0.0, 0.0, 40241.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 30888.0, 0.0, 0.0, 0.0, 4956.0, 0.0, 50379.0, 0.0, 0.0, 0.0, 0.0, 85689.0, 78.0, 0.0], [0.0, 0.0, 0.0, 1.0, 58723.0, 0.0, 0.0, 797261.0, 1418.0, 1861.0, 0.0, 6522.0, 49.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 3908.0, 6672.0, 132.0, 4463.0, 1212279.0, 31530.0, 455.0, 20594.0, 4773.0, 1529.0, 553.0, 0.0, 199.0, 0.0, 0.0, 269.0], [0.0, 0.0, 0.0, 26.0, 5388.0, 3042.0, 10680.0, 3508.0, 5634.0, 244621.0, 21.0, 49747.0, 1132.0, 1321.0, 1041.0, 28789.0, 0.0, 10941.0, 1569.0, 1232.0], [24.0, 729.0, 0.0, 194271.0, 15928.0, 2981.0, 1202.0, 19683.0, 33839.0, 2689.0, 1891.0, 92246.0, 54017.0, 10629.0, 3295.0, 39744.0, 329.0, 17681.0, 2594.0, 3033.0], [132.0, 129.0, 0.0, 4680.0, 36979.0, 20090.0, 7645.0, 57312.0, 113543.0, 13579.0, 33554.0, 90317.0, 33773.0, 4776.0, 5104.0, 3231.0, 375.0, 28083.0, 3120.0, 2610.0], [72.0, 3357.0, 0.0, 58419.0, 40729.0, 29078.0, 8622.0, 46974.0, 31722.0, 10566.0, 10012.0, 57169.0, 18412.0, 8337.0, 3485.0, 886.0, 1161.0, 23293.0, 3813.0, 2019.0], [49.0, 50.0, 0.0, 2850.0, 7790.0, 2500.0, 270.0, 1034.0, 2477.0, 233.0, 0.0, 30789.0, 86651.0, 1401.0, 2551.0, 0.0, 0.0, 3912.0, 115.0, 218.0], [87.0, 666.0, 0.0, 15626.0, 18900.0, 10670.0, 4353.0, 9675.0, 20109.0, 7411.0, 6891.0, 58496.0, 25186.0, 690.0, 981443.0, 14176.0, 997.0, 135991.0, 19869.0, 9730.0], [8.0, 41.0, 0.0, 4410.0, 7102.0, 2112.0, 6335.0, 13161.0, 30827.0, 17153.0, 25341.0, 234931.0, 9318.0, 21069.0, 10152.0, 0.0, 3126.0, 26944.0, 8811.0, 4045.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 500.0, 393.0, 185.0, 0.0, 280.0, 600.0, 0.0, 0.0, 5420.0, 3.0, 0.0, 0.0, 0.0, 0.0, 52063.0, 889.0, 1765.0], [0.0, 0.0, 0.0, 0.0, 215.0, 215.0, 0.0, 0.0, 0.0, 0.0, 0.0, 646.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 658.0, 0.0], [0.0, 0.0, 0.0, 0.0, 92.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]]","[14, 10, 15, 13, 0, 12, 1, 11, 4, 8, 3, 9, 6, 5, 17, 19, 18, 16, 7, 2]",2970983.0,"{'problem_type': 'LOP', 'num_nodes': 20, 'nodes': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S', 'T'], 'pairs': [{'from_id': 'A', 'to_id': 'B', 'weight': 71241.0}, {'from_id': 'A', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'D', 'weight': 1141.0}, {'from_id': 'A', 'to_id': 'E', 'weight': 1539.0}, {'from_id': 'A', 'to_id': 'F', 'weight': 6.0}, {'from_id': 'A', 'to_id': 'G', 'weight': 19.0}, {'from_id': 'A', 'to_id': 'H', 'weight': 93.0}, {'from_id': 'A', 'to_id': 'I', 'weight': 45.0}, {'from_id': 'A', 'to_id': 'J', 'weight': 10.0}, {'from_id': 'A', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'L', 'weight': 1221.0}, {'from_id': 'A', 'to_id': 'M', 'weight': 6589.0}, {'from_id': 'A', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'O', 'weight': 78.0}, {'from_id': 'A', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'R', 'weight': 3347.0}, {'from_id': 'A', 'to_id': 'S', 'weight': 298.0}, {'from_id': 'A', 'to_id': 'T', 'weight': 394.0}, {'from_id': 'B', 'to_id': 'A', 'weight': 2.0}, {'from_id': 'B', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'D', 'weight': 54816.0}, {'from_id': 'B', 'to_id': 'E', 'weight': 10345.0}, {'from_id': 'B', 'to_id': 'F', 'weight': 61.0}, {'from_id': 'B', 'to_id': 'G', 'weight': 163.0}, {'from_id': 'B', 'to_id': 'H', 'weight': 1043.0}, {'from_id': 'B', 'to_id': 'I', 'weight': 138.0}, {'from_id': 'B', 'to_id': 'J', 'weight': 35.0}, {'from_id': 'B', 'to_id': 'K', 'weight': 309.0}, {'from_id': 'B', 'to_id': 'L', 'weight': 1406.0}, {'from_id': 'B', 'to_id': 'M', 'weight': 19.0}, {'from_id': 'B', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'O', 'weight': 155.0}, {'from_id': 'B', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'R', 'weight': 5305.0}, {'from_id': 'B', 'to_id': 'S', 'weight': 327.0}, {'from_id': 'B', 'to_id': 'T', 'weight': 880.0}, {'from_id': 'C', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'S', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'T', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'A', 'weight': 162.0}, {'from_id': 'D', 'to_id': 'B', 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{'from_id': 'F', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'K', 'weight': 70788.0}, {'from_id': 'F', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'M', 'weight': 3278.0}, {'from_id': 'F', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'R', 'weight': 40241.0}, {'from_id': 'F', 'to_id': 'S', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'T', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'F', 'weight': 0.0}, 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'weight': 0.0}, {'from_id': 'S', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'E', 'weight': 215.0}, {'from_id': 'S', 'to_id': 'F', 'weight': 215.0}, {'from_id': 'S', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'L', 'weight': 646.0}, {'from_id': 'S', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'T', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'E', 'weight': 92.0}, {'from_id': 'T', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'S', 'weight': 0.0}]}","['O', 'K', 'P', 'N', 'A', 'M', 'B', 'L', 'E', 'I', 'D', 'J', 'G', 'F', 'R', 'T', 'S', 'Q', 'H', 'C']",1,markdown_table,names
LOP,LOP,"Imagine a DJ lining up a deck of tracks and being told each song must appear once in a single, uninterrupted set. For every two different songs there’s a little energy value tied to playing one before the other, and the playlist’s magic comes from summing all those values for every ordered pair in the lineup. The challenge is to choose the order that makes that total sum as large as possible, with all songs included and no duplicates. The specific track names and the pairwise energy numbers are provided below.

There are 19 distinct tracks to place into one continuous set: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18.

*Meaning: this is a directed energy_gain matrix over items in track_ids (19 total). Entry at row preceding_track_id=i and column following_track_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).*

**energy_gain)**

| preceding_track_id\following_track_id | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 125900.0 | 0.0 | 0.0 | 0.0 | 22100.0 | 0.0 | 0.0 | 0.0 | 0.0 | 640200.0 | 111400.0 | 8700.0 | 15800.0 | 36200.0 | 600.0 | 0.0 | 0.0 | 73200.0 | 0.0 |
| 1 | 500.0 | 26500.0 | 0.0 | 33900.0 | 23600.0 | 10400.0 | 2700.0 | 200.0 | 1500.0 | 7000.0 | 0.0 | 800.0 | 700.0 | 3200.0 | 500.0 | 0.0 | 0.0 | 35400.0 | 0.0 |
| 2 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 3 | 800.0 | 38700.0 | 0.0 | 1528500.0 | 85200.0 | 665000.0 | 529100.0 | 41400.0 | 272200.0 | 19500.0 | 4800.0 | 500.0 | 14600.0 | 14100.0 | 17800.0 | 0.0 | 0.0 | 7600.0 | 0.0 |
| 4 | 134900.0 | 6900.0 | 0.0 | 38800.0 | 726900.0 | 52400.0 | 30500.0 | 7300.0 | 30000.0 | 68500.0 | 4200.0 | 19400.0 | 37000.0 | 240000.0 | 21100.0 | 0.0 | 0.0 | 245100.0 | 0.0 |
| 5 | 8100.0 | 21900.0 | 0.0 | 291800.0 | 86100.0 | 121700.0 | 458600.0 | 47400.0 | 288500.0 | 50600.0 | 4700.0 | 16100.0 | 43400.0 | 36200.0 | 17900.0 | 0.0 | 0.0 | 45300.0 | 0.0 |
| 6 | 29300.0 | 48800.0 | 0.0 | 79300.0 | 51700.0 | 84900.0 | 345000.0 | 26200.0 | 81000.0 | 21500.0 | 3400.0 | 2000.0 | 5500.0 | 17700.0 | 6000.0 | 0.0 | 0.0 | 97500.0 | 0.0 |
| 7 | 0.0 | 100.0 | 0.0 | 1400.0 | 1400.0 | 9000.0 | 41200.0 | 36200.0 | 9800.0 | 600.0 | 300.0 | 400.0 | 0.0 | 600.0 | 200.0 | 0.0 | 0.0 | 117800.0 | 0.0 |
| 8 | 4200.0 | 0.0 | 0.0 | 13700.0 | 300.0 | 3300.0 | 81000.0 | 0.0 | 25500.0 | 900.0 | 0.0 | 100.0 | 300.0 | 200.0 | 0.0 | 0.0 | 0.0 | 49800.0 | 0.0 |
| 9 | 498900.0 | 0.0 | 0.0 | 0.0 | 60600.0 | 0.0 | 0.0 | 0.0 | 0.0 | 575300.0 | 0.0 | 0.0 | 200.0 | 100.0 | 1200.0 | 0.0 | 0.0 | 62500.0 | 0.0 |
| 10 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 400.0 | 0.0 |
| 11 | 0.0 | 0.0 | 0.0 | 0.0 | 900.0 | 900.0 | 0.0 | 500.0 | 2600.0 | 0.0 | 0.0 | 60500.0 | 300.0 | 2400.0 | 500.0 | 0.0 | 0.0 | 4800.0 | 0.0 |
| 12 | 26900.0 | 10700.0 | 0.0 | 4700.0 | 9300.0 | 10500.0 | 16000.0 | 4000.0 | 20100.0 | 3500.0 | 300.0 | 3200.0 | 280500.0 | 2600.0 | 4100.0 | 0.0 | 0.0 | 31900.0 | 0.0 |
| 13 | 9700.0 | 6600.0 | 0.0 | 19400.0 | 59000.0 | 33300.0 | 43300.0 | 15300.0 | 148700.0 | 13400.0 | 800.0 | 18000.0 | 13300.0 | 5500.0 | 6000.0 | 0.0 | 0.0 | 7500.0 | 0.0 |
| 14 | 0.0 | 100.0 | 0.0 | 500.0 | 2000.0 | 18800.0 | 7600.0 | 2300.0 | 6700.0 | 200.0 | 100.0 | 2900.0 | 14200.0 | 500.0 | 25400.0 | 0.0 | 0.0 | 17500.0 | 0.0 |
| 15 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 16 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 17 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 18 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Order the set to maximize the playlist's total energy from all ordered pairs.

When you send back the final ordering, toss it into a tiny JSON snippet like this — just a single ""solution"" key with the playlist listed from first to last:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of that ""solution"" array as the exact play order: the first entry is the opening track, the next is what follows, and so on until the last one. This JSON is just a sketch of the shape I want — not the real answer — so replace those placeholders with the actual track identifiers from the instance.

Please use the identifiers exactly as they appear in the 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”.","[[125900.0, 0.0, 0.0, 0.0, 22100.0, 0.0, 0.0, 0.0, 0.0, 640200.0, 111400.0, 8700.0, 15800.0, 36200.0, 600.0, 0.0, 0.0, 73200.0, 0.0], [500.0, 26500.0, 0.0, 33900.0, 23600.0, 10400.0, 2700.0, 200.0, 1500.0, 7000.0, 0.0, 800.0, 700.0, 3200.0, 500.0, 0.0, 0.0, 35400.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [800.0, 38700.0, 0.0, 1528500.0, 85200.0, 665000.0, 529100.0, 41400.0, 272200.0, 19500.0, 4800.0, 500.0, 14600.0, 14100.0, 17800.0, 0.0, 0.0, 7600.0, 0.0], [134900.0, 6900.0, 0.0, 38800.0, 726900.0, 52400.0, 30500.0, 7300.0, 30000.0, 68500.0, 4200.0, 19400.0, 37000.0, 240000.0, 21100.0, 0.0, 0.0, 245100.0, 0.0], [8100.0, 21900.0, 0.0, 291800.0, 86100.0, 121700.0, 458600.0, 47400.0, 288500.0, 50600.0, 4700.0, 16100.0, 43400.0, 36200.0, 17900.0, 0.0, 0.0, 45300.0, 0.0], [29300.0, 48800.0, 0.0, 79300.0, 51700.0, 84900.0, 345000.0, 26200.0, 81000.0, 21500.0, 3400.0, 2000.0, 5500.0, 17700.0, 6000.0, 0.0, 0.0, 97500.0, 0.0], [0.0, 100.0, 0.0, 1400.0, 1400.0, 9000.0, 41200.0, 36200.0, 9800.0, 600.0, 300.0, 400.0, 0.0, 600.0, 200.0, 0.0, 0.0, 117800.0, 0.0], [4200.0, 0.0, 0.0, 13700.0, 300.0, 3300.0, 81000.0, 0.0, 25500.0, 900.0, 0.0, 100.0, 300.0, 200.0, 0.0, 0.0, 0.0, 49800.0, 0.0], [498900.0, 0.0, 0.0, 0.0, 60600.0, 0.0, 0.0, 0.0, 0.0, 575300.0, 0.0, 0.0, 200.0, 100.0, 1200.0, 0.0, 0.0, 62500.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 400.0, 0.0], [0.0, 0.0, 0.0, 0.0, 900.0, 900.0, 0.0, 500.0, 2600.0, 0.0, 0.0, 60500.0, 300.0, 2400.0, 500.0, 0.0, 0.0, 4800.0, 0.0], [26900.0, 10700.0, 0.0, 4700.0, 9300.0, 10500.0, 16000.0, 4000.0, 20100.0, 3500.0, 300.0, 3200.0, 280500.0, 2600.0, 4100.0, 0.0, 0.0, 31900.0, 0.0], [9700.0, 6600.0, 0.0, 19400.0, 59000.0, 33300.0, 43300.0, 15300.0, 148700.0, 13400.0, 800.0, 18000.0, 13300.0, 5500.0, 6000.0, 0.0, 0.0, 7500.0, 0.0], [0.0, 100.0, 0.0, 500.0, 2000.0, 18800.0, 7600.0, 2300.0, 6700.0, 200.0, 100.0, 2900.0, 14200.0, 500.0, 25400.0, 0.0, 0.0, 17500.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]]","[18, 3, 5, 4, 13, 14, 12, 7, 6, 1, 8, 0, 11, 10, 9, 17, 16, 15, 2]",5586900.0,"{'problem_type': 'LOP', 'num_nodes': 19, 'nodes': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], 'pairs': [{'from_id': 0, 'to_id': 1, 'weight': 0.0}, {'from_id': 0, 'to_id': 2, 'weight': 0.0}, {'from_id': 0, 'to_id': 3, 'weight': 0.0}, {'from_id': 0, 'to_id': 4, 'weight': 22100.0}, {'from_id': 0, 'to_id': 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500.0}, {'from_id': 14, 'to_id': 15, 'weight': 0.0}, {'from_id': 14, 'to_id': 16, 'weight': 0.0}, {'from_id': 14, 'to_id': 17, 'weight': 17500.0}, {'from_id': 14, 'to_id': 18, 'weight': 0.0}, {'from_id': 15, 'to_id': 0, 'weight': 0.0}, {'from_id': 15, 'to_id': 1, 'weight': 0.0}, {'from_id': 15, 'to_id': 2, 'weight': 0.0}, {'from_id': 15, 'to_id': 3, 'weight': 0.0}, {'from_id': 15, 'to_id': 4, 'weight': 0.0}, {'from_id': 15, 'to_id': 5, 'weight': 0.0}, {'from_id': 15, 'to_id': 6, 'weight': 0.0}, {'from_id': 15, 'to_id': 7, 'weight': 0.0}, {'from_id': 15, 'to_id': 8, 'weight': 0.0}, {'from_id': 15, 'to_id': 9, 'weight': 0.0}, {'from_id': 15, 'to_id': 10, 'weight': 0.0}, {'from_id': 15, 'to_id': 11, 'weight': 0.0}, {'from_id': 15, 'to_id': 12, 'weight': 0.0}, {'from_id': 15, 'to_id': 13, 'weight': 0.0}, {'from_id': 15, 'to_id': 14, 'weight': 0.0}, {'from_id': 15, 'to_id': 16, 'weight': 0.0}, {'from_id': 15, 'to_id': 17, 'weight': 0.0}, {'from_id': 15, 'to_id': 18, 'weight': 0.0}, {'from_id': 16, 'to_id': 0, 'weight': 0.0}, {'from_id': 16, 'to_id': 1, 'weight': 0.0}, {'from_id': 16, 'to_id': 2, 'weight': 0.0}, {'from_id': 16, 'to_id': 3, 'weight': 0.0}, {'from_id': 16, 'to_id': 4, 'weight': 0.0}, {'from_id': 16, 'to_id': 5, 'weight': 0.0}, {'from_id': 16, 'to_id': 6, 'weight': 0.0}, {'from_id': 16, 'to_id': 7, 'weight': 0.0}, {'from_id': 16, 'to_id': 8, 'weight': 0.0}, {'from_id': 16, 'to_id': 9, 'weight': 0.0}, {'from_id': 16, 'to_id': 10, 'weight': 0.0}, {'from_id': 16, 'to_id': 11, 'weight': 0.0}, {'from_id': 16, 'to_id': 12, 'weight': 0.0}, {'from_id': 16, 'to_id': 13, 'weight': 0.0}, {'from_id': 16, 'to_id': 14, 'weight': 0.0}, {'from_id': 16, 'to_id': 15, 'weight': 0.0}, {'from_id': 16, 'to_id': 17, 'weight': 0.0}, {'from_id': 16, 'to_id': 18, 'weight': 0.0}, {'from_id': 17, 'to_id': 0, 'weight': 0.0}, {'from_id': 17, 'to_id': 1, 'weight': 0.0}, {'from_id': 17, 'to_id': 2, 'weight': 0.0}, {'from_id': 17, 'to_id': 3, 'weight': 0.0}, {'from_id': 17, 'to_id': 4, 'weight': 0.0}, {'from_id': 17, 'to_id': 5, 'weight': 0.0}, {'from_id': 17, 'to_id': 6, 'weight': 0.0}, {'from_id': 17, 'to_id': 7, 'weight': 0.0}, {'from_id': 17, 'to_id': 8, 'weight': 0.0}, {'from_id': 17, 'to_id': 9, 'weight': 0.0}, {'from_id': 17, 'to_id': 10, 'weight': 0.0}, {'from_id': 17, 'to_id': 11, 'weight': 0.0}, {'from_id': 17, 'to_id': 12, 'weight': 0.0}, {'from_id': 17, 'to_id': 13, 'weight': 0.0}, {'from_id': 17, 'to_id': 14, 'weight': 0.0}, {'from_id': 17, 'to_id': 15, 'weight': 0.0}, {'from_id': 17, 'to_id': 16, 'weight': 0.0}, {'from_id': 17, 'to_id': 18, 'weight': 0.0}, {'from_id': 18, 'to_id': 0, 'weight': 0.0}, {'from_id': 18, 'to_id': 1, 'weight': 0.0}, {'from_id': 18, 'to_id': 2, 'weight': 0.0}, {'from_id': 18, 'to_id': 3, 'weight': 0.0}, {'from_id': 18, 'to_id': 4, 'weight': 0.0}, {'from_id': 18, 'to_id': 5, 'weight': 0.0}, {'from_id': 18, 'to_id': 6, 'weight': 0.0}, {'from_id': 18, 'to_id': 7, 'weight': 0.0}, {'from_id': 18, 'to_id': 8, 'weight': 0.0}, {'from_id': 18, 'to_id': 9, 'weight': 0.0}, {'from_id': 18, 'to_id': 10, 'weight': 0.0}, {'from_id': 18, 'to_id': 11, 'weight': 0.0}, {'from_id': 18, 'to_id': 12, 'weight': 0.0}, {'from_id': 18, 'to_id': 13, 'weight': 0.0}, {'from_id': 18, 'to_id': 14, 'weight': 0.0}, {'from_id': 18, 'to_id': 15, 'weight': 0.0}, {'from_id': 18, 'to_id': 16, 'weight': 0.0}, {'from_id': 18, 'to_id': 17, 'weight': 0.0}]}","[18, 3, 5, 4, 13, 14, 12, 7, 6, 1, 8, 0, 11, 10, 9, 17, 16, 15, 2]",2,markdown_table,0
LOP,LOP,"There’s a presenter trying to make a one-shot run of slides that tells a smooth story, which means every slide must appear once and only once. For every two different slides there’s a number that says how much clarity you get if one precedes the other, and the overall clarity of a particular lineup is just the sum of those numbers for every pair in that order — so the job is to find the lineup that makes that total as high as it can be. The exact slide items and their pairwise clarity values are listed below.

They are the 23 slides to place, listed as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22.
Meaning: this is a directed pairwise_clarity_gain matrix over items in slide_ids (23 total). Entry at row preceding_slide_id=i and column following_slide_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
pairwise_clarity_gain:
| preceding_slide_id\following_slide_id | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 4161.0 | 0.0 | 0.0 | 0.0 | 8010.0 | 13877.0 | 35.0 | 331.0 | 520.0 | 7.0 | 71.0 | 129.0 | 77.0 | 450.0 | 56.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2657.0 | 1409.0 |
| 1 | 1019.0 | 6745583.0 | 1131915.0 | 0.0 | 331658.0 | 1119473.0 | 7729.0 | 31839.0 | 11401.0 | 23284.0 | 100437.0 | 66189.0 | 47922.0 | 37878.0 | 663490.0 | 379487.0 | 16953.0 | 290020.0 | 0.0 | 10322.0 | 88788.0 | 155368.0 | 98510.0 |
| 2 | 58375.0 | 259445.0 | 883005.0 | 0.0 | 325992.0 | 493456.0 | 17811.0 | 64386.0 | 23810.0 | 9456.0 | 152283.0 | 111407.0 | 70692.0 | 28774.0 | 78806.0 | 9519.0 | 14348.0 | 51963.0 | 74030.0 | 5595.0 | 44170.0 | 76892.0 | 23810.0 |
| 3 | 0.0 | 0.0 | 10359.0 | 0.0 | 0.0 | 2.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 140.0 | 0.0 | 0.0 | 0.0 | 0.0 | 37.0 | 0.0 | 0.0 | 0.0 |
| 4 | 528.0 | 3918.0 | 10193.0 | 0.0 | 938503.0 | 307728.0 | 19636.0 | 65446.0 | 13542.0 | 67954.0 | 2887.0 | 1423.0 | 3946.0 | 9681.0 | 1880.0 | 4506.0 | 0.0 | 2494.0 | 3221.0 | 1165.0 | 0.0 | 19867.0 | 5871.0 |
| 5 | 6683.0 | 85830.0 | 34202.0 | 0.0 | 193560.0 | 3212827.0 | 5879.0 | 120819.0 | 20931.0 | 41932.0 | 859403.0 | 197900.0 | 1169695.0 | 80552.0 | 7197.0 | 4235.0 | 1474.0 | 30732.0 | 19740.0 | 1075.0 | 140881.0 | 71172.0 | 314190.0 |
| 6 | 625.0 | 2517.0 | 6923.0 | 0.0 | 0.0 | 2086.0 | 46024.0 | 22590.0 | 27284.0 | 0.0 | 0.0 | 0.0 | 0.0 | 37471.0 | 8097.0 | 6485.0 | 7385.0 | 9020.0 | 0.0 | 200.0 | 6187.0 | 21886.0 | 13456.0 |
| 7 | 3320.0 | 3921.0 | 143225.0 | 0.0 | 15915.0 | 23603.0 | 77764.0 | 556323.0 | 77501.0 | 1108.0 | 11880.0 | 11944.0 | 6593.0 | 256053.0 | 13021.0 | 2244.0 | 39229.0 | 11540.0 | 7092.0 | 223.0 | 19610.0 | 174444.0 | 7555.0 |
| 8 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 138075.0 | 0.0 | 0.0 | 0.0 | 0.0 | 49503.0 | 29788.0 | 113007.0 | 0.0 | 445.0 | 0.0 | 0.0 | 250.0 | 208937.0 | 0.0 |
| 9 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 3955.0 | 0.0 | 0.0 | 0.0 | 22880.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2642.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2913.0 | 1353.0 |
| 10 | 196.0 | 482.0 | 214.0 | 0.0 | 2302.0 | 13753.0 | 1140.0 | 4944.0 | 502.0 | 129.0 | 4378815.0 | 31191.0 | 102892.0 | 1731.0 | 14732.0 | 2977.0 | 12094.0 | 4177.0 | 0.0 | 842.0 | 10831.0 | 5543.0 | 29781.0 |
| 11 | 1710.0 | 4767.0 | 22091.0 | 0.0 | 115136.0 | 275737.0 | 18210.0 | 96961.0 | 16248.0 | 38320.0 | 111165.0 | 2118300.0 | 25504.0 | 2848.0 | 16598.0 | 25799.0 | 22988.0 | 121214.0 | 11593.0 | 14921.0 | 95186.0 | 121800.0 | 30846.0 |
| 12 | 0.0 | 1657.0 | 13157.0 | 0.0 | 12188.0 | 117199.0 | 27939.0 | 213207.0 | 42243.0 | 18057.0 | 66489.0 | 10957.0 | 93453.0 | 83384.0 | 197106.0 | 2363.0 | 2341.0 | 28878.0 | 3555.0 | 254.0 | 12886.0 | 33031.0 | 38608.0 |
| 13 | 2767.0 | 7359.0 | 18119.0 | 0.0 | 93997.0 | 79464.0 | 7422.0 | 38939.0 | 5964.0 | 17169.0 | 97908.0 | 103155.0 | 29400.0 | 3624.0 | 275909.0 | 9064.0 | 44895.0 | 82303.0 | 196311.0 | 7246.0 | 28284.0 | 66397.0 | 50727.0 |
| 14 | 22703.0 | 58518.0 | 38585.0 | 0.0 | 146890.0 | 115541.0 | 13411.0 | 60811.0 | 27451.0 | 22288.0 | 114002.0 | 55166.0 | 44423.0 | 38065.0 | 146401.0 | 17117.0 | 64570.0 | 46844.0 | 3401.0 | 1859.0 | 51351.0 | 64878.0 | 30267.0 |
| 15 | 2542.0 | 9565.0 | 3493.0 | 0.0 | 11879.0 | 28998.0 | 3004.0 | 11563.0 | 3281.0 | 3642.0 | 8320.0 | 4696.0 | 3765.0 | 0.0 | 19564.0 | 0.0 | 20124.0 | 14980.0 | 171.0 | 122.0 | 2104.0 | 24430.0 | 1567.0 |
| 16 | 129.0 | 13401.0 | 18675.0 | 0.0 | 21980.0 | 128975.0 | 4221.0 | 38826.0 | 11959.0 | 19478.0 | 37583.0 | 74751.0 | 9656.0 | 9318.0 | 22657.0 | 18452.0 | 25900.0 | 63056.0 | 3398.0 | 4224.0 | 32988.0 | 81521.0 | 15403.0 |
| 17 | 828.0 | 15741.0 | 26828.0 | 0.0 | 45194.0 | 262992.0 | 52972.0 | 122626.0 | 50006.0 | 18737.0 | 76311.0 | 100666.0 | 44007.0 | 6568.0 | 23238.0 | 32387.0 | 41774.0 | 49009.0 | 319233.0 | 38208.0 | 84498.0 | 592507.0 | 71397.0 |
| 18 | 90.0 | 6419.0 | 13390.0 | 0.0 | 18711.0 | 43638.0 | 7498.0 | 19321.0 | 2208.0 | 2921.0 | 27338.0 | 25181.0 | 9823.0 | 67900.0 | 45244.0 | 7617.0 | 17437.0 | 121350.0 | 0.0 | 8862.0 | 158685.0 | 57962.0 | 19152.0 |
| 19 | 0.0 | 1764.0 | 5263.0 | 0.0 | 1521.0 | 99969.0 | 25143.0 | 25939.0 | 7483.0 | 0.0 | 77.0 | 7581.0 | 26812.0 | 0.0 | 0.0 | 0.0 | 712.0 | 0.0 | 0.0 | 0.0 | 1830.0 | 0.0 | 0.0 |
| 20 | 0.0 | 3306.0 | 3588.0 | 0.0 | 4077.0 | 82002.0 | 560.0 | 15986.0 | 3689.0 | 17407.0 | 6436.0 | 38997.0 | 8360.0 | 354.0 | 3035.0 | 4918.0 | 15407.0 | 19497.0 | 20031.0 | 1706.0 | 128877.0 | 204721.0 | 36382.0 |
| 21 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 99499.0 | 16963.0 |
| 22 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 4386.0 | 0.0 |
Arrange the 23 slides to maximize the total clarity contributed by the listed pairwise gains.

Oh, and when you give the final lineup, just drop it into a tiny JSON object like this:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

""solution"" is just the ordered list of slide identifiers from first to last — basically the sequence you want the presenter to run through. Consider this JSON a simple sketch of the shape I expect, not the actual ordering answer.

Please use the exact identifiers as they appear in the instance input — don't rename them or make up 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”.""","[[4161.0, 0.0, 0.0, 0.0, 8010.0, 13877.0, 35.0, 331.0, 520.0, 7.0, 71.0, 129.0, 77.0, 450.0, 56.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2657.0, 1409.0], [1019.0, 6745583.0, 1131915.0, 0.0, 331658.0, 1119473.0, 7729.0, 31839.0, 11401.0, 23284.0, 100437.0, 66189.0, 47922.0, 37878.0, 663490.0, 379487.0, 16953.0, 290020.0, 0.0, 10322.0, 88788.0, 155368.0, 98510.0], [58375.0, 259445.0, 883005.0, 0.0, 325992.0, 493456.0, 17811.0, 64386.0, 23810.0, 9456.0, 152283.0, 111407.0, 70692.0, 28774.0, 78806.0, 9519.0, 14348.0, 51963.0, 74030.0, 5595.0, 44170.0, 76892.0, 23810.0], [0.0, 0.0, 10359.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 140.0, 0.0, 0.0, 0.0, 0.0, 37.0, 0.0, 0.0, 0.0], [528.0, 3918.0, 10193.0, 0.0, 938503.0, 307728.0, 19636.0, 65446.0, 13542.0, 67954.0, 2887.0, 1423.0, 3946.0, 9681.0, 1880.0, 4506.0, 0.0, 2494.0, 3221.0, 1165.0, 0.0, 19867.0, 5871.0], [6683.0, 85830.0, 34202.0, 0.0, 193560.0, 3212827.0, 5879.0, 120819.0, 20931.0, 41932.0, 859403.0, 197900.0, 1169695.0, 80552.0, 7197.0, 4235.0, 1474.0, 30732.0, 19740.0, 1075.0, 140881.0, 71172.0, 314190.0], [625.0, 2517.0, 6923.0, 0.0, 0.0, 2086.0, 46024.0, 22590.0, 27284.0, 0.0, 0.0, 0.0, 0.0, 37471.0, 8097.0, 6485.0, 7385.0, 9020.0, 0.0, 200.0, 6187.0, 21886.0, 13456.0], [3320.0, 3921.0, 143225.0, 0.0, 15915.0, 23603.0, 77764.0, 556323.0, 77501.0, 1108.0, 11880.0, 11944.0, 6593.0, 256053.0, 13021.0, 2244.0, 39229.0, 11540.0, 7092.0, 223.0, 19610.0, 174444.0, 7555.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 138075.0, 0.0, 0.0, 0.0, 0.0, 49503.0, 29788.0, 113007.0, 0.0, 445.0, 0.0, 0.0, 250.0, 208937.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 3955.0, 0.0, 0.0, 0.0, 22880.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2642.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2913.0, 1353.0], [196.0, 482.0, 214.0, 0.0, 2302.0, 13753.0, 1140.0, 4944.0, 502.0, 129.0, 4378815.0, 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LOP,LOP,"Backstage at the museum the crew debated the running order, agreeing it should be one continuous route that lists each painting just once so the visitors don’t miss or see anything twice. Someone had already scored every possible pair with how much better it is to see one painting before another, and the total value of any order comes from adding those pairwise scores wherever the earlier painting precedes the later one. The preferable lineup is simply the one with the biggest total sum under that rule. The concrete details are shown below.

They listed 21 paintings and gave their identifiers as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20.

*Meaning: this is a directed continuity_gain matrix over items in painting_ids (21 total). Entry at row preceding_painting_id=i and column succeeding_painting_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).*

**continuity_gain)**

| preceding_painting_id\succeeding_painting_id | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 525.0 | 3464.0 | 98.0 | 16.0 | 2.0 | 22.0 | 64.0 | 15.0 | 21.0 | 1.0 | 19.0 | 7.0 | 5.0 | 3.0 | 4.0 | 29.0 | 2.0 | 1.0 | 3.0 | 2.0 | 1.0 |
| 1 | 369.0 | 2109.0 | 1178.0 | 627.0 | 87.0 | 88.0 | 391.0 | 41.0 | 381.0 | 21.0 | 723.0 | 165.0 | 258.0 | 146.0 | 210.0 | 558.0 | 108.0 | 30.0 | 330.0 | 228.0 | 121.0 |
| 2 | 55.0 | 56.0 | 6024.0 | 141.0 | 13.0 | 96.0 | 353.0 | 12.0 | 133.0 | 116.0 | 15310.0 | 103.0 | 169.0 | 2.0 | 0.0 | 7.0 | 144.0 | 6.0 | 100.0 | 109.0 | 49.0 |
| 3 | 423.0 | 276.0 | 157.0 | 9394.0 | 678.0 | 28.0 | 58.0 | 29.0 | 157.0 | 4.0 | 680.0 | 93.0 | 24.0 | 11.0 | 39.0 | 74.0 | 2.0 | 0.0 | 19.0 | 2.0 | 1.0 |
| 4 | 0.0 | 0.0 | 0.0 | 0.0 | 878.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 9.0 | 55.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 39.0 |
| 5 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2239.0 | 624.0 | 340.0 | 0.0 | 4.0 | 0.0 | 0.0 | 1289.0 | 0.0 | 0.0 | 0.0 | 49.0 | 35.0 | 122.0 | 193.0 | 72.0 |
| 6 | 5.0 | 4.0 | 3.0 | 12.0 | 0.0 | 311.0 | 11779.0 | 5.0 | 43.0 | 0.0 | 13.0 | 9.0 | 2858.0 | 0.0 | 9.0 | 388.0 | 124.0 | 46.0 | 94.0 | 339.0 | 171.0 |
| 7 | 10.0 | 0.0 | 6.0 | 39.0 | 4.0 | 2.0 | 2.0 | 994.0 | 9.0 | 6.0 | 10.0 | 75.0 | 0.0 | 52.0 | 0.0 | 26.0 | 36.0 | 0.0 | 27.0 | 27.0 | 6.0 |
| 8 | 59.0 | 74.0 | 242.0 | 1295.0 | 104.0 | 138.0 | 355.0 | 501.0 | 1686.0 | 65.0 | 1905.0 | 282.0 | 44.0 | 12.0 | 15.0 | 157.0 | 187.0 | 4.0 | 312.0 | 148.0 | 13.0 |
| 9 | 0.0 | 7.0 | 0.0 | 3.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 335.0 | 10.0 | 26.0 | 8.0 | 1.0 | 18.0 | 35.0 | 3.0 | 0.0 | 83.0 | 4.0 | 4.0 |
| 10 | 413.0 | 81.0 | 25.0 | 32.0 | 17.0 | 12.0 | 25.0 | 3.0 | 10.0 | 0.0 | 390.0 | 18.0 | 10.0 | 31.0 | 32.0 | 3.0 | 21.0 | 0.0 | 213.0 | 39.0 | 0.0 |
| 11 | 38.0 | 115.0 | 169.0 | 164.0 | 35.0 | 40.0 | 78.0 | 19.0 | 127.0 | 3.0 | 162.0 | 237.0 | 76.0 | 49.0 | 168.0 | 654.0 | 95.0 | 8.0 | 113.0 | 77.0 | 7.0 |
| 12 | 66.0 | 9.0 | 173.0 | 373.0 | 18.0 | 50.0 | 146.0 | 33.0 | 106.0 | 14.0 | 85.0 | 465.0 | 60.0 | 38.0 | 130.0 | 740.0 | 3.0 | 63.0 | 4.0 | 7.0 | 86.0 |
| 13 | 152.0 | 68.0 | 148.0 | 363.0 | 36.0 | 39.0 | 93.0 | 72.0 | 81.0 | 5.0 | 267.0 | 105.0 | 153.0 | 161.0 | 620.0 | 249.0 | 93.0 | 15.0 | 221.0 | 56.0 | 50.0 |
| 14 | 36.0 | 73.0 | 95.0 | 114.0 | 12.0 | 24.0 | 64.0 | 15.0 | 38.0 | 2.0 | 255.0 | 83.0 | 45.0 | 64.0 | 16556.0 | 236.0 | 33.0 | 7.0 | 117.0 | 27.0 | 32.0 |
| 15 | 464.0 | 260.0 | 552.0 | 1332.0 | 47.0 | 114.0 | 313.0 | 88.0 | 287.0 | 14.0 | 6009.0 | 554.0 | 596.0 | 206.0 | 3681.0 | 3287.0 | 374.0 | 144.0 | 486.0 | 228.0 | 36.0 |
| 16 | 2.0 | 11.0 | 8.0 | 8.0 | 0.0 | 6.0 | 5.0 | 1.0 | 4.0 | 1.0 | 10.0 | 3.0 | 2.0 | 5.0 | 115.0 | 27.0 | 60.0 | 0.0 | 24.0 | 5.0 | 0.0 |
| 17 | 12.0 | 7.0 | 14.0 | 38.0 | 1.0 | 4.0 | 17.0 | 4.0 | 9.0 | 1.0 | 30.0 | 17.0 | 9.0 | 7.0 | 50.0 | 144.0 | 9.0 | 0.0 | 0.0 | 4.0 | 1.0 |
| 18 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 19 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 20 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 32.0 | 64.0 | 0.0 | 0.0 | 8.0 | 0.0 |

Those pairwise gains then add up to determine which single continuous route yields the highest total continuity.

Oh, and before you send the final ordering, please put it into this little JSON shape so it's easy to read and process:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of ""solution"" as the single, continuous route through the gallery — just list the paintings in order from the one visitors see first to the one they see last. This JSON is just a sketch of the shape I want, not the actual answer itself.

Please use the exact identifiers from the problem input — don't 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”.""","[[525.0, 3464.0, 98.0, 16.0, 2.0, 22.0, 64.0, 15.0, 21.0, 1.0, 19.0, 7.0, 5.0, 3.0, 4.0, 29.0, 2.0, 1.0, 3.0, 2.0, 1.0], [369.0, 2109.0, 1178.0, 627.0, 87.0, 88.0, 391.0, 41.0, 381.0, 21.0, 723.0, 165.0, 258.0, 146.0, 210.0, 558.0, 108.0, 30.0, 330.0, 228.0, 121.0], [55.0, 56.0, 6024.0, 141.0, 13.0, 96.0, 353.0, 12.0, 133.0, 116.0, 15310.0, 103.0, 169.0, 2.0, 0.0, 7.0, 144.0, 6.0, 100.0, 109.0, 49.0], [423.0, 276.0, 157.0, 9394.0, 678.0, 28.0, 58.0, 29.0, 157.0, 4.0, 680.0, 93.0, 24.0, 11.0, 39.0, 74.0, 2.0, 0.0, 19.0, 2.0, 1.0], [0.0, 0.0, 0.0, 0.0, 878.0, 0.0, 0.0, 0.0, 0.0, 0.0, 9.0, 55.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 39.0], [0.0, 0.0, 0.0, 0.0, 0.0, 2239.0, 624.0, 340.0, 0.0, 4.0, 0.0, 0.0, 1289.0, 0.0, 0.0, 0.0, 49.0, 35.0, 122.0, 193.0, 72.0], [5.0, 4.0, 3.0, 12.0, 0.0, 311.0, 11779.0, 5.0, 43.0, 0.0, 13.0, 9.0, 2858.0, 0.0, 9.0, 388.0, 124.0, 46.0, 94.0, 339.0, 171.0], [10.0, 0.0, 6.0, 39.0, 4.0, 2.0, 2.0, 994.0, 9.0, 6.0, 10.0, 75.0, 0.0, 52.0, 0.0, 26.0, 36.0, 0.0, 27.0, 27.0, 6.0], [59.0, 74.0, 242.0, 1295.0, 104.0, 138.0, 355.0, 501.0, 1686.0, 65.0, 1905.0, 282.0, 44.0, 12.0, 15.0, 157.0, 187.0, 4.0, 312.0, 148.0, 13.0], [0.0, 7.0, 0.0, 3.0, 0.0, 0.0, 0.0, 1.0, 0.0, 335.0, 10.0, 26.0, 8.0, 1.0, 18.0, 35.0, 3.0, 0.0, 83.0, 4.0, 4.0], [413.0, 81.0, 25.0, 32.0, 17.0, 12.0, 25.0, 3.0, 10.0, 0.0, 390.0, 18.0, 10.0, 31.0, 32.0, 3.0, 21.0, 0.0, 213.0, 39.0, 0.0], [38.0, 115.0, 169.0, 164.0, 35.0, 40.0, 78.0, 19.0, 127.0, 3.0, 162.0, 237.0, 76.0, 49.0, 168.0, 654.0, 95.0, 8.0, 113.0, 77.0, 7.0], [66.0, 9.0, 173.0, 373.0, 18.0, 50.0, 146.0, 33.0, 106.0, 14.0, 85.0, 465.0, 60.0, 38.0, 130.0, 740.0, 3.0, 63.0, 4.0, 7.0, 86.0], [152.0, 68.0, 148.0, 363.0, 36.0, 39.0, 93.0, 72.0, 81.0, 5.0, 267.0, 105.0, 153.0, 161.0, 620.0, 249.0, 93.0, 15.0, 221.0, 56.0, 50.0], [36.0, 73.0, 95.0, 114.0, 12.0, 24.0, 64.0, 15.0, 38.0, 2.0, 255.0, 83.0, 45.0, 64.0, 16556.0, 236.0, 33.0, 7.0, 117.0, 27.0, 32.0], [464.0, 260.0, 552.0, 1332.0, 47.0, 114.0, 313.0, 88.0, 287.0, 14.0, 6009.0, 554.0, 596.0, 206.0, 3681.0, 3287.0, 374.0, 144.0, 486.0, 228.0, 36.0], [2.0, 11.0, 8.0, 8.0, 0.0, 6.0, 5.0, 1.0, 4.0, 1.0, 10.0, 3.0, 2.0, 5.0, 115.0, 27.0, 60.0, 0.0, 24.0, 5.0, 0.0], [12.0, 7.0, 14.0, 38.0, 1.0, 4.0, 17.0, 4.0, 9.0, 1.0, 30.0, 17.0, 9.0, 7.0, 50.0, 144.0, 9.0, 0.0, 0.0, 4.0, 1.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 32.0, 64.0, 0.0, 0.0, 8.0, 0.0]]","[13, 15, 0, 1, 8, 2, 5, 6, 12, 17, 7, 9, 11, 16, 14, 3, 10, 4, 20, 19, 18]",60991.0,"{'problem_type': 'LOP', 'num_nodes': 21, 'nodes': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], 'pairs': [{'from_id': 0, 'to_id': 1, 'weight': 3464.0}, {'from_id': 0, 'to_id': 2, 'weight': 98.0}, {'from_id': 0, 'to_id': 3, 'weight': 16.0}, {'from_id': 0, 'to_id': 4, 'weight': 2.0}, {'from_id': 0, 'to_id': 5, 'weight': 22.0}, {'from_id': 0, 'to_id': 6, 'weight': 64.0}, {'from_id': 0, 'to_id': 7, 'weight': 15.0}, {'from_id': 0, 'to_id': 8, 'weight': 21.0}, {'from_id': 0, 'to_id': 9, 'weight': 1.0}, {'from_id': 0, 'to_id': 10, 'weight': 19.0}, {'from_id': 0, 'to_id': 11, 'weight': 7.0}, {'from_id': 0, 'to_id': 12, 'weight': 5.0}, {'from_id': 0, 'to_id': 13, 'weight': 3.0}, {'from_id': 0, 'to_id': 14, 'weight': 4.0}, {'from_id': 0, 'to_id': 15, 'weight': 29.0}, {'from_id': 0, 'to_id': 16, 'weight': 2.0}, {'from_id': 0, 'to_id': 17, 'weight': 1.0}, {'from_id': 0, 'to_id': 18, 'weight': 3.0}, {'from_id': 0, 'to_id': 19, 'weight': 2.0}, {'from_id': 0, 'to_id': 20, 'weight': 1.0}, {'from_id': 1, 'to_id': 0, 'weight': 369.0}, {'from_id': 1, 'to_id': 2, 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'weight': 38.0}, {'from_id': 14, 'to_id': 9, 'weight': 2.0}, {'from_id': 14, 'to_id': 10, 'weight': 255.0}, {'from_id': 14, 'to_id': 11, 'weight': 83.0}, {'from_id': 14, 'to_id': 12, 'weight': 45.0}, {'from_id': 14, 'to_id': 13, 'weight': 64.0}, {'from_id': 14, 'to_id': 15, 'weight': 236.0}, {'from_id': 14, 'to_id': 16, 'weight': 33.0}, {'from_id': 14, 'to_id': 17, 'weight': 7.0}, {'from_id': 14, 'to_id': 18, 'weight': 117.0}, {'from_id': 14, 'to_id': 19, 'weight': 27.0}, {'from_id': 14, 'to_id': 20, 'weight': 32.0}, {'from_id': 15, 'to_id': 0, 'weight': 464.0}, {'from_id': 15, 'to_id': 1, 'weight': 260.0}, {'from_id': 15, 'to_id': 2, 'weight': 552.0}, {'from_id': 15, 'to_id': 3, 'weight': 1332.0}, {'from_id': 15, 'to_id': 4, 'weight': 47.0}, {'from_id': 15, 'to_id': 5, 'weight': 114.0}, {'from_id': 15, 'to_id': 6, 'weight': 313.0}, {'from_id': 15, 'to_id': 7, 'weight': 88.0}, {'from_id': 15, 'to_id': 8, 'weight': 287.0}, {'from_id': 15, 'to_id': 9, 'weight': 14.0}, {'from_id': 15, 'to_id': 10, 'weight': 6009.0}, {'from_id': 15, 'to_id': 11, 'weight': 554.0}, {'from_id': 15, 'to_id': 12, 'weight': 596.0}, {'from_id': 15, 'to_id': 13, 'weight': 206.0}, {'from_id': 15, 'to_id': 14, 'weight': 3681.0}, {'from_id': 15, 'to_id': 16, 'weight': 374.0}, {'from_id': 15, 'to_id': 17, 'weight': 144.0}, {'from_id': 15, 'to_id': 18, 'weight': 486.0}, {'from_id': 15, 'to_id': 19, 'weight': 228.0}, {'from_id': 15, 'to_id': 20, 'weight': 36.0}, {'from_id': 16, 'to_id': 0, 'weight': 2.0}, {'from_id': 16, 'to_id': 1, 'weight': 11.0}, {'from_id': 16, 'to_id': 2, 'weight': 8.0}, {'from_id': 16, 'to_id': 3, 'weight': 8.0}, {'from_id': 16, 'to_id': 4, 'weight': 0.0}, {'from_id': 16, 'to_id': 5, 'weight': 6.0}, {'from_id': 16, 'to_id': 6, 'weight': 5.0}, {'from_id': 16, 'to_id': 7, 'weight': 1.0}, {'from_id': 16, 'to_id': 8, 'weight': 4.0}, {'from_id': 16, 'to_id': 9, 'weight': 1.0}, {'from_id': 16, 'to_id': 10, 'weight': 10.0}, {'from_id': 16, 'to_id': 11, 'weight': 3.0}, {'from_id': 16, 'to_id': 12, 'weight': 2.0}, {'from_id': 16, 'to_id': 13, 'weight': 5.0}, {'from_id': 16, 'to_id': 14, 'weight': 115.0}, {'from_id': 16, 'to_id': 15, 'weight': 27.0}, {'from_id': 16, 'to_id': 17, 'weight': 0.0}, {'from_id': 16, 'to_id': 18, 'weight': 24.0}, {'from_id': 16, 'to_id': 19, 'weight': 5.0}, {'from_id': 16, 'to_id': 20, 'weight': 0.0}, {'from_id': 17, 'to_id': 0, 'weight': 12.0}, {'from_id': 17, 'to_id': 1, 'weight': 7.0}, {'from_id': 17, 'to_id': 2, 'weight': 14.0}, {'from_id': 17, 'to_id': 3, 'weight': 38.0}, {'from_id': 17, 'to_id': 4, 'weight': 1.0}, {'from_id': 17, 'to_id': 5, 'weight': 4.0}, {'from_id': 17, 'to_id': 6, 'weight': 17.0}, {'from_id': 17, 'to_id': 7, 'weight': 4.0}, {'from_id': 17, 'to_id': 8, 'weight': 9.0}, {'from_id': 17, 'to_id': 9, 'weight': 1.0}, {'from_id': 17, 'to_id': 10, 'weight': 30.0}, {'from_id': 17, 'to_id': 11, 'weight': 17.0}, {'from_id': 17, 'to_id': 12, 'weight': 9.0}, {'from_id': 17, 'to_id': 13, 'weight': 7.0}, {'from_id': 17, 'to_id': 14, 'weight': 50.0}, {'from_id': 17, 'to_id': 15, 'weight': 144.0}, {'from_id': 17, 'to_id': 16, 'weight': 9.0}, {'from_id': 17, 'to_id': 18, 'weight': 0.0}, {'from_id': 17, 'to_id': 19, 'weight': 4.0}, {'from_id': 17, 'to_id': 20, 'weight': 1.0}, {'from_id': 18, 'to_id': 0, 'weight': 0.0}, {'from_id': 18, 'to_id': 1, 'weight': 0.0}, {'from_id': 18, 'to_id': 2, 'weight': 0.0}, {'from_id': 18, 'to_id': 3, 'weight': 0.0}, {'from_id': 18, 'to_id': 4, 'weight': 0.0}, {'from_id': 18, 'to_id': 5, 'weight': 0.0}, {'from_id': 18, 'to_id': 6, 'weight': 0.0}, {'from_id': 18, 'to_id': 7, 'weight': 0.0}, {'from_id': 18, 'to_id': 8, 'weight': 0.0}, {'from_id': 18, 'to_id': 9, 'weight': 0.0}, {'from_id': 18, 'to_id': 10, 'weight': 0.0}, {'from_id': 18, 'to_id': 11, 'weight': 0.0}, {'from_id': 18, 'to_id': 12, 'weight': 0.0}, {'from_id': 18, 'to_id': 13, 'weight': 0.0}, {'from_id': 18, 'to_id': 14, 'weight': 0.0}, {'from_id': 18, 'to_id': 15, 'weight': 0.0}, {'from_id': 18, 'to_id': 16, 'weight': 0.0}, {'from_id': 18, 'to_id': 17, 'weight': 0.0}, {'from_id': 18, 'to_id': 19, 'weight': 0.0}, {'from_id': 18, 'to_id': 20, 'weight': 0.0}, {'from_id': 19, 'to_id': 0, 'weight': 0.0}, {'from_id': 19, 'to_id': 1, 'weight': 0.0}, {'from_id': 19, 'to_id': 2, 'weight': 0.0}, {'from_id': 19, 'to_id': 3, 'weight': 0.0}, {'from_id': 19, 'to_id': 4, 'weight': 0.0}, {'from_id': 19, 'to_id': 5, 'weight': 0.0}, {'from_id': 19, 'to_id': 6, 'weight': 0.0}, {'from_id': 19, 'to_id': 7, 'weight': 0.0}, {'from_id': 19, 'to_id': 8, 'weight': 0.0}, {'from_id': 19, 'to_id': 9, 'weight': 0.0}, {'from_id': 19, 'to_id': 10, 'weight': 0.0}, {'from_id': 19, 'to_id': 11, 'weight': 0.0}, {'from_id': 19, 'to_id': 12, 'weight': 0.0}, {'from_id': 19, 'to_id': 13, 'weight': 0.0}, {'from_id': 19, 'to_id': 14, 'weight': 0.0}, {'from_id': 19, 'to_id': 15, 'weight': 0.0}, {'from_id': 19, 'to_id': 16, 'weight': 0.0}, {'from_id': 19, 'to_id': 17, 'weight': 0.0}, {'from_id': 19, 'to_id': 18, 'weight': 0.0}, {'from_id': 19, 'to_id': 20, 'weight': 0.0}, {'from_id': 20, 'to_id': 0, 'weight': 0.0}, {'from_id': 20, 'to_id': 1, 'weight': 0.0}, {'from_id': 20, 'to_id': 2, 'weight': 0.0}, {'from_id': 20, 'to_id': 3, 'weight': 0.0}, {'from_id': 20, 'to_id': 4, 'weight': 0.0}, {'from_id': 20, 'to_id': 5, 'weight': 0.0}, {'from_id': 20, 'to_id': 6, 'weight': 0.0}, {'from_id': 20, 'to_id': 7, 'weight': 0.0}, {'from_id': 20, 'to_id': 8, 'weight': 0.0}, {'from_id': 20, 'to_id': 9, 'weight': 0.0}, {'from_id': 20, 'to_id': 10, 'weight': 0.0}, {'from_id': 20, 'to_id': 11, 'weight': 0.0}, {'from_id': 20, 'to_id': 12, 'weight': 0.0}, {'from_id': 20, 'to_id': 13, 'weight': 0.0}, {'from_id': 20, 'to_id': 14, 'weight': 0.0}, {'from_id': 20, 'to_id': 15, 'weight': 32.0}, {'from_id': 20, 'to_id': 16, 'weight': 64.0}, {'from_id': 20, 'to_id': 17, 'weight': 0.0}, {'from_id': 20, 'to_id': 18, 'weight': 0.0}, {'from_id': 20, 'to_id': 19, 'weight': 8.0}]}","[13, 15, 0, 1, 8, 2, 5, 6, 12, 17, 7, 9, 11, 16, 14, 3, 10, 4, 20, 19, 18]",4,markdown_table,0
LOP,LOP,"Imagine arranging a one-shot tour that starts and ends smoothly: every stop goes into the loop once and only once, and the order matters because each time one stop comes before another it adds a specific bit of extra interest. Score any proposed order by summing all those one-before-the-other interest amounts for every pair, and the order with the highest total is the one to use. The concrete pairwise interest details are shown below.

# num_stops=21
# stop_ids=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21

# Meaning: this is a directed interest_gain matrix over items in stop_ids (21 total). Entry at row from_stop=i and column to_stop=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# interest_gain
| from_stop\to_stop | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 20300.0 | 73000.0 | 47500.0 | 45900.0 | 1800.0 | 25900.0 | 3800.0 | 39000.0 | 18600.0 | 17400.0 | 32000.0 | 40100.0 | 5500.0 | 32000.0 | 1700.0 | 13300.0 | 238100.0 | 46000.0 | 0.0 | 0.0 | 900.0 |
| 2 | 1200.0 | 535600.0 | 103500.0 | 30100.0 | 2000.0 | 27600.0 | 8400.0 | 16900.0 | 3400.0 | 75200.0 | 2900.0 | 6900.0 | 5500.0 | 1300.0 | 600.0 | 300.0 | 0.0 | 300.0 | 0.0 | 0.0 | 0.0 |
| 3 | 8200.0 | 63600.0 | 886900.0 | 107100.0 | 17500.0 | 118000.0 | 10600.0 | 13300.0 | 46400.0 | 16100.0 | 113300.0 | 117400.0 | 59300.0 | 1700.0 | 10300.0 | 5900.0 | 1500.0 | 88000.0 | 0.0 | 0.0 | 900.0 |
| 4 | 13600.0 | 68500.0 | 93000.0 | 562700.0 | 33900.0 | 260100.0 | 25200.0 | 207900.0 | 15500.0 | 98100.0 | 35600.0 | 55500.0 | 11200.0 | 1600.0 | 8000.0 | 5700.0 | 800.0 | 8300.0 | 0.0 | 0.0 | 100.0 |
| 5 | 2000.0 | 2300.0 | 2100.0 | 10400.0 | 24900.0 | 42000.0 | 5200.0 | 1500.0 | 600.0 | 700.0 | 1300.0 | 22600.0 | 300.0 | 400.0 | 24600.0 | 2000.0 | 600.0 | 41400.0 | 0.0 | 0.0 | 0.0 |
| 6 | 24400.0 | 33200.0 | 16700.0 | 35300.0 | 26600.0 | 540400.0 | 6700.0 | 16600.0 | 4100.0 | 4700.0 | 9800.0 | 17500.0 | 2100.0 | 2500.0 | 25800.0 | 8300.0 | 800.0 | 90900.0 | 0.0 | 0.0 | 200.0 |
| 7 | 2500.0 | 3100.0 | 1500.0 | 1500.0 | 200.0 | 800.0 | 204800.0 | 1700.0 | 400.0 | 400.0 | 900.0 | 1500.0 | 200.0 | 200.0 | 1200.0 | 700.0 | 0.0 | 700.0 | 0.0 | 0.0 | 0.0 |
| 8 | 0.0 | 0.0 | 31600.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2177400.0 | 0.0 | 100.0 | 32300.0 | 0.0 | 900.0 | 316300.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 9 | 0.0 | 0.0 | 8600.0 | 0.0 | 0.0 | 0.0 | 0.0 | 5200.0 | 280700.0 | 0.0 | 0.0 | 0.0 | 0.0 | 114300.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 10 | 1300.0 | 1100.0 | 37100.0 | 1900.0 | 400.0 | 2400.0 | 200.0 | 1800.0 | 1400.0 | 32600.0 | 2500.0 | 6000.0 | 500.0 | 673200.0 | 3900.0 | 5800.0 | 2200.0 | 2400.0 | 0.0 | 0.0 | 100.0 |
| 11 | 1100.0 | 7700.0 | 5000.0 | 1800.0 | 200.0 | 13500.0 | 600.0 | 1100.0 | 700.0 | 800.0 | 1983000.0 | 25400.0 | 13500.0 | 8100.0 | 2000.0 | 2800.0 | 1000.0 | 6700.0 | 0.0 | 0.0 | 0.0 |
| 12 | 18700.0 | 78100.0 | 96100.0 | 54900.0 | 8400.0 | 71400.0 | 3700.0 | 65100.0 | 99500.0 | 42800.0 | 67500.0 | 2111600.0 | 33600.0 | 26000.0 | 49800.0 | 106400.0 | 30600.0 | 43900.0 | 0.0 | 0.0 | 1200.0 |
| 13 | 1600.0 | 1800.0 | 2200.0 | 1700.0 | 1200.0 | 1700.0 | 200.0 | 1400.0 | 700.0 | 800.0 | 28400.0 | 3000.0 | 16700.0 | 800.0 | 2000.0 | 2500.0 | 800.0 | 6900.0 | 0.0 | 0.0 | 0.0 |
| 14 | 11700.0 | 10800.0 | 27600.0 | 13300.0 | 2500.0 | 17400.0 | 1600.0 | 14200.0 | 10100.0 | 12800.0 | 18200.0 | 36000.0 | 3200.0 | 23400.0 | 29000.0 | 99800.0 | 19400.0 | 21800.0 | 0.0 | 0.0 | 800.0 |
| 15 | 1500.0 | 11900.0 | 19700.0 | 8600.0 | 700.0 | 4700.0 | 500.0 | 25200.0 | 11000.0 | 24900.0 | 4400.0 | 13000.0 | 1000.0 | 2800.0 | 315700.0 | 11900.0 | 2600.0 | 2600.0 | 0.0 | 0.0 | 100.0 |
| 16 | 5900.0 | 6000.0 | 12200.0 | 7100.0 | 1200.0 | 8500.0 | 900.0 | 8600.0 | 5800.0 | 6000.0 | 9300.0 | 17200.0 | 1700.0 | 11100.0 | 56300.0 | 3301500.0 | 114700.0 | 12000.0 | 0.0 | 0.0 | 300.0 |
| 17 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 18 | 5600.0 | 4900.0 | 12300.0 | 5900.0 | 1100.0 | 7700.0 | 700.0 | 6200.0 | 4500.0 | 5700.0 | 8100.0 | 49600.0 | 1400.0 | 57100.0 | 13000.0 | 44600.0 | 332900.0 | 140000.0 | 0.0 | 0.0 | 1700.0 |
| 19 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 20 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 21 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Also, when you send your final ordering back, please use this little JSON layout so everything's neat and easy to read:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of ""solution"" as the ordered list of stops for the tour. The array is read left-to-right: the first entry is where you start, and the last entry is where you finish. The angle-bracket placeholders are just showing the shape — replace them with the actual identifiers from the instance when you submit the real order.

This JSON block is only a sketch of the expected shape, not the actual answer — don't paste it verbatim as your solution.

Please make sure you use the exact identifiers from the instance input, with no renaming and no extra 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”.","[[20300.0, 73000.0, 47500.0, 45900.0, 1800.0, 25900.0, 3800.0, 39000.0, 18600.0, 17400.0, 32000.0, 40100.0, 5500.0, 32000.0, 1700.0, 13300.0, 238100.0, 46000.0, 0.0, 0.0, 900.0], [1200.0, 535600.0, 103500.0, 30100.0, 2000.0, 27600.0, 8400.0, 16900.0, 3400.0, 75200.0, 2900.0, 6900.0, 5500.0, 1300.0, 600.0, 300.0, 0.0, 300.0, 0.0, 0.0, 0.0], [8200.0, 63600.0, 886900.0, 107100.0, 17500.0, 118000.0, 10600.0, 13300.0, 46400.0, 16100.0, 113300.0, 117400.0, 59300.0, 1700.0, 10300.0, 5900.0, 1500.0, 88000.0, 0.0, 0.0, 900.0], [13600.0, 68500.0, 93000.0, 562700.0, 33900.0, 260100.0, 25200.0, 207900.0, 15500.0, 98100.0, 35600.0, 55500.0, 11200.0, 1600.0, 8000.0, 5700.0, 800.0, 8300.0, 0.0, 0.0, 100.0], [2000.0, 2300.0, 2100.0, 10400.0, 24900.0, 42000.0, 5200.0, 1500.0, 600.0, 700.0, 1300.0, 22600.0, 300.0, 400.0, 24600.0, 2000.0, 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'to_id': 17, 'weight': 0.0}, {'from_id': 21, 'to_id': 18, 'weight': 0.0}, {'from_id': 21, 'to_id': 19, 'weight': 0.0}, {'from_id': 21, 'to_id': 20, 'weight': 0.0}]}","[1, 3, 4, 5, 12, 6, 13, 18, 2, 7, 10, 9, 8, 14, 16, 15, 21, 20, 19, 11, 17]",5,csv,1
LOP,LOP,"I was asked to put together a lineup of candidate interviews for a hiring round — everyone has to go once and only once, no skipping or repeats. For every pair of people there’s a little piece of insight you get if one is interviewed before the other, and the plan is to order the whole list so that, when you look at each pair and take whichever “before” benefit applies, the total of those benefits is as big as it can be. The lineup itself is the decision: which candidate goes first, second, third, and so on. The actual candidate names and the pairwise insight numbers are shown below.

B) We have a stack of candidates to interview and the job is to make a single run-through where each person is seen exactly once — nobody appears twice and nobody is left out. Between any two candidates there’s a known advantage to interviewing one before the other, and the trick is to pick an order so that, if you add up those pairwise advantages according to who comes earlier in your list, the sum is as high as possible. In short, choose the interview order that gives the biggest total comparative insight. The specific list of candidates and their pairwise values are given below.

C) There’s a hiring day coming up and the goal is to arrange the interviews into one clean sequence with every candidate interviewed one time only. For each pair of candidates someone already estimated how much more you’d learn if candidate A goes before candidate B, so the way to judge any ordering is to go through every pair and add the appropriate value for whoever is earlier. A better order is simply one that leads to a larger total when you add up those pairwise gains. The concrete candidate names and those pairwise numbers are shown below.

D) Someone in HR handed over a pile of profiles and a small table saying, for each pair, how much more clarity you get if person X is seen before person Y. The job is to make a single schedule where each candidate shows up exactly once, and then score that schedule by summing the pairwise benefits for every pair according to who’s earlier. The aim is to pick the order that ends up with the highest total sum. The exact candidates and their pairwise insights are listed below.

E) Recently the hiring lead wanted a single interview run where everyone gets one turn — no repeats, no misses. They’d already rated every head-to-head ordering with a number that tells how much comparative learning you get if one candidate comes before another; to compare schedules you just add up the appropriate numbers for all pairs based on the order you chose. The better schedule is the one with the bigger total. The specific candidate list and pairwise numbers follow below.

F) Many teams face the same little puzzle: line up a set of applicants so each is interviewed exactly once, and use the known “if A is before B” benefits to judge the lineup. For every pair there’s a number saying how much more useful it is to see one before the other, and you score a full schedule by summing those numbers for all pairs according to their positions. The best lineup is simply the one that gives the largest total sum. The actual candidates and the pairwise benefit values are shown below.

G) Back at the office the hiring coordinator scribbled down a rule: every candidate gets one interview, listed in one order, and for every pair there’s a recorded benefit if one comes before the other. To pick the order you mentally go through each pair, pick the benefit that corresponds to whoever is earlier, and add all those numbers up — the order with the biggest total is the one to run with. The candidate names and the full table of pairwise numbers are provided below.

H) On a busy Monday the team needed a single-line interview schedule where nobody is repeated and nobody is left off the list. They also had a cheat-sheet that says, for each pair, how much comparative insight results from interviewing person A before person B; a schedule is evaluated by summing those pairwise values for all pairs according to who’s earlier, and the schedule that yields the highest total sum is the one they want. The detailed list of candidates and the pairwise insight values appears below.

{
  ""total_candidates"": 24,
  ""candidate_ids"": [
    0,
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14,
    15,
    16,
    17,
    18,
    19,
    20,
    21,
    22,
    23
  ]
}

# Meaning: this is a directed insight_gain matrix over items in candidate_ids (24 total). Entry at row candidate_before_id=i and column candidate_after_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# insight_gain 
| candidate_before_id\candidate_after_id | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 17715.0 | 0.0 | 0.0 | 8.0 | 575.0 | 4.0 | 2.0 | 10653.0 | 16545.0 | 1945.0 | 53.0 | 761.0 | 568.0 | 6.0 | 50.0 | 1784.0 | 8.0 | 3.0 | 696.0 | 830.0 | 0.0 | 779.0 | 77.0 | 71.0 |
| 1 | 2307.0 | 26960.0 | 5409.0 | 2128.0 | 5274.0 | 704.0 | 685.0 | 166.0 | 691.0 | 989.0 | 167.0 | 746.0 | 296.0 | 212.0 | 2909.0 | 514.0 | 2920.0 | 399.0 | 1045.0 | 23.0 | 33.0 | 2168.0 | 242.0 | 103.0 |
| 2 | 993.0 | 686.0 | 3308.0 | 1731.0 | 3442.0 | 1185.0 | 885.0 | 141.0 | 553.0 | 850.0 | 57.0 | 1059.0 | 528.0 | 230.0 | 1951.0 | 405.0 | 721.0 | 182.0 | 669.0 | 2092.0 | 43.0 | 1107.0 | 485.0 | 178.0 |
| 3 | 435.0 | 0.0 | 38.0 | 7571.0 | 1483.0 | 713.0 | 190.0 | 88.0 | 491.0 | 0.0 | 12.0 | 121.0 | 189.0 | 139.0 | 7.0 | 263.0 | 80.0 | 2.0 | 9.0 | 10.0 | 9.0 | 648.0 | 163.0 | 64.0 |
| 4 | 2076.0 | 419.0 | 280.0 | 921.0 | 33624.0 | 1451.0 | 823.0 | 90.0 | 413.0 | 7666.0 | 155.0 | 2581.0 | 8171.0 | 135.0 | 1467.0 | 237.0 | 400.0 | 91.0 | 2570.0 | 111.0 | 122.0 | 6717.0 | 867.0 | 63.0 |
| 5 | 905.0 | 298.0 | 1271.0 | 505.0 | 3136.0 | 8428.0 | 8403.0 | 279.0 | 793.0 | 450.0 | 276.0 | 753.0 | 533.0 | 677.0 | 394.0 | 170.0 | 1014.0 | 79.0 | 541.0 | 107.0 | 0.0 | 2717.0 | 170.0 | 27.0 |
| 6 | 720.0 | 201.0 | 223.0 | 220.0 | 383.0 | 1943.0 | 13320.0 | 14.0 | 81.0 | 287.0 | 33.0 | 280.0 | 325.0 | 102.0 | 309.0 | 32.0 | 158.0 | 14.0 | 119.0 | 21.0 | 0.0 | 995.0 | 28.0 | 1.0 |
| 7 | 765.0 | 0.0 | 0.0 | 0.0 | 73.0 | 0.0 | 0.0 | 1836.0 | 1838.0 | 0.0 | 0.0 | 8.0 | 0.0 | 0.0 | 0.0 | 781.0 | 0.0 | 0.0 | 0.0 | 0.0 | 157.0 | 156.0 | 25.0 | 163.0 |
| 8 | 6428.0 | 23.0 | 4.0 | 3.0 | 1503.0 | 14.0 | 19.0 | 45.0 | 16479.0 | 5.0 | 5.0 | 23.0 | 96.0 | 10.0 | 115.0 | 3600.0 | 0.0 | 0.0 | 669.0 | 3.0 | 61.0 | 1026.0 | 137.0 | 229.0 |
| 9 | 157.0 | 0.0 | 31.0 | 55.0 | 267.0 | 129.0 | 69.0 | 5.0 | 50.0 | 16080.0 | 315.0 | 390.0 | 1468.0 | 301.0 | 625.0 | 200.0 | 144.0 | 76.0 | 682.0 | 26.0 | 100.0 | 1759.0 | 384.0 | 143.0 |
| 10 | 73.0 | 0.0 | 0.0 | 7.0 | 15.0 | 76.0 | 51.0 | 0.0 | 2.0 | 256.0 | 973.0 | 21.0 | 11.0 | 107.0 | 23.0 | 0.0 | 38.0 | 66.0 | 43.0 | 0.0 | 0.0 | 620.0 | 42.0 | 7.0 |
| 11 | 293.0 | 46.0 | 224.0 | 1716.0 | 3143.0 | 931.0 | 910.0 | 414.0 | 1838.0 | 557.0 | 183.0 | 15440.0 | 463.0 | 239.0 | 5821.0 | 296.0 | 695.0 | 314.0 | 1614.0 | 97.0 | 42.0 | 3320.0 | 1071.0 | 139.0 |
| 12 | 311.0 | 48.0 | 88.0 | 398.0 | 2152.0 | 894.0 | 1822.0 | 168.0 | 653.0 | 371.0 | 612.0 | 324.0 | 2820.0 | 365.0 | 1237.0 | 63.0 | 594.0 | 15.0 | 262.0 | 8.0 | 4.0 | 995.0 | 603.0 | 18.0 |
| 13 | 154.0 | 9.0 | 131.0 | 226.0 | 198.0 | 906.0 | 152.0 | 17.0 | 100.0 | 55.0 | 20.0 | 400.0 | 151.0 | 102.0 | 1261.0 | 78.0 | 1000.0 | 57.0 | 1267.0 | 137.0 | 10.0 | 454.0 | 176.0 | 8.0 |
| 14 | 2938.0 | 231.0 | 912.0 | 1390.0 | 3144.0 | 2887.0 | 1982.0 | 877.0 | 2438.0 | 1593.0 | 250.0 | 1701.0 | 1008.0 | 656.0 | 3496.0 | 2465.0 | 813.0 | 75.0 | 1846.0 | 208.0 | 81.0 | 3300.0 | 513.0 | 238.0 |
| 15 | 82.0 | 49.0 | 13.0 | 277.0 | 114.0 | 414.0 | 505.0 | 14.0 | 233.0 | 141.0 | 38.0 | 859.0 | 158.0 | 173.0 | 4291.0 | 80.0 | 776.0 | 47.0 | 1548.0 | 78.0 | 84.0 | 1309.0 | 6.0 | 117.0 |
| 16 | 1064.0 | 185.0 | 356.0 | 1543.0 | 1883.0 | 1379.0 | 839.0 | 307.0 | 1169.0 | 586.0 | 52.0 | 1770.0 | 522.0 | 125.0 | 763.0 | 443.0 | 257.0 | 83.0 | 391.0 | 64.0 | 15.0 | 895.0 | 211.0 | 75.0 |
| 17 | 239.0 | 10.0 | 85.0 | 246.0 | 420.0 | 499.0 | 634.0 | 47.0 | 149.0 | 301.0 | 93.0 | 389.0 | 126.0 | 88.0 | 3016.0 | 222.0 | 859.0 | 175.0 | 452.0 | 11.0 | 21.0 | 2100.0 | 507.0 | 73.0 |
| 18 | 1132.0 | 115.0 | 383.0 | 1069.0 | 2223.0 | 1897.0 | 2569.0 | 213.0 | 611.0 | 656.0 | 124.0 | 3149.0 | 617.0 | 824.0 | 14492.0 | 1267.0 | 1098.0 | 411.0 | 8365.0 | 6890.0 | 231.0 | 1868.0 | 1262.0 | 58.0 |
| 19 | 16.0 | 3.0 | 20.0 | 47.0 | 45.0 | 86.0 | 92.0 | 2.0 | 74.0 | 33.0 | 12.0 | 57.0 | 27.0 | 49.0 | 4235.0 | 33.0 | 103.0 | 31.0 | 447.0 | 6.0 | 0.0 | 1425.0 | 100.0 | 27.0 |
| 20 | 21.0 | 1.0 | 9.0 | 18.0 | 99.0 | 47.0 | 56.0 | 2.0 | 21.0 | 13.0 | 4.0 | 88.0 | 14.0 | 22.0 | 300.0 | 12.0 | 37.0 | 9.0 | 302.0 | 179.0 | 0.0 | 0.0 | 0.0 | 1.0 |
| 21 | 128.0 | 14.0 | 97.0 | 98.0 | 300.0 | 226.0 | 247.0 | 56.0 | 180.0 | 111.0 | 23.0 | 150.0 | 60.0 | 30.0 | 536.0 | 71.0 | 164.0 | 63.0 | 342.0 | 1469.0 | 5.0 | 865.0 | 201.0 | 21.0 |
| 22 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 135.0 | 0.0 | 0.0 |
| 23 | 27.0 | 0.0 | 0.0 | 0.0 | 13.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 68.0 | 0.0 | 0.0 | 3521.0 | 0.0 | 0.0 |

If you want the result in a simple, machine-friendly form I can return it like this:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

The ""solution"" entry is just the interview lineup from first to last — put the candidate identifiers in that array in the order you want them to be interviewed. This JSON is only a sketch of the shape I'll use, not the actual answer itself.

Please make sure any identifiers are used exactly as they appear in the instance input — no renaming and no 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”.""","[[17715.0, 0.0, 0.0, 8.0, 575.0, 4.0, 2.0, 10653.0, 16545.0, 1945.0, 53.0, 761.0, 568.0, 6.0, 50.0, 1784.0, 8.0, 3.0, 696.0, 830.0, 0.0, 779.0, 77.0, 71.0], [2307.0, 26960.0, 5409.0, 2128.0, 5274.0, 704.0, 685.0, 166.0, 691.0, 989.0, 167.0, 746.0, 296.0, 212.0, 2909.0, 514.0, 2920.0, 399.0, 1045.0, 23.0, 33.0, 2168.0, 242.0, 103.0], [993.0, 686.0, 3308.0, 1731.0, 3442.0, 1185.0, 885.0, 141.0, 553.0, 850.0, 57.0, 1059.0, 528.0, 230.0, 1951.0, 405.0, 721.0, 182.0, 669.0, 2092.0, 43.0, 1107.0, 485.0, 178.0], [435.0, 0.0, 38.0, 7571.0, 1483.0, 713.0, 190.0, 88.0, 491.0, 0.0, 12.0, 121.0, 189.0, 139.0, 7.0, 263.0, 80.0, 2.0, 9.0, 10.0, 9.0, 648.0, 163.0, 64.0], [2076.0, 419.0, 280.0, 921.0, 33624.0, 1451.0, 823.0, 90.0, 413.0, 7666.0, 155.0, 2581.0, 8171.0, 135.0, 1467.0, 237.0, 400.0, 91.0, 2570.0, 111.0, 122.0, 6717.0, 867.0, 63.0], [905.0, 298.0, 1271.0, 505.0, 3136.0, 8428.0, 8403.0, 279.0, 793.0, 450.0, 276.0, 753.0, 533.0, 677.0, 394.0, 170.0, 1014.0, 79.0, 541.0, 107.0, 0.0, 2717.0, 170.0, 27.0], [720.0, 201.0, 223.0, 220.0, 383.0, 1943.0, 13320.0, 14.0, 81.0, 287.0, 33.0, 280.0, 325.0, 102.0, 309.0, 32.0, 158.0, 14.0, 119.0, 21.0, 0.0, 995.0, 28.0, 1.0], [765.0, 0.0, 0.0, 0.0, 73.0, 0.0, 0.0, 1836.0, 1838.0, 0.0, 0.0, 8.0, 0.0, 0.0, 0.0, 781.0, 0.0, 0.0, 0.0, 0.0, 157.0, 156.0, 25.0, 163.0], [6428.0, 23.0, 4.0, 3.0, 1503.0, 14.0, 19.0, 45.0, 16479.0, 5.0, 5.0, 23.0, 96.0, 10.0, 115.0, 3600.0, 0.0, 0.0, 669.0, 3.0, 61.0, 1026.0, 137.0, 229.0], [157.0, 0.0, 31.0, 55.0, 267.0, 129.0, 69.0, 5.0, 50.0, 16080.0, 315.0, 390.0, 1468.0, 301.0, 625.0, 200.0, 144.0, 76.0, 682.0, 26.0, 100.0, 1759.0, 384.0, 143.0], [73.0, 0.0, 0.0, 7.0, 15.0, 76.0, 51.0, 0.0, 2.0, 256.0, 973.0, 21.0, 11.0, 107.0, 23.0, 0.0, 38.0, 66.0, 43.0, 0.0, 0.0, 620.0, 42.0, 7.0], [293.0, 46.0, 224.0, 1716.0, 3143.0, 931.0, 910.0, 414.0, 1838.0, 557.0, 183.0, 15440.0, 463.0, 239.0, 5821.0, 296.0, 695.0, 314.0, 1614.0, 97.0, 42.0, 3320.0, 1071.0, 139.0], [311.0, 48.0, 88.0, 398.0, 2152.0, 894.0, 1822.0, 168.0, 653.0, 371.0, 612.0, 324.0, 2820.0, 365.0, 1237.0, 63.0, 594.0, 15.0, 262.0, 8.0, 4.0, 995.0, 603.0, 18.0], [154.0, 9.0, 131.0, 226.0, 198.0, 906.0, 152.0, 17.0, 100.0, 55.0, 20.0, 400.0, 151.0, 102.0, 1261.0, 78.0, 1000.0, 57.0, 1267.0, 137.0, 10.0, 454.0, 176.0, 8.0], [2938.0, 231.0, 912.0, 1390.0, 3144.0, 2887.0, 1982.0, 877.0, 2438.0, 1593.0, 250.0, 1701.0, 1008.0, 656.0, 3496.0, 2465.0, 813.0, 75.0, 1846.0, 208.0, 81.0, 3300.0, 513.0, 238.0], [82.0, 49.0, 13.0, 277.0, 114.0, 414.0, 505.0, 14.0, 233.0, 141.0, 38.0, 859.0, 158.0, 173.0, 4291.0, 80.0, 776.0, 47.0, 1548.0, 78.0, 84.0, 1309.0, 6.0, 117.0], [1064.0, 185.0, 356.0, 1543.0, 1883.0, 1379.0, 839.0, 307.0, 1169.0, 586.0, 52.0, 1770.0, 522.0, 125.0, 763.0, 443.0, 257.0, 83.0, 391.0, 64.0, 15.0, 895.0, 211.0, 75.0], [239.0, 10.0, 85.0, 246.0, 420.0, 499.0, 634.0, 47.0, 149.0, 301.0, 93.0, 389.0, 126.0, 88.0, 3016.0, 222.0, 859.0, 175.0, 452.0, 11.0, 21.0, 2100.0, 507.0, 73.0], [1132.0, 115.0, 383.0, 1069.0, 2223.0, 1897.0, 2569.0, 213.0, 611.0, 656.0, 124.0, 3149.0, 617.0, 824.0, 14492.0, 1267.0, 1098.0, 411.0, 8365.0, 6890.0, 231.0, 1868.0, 1262.0, 58.0], [16.0, 3.0, 20.0, 47.0, 45.0, 86.0, 92.0, 2.0, 74.0, 33.0, 12.0, 57.0, 27.0, 49.0, 4235.0, 33.0, 103.0, 31.0, 447.0, 6.0, 0.0, 1425.0, 100.0, 27.0], [21.0, 1.0, 9.0, 18.0, 99.0, 47.0, 56.0, 2.0, 21.0, 13.0, 4.0, 88.0, 14.0, 22.0, 300.0, 12.0, 37.0, 9.0, 302.0, 179.0, 0.0, 0.0, 0.0, 1.0], [128.0, 14.0, 97.0, 98.0, 300.0, 226.0, 247.0, 56.0, 180.0, 111.0, 23.0, 150.0, 60.0, 30.0, 536.0, 71.0, 164.0, 63.0, 342.0, 1469.0, 5.0, 865.0, 201.0, 21.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 135.0, 0.0, 0.0], [27.0, 0.0, 0.0, 0.0, 13.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 68.0, 0.0, 0.0, 3521.0, 0.0, 0.0]]","[1, 2, 17, 20, 13, 18, 16, 11, 19, 14, 3, 5, 4, 0, 9, 12, 7, 8, 15, 10, 6, 23, 21, 22]",282434.0,"{'problem_type': 'LOP', 'num_nodes': 24, 'nodes': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], 'pairs': [{'from_id': 0, 'to_id': 1, 'weight': 0.0}, {'from_id': 0, 'to_id': 2, 'weight': 0.0}, {'from_id': 0, 'to_id': 3, 'weight': 8.0}, {'from_id': 0, 'to_id': 4, 'weight': 575.0}, {'from_id': 0, 'to_id': 5, 'weight': 4.0}, {'from_id': 0, 'to_id': 6, 'weight': 2.0}, {'from_id': 0, 'to_id': 7, 'weight': 10653.0}, {'from_id': 0, 'to_id': 8, 'weight': 16545.0}, {'from_id': 0, 'to_id': 9, 'weight': 1945.0}, {'from_id': 0, 'to_id': 10, 'weight': 53.0}, {'from_id': 0, 'to_id': 11, 'weight': 761.0}, {'from_id': 0, 'to_id': 12, 'weight': 568.0}, {'from_id': 0, 'to_id': 13, 'weight': 6.0}, {'from_id': 0, 'to_id': 14, 'weight': 50.0}, {'from_id': 0, 'to_id': 15, 'weight': 1784.0}, {'from_id': 0, 'to_id': 16, 'weight': 8.0}, 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LOP,LOP,"Picture a small shop with a single long shelf that must show every product in a line, each item occupying one spot and only one. Between every two different products there’s a known cross-sell value for putting one in front of the other, and the overall score for any lineup is just the sum of those values for all pairs in that order. The point is to pick the sequence that gives the highest total score from those pair values. Nothing can be left off the shelf or shown twice. The full item list and the pairwise scores are shown below.

{
  ""total_products"": 22,
  ""product_ids"": [
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14,
    15,
    16,
    17,
    18,
    19,
    20,
    21,
    22
  ]
}

# Meaning: this is a directed cross_sell_value matrix over items in product_ids (22 total). Entry at row leading_product_id=i and column following_product_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# cross_sell_value 
| leading_product_id\following_product_id | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 47845.0 | 520.0 | 1167.0 | 817.0 | 2778.0 | 2.0 | 0.0 | 0.0 | 0.0 | 494532.0 | 274152.0 | 25522.0 | 18418.0 | 0.0 | 128.0 | 42.0 | 0.0 | 0.0 | 8352.0 | 8917.0 | 0.0 | 0.0 |
| 2 | 435.0 | 4.0 | 0.0 | 833.0 | 12080.0 | 211.0 | 171.0 | 12.0 | 36.0 | 32.0 | 10.0 | 106.0 | 138.0 | 406.0 | 0.0 | 313.0 | 0.0 | 0.0 | 839.0 | 3787.0 | 0.0 | 0.0 |
| 3 | 29292.0 | 88.0 | 302775.0 | 7816.0 | 29404.0 | 3505.0 | 6856.0 | 302.0 | 8386.0 | 886.0 | 1695.0 | 498.0 | 8452.0 | 35394.0 | 4901.0 | 1506.0 | 0.0 | 0.0 | 53193.0 | 32434.0 | 0.0 | 0.0 |
| 4 | 4605.0 | 412.0 | 1982.0 | 285765.0 | 8932.0 | 181560.0 | 99824.0 | 21374.0 | 121111.0 | 143.0 | 833.0 | 0.0 | 1354.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 3935.0 | 0.0 | 0.0 |
| 5 | 5511.0 | 0.0 | 2521.0 | 3167.0 | 70629.0 | 6677.0 | 2763.0 | 3099.0 | 5795.0 | 0.0 | 450.0 | 0.0 | 83.0 | 207.0 | 568.0 | 0.0 | 4000.0 | 0.0 | 5101.0 | 4435.0 | 0.0 | 0.0 |
| 6 | 5547.0 | 40.0 | 1919.0 | 134.0 | 2702.0 | 30403.0 | 85675.0 | 1336.0 | 21837.0 | 834.0 | 50.0 | 0.0 | 2220.0 | 4622.0 | 208.0 | 236.0 | 0.0 | 0.0 | 571.0 | 18681.0 | 0.0 | 0.0 |
| 7 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2350.0 | 6727.0 | 283.0 | 1235.0 | 0.0 | 0.0 | 0.0 | 0.0 | 682.0 | 19.0 | 0.0 | 0.0 | 0.0 | 24.0 | 1436.0 | 0.0 | 0.0 |
| 8 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 879.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 806.0 | 100.0 | 0.0 | 0.0 | 0.0 | 0.0 | 5836.0 | 0.0 | 0.0 |
| 9 | 634.0 | 0.0 | 0.0 | 0.0 | 0.0 | 436.0 | 31069.0 | 373.0 | 13408.0 | 4.0 | 12.0 | 0.0 | 0.0 | 1901.0 | 303.0 | 96.0 | 0.0 | 0.0 | 1411.0 | 14623.0 | 0.0 | 0.0 |
| 10 | 0.0 | 0.0 | 5374.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 29261.0 | 999.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 11 | 4646.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 28.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 12 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 8657.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 13 | 2168.0 | 0.0 | 1734.0 | 2643.0 | 15609.0 | 3032.0 | 8451.0 | 1402.0 | 10404.0 | 1199.0 | 3104.0 | 4084.0 | 125267.0 | 2389.0 | 2770.0 | 17374.0 | 1023.0 | 0.0 | 48784.0 | 36675.0 | 0.0 | 0.0 |
| 14 | 1013.0 | 19.0 | 1084.0 | 2391.0 | 2373.0 | 2373.0 | 1826.0 | 165.0 | 1878.0 | 478.0 | 633.0 | 52.0 | 1831.0 | 0.0 | 56.0 | 25.0 | 34.0 | 0.0 | 1526.0 | 5252.0 | 0.0 | 0.0 |
| 15 | 13.0 | 14.0 | 30.0 | 1384.0 | 771.0 | 1168.0 | 0.0 | 0.0 | 0.0 | 121.0 | 0.0 | 0.0 | 427.0 | 68126.0 | 0.0 | 228.0 | 0.0 | 0.0 | 1729.0 | 1907.0 | 0.0 | 0.0 |
| 16 | 61242.0 | 1903.0 | 16017.0 | 23157.0 | 18712.0 | 9862.0 | 13114.0 | 2236.0 | 9777.0 | 4896.0 | 2546.0 | 1455.0 | 7887.0 | 15354.0 | 1198.0 | 0.0 | 5649.0 | 0.0 | 10217.0 | 31273.0 | 0.0 | 0.0 |
| 17 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 18 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 19 | 16973.0 | 99.0 | 3624.0 | 5142.0 | 4537.0 | 6018.0 | 8140.0 | 566.0 | 6467.0 | 730.0 | 1458.0 | 2836.0 | 5860.0 | 3753.0 | 1791.0 | 11924.0 | 39403.0 | 0.0 | 1292.0 | 70861.0 | 0.0 | 0.0 |
| 20 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 21 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 22 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Also, when you send back the final lineup, just tuck it into this little JSON sketch so it's easy to read and parse:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Here ""solution"" should be the ordered list of every product from left to right on the shelf — first entry is the leftmost product, last entry is the rightmost. Think of it like filling out a simple form: one list, all items once, in the order you want them displayed. This is just the shape I expect, not the actual answer.

Please be sure to use the exact identifiers given in the instance input — don't 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”.""","[[47845.0, 520.0, 1167.0, 817.0, 2778.0, 2.0, 0.0, 0.0, 0.0, 494532.0, 274152.0, 25522.0, 18418.0, 0.0, 128.0, 42.0, 0.0, 0.0, 8352.0, 8917.0, 0.0, 0.0], [435.0, 4.0, 0.0, 833.0, 12080.0, 211.0, 171.0, 12.0, 36.0, 32.0, 10.0, 106.0, 138.0, 406.0, 0.0, 313.0, 0.0, 0.0, 839.0, 3787.0, 0.0, 0.0], [29292.0, 88.0, 302775.0, 7816.0, 29404.0, 3505.0, 6856.0, 302.0, 8386.0, 886.0, 1695.0, 498.0, 8452.0, 35394.0, 4901.0, 1506.0, 0.0, 0.0, 53193.0, 32434.0, 0.0, 0.0], [4605.0, 412.0, 1982.0, 285765.0, 8932.0, 181560.0, 99824.0, 21374.0, 121111.0, 143.0, 833.0, 0.0, 1354.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 3935.0, 0.0, 0.0], [5511.0, 0.0, 2521.0, 3167.0, 70629.0, 6677.0, 2763.0, 3099.0, 5795.0, 0.0, 450.0, 0.0, 83.0, 207.0, 568.0, 0.0, 4000.0, 0.0, 5101.0, 4435.0, 0.0, 0.0], [5547.0, 40.0, 1919.0, 134.0, 2702.0, 30403.0, 85675.0, 1336.0, 21837.0, 834.0, 50.0, 0.0, 2220.0, 4622.0, 208.0, 236.0, 0.0, 0.0, 571.0, 18681.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 2350.0, 6727.0, 283.0, 1235.0, 0.0, 0.0, 0.0, 0.0, 682.0, 19.0, 0.0, 0.0, 0.0, 24.0, 1436.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 879.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 806.0, 100.0, 0.0, 0.0, 0.0, 0.0, 5836.0, 0.0, 0.0], [634.0, 0.0, 0.0, 0.0, 0.0, 436.0, 31069.0, 373.0, 13408.0, 4.0, 12.0, 0.0, 0.0, 1901.0, 303.0, 96.0, 0.0, 0.0, 1411.0, 14623.0, 0.0, 0.0], [0.0, 0.0, 5374.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 29261.0, 999.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [4646.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 28.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 8657.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [2168.0, 0.0, 1734.0, 2643.0, 15609.0, 3032.0, 8451.0, 1402.0, 10404.0, 1199.0, 3104.0, 4084.0, 125267.0, 2389.0, 2770.0, 17374.0, 1023.0, 0.0, 48784.0, 36675.0, 0.0, 0.0], [1013.0, 19.0, 1084.0, 2391.0, 2373.0, 2373.0, 1826.0, 165.0, 1878.0, 478.0, 633.0, 52.0, 1831.0, 0.0, 56.0, 25.0, 34.0, 0.0, 1526.0, 5252.0, 0.0, 0.0], [13.0, 14.0, 30.0, 1384.0, 771.0, 1168.0, 0.0, 0.0, 0.0, 121.0, 0.0, 0.0, 427.0, 68126.0, 0.0, 228.0, 0.0, 0.0, 1729.0, 1907.0, 0.0, 0.0], [61242.0, 1903.0, 16017.0, 23157.0, 18712.0, 9862.0, 13114.0, 2236.0, 9777.0, 4896.0, 2546.0, 1455.0, 7887.0, 15354.0, 1198.0, 0.0, 5649.0, 0.0, 10217.0, 31273.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [16973.0, 99.0, 3624.0, 5142.0, 4537.0, 6018.0, 8140.0, 566.0, 6467.0, 730.0, 1458.0, 2836.0, 5860.0, 3753.0, 1791.0, 11924.0, 39403.0, 0.0, 1292.0, 70861.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]]","[21, 20, 12, 15, 2, 1, 18, 14, 13, 3, 4, 5, 8, 7, 6, 0, 19, 17, 16, 11, 9, 10]",2353689.0,"{'problem_type': 'LOP', 'num_nodes': 22, 'nodes': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], 'pairs': [{'from_id': 1, 'to_id': 2, 'weight': 520.0}, {'from_id': 1, 'to_id': 3, 'weight': 1167.0}, {'from_id': 1, 'to_id': 4, 'weight': 817.0}, {'from_id': 1, 'to_id': 5, 'weight': 2778.0}, {'from_id': 1, 'to_id': 6, 'weight': 2.0}, {'from_id': 1, 'to_id': 7, 'weight': 0.0}, {'from_id': 1, 'to_id': 8, 'weight': 0.0}, {'from_id': 1, 'to_id': 9, 'weight': 0.0}, {'from_id': 1, 'to_id': 10, 'weight': 494532.0}, {'from_id': 1, 'to_id': 11, 'weight': 274152.0}, {'from_id': 1, 'to_id': 12, 'weight': 25522.0}, {'from_id': 1, 'to_id': 13, 'weight': 18418.0}, {'from_id': 1, 'to_id': 14, 'weight': 0.0}, {'from_id': 1, 'to_id': 15, 'weight': 128.0}, {'from_id': 1, 'to_id': 16, 'weight': 42.0}, {'from_id': 1, 'to_id': 17, 'weight': 0.0}, {'from_id': 1, 'to_id': 18, 'weight': 0.0}, {'from_id': 1, 'to_id': 19, 'weight': 8352.0}, {'from_id': 1, 'to_id': 20, 'weight': 8917.0}, {'from_id': 1, 'to_id': 21, 'weight': 0.0}, {'from_id': 1, 'to_id': 22, 'weight': 0.0}, {'from_id': 2, 'to_id': 1, 'weight': 435.0}, {'from_id': 2, 'to_id': 3, 'weight': 0.0}, {'from_id': 2, 'to_id': 4, 'weight': 833.0}, {'from_id': 2, 'to_id': 5, 'weight': 12080.0}, {'from_id': 2, 'to_id': 6, 'weight': 211.0}, {'from_id': 2, 'to_id': 7, 'weight': 171.0}, {'from_id': 2, 'to_id': 8, 'weight': 12.0}, {'from_id': 2, 'to_id': 9, 'weight': 36.0}, {'from_id': 2, 'to_id': 10, 'weight': 32.0}, {'from_id': 2, 'to_id': 11, 'weight': 10.0}, {'from_id': 2, 'to_id': 12, 'weight': 106.0}, {'from_id': 2, 'to_id': 13, 'weight': 138.0}, {'from_id': 2, 'to_id': 14, 'weight': 406.0}, {'from_id': 2, 'to_id': 15, 'weight': 0.0}, {'from_id': 2, 'to_id': 16, 'weight': 313.0}, {'from_id': 2, 'to_id': 17, 'weight': 0.0}, {'from_id': 2, 'to_id': 18, 'weight': 0.0}, {'from_id': 2, 'to_id': 19, 'weight': 839.0}, {'from_id': 2, 'to_id': 20, 'weight': 3787.0}, {'from_id': 2, 'to_id': 21, 'weight': 0.0}, {'from_id': 2, 'to_id': 22, 'weight': 0.0}, {'from_id': 3, 'to_id': 1, 'weight': 29292.0}, {'from_id': 3, 'to_id': 2, 'weight': 88.0}, {'from_id': 3, 'to_id': 4, 'weight': 7816.0}, {'from_id': 3, 'to_id': 5, 'weight': 29404.0}, {'from_id': 3, 'to_id': 6, 'weight': 3505.0}, {'from_id': 3, 'to_id': 7, 'weight': 6856.0}, {'from_id': 3, 'to_id': 8, 'weight': 302.0}, {'from_id': 3, 'to_id': 9, 'weight': 8386.0}, {'from_id': 3, 'to_id': 10, 'weight': 886.0}, {'from_id': 3, 'to_id': 11, 'weight': 1695.0}, {'from_id': 3, 'to_id': 12, 'weight': 498.0}, {'from_id': 3, 'to_id': 13, 'weight': 8452.0}, {'from_id': 3, 'to_id': 14, 'weight': 35394.0}, {'from_id': 3, 'to_id': 15, 'weight': 4901.0}, {'from_id': 3, 'to_id': 16, 'weight': 1506.0}, {'from_id': 3, 'to_id': 17, 'weight': 0.0}, {'from_id': 3, 'to_id': 18, 'weight': 0.0}, {'from_id': 3, 'to_id': 19, 'weight': 53193.0}, {'from_id': 3, 'to_id': 20, 'weight': 32434.0}, {'from_id': 3, 'to_id': 21, 'weight': 0.0}, {'from_id': 3, 'to_id': 22, 'weight': 0.0}, {'from_id': 4, 'to_id': 1, 'weight': 4605.0}, {'from_id': 4, 'to_id': 2, 'weight': 412.0}, {'from_id': 4, 'to_id': 3, 'weight': 1982.0}, {'from_id': 4, 'to_id': 5, 'weight': 8932.0}, {'from_id': 4, 'to_id': 6, 'weight': 181560.0}, {'from_id': 4, 'to_id': 7, 'weight': 99824.0}, {'from_id': 4, 'to_id': 8, 'weight': 21374.0}, {'from_id': 4, 'to_id': 9, 'weight': 121111.0}, {'from_id': 4, 'to_id': 10, 'weight': 143.0}, {'from_id': 4, 'to_id': 11, 'weight': 833.0}, {'from_id': 4, 'to_id': 12, 'weight': 0.0}, {'from_id': 4, 'to_id': 13, 'weight': 1354.0}, {'from_id': 4, 'to_id': 14, 'weight': 0.0}, {'from_id': 4, 'to_id': 15, 'weight': 0.0}, {'from_id': 4, 'to_id': 16, 'weight': 0.0}, {'from_id': 4, 'to_id': 17, 'weight': 0.0}, {'from_id': 4, 'to_id': 18, 'weight': 0.0}, {'from_id': 4, 'to_id': 19, 'weight': 0.0}, {'from_id': 4, 'to_id': 20, 'weight': 3935.0}, {'from_id': 4, 'to_id': 21, 'weight': 0.0}, {'from_id': 4, 'to_id': 22, 'weight': 0.0}, {'from_id': 5, 'to_id': 1, 'weight': 5511.0}, {'from_id': 5, 'to_id': 2, 'weight': 0.0}, {'from_id': 5, 'to_id': 3, 'weight': 2521.0}, {'from_id': 5, 'to_id': 4, 'weight': 3167.0}, {'from_id': 5, 'to_id': 6, 'weight': 6677.0}, {'from_id': 5, 'to_id': 7, 'weight': 2763.0}, {'from_id': 5, 'to_id': 8, 'weight': 3099.0}, {'from_id': 5, 'to_id': 9, 'weight': 5795.0}, {'from_id': 5, 'to_id': 10, 'weight': 0.0}, {'from_id': 5, 'to_id': 11, 'weight': 450.0}, {'from_id': 5, 'to_id': 12, 'weight': 0.0}, {'from_id': 5, 'to_id': 13, 'weight': 83.0}, {'from_id': 5, 'to_id': 14, 'weight': 207.0}, {'from_id': 5, 'to_id': 15, 'weight': 568.0}, {'from_id': 5, 'to_id': 16, 'weight': 0.0}, {'from_id': 5, 'to_id': 17, 'weight': 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LOP,LOP,"Recently the magazine experimented with ordering algorithms, but the human task is familiar: arrange all the articles into one sequence so every article shows up exactly once. There’s a number tied to every ordered pair that says how much engagement is gained when article A precedes article B, and a full ordering gets a score by adding up those numbers for all pairs that follow that order. The aim is to get the ordering with the biggest total, and the exact article list and pair scores are laid out below.

Articles (19 total): A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S.
Meaning: this is a directed engagement_gain matrix over items in article_ids (19 total). Entry at row preceding_article=i and column following_article=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
engagement_gain:
| preceding_article\following_article | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 1602.0 | 0.0 | 433.0 | 1208.0 | 75.0 | 26.0 | 100.0 | 306.0 | 54.0 | 206.0 | 62.0 | 93.0 | 153.0 | 188.0 | 16.0 | 48.0 | 161.0 | 73.0 | 112.0 |
| B | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| C | 24.0 | 0.0 | 1812.0 | 27.0 | 901.0 | 203.0 | 7.0 | 15.0 | 9.0 | 28.0 | 18.0 | 81.0 | 1.0 | 2.0 | 0.0 | 1.0 | 1.0 | 1.0 | 3.0 |
| D | 113.0 | 0.0 | 677.0 | 4965.0 | 59.0 | 104.0 | 59.0 | 243.0 | 99.0 | 328.0 | 886.0 | 81.0 | 19.0 | 41.0 | 21.0 | 43.0 | 28.0 | 83.0 | 232.0 |
| E | 42.0 | 0.0 | 33.0 | 37.0 | 1353.0 | 218.0 | 4.0 | 25.0 | 28.0 | 63.0 | 24.0 | 231.0 | 5.0 | 41.0 | 15.0 | 14.0 | 51.0 | 6.0 | 11.0 |
| F | 0.0 | 0.0 | 5.0 | 7.0 | 3.0 | 612.0 | 3.0 | 6.0 | 0.0 | 0.0 | 1.0 | 708.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | -1.0 | -1.0 |
| G | 0.0 | 0.0 | 0.0 | 3.0 | 0.0 | 0.0 | 120.0 | 852.0 | 0.0 | 0.0 | 5.0 | 0.0 | 89.0 | 0.0 | 35.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| H | 2.0 | 0.0 | 5.0 | 208.0 | 0.0 | 0.0 | 194.0 | 4944.0 | 9.0 | 58.0 | 0.0 | 0.0 | 237.0 | 0.0 | 68.0 | 0.0 | 0.0 | 6.0 | 19.0 |
| I | 1.0 | 0.0 | 1.0 | 33.0 | 25.0 | 13.0 | 3.0 | 51.0 | 931.0 | 23.0 | 14.0 | 14.0 | 0.0 | 41.0 | 3.0 | 7.0 | 0.0 | 0.0 | 1.0 |
| J | 31.0 | 0.0 | 44.0 | 525.0 | 58.0 | 23.0 | 193.0 | 713.0 | 48.0 | 4722.0 | 97.0 | 51.0 | 80.0 | 58.0 | 47.0 | 75.0 | 304.0 | 191.0 | 126.0 |
| K | 27.0 | 0.0 | 22.0 | 208.0 | 59.0 | 79.0 | 60.0 | 106.0 | 100.0 | 110.0 | 144.0 | 23.0 | 26.0 | 118.0 | 13.0 | 15.0 | 14.0 | 32.0 | 40.0 |
| L | 22.0 | 0.0 | 169.0 | 52.0 | 25.0 | 7.0 | 41.0 | 89.0 | 29.0 | 137.0 | 19.0 | 13.0 | 5.0 | 421.0 | 11.0 | 26.0 | 53.0 | 24.0 | 29.0 |
| M | 8.0 | 0.0 | 8.0 | 31.0 | 14.0 | 3.0 | 2.0 | 14.0 | 13.0 | 34.0 | 8.0 | 7.0 | 0.0 | 32.0 | 112.0 | 9.0 | 40.0 | 16.0 | 11.0 |
| N | 11.0 | 0.0 | 8.0 | 53.0 | 16.0 | 5.0 | 65.0 | 89.0 | 14.0 | 84.0 | 8.0 | 19.0 | 0.0 | 502.0 | 7.0 | 41.0 | 79.0 | 32.0 | 26.0 |
| O | 3.0 | 0.0 | 2.0 | 19.0 | 8.0 | 2.0 | 1.0 | 9.0 | 2.0 | 15.0 | 2.0 | 5.0 | 0.0 | 14.0 | 653.0 | 24.0 | 24.0 | 27.0 | 20.0 |
| P | 23.0 | 0.0 | 21.0 | 68.0 | 19.0 | 8.0 | 20.0 | 87.0 | 33.0 | 115.0 | 11.0 | 15.0 | 28.0 | 79.0 | 68.0 | 0.0 | 549.0 | 69.0 | 62.0 |
| Q | 18.0 | 0.0 | 47.0 | 113.0 | 32.0 | 14.0 | 21.0 | 75.0 | 21.0 | 46.0 | 18.0 | 14.0 | 11.0 | 134.0 | 48.0 | 1.0 | 8178.0 | 22.0 | 15.0 |
| R | 61.0 | 0.0 | 24.0 | 282.0 | 97.0 | 21.0 | 29.0 | 117.0 | 49.0 | 396.0 | 46.0 | 31.0 | 89.0 | 150.0 | 98.0 | 69.0 | 717.0 | 435.0 | 504.0 |
| S | 36.0 | 0.0 | 3.0 | 199.0 | 67.0 | 21.0 | 10.0 | 80.0 | 7.0 | 154.0 | 21.0 | 22.0 | 48.0 | 63.0 | 51.0 | 51.0 | 408.0 | 402.0 | 466.0 |
Aim for the ordering with the highest total engagement.

When you're ready, just drop the final ordering into a tiny JSON snippet like this:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Here ""solution"" is simply the list of articles in the order you want them to appear, from first to last. The angle-bracket bits are just placeholders showing the shape — replace them with the real article identifiers from the instance input when you answer. This JSON is only a sketch of the expected shape, not the actual ordering itself.

Please be sure to use the exact identifiers as they appear in the input — no renaming and no 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”.""","[[1602.0, 0.0, 433.0, 1208.0, 75.0, 26.0, 100.0, 306.0, 54.0, 206.0, 62.0, 93.0, 153.0, 188.0, 16.0, 48.0, 161.0, 73.0, 112.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [24.0, 0.0, 1812.0, 27.0, 901.0, 203.0, 7.0, 15.0, 9.0, 28.0, 18.0, 81.0, 1.0, 2.0, 0.0, 1.0, 1.0, 1.0, 3.0], [113.0, 0.0, 677.0, 4965.0, 59.0, 104.0, 59.0, 243.0, 99.0, 328.0, 886.0, 81.0, 19.0, 41.0, 21.0, 43.0, 28.0, 83.0, 232.0], [42.0, 0.0, 33.0, 37.0, 1353.0, 218.0, 4.0, 25.0, 28.0, 63.0, 24.0, 231.0, 5.0, 41.0, 15.0, 14.0, 51.0, 6.0, 11.0], [0.0, 0.0, 5.0, 7.0, 3.0, 612.0, 3.0, 6.0, 0.0, 0.0, 1.0, 708.0, 0.0, 1.0, 0.0, 0.0, 0.0, -1.0, -1.0], [0.0, 0.0, 0.0, 3.0, 0.0, 0.0, 120.0, 852.0, 0.0, 0.0, 5.0, 0.0, 89.0, 0.0, 35.0, 0.0, 0.0, 0.0, 0.0], [2.0, 0.0, 5.0, 208.0, 0.0, 0.0, 194.0, 4944.0, 9.0, 58.0, 0.0, 0.0, 237.0, 0.0, 68.0, 0.0, 0.0, 6.0, 19.0], [1.0, 0.0, 1.0, 33.0, 25.0, 13.0, 3.0, 51.0, 931.0, 23.0, 14.0, 14.0, 0.0, 41.0, 3.0, 7.0, 0.0, 0.0, 1.0], [31.0, 0.0, 44.0, 525.0, 58.0, 23.0, 193.0, 713.0, 48.0, 4722.0, 97.0, 51.0, 80.0, 58.0, 47.0, 75.0, 304.0, 191.0, 126.0], [27.0, 0.0, 22.0, 208.0, 59.0, 79.0, 60.0, 106.0, 100.0, 110.0, 144.0, 23.0, 26.0, 118.0, 13.0, 15.0, 14.0, 32.0, 40.0], [22.0, 0.0, 169.0, 52.0, 25.0, 7.0, 41.0, 89.0, 29.0, 137.0, 19.0, 13.0, 5.0, 421.0, 11.0, 26.0, 53.0, 24.0, 29.0], [8.0, 0.0, 8.0, 31.0, 14.0, 3.0, 2.0, 14.0, 13.0, 34.0, 8.0, 7.0, 0.0, 32.0, 112.0, 9.0, 40.0, 16.0, 11.0], [11.0, 0.0, 8.0, 53.0, 16.0, 5.0, 65.0, 89.0, 14.0, 84.0, 8.0, 19.0, 0.0, 502.0, 7.0, 41.0, 79.0, 32.0, 26.0], [3.0, 0.0, 2.0, 19.0, 8.0, 2.0, 1.0, 9.0, 2.0, 15.0, 2.0, 5.0, 0.0, 14.0, 653.0, 24.0, 24.0, 27.0, 20.0], [23.0, 0.0, 21.0, 68.0, 19.0, 8.0, 20.0, 87.0, 33.0, 115.0, 11.0, 15.0, 28.0, 79.0, 68.0, 0.0, 549.0, 69.0, 62.0], [18.0, 0.0, 47.0, 113.0, 32.0, 14.0, 21.0, 75.0, 21.0, 46.0, 18.0, 14.0, 11.0, 134.0, 48.0, 1.0, 8178.0, 22.0, 15.0], [61.0, 0.0, 24.0, 282.0, 97.0, 21.0, 29.0, 117.0, 49.0, 396.0, 46.0, 31.0, 89.0, 150.0, 98.0, 69.0, 717.0, 435.0, 504.0], [36.0, 0.0, 3.0, 199.0, 67.0, 21.0, 10.0, 80.0, 7.0, 154.0, 21.0, 22.0, 48.0, 63.0, 51.0, 51.0, 408.0, 402.0, 466.0]]","[0, 17, 15, 18, 9, 16, 3, 10, 2, 4, 5, 11, 8, 13, 6, 7, 12, 14, 1]",18806.0,"{'problem_type': 'LOP', 'num_nodes': 19, 'nodes': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S'], 'pairs': [{'from_id': 'A', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'C', 'weight': 433.0}, {'from_id': 'A', 'to_id': 'D', 'weight': 1208.0}, {'from_id': 'A', 'to_id': 'E', 'weight': 75.0}, {'from_id': 'A', 'to_id': 'F', 'weight': 26.0}, {'from_id': 'A', 'to_id': 'G', 'weight': 100.0}, {'from_id': 'A', 'to_id': 'H', 'weight': 306.0}, {'from_id': 'A', 'to_id': 'I', 'weight': 54.0}, {'from_id': 'A', 'to_id': 'J', 'weight': 206.0}, {'from_id': 'A', 'to_id': 'K', 'weight': 62.0}, {'from_id': 'A', 'to_id': 'L', 'weight': 93.0}, {'from_id': 'A', 'to_id': 'M', 'weight': 153.0}, {'from_id': 'A', 'to_id': 'N', 'weight': 188.0}, {'from_id': 'A', 'to_id': 'O', 'weight': 16.0}, {'from_id': 'A', 'to_id': 'P', 'weight': 48.0}, {'from_id': 'A', 'to_id': 'Q', 'weight': 161.0}, {'from_id': 'A', 'to_id': 'R', 'weight': 73.0}, {'from_id': 'A', 'to_id': 'S', 'weight': 112.0}, {'from_id': 'B', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'S', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'A', 'weight': 24.0}, {'from_id': 'C', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'D', 'weight': 27.0}, {'from_id': 'C', 'to_id': 'E', 'weight': 901.0}, {'from_id': 'C', 'to_id': 'F', 'weight': 203.0}, {'from_id': 'C', 'to_id': 'G', 'weight': 7.0}, {'from_id': 'C', 'to_id': 'H', 'weight': 15.0}, {'from_id': 'C', 'to_id': 'I', 'weight': 9.0}, {'from_id': 'C', 'to_id': 'J', 'weight': 28.0}, {'from_id': 'C', 'to_id': 'K', 'weight': 18.0}, {'from_id': 'C', 'to_id': 'L', 'weight': 81.0}, {'from_id': 'C', 'to_id': 'M', 'weight': 1.0}, {'from_id': 'C', 'to_id': 'N', 'weight': 2.0}, {'from_id': 'C', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'P', 'weight': 1.0}, {'from_id': 'C', 'to_id': 'Q', 'weight': 1.0}, {'from_id': 'C', 'to_id': 'R', 'weight': 1.0}, {'from_id': 'C', 'to_id': 'S', 'weight': 3.0}, {'from_id': 'D', 'to_id': 'A', 'weight': 113.0}, {'from_id': 'D', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'C', 'weight': 677.0}, {'from_id': 'D', 'to_id': 'E', 'weight': 59.0}, {'from_id': 'D', 'to_id': 'F', 'weight': 104.0}, {'from_id': 'D', 'to_id': 'G', 'weight': 59.0}, {'from_id': 'D', 'to_id': 'H', 'weight': 243.0}, {'from_id': 'D', 'to_id': 'I', 'weight': 99.0}, {'from_id': 'D', 'to_id': 'J', 'weight': 328.0}, {'from_id': 'D', 'to_id': 'K', 'weight': 886.0}, {'from_id': 'D', 'to_id': 'L', 'weight': 81.0}, {'from_id': 'D', 'to_id': 'M', 'weight': 19.0}, {'from_id': 'D', 'to_id': 'N', 'weight': 41.0}, {'from_id': 'D', 'to_id': 'O', 'weight': 21.0}, {'from_id': 'D', 'to_id': 'P', 'weight': 43.0}, {'from_id': 'D', 'to_id': 'Q', 'weight': 28.0}, {'from_id': 'D', 'to_id': 'R', 'weight': 83.0}, {'from_id': 'D', 'to_id': 'S', 'weight': 232.0}, {'from_id': 'E', 'to_id': 'A', 'weight': 42.0}, {'from_id': 'E', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'C', 'weight': 33.0}, {'from_id': 'E', 'to_id': 'D', 'weight': 37.0}, {'from_id': 'E', 'to_id': 'F', 'weight': 218.0}, {'from_id': 'E', 'to_id': 'G', 'weight': 4.0}, {'from_id': 'E', 'to_id': 'H', 'weight': 25.0}, {'from_id': 'E', 'to_id': 'I', 'weight': 28.0}, {'from_id': 'E', 'to_id': 'J', 'weight': 63.0}, {'from_id': 'E', 'to_id': 'K', 'weight': 24.0}, {'from_id': 'E', 'to_id': 'L', 'weight': 231.0}, {'from_id': 'E', 'to_id': 'M', 'weight': 5.0}, {'from_id': 'E', 'to_id': 'N', 'weight': 41.0}, {'from_id': 'E', 'to_id': 'O', 'weight': 15.0}, {'from_id': 'E', 'to_id': 'P', 'weight': 14.0}, {'from_id': 'E', 'to_id': 'Q', 'weight': 51.0}, {'from_id': 'E', 'to_id': 'R', 'weight': 6.0}, {'from_id': 'E', 'to_id': 'S', 'weight': 11.0}, {'from_id': 'F', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'C', 'weight': 5.0}, {'from_id': 'F', 'to_id': 'D', 'weight': 7.0}, {'from_id': 'F', 'to_id': 'E', 'weight': 3.0}, {'from_id': 'F', 'to_id': 'G', 'weight': 3.0}, {'from_id': 'F', 'to_id': 'H', 'weight': 6.0}, {'from_id': 'F', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'K', 'weight': 1.0}, {'from_id': 'F', 'to_id': 'L', 'weight': 708.0}, {'from_id': 'F', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'N', 'weight': 1.0}, {'from_id': 'F', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'R', 'weight': -1.0}, {'from_id': 'F', 'to_id': 'S', 'weight': -1.0}, {'from_id': 'G', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'D', 'weight': 3.0}, {'from_id': 'G', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'H', 'weight': 852.0}, {'from_id': 'G', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'K', 'weight': 5.0}, {'from_id': 'G', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'M', 'weight': 89.0}, {'from_id': 'G', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'O', 'weight': 35.0}, {'from_id': 'G', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'S', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'A', 'weight': 2.0}, {'from_id': 'H', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'C', 'weight': 5.0}, {'from_id': 'H', 'to_id': 'D', 'weight': 208.0}, {'from_id': 'H', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'G', 'weight': 194.0}, {'from_id': 'H', 'to_id': 'I', 'weight': 9.0}, {'from_id': 'H', 'to_id': 'J', 'weight': 58.0}, {'from_id': 'H', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'M', 'weight': 237.0}, {'from_id': 'H', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'O', 'weight': 68.0}, {'from_id': 'H', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'R', 'weight': 6.0}, {'from_id': 'H', 'to_id': 'S', 'weight': 19.0}, {'from_id': 'I', 'to_id': 'A', 'weight': 1.0}, {'from_id': 'I', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'C', 'weight': 1.0}, {'from_id': 'I', 'to_id': 'D', 'weight': 33.0}, {'from_id': 'I', 'to_id': 'E', 'weight': 25.0}, {'from_id': 'I', 'to_id': 'F', 'weight': 13.0}, {'from_id': 'I', 'to_id': 'G', 'weight': 3.0}, {'from_id': 'I', 'to_id': 'H', 'weight': 51.0}, {'from_id': 'I', 'to_id': 'J', 'weight': 23.0}, {'from_id': 'I', 'to_id': 'K', 'weight': 14.0}, {'from_id': 'I', 'to_id': 'L', 'weight': 14.0}, {'from_id': 'I', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'N', 'weight': 41.0}, {'from_id': 'I', 'to_id': 'O', 'weight': 3.0}, {'from_id': 'I', 'to_id': 'P', 'weight': 7.0}, {'from_id': 'I', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'S', 'weight': 1.0}, {'from_id': 'J', 'to_id': 'A', 'weight': 31.0}, {'from_id': 'J', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'C', 'weight': 44.0}, {'from_id': 'J', 'to_id': 'D', 'weight': 525.0}, {'from_id': 'J', 'to_id': 'E', 'weight': 58.0}, {'from_id': 'J', 'to_id': 'F', 'weight': 23.0}, {'from_id': 'J', 'to_id': 'G', 'weight': 193.0}, {'from_id': 'J', 'to_id': 'H', 'weight': 713.0}, {'from_id': 'J', 'to_id': 'I', 'weight': 48.0}, {'from_id': 'J', 'to_id': 'K', 'weight': 97.0}, {'from_id': 'J', 'to_id': 'L', 'weight': 51.0}, {'from_id': 'J', 'to_id': 'M', 'weight': 80.0}, {'from_id': 'J', 'to_id': 'N', 'weight': 58.0}, {'from_id': 'J', 'to_id': 'O', 'weight': 47.0}, {'from_id': 'J', 'to_id': 'P', 'weight': 75.0}, {'from_id': 'J', 'to_id': 'Q', 'weight': 304.0}, {'from_id': 'J', 'to_id': 'R', 'weight': 191.0}, {'from_id': 'J', 'to_id': 'S', 'weight': 126.0}, {'from_id': 'K', 'to_id': 'A', 'weight': 27.0}, {'from_id': 'K', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'C', 'weight': 22.0}, {'from_id': 'K', 'to_id': 'D', 'weight': 208.0}, {'from_id': 'K', 'to_id': 'E', 'weight': 59.0}, {'from_id': 'K', 'to_id': 'F', 'weight': 79.0}, {'from_id': 'K', 'to_id': 'G', 'weight': 60.0}, {'from_id': 'K', 'to_id': 'H', 'weight': 106.0}, {'from_id': 'K', 'to_id': 'I', 'weight': 100.0}, {'from_id': 'K', 'to_id': 'J', 'weight': 110.0}, {'from_id': 'K', 'to_id': 'L', 'weight': 23.0}, {'from_id': 'K', 'to_id': 'M', 'weight': 26.0}, {'from_id': 'K', 'to_id': 'N', 'weight': 118.0}, {'from_id': 'K', 'to_id': 'O', 'weight': 13.0}, {'from_id': 'K', 'to_id': 'P', 'weight': 15.0}, {'from_id': 'K', 'to_id': 'Q', 'weight': 14.0}, {'from_id': 'K', 'to_id': 'R', 'weight': 32.0}, {'from_id': 'K', 'to_id': 'S', 'weight': 40.0}, {'from_id': 'L', 'to_id': 'A', 'weight': 22.0}, {'from_id': 'L', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'C', 'weight': 169.0}, {'from_id': 'L', 'to_id': 'D', 'weight': 52.0}, {'from_id': 'L', 'to_id': 'E', 'weight': 25.0}, {'from_id': 'L', 'to_id': 'F', 'weight': 7.0}, {'from_id': 'L', 'to_id': 'G', 'weight': 41.0}, {'from_id': 'L', 'to_id': 'H', 'weight': 89.0}, {'from_id': 'L', 'to_id': 'I', 'weight': 29.0}, {'from_id': 'L', 'to_id': 'J', 'weight': 137.0}, {'from_id': 'L', 'to_id': 'K', 'weight': 19.0}, {'from_id': 'L', 'to_id': 'M', 'weight': 5.0}, {'from_id': 'L', 'to_id': 'N', 'weight': 421.0}, {'from_id': 'L', 'to_id': 'O', 'weight': 11.0}, {'from_id': 'L', 'to_id': 'P', 'weight': 26.0}, {'from_id': 'L', 'to_id': 'Q', 'weight': 53.0}, {'from_id': 'L', 'to_id': 'R', 'weight': 24.0}, {'from_id': 'L', 'to_id': 'S', 'weight': 29.0}, {'from_id': 'M', 'to_id': 'A', 'weight': 8.0}, {'from_id': 'M', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'M', 'to_id': 'C', 'weight': 8.0}, {'from_id': 'M', 'to_id': 'D', 'weight': 31.0}, {'from_id': 'M', 'to_id': 'E', 'weight': 14.0}, {'from_id': 'M', 'to_id': 'F', 'weight': 3.0}, {'from_id': 'M', 'to_id': 'G', 'weight': 2.0}, {'from_id': 'M', 'to_id': 'H', 'weight': 14.0}, {'from_id': 'M', 'to_id': 'I', 'weight': 13.0}, {'from_id': 'M', 'to_id': 'J', 'weight': 34.0}, {'from_id': 'M', 'to_id': 'K', 'weight': 8.0}, {'from_id': 'M', 'to_id': 'L', 'weight': 7.0}, {'from_id': 'M', 'to_id': 'N', 'weight': 32.0}, {'from_id': 'M', 'to_id': 'O', 'weight': 112.0}, {'from_id': 'M', 'to_id': 'P', 'weight': 9.0}, {'from_id': 'M', 'to_id': 'Q', 'weight': 40.0}, {'from_id': 'M', 'to_id': 'R', 'weight': 16.0}, {'from_id': 'M', 'to_id': 'S', 'weight': 11.0}, {'from_id': 'N', 'to_id': 'A', 'weight': 11.0}, {'from_id': 'N', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'C', 'weight': 8.0}, {'from_id': 'N', 'to_id': 'D', 'weight': 53.0}, {'from_id': 'N', 'to_id': 'E', 'weight': 16.0}, {'from_id': 'N', 'to_id': 'F', 'weight': 5.0}, {'from_id': 'N', 'to_id': 'G', 'weight': 65.0}, {'from_id': 'N', 'to_id': 'H', 'weight': 89.0}, {'from_id': 'N', 'to_id': 'I', 'weight': 14.0}, {'from_id': 'N', 'to_id': 'J', 'weight': 84.0}, {'from_id': 'N', 'to_id': 'K', 'weight': 8.0}, {'from_id': 'N', 'to_id': 'L', 'weight': 19.0}, {'from_id': 'N', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'O', 'weight': 7.0}, {'from_id': 'N', 'to_id': 'P', 'weight': 41.0}, {'from_id': 'N', 'to_id': 'Q', 'weight': 79.0}, {'from_id': 'N', 'to_id': 'R', 'weight': 32.0}, {'from_id': 'N', 'to_id': 'S', 'weight': 26.0}, {'from_id': 'O', 'to_id': 'A', 'weight': 3.0}, {'from_id': 'O', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'C', 'weight': 2.0}, {'from_id': 'O', 'to_id': 'D', 'weight': 19.0}, {'from_id': 'O', 'to_id': 'E', 'weight': 8.0}, {'from_id': 'O', 'to_id': 'F', 'weight': 2.0}, {'from_id': 'O', 'to_id': 'G', 'weight': 1.0}, {'from_id': 'O', 'to_id': 'H', 'weight': 9.0}, {'from_id': 'O', 'to_id': 'I', 'weight': 2.0}, {'from_id': 'O', 'to_id': 'J', 'weight': 15.0}, {'from_id': 'O', 'to_id': 'K', 'weight': 2.0}, {'from_id': 'O', 'to_id': 'L', 'weight': 5.0}, {'from_id': 'O', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'N', 'weight': 14.0}, {'from_id': 'O', 'to_id': 'P', 'weight': 24.0}, {'from_id': 'O', 'to_id': 'Q', 'weight': 24.0}, {'from_id': 'O', 'to_id': 'R', 'weight': 27.0}, {'from_id': 'O', 'to_id': 'S', 'weight': 20.0}, {'from_id': 'P', 'to_id': 'A', 'weight': 23.0}, {'from_id': 'P', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'C', 'weight': 21.0}, {'from_id': 'P', 'to_id': 'D', 'weight': 68.0}, {'from_id': 'P', 'to_id': 'E', 'weight': 19.0}, {'from_id': 'P', 'to_id': 'F', 'weight': 8.0}, {'from_id': 'P', 'to_id': 'G', 'weight': 20.0}, {'from_id': 'P', 'to_id': 'H', 'weight': 87.0}, {'from_id': 'P', 'to_id': 'I', 'weight': 33.0}, {'from_id': 'P', 'to_id': 'J', 'weight': 115.0}, {'from_id': 'P', 'to_id': 'K', 'weight': 11.0}, {'from_id': 'P', 'to_id': 'L', 'weight': 15.0}, {'from_id': 'P', 'to_id': 'M', 'weight': 28.0}, {'from_id': 'P', 'to_id': 'N', 'weight': 79.0}, {'from_id': 'P', 'to_id': 'O', 'weight': 68.0}, {'from_id': 'P', 'to_id': 'Q', 'weight': 549.0}, {'from_id': 'P', 'to_id': 'R', 'weight': 69.0}, {'from_id': 'P', 'to_id': 'S', 'weight': 62.0}, {'from_id': 'Q', 'to_id': 'A', 'weight': 18.0}, {'from_id': 'Q', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'C', 'weight': 47.0}, {'from_id': 'Q', 'to_id': 'D', 'weight': 113.0}, {'from_id': 'Q', 'to_id': 'E', 'weight': 32.0}, {'from_id': 'Q', 'to_id': 'F', 'weight': 14.0}, {'from_id': 'Q', 'to_id': 'G', 'weight': 21.0}, {'from_id': 'Q', 'to_id': 'H', 'weight': 75.0}, {'from_id': 'Q', 'to_id': 'I', 'weight': 21.0}, {'from_id': 'Q', 'to_id': 'J', 'weight': 46.0}, {'from_id': 'Q', 'to_id': 'K', 'weight': 18.0}, {'from_id': 'Q', 'to_id': 'L', 'weight': 14.0}, {'from_id': 'Q', 'to_id': 'M', 'weight': 11.0}, {'from_id': 'Q', 'to_id': 'N', 'weight': 134.0}, {'from_id': 'Q', 'to_id': 'O', 'weight': 48.0}, {'from_id': 'Q', 'to_id': 'P', 'weight': 1.0}, {'from_id': 'Q', 'to_id': 'R', 'weight': 22.0}, {'from_id': 'Q', 'to_id': 'S', 'weight': 15.0}, {'from_id': 'R', 'to_id': 'A', 'weight': 61.0}, {'from_id': 'R', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'C', 'weight': 24.0}, {'from_id': 'R', 'to_id': 'D', 'weight': 282.0}, {'from_id': 'R', 'to_id': 'E', 'weight': 97.0}, {'from_id': 'R', 'to_id': 'F', 'weight': 21.0}, {'from_id': 'R', 'to_id': 'G', 'weight': 29.0}, {'from_id': 'R', 'to_id': 'H', 'weight': 117.0}, {'from_id': 'R', 'to_id': 'I', 'weight': 49.0}, {'from_id': 'R', 'to_id': 'J', 'weight': 396.0}, {'from_id': 'R', 'to_id': 'K', 'weight': 46.0}, {'from_id': 'R', 'to_id': 'L', 'weight': 31.0}, {'from_id': 'R', 'to_id': 'M', 'weight': 89.0}, {'from_id': 'R', 'to_id': 'N', 'weight': 150.0}, {'from_id': 'R', 'to_id': 'O', 'weight': 98.0}, {'from_id': 'R', 'to_id': 'P', 'weight': 69.0}, {'from_id': 'R', 'to_id': 'Q', 'weight': 717.0}, {'from_id': 'R', 'to_id': 'S', 'weight': 504.0}, {'from_id': 'S', 'to_id': 'A', 'weight': 36.0}, {'from_id': 'S', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'C', 'weight': 3.0}, {'from_id': 'S', 'to_id': 'D', 'weight': 199.0}, {'from_id': 'S', 'to_id': 'E', 'weight': 67.0}, {'from_id': 'S', 'to_id': 'F', 'weight': 21.0}, {'from_id': 'S', 'to_id': 'G', 'weight': 10.0}, {'from_id': 'S', 'to_id': 'H', 'weight': 80.0}, {'from_id': 'S', 'to_id': 'I', 'weight': 7.0}, {'from_id': 'S', 'to_id': 'J', 'weight': 154.0}, {'from_id': 'S', 'to_id': 'K', 'weight': 21.0}, {'from_id': 'S', 'to_id': 'L', 'weight': 22.0}, {'from_id': 'S', 'to_id': 'M', 'weight': 48.0}, {'from_id': 'S', 'to_id': 'N', 'weight': 63.0}, {'from_id': 'S', 'to_id': 'O', 'weight': 51.0}, {'from_id': 'S', 'to_id': 'P', 'weight': 51.0}, {'from_id': 'S', 'to_id': 'Q', 'weight': 408.0}, {'from_id': 'S', 'to_id': 'R', 'weight': 402.0}]}","['A', 'R', 'P', 'S', 'J', 'Q', 'D', 'K', 'C', 'E', 'F', 'L', 'I', 'N', 'G', 'H', 'M', 'O', 'B']",8,nl,names
LOP,LOP,"I remember juggling a similar reception order where the secret was how one speech set up the next; some combinations lit up the room and others didn’t. The task now is to arrange all the speeches in a single sequence, using each speaker one time, so that the sum of those pairwise emotional lifts is as high as possible. The measurement is straightforward: tally each transition’s contribution and total them. The full list of speakers and the transition details are provided below.

I'll be arranging 20 speeches — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 — into that single sequence.
Meaning: this is a directed emotional_lift matrix over items in speaker_ids (20 total). Entry at row preceding_speaker_id=i and column following_speaker_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
emotional_lift:
| preceding_speaker_id\following_speaker_id | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1978900.0 | 0.0 | 0.0 | 0.0 | 1000.0 | 0.0 | 15300.0 | 6648400.0 | 131900.0 | 2700.0 | 280800.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 139600.0 | 0.0 | 0.0 |
| 2 | 0.0 | 0.0 | 0.0 | 189900.0 | 15900.0 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 200.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 3 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 4 | 157500.0 | 0.0 | 0.0 | 20300.0 | 73000.0 | 25600.0 | 23100.0 | 39000.0 | 32000.0 | 3400.0 | 31500.0 | 113900.0 | 8000.0 | 13300.0 | 13600.0 | 200.0 | 7000.0 | 127700.0 | 0.0 | 900.0 |
| 5 | 17000.0 | 0.0 | 0.0 | 1200.0 | 535600.0 | 26400.0 | 31900.0 | 16900.0 | 2900.0 | 0.0 | 36800.0 | 51500.0 | 500.0 | 300.0 | 100.0 | 0.0 | 100.0 | 50000.0 | 0.0 | 0.0 |
| 6 | 36100.0 | 0.0 | 0.0 | 24300.0 | 30200.0 | 403400.0 | 189500.0 | 16700.0 | 10200.0 | 1000.0 | 10100.0 | 93800.0 | 12100.0 | 9600.0 | 5500.0 | 500.0 | 1900.0 | 122800.0 | 0.0 | 200.0 |
| 7 | 83200.0 | 0.0 | 0.0 | 1000.0 | 1200.0 | 80300.0 | 325300.0 | 700.0 | 400.0 | 0.0 | 400.0 | 2600.0 | 500.0 | 300.0 | 100.0 | 0.0 | 100.0 | 193900.0 | 0.0 | 0.0 |
| 8 | 17900.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2177400.0 | 32300.0 | 44400.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 119800.0 | 0.0 | 0.0 |
| 9 | 33400.0 | 0.0 | 0.0 | 1100.0 | 7700.0 | 11000.0 | 35200.0 | 1100.0 | 1983000.0 | 9700.0 | 177100.0 | 61400.0 | 900.0 | 2800.0 | 3200.0 | 100.0 | 500.0 | 208400.0 | 0.0 | 0.0 |
| 10 | 900.0 | 0.0 | 0.0 | 100.0 | 100.0 | 400.0 | 0.0 | 100.0 | 38700.0 | 144800.0 | 24700.0 | 1000.0 | 0.0 | 100.0 | 100.0 | 0.0 | 0.0 | 24000.0 | 0.0 | 0.0 |
| 11 | 20100.0 | 0.0 | 0.0 | 300.0 | 21100.0 | 300.0 | 72700.0 | 200.0 | 7800.0 | 38400.0 | 703600.0 | 22700.0 | 200.0 | 100.0 | 0.0 | 0.0 | 0.0 | 3500.0 | 0.0 | 0.0 |
| 12 | 993200.0 | 0.0 | 0.0 | 46200.0 | 151400.0 | 319100.0 | 106500.0 | 45200.0 | 122000.0 | 21900.0 | 168000.0 | 332600.0 | 17100.0 | 22500.0 | 21600.0 | 3600.0 | 13700.0 | 713500.0 | 0.0 | 600.0 |
| 13 | 56600.0 | 0.0 | 0.0 | 22100.0 | 19300.0 | 39700.0 | 11500.0 | 24600.0 | 31900.0 | 4000.0 | 23300.0 | 211300.0 | 21800.0 | 175500.0 | 122300.0 | 2700.0 | 10900.0 | 553900.0 | 0.0 | 1300.0 |
| 14 | 42200.0 | 0.0 | 0.0 | 5900.0 | 6000.0 | 10800.0 | 3400.0 | 8600.0 | 9300.0 | 1200.0 | 6800.0 | 76000.0 | 6500.0 | 3301500.0 | 29700.0 | 700.0 | 3200.0 | 66100.0 | 0.0 | 300.0 |
| 15 | 109400.0 | 0.0 | 0.0 | 42700.0 | 37300.0 | 76800.0 | 22200.0 | 47700.0 | 61700.0 | 7700.0 | 52600.0 | 416500.0 | 42100.0 | 339500.0 | 255300.0 | 5300.0 | 21100.0 | 583600.0 | 0.0 | 2600.0 |
| 16 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 17 | 115700.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 64100.0 | 898700.0 | 0.0 | 0.0 |
| 18 | 13900.0 | 0.0 | 0.0 | 5400.0 | 4700.0 | 9800.0 | 2800.0 | 6100.0 | 7800.0 | 1000.0 | 5700.0 | 51800.0 | 5300.0 | 43400.0 | 30100.0 | 700.0 | 2700.0 | 66200.0 | 0.0 | 300.0 |
| 19 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 20 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
I'll tally these contributions to pick the sequence with the highest total emotional lift.

Oh, and when you hand me the final ordering, please follow this simple JSON layout so it's easy to read and parse:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Here ""solution"" is just the single ordered list of speakers (from the one who goes first to the one who goes last). The angle-bracket names show the slots where each actual speaker identifier goes — think of it like the blanks on a form. This is just a sketch of the expected shape, not the actual final answer.

Please 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"".","[[1978900.0, 0.0, 0.0, 0.0, 1000.0, 0.0, 15300.0, 6648400.0, 131900.0, 2700.0, 280800.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 139600.0, 0.0, 0.0], [0.0, 0.0, 0.0, 189900.0, 15900.0, 0.0, 0.0, 0.0, 100.0, 100.0, 200.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [157500.0, 0.0, 0.0, 20300.0, 73000.0, 25600.0, 23100.0, 39000.0, 32000.0, 3400.0, 31500.0, 113900.0, 8000.0, 13300.0, 13600.0, 200.0, 7000.0, 127700.0, 0.0, 900.0], [17000.0, 0.0, 0.0, 1200.0, 535600.0, 26400.0, 31900.0, 16900.0, 2900.0, 0.0, 36800.0, 51500.0, 500.0, 300.0, 100.0, 0.0, 100.0, 50000.0, 0.0, 0.0], [36100.0, 0.0, 0.0, 24300.0, 30200.0, 403400.0, 189500.0, 16700.0, 10200.0, 1000.0, 10100.0, 93800.0, 12100.0, 9600.0, 5500.0, 500.0, 1900.0, 122800.0, 0.0, 200.0], [83200.0, 0.0, 0.0, 1000.0, 1200.0, 80300.0, 325300.0, 700.0, 400.0, 0.0, 400.0, 2600.0, 500.0, 300.0, 100.0, 0.0, 100.0, 193900.0, 0.0, 0.0], [17900.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2177400.0, 32300.0, 44400.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 119800.0, 0.0, 0.0], [33400.0, 0.0, 0.0, 1100.0, 7700.0, 11000.0, 35200.0, 1100.0, 1983000.0, 9700.0, 177100.0, 61400.0, 900.0, 2800.0, 3200.0, 100.0, 500.0, 208400.0, 0.0, 0.0], [900.0, 0.0, 0.0, 100.0, 100.0, 400.0, 0.0, 100.0, 38700.0, 144800.0, 24700.0, 1000.0, 0.0, 100.0, 100.0, 0.0, 0.0, 24000.0, 0.0, 0.0], [20100.0, 0.0, 0.0, 300.0, 21100.0, 300.0, 72700.0, 200.0, 7800.0, 38400.0, 703600.0, 22700.0, 200.0, 100.0, 0.0, 0.0, 0.0, 3500.0, 0.0, 0.0], [993200.0, 0.0, 0.0, 46200.0, 151400.0, 319100.0, 106500.0, 45200.0, 122000.0, 21900.0, 168000.0, 332600.0, 17100.0, 22500.0, 21600.0, 3600.0, 13700.0, 713500.0, 0.0, 600.0], [56600.0, 0.0, 0.0, 22100.0, 19300.0, 39700.0, 11500.0, 24600.0, 31900.0, 4000.0, 23300.0, 211300.0, 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14, 'weight': 0.0}, {'from_id': 16, 'to_id': 15, 'weight': 0.0}, {'from_id': 16, 'to_id': 17, 'weight': 0.0}, {'from_id': 16, 'to_id': 18, 'weight': 0.0}, {'from_id': 16, 'to_id': 19, 'weight': 0.0}, {'from_id': 16, 'to_id': 20, 'weight': 0.0}, {'from_id': 17, 'to_id': 1, 'weight': 115700.0}, {'from_id': 17, 'to_id': 2, 'weight': 0.0}, {'from_id': 17, 'to_id': 3, 'weight': 0.0}, {'from_id': 17, 'to_id': 4, 'weight': 0.0}, {'from_id': 17, 'to_id': 5, 'weight': 0.0}, {'from_id': 17, 'to_id': 6, 'weight': 0.0}, {'from_id': 17, 'to_id': 7, 'weight': 0.0}, {'from_id': 17, 'to_id': 8, 'weight': 0.0}, {'from_id': 17, 'to_id': 9, 'weight': 0.0}, {'from_id': 17, 'to_id': 10, 'weight': 0.0}, {'from_id': 17, 'to_id': 11, 'weight': 0.0}, {'from_id': 17, 'to_id': 12, 'weight': 0.0}, {'from_id': 17, 'to_id': 13, 'weight': 0.0}, {'from_id': 17, 'to_id': 14, 'weight': 0.0}, {'from_id': 17, 'to_id': 15, 'weight': 0.0}, {'from_id': 17, 'to_id': 16, 'weight': 0.0}, {'from_id': 17, 'to_id': 18, 'weight': 898700.0}, {'from_id': 17, 'to_id': 19, 'weight': 0.0}, {'from_id': 17, 'to_id': 20, 'weight': 0.0}, {'from_id': 18, 'to_id': 1, 'weight': 13900.0}, {'from_id': 18, 'to_id': 2, 'weight': 0.0}, {'from_id': 18, 'to_id': 3, 'weight': 0.0}, {'from_id': 18, 'to_id': 4, 'weight': 5400.0}, {'from_id': 18, 'to_id': 5, 'weight': 4700.0}, {'from_id': 18, 'to_id': 6, 'weight': 9800.0}, {'from_id': 18, 'to_id': 7, 'weight': 2800.0}, {'from_id': 18, 'to_id': 8, 'weight': 6100.0}, {'from_id': 18, 'to_id': 9, 'weight': 7800.0}, {'from_id': 18, 'to_id': 10, 'weight': 1000.0}, {'from_id': 18, 'to_id': 11, 'weight': 5700.0}, {'from_id': 18, 'to_id': 12, 'weight': 51800.0}, {'from_id': 18, 'to_id': 13, 'weight': 5300.0}, {'from_id': 18, 'to_id': 14, 'weight': 43400.0}, {'from_id': 18, 'to_id': 15, 'weight': 30100.0}, {'from_id': 18, 'to_id': 16, 'weight': 700.0}, {'from_id': 18, 'to_id': 17, 'weight': 2700.0}, {'from_id': 18, 'to_id': 19, 'weight': 0.0}, {'from_id': 18, 'to_id': 20, 'weight': 300.0}, {'from_id': 19, 'to_id': 1, 'weight': 0.0}, {'from_id': 19, 'to_id': 2, 'weight': 0.0}, {'from_id': 19, 'to_id': 3, 'weight': 0.0}, {'from_id': 19, 'to_id': 4, 'weight': 0.0}, {'from_id': 19, 'to_id': 5, 'weight': 0.0}, {'from_id': 19, 'to_id': 6, 'weight': 0.0}, {'from_id': 19, 'to_id': 7, 'weight': 0.0}, {'from_id': 19, 'to_id': 8, 'weight': 0.0}, {'from_id': 19, 'to_id': 9, 'weight': 0.0}, {'from_id': 19, 'to_id': 10, 'weight': 0.0}, {'from_id': 19, 'to_id': 11, 'weight': 0.0}, {'from_id': 19, 'to_id': 12, 'weight': 0.0}, {'from_id': 19, 'to_id': 13, 'weight': 0.0}, {'from_id': 19, 'to_id': 14, 'weight': 0.0}, {'from_id': 19, 'to_id': 15, 'weight': 0.0}, {'from_id': 19, 'to_id': 16, 'weight': 0.0}, {'from_id': 19, 'to_id': 17, 'weight': 0.0}, {'from_id': 19, 'to_id': 18, 'weight': 0.0}, {'from_id': 19, 'to_id': 20, 'weight': 0.0}, {'from_id': 20, 'to_id': 1, 'weight': 0.0}, {'from_id': 20, 'to_id': 2, 'weight': 0.0}, {'from_id': 20, 'to_id': 3, 'weight': 0.0}, {'from_id': 20, 'to_id': 4, 'weight': 0.0}, {'from_id': 20, 'to_id': 5, 'weight': 0.0}, {'from_id': 20, 'to_id': 6, 'weight': 0.0}, {'from_id': 20, 'to_id': 7, 'weight': 0.0}, {'from_id': 20, 'to_id': 8, 'weight': 0.0}, {'from_id': 20, 'to_id': 9, 'weight': 0.0}, {'from_id': 20, 'to_id': 10, 'weight': 0.0}, {'from_id': 20, 'to_id': 11, 'weight': 0.0}, {'from_id': 20, 'to_id': 12, 'weight': 0.0}, {'from_id': 20, 'to_id': 13, 'weight': 0.0}, {'from_id': 20, 'to_id': 14, 'weight': 0.0}, {'from_id': 20, 'to_id': 15, 'weight': 0.0}, {'from_id': 20, 'to_id': 16, 'weight': 0.0}, {'from_id': 20, 'to_id': 17, 'weight': 0.0}, {'from_id': 20, 'to_id': 18, 'weight': 0.0}, {'from_id': 20, 'to_id': 19, 'weight': 0.0}]}","[13, 15, 2, 4, 14, 12, 6, 5, 17, 1, 8, 10, 9, 11, 7, 18, 20, 19, 16, 3]",9,nl,1
LOP,LOP,"Back when the team experimented with segment order, they noticed a pattern: for any two distinct pieces, placing A before B gives a measurable retention gain (those gains are recorded). The constraint is straightforward — every piece must appear exactly once in the final lineup, no repeats, no omissions — and the episode’s success is measured by adding up the gains for all ordered pairs that occur in the chosen sequence. The aim is to pick the sequence with the highest total. The specific segment names and pairwise gains follow below.

They had 17 segments to include, listed as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16.
Meaning: this is a directed pairwise_retention_gain matrix over items in segments_list (17 total). Entry at row preceding_segment_id=i and column following_segment_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
pairwise_retention_gain:
| preceding_segment_id\following_segment_id | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 24967.0 | 3012.0 | 580.0 | 448.0 | 319.0 | 679.0 | 101.0 | 2278.0 | 2799.0 | 2105.0 | 401.0 | 244.0 | 822.0 | 33.0 | 19.0 | 1752.0 | 231.0 |
| 1 | 731.0 | 34044.0 | 1429.0 | 1876.0 | 213.0 | 455.0 | 208.0 | 430.0 | 359.0 | 28.0 | 70.0 | 284.0 | 2879.0 | 104.0 | 13.0 | 7044.0 | 1652.0 |
| 2 | 20.0 | 272.0 | 16171.0 | 4257.0 | 9.0 | 8.0 | 4.0 | 3871.0 | 412.0 | 48.0 | 347.0 | 481.0 | 1659.0 | 69.0 | 3.0 | 772.0 | 120.0 |
| 3 | 0.0 | 15.0 | 13.0 | 13637.0 | 2.0 | 2.0 | 1.0 | 25.0 | 2410.0 | 86.0 | 33.0 | 15.0 | 50.0 | 3.0 | 0.0 | 3341.0 | 7.0 |
| 4 | 0.0 | 77.0 | 0.0 | 0.0 | 1754.0 | 470.0 | 282.0 | 0.0 | 0.0 | 69.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 631.0 | 367.0 |
| 5 | 6.0 | 702.0 | 22.0 | 21.0 | 424.0 | 15992.0 | 5.0 | 13.0 | 0.0 | 459.0 | 0.0 | 12.0 | 647.0 | 2.0 | 0.0 | 1198.0 | 778.0 |
| 6 | 2.0 | 14.0 | 8.0 | 46.0 | 3.0 | 1.0 | 1138.0 | 9.0 | 37.0 | 6.0 | 52.0 | 0.0 | 46.0 | 0.0 | 0.0 | 685.0 | 44.0 |
| 7 | 12.0 | 145.0 | 90.0 | 49.0 | 25.0 | 75.0 | 3.0 | 3727.0 | 686.0 | 14.0 | 46.0 | 137.0 | 122.0 | 2192.0 | 3.0 | 1749.0 | 91.0 |
| 8 | 512.0 | 1930.0 | 696.0 | 919.0 | 350.0 | 956.0 | 51.0 | 2657.0 | 211.0 | 82.0 | 51.0 | 126.0 | 375.0 | 55.0 | 3.0 | 829.0 | 133.0 |
| 9 | 52.0 | 318.0 | 153.0 | 125.0 | 80.0 | 123.0 | 20.0 | 205.0 | 45.0 | 2305.0 | 5.0 | 82.0 | 78.0 | 31.0 | 1.0 | 420.0 | 38.0 |
| 10 | 8.0 | 450.0 | 289.0 | 407.0 | 58.0 | 143.0 | 113.0 | 531.0 | 535.0 | 48.0 | 80.0 | 1215.0 | 554.0 | 12.0 | 12.0 | 2491.0 | 158.0 |
| 11 | 18.0 | 361.0 | 180.0 | 247.0 | 41.0 | 94.0 | 24.0 | 670.0 | 1075.0 | 71.0 | 1849.0 | 35723.0 | 724.0 | 873.0 | 4.0 | 1414.0 | 91.0 |
| 12 | 126.0 | 2347.0 | 2120.0 | 2238.0 | 185.0 | 594.0 | 146.0 | 9213.0 | 1027.0 | 108.0 | 188.0 | 9108.0 | 9476.0 | 7322.0 | 33.0 | 2342.0 | 912.0 |
| 13 | 1.0 | 47.0 | 101.0 | 59.0 | 11.0 | 60.0 | 12.0 | 133.0 | 96.0 | 7.0 | 23.0 | 55.0 | 516.0 | 6.0 | 3.0 | 1578.0 | 70.0 |
| 14 | 0.0 | 0.0 | 26.0 | 5.0 | 1.0 | 2.0 | 0.0 | 5.0 | 32.0 | 0.0 | 2.0 | 18.0 | 41.0 | 6.0 | 0.0 | 359.0 | 10.0 |
| 15 | 11.0 | 329.0 | 264.0 | 217.0 | 87.0 | 194.0 | 30.0 | 345.0 | 178.0 | 44.0 | 55.0 | 202.0 | 458.0 | 1765.0 | 44.0 | 1114.0 | 219.0 |
| 16 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 18347.0 | 0.0 |
They will choose the sequence of 17 segments that maximizes the total of these recorded pairwise gains.

When you’re ready to send back the chosen lineup, just use this simple JSON shape so it’s easy to read:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

This means ""solution"" holds the final sequence from the piece that plays first to the one that plays last — just list the segment identifiers in order. Think of it like filling out a quick form: one array, ordered exactly how you want the episode to flow. This JSON is only a sketch of the expected shape, not the actual answer.

Please use the identifiers exactly as they appear in the instance input — don’t rename them or add 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”.""","[[24967.0, 3012.0, 580.0, 448.0, 319.0, 679.0, 101.0, 2278.0, 2799.0, 2105.0, 401.0, 244.0, 822.0, 33.0, 19.0, 1752.0, 231.0], [731.0, 34044.0, 1429.0, 1876.0, 213.0, 455.0, 208.0, 430.0, 359.0, 28.0, 70.0, 284.0, 2879.0, 104.0, 13.0, 7044.0, 1652.0], [20.0, 272.0, 16171.0, 4257.0, 9.0, 8.0, 4.0, 3871.0, 412.0, 48.0, 347.0, 481.0, 1659.0, 69.0, 3.0, 772.0, 120.0], [0.0, 15.0, 13.0, 13637.0, 2.0, 2.0, 1.0, 25.0, 2410.0, 86.0, 33.0, 15.0, 50.0, 3.0, 0.0, 3341.0, 7.0], [0.0, 77.0, 0.0, 0.0, 1754.0, 470.0, 282.0, 0.0, 0.0, 69.0, 0.0, 0.0, 0.0, 0.0, 0.0, 631.0, 367.0], [6.0, 702.0, 22.0, 21.0, 424.0, 15992.0, 5.0, 13.0, 0.0, 459.0, 0.0, 12.0, 647.0, 2.0, 0.0, 1198.0, 778.0], [2.0, 14.0, 8.0, 46.0, 3.0, 1.0, 1138.0, 9.0, 37.0, 6.0, 52.0, 0.0, 46.0, 0.0, 0.0, 685.0, 44.0], [12.0, 145.0, 90.0, 49.0, 25.0, 75.0, 3.0, 3727.0, 686.0, 14.0, 46.0, 137.0, 122.0, 2192.0, 3.0, 1749.0, 91.0], [512.0, 1930.0, 696.0, 919.0, 350.0, 956.0, 51.0, 2657.0, 211.0, 82.0, 51.0, 126.0, 375.0, 55.0, 3.0, 829.0, 133.0], [52.0, 318.0, 153.0, 125.0, 80.0, 123.0, 20.0, 205.0, 45.0, 2305.0, 5.0, 82.0, 78.0, 31.0, 1.0, 420.0, 38.0], [8.0, 450.0, 289.0, 407.0, 58.0, 143.0, 113.0, 531.0, 535.0, 48.0, 80.0, 1215.0, 554.0, 12.0, 12.0, 2491.0, 158.0], [18.0, 361.0, 180.0, 247.0, 41.0, 94.0, 24.0, 670.0, 1075.0, 71.0, 1849.0, 35723.0, 724.0, 873.0, 4.0, 1414.0, 91.0], [126.0, 2347.0, 2120.0, 2238.0, 185.0, 594.0, 146.0, 9213.0, 1027.0, 108.0, 188.0, 9108.0, 9476.0, 7322.0, 33.0, 2342.0, 912.0], [1.0, 47.0, 101.0, 59.0, 11.0, 60.0, 12.0, 133.0, 96.0, 7.0, 23.0, 55.0, 516.0, 6.0, 3.0, 1578.0, 70.0], [0.0, 0.0, 26.0, 5.0, 1.0, 2.0, 0.0, 5.0, 32.0, 0.0, 2.0, 18.0, 41.0, 6.0, 0.0, 359.0, 10.0], [11.0, 329.0, 264.0, 217.0, 87.0, 194.0, 30.0, 345.0, 178.0, 44.0, 55.0, 202.0, 458.0, 1765.0, 44.0, 1114.0, 219.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 18347.0, 0.0]]","[0, 14, 12, 11, 10, 8, 5, 9, 1, 2, 7, 4, 6, 3, 16, 15, 13]",131762.0,"{'problem_type': 'LOP', 'num_nodes': 17, 'nodes': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 'pairs': [{'from_id': 0, 'to_id': 1, 'weight': 3012.0}, {'from_id': 0, 'to_id': 2, 'weight': 580.0}, {'from_id': 0, 'to_id': 3, 'weight': 448.0}, {'from_id': 0, 'to_id': 4, 'weight': 319.0}, {'from_id': 0, 'to_id': 5, 'weight': 679.0}, {'from_id': 0, 'to_id': 6, 'weight': 101.0}, {'from_id': 0, 'to_id': 7, 'weight': 2278.0}, {'from_id': 0, 'to_id': 8, 'weight': 2799.0}, {'from_id': 0, 'to_id': 9, 'weight': 2105.0}, {'from_id': 0, 'to_id': 10, 'weight': 401.0}, {'from_id': 0, 'to_id': 11, 'weight': 244.0}, {'from_id': 0, 'to_id': 12, 'weight': 822.0}, {'from_id': 0, 'to_id': 13, 'weight': 33.0}, {'from_id': 0, 'to_id': 14, 'weight': 19.0}, {'from_id': 0, 'to_id': 15, 'weight': 1752.0}, {'from_id': 0, 'to_id': 16, 'weight': 231.0}, {'from_id': 1, 'to_id': 0, 'weight': 731.0}, {'from_id': 1, 'to_id': 2, 'weight': 1429.0}, {'from_id': 1, 'to_id': 3, 'weight': 1876.0}, {'from_id': 1, 'to_id': 4, 'weight': 213.0}, {'from_id': 1, 'to_id': 5, 'weight': 455.0}, {'from_id': 1, 'to_id': 6, 'weight': 208.0}, {'from_id': 1, 'to_id': 7, 'weight': 430.0}, {'from_id': 1, 'to_id': 8, 'weight': 359.0}, {'from_id': 1, 'to_id': 9, 'weight': 28.0}, {'from_id': 1, 'to_id': 10, 'weight': 70.0}, {'from_id': 1, 'to_id': 11, 'weight': 284.0}, {'from_id': 1, 'to_id': 12, 'weight': 2879.0}, {'from_id': 1, 'to_id': 13, 'weight': 104.0}, {'from_id': 1, 'to_id': 14, 'weight': 13.0}, {'from_id': 1, 'to_id': 15, 'weight': 7044.0}, {'from_id': 1, 'to_id': 16, 'weight': 1652.0}, {'from_id': 2, 'to_id': 0, 'weight': 20.0}, {'from_id': 2, 'to_id': 1, 'weight': 272.0}, {'from_id': 2, 'to_id': 3, 'weight': 4257.0}, {'from_id': 2, 'to_id': 4, 'weight': 9.0}, {'from_id': 2, 'to_id': 5, 'weight': 8.0}, {'from_id': 2, 'to_id': 6, 'weight': 4.0}, {'from_id': 2, 'to_id': 7, 'weight': 3871.0}, {'from_id': 2, 'to_id': 8, 'weight': 412.0}, {'from_id': 2, 'to_id': 9, 'weight': 48.0}, {'from_id': 2, 'to_id': 10, 'weight': 347.0}, {'from_id': 2, 'to_id': 11, 'weight': 481.0}, {'from_id': 2, 'to_id': 12, 'weight': 1659.0}, {'from_id': 2, 'to_id': 13, 'weight': 69.0}, {'from_id': 2, 'to_id': 14, 'weight': 3.0}, {'from_id': 2, 'to_id': 15, 'weight': 772.0}, {'from_id': 2, 'to_id': 16, 'weight': 120.0}, {'from_id': 3, 'to_id': 0, 'weight': 0.0}, {'from_id': 3, 'to_id': 1, 'weight': 15.0}, {'from_id': 3, 'to_id': 2, 'weight': 13.0}, {'from_id': 3, 'to_id': 4, 'weight': 2.0}, {'from_id': 3, 'to_id': 5, 'weight': 2.0}, {'from_id': 3, 'to_id': 6, 'weight': 1.0}, {'from_id': 3, 'to_id': 7, 'weight': 25.0}, {'from_id': 3, 'to_id': 8, 'weight': 2410.0}, {'from_id': 3, 'to_id': 9, 'weight': 86.0}, {'from_id': 3, 'to_id': 10, 'weight': 33.0}, {'from_id': 3, 'to_id': 11, 'weight': 15.0}, {'from_id': 3, 'to_id': 12, 'weight': 50.0}, {'from_id': 3, 'to_id': 13, 'weight': 3.0}, {'from_id': 3, 'to_id': 14, 'weight': 0.0}, {'from_id': 3, 'to_id': 15, 'weight': 3341.0}, {'from_id': 3, 'to_id': 16, 'weight': 7.0}, {'from_id': 4, 'to_id': 0, 'weight': 0.0}, {'from_id': 4, 'to_id': 1, 'weight': 77.0}, {'from_id': 4, 'to_id': 2, 'weight': 0.0}, {'from_id': 4, 'to_id': 3, 'weight': 0.0}, {'from_id': 4, 'to_id': 5, 'weight': 470.0}, {'from_id': 4, 'to_id': 6, 'weight': 282.0}, {'from_id': 4, 'to_id': 7, 'weight': 0.0}, {'from_id': 4, 'to_id': 8, 'weight': 0.0}, {'from_id': 4, 'to_id': 9, 'weight': 69.0}, {'from_id': 4, 'to_id': 10, 'weight': 0.0}, {'from_id': 4, 'to_id': 11, 'weight': 0.0}, {'from_id': 4, 'to_id': 12, 'weight': 0.0}, {'from_id': 4, 'to_id': 13, 'weight': 0.0}, {'from_id': 4, 'to_id': 14, 'weight': 0.0}, {'from_id': 4, 'to_id': 15, 'weight': 631.0}, {'from_id': 4, 'to_id': 16, 'weight': 367.0}, {'from_id': 5, 'to_id': 0, 'weight': 6.0}, {'from_id': 5, 'to_id': 1, 'weight': 702.0}, {'from_id': 5, 'to_id': 2, 'weight': 22.0}, {'from_id': 5, 'to_id': 3, 'weight': 21.0}, {'from_id': 5, 'to_id': 4, 'weight': 424.0}, {'from_id': 5, 'to_id': 6, 'weight': 5.0}, {'from_id': 5, 'to_id': 7, 'weight': 13.0}, {'from_id': 5, 'to_id': 8, 'weight': 0.0}, {'from_id': 5, 'to_id': 9, 'weight': 459.0}, {'from_id': 5, 'to_id': 10, 'weight': 0.0}, {'from_id': 5, 'to_id': 11, 'weight': 12.0}, {'from_id': 5, 'to_id': 12, 'weight': 647.0}, {'from_id': 5, 'to_id': 13, 'weight': 2.0}, {'from_id': 5, 'to_id': 14, 'weight': 0.0}, {'from_id': 5, 'to_id': 15, 'weight': 1198.0}, {'from_id': 5, 'to_id': 16, 'weight': 778.0}, {'from_id': 6, 'to_id': 0, 'weight': 2.0}, {'from_id': 6, 'to_id': 1, 'weight': 14.0}, {'from_id': 6, 'to_id': 2, 'weight': 8.0}, {'from_id': 6, 'to_id': 3, 'weight': 46.0}, {'from_id': 6, 'to_id': 4, 'weight': 3.0}, {'from_id': 6, 'to_id': 5, 'weight': 1.0}, {'from_id': 6, 'to_id': 7, 'weight': 9.0}, {'from_id': 6, 'to_id': 8, 'weight': 37.0}, {'from_id': 6, 'to_id': 9, 'weight': 6.0}, {'from_id': 6, 'to_id': 10, 'weight': 52.0}, {'from_id': 6, 'to_id': 11, 'weight': 0.0}, {'from_id': 6, 'to_id': 12, 'weight': 46.0}, {'from_id': 6, 'to_id': 13, 'weight': 0.0}, {'from_id': 6, 'to_id': 14, 'weight': 0.0}, {'from_id': 6, 'to_id': 15, 'weight': 685.0}, {'from_id': 6, 'to_id': 16, 'weight': 44.0}, {'from_id': 7, 'to_id': 0, 'weight': 12.0}, {'from_id': 7, 'to_id': 1, 'weight': 145.0}, {'from_id': 7, 'to_id': 2, 'weight': 90.0}, {'from_id': 7, 'to_id': 3, 'weight': 49.0}, {'from_id': 7, 'to_id': 4, 'weight': 25.0}, {'from_id': 7, 'to_id': 5, 'weight': 75.0}, {'from_id': 7, 'to_id': 6, 'weight': 3.0}, {'from_id': 7, 'to_id': 8, 'weight': 686.0}, {'from_id': 7, 'to_id': 9, 'weight': 14.0}, {'from_id': 7, 'to_id': 10, 'weight': 46.0}, {'from_id': 7, 'to_id': 11, 'weight': 137.0}, {'from_id': 7, 'to_id': 12, 'weight': 122.0}, {'from_id': 7, 'to_id': 13, 'weight': 2192.0}, {'from_id': 7, 'to_id': 14, 'weight': 3.0}, {'from_id': 7, 'to_id': 15, 'weight': 1749.0}, {'from_id': 7, 'to_id': 16, 'weight': 91.0}, {'from_id': 8, 'to_id': 0, 'weight': 512.0}, {'from_id': 8, 'to_id': 1, 'weight': 1930.0}, {'from_id': 8, 'to_id': 2, 'weight': 696.0}, {'from_id': 8, 'to_id': 3, 'weight': 919.0}, {'from_id': 8, 'to_id': 4, 'weight': 350.0}, {'from_id': 8, 'to_id': 5, 'weight': 956.0}, {'from_id': 8, 'to_id': 6, 'weight': 51.0}, {'from_id': 8, 'to_id': 7, 'weight': 2657.0}, {'from_id': 8, 'to_id': 9, 'weight': 82.0}, {'from_id': 8, 'to_id': 10, 'weight': 51.0}, {'from_id': 8, 'to_id': 11, 'weight': 126.0}, {'from_id': 8, 'to_id': 12, 'weight': 375.0}, {'from_id': 8, 'to_id': 13, 'weight': 55.0}, {'from_id': 8, 'to_id': 14, 'weight': 3.0}, {'from_id': 8, 'to_id': 15, 'weight': 829.0}, {'from_id': 8, 'to_id': 16, 'weight': 133.0}, {'from_id': 9, 'to_id': 0, 'weight': 52.0}, {'from_id': 9, 'to_id': 1, 'weight': 318.0}, {'from_id': 9, 'to_id': 2, 'weight': 153.0}, {'from_id': 9, 'to_id': 3, 'weight': 125.0}, {'from_id': 9, 'to_id': 4, 'weight': 80.0}, {'from_id': 9, 'to_id': 5, 'weight': 123.0}, {'from_id': 9, 'to_id': 6, 'weight': 20.0}, {'from_id': 9, 'to_id': 7, 'weight': 205.0}, {'from_id': 9, 'to_id': 8, 'weight': 45.0}, {'from_id': 9, 'to_id': 10, 'weight': 5.0}, {'from_id': 9, 'to_id': 11, 'weight': 82.0}, {'from_id': 9, 'to_id': 12, 'weight': 78.0}, {'from_id': 9, 'to_id': 13, 'weight': 31.0}, {'from_id': 9, 'to_id': 14, 'weight': 1.0}, {'from_id': 9, 'to_id': 15, 'weight': 420.0}, {'from_id': 9, 'to_id': 16, 'weight': 38.0}, {'from_id': 10, 'to_id': 0, 'weight': 8.0}, {'from_id': 10, 'to_id': 1, 'weight': 450.0}, {'from_id': 10, 'to_id': 2, 'weight': 289.0}, {'from_id': 10, 'to_id': 3, 'weight': 407.0}, {'from_id': 10, 'to_id': 4, 'weight': 58.0}, {'from_id': 10, 'to_id': 5, 'weight': 143.0}, {'from_id': 10, 'to_id': 6, 'weight': 113.0}, {'from_id': 10, 'to_id': 7, 'weight': 531.0}, {'from_id': 10, 'to_id': 8, 'weight': 535.0}, {'from_id': 10, 'to_id': 9, 'weight': 48.0}, {'from_id': 10, 'to_id': 11, 'weight': 1215.0}, {'from_id': 10, 'to_id': 12, 'weight': 554.0}, {'from_id': 10, 'to_id': 13, 'weight': 12.0}, {'from_id': 10, 'to_id': 14, 'weight': 12.0}, {'from_id': 10, 'to_id': 15, 'weight': 2491.0}, {'from_id': 10, 'to_id': 16, 'weight': 158.0}, {'from_id': 11, 'to_id': 0, 'weight': 18.0}, {'from_id': 11, 'to_id': 1, 'weight': 361.0}, {'from_id': 11, 'to_id': 2, 'weight': 180.0}, {'from_id': 11, 'to_id': 3, 'weight': 247.0}, {'from_id': 11, 'to_id': 4, 'weight': 41.0}, {'from_id': 11, 'to_id': 5, 'weight': 94.0}, {'from_id': 11, 'to_id': 6, 'weight': 24.0}, {'from_id': 11, 'to_id': 7, 'weight': 670.0}, {'from_id': 11, 'to_id': 8, 'weight': 1075.0}, {'from_id': 11, 'to_id': 9, 'weight': 71.0}, {'from_id': 11, 'to_id': 10, 'weight': 1849.0}, {'from_id': 11, 'to_id': 12, 'weight': 724.0}, {'from_id': 11, 'to_id': 13, 'weight': 873.0}, {'from_id': 11, 'to_id': 14, 'weight': 4.0}, {'from_id': 11, 'to_id': 15, 'weight': 1414.0}, {'from_id': 11, 'to_id': 16, 'weight': 91.0}, {'from_id': 12, 'to_id': 0, 'weight': 126.0}, {'from_id': 12, 'to_id': 1, 'weight': 2347.0}, {'from_id': 12, 'to_id': 2, 'weight': 2120.0}, {'from_id': 12, 'to_id': 3, 'weight': 2238.0}, {'from_id': 12, 'to_id': 4, 'weight': 185.0}, {'from_id': 12, 'to_id': 5, 'weight': 594.0}, {'from_id': 12, 'to_id': 6, 'weight': 146.0}, {'from_id': 12, 'to_id': 7, 'weight': 9213.0}, {'from_id': 12, 'to_id': 8, 'weight': 1027.0}, {'from_id': 12, 'to_id': 9, 'weight': 108.0}, {'from_id': 12, 'to_id': 10, 'weight': 188.0}, {'from_id': 12, 'to_id': 11, 'weight': 9108.0}, {'from_id': 12, 'to_id': 13, 'weight': 7322.0}, {'from_id': 12, 'to_id': 14, 'weight': 33.0}, {'from_id': 12, 'to_id': 15, 'weight': 2342.0}, {'from_id': 12, 'to_id': 16, 'weight': 912.0}, {'from_id': 13, 'to_id': 0, 'weight': 1.0}, {'from_id': 13, 'to_id': 1, 'weight': 47.0}, {'from_id': 13, 'to_id': 2, 'weight': 101.0}, {'from_id': 13, 'to_id': 3, 'weight': 59.0}, {'from_id': 13, 'to_id': 4, 'weight': 11.0}, {'from_id': 13, 'to_id': 5, 'weight': 60.0}, {'from_id': 13, 'to_id': 6, 'weight': 12.0}, {'from_id': 13, 'to_id': 7, 'weight': 133.0}, {'from_id': 13, 'to_id': 8, 'weight': 96.0}, {'from_id': 13, 'to_id': 9, 'weight': 7.0}, {'from_id': 13, 'to_id': 10, 'weight': 23.0}, {'from_id': 13, 'to_id': 11, 'weight': 55.0}, {'from_id': 13, 'to_id': 12, 'weight': 516.0}, {'from_id': 13, 'to_id': 14, 'weight': 3.0}, {'from_id': 13, 'to_id': 15, 'weight': 1578.0}, {'from_id': 13, 'to_id': 16, 'weight': 70.0}, {'from_id': 14, 'to_id': 0, 'weight': 0.0}, {'from_id': 14, 'to_id': 1, 'weight': 0.0}, {'from_id': 14, 'to_id': 2, 'weight': 26.0}, {'from_id': 14, 'to_id': 3, 'weight': 5.0}, {'from_id': 14, 'to_id': 4, 'weight': 1.0}, {'from_id': 14, 'to_id': 5, 'weight': 2.0}, {'from_id': 14, 'to_id': 6, 'weight': 0.0}, {'from_id': 14, 'to_id': 7, 'weight': 5.0}, {'from_id': 14, 'to_id': 8, 'weight': 32.0}, {'from_id': 14, 'to_id': 9, 'weight': 0.0}, {'from_id': 14, 'to_id': 10, 'weight': 2.0}, {'from_id': 14, 'to_id': 11, 'weight': 18.0}, {'from_id': 14, 'to_id': 12, 'weight': 41.0}, {'from_id': 14, 'to_id': 13, 'weight': 6.0}, {'from_id': 14, 'to_id': 15, 'weight': 359.0}, {'from_id': 14, 'to_id': 16, 'weight': 10.0}, {'from_id': 15, 'to_id': 0, 'weight': 11.0}, {'from_id': 15, 'to_id': 1, 'weight': 329.0}, {'from_id': 15, 'to_id': 2, 'weight': 264.0}, {'from_id': 15, 'to_id': 3, 'weight': 217.0}, {'from_id': 15, 'to_id': 4, 'weight': 87.0}, {'from_id': 15, 'to_id': 5, 'weight': 194.0}, {'from_id': 15, 'to_id': 6, 'weight': 30.0}, {'from_id': 15, 'to_id': 7, 'weight': 345.0}, {'from_id': 15, 'to_id': 8, 'weight': 178.0}, {'from_id': 15, 'to_id': 9, 'weight': 44.0}, {'from_id': 15, 'to_id': 10, 'weight': 55.0}, {'from_id': 15, 'to_id': 11, 'weight': 202.0}, {'from_id': 15, 'to_id': 12, 'weight': 458.0}, {'from_id': 15, 'to_id': 13, 'weight': 1765.0}, {'from_id': 15, 'to_id': 14, 'weight': 44.0}, {'from_id': 15, 'to_id': 16, 'weight': 219.0}, {'from_id': 16, 'to_id': 0, 'weight': 0.0}, {'from_id': 16, 'to_id': 1, 'weight': 0.0}, {'from_id': 16, 'to_id': 2, 'weight': 0.0}, {'from_id': 16, 'to_id': 3, 'weight': 0.0}, {'from_id': 16, 'to_id': 4, 'weight': 0.0}, {'from_id': 16, 'to_id': 5, 'weight': 0.0}, {'from_id': 16, 'to_id': 6, 'weight': 0.0}, {'from_id': 16, 'to_id': 7, 'weight': 0.0}, {'from_id': 16, 'to_id': 8, 'weight': 0.0}, {'from_id': 16, 'to_id': 9, 'weight': 0.0}, {'from_id': 16, 'to_id': 10, 'weight': 0.0}, {'from_id': 16, 'to_id': 11, 'weight': 0.0}, {'from_id': 16, 'to_id': 12, 'weight': 0.0}, {'from_id': 16, 'to_id': 13, 'weight': 0.0}, {'from_id': 16, 'to_id': 14, 'weight': 0.0}, {'from_id': 16, 'to_id': 15, 'weight': 18347.0}]}","[0, 14, 12, 11, 10, 8, 5, 9, 1, 2, 7, 4, 6, 3, 16, 15, 13]",10,nl,0
LOP,LOP,"We found ourselves sorting through a set of class modules and needed to decide a single teaching order that uses each module exactly one time. Between every two modules there’s a benefit value that applies only if the first module comes before the second, so the overall effectiveness is just the sum of all those chosen-before benefits across every pair. The trick is picking the lineup that makes that sum as large as it can be, while making sure no topic is left out or taught twice. The concrete topic names and the benefit numbers appear below.

{
  ""total_topics"": 21,
  ""topic_identifiers"": [
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14,
    15,
    16,
    17,
    18,
    19,
    20,
    21
  ]
}

# Meaning: this is a directed pair_learning_gain matrix over items in topic_identifiers (21 total). Entry at row earlier_topic_id=i and column later_topic_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# pair_learning_gain 
| earlier_topic_id\later_topic_id | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 2 | 0.0 | 3997.0 | 0.0 | 113.0 | 64.0 | 504.0 | 44.0 | 5.0 | 13.0 | 24.0 | 156.0 | 5.0 | 6.0 | 30.0 | 342.0 | 23.0 | 658.0 | 55.0 | 191.0 | 48.0 | 0.0 |
| 3 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 4 | 0.0 | 12.0 | 0.0 | 1103.0 | 104.0 | 32.0 | 774.0 | 153.0 | 376.0 | 1.0 | 15.0 | 1.0 | 12.0 | 60.0 | 6.0 | 1.0 | 1.0 | 0.0 | 39.0 | 0.0 | 0.0 |
| 5 | 0.0 | 0.0 | 0.0 | 66.0 | 600.0 | 184.0 | 87.0 | 14.0 | 12.0 | 7.0 | 52.0 | 1.0 | 1.0 | 8.0 | 4.0 | 8.0 | 0.0 | 0.0 | 99.0 | 14.0 | 0.0 |
| 6 | 0.0 | 359.0 | 0.0 | 74.0 | 50.0 | 2650.0 | 180.0 | 51.0 | 41.0 | 23.0 | 128.0 | 53.0 | 12.0 | 45.0 | 142.0 | 17.0 | 14.0 | 25.0 | 85.0 | 31.0 | 0.0 |
| 7 | 0.0 | 12.0 | 0.0 | 56.0 | 2.0 | 21.0 | 2184.0 | 144.0 | 120.0 | 2.0 | 14.0 | 1.0 | 99.0 | 154.0 | 22.0 | 2.0 | 0.0 | 7.0 | 212.0 | 14.0 | 0.0 |
| 8 | 0.0 | 2.0 | 0.0 | 4.0 | 2.0 | 4.0 | 9.0 | 393.0 | 10.0 | 1.0 | 7.0 | 0.0 | 0.0 | 438.0 | 40.0 | 0.0 | 0.0 | 0.0 | 13.0 | 0.0 | 0.0 |
| 9 | 0.0 | 1.0 | 0.0 | 13.0 | 3.0 | 2.0 | 7.0 | 0.0 | 308.0 | 0.0 | 1.0 | 0.0 | 0.0 | 8.0 | 2.0 | 1.0 | 155.0 | 0.0 | 347.0 | 2.0 | 0.0 |
| 10 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 18.0 | 0.0 | 0.0 | 0.0 | 82.0 | 169.0 | 77.0 | 0.0 | 0.0 | 0.0 | 104.0 | 44.0 | 0.0 | 4.0 | 2.0 | 0.0 |
| 11 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 147.0 | 0.0 | 0.0 | 0.0 | 20.0 | 3190.0 | 3.0 | 0.0 | 0.0 | 0.0 | 177.0 | 27.0 | 0.0 | 15.0 | 3.0 | 0.0 |
| 12 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 2.0 | 2.0 | 5.0 | 1.0 | 0.0 | 1.0 | 146.0 | 0.0 | 7.0 | 0.0 | 0.0 | 0.0 | 0.0 | 3.0 | 1.0 | 0.0 |
| 13 | 0.0 | 6.0 | 0.0 | 0.0 | 1.0 | 10.0 | 65.0 | 4.0 | 6.0 | 0.0 | 2.0 | 1.0 | 128.0 | 2.0 | 5.0 | 2.0 | 1.0 | 1.0 | 21.0 | 12.0 | 0.0 |
| 14 | 0.0 | 23.0 | 0.0 | 240.0 | 32.0 | 23.0 | 20.0 | 2.0 | 7.0 | 15.0 | 49.0 | 4.0 | 2.0 | 8.0 | 203.0 | 2.0 | 8.0 | 30.0 | 74.0 | 1.0 | 0.0 |
| 15 | 0.0 | 22.0 | 0.0 | 89.0 | 199.0 | 248.0 | 93.0 | 44.0 | 58.0 | 119.0 | 559.0 | 26.0 | 12.0 | 83.0 | 1222.0 | 87.0 | 199.0 | 87.0 | 95.0 | 15.0 | 0.0 |
| 16 | 0.0 | 7.0 | 0.0 | 5.0 | 8.0 | 23.0 | 18.0 | 2.0 | 5.0 | 3.0 | 11.0 | 2.0 | 1.0 | 5.0 | 67.0 | 0.0 | 81.0 | 29.0 | 64.0 | 17.0 | 0.0 |
| 17 | 0.0 | 4.0 | 0.0 | 2.0 | 2.0 | 9.0 | 6.0 | 1.0 | 2.0 | 1.0 | 3.0 | 1.0 | 0.0 | 1.0 | 25.0 | 0.0 | 1095.0 | 11.0 | 27.0 | 2.0 | 0.0 |
| 18 | 0.0 | 16.0 | 0.0 | 27.0 | 13.0 | 52.0 | 33.0 | 6.0 | 17.0 | 5.0 | 35.0 | 3.0 | 3.0 | 9.0 | 132.0 | 4.0 | 24.0 | 3645.0 | 52.0 | 2.0 | 0.0 |
| 19 | 0.0 | 2.0 | 0.0 | 0.0 | 10.0 | 3.0 | 1.0 | 1.0 | 1.0 | 6.0 | 7.0 | 0.0 | 3.0 | 0.0 | 16.0 | 2.0 | 5.0 | 7.0 | 45.0 | 0.0 | 0.0 |
| 20 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 10.0 | 0.0 |
| 21 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

When you send back the chosen lineup, just use this simple JSON shape so it's easy to read and parse:

{
  ""solution"": [<first_module>, <second_module>, ..., <last_module>]
}

Here ""solution"" is just a list that shows the teaching order from the very first module to the very last one — each module appears exactly once. The angle-bracket items are placeholders showing the shape I expect; think of them like blanks on a form you fill in with the real module identifiers. This JSON is only a sketch of the expected shape, not the actual answer.

Please make sure you use the exact identifiers from 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”.""","[[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 3997.0, 0.0, 113.0, 64.0, 504.0, 44.0, 5.0, 13.0, 24.0, 156.0, 5.0, 6.0, 30.0, 342.0, 23.0, 658.0, 55.0, 191.0, 48.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 12.0, 0.0, 1103.0, 104.0, 32.0, 774.0, 153.0, 376.0, 1.0, 15.0, 1.0, 12.0, 60.0, 6.0, 1.0, 1.0, 0.0, 39.0, 0.0, 0.0], [0.0, 0.0, 0.0, 66.0, 600.0, 184.0, 87.0, 14.0, 12.0, 7.0, 52.0, 1.0, 1.0, 8.0, 4.0, 8.0, 0.0, 0.0, 99.0, 14.0, 0.0], [0.0, 359.0, 0.0, 74.0, 50.0, 2650.0, 180.0, 51.0, 41.0, 23.0, 128.0, 53.0, 12.0, 45.0, 142.0, 17.0, 14.0, 25.0, 85.0, 31.0, 0.0], [0.0, 12.0, 0.0, 56.0, 2.0, 21.0, 2184.0, 144.0, 120.0, 2.0, 14.0, 1.0, 99.0, 154.0, 22.0, 2.0, 0.0, 7.0, 212.0, 14.0, 0.0], [0.0, 2.0, 0.0, 4.0, 2.0, 4.0, 9.0, 393.0, 10.0, 1.0, 7.0, 0.0, 0.0, 438.0, 40.0, 0.0, 0.0, 0.0, 13.0, 0.0, 0.0], [0.0, 1.0, 0.0, 13.0, 3.0, 2.0, 7.0, 0.0, 308.0, 0.0, 1.0, 0.0, 0.0, 8.0, 2.0, 1.0, 155.0, 0.0, 347.0, 2.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 18.0, 0.0, 0.0, 0.0, 82.0, 169.0, 77.0, 0.0, 0.0, 0.0, 104.0, 44.0, 0.0, 4.0, 2.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 147.0, 0.0, 0.0, 0.0, 20.0, 3190.0, 3.0, 0.0, 0.0, 0.0, 177.0, 27.0, 0.0, 15.0, 3.0, 0.0], [0.0, 1.0, 0.0, 0.0, 0.0, 2.0, 2.0, 5.0, 1.0, 0.0, 1.0, 146.0, 0.0, 7.0, 0.0, 0.0, 0.0, 0.0, 3.0, 1.0, 0.0], [0.0, 6.0, 0.0, 0.0, 1.0, 10.0, 65.0, 4.0, 6.0, 0.0, 2.0, 1.0, 128.0, 2.0, 5.0, 2.0, 1.0, 1.0, 21.0, 12.0, 0.0], [0.0, 23.0, 0.0, 240.0, 32.0, 23.0, 20.0, 2.0, 7.0, 15.0, 49.0, 4.0, 2.0, 8.0, 203.0, 2.0, 8.0, 30.0, 74.0, 1.0, 0.0], [0.0, 22.0, 0.0, 89.0, 199.0, 248.0, 93.0, 44.0, 58.0, 119.0, 559.0, 26.0, 12.0, 83.0, 1222.0, 87.0, 199.0, 87.0, 95.0, 15.0, 0.0], [0.0, 7.0, 0.0, 5.0, 8.0, 23.0, 18.0, 2.0, 5.0, 3.0, 11.0, 2.0, 1.0, 5.0, 67.0, 0.0, 81.0, 29.0, 64.0, 17.0, 0.0], [0.0, 4.0, 0.0, 2.0, 2.0, 9.0, 6.0, 1.0, 2.0, 1.0, 3.0, 1.0, 0.0, 1.0, 25.0, 0.0, 1095.0, 11.0, 27.0, 2.0, 0.0], [0.0, 16.0, 0.0, 27.0, 13.0, 52.0, 33.0, 6.0, 17.0, 5.0, 35.0, 3.0, 3.0, 9.0, 132.0, 4.0, 24.0, 3645.0, 52.0, 2.0, 0.0], [0.0, 2.0, 0.0, 0.0, 10.0, 3.0, 1.0, 1.0, 1.0, 6.0, 7.0, 0.0, 3.0, 0.0, 16.0, 2.0, 5.0, 7.0, 45.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 10.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]]","[20, 1, 17, 14, 4, 5, 3, 6, 12, 7, 13, 9, 10, 15, 8, 16, 11, 19, 18, 2, 0]",10138.0,"{'problem_type': 'LOP', 'num_nodes': 21, 'nodes': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], 'pairs': [{'from_id': 1, 'to_id': 2, 'weight': 0.0}, {'from_id': 1, 'to_id': 3, 'weight': 0.0}, {'from_id': 1, 'to_id': 4, 'weight': 0.0}, {'from_id': 1, 'to_id': 5, 'weight': 0.0}, {'from_id': 1, 'to_id': 6, 'weight': 0.0}, {'from_id': 1, 'to_id': 7, 'weight': 0.0}, {'from_id': 1, 'to_id': 8, 'weight': 0.0}, {'from_id': 1, 'to_id': 9, 'weight': 0.0}, {'from_id': 1, 'to_id': 10, 'weight': 0.0}, {'from_id': 1, 'to_id': 11, 'weight': 0.0}, {'from_id': 1, 'to_id': 12, 'weight': 0.0}, {'from_id': 1, 'to_id': 13, 'weight': 0.0}, {'from_id': 1, 'to_id': 14, 'weight': 0.0}, {'from_id': 1, 'to_id': 15, 'weight': 0.0}, {'from_id': 1, 'to_id': 16, 'weight': 0.0}, {'from_id': 1, 'to_id': 17, 'weight': 0.0}, {'from_id': 1, 'to_id': 18, 'weight': 0.0}, {'from_id': 1, 'to_id': 19, 'weight': 0.0}, {'from_id': 1, 'to_id': 20, 'weight': 0.0}, {'from_id': 1, 'to_id': 21, 'weight': 0.0}, {'from_id': 2, 'to_id': 1, 'weight': 0.0}, {'from_id': 2, 'to_id': 3, 'weight': 0.0}, {'from_id': 2, 'to_id': 4, 'weight': 113.0}, {'from_id': 2, 'to_id': 5, 'weight': 64.0}, {'from_id': 2, 'to_id': 6, 'weight': 504.0}, {'from_id': 2, 'to_id': 7, 'weight': 44.0}, {'from_id': 2, 'to_id': 8, 'weight': 5.0}, {'from_id': 2, 'to_id': 9, 'weight': 13.0}, {'from_id': 2, 'to_id': 10, 'weight': 24.0}, {'from_id': 2, 'to_id': 11, 'weight': 156.0}, {'from_id': 2, 'to_id': 12, 'weight': 5.0}, {'from_id': 2, 'to_id': 13, 'weight': 6.0}, {'from_id': 2, 'to_id': 14, 'weight': 30.0}, {'from_id': 2, 'to_id': 15, 'weight': 342.0}, {'from_id': 2, 'to_id': 16, 'weight': 23.0}, {'from_id': 2, 'to_id': 17, 'weight': 658.0}, {'from_id': 2, 'to_id': 18, 'weight': 55.0}, {'from_id': 2, 'to_id': 19, 'weight': 191.0}, {'from_id': 2, 'to_id': 20, 'weight': 48.0}, {'from_id': 2, 'to_id': 21, 'weight': 0.0}, {'from_id': 3, 'to_id': 1, 'weight': 0.0}, {'from_id': 3, 'to_id': 2, 'weight': 0.0}, {'from_id': 3, 'to_id': 4, 'weight': 0.0}, {'from_id': 3, 'to_id': 5, 'weight': 0.0}, {'from_id': 3, 'to_id': 6, 'weight': 0.0}, {'from_id': 3, 'to_id': 7, 'weight': 0.0}, {'from_id': 3, 'to_id': 8, 'weight': 0.0}, {'from_id': 3, 'to_id': 9, 'weight': 0.0}, {'from_id': 3, 'to_id': 10, 'weight': 0.0}, {'from_id': 3, 'to_id': 11, 'weight': 0.0}, {'from_id': 3, 'to_id': 12, 'weight': 0.0}, {'from_id': 3, 'to_id': 13, 'weight': 0.0}, {'from_id': 3, 'to_id': 14, 'weight': 0.0}, {'from_id': 3, 'to_id': 15, 'weight': 0.0}, {'from_id': 3, 'to_id': 16, 'weight': 0.0}, {'from_id': 3, 'to_id': 17, 'weight': 0.0}, {'from_id': 3, 'to_id': 18, 'weight': 0.0}, {'from_id': 3, 'to_id': 19, 'weight': 0.0}, {'from_id': 3, 'to_id': 20, 'weight': 0.0}, {'from_id': 3, 'to_id': 21, 'weight': 0.0}, {'from_id': 4, 'to_id': 1, 'weight': 0.0}, {'from_id': 4, 'to_id': 2, 'weight': 12.0}, {'from_id': 4, 'to_id': 3, 'weight': 0.0}, {'from_id': 4, 'to_id': 5, 'weight': 104.0}, {'from_id': 4, 'to_id': 6, 'weight': 32.0}, {'from_id': 4, 'to_id': 7, 'weight': 774.0}, {'from_id': 4, 'to_id': 8, 'weight': 153.0}, {'from_id': 4, 'to_id': 9, 'weight': 376.0}, {'from_id': 4, 'to_id': 10, 'weight': 1.0}, {'from_id': 4, 'to_id': 11, 'weight': 15.0}, {'from_id': 4, 'to_id': 12, 'weight': 1.0}, {'from_id': 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15, 'weight': 203.0}, {'from_id': 14, 'to_id': 16, 'weight': 2.0}, {'from_id': 14, 'to_id': 17, 'weight': 8.0}, {'from_id': 14, 'to_id': 18, 'weight': 30.0}, {'from_id': 14, 'to_id': 19, 'weight': 74.0}, {'from_id': 14, 'to_id': 20, 'weight': 1.0}, {'from_id': 14, 'to_id': 21, 'weight': 0.0}, {'from_id': 15, 'to_id': 1, 'weight': 0.0}, {'from_id': 15, 'to_id': 2, 'weight': 22.0}, {'from_id': 15, 'to_id': 3, 'weight': 0.0}, {'from_id': 15, 'to_id': 4, 'weight': 89.0}, {'from_id': 15, 'to_id': 5, 'weight': 199.0}, {'from_id': 15, 'to_id': 6, 'weight': 248.0}, {'from_id': 15, 'to_id': 7, 'weight': 93.0}, {'from_id': 15, 'to_id': 8, 'weight': 44.0}, {'from_id': 15, 'to_id': 9, 'weight': 58.0}, {'from_id': 15, 'to_id': 10, 'weight': 119.0}, {'from_id': 15, 'to_id': 11, 'weight': 559.0}, {'from_id': 15, 'to_id': 12, 'weight': 26.0}, {'from_id': 15, 'to_id': 13, 'weight': 12.0}, {'from_id': 15, 'to_id': 14, 'weight': 83.0}, {'from_id': 15, 'to_id': 16, 'weight': 87.0}, {'from_id': 15, 'to_id': 17, 'weight': 199.0}, {'from_id': 15, 'to_id': 18, 'weight': 87.0}, {'from_id': 15, 'to_id': 19, 'weight': 95.0}, {'from_id': 15, 'to_id': 20, 'weight': 15.0}, {'from_id': 15, 'to_id': 21, 'weight': 0.0}, {'from_id': 16, 'to_id': 1, 'weight': 0.0}, {'from_id': 16, 'to_id': 2, 'weight': 7.0}, {'from_id': 16, 'to_id': 3, 'weight': 0.0}, {'from_id': 16, 'to_id': 4, 'weight': 5.0}, {'from_id': 16, 'to_id': 5, 'weight': 8.0}, {'from_id': 16, 'to_id': 6, 'weight': 23.0}, {'from_id': 16, 'to_id': 7, 'weight': 18.0}, {'from_id': 16, 'to_id': 8, 'weight': 2.0}, {'from_id': 16, 'to_id': 9, 'weight': 5.0}, {'from_id': 16, 'to_id': 10, 'weight': 3.0}, {'from_id': 16, 'to_id': 11, 'weight': 11.0}, {'from_id': 16, 'to_id': 12, 'weight': 2.0}, {'from_id': 16, 'to_id': 13, 'weight': 1.0}, {'from_id': 16, 'to_id': 14, 'weight': 5.0}, {'from_id': 16, 'to_id': 15, 'weight': 67.0}, {'from_id': 16, 'to_id': 17, 'weight': 81.0}, {'from_id': 16, 'to_id': 18, 'weight': 29.0}, {'from_id': 16, 'to_id': 19, 'weight': 64.0}, {'from_id': 16, 'to_id': 20, 'weight': 17.0}, {'from_id': 16, 'to_id': 21, 'weight': 0.0}, {'from_id': 17, 'to_id': 1, 'weight': 0.0}, {'from_id': 17, 'to_id': 2, 'weight': 4.0}, {'from_id': 17, 'to_id': 3, 'weight': 0.0}, {'from_id': 17, 'to_id': 4, 'weight': 2.0}, {'from_id': 17, 'to_id': 5, 'weight': 2.0}, {'from_id': 17, 'to_id': 6, 'weight': 9.0}, {'from_id': 17, 'to_id': 7, 'weight': 6.0}, {'from_id': 17, 'to_id': 8, 'weight': 1.0}, {'from_id': 17, 'to_id': 9, 'weight': 2.0}, {'from_id': 17, 'to_id': 10, 'weight': 1.0}, {'from_id': 17, 'to_id': 11, 'weight': 3.0}, {'from_id': 17, 'to_id': 12, 'weight': 1.0}, {'from_id': 17, 'to_id': 13, 'weight': 0.0}, {'from_id': 17, 'to_id': 14, 'weight': 1.0}, {'from_id': 17, 'to_id': 15, 'weight': 25.0}, {'from_id': 17, 'to_id': 16, 'weight': 0.0}, {'from_id': 17, 'to_id': 18, 'weight': 11.0}, {'from_id': 17, 'to_id': 19, 'weight': 27.0}, {'from_id': 17, 'to_id': 20, 'weight': 2.0}, {'from_id': 17, 'to_id': 21, 'weight': 0.0}, {'from_id': 18, 'to_id': 1, 'weight': 0.0}, {'from_id': 18, 'to_id': 2, 'weight': 16.0}, {'from_id': 18, 'to_id': 3, 'weight': 0.0}, {'from_id': 18, 'to_id': 4, 'weight': 27.0}, {'from_id': 18, 'to_id': 5, 'weight': 13.0}, {'from_id': 18, 'to_id': 6, 'weight': 52.0}, {'from_id': 18, 'to_id': 7, 'weight': 33.0}, {'from_id': 18, 'to_id': 8, 'weight': 6.0}, {'from_id': 18, 'to_id': 9, 'weight': 17.0}, {'from_id': 18, 'to_id': 10, 'weight': 5.0}, {'from_id': 18, 'to_id': 11, 'weight': 35.0}, {'from_id': 18, 'to_id': 12, 'weight': 3.0}, {'from_id': 18, 'to_id': 13, 'weight': 3.0}, {'from_id': 18, 'to_id': 14, 'weight': 9.0}, {'from_id': 18, 'to_id': 15, 'weight': 132.0}, {'from_id': 18, 'to_id': 16, 'weight': 4.0}, {'from_id': 18, 'to_id': 17, 'weight': 24.0}, {'from_id': 18, 'to_id': 19, 'weight': 52.0}, {'from_id': 18, 'to_id': 20, 'weight': 2.0}, {'from_id': 18, 'to_id': 21, 'weight': 0.0}, {'from_id': 19, 'to_id': 1, 'weight': 0.0}, {'from_id': 19, 'to_id': 2, 'weight': 2.0}, {'from_id': 19, 'to_id': 3, 'weight': 0.0}, {'from_id': 19, 'to_id': 4, 'weight': 0.0}, {'from_id': 19, 'to_id': 5, 'weight': 10.0}, {'from_id': 19, 'to_id': 6, 'weight': 3.0}, {'from_id': 19, 'to_id': 7, 'weight': 1.0}, {'from_id': 19, 'to_id': 8, 'weight': 1.0}, {'from_id': 19, 'to_id': 9, 'weight': 1.0}, {'from_id': 19, 'to_id': 10, 'weight': 6.0}, {'from_id': 19, 'to_id': 11, 'weight': 7.0}, {'from_id': 19, 'to_id': 12, 'weight': 0.0}, {'from_id': 19, 'to_id': 13, 'weight': 3.0}, {'from_id': 19, 'to_id': 14, 'weight': 0.0}, {'from_id': 19, 'to_id': 15, 'weight': 16.0}, {'from_id': 19, 'to_id': 16, 'weight': 2.0}, {'from_id': 19, 'to_id': 17, 'weight': 5.0}, {'from_id': 19, 'to_id': 18, 'weight': 7.0}, {'from_id': 19, 'to_id': 20, 'weight': 0.0}, {'from_id': 19, 'to_id': 21, 'weight': 0.0}, {'from_id': 20, 'to_id': 1, 'weight': 0.0}, {'from_id': 20, 'to_id': 2, 'weight': 0.0}, {'from_id': 20, 'to_id': 3, 'weight': 0.0}, {'from_id': 20, 'to_id': 4, 'weight': 0.0}, {'from_id': 20, 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0.0}, {'from_id': 21, 'to_id': 8, 'weight': 0.0}, {'from_id': 21, 'to_id': 9, 'weight': 0.0}, {'from_id': 21, 'to_id': 10, 'weight': 0.0}, {'from_id': 21, 'to_id': 11, 'weight': 0.0}, {'from_id': 21, 'to_id': 12, 'weight': 0.0}, {'from_id': 21, 'to_id': 13, 'weight': 0.0}, {'from_id': 21, 'to_id': 14, 'weight': 0.0}, {'from_id': 21, 'to_id': 15, 'weight': 0.0}, {'from_id': 21, 'to_id': 16, 'weight': 0.0}, {'from_id': 21, 'to_id': 17, 'weight': 0.0}, {'from_id': 21, 'to_id': 18, 'weight': 0.0}, {'from_id': 21, 'to_id': 19, 'weight': 0.0}, {'from_id': 21, 'to_id': 20, 'weight': 0.0}]}","[21, 2, 18, 15, 5, 6, 4, 7, 13, 8, 14, 10, 11, 16, 9, 17, 12, 20, 19, 3, 1]",11,json,1
LOP,LOP,"Picture this: the festival has one continuous short-film set and every possible head-to-head of films has a little preference score for which should come first. The job is to arrange every short exactly once into a single sequence, and then total all the scores for each pair where the first film is placed before the second — that total tells how well the sequence works for the audience. The full list of films and those pairwise scores is shown below.

# num_films=24
# film_ids=A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X

# Meaning: this is a directed pairwise_audience_score matrix over items in film_ids (24 total). Entry at row preceding_film_id=i and column succeeding_film_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# pairwise_audience_score
| preceding_film_id\succeeding_film_id | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 4742.0 | 11.0 | 767.0 | 0.0 | 19.0 | 8.0 | 259.0 | 0.0 | 47.0 | 0.0 | 0.0 | 258.0 | 3.0 | 25.0 | 100.0 | 58.0 | 0.0 | 0.0 | 0.0 | 168.0 | 0.0 | 53.0 | 0.0 | 0.0 |
| B | 12.0 | 0.0 | 24.0 | 0.0 | 6.0 | 13.0 | 88.0 | 49.0 | 22.0 | 108.0 | 2.0 | 0.0 | 21.0 | 11.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2.0 | 2.0 | 0.0 | 48.0 | 0.0 | 0.0 |
| C | 0.0 | 0.0 | 322.0 | 0.0 | 1.0 | 0.0 | 67.0 | 0.0 | 7.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2.0 | 0.0 | 0.0 | 4.0 | 0.0 | 0.0 | 47.0 | 0.0 | 0.0 |
| D | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 1.0 | 17.0 | 49.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 12.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 2.0 | 0.0 | 0.0 |
| E | 0.0 | 0.0 | 175.0 | 0.0 | 870.0 | 0.0 | 49.0 | 5.0 | 38.0 | 0.0 | 2.0 | 0.0 | 0.0 | 66.0 | 6.0 | 9.0 | 0.0 | 0.0 | 4.0 | 6.0 | 1.0 | 78.0 | 0.0 | 0.0 |
| F | 0.0 | 5.0 | 203.0 | 0.0 | 1.0 | 50.0 | 147.0 | 11.0 | 47.0 | 0.0 | 2.0 | 20.0 | 0.0 | 23.0 | 17.0 | 26.0 | 2.0 | 0.0 | 7.0 | 55.0 | 0.0 | 122.0 | 0.0 | 0.0 |
| G | 0.0 | 14.0 | 168.0 | 0.0 | 76.0 | 1.0 | 1901.0 | 112.0 | 536.0 | 14.0 | 7.0 | 142.0 | 11.0 | 75.0 | 43.0 | 278.0 | 29.0 | 0.0 | 6.0 | 38.0 | 5.0 | 351.0 | 0.0 | 0.0 |
| H | 0.0 | 0.0 | 3.0 | 0.0 | 2.0 | 0.0 | 86.0 | 109.0 | 619.0 | 0.0 | 0.0 | 2.0 | 7.0 | 2.0 | 2.0 | 25.0 | 41.0 | 0.0 | 0.0 | 0.0 | 0.0 | 26.0 | 0.0 | 0.0 |
| I | 0.0 | 0.0 | 62.0 | 0.0 | 214.0 | 1.0 | 780.0 | 174.0 | 14394.0 | 6.0 | 6.0 | 44.0 | 0.0 | 86.0 | 28.0 | 1053.0 | 167.0 | 0.0 | 6.0 | 15.0 | 8.0 | 385.0 | 0.0 | 0.0 |
| J | 0.0 | 0.0 | 32.0 | 0.0 | 2.0 | 0.0 | 302.0 | 10.0 | 565.0 | 1094.0 | 2.0 | 33.0 | 1.0 | 16.0 | 102.0 | 24.0 | 18.0 | 0.0 | 1.0 | 9.0 | 1.0 | 66.0 | 0.0 | 0.0 |
| K | 0.0 | 0.0 | 61.0 | 0.0 | 1.0 | 1.0 | 39.0 | 33.0 | 3.0 | 0.0 | 4598.0 | 6632.0 | 313.0 | 1.0 | 0.0 | 198.0 | 0.0 | 0.0 | 1.0 | 0.0 | 5.0 | 161.0 | 0.0 | 0.0 |
| L | 777.0 | 0.0 | 197.0 | 0.0 | 41.0 | 2.0 | 141.0 | 31.0 | 53.0 | 0.0 | 18.0 | 11253.0 | 218.0 | 8.0 | 19.0 | 84.0 | 0.0 | 0.0 | 72.0 | 7.0 | 15.0 | 243.0 | 0.0 | 0.0 |
| M | 764.0 | 0.0 | 17.0 | 0.0 | 0.0 | 0.0 | 43.0 | 0.0 | 3.0 | 0.0 | 1.0 | 193.0 | 901.0 | 0.0 | 17.0 | 26.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 13.0 | 0.0 | 0.0 |
| N | 1270.0 | 0.0 | 133.0 | 0.0 | 76.0 | 0.0 | 564.0 | 14.0 | 74.0 | 0.0 | 4.0 | 1125.0 | 115.0 | 2300.0 | 450.0 | 422.0 | 2.0 | 0.0 | 3.0 | 12.0 | 5.0 | 166.0 | 0.0 | 0.0 |
| O | 0.0 | 0.0 | 22.0 | 0.0 | 1.0 | 0.0 | 151.0 | 4.0 | 4.0 | 0.0 | 1.0 | 248.0 | 0.0 | 0.0 | 49.0 | 12.0 | 0.0 | 0.0 | 1.0 | 2.0 | 0.0 | 49.0 | 0.0 | 0.0 |
| P | 0.0 | 0.0 | 51.0 | 0.0 | 12.0 | 0.0 | 143.0 | 10.0 | 33.0 | 0.0 | 2.0 | 144.0 | 0.0 | 4.0 | 2.0 | 214.0 | 0.0 | 0.0 | 3.0 | 17.0 | 2.0 | 68.0 | 0.0 | 0.0 |
| Q | 30.0 | 0.0 | 14.0 | 0.0 | 37.0 | 0.0 | 26.0 | 2.0 | 9.0 | 0.0 | 0.0 | 127.0 | 28.0 | 27.0 | 10.0 | 233.0 | 4202.0 | 0.0 | 4.0 | 2.0 | 1.0 | 26.0 | 0.0 | 0.0 |
| R | 0.0 | 2.0 | 676.0 | 0.0 | 617.0 | 1605.0 | 3265.0 | 0.0 | 926.0 | 12.0 | 10.0 | 259.0 | 29.0 | 2799.0 | 552.0 | 2737.0 | 28.0 | 0.0 | 0.0 | 120.0 | 0.0 | 38.0 | 0.0 | 0.0 |
| S | 0.0 | 0.0 | 16.0 | 0.0 | 0.0 | 0.0 | 6.0 | 61.0 | 129.0 | 0.0 | 2.0 | 6.0 | 128.0 | 1.0 | 168.0 | 0.0 | 0.0 | 0.0 | 1.0 | 14.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| T | 0.0 | 0.0 | 202.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 3.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| U | 0.0 | 3.0 | 63.0 | 0.0 | 0.0 | 0.0 | 128.0 | 4.0 | 2.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1456.0 | 2254.0 | 0.0 | 0.0 |
| V | 486.0 | 17.0 | 814.0 | 0.0 | 22.0 | 20.0 | 29.0 | 0.0 | 0.0 | 0.0 | 8.0 | 136.0 | 40.0 | 372.0 | 20.0 | 757.0 | 53.0 | 0.0 | 0.0 | 13.0 | 9.0 | 458.0 | 0.0 | 0.0 |
| W | 120.0 | 5.0 | 258.0 | 0.0 | 8.0 | 9.0 | 104.0 | 480.0 | 183.0 | 0.0 | 8.0 | 242.0 | 12.0 | 0.0 | 157.0 | 142.0 | 0.0 | 0.0 | 0.0 | 9.0 | 0.0 | 148.0 | 0.0 | 0.0 |
| X | 42.0 | 13.0 | 808.0 | 0.0 | 14.0 | 11.0 | 335.0 | 403.0 | 1404.0 | 2781.0 | 0.0 | 155.0 | 68.0 | 28.0 | 74.0 | 12.0 | 45.0 | 45.0 | 0.0 | 38.0 | 0.0 | 609.0 | 0.0 | 0.0 |

When you're ready to give the film order back, just pop it into this little JSON layout so it's easy to read and parse:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Here ""solution"" is the single list that shows the screening order from the very first short to the very last one. The angle-bracket placeholders are just showing where each film's identifier goes — replace each placeholder with the exact film ID from the instance. This is only a sketch of the shape I want, not the actual answer itself.

Please use the identifiers exactly as they appear in the instance input — don't 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”.""","[[4742.0, 11.0, 767.0, 0.0, 19.0, 8.0, 259.0, 0.0, 47.0, 0.0, 0.0, 258.0, 3.0, 25.0, 100.0, 58.0, 0.0, 0.0, 0.0, 168.0, 0.0, 53.0, 0.0, 0.0], [12.0, 0.0, 24.0, 0.0, 6.0, 13.0, 88.0, 49.0, 22.0, 108.0, 2.0, 0.0, 21.0, 11.0, 0.0, 0.0, 0.0, 0.0, 2.0, 2.0, 0.0, 48.0, 0.0, 0.0], [0.0, 0.0, 322.0, 0.0, 1.0, 0.0, 67.0, 0.0, 7.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 4.0, 0.0, 0.0, 47.0, 0.0, 0.0], [0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 17.0, 49.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 12.0, 0.0, 0.0, 1.0, 0.0, 0.0, 2.0, 0.0, 0.0], [0.0, 0.0, 175.0, 0.0, 870.0, 0.0, 49.0, 5.0, 38.0, 0.0, 2.0, 0.0, 0.0, 66.0, 6.0, 9.0, 0.0, 0.0, 4.0, 6.0, 1.0, 78.0, 0.0, 0.0], [0.0, 5.0, 203.0, 0.0, 1.0, 50.0, 147.0, 11.0, 47.0, 0.0, 2.0, 20.0, 0.0, 23.0, 17.0, 26.0, 2.0, 0.0, 7.0, 55.0, 0.0, 122.0, 0.0, 0.0], [0.0, 14.0, 168.0, 0.0, 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0.0, 0.0, 1456.0, 2254.0, 0.0, 0.0], [486.0, 17.0, 814.0, 0.0, 22.0, 20.0, 29.0, 0.0, 0.0, 0.0, 8.0, 136.0, 40.0, 372.0, 20.0, 757.0, 53.0, 0.0, 0.0, 13.0, 9.0, 458.0, 0.0, 0.0], [120.0, 5.0, 258.0, 0.0, 8.0, 9.0, 104.0, 480.0, 183.0, 0.0, 8.0, 242.0, 12.0, 0.0, 157.0, 142.0, 0.0, 0.0, 0.0, 9.0, 0.0, 148.0, 0.0, 0.0], [42.0, 13.0, 808.0, 0.0, 14.0, 11.0, 335.0, 403.0, 1404.0, 2781.0, 0.0, 155.0, 68.0, 28.0, 74.0, 12.0, 45.0, 45.0, 0.0, 38.0, 0.0, 609.0, 0.0, 0.0]]","[23, 22, 17, 1, 9, 3, 5, 18, 10, 7, 8, 20, 21, 16, 13, 14, 11, 12, 0, 6, 15, 4, 19, 2]",51372.0,"{'problem_type': 'LOP', 'num_nodes': 24, 'nodes': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S', 'T', 'U', 'V', 'W', 'X'], 'pairs': [{'from_id': 'A', 'to_id': 'B', 'weight': 11.0}, {'from_id': 'A', 'to_id': 'C', 'weight': 767.0}, {'from_id': 'A', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'E', 'weight': 19.0}, {'from_id': 'A', 'to_id': 'F', 'weight': 8.0}, {'from_id': 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'A', 'weight': 42.0}, {'from_id': 'X', 'to_id': 'B', 'weight': 13.0}, {'from_id': 'X', 'to_id': 'C', 'weight': 808.0}, {'from_id': 'X', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'X', 'to_id': 'E', 'weight': 14.0}, {'from_id': 'X', 'to_id': 'F', 'weight': 11.0}, {'from_id': 'X', 'to_id': 'G', 'weight': 335.0}, {'from_id': 'X', 'to_id': 'H', 'weight': 403.0}, {'from_id': 'X', 'to_id': 'I', 'weight': 1404.0}, {'from_id': 'X', 'to_id': 'J', 'weight': 2781.0}, {'from_id': 'X', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'X', 'to_id': 'L', 'weight': 155.0}, {'from_id': 'X', 'to_id': 'M', 'weight': 68.0}, {'from_id': 'X', 'to_id': 'N', 'weight': 28.0}, {'from_id': 'X', 'to_id': 'O', 'weight': 74.0}, {'from_id': 'X', 'to_id': 'P', 'weight': 12.0}, {'from_id': 'X', 'to_id': 'Q', 'weight': 45.0}, {'from_id': 'X', 'to_id': 'R', 'weight': 45.0}, {'from_id': 'X', 'to_id': 'S', 'weight': 0.0}, {'from_id': 'X', 'to_id': 'T', 'weight': 38.0}, {'from_id': 'X', 'to_id': 'U', 'weight': 0.0}, {'from_id': 'X', 'to_id': 'V', 'weight': 609.0}, {'from_id': 'X', 'to_id': 'W', 'weight': 0.0}]}","['X', 'W', 'R', 'B', 'J', 'D', 'F', 'S', 'K', 'H', 'I', 'U', 'V', 'Q', 'N', 'O', 'L', 'M', 'A', 'G', 'P', 'E', 'T', 'C']",12,csv,names
LOP,LOP,"I’m putting together a single announcement that lists every new feature, and the trick is choosing the order. The plan is to list each feature once, no repeats, so that when a feature appears before another it can give a certain adoption lift — those lifts are different for each pair. The aim is to pick the sequence that makes the total adoption lift as big as it can be, by adding up the lift values for every pair where the earlier feature boosts the later one. The concrete numbers and details will be shown below.

{
  ""total_features_count"": 16,
  ""feature_identifiers"": [
    0,
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14,
    15
  ]
}

# Meaning: this is a directed adoption_lift_value matrix over items in feature_identifiers (16 total). Entry at row preceding_feature_id=i and column succeeding_feature_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# adoption_lift_value 
| preceding_feature_id\succeeding_feature_id | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 46854.0 | 0.0 | 1066115.0 | 0.0 | 27950.0 | 2141.0 | 67.0 | 49.0 | 992.0 | 0.0 | 166.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1500.0 |
| 1 | 22440.0 | 4766171.0 | 930501.0 | 0.0 | 314759.0 | 128667.0 | 12371.0 | 43991.0 | 112195.0 | 4477.0 | 6521.0 | 0.0 | 13563.0 | 14916.0 | 13082.0 | 6900.0 |
| 2 | 83987.0 | 16591.0 | 126907.0 | 0.0 | 140981.0 | 96594.0 | 12670.0 | 16916.0 | 39592.0 | 3717.0 | 5460.0 | 0.0 | 33405.0 | 52788.0 | 13514.0 | 15200.0 |
| 3 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 4 | 3477.0 | 226.0 | 0.0 | 0.0 | 295486.0 | 33568.0 | 5695.0 | 2009.0 | 53953.0 | 38.0 | 2891.0 | 0.0 | 3369.0 | 0.0 | 0.0 | 1200.0 |
| 5 | 105592.0 | 33300.0 | 70271.0 | 0.0 | 74791.0 | 1222814.0 | 43907.0 | 5849.0 | 12052.0 | 1985.0 | 30226.0 | 0.0 | 31.0 | 0.0 | 0.0 | 7400.0 |
| 6 | 313.0 | 847.0 | 3796.0 | 0.0 | 343.0 | 28991.0 | 89015.0 | 29.0 | 254.0 | 21.0 | 98.0 | 0.0 | 0.0 | 8333.0 | 9.0 | 4500.0 |
| 7 | 22.0 | 564.0 | 7.0 | 0.0 | 123.0 | 13.0 | 3.0 | 65946.0 | 25.0 | 0.0 | 120.0 | 0.0 | 0.0 | 0.0 | 0.0 | 50000.0 |
| 8 | 369.0 | 12529.0 | 3345.0 | 0.0 | 6599.0 | 2895.0 | 428.0 | 414.0 | 254099.0 | 96.0 | 997.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 9 | 0.0 | 0.0 | 3.0 | 0.0 | 0.0 | 5.0 | 1.0 | 1.0 | 6.0 | 3201.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 10 | 1.0 | 8.0 | 206.0 | 0.0 | 6423.0 | 19925.0 | 12979.0 | 0.0 | 2110.0 | 141.0 | 131993.0 | 0.0 | 1109.0 | 1777.0 | 317.0 | 2800.0 |
| 11 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 12 | 2746.0 | 1198.0 | 24564.0 | 0.0 | 10274.0 | 37604.0 | 10365.0 | 2192.0 | 802.0 | 1569.0 | 2976.0 | 0.0 | 253163.0 | 382139.0 | 54656.0 | 30700.0 |
| 13 | 41.0 | 5011.0 | 3153.0 | 0.0 | 5435.0 | 22785.0 | 3616.0 | 914.0 | 410.0 | 1243.0 | 5208.0 | 0.0 | 30373.0 | 4692408.0 | 9788.0 | 1100.0 |
| 14 | 3922.0 | 2068.0 | 8767.0 | 0.0 | 20064.0 | 28560.0 | 7665.0 | 3760.0 | 2889.0 | 899.0 | 1443.0 | 0.0 | 3194.0 | 97585.0 | 14197.0 | 2100.0 |
| 15 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

When you’re ready to send back the ordering, please use this simple JSON shape so everything’s easy to parse and check:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

""solution"" is just the list of features in the exact order you want them announced. The angle-bracket names are placeholders showing where each real feature identifier should go — think of it like filling in a form rather than writing a paragraph. This block is only a sketch of the expected shape, not the actual answer.

Please 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”.","[[46854.0, 0.0, 1066115.0, 0.0, 27950.0, 2141.0, 67.0, 49.0, 992.0, 0.0, 166.0, 0.0, 0.0, 0.0, 0.0, 1500.0], [22440.0, 4766171.0, 930501.0, 0.0, 314759.0, 128667.0, 12371.0, 43991.0, 112195.0, 4477.0, 6521.0, 0.0, 13563.0, 14916.0, 13082.0, 6900.0], [83987.0, 16591.0, 126907.0, 0.0, 140981.0, 96594.0, 12670.0, 16916.0, 39592.0, 3717.0, 5460.0, 0.0, 33405.0, 52788.0, 13514.0, 15200.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [3477.0, 226.0, 0.0, 0.0, 295486.0, 33568.0, 5695.0, 2009.0, 53953.0, 38.0, 2891.0, 0.0, 3369.0, 0.0, 0.0, 1200.0], [105592.0, 33300.0, 70271.0, 0.0, 74791.0, 1222814.0, 43907.0, 5849.0, 12052.0, 1985.0, 30226.0, 0.0, 31.0, 0.0, 0.0, 7400.0], [313.0, 847.0, 3796.0, 0.0, 343.0, 28991.0, 89015.0, 29.0, 254.0, 21.0, 98.0, 0.0, 0.0, 8333.0, 9.0, 4500.0], [22.0, 564.0, 7.0, 0.0, 123.0, 13.0, 3.0, 65946.0, 25.0, 0.0, 120.0, 0.0, 0.0, 0.0, 0.0, 50000.0], [369.0, 12529.0, 3345.0, 0.0, 6599.0, 2895.0, 428.0, 414.0, 254099.0, 96.0, 997.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 3.0, 0.0, 0.0, 5.0, 1.0, 1.0, 6.0, 3201.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0], [1.0, 8.0, 206.0, 0.0, 6423.0, 19925.0, 12979.0, 0.0, 2110.0, 141.0, 131993.0, 0.0, 1109.0, 1777.0, 317.0, 2800.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [2746.0, 1198.0, 24564.0, 0.0, 10274.0, 37604.0, 10365.0, 2192.0, 802.0, 1569.0, 2976.0, 0.0, 253163.0, 382139.0, 54656.0, 30700.0], [41.0, 5011.0, 3153.0, 0.0, 5435.0, 22785.0, 3616.0, 914.0, 410.0, 1243.0, 5208.0, 0.0, 30373.0, 4692408.0, 9788.0, 1100.0], [3922.0, 2068.0, 8767.0, 0.0, 20064.0, 28560.0, 7665.0, 3760.0, 2889.0, 899.0, 1443.0, 0.0, 3194.0, 97585.0, 14197.0, 2100.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]]","[1, 12, 14, 5, 0, 2, 13, 10, 4, 8, 6, 9, 7, 15, 11, 3]",4259623.0,"{'problem_type': 'LOP', 'num_nodes': 16, 'nodes': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], 'pairs': [{'from_id': 0, 'to_id': 1, 'weight': 0.0}, {'from_id': 0, 'to_id': 2, 'weight': 1066115.0}, {'from_id': 0, 'to_id': 3, 'weight': 0.0}, {'from_id': 0, 'to_id': 4, 'weight': 27950.0}, {'from_id': 0, 'to_id': 5, 'weight': 2141.0}, {'from_id': 0, 'to_id': 6, 'weight': 67.0}, {'from_id': 0, 'to_id': 7, 'weight': 49.0}, {'from_id': 0, 'to_id': 8, 'weight': 992.0}, {'from_id': 0, 'to_id': 9, 'weight': 0.0}, {'from_id': 0, 'to_id': 10, 'weight': 166.0}, {'from_id': 0, 'to_id': 11, 'weight': 0.0}, {'from_id': 0, 'to_id': 12, 'weight': 0.0}, {'from_id': 0, 'to_id': 13, 'weight': 0.0}, {'from_id': 0, 'to_id': 14, 'weight': 0.0}, {'from_id': 0, 'to_id': 15, 'weight': 1500.0}, {'from_id': 1, 'to_id': 0, 'weight': 22440.0}, {'from_id': 1, 'to_id': 2, 'weight': 930501.0}, {'from_id': 1, 'to_id': 3, 'weight': 0.0}, {'from_id': 1, 'to_id': 4, 'weight': 314759.0}, {'from_id': 1, 'to_id': 5, 'weight': 128667.0}, {'from_id': 1, 'to_id': 6, 'weight': 12371.0}, {'from_id': 1, 'to_id': 7, 'weight': 43991.0}, {'from_id': 1, 'to_id': 8, 'weight': 112195.0}, {'from_id': 1, 'to_id': 9, 'weight': 4477.0}, {'from_id': 1, 'to_id': 10, 'weight': 6521.0}, {'from_id': 1, 'to_id': 11, 'weight': 0.0}, {'from_id': 1, 'to_id': 12, 'weight': 13563.0}, {'from_id': 1, 'to_id': 13, 'weight': 14916.0}, {'from_id': 1, 'to_id': 14, 'weight': 13082.0}, {'from_id': 1, 'to_id': 15, 'weight': 6900.0}, {'from_id': 2, 'to_id': 0, 'weight': 83987.0}, {'from_id': 2, 'to_id': 1, 'weight': 16591.0}, {'from_id': 2, 'to_id': 3, 'weight': 0.0}, {'from_id': 2, 'to_id': 4, 'weight': 140981.0}, {'from_id': 2, 'to_id': 5, 'weight': 96594.0}, {'from_id': 2, 'to_id': 6, 'weight': 12670.0}, {'from_id': 2, 'to_id': 7, 'weight': 16916.0}, {'from_id': 2, 'to_id': 8, 'weight': 39592.0}, {'from_id': 2, 'to_id': 9, 'weight': 3717.0}, {'from_id': 2, 'to_id': 10, 'weight': 5460.0}, {'from_id': 2, 'to_id': 11, 'weight': 0.0}, {'from_id': 2, 'to_id': 12, 'weight': 33405.0}, {'from_id': 2, 'to_id': 13, 'weight': 52788.0}, {'from_id': 2, 'to_id': 14, 'weight': 13514.0}, {'from_id': 2, 'to_id': 15, 'weight': 15200.0}, {'from_id': 3, 'to_id': 0, 'weight': 0.0}, {'from_id': 3, 'to_id': 1, 'weight': 0.0}, {'from_id': 3, 'to_id': 2, 'weight': 0.0}, {'from_id': 3, 'to_id': 4, 'weight': 0.0}, {'from_id': 3, 'to_id': 5, 'weight': 0.0}, {'from_id': 3, 'to_id': 6, 'weight': 0.0}, {'from_id': 3, 'to_id': 7, 'weight': 0.0}, {'from_id': 3, 'to_id': 8, 'weight': 0.0}, {'from_id': 3, 'to_id': 9, 'weight': 0.0}, {'from_id': 3, 'to_id': 10, 'weight': 0.0}, {'from_id': 3, 'to_id': 11, 'weight': 0.0}, {'from_id': 3, 'to_id': 12, 'weight': 0.0}, {'from_id': 3, 'to_id': 13, 'weight': 0.0}, {'from_id': 3, 'to_id': 14, 'weight': 0.0}, {'from_id': 3, 'to_id': 15, 'weight': 0.0}, {'from_id': 4, 'to_id': 0, 'weight': 3477.0}, {'from_id': 4, 'to_id': 1, 'weight': 226.0}, {'from_id': 4, 'to_id': 2, 'weight': 0.0}, {'from_id': 4, 'to_id': 3, 'weight': 0.0}, {'from_id': 4, 'to_id': 5, 'weight': 33568.0}, {'from_id': 4, 'to_id': 6, 'weight': 5695.0}, {'from_id': 4, 'to_id': 7, 'weight': 2009.0}, {'from_id': 4, 'to_id': 8, 'weight': 53953.0}, {'from_id': 4, 'to_id': 9, 'weight': 38.0}, {'from_id': 4, 'to_id': 10, 'weight': 2891.0}, {'from_id': 4, 'to_id': 11, 'weight': 0.0}, {'from_id': 4, 'to_id': 12, 'weight': 3369.0}, {'from_id': 4, 'to_id': 13, 'weight': 0.0}, {'from_id': 4, 'to_id': 14, 'weight': 0.0}, {'from_id': 4, 'to_id': 15, 'weight': 1200.0}, {'from_id': 5, 'to_id': 0, 'weight': 105592.0}, {'from_id': 5, 'to_id': 1, 'weight': 33300.0}, {'from_id': 5, 'to_id': 2, 'weight': 70271.0}, {'from_id': 5, 'to_id': 3, 'weight': 0.0}, {'from_id': 5, 'to_id': 4, 'weight': 74791.0}, {'from_id': 5, 'to_id': 6, 'weight': 43907.0}, {'from_id': 5, 'to_id': 7, 'weight': 5849.0}, {'from_id': 5, 'to_id': 8, 'weight': 12052.0}, {'from_id': 5, 'to_id': 9, 'weight': 1985.0}, {'from_id': 5, 'to_id': 10, 'weight': 30226.0}, {'from_id': 5, 'to_id': 11, 'weight': 0.0}, {'from_id': 5, 'to_id': 12, 'weight': 31.0}, {'from_id': 5, 'to_id': 13, 'weight': 0.0}, {'from_id': 5, 'to_id': 14, 'weight': 0.0}, {'from_id': 5, 'to_id': 15, 'weight': 7400.0}, {'from_id': 6, 'to_id': 0, 'weight': 313.0}, {'from_id': 6, 'to_id': 1, 'weight': 847.0}, {'from_id': 6, 'to_id': 2, 'weight': 3796.0}, {'from_id': 6, 'to_id': 3, 'weight': 0.0}, {'from_id': 6, 'to_id': 4, 'weight': 343.0}, {'from_id': 6, 'to_id': 5, 'weight': 28991.0}, {'from_id': 6, 'to_id': 7, 'weight': 29.0}, {'from_id': 6, 'to_id': 8, 'weight': 254.0}, {'from_id': 6, 'to_id': 9, 'weight': 21.0}, {'from_id': 6, 'to_id': 10, 'weight': 98.0}, {'from_id': 6, 'to_id': 11, 'weight': 0.0}, {'from_id': 6, 'to_id': 12, 'weight': 0.0}, {'from_id': 6, 'to_id': 13, 'weight': 8333.0}, {'from_id': 6, 'to_id': 14, 'weight': 9.0}, {'from_id': 6, 'to_id': 15, 'weight': 4500.0}, {'from_id': 7, 'to_id': 0, 'weight': 22.0}, {'from_id': 7, 'to_id': 1, 'weight': 564.0}, {'from_id': 7, 'to_id': 2, 'weight': 7.0}, {'from_id': 7, 'to_id': 3, 'weight': 0.0}, {'from_id': 7, 'to_id': 4, 'weight': 123.0}, {'from_id': 7, 'to_id': 5, 'weight': 13.0}, {'from_id': 7, 'to_id': 6, 'weight': 3.0}, {'from_id': 7, 'to_id': 8, 'weight': 25.0}, {'from_id': 7, 'to_id': 9, 'weight': 0.0}, {'from_id': 7, 'to_id': 10, 'weight': 120.0}, {'from_id': 7, 'to_id': 11, 'weight': 0.0}, {'from_id': 7, 'to_id': 12, 'weight': 0.0}, {'from_id': 7, 'to_id': 13, 'weight': 0.0}, {'from_id': 7, 'to_id': 14, 'weight': 0.0}, {'from_id': 7, 'to_id': 15, 'weight': 50000.0}, {'from_id': 8, 'to_id': 0, 'weight': 369.0}, {'from_id': 8, 'to_id': 1, 'weight': 12529.0}, {'from_id': 8, 'to_id': 2, 'weight': 3345.0}, {'from_id': 8, 'to_id': 3, 'weight': 0.0}, {'from_id': 8, 'to_id': 4, 'weight': 6599.0}, {'from_id': 8, 'to_id': 5, 'weight': 2895.0}, {'from_id': 8, 'to_id': 6, 'weight': 428.0}, {'from_id': 8, 'to_id': 7, 'weight': 414.0}, {'from_id': 8, 'to_id': 9, 'weight': 96.0}, {'from_id': 8, 'to_id': 10, 'weight': 997.0}, {'from_id': 8, 'to_id': 11, 'weight': 0.0}, {'from_id': 8, 'to_id': 12, 'weight': 0.0}, {'from_id': 8, 'to_id': 13, 'weight': 0.0}, {'from_id': 8, 'to_id': 14, 'weight': 0.0}, {'from_id': 8, 'to_id': 15, 'weight': 0.0}, {'from_id': 9, 'to_id': 0, 'weight': 0.0}, {'from_id': 9, 'to_id': 1, 'weight': 0.0}, {'from_id': 9, 'to_id': 2, 'weight': 3.0}, {'from_id': 9, 'to_id': 3, 'weight': 0.0}, {'from_id': 9, 'to_id': 4, 'weight': 0.0}, {'from_id': 9, 'to_id': 5, 'weight': 5.0}, {'from_id': 9, 'to_id': 6, 'weight': 1.0}, {'from_id': 9, 'to_id': 7, 'weight': 1.0}, {'from_id': 9, 'to_id': 8, 'weight': 6.0}, {'from_id': 9, 'to_id': 10, 'weight': 1.0}, {'from_id': 9, 'to_id': 11, 'weight': 0.0}, {'from_id': 9, 'to_id': 12, 'weight': 0.0}, {'from_id': 9, 'to_id': 13, 'weight': 0.0}, {'from_id': 9, 'to_id': 14, 'weight': 0.0}, {'from_id': 9, 'to_id': 15, 'weight': 0.0}, {'from_id': 10, 'to_id': 0, 'weight': 1.0}, {'from_id': 10, 'to_id': 1, 'weight': 8.0}, {'from_id': 10, 'to_id': 2, 'weight': 206.0}, {'from_id': 10, 'to_id': 3, 'weight': 0.0}, {'from_id': 10, 'to_id': 4, 'weight': 6423.0}, {'from_id': 10, 'to_id': 5, 'weight': 19925.0}, {'from_id': 10, 'to_id': 6, 'weight': 12979.0}, {'from_id': 10, 'to_id': 7, 'weight': 0.0}, {'from_id': 10, 'to_id': 8, 'weight': 2110.0}, {'from_id': 10, 'to_id': 9, 'weight': 141.0}, {'from_id': 10, 'to_id': 11, 'weight': 0.0}, {'from_id': 10, 'to_id': 12, 'weight': 1109.0}, {'from_id': 10, 'to_id': 13, 'weight': 1777.0}, {'from_id': 10, 'to_id': 14, 'weight': 317.0}, {'from_id': 10, 'to_id': 15, 'weight': 2800.0}, {'from_id': 11, 'to_id': 0, 'weight': 0.0}, {'from_id': 11, 'to_id': 1, 'weight': 0.0}, {'from_id': 11, 'to_id': 2, 'weight': 0.0}, {'from_id': 11, 'to_id': 3, 'weight': 0.0}, {'from_id': 11, 'to_id': 4, 'weight': 0.0}, {'from_id': 11, 'to_id': 5, 'weight': 0.0}, {'from_id': 11, 'to_id': 6, 'weight': 0.0}, {'from_id': 11, 'to_id': 7, 'weight': 0.0}, {'from_id': 11, 'to_id': 8, 'weight': 0.0}, {'from_id': 11, 'to_id': 9, 'weight': 0.0}, {'from_id': 11, 'to_id': 10, 'weight': 0.0}, {'from_id': 11, 'to_id': 12, 'weight': 0.0}, {'from_id': 11, 'to_id': 13, 'weight': 0.0}, {'from_id': 11, 'to_id': 14, 'weight': 0.0}, {'from_id': 11, 'to_id': 15, 'weight': 0.0}, {'from_id': 12, 'to_id': 0, 'weight': 2746.0}, {'from_id': 12, 'to_id': 1, 'weight': 1198.0}, {'from_id': 12, 'to_id': 2, 'weight': 24564.0}, {'from_id': 12, 'to_id': 3, 'weight': 0.0}, {'from_id': 12, 'to_id': 4, 'weight': 10274.0}, {'from_id': 12, 'to_id': 5, 'weight': 37604.0}, {'from_id': 12, 'to_id': 6, 'weight': 10365.0}, {'from_id': 12, 'to_id': 7, 'weight': 2192.0}, {'from_id': 12, 'to_id': 8, 'weight': 802.0}, {'from_id': 12, 'to_id': 9, 'weight': 1569.0}, {'from_id': 12, 'to_id': 10, 'weight': 2976.0}, {'from_id': 12, 'to_id': 11, 'weight': 0.0}, {'from_id': 12, 'to_id': 13, 'weight': 382139.0}, {'from_id': 12, 'to_id': 14, 'weight': 54656.0}, {'from_id': 12, 'to_id': 15, 'weight': 30700.0}, {'from_id': 13, 'to_id': 0, 'weight': 41.0}, {'from_id': 13, 'to_id': 1, 'weight': 5011.0}, {'from_id': 13, 'to_id': 2, 'weight': 3153.0}, {'from_id': 13, 'to_id': 3, 'weight': 0.0}, {'from_id': 13, 'to_id': 4, 'weight': 5435.0}, {'from_id': 13, 'to_id': 5, 'weight': 22785.0}, {'from_id': 13, 'to_id': 6, 'weight': 3616.0}, {'from_id': 13, 'to_id': 7, 'weight': 914.0}, {'from_id': 13, 'to_id': 8, 'weight': 410.0}, {'from_id': 13, 'to_id': 9, 'weight': 1243.0}, {'from_id': 13, 'to_id': 10, 'weight': 5208.0}, {'from_id': 13, 'to_id': 11, 'weight': 0.0}, {'from_id': 13, 'to_id': 12, 'weight': 30373.0}, {'from_id': 13, 'to_id': 14, 'weight': 9788.0}, {'from_id': 13, 'to_id': 15, 'weight': 1100.0}, {'from_id': 14, 'to_id': 0, 'weight': 3922.0}, {'from_id': 14, 'to_id': 1, 'weight': 2068.0}, {'from_id': 14, 'to_id': 2, 'weight': 8767.0}, {'from_id': 14, 'to_id': 3, 'weight': 0.0}, {'from_id': 14, 'to_id': 4, 'weight': 20064.0}, {'from_id': 14, 'to_id': 5, 'weight': 28560.0}, {'from_id': 14, 'to_id': 6, 'weight': 7665.0}, {'from_id': 14, 'to_id': 7, 'weight': 3760.0}, {'from_id': 14, 'to_id': 8, 'weight': 2889.0}, {'from_id': 14, 'to_id': 9, 'weight': 899.0}, {'from_id': 14, 'to_id': 10, 'weight': 1443.0}, {'from_id': 14, 'to_id': 11, 'weight': 0.0}, {'from_id': 14, 'to_id': 12, 'weight': 3194.0}, {'from_id': 14, 'to_id': 13, 'weight': 97585.0}, {'from_id': 14, 'to_id': 15, 'weight': 2100.0}, {'from_id': 15, 'to_id': 0, 'weight': 0.0}, {'from_id': 15, 'to_id': 1, 'weight': 0.0}, {'from_id': 15, 'to_id': 2, 'weight': 0.0}, {'from_id': 15, 'to_id': 3, 'weight': 0.0}, {'from_id': 15, 'to_id': 4, 'weight': 0.0}, {'from_id': 15, 'to_id': 5, 'weight': 0.0}, {'from_id': 15, 'to_id': 6, 'weight': 0.0}, {'from_id': 15, 'to_id': 7, 'weight': 0.0}, {'from_id': 15, 'to_id': 8, 'weight': 0.0}, {'from_id': 15, 'to_id': 9, 'weight': 0.0}, {'from_id': 15, 'to_id': 10, 'weight': 0.0}, {'from_id': 15, 'to_id': 11, 'weight': 0.0}, {'from_id': 15, 'to_id': 12, 'weight': 0.0}, {'from_id': 15, 'to_id': 13, 'weight': 0.0}, {'from_id': 15, 'to_id': 14, 'weight': 0.0}]}","[1, 12, 14, 5, 0, 2, 13, 10, 4, 8, 6, 9, 7, 15, 11, 3]",13,json,0
LOP,LOP,"Recently the test kitchen handed over a stack of recipes and the editor’s job became arranging them into one smooth chapter. Each time Recipe A precedes Recipe B there’s a known bit of extra menu-compatibility, and the chapter’s quality is just the sum of all those bits across every ordered pair — the goal is to make that sum as large as possible. Nothing can be left out or slotted in twice, and the concrete details follow below.

# total_recipes=25
# recipe_ids=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25

# Meaning: this is a directed compatibility_gain matrix over items in recipe_ids (25 total). Entry at row prior_recipe_id=i and column subsequent_recipe_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# compatibility_gain
| prior_recipe_id\subsequent_recipe_id | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 15660.0 | 1.0 | 2.0 | 0.0 | 33.0 | 37.0 | 179.0 | 11.0 | 1.0 | 64.0 | 80.0 | 24.0 | 739.0 | 30.0 | 623.0 | 580.0 | 69.0 | 38.0 | 420.0 | 47.0 | 32.0 | 8.0 | 22.0 | 13.0 | 123.0 |
| 2 | 0.0 | 4.0 | 1997.0 | 1456.0 | 0.0 | 0.0 | 0.0 | 6.0 | 240.0 | 5.0 | 0.0 | 1.0 | 0.0 | 0.0 | 4.0 | 156.0 | 64.0 | 9.0 | 2.0 | 9.0 | 5.0 | 7.0 | 1.0 | 0.0 | 41.0 |
| 3 | 10.0 | 987.0 | 6057.0 | 0.0 | 51.0 | 19.0 | 0.0 | 31.0 | 109.0 | 164.0 | 236.0 | 13.0 | 0.0 | 0.0 | 713.0 | 208.0 | 46.0 | 144.0 | 33.0 | 82.0 | 59.0 | 37.0 | 38.0 | 0.0 | 58.0 |
| 4 | 0.0 | 14.0 | 0.0 | 183.0 | 0.0 | 0.0 | 0.0 | 0.0 | 120.0 | 13.0 | 0.0 | 0.0 | 0.0 | 0.0 | 45.0 | 22.0 | 2.0 | 0.0 | 2.0 | 5.0 | 6.0 | 1.0 | 0.0 | 0.0 | 2.0 |
| 5 | 3.0 | 53.0 | 18.0 | 0.0 | 508.0 | 35.0 | 2.0 | 11.0 | 1.0 | 15.0 | 43.0 | 7.0 | 0.0 | 0.0 | 11.0 | 237.0 | 40.0 | 84.0 | 23.0 | 155.0 | 52.0 | 17.0 | 11.0 | 0.0 | 27.0 |
| 6 | 0.0 | 92.0 | 5.0 | 0.0 | 26.0 | 20.0 | 2.0 | 1.0 | 5.0 | 1.0 | 38.0 | 14.0 | 0.0 | 0.0 | 10.0 | 38.0 | 3.0 | 2.0 | 11.0 | 20.0 | 11.0 | 5.0 | 3.0 | 0.0 | 7.0 |
| 7 | 0.0 | 142.0 | 1.0 | 0.0 | 84.0 | 25.0 | 1136.0 | 37.0 | 0.0 | 2.0 | 147.0 | 5.0 | 0.0 | 0.0 | 7.0 | 158.0 | 27.0 | 3.0 | 12.0 | 35.0 | 17.0 | 8.0 | 3.0 | 0.0 | 16.0 |
| 8 | 2.0 | 73.0 | 150.0 | 0.0 | 121.0 | 17.0 | 0.0 | 859.0 | 16.0 | 35.0 | 65.0 | 0.0 | 0.0 | 0.0 | 8.0 | 896.0 | 32.0 | 1.0 | 9.0 | 39.0 | 39.0 | 12.0 | 6.0 | 0.0 | 39.0 |
| 9 | 0.0 | 19.0 | 12.0 | 0.0 | 45.0 | 73.0 | 55.0 | 417.0 | 2277.0 | 242.0 | 30.0 | 8.0 | 0.0 | 0.0 | 30.0 | 429.0 | 71.0 | 2.0 | 44.0 | 145.0 | 72.0 | 32.0 | 19.0 | 0.0 | 50.0 |
| 10 | 12.0 | 40.0 | 32.0 | 0.0 | 43.0 | 230.0 | 690.0 | 1386.0 | 8.0 | 13073.0 | 535.0 | 122.0 | 0.0 | 0.0 | 81.0 | 968.0 | 206.0 | 4.0 | 15.0 | 809.0 | 161.0 | 100.0 | 91.0 | 0.0 | 223.0 |
| 11 | 15.0 | 26.0 | 29.0 | 0.0 | 6.0 | 102.0 | 136.0 | 2.0 | 21.0 | 94.0 | 7832.0 | 642.0 | 0.0 | 3.0 | 16.0 | 1120.0 | 149.0 | 4.0 | 82.0 | 194.0 | 84.0 | 29.0 | 21.0 | 0.0 | 98.0 |
| 12 | 51.0 | 46.0 | 50.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 4.0 | 1.0 | 93.0 | 3816.0 | 0.0 | 15.0 | 10.0 | 268.0 | 63.0 | 1.0 | 27.0 | 69.0 | 46.0 | 10.0 | 6.0 | 0.0 | 62.0 |
| 13 | 10653.0 | 7.0 | 7.0 | 0.0 | 0.0 | 0.0 | 75.0 | 0.0 | 0.0 | 0.0 | 17.0 | 3.0 | 1597.0 | 0.0 | 11.0 | 317.0 | 282.0 | 7.0 | 16.0 | 11.0 | 22.0 | 2.0 | 2.0 | 0.0 | 56.0 |
| 14 | 14111.0 | 17.0 | 23.0 | 0.0 | 2.0 | 0.0 | 80.0 | 0.0 | 1.0 | 9.0 | 9.0 | 16.0 | 0.0 | 2248.0 | 25.0 | 501.0 | 395.0 | 12.0 | 23.0 | 71.0 | 38.0 | 13.0 | 7.0 | 0.0 | 82.0 |
| 15 | 4.0 | 75.0 | 56.0 | 0.0 | 6147.0 | 321.0 | 492.0 | 690.0 | 13702.0 | 3594.0 | 10186.0 | 37.0 | 0.0 | 0.0 | 1068.0 | 3222.0 | 482.0 | 106.0 | 506.0 | 98.0 | 61.0 | 127.0 | 48.0 | 0.0 | 280.0 |
| 16 | 7.0 | 102.0 | 27.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 1.0 | 91.0 | 17.0 | 13.0 | 0.0 | 0.0 | 205.0 | 123.0 | 0.0 | 15.0 | 347.0 | 2755.0 | 514.0 | 85.0 | 54.0 | 0.0 | 233.0 |
| 17 | 4.0 | 18.0 | 3.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 7.0 | 139.0 | 0.0 | 36.0 | 0.0 | 0.0 | 74.0 | 63.0 | 1066.0 | 1.0 | 202.0 | 1099.0 | 93.0 | 85.0 | 37.0 | 0.0 | 68.0 |
| 18 | 4.0 | 3.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 32.0 | 19.0 | 2.0 | 32.0 | 45.0 | 39.0 | 14.0 | 71.0 | 60.0 | 512.0 | 50.0 | 47.0 | 20.0 | 8.0 | 5.0 | 0.0 | 29.0 |
| 19 | 0.0 | 7.0 | 3.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 167.0 | 19.0 | 22.0 | 0.0 | 0.0 | 32.0 | 26.0 | 7.0 | 0.0 | 221.0 | 79.0 | 145.0 | 18.0 | 54.0 | 0.0 | 79.0 |
| 20 | 780.0 | 50.0 | 19.0 | 0.0 | 0.0 | 49.0 | 138.0 | 0.0 | 2.0 | 75.0 | 6.0 | 50.0 | 666.0 | 1414.0 | 15.0 | 797.0 | 425.0 | 2.0 | 32.0 | 80.0 | 72.0 | 33.0 | 18.0 | 0.0 | 71.0 |
| 21 | 1.0 | 3.0 | 2.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 5.0 | 19.0 | 1.0 | 27.0 | 0.0 | 0.0 | 38.0 | 97.0 | 30.0 | 0.0 | 112.0 | 565.0 | 491.0 | 12.0 | 37.0 | 0.0 | 43.0 |
| 22 | 830.0 | 3.0 | 2.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 58.0 | 71.0 | 7.0 | 14.0 | 0.0 | 0.0 | 1421.0 | 61.0 | 27.0 | 1.0 | 240.0 | 78.0 | 19.0 | 6.0 | 186.0 | 0.0 | 1469.0 |
| 23 | 0.0 | 4.0 | 3.0 | 0.0 | 0.0 | 9.0 | 0.0 | 0.0 | 0.0 | 26.0 | 0.0 | 7.0 | 151.0 | 34.0 | 3.0 | 18.0 | 7.0 | 0.0 | 8.0 | 88.0 | 12.0 | 3.0 | 0.0 | 32.0 | 14.0 |
| 24 | 84.0 | 19.0 | 10.0 | 0.0 | 58.0 | 5.0 | 2.0 | 0.0 | 9.0 | 36.0 | 35.0 | 72.0 | 230.0 | 255.0 | 44.0 | 121.0 | 59.0 | 1.0 | 39.0 | 120.0 | 65.0 | 65.0 | 8.0 | 0.0 | 31.0 |
| 25 | 754.0 | 401.0 | 264.0 | 0.0 | 249.0 | 315.0 | 200.0 | 8.0 | 650.0 | 714.0 | 224.0 | 1246.0 | 441.0 | 1039.0 | 2201.0 | 1446.0 | 508.0 | 11.0 | 749.0 | 903.0 | 1362.0 | 1526.0 | 482.0 | 10838.0 | 10746.0 |

When you hand back the chapter order, a simple JSON reply is perfect — just a single list that runs from the first recipe to the last. Here’s the shape I’m expecting:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of ""solution"" as the ordered list of recipes in the chapter, from the one that should appear first through to the one that should appear last. This JSON is just a sketch of the shape I want you to follow, not the actual answer itself.

Please make sure to use the exact identifiers from the instance input — no renaming and no 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”.""","[[15660.0, 1.0, 2.0, 0.0, 33.0, 37.0, 179.0, 11.0, 1.0, 64.0, 80.0, 24.0, 739.0, 30.0, 623.0, 580.0, 69.0, 38.0, 420.0, 47.0, 32.0, 8.0, 22.0, 13.0, 123.0], [0.0, 4.0, 1997.0, 1456.0, 0.0, 0.0, 0.0, 6.0, 240.0, 5.0, 0.0, 1.0, 0.0, 0.0, 4.0, 156.0, 64.0, 9.0, 2.0, 9.0, 5.0, 7.0, 1.0, 0.0, 41.0], [10.0, 987.0, 6057.0, 0.0, 51.0, 19.0, 0.0, 31.0, 109.0, 164.0, 236.0, 13.0, 0.0, 0.0, 713.0, 208.0, 46.0, 144.0, 33.0, 82.0, 59.0, 37.0, 38.0, 0.0, 58.0], [0.0, 14.0, 0.0, 183.0, 0.0, 0.0, 0.0, 0.0, 120.0, 13.0, 0.0, 0.0, 0.0, 0.0, 45.0, 22.0, 2.0, 0.0, 2.0, 5.0, 6.0, 1.0, 0.0, 0.0, 2.0], [3.0, 53.0, 18.0, 0.0, 508.0, 35.0, 2.0, 11.0, 1.0, 15.0, 43.0, 7.0, 0.0, 0.0, 11.0, 237.0, 40.0, 84.0, 23.0, 155.0, 52.0, 17.0, 11.0, 0.0, 27.0], [0.0, 92.0, 5.0, 0.0, 26.0, 20.0, 2.0, 1.0, 5.0, 1.0, 38.0, 14.0, 0.0, 0.0, 10.0, 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{'from_id': 19, 'to_id': 9, 'weight': 1.0}, {'from_id': 19, 'to_id': 10, 'weight': 167.0}, {'from_id': 19, 'to_id': 11, 'weight': 19.0}, {'from_id': 19, 'to_id': 12, 'weight': 22.0}, {'from_id': 19, 'to_id': 13, 'weight': 0.0}, {'from_id': 19, 'to_id': 14, 'weight': 0.0}, {'from_id': 19, 'to_id': 15, 'weight': 32.0}, {'from_id': 19, 'to_id': 16, 'weight': 26.0}, {'from_id': 19, 'to_id': 17, 'weight': 7.0}, {'from_id': 19, 'to_id': 18, 'weight': 0.0}, {'from_id': 19, 'to_id': 20, 'weight': 79.0}, {'from_id': 19, 'to_id': 21, 'weight': 145.0}, {'from_id': 19, 'to_id': 22, 'weight': 18.0}, {'from_id': 19, 'to_id': 23, 'weight': 54.0}, {'from_id': 19, 'to_id': 24, 'weight': 0.0}, {'from_id': 19, 'to_id': 25, 'weight': 79.0}, {'from_id': 20, 'to_id': 1, 'weight': 780.0}, {'from_id': 20, 'to_id': 2, 'weight': 50.0}, {'from_id': 20, 'to_id': 3, 'weight': 19.0}, {'from_id': 20, 'to_id': 4, 'weight': 0.0}, {'from_id': 20, 'to_id': 5, 'weight': 0.0}, {'from_id': 20, 'to_id': 6, 'weight': 49.0}, {'from_id': 20, 'to_id': 7, 'weight': 138.0}, {'from_id': 20, 'to_id': 8, 'weight': 0.0}, {'from_id': 20, 'to_id': 9, 'weight': 2.0}, {'from_id': 20, 'to_id': 10, 'weight': 75.0}, {'from_id': 20, 'to_id': 11, 'weight': 6.0}, {'from_id': 20, 'to_id': 12, 'weight': 50.0}, {'from_id': 20, 'to_id': 13, 'weight': 666.0}, {'from_id': 20, 'to_id': 14, 'weight': 1414.0}, {'from_id': 20, 'to_id': 15, 'weight': 15.0}, {'from_id': 20, 'to_id': 16, 'weight': 797.0}, {'from_id': 20, 'to_id': 17, 'weight': 425.0}, {'from_id': 20, 'to_id': 18, 'weight': 2.0}, {'from_id': 20, 'to_id': 19, 'weight': 32.0}, {'from_id': 20, 'to_id': 21, 'weight': 72.0}, {'from_id': 20, 'to_id': 22, 'weight': 33.0}, {'from_id': 20, 'to_id': 23, 'weight': 18.0}, {'from_id': 20, 'to_id': 24, 'weight': 0.0}, {'from_id': 20, 'to_id': 25, 'weight': 71.0}, {'from_id': 21, 'to_id': 1, 'weight': 1.0}, {'from_id': 21, 'to_id': 2, 'weight': 3.0}, {'from_id': 21, 'to_id': 3, 'weight': 2.0}, {'from_id': 21, 'to_id': 4, 'weight': 0.0}, {'from_id': 21, 'to_id': 5, 'weight': 0.0}, {'from_id': 21, 'to_id': 6, 'weight': 0.0}, {'from_id': 21, 'to_id': 7, 'weight': 1.0}, {'from_id': 21, 'to_id': 8, 'weight': 0.0}, {'from_id': 21, 'to_id': 9, 'weight': 5.0}, {'from_id': 21, 'to_id': 10, 'weight': 19.0}, {'from_id': 21, 'to_id': 11, 'weight': 1.0}, {'from_id': 21, 'to_id': 12, 'weight': 27.0}, {'from_id': 21, 'to_id': 13, 'weight': 0.0}, {'from_id': 21, 'to_id': 14, 'weight': 0.0}, {'from_id': 21, 'to_id': 15, 'weight': 38.0}, {'from_id': 21, 'to_id': 16, 'weight': 97.0}, {'from_id': 21, 'to_id': 17, 'weight': 30.0}, {'from_id': 21, 'to_id': 18, 'weight': 0.0}, {'from_id': 21, 'to_id': 19, 'weight': 112.0}, {'from_id': 21, 'to_id': 20, 'weight': 565.0}, {'from_id': 21, 'to_id': 22, 'weight': 12.0}, {'from_id': 21, 'to_id': 23, 'weight': 37.0}, {'from_id': 21, 'to_id': 24, 'weight': 0.0}, {'from_id': 21, 'to_id': 25, 'weight': 43.0}, {'from_id': 22, 'to_id': 1, 'weight': 830.0}, {'from_id': 22, 'to_id': 2, 'weight': 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1469.0}, {'from_id': 23, 'to_id': 1, 'weight': 0.0}, {'from_id': 23, 'to_id': 2, 'weight': 4.0}, {'from_id': 23, 'to_id': 3, 'weight': 3.0}, {'from_id': 23, 'to_id': 4, 'weight': 0.0}, {'from_id': 23, 'to_id': 5, 'weight': 0.0}, {'from_id': 23, 'to_id': 6, 'weight': 9.0}, {'from_id': 23, 'to_id': 7, 'weight': 0.0}, {'from_id': 23, 'to_id': 8, 'weight': 0.0}, {'from_id': 23, 'to_id': 9, 'weight': 0.0}, {'from_id': 23, 'to_id': 10, 'weight': 26.0}, {'from_id': 23, 'to_id': 11, 'weight': 0.0}, {'from_id': 23, 'to_id': 12, 'weight': 7.0}, {'from_id': 23, 'to_id': 13, 'weight': 151.0}, {'from_id': 23, 'to_id': 14, 'weight': 34.0}, {'from_id': 23, 'to_id': 15, 'weight': 3.0}, {'from_id': 23, 'to_id': 16, 'weight': 18.0}, {'from_id': 23, 'to_id': 17, 'weight': 7.0}, {'from_id': 23, 'to_id': 18, 'weight': 0.0}, {'from_id': 23, 'to_id': 19, 'weight': 8.0}, {'from_id': 23, 'to_id': 20, 'weight': 88.0}, {'from_id': 23, 'to_id': 21, 'weight': 12.0}, {'from_id': 23, 'to_id': 22, 'weight': 3.0}, {'from_id': 23, 'to_id': 24, 'weight': 32.0}, {'from_id': 23, 'to_id': 25, 'weight': 14.0}, {'from_id': 24, 'to_id': 1, 'weight': 84.0}, {'from_id': 24, 'to_id': 2, 'weight': 19.0}, {'from_id': 24, 'to_id': 3, 'weight': 10.0}, {'from_id': 24, 'to_id': 4, 'weight': 0.0}, {'from_id': 24, 'to_id': 5, 'weight': 58.0}, {'from_id': 24, 'to_id': 6, 'weight': 5.0}, {'from_id': 24, 'to_id': 7, 'weight': 2.0}, {'from_id': 24, 'to_id': 8, 'weight': 0.0}, {'from_id': 24, 'to_id': 9, 'weight': 9.0}, {'from_id': 24, 'to_id': 10, 'weight': 36.0}, {'from_id': 24, 'to_id': 11, 'weight': 35.0}, {'from_id': 24, 'to_id': 12, 'weight': 72.0}, {'from_id': 24, 'to_id': 13, 'weight': 230.0}, {'from_id': 24, 'to_id': 14, 'weight': 255.0}, {'from_id': 24, 'to_id': 15, 'weight': 44.0}, {'from_id': 24, 'to_id': 16, 'weight': 121.0}, {'from_id': 24, 'to_id': 17, 'weight': 59.0}, {'from_id': 24, 'to_id': 18, 'weight': 1.0}, {'from_id': 24, 'to_id': 19, 'weight': 39.0}, {'from_id': 24, 'to_id': 20, 'weight': 120.0}, {'from_id': 24, 'to_id': 21, 'weight': 65.0}, {'from_id': 24, 'to_id': 22, 'weight': 65.0}, {'from_id': 24, 'to_id': 23, 'weight': 8.0}, {'from_id': 24, 'to_id': 25, 'weight': 31.0}, {'from_id': 25, 'to_id': 1, 'weight': 754.0}, {'from_id': 25, 'to_id': 2, 'weight': 401.0}, {'from_id': 25, 'to_id': 3, 'weight': 264.0}, {'from_id': 25, 'to_id': 4, 'weight': 0.0}, {'from_id': 25, 'to_id': 5, 'weight': 249.0}, {'from_id': 25, 'to_id': 6, 'weight': 315.0}, {'from_id': 25, 'to_id': 7, 'weight': 200.0}, {'from_id': 25, 'to_id': 8, 'weight': 8.0}, {'from_id': 25, 'to_id': 9, 'weight': 650.0}, {'from_id': 25, 'to_id': 10, 'weight': 714.0}, {'from_id': 25, 'to_id': 11, 'weight': 224.0}, {'from_id': 25, 'to_id': 12, 'weight': 1246.0}, {'from_id': 25, 'to_id': 13, 'weight': 441.0}, {'from_id': 25, 'to_id': 14, 'weight': 1039.0}, {'from_id': 25, 'to_id': 15, 'weight': 2201.0}, {'from_id': 25, 'to_id': 16, 'weight': 1446.0}, {'from_id': 25, 'to_id': 17, 'weight': 508.0}, {'from_id': 25, 'to_id': 18, 'weight': 11.0}, {'from_id': 25, 'to_id': 19, 'weight': 749.0}, {'from_id': 25, 'to_id': 20, 'weight': 903.0}, {'from_id': 25, 'to_id': 21, 'weight': 1362.0}, {'from_id': 25, 'to_id': 22, 'weight': 1526.0}, {'from_id': 25, 'to_id': 23, 'weight': 482.0}, {'from_id': 25, 'to_id': 24, 'weight': 10838.0}]}","[25, 24, 14, 13, 2, 4, 3, 22, 1, 15, 9, 10, 7, 8, 5, 11, 6, 18, 12, 17, 16, 19, 21, 23, 20]",14,csv,1
LOP,LOP,"There’s a tiny pressure-cooker moment before a show where the order of looks gets locked in—every dress has to appear once, no repeats, no skips. For every possible pair of different looks, putting one before the other changes the show’s momentum by a certain amount, and those changes are known ahead of time. The aim is to pick the single lineup so that, if you tally up the effect from every ordered pair (every time an outfit appears before another), the sum is as large as it can be. The exact outfits and those pairwise values are presented below.

{
  ""total_outfits"": 24,
  ""outfit_ids"": [
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14,
    15,
    16,
    17,
    18,
    19,
    20,
    21,
    22,
    23,
    24
  ]
}

# Meaning: this is a directed visual_impact_delta matrix over items in outfit_ids (24 total). Entry at row preceding_outfit_id=i and column following_outfit_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# visual_impact_delta 
| preceding_outfit_id\following_outfit_id | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 7410.0 | 6.0 | 1.0 | 588.0 | 1.0 | 11676.0 | 9061.0 | 921.0 | 971.0 | 410.0 | 1440.0 | 470.0 | 37.0 | 326.0 | 9.0 | 10.0 | 510.0 | 0.0 | 1.0 | 0.0 | 0.0 | 10.0 | 48.0 | 70.0 |
| 2 | 726.0 | 4637.0 | 374.0 | 586.0 | 353.0 | 29.0 | 312.0 | 256.0 | 21.0 | 38.0 | 183.0 | 151.0 | 19.0 | 569.0 | 63.0 | 444.0 | 170.0 | 80.0 | 78.0 | 0.0 | 0.0 | 297.0 | 293.0 | 40.0 |
| 3 | 423.0 | 481.0 | 1792.0 | 1538.0 | 700.0 | 105.0 | 342.0 | 151.0 | 21.0 | 44.0 | 294.0 | 506.0 | 40.0 | 183.0 | 123.0 | 969.0 | 191.0 | 115.0 | 133.0 | 0.0 | 42.0 | 264.0 | 148.0 | 38.0 |
| 4 | 1598.0 | 833.0 | 193.0 | 9235.0 | 554.0 | 20.0 | 447.0 | 192.0 | 34.0 | 227.0 | 635.0 | 553.0 | 161.0 | 172.0 | 155.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 527.0 | 453.0 | 56.0 |
| 5 | 1805.0 | 14.0 | 33.0 | 693.0 | 3028.0 | 12.0 | 44.0 | 22.0 | 8.0 | 7.0 | 23.0 | 95.0 | 7.0 | 1470.0 | 71.0 | 284.0 | 61.0 | 3.0 | 41.0 | 0.0 | 18.0 | 58.0 | 18.0 | 1.0 |
| 6 | 2.0 | 0.0 | 0.0 | 105.0 | 0.0 | 236.0 | 235.0 | 0.0 | 0.0 | 560.0 | 0.0 | 0.0 | 19.0 | 0.0 | 0.0 | 0.0 | 278.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 105.0 | 28.0 |
| 7 | 3870.0 | 4.0 | 20.0 | 700.0 | 32.0 | 47.0 | 5213.0 | 548.0 | 13.0 | 16.0 | 2.0 | 14.0 | 1.0 | 37.0 | 7.0 | 39.0 | 3677.0 | 0.0 | 4.0 | 0.0 | 1.0 | 15.0 | 148.0 | 85.0 |
| 8 | 47.0 | 6.0 | 8.0 | 33.0 | 12.0 | 5.0 | 265.0 | 1079.0 | 3.0 | 3.0 | 5.0 | 9.0 | 1.0 | 21.0 | 3.0 | 98.0 | 5161.0 | 1.0 | 10.0 | 0.0 | 3.0 | 16.0 | 10.0 | 1.0 |
| 9 | 4.0 | 4.0 | 8.0 | 13.0 | 11.0 | 4.0 | 11.0 | 5.0 | 1.0 | 3.0 | 4.0 | 7.0 | 1.0 | 18.0 | 3.0 | 83.0 | 2190.0 | 1.0 | 9.0 | 0.0 | 3.0 | 14.0 | 0.0 | 1.0 |
| 10 | 1.0 | 0.0 | 0.0 | 2.0 | 51.0 | 1.0 | 0.0 | 0.0 | 0.0 | 1144.0 | 42.0 | 114.0 | 19.0 | 0.0 | 14.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 12.0 | 0.0 |
| 11 | 87.0 | 16.0 | 74.0 | 34.0 | 552.0 | 3.0 | 56.0 | 78.0 | 42.0 | 72.0 | 3644.0 | 93.0 | 141.0 | 4723.0 | 206.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 5.0 | 23.0 |
| 12 | 161.0 | 117.0 | 286.0 | 1107.0 | 465.0 | 137.0 | 807.0 | 199.0 | 268.0 | 128.0 | 242.0 | 4981.0 | 71.0 | 712.0 | 100.0 | 2681.0 | 326.0 | 95.0 | 368.0 | 0.0 | 85.0 | 554.0 | 76.0 | 250.0 |
| 13 | 5.0 | 4.0 | 8.0 | 35.0 | 14.0 | 6.0 | 17.0 | 5.0 | 3.0 | 4.0 | 16.0 | 9.0 | 394.0 | 27.0 | 8.0 | 101.0 | 13.0 | 1.0 | 13.0 | 0.0 | 3.0 | 17.0 | 0.0 | 3.0 |
| 14 | 368.0 | 107.0 | 8.0 | 11.0 | 10.0 | 3.0 | 6.0 | 3.0 | 1.0 | 0.0 | 3.0 | 4.0 | 1.0 | 258.0 | 1.0 | 46.0 | 6.0 | 1.0 | 6.0 | 0.0 | 3288.0 | 8.0 | 2.0 | 135.0 |
| 15 | 29.0 | 28.0 | 53.0 | 150.0 | 83.0 | 35.0 | 71.0 | 37.0 | 17.0 | 10.0 | 35.0 | 193.0 | 14.0 | 144.0 | 21.0 | 618.0 | 57.0 | 9.0 | 87.0 | 0.0 | 22.0 | 314.0 | 0.0 | 0.0 |
| 16 | 1620.0 | 236.0 | 460.0 | 1776.0 | 1346.0 | 209.0 | 1346.0 | 471.0 | 104.0 | 295.0 | 765.0 | 1012.0 | 60.0 | 3896.0 | 229.0 | 0.0 | 2391.0 | 55.0 | 380.0 | 0.0 | 277.0 | 922.0 | 716.0 | 145.0 |
| 17 | 100.0 | 93.0 | 185.0 | 549.0 | 363.0 | 116.0 | 255.0 | 127.0 | 61.0 | 26.0 | 123.0 | 244.0 | 37.0 | 496.0 | 70.0 | 2229.0 | 1.0 | 31.0 | 303.0 | 0.0 | 76.0 | 799.0 | 1.0 | 0.0 |
| 18 | 149.0 | 8.0 | 58.0 | 116.0 | 270.0 | 48.0 | 104.0 | 51.0 | 12.0 | 63.0 | 149.0 | 152.0 | 20.0 | 512.0 | 30.0 | 2046.0 | 245.0 | 64.0 | 356.0 | 0.0 | 7.0 | 321.0 | 28.0 | 54.0 |
| 19 | 121.0 | 45.0 | 74.0 | 162.0 | 151.0 | 39.0 | 86.0 | 50.0 | 35.0 | 19.0 | 59.0 | 88.0 | 13.0 | 280.0 | 22.0 | 736.0 | 71.0 | 40.0 | 9259.0 | 0.0 | 92.0 | 158.0 | 0.0 | 33.0 |
| 20 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 21 | 16.0 | 16.0 | 30.0 | 89.0 | 49.0 | 20.0 | 43.0 | 22.0 | 10.0 | 6.0 | 21.0 | 36.0 | 5.0 | 84.0 | 11.0 | 364.0 | 34.0 | 5.0 | 49.0 | 0.0 | 12.0 | 65.0 | 0.0 | 0.0 |
| 22 | 108.0 | 90.0 | 161.0 | 519.0 | 343.0 | 109.0 | 232.0 | 116.0 | 55.0 | 27.0 | 115.0 | 931.0 | 35.0 | 3192.0 | 73.0 | 2209.0 | 192.0 | 42.0 | 1945.0 | 0.0 | 2686.0 | 1717.0 | 144.0 | 139.0 |
| 23 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 24 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

If you want to hand me the final lineup in a neat, machine-friendly way, a tiny JSON snippet works nicely. Just follow this shape:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of ""solution"" as the full run order: the first placeholder is the very first look on the runway, the second is the one after that, and so on down to the last. This block is just a sketch of the shape I expect — you'll replace those placeholders with the actual outfit identifiers from the instance when you send the answer.

Please 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”.","[[7410.0, 6.0, 1.0, 588.0, 1.0, 11676.0, 9061.0, 921.0, 971.0, 410.0, 1440.0, 470.0, 37.0, 326.0, 9.0, 10.0, 510.0, 0.0, 1.0, 0.0, 0.0, 10.0, 48.0, 70.0], [726.0, 4637.0, 374.0, 586.0, 353.0, 29.0, 312.0, 256.0, 21.0, 38.0, 183.0, 151.0, 19.0, 569.0, 63.0, 444.0, 170.0, 80.0, 78.0, 0.0, 0.0, 297.0, 293.0, 40.0], [423.0, 481.0, 1792.0, 1538.0, 700.0, 105.0, 342.0, 151.0, 21.0, 44.0, 294.0, 506.0, 40.0, 183.0, 123.0, 969.0, 191.0, 115.0, 133.0, 0.0, 42.0, 264.0, 148.0, 38.0], [1598.0, 833.0, 193.0, 9235.0, 554.0, 20.0, 447.0, 192.0, 34.0, 227.0, 635.0, 553.0, 161.0, 172.0, 155.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 527.0, 453.0, 56.0], [1805.0, 14.0, 33.0, 693.0, 3028.0, 12.0, 44.0, 22.0, 8.0, 7.0, 23.0, 95.0, 7.0, 1470.0, 71.0, 284.0, 61.0, 3.0, 41.0, 0.0, 18.0, 58.0, 18.0, 1.0], [2.0, 0.0, 0.0, 105.0, 0.0, 236.0, 235.0, 0.0, 0.0, 560.0, 0.0, 0.0, 19.0, 0.0, 0.0, 0.0, 278.0, 0.0, 0.0, 0.0, 0.0, 0.0, 105.0, 28.0], [3870.0, 4.0, 20.0, 700.0, 32.0, 47.0, 5213.0, 548.0, 13.0, 16.0, 2.0, 14.0, 1.0, 37.0, 7.0, 39.0, 3677.0, 0.0, 4.0, 0.0, 1.0, 15.0, 148.0, 85.0], [47.0, 6.0, 8.0, 33.0, 12.0, 5.0, 265.0, 1079.0, 3.0, 3.0, 5.0, 9.0, 1.0, 21.0, 3.0, 98.0, 5161.0, 1.0, 10.0, 0.0, 3.0, 16.0, 10.0, 1.0], [4.0, 4.0, 8.0, 13.0, 11.0, 4.0, 11.0, 5.0, 1.0, 3.0, 4.0, 7.0, 1.0, 18.0, 3.0, 83.0, 2190.0, 1.0, 9.0, 0.0, 3.0, 14.0, 0.0, 1.0], [1.0, 0.0, 0.0, 2.0, 51.0, 1.0, 0.0, 0.0, 0.0, 1144.0, 42.0, 114.0, 19.0, 0.0, 14.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 12.0, 0.0], [87.0, 16.0, 74.0, 34.0, 552.0, 3.0, 56.0, 78.0, 42.0, 72.0, 3644.0, 93.0, 141.0, 4723.0, 206.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 5.0, 23.0], [161.0, 117.0, 286.0, 1107.0, 465.0, 137.0, 807.0, 199.0, 268.0, 128.0, 242.0, 4981.0, 71.0, 712.0, 100.0, 2681.0, 326.0, 95.0, 368.0, 0.0, 85.0, 554.0, 76.0, 250.0], [5.0, 4.0, 8.0, 35.0, 14.0, 6.0, 17.0, 5.0, 3.0, 4.0, 16.0, 9.0, 394.0, 27.0, 8.0, 101.0, 13.0, 1.0, 13.0, 0.0, 3.0, 17.0, 0.0, 3.0], [368.0, 107.0, 8.0, 11.0, 10.0, 3.0, 6.0, 3.0, 1.0, 0.0, 3.0, 4.0, 1.0, 258.0, 1.0, 46.0, 6.0, 1.0, 6.0, 0.0, 3288.0, 8.0, 2.0, 135.0], [29.0, 28.0, 53.0, 150.0, 83.0, 35.0, 71.0, 37.0, 17.0, 10.0, 35.0, 193.0, 14.0, 144.0, 21.0, 618.0, 57.0, 9.0, 87.0, 0.0, 22.0, 314.0, 0.0, 0.0], [1620.0, 236.0, 460.0, 1776.0, 1346.0, 209.0, 1346.0, 471.0, 104.0, 295.0, 765.0, 1012.0, 60.0, 3896.0, 229.0, 0.0, 2391.0, 55.0, 380.0, 0.0, 277.0, 922.0, 716.0, 145.0], [100.0, 93.0, 185.0, 549.0, 363.0, 116.0, 255.0, 127.0, 61.0, 26.0, 123.0, 244.0, 37.0, 496.0, 70.0, 2229.0, 1.0, 31.0, 303.0, 0.0, 76.0, 799.0, 1.0, 0.0], [149.0, 8.0, 58.0, 116.0, 270.0, 48.0, 104.0, 51.0, 12.0, 63.0, 149.0, 152.0, 20.0, 512.0, 30.0, 2046.0, 245.0, 64.0, 356.0, 0.0, 7.0, 321.0, 28.0, 54.0], [121.0, 45.0, 74.0, 162.0, 151.0, 39.0, 86.0, 50.0, 35.0, 19.0, 59.0, 88.0, 13.0, 280.0, 22.0, 736.0, 71.0, 40.0, 9259.0, 0.0, 92.0, 158.0, 0.0, 33.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [16.0, 16.0, 30.0, 89.0, 49.0, 20.0, 43.0, 22.0, 10.0, 6.0, 21.0, 36.0, 5.0, 84.0, 11.0, 364.0, 34.0, 5.0, 49.0, 0.0, 12.0, 65.0, 0.0, 0.0], [108.0, 90.0, 161.0, 519.0, 343.0, 109.0, 232.0, 116.0, 55.0, 27.0, 115.0, 931.0, 35.0, 3192.0, 73.0, 2209.0, 192.0, 42.0, 1945.0, 0.0, 2686.0, 1717.0, 144.0, 139.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]]","[2, 1, 17, 14, 21, 11, 18, 15, 4, 3, 0, 10, 8, 5, 6, 7, 16, 9, 12, 13, 23, 22, 20, 19]",112848.0,"{'problem_type': 'LOP', 'num_nodes': 24, 'nodes': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], 'pairs': [{'from_id': 1, 'to_id': 2, 'weight': 6.0}, {'from_id': 1, 'to_id': 3, 'weight': 1.0}, {'from_id': 1, 'to_id': 4, 'weight': 588.0}, {'from_id': 1, 'to_id': 5, 'weight': 1.0}, {'from_id': 1, 'to_id': 6, 'weight': 11676.0}, {'from_id': 1, 'to_id': 7, 'weight': 9061.0}, {'from_id': 1, 'to_id': 8, 'weight': 921.0}, {'from_id': 1, 'to_id': 9, 'weight': 971.0}, {'from_id': 1, 'to_id': 10, 'weight': 410.0}, {'from_id': 1, 'to_id': 11, 'weight': 1440.0}, {'from_id': 1, 'to_id': 12, 'weight': 470.0}, {'from_id': 1, 'to_id': 13, 'weight': 37.0}, {'from_id': 1, 'to_id': 14, 'weight': 326.0}, {'from_id': 1, 'to_id': 15, 'weight': 9.0}, {'from_id': 1, 'to_id': 16, 'weight': 10.0}, {'from_id': 1, 'to_id': 17, 'weight': 510.0}, {'from_id': 1, 'to_id': 18, 'weight': 0.0}, {'from_id': 1, 'to_id': 19, 'weight': 1.0}, {'from_id': 1, 'to_id': 20, 'weight': 0.0}, {'from_id': 1, 'to_id': 21, 'weight': 0.0}, {'from_id': 1, 'to_id': 22, 'weight': 10.0}, {'from_id': 1, 'to_id': 23, 'weight': 48.0}, {'from_id': 1, 'to_id': 24, 'weight': 70.0}, {'from_id': 2, 'to_id': 1, 'weight': 726.0}, {'from_id': 2, 'to_id': 3, 'weight': 374.0}, {'from_id': 2, 'to_id': 4, 'weight': 586.0}, {'from_id': 2, 'to_id': 5, 'weight': 353.0}, {'from_id': 2, 'to_id': 6, 'weight': 29.0}, {'from_id': 2, 'to_id': 7, 'weight': 312.0}, {'from_id': 2, 'to_id': 8, 'weight': 256.0}, {'from_id': 2, 'to_id': 9, 'weight': 21.0}, {'from_id': 2, 'to_id': 10, 'weight': 38.0}, {'from_id': 2, 'to_id': 11, 'weight': 183.0}, {'from_id': 2, 'to_id': 12, 'weight': 151.0}, {'from_id': 2, 'to_id': 13, 'weight': 19.0}, {'from_id': 2, 'to_id': 14, 'weight': 569.0}, {'from_id': 2, 'to_id': 15, 'weight': 63.0}, {'from_id': 2, 'to_id': 16, 'weight': 444.0}, {'from_id': 2, 'to_id': 17, 'weight': 170.0}, {'from_id': 2, 'to_id': 18, 'weight': 80.0}, {'from_id': 2, 'to_id': 19, 'weight': 78.0}, {'from_id': 2, 'to_id': 20, 'weight': 0.0}, {'from_id': 2, 'to_id': 21, 'weight': 0.0}, {'from_id': 2, 'to_id': 22, 'weight': 297.0}, {'from_id': 2, 'to_id': 23, 'weight': 293.0}, {'from_id': 2, 'to_id': 24, 'weight': 40.0}, {'from_id': 3, 'to_id': 1, 'weight': 423.0}, {'from_id': 3, 'to_id': 2, 'weight': 481.0}, {'from_id': 3, 'to_id': 4, 'weight': 1538.0}, {'from_id': 3, 'to_id': 5, 'weight': 700.0}, {'from_id': 3, 'to_id': 6, 'weight': 105.0}, {'from_id': 3, 'to_id': 7, 'weight': 342.0}, {'from_id': 3, 'to_id': 8, 'weight': 151.0}, {'from_id': 3, 'to_id': 9, 'weight': 21.0}, {'from_id': 3, 'to_id': 10, 'weight': 44.0}, {'from_id': 3, 'to_id': 11, 'weight': 294.0}, {'from_id': 3, 'to_id': 12, 'weight': 506.0}, {'from_id': 3, 'to_id': 13, 'weight': 40.0}, {'from_id': 3, 'to_id': 14, 'weight': 183.0}, {'from_id': 3, 'to_id': 15, 'weight': 123.0}, {'from_id': 3, 'to_id': 16, 'weight': 969.0}, {'from_id': 3, 'to_id': 17, 'weight': 191.0}, {'from_id': 3, 'to_id': 18, 'weight': 115.0}, {'from_id': 3, 'to_id': 19, 'weight': 133.0}, {'from_id': 3, 'to_id': 20, 'weight': 0.0}, {'from_id': 3, 'to_id': 21, 'weight': 42.0}, {'from_id': 3, 'to_id': 22, 'weight': 264.0}, {'from_id': 3, 'to_id': 23, 'weight': 148.0}, {'from_id': 3, 'to_id': 24, 'weight': 38.0}, {'from_id': 4, 'to_id': 1, 'weight': 1598.0}, {'from_id': 4, 'to_id': 2, 'weight': 833.0}, {'from_id': 4, 'to_id': 3, 'weight': 193.0}, {'from_id': 4, 'to_id': 5, 'weight': 554.0}, {'from_id': 4, 'to_id': 6, 'weight': 20.0}, {'from_id': 4, 'to_id': 7, 'weight': 447.0}, {'from_id': 4, 'to_id': 8, 'weight': 192.0}, {'from_id': 4, 'to_id': 9, 'weight': 34.0}, {'from_id': 4, 'to_id': 10, 'weight': 227.0}, {'from_id': 4, 'to_id': 11, 'weight': 635.0}, {'from_id': 4, 'to_id': 12, 'weight': 553.0}, {'from_id': 4, 'to_id': 13, 'weight': 161.0}, {'from_id': 4, 'to_id': 14, 'weight': 172.0}, {'from_id': 4, 'to_id': 15, 'weight': 155.0}, {'from_id': 4, 'to_id': 16, 'weight': 0.0}, {'from_id': 4, 'to_id': 17, 'weight': 0.0}, {'from_id': 4, 'to_id': 18, 'weight': 0.0}, {'from_id': 4, 'to_id': 19, 'weight': 0.0}, {'from_id': 4, 'to_id': 20, 'weight': 0.0}, {'from_id': 4, 'to_id': 21, 'weight': 0.0}, {'from_id': 4, 'to_id': 22, 'weight': 527.0}, 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21, 'to_id': 6, 'weight': 20.0}, {'from_id': 21, 'to_id': 7, 'weight': 43.0}, {'from_id': 21, 'to_id': 8, 'weight': 22.0}, {'from_id': 21, 'to_id': 9, 'weight': 10.0}, {'from_id': 21, 'to_id': 10, 'weight': 6.0}, {'from_id': 21, 'to_id': 11, 'weight': 21.0}, {'from_id': 21, 'to_id': 12, 'weight': 36.0}, {'from_id': 21, 'to_id': 13, 'weight': 5.0}, {'from_id': 21, 'to_id': 14, 'weight': 84.0}, {'from_id': 21, 'to_id': 15, 'weight': 11.0}, {'from_id': 21, 'to_id': 16, 'weight': 364.0}, {'from_id': 21, 'to_id': 17, 'weight': 34.0}, {'from_id': 21, 'to_id': 18, 'weight': 5.0}, {'from_id': 21, 'to_id': 19, 'weight': 49.0}, {'from_id': 21, 'to_id': 20, 'weight': 0.0}, {'from_id': 21, 'to_id': 22, 'weight': 65.0}, {'from_id': 21, 'to_id': 23, 'weight': 0.0}, {'from_id': 21, 'to_id': 24, 'weight': 0.0}, {'from_id': 22, 'to_id': 1, 'weight': 108.0}, {'from_id': 22, 'to_id': 2, 'weight': 90.0}, {'from_id': 22, 'to_id': 3, 'weight': 161.0}, {'from_id': 22, 'to_id': 4, 'weight': 519.0}, 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{'from_id': 24, 'to_id': 3, 'weight': 0.0}, {'from_id': 24, 'to_id': 4, 'weight': 0.0}, {'from_id': 24, 'to_id': 5, 'weight': 0.0}, {'from_id': 24, 'to_id': 6, 'weight': 0.0}, {'from_id': 24, 'to_id': 7, 'weight': 0.0}, {'from_id': 24, 'to_id': 8, 'weight': 0.0}, {'from_id': 24, 'to_id': 9, 'weight': 0.0}, {'from_id': 24, 'to_id': 10, 'weight': 0.0}, {'from_id': 24, 'to_id': 11, 'weight': 0.0}, {'from_id': 24, 'to_id': 12, 'weight': 0.0}, {'from_id': 24, 'to_id': 13, 'weight': 0.0}, {'from_id': 24, 'to_id': 14, 'weight': 0.0}, {'from_id': 24, 'to_id': 15, 'weight': 0.0}, {'from_id': 24, 'to_id': 16, 'weight': 0.0}, {'from_id': 24, 'to_id': 17, 'weight': 0.0}, {'from_id': 24, 'to_id': 18, 'weight': 0.0}, {'from_id': 24, 'to_id': 19, 'weight': 0.0}, {'from_id': 24, 'to_id': 20, 'weight': 0.0}, {'from_id': 24, 'to_id': 21, 'weight': 0.0}, {'from_id': 24, 'to_id': 22, 'weight': 0.0}, {'from_id': 24, 'to_id': 23, 'weight': 0.0}]}","[3, 2, 18, 15, 22, 12, 19, 16, 5, 4, 1, 11, 9, 6, 7, 8, 17, 10, 13, 14, 24, 23, 21, 20]",15,json,1
LOP,LOP,"On a busy conference morning the schedule has to be set so each panel appears once and only once, and the lineup is judged by how well earlier panels help later ones hold the audience. For each possible pair there’s a number saying how much audience retention improves when that specific session comes first, and the best lineup is the one with the highest total when all those pairwise numbers are added together. The concrete list of sessions and their pairwise retention values are shown below.

# num_sessions=20
# session_ids=A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T

# Meaning: this is a directed retention_gain matrix over items in session_ids (20 total). Entry at row preceding_session_id=i and column following_session_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# retention_gain
| preceding_session_id\following_session_id | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 130.0 | 12.0 | 67.0 | 147.0 | 48.0 | 0.0 | 705.0 | 20.0 | 11.0 | 0.0 | 6.0 | 17.0 | 10.0 | 9.0 | 1042.0 | 287.0 | 77.0 | 53.0 | 578.0 | 93.0 |
| B | 4.0 | 169.0 | 864.0 | 0.0 | 11.0 | 1.0 | 2.0 | 2.0 | 3.0 | 0.0 | 2.0 | 6.0 | 16.0 | 1.0 | 23.0 | 45.0 | 100.0 | 17.0 | 11.0 | 7.0 |
| C | 30.0 | 126.0 | 296.0 | 0.0 | 101.0 | 41.0 | 170.0 | 227.0 | 17.0 | 0.0 | 0.0 | 7.0 | 16.0 | 1.0 | 131.0 | 125.0 | 220.0 | 39.0 | 38.0 | 16.0 |
| D | 0.0 | 1.0 | 26.0 | 2.0 | 2.0 | 2.0 | 21.0 | 2.0 | 0.0 | 0.0 | 0.0 | 2.0 | 1.0 | 0.0 | 1.0 | 2.0 | 3.0 | 3.0 | 2.0 | 2.0 |
| E | 8.0 | 10.0 | 265.0 | 0.0 | 478.0 | 6.0 | 27.0 | 26.0 | 7.0 | 0.0 | 13.0 | 69.0 | 7.0 | 1.0 | 79.0 | 194.0 | 95.0 | 23.0 | 20.0 | 10.0 |
| F | 6.0 | 28.0 | 71.0 | 0.0 | 27.0 | 78.0 | 73.0 | 0.0 | 2.0 | 0.0 | 0.0 | 2.0 | 9.0 | 1.0 | 27.0 | 95.0 | 32.0 | 13.0 | 8.0 | 4.0 |
| G | 130.0 | 194.0 | 594.0 | 94.0 | 98.0 | 27.0 | 2311.0 | 17.0 | 80.0 | 0.0 | 11.0 | 171.0 | 27.0 | 3.0 | 357.0 | 532.0 | 357.0 | 67.0 | 41.0 | 33.0 |
| H | 860.0 | 2.0 | 5.0 | 0.0 | 0.0 | 0.0 | 39.0 | 126.0 | 5.0 | 0.0 | 0.0 | 2.0 | 6.0 | 0.0 | 36.0 | 109.0 | 83.0 | 11.0 | 6.0 | 4.0 |
| I | 133.0 | 3.0 | 93.0 | 0.0 | 31.0 | 0.0 | 79.0 | 144.0 | 326.0 | 0.0 | 0.0 | 21.0 | 35.0 | 0.0 | 39.0 | 108.0 | 31.0 | 14.0 | 6.0 | 4.0 |
| J | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| K | 4.0 | 4.0 | 20.0 | 0.0 | 64.0 | 2.0 | 50.0 | 0.0 | 7.0 | 0.0 | 46.0 | 6.0 | 8.0 | 1.0 | 19.0 | 44.0 | 4.0 | 4.0 | 3.0 | 2.0 |
| L | 3.0 | 2.0 | 7.0 | 0.0 | 8.0 | 0.0 | 50.0 | 5.0 | 699.0 | 0.0 | 1.0 | 31.0 | 30.0 | 0.0 | 17.0 | 48.0 | 8.0 | 8.0 | 7.0 | 5.0 |
| M | 5.0 | 3.0 | 6.0 | 0.0 | 1.0 | 0.0 | 87.0 | 4.0 | 635.0 | 0.0 | 1.0 | 35.0 | 60.0 | 1.0 | 39.0 | 57.0 | 10.0 | 11.0 | 10.0 | 6.0 |
| N | 262.0 | 9.0 | 32.0 | 1.0 | 1.0 | 0.0 | 3.0 | 0.0 | 5.0 | 0.0 | 3.0 | 3.0 | 6.0 | 226.0 | 32.0 | 42.0 | 7.0 | 12.0 | 20.0 | 12.0 |
| O | 5348.0 | 105.0 | 132.0 | 1.0 | 190.0 | 110.0 | 278.0 | 250.0 | 237.0 | 0.0 | 22.0 | 49.0 | 41.0 | 34.0 | 4294.0 | 743.0 | 130.0 | 161.0 | 186.0 | 92.0 |
| P | 129.0 | 26.0 | 16.0 | 1.0 | 54.0 | 19.0 | 89.0 | 15.0 | 46.0 | 0.0 | 5.0 | 93.0 | 71.0 | 7.0 | 551.0 | 996.0 | 384.0 | 545.0 | 242.0 | 48.0 |
| Q | 6.0 | 20.0 | 504.0 | 0.0 | 15.0 | 16.0 | 28.0 | 52.0 | 6.0 | 0.0 | 3.0 | 7.0 | 8.0 | 0.0 | 33.0 | 48.0 | 3.0 | 11.0 | 42.0 | 18.0 |
| R | 0.0 | 2.0 | 13.0 | 0.0 | 2.0 | 1.0 | 11.0 | 2.0 | 2.0 | 0.0 | 3.0 | 6.0 | 14.0 | 0.0 | 12.0 | 25.0 | 144.0 | 0.0 | 18.0 | 19.0 |
| S | 307.0 | 10.0 | 10.0 | 2.0 | 0.0 | 0.0 | 25.0 | 0.0 | 11.0 | 0.0 | 4.0 | 29.0 | 133.0 | 1.0 | 110.0 | 45.0 | 15.0 | 153.0 | 568.0 | 17.0 |
| T | 161.0 | 204.0 | 297.0 | 16.0 | 32.0 | 6.0 | 521.0 | 40.0 | 12.0 | 2.0 | 17.0 | 67.0 | 174.0 | 19.0 | 768.0 | 558.0 | 361.0 | 371.0 | 377.0 | 580.0 |

When you’re ready to give the final lineup, just use this simple JSON shape so it’s easy to check and plug in:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of ""solution"" as the single ordered list of sessions from the one you want first to the one you want last. Each entry in that list should be the exact session identifier from the instance (not a description, not a new nickname) — it’s just a straight sequence of the session IDs in the order you prefer. This JSON is just a sketch of the shape I’m expecting, not the actual answer itself.

Please make sure all identifiers are used exactly as they appear in the instance input — no renaming and no 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”.""","[[130.0, 12.0, 67.0, 147.0, 48.0, 0.0, 705.0, 20.0, 11.0, 0.0, 6.0, 17.0, 10.0, 9.0, 1042.0, 287.0, 77.0, 53.0, 578.0, 93.0], [4.0, 169.0, 864.0, 0.0, 11.0, 1.0, 2.0, 2.0, 3.0, 0.0, 2.0, 6.0, 16.0, 1.0, 23.0, 45.0, 100.0, 17.0, 11.0, 7.0], [30.0, 126.0, 296.0, 0.0, 101.0, 41.0, 170.0, 227.0, 17.0, 0.0, 0.0, 7.0, 16.0, 1.0, 131.0, 125.0, 220.0, 39.0, 38.0, 16.0], [0.0, 1.0, 26.0, 2.0, 2.0, 2.0, 21.0, 2.0, 0.0, 0.0, 0.0, 2.0, 1.0, 0.0, 1.0, 2.0, 3.0, 3.0, 2.0, 2.0], [8.0, 10.0, 265.0, 0.0, 478.0, 6.0, 27.0, 26.0, 7.0, 0.0, 13.0, 69.0, 7.0, 1.0, 79.0, 194.0, 95.0, 23.0, 20.0, 10.0], [6.0, 28.0, 71.0, 0.0, 27.0, 78.0, 73.0, 0.0, 2.0, 0.0, 0.0, 2.0, 9.0, 1.0, 27.0, 95.0, 32.0, 13.0, 8.0, 4.0], [130.0, 194.0, 594.0, 94.0, 98.0, 27.0, 2311.0, 17.0, 80.0, 0.0, 11.0, 171.0, 27.0, 3.0, 357.0, 532.0, 357.0, 67.0, 41.0, 33.0], [860.0, 2.0, 5.0, 0.0, 0.0, 0.0, 39.0, 126.0, 5.0, 0.0, 0.0, 2.0, 6.0, 0.0, 36.0, 109.0, 83.0, 11.0, 6.0, 4.0], [133.0, 3.0, 93.0, 0.0, 31.0, 0.0, 79.0, 144.0, 326.0, 0.0, 0.0, 21.0, 35.0, 0.0, 39.0, 108.0, 31.0, 14.0, 6.0, 4.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [4.0, 4.0, 20.0, 0.0, 64.0, 2.0, 50.0, 0.0, 7.0, 0.0, 46.0, 6.0, 8.0, 1.0, 19.0, 44.0, 4.0, 4.0, 3.0, 2.0], [3.0, 2.0, 7.0, 0.0, 8.0, 0.0, 50.0, 5.0, 699.0, 0.0, 1.0, 31.0, 30.0, 0.0, 17.0, 48.0, 8.0, 8.0, 7.0, 5.0], [5.0, 3.0, 6.0, 0.0, 1.0, 0.0, 87.0, 4.0, 635.0, 0.0, 1.0, 35.0, 60.0, 1.0, 39.0, 57.0, 10.0, 11.0, 10.0, 6.0], [262.0, 9.0, 32.0, 1.0, 1.0, 0.0, 3.0, 0.0, 5.0, 0.0, 3.0, 3.0, 6.0, 226.0, 32.0, 42.0, 7.0, 12.0, 20.0, 12.0], [5348.0, 105.0, 132.0, 1.0, 190.0, 110.0, 278.0, 250.0, 237.0, 0.0, 22.0, 49.0, 41.0, 34.0, 4294.0, 743.0, 130.0, 161.0, 186.0, 92.0], [129.0, 26.0, 16.0, 1.0, 54.0, 19.0, 89.0, 15.0, 46.0, 0.0, 5.0, 93.0, 71.0, 7.0, 551.0, 996.0, 384.0, 545.0, 242.0, 48.0], [6.0, 20.0, 504.0, 0.0, 15.0, 16.0, 28.0, 52.0, 6.0, 0.0, 3.0, 7.0, 8.0, 0.0, 33.0, 48.0, 3.0, 11.0, 42.0, 18.0], [0.0, 2.0, 13.0, 0.0, 2.0, 1.0, 11.0, 2.0, 2.0, 0.0, 3.0, 6.0, 14.0, 0.0, 12.0, 25.0, 144.0, 0.0, 18.0, 19.0], [307.0, 10.0, 10.0, 2.0, 0.0, 0.0, 25.0, 0.0, 11.0, 0.0, 4.0, 29.0, 133.0, 1.0, 110.0, 45.0, 15.0, 153.0, 568.0, 17.0], [161.0, 204.0, 297.0, 16.0, 32.0, 6.0, 521.0, 40.0, 12.0, 2.0, 17.0, 67.0, 174.0, 19.0, 768.0, 558.0, 361.0, 371.0, 377.0, 580.0]]","[19, 14, 5, 13, 7, 0, 10, 6, 3, 1, 4, 15, 18, 12, 11, 8, 17, 16, 9, 2]",24169.0,"{'problem_type': 'LOP', 'num_nodes': 20, 'nodes': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S', 'T'], 'pairs': [{'from_id': 'A', 'to_id': 'B', 'weight': 12.0}, {'from_id': 'A', 'to_id': 'C', 'weight': 67.0}, {'from_id': 'A', 'to_id': 'D', 'weight': 147.0}, {'from_id': 'A', 'to_id': 'E', 'weight': 48.0}, {'from_id': 'A', 'to_id': 'F', 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'Q', 'weight': 144.0}, {'from_id': 'R', 'to_id': 'S', 'weight': 18.0}, {'from_id': 'R', 'to_id': 'T', 'weight': 19.0}, {'from_id': 'S', 'to_id': 'A', 'weight': 307.0}, {'from_id': 'S', 'to_id': 'B', 'weight': 10.0}, {'from_id': 'S', 'to_id': 'C', 'weight': 10.0}, {'from_id': 'S', 'to_id': 'D', 'weight': 2.0}, {'from_id': 'S', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'G', 'weight': 25.0}, {'from_id': 'S', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'I', 'weight': 11.0}, {'from_id': 'S', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'K', 'weight': 4.0}, {'from_id': 'S', 'to_id': 'L', 'weight': 29.0}, {'from_id': 'S', 'to_id': 'M', 'weight': 133.0}, {'from_id': 'S', 'to_id': 'N', 'weight': 1.0}, {'from_id': 'S', 'to_id': 'O', 'weight': 110.0}, {'from_id': 'S', 'to_id': 'P', 'weight': 45.0}, {'from_id': 'S', 'to_id': 'Q', 'weight': 15.0}, {'from_id': 'S', 'to_id': 'R', 'weight': 153.0}, {'from_id': 'S', 'to_id': 'T', 'weight': 17.0}, {'from_id': 'T', 'to_id': 'A', 'weight': 161.0}, {'from_id': 'T', 'to_id': 'B', 'weight': 204.0}, {'from_id': 'T', 'to_id': 'C', 'weight': 297.0}, {'from_id': 'T', 'to_id': 'D', 'weight': 16.0}, {'from_id': 'T', 'to_id': 'E', 'weight': 32.0}, {'from_id': 'T', 'to_id': 'F', 'weight': 6.0}, {'from_id': 'T', 'to_id': 'G', 'weight': 521.0}, {'from_id': 'T', 'to_id': 'H', 'weight': 40.0}, {'from_id': 'T', 'to_id': 'I', 'weight': 12.0}, {'from_id': 'T', 'to_id': 'J', 'weight': 2.0}, {'from_id': 'T', 'to_id': 'K', 'weight': 17.0}, {'from_id': 'T', 'to_id': 'L', 'weight': 67.0}, {'from_id': 'T', 'to_id': 'M', 'weight': 174.0}, {'from_id': 'T', 'to_id': 'N', 'weight': 19.0}, {'from_id': 'T', 'to_id': 'O', 'weight': 768.0}, {'from_id': 'T', 'to_id': 'P', 'weight': 558.0}, {'from_id': 'T', 'to_id': 'Q', 'weight': 361.0}, {'from_id': 'T', 'to_id': 'R', 'weight': 371.0}, {'from_id': 'T', 'to_id': 'S', 'weight': 377.0}]}","['T', 'O', 'F', 'N', 'H', 'A', 'K', 'G', 'D', 'B', 'E', 'P', 'S', 'M', 'L', 'I', 'R', 'Q', 'J', 'C']",16,csv,names
LOP,LOP,"Recently the team had to stitch together one continuous presentation from a pile of product pitches, and the rule was simple: every product gets exactly one spot, nothing left out, nothing repeated. Each time a product precedes another there’s a known persuasion boost for that ordering, and the whole show is judged by adding up all those boosts — the goal is to make that total as big as it can be. The detailed product names and their pairwise boosts are listed below.

{
  ""total_products"": 23,
  ""product_ids"": [
    0,
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14,
    15,
    16,
    17,
    18,
    19,
    20,
    21,
    22
  ]
}

# Meaning: this is a directed persuasion_boost matrix over items in product_ids (23 total). Entry at row preceding_product=i and column succeeding_product=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# persuasion_boost 
| preceding_product\succeeding_product | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 3274.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 48.0 | 3.0 | 0.0 | 6.0 | 27.0 | 2.0 | 1.0 | 11.0 | 242.0 | 2.0 | 3.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 |
| 1 | 0.0 | 0.0 | 0.0 | 654.0 | 0.0 | 101.0 | 0.0 | 0.0 | 0.0 | 0.0 | 14.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 2 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 3 | 307.0 | 0.0 | 0.0 | 55.0 | 53.0 | 126.0 | 113.0 | 84.0 | 9.0 | 6.0 | 80.0 | 13.0 | 29.0 | 344.0 | 96.0 | 55.0 | 60.0 | 271.0 | 0.0 | 25.0 | 137.0 | 0.0 | 2.0 |
| 4 | 0.0 | 0.0 | 0.0 | 0.0 | 451.0 | 148.0 | 41.0 | 1367.0 | 26.0 | 0.0 | 36.0 | 78.0 | 2.0 | 17.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 23.0 | 0.0 | 0.0 | 0.0 |
| 5 | 24.0 | 0.0 | 0.0 | 4.0 | 68.0 | 892.0 | 192.0 | 41.0 | 3.0 | 0.0 | 8.0 | 8.0 | 16.0 | 62.0 | 5.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 |
| 6 | 1954.0 | 0.0 | 0.0 | 10.0 | 19.0 | 160.0 | 1908.0 | 190.0 | 32.0 | 58.0 | 196.0 | 97.0 | 76.0 | 83.0 | 16.0 | 12.0 | 17.0 | 3.0 | 0.0 | 15.0 | 159.0 | 0.0 | 1.0 |
| 7 | 21.0 | 0.0 | 0.0 | 21.0 | 63.0 | 96.0 | 158.0 | 715.0 | 32.0 | 17.0 | 94.0 | 35.0 | 67.0 | 158.0 | 2.0 | 9.0 | 7.0 | 1.0 | 0.0 | 2.0 | 11.0 | 0.0 | 0.0 |
| 8 | 4.0 | 0.0 | 0.0 | 5.0 | 2.0 | 6.0 | 5.0 | 16.0 | 101.0 | 0.0 | 51.0 | 3.0 | 10.0 | 24.0 | 1.0 | 4.0 | 5.0 | 1.0 | 0.0 | 26.0 | 54.0 | 0.0 | 0.0 |
| 9 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 147.0 | 1.0 | 7.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 8.0 | 0.0 | 0.0 |
| 10 | 74.0 | 0.0 | 0.0 | 24.0 | 5.0 | 147.0 | 247.0 | 100.0 | 26.0 | 26.0 | 3757.0 | 85.0 | 42.0 | 1176.0 | 40.0 | 174.0 | 1397.0 | 38.0 | 1.0 | 21.0 | 130.0 | 0.0 | 3.0 |
| 11 | 2.0 | 0.0 | 0.0 | 2.0 | 0.0 | 3.0 | 5.0 | 3.0 | 3.0 | 4.0 | 5.0 | 29.0 | 1.0 | 21.0 | 1.0 | 3.0 | 5.0 | 1.0 | 0.0 | 1.0 | 11.0 | 0.0 | 0.0 |
| 12 | 171.0 | 0.0 | 0.0 | 5.0 | 1.0 | 10.0 | 7.0 | 10.0 | 2.0 | 1.0 | 10.0 | 2.0 | 19.0 | 618.0 | 7.0 | 6.0 | 13.0 | 0.0 | 16.0 | 6.0 | 131.0 | 0.0 | 1.0 |
| 13 | 1748.0 | 0.0 | 0.0 | 115.0 | 162.0 | 293.0 | 180.0 | 454.0 | 55.0 | 32.0 | 307.0 | 85.0 | 1058.0 | 485.0 | 397.0 | 31.0 | 44.0 | 7.0 | 10.0 | 24.0 | 136.0 | 0.0 | 1.0 |
| 14 | 49.0 | 0.0 | 0.0 | 17.0 | 2.0 | 35.0 | 57.0 | 27.0 | 11.0 | 2.0 | 57.0 | 6.0 | 21.0 | 205.0 | 36.0 | 190.0 | 171.0 | 31.0 | 1.0 | 13.0 | 58.0 | 0.0 | 2.0 |
| 15 | 97.0 | 0.0 | 0.0 | 10.0 | 2.0 | 18.0 | 26.0 | 15.0 | 5.0 | 2.0 | 30.0 | 3.0 | 12.0 | 195.0 | 20.0 | 7909.0 | 73.0 | 174.0 | 2.0 | 7.0 | 38.0 | 0.0 | 1.0 |
| 16 | 195.0 | 0.0 | 0.0 | 66.0 | 9.0 | 139.0 | 226.0 | 105.0 | 44.0 | 9.0 | 250.0 | 23.0 | 85.0 | 828.0 | 144.0 | 750.0 | 727.0 | 135.0 | 3.0 | 53.0 | 235.0 | 0.0 | 9.0 |
| 17 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 18 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 2.0 | 0.0 | 2.0 | 2.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 |
| 19 | 182.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 122.0 | 0.0 | 0.0 | 0.0 |
| 20 | 32.0 | 0.0 | 0.0 | 16.0 | 1.0 | 23.0 | 38.0 | 18.0 | 7.0 | 1.0 | 73.0 | 4.0 | 14.0 | 136.0 | 112.0 | 125.0 | 194.0 | 541.0 | 1.0 | 26.0 | 237.0 | 0.0 | 4.0 |
| 21 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 22 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

If you want to hand your final order back in a tidy way, just drop it into this little JSON shape so it's easy to read and parse:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of ""solution"" as the single, ordered playlist — from the very first pitch to the one that closes the show. Each placeholder in the array stands for one of the items (use the exact item labels from the instance). This JSON is just the expected shape — a sketch of how I’d like to receive the ordering, not the actual answer itself.

Please use the exact identifiers from the input — don’t 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”.""","[[3274.0, 0.0, 0.0, 0.0, 0.0, 1.0, 48.0, 3.0, 0.0, 6.0, 27.0, 2.0, 1.0, 11.0, 242.0, 2.0, 3.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0], [0.0, 0.0, 0.0, 654.0, 0.0, 101.0, 0.0, 0.0, 0.0, 0.0, 14.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [307.0, 0.0, 0.0, 55.0, 53.0, 126.0, 113.0, 84.0, 9.0, 6.0, 80.0, 13.0, 29.0, 344.0, 96.0, 55.0, 60.0, 271.0, 0.0, 25.0, 137.0, 0.0, 2.0], [0.0, 0.0, 0.0, 0.0, 451.0, 148.0, 41.0, 1367.0, 26.0, 0.0, 36.0, 78.0, 2.0, 17.0, 0.0, 0.0, 0.0, 0.0, 0.0, 23.0, 0.0, 0.0, 0.0], [24.0, 0.0, 0.0, 4.0, 68.0, 892.0, 192.0, 41.0, 3.0, 0.0, 8.0, 8.0, 16.0, 62.0, 5.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0], [1954.0, 0.0, 0.0, 10.0, 19.0, 160.0, 1908.0, 190.0, 32.0, 58.0, 196.0, 97.0, 76.0, 83.0, 16.0, 12.0, 17.0, 3.0, 0.0, 15.0, 159.0, 0.0, 1.0], [21.0, 0.0, 0.0, 21.0, 63.0, 96.0, 158.0, 715.0, 32.0, 17.0, 94.0, 35.0, 67.0, 158.0, 2.0, 9.0, 7.0, 1.0, 0.0, 2.0, 11.0, 0.0, 0.0], [4.0, 0.0, 0.0, 5.0, 2.0, 6.0, 5.0, 16.0, 101.0, 0.0, 51.0, 3.0, 10.0, 24.0, 1.0, 4.0, 5.0, 1.0, 0.0, 26.0, 54.0, 0.0, 0.0], [1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 147.0, 1.0, 7.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 8.0, 0.0, 0.0], [74.0, 0.0, 0.0, 24.0, 5.0, 147.0, 247.0, 100.0, 26.0, 26.0, 3757.0, 85.0, 42.0, 1176.0, 40.0, 174.0, 1397.0, 38.0, 1.0, 21.0, 130.0, 0.0, 3.0], [2.0, 0.0, 0.0, 2.0, 0.0, 3.0, 5.0, 3.0, 3.0, 4.0, 5.0, 29.0, 1.0, 21.0, 1.0, 3.0, 5.0, 1.0, 0.0, 1.0, 11.0, 0.0, 0.0], [171.0, 0.0, 0.0, 5.0, 1.0, 10.0, 7.0, 10.0, 2.0, 1.0, 10.0, 2.0, 19.0, 618.0, 7.0, 6.0, 13.0, 0.0, 16.0, 6.0, 131.0, 0.0, 1.0], [1748.0, 0.0, 0.0, 115.0, 162.0, 293.0, 180.0, 454.0, 55.0, 32.0, 307.0, 85.0, 1058.0, 485.0, 397.0, 31.0, 44.0, 7.0, 10.0, 24.0, 136.0, 0.0, 1.0], [49.0, 0.0, 0.0, 17.0, 2.0, 35.0, 57.0, 27.0, 11.0, 2.0, 57.0, 6.0, 21.0, 205.0, 36.0, 190.0, 171.0, 31.0, 1.0, 13.0, 58.0, 0.0, 2.0], [97.0, 0.0, 0.0, 10.0, 2.0, 18.0, 26.0, 15.0, 5.0, 2.0, 30.0, 3.0, 12.0, 195.0, 20.0, 7909.0, 73.0, 174.0, 2.0, 7.0, 38.0, 0.0, 1.0], [195.0, 0.0, 0.0, 66.0, 9.0, 139.0, 226.0, 105.0, 44.0, 9.0, 250.0, 23.0, 85.0, 828.0, 144.0, 750.0, 727.0, 135.0, 3.0, 53.0, 235.0, 0.0, 9.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 2.0, 0.0, 2.0, 2.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0], [182.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 122.0, 0.0, 0.0, 0.0], [32.0, 0.0, 0.0, 16.0, 1.0, 23.0, 38.0, 18.0, 7.0, 1.0, 73.0, 4.0, 14.0, 136.0, 112.0, 125.0, 194.0, 541.0, 1.0, 26.0, 237.0, 0.0, 4.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 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'weight': 0.0}]}","[1, 3, 10, 16, 15, 13, 4, 6, 7, 8, 12, 20, 19, 0, 14, 22, 21, 18, 9, 5, 11, 17, 2]",17,json,0
LOP,LOP,"A friend who edits anthologies described the job like arranging people at a dinner: everyone gets one seat, no one sits twice, and the conversation carries better depending on who sits next to whom and who’s before whom in the course of the meal. Each time one chapter comes before another there’s a little compatibility score, and the overall harmony of the table of contents is just the total of all those scores across the whole seating plan — the higher that total, the better the flow from start to finish. The exact chapter list and all the compatibility numbers are listed below.

# num_chapters=17
# chapter_ids=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

# Meaning: this is a directed compatibility_score matrix over items in chapter_ids (17 total). Entry at row earlier_chapter_id=i and column later_chapter_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# compatibility_score
| earlier_chapter_id\later_chapter_id | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 525.0 | 30.0 | 50.0 | 16.0 | 7.0 | 18.0 | 22.0 | 64.0 | 15.0 | 28.0 | 1.0 | 19.0 | 2.0 | 1.0 | 140.0 | 3.0 | 1.0 |
| 1 | 15.0 | 8452.0 | 322.0 | 320.0 | 55.0 | 253.0 | 175.0 | 353.0 | 51.0 | 271.0 | 17.0 | 1178.0 | 140.0 | 15.0 | 1053.0 | 99.0 | 44.0 |
| 2 | 690.0 | 727.0 | 5812.0 | 7005.0 | 560.0 | 2995.0 | 391.0 | 547.0 | 259.0 | 1196.0 | 76.0 | 13827.0 | 31.0 | 0.0 | 1906.0 | 107.0 | 19.0 |
| 3 | 423.0 | 265.0 | 1273.0 | 9394.0 | 200.0 | 681.0 | 28.0 | 58.0 | 29.0 | 47.0 | 4.0 | 680.0 | 2.0 | 0.0 | 811.0 | 19.0 | 1.0 |
| 4 | 12.0 | 32.0 | 145.0 | 237.0 | 1259.0 | 224.0 | 14.0 | 19.0 | 4.0 | 22.0 | 11.0 | 18.0 | 102.0 | 0.0 | 857.0 | 143.0 | 6.0 |
| 5 | 124.0 | 75.0 | 463.0 | 2715.0 | 674.0 | 10804.0 | 7.0 | 9.0 | 5.0 | 66.0 | 25.0 | 3079.0 | 32.0 | 16.0 | 633.0 | 46.0 | 19.0 |
| 6 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2239.0 | 624.0 | 340.0 | 3.0 | 4.0 | 0.0 | 49.0 | 35.0 | 443.0 | 122.0 | 72.0 |
| 7 | 5.0 | 15.0 | 12.0 | 12.0 | 1.0 | 16.0 | 311.0 | 11779.0 | 5.0 | 3.0 | 0.0 | 13.0 | 124.0 | 46.0 | 787.0 | 94.0 | 171.0 |
| 8 | 10.0 | 0.0 | 59.0 | 39.0 | 45.0 | 5.0 | 2.0 | 2.0 | 994.0 | 45.0 | 6.0 | 10.0 | 36.0 | 0.0 | 443.0 | 27.0 | 6.0 |
| 9 | 207.0 | 9.0 | 392.0 | 466.0 | 45.0 | 404.0 | 7.0 | 30.0 | 65.0 | 6070.0 | 39.0 | 7746.0 | 7.0 | 0.0 | 107.0 | 19.0 | 12.0 |
| 10 | 0.0 | 0.0 | 22.0 | 3.0 | 17.0 | 17.0 | 0.0 | 0.0 | 1.0 | 1.0 | 335.0 | 10.0 | 3.0 | 0.0 | 73.0 | 83.0 | 4.0 |
| 11 | 413.0 | 64.0 | 39.0 | 32.0 | 0.0 | 43.0 | 12.0 | 25.0 | 3.0 | 11.0 | 0.0 | 390.0 | 21.0 | 0.0 | 864.0 | 213.0 | 0.0 |
| 12 | 2.0 | 5.0 | 13.0 | 8.0 | 6.0 | 13.0 | 6.0 | 5.0 | 1.0 | 6.0 | 1.0 | 10.0 | 60.0 | 0.0 | 9487.0 | 24.0 | 0.0 |
| 13 | 12.0 | 8.0 | 31.0 | 38.0 | 8.0 | 39.0 | 4.0 | 17.0 | 4.0 | 10.0 | 1.0 | 30.0 | 9.0 | 0.0 | 0.0 | 0.0 | 1.0 |
| 14 | 28.0 | 35.0 | 117.0 | 132.0 | 26.0 | 108.0 | 38.0 | 81.0 | 15.0 | 44.0 | 6.0 | 138.0 | 34.0 | 4.0 | 271.0 | 81.0 | 14.0 |
| 15 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 16 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 64.0 | 0.0 | 1729.0 | 0.0 | 0.0 |

If you'd like the answer in a tidy, machine-friendly shape, I usually lay it out like this:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of this as a simple form: the ""solution"" field is the sequence from the very first chapter to the very last chapter — just list each chapter identifier in the order you want them to appear. It's just an example of the shape I expect, not the actual finished ordering.

Please use the exact identifiers from the instance input — don't 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”.""","[[525.0, 30.0, 50.0, 16.0, 7.0, 18.0, 22.0, 64.0, 15.0, 28.0, 1.0, 19.0, 2.0, 1.0, 140.0, 3.0, 1.0], [15.0, 8452.0, 322.0, 320.0, 55.0, 253.0, 175.0, 353.0, 51.0, 271.0, 17.0, 1178.0, 140.0, 15.0, 1053.0, 99.0, 44.0], [690.0, 727.0, 5812.0, 7005.0, 560.0, 2995.0, 391.0, 547.0, 259.0, 1196.0, 76.0, 13827.0, 31.0, 0.0, 1906.0, 107.0, 19.0], [423.0, 265.0, 1273.0, 9394.0, 200.0, 681.0, 28.0, 58.0, 29.0, 47.0, 4.0, 680.0, 2.0, 0.0, 811.0, 19.0, 1.0], [12.0, 32.0, 145.0, 237.0, 1259.0, 224.0, 14.0, 19.0, 4.0, 22.0, 11.0, 18.0, 102.0, 0.0, 857.0, 143.0, 6.0], [124.0, 75.0, 463.0, 2715.0, 674.0, 10804.0, 7.0, 9.0, 5.0, 66.0, 25.0, 3079.0, 32.0, 16.0, 633.0, 46.0, 19.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2239.0, 624.0, 340.0, 3.0, 4.0, 0.0, 49.0, 35.0, 443.0, 122.0, 72.0], [5.0, 15.0, 12.0, 12.0, 1.0, 16.0, 311.0, 11779.0, 5.0, 3.0, 0.0, 13.0, 124.0, 46.0, 787.0, 94.0, 171.0], [10.0, 0.0, 59.0, 39.0, 45.0, 5.0, 2.0, 2.0, 994.0, 45.0, 6.0, 10.0, 36.0, 0.0, 443.0, 27.0, 6.0], [207.0, 9.0, 392.0, 466.0, 45.0, 404.0, 7.0, 30.0, 65.0, 6070.0, 39.0, 7746.0, 7.0, 0.0, 107.0, 19.0, 12.0], [0.0, 0.0, 22.0, 3.0, 17.0, 17.0, 0.0, 0.0, 1.0, 1.0, 335.0, 10.0, 3.0, 0.0, 73.0, 83.0, 4.0], [413.0, 64.0, 39.0, 32.0, 0.0, 43.0, 12.0, 25.0, 3.0, 11.0, 0.0, 390.0, 21.0, 0.0, 864.0, 213.0, 0.0], [2.0, 5.0, 13.0, 8.0, 6.0, 13.0, 6.0, 5.0, 1.0, 6.0, 1.0, 10.0, 60.0, 0.0, 9487.0, 24.0, 0.0], [12.0, 8.0, 31.0, 38.0, 8.0, 39.0, 4.0, 17.0, 4.0, 10.0, 1.0, 30.0, 9.0, 0.0, 0.0, 0.0, 1.0], [28.0, 35.0, 117.0, 132.0, 26.0, 108.0, 38.0, 81.0, 15.0, 44.0, 6.0, 138.0, 34.0, 4.0, 271.0, 81.0, 14.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 64.0, 0.0, 1729.0, 0.0, 0.0]]","[13, 2, 1, 9, 5, 10, 4, 3, 11, 0, 6, 7, 8, 16, 12, 14, 15]",71277.0,"{'problem_type': 'LOP', 'num_nodes': 17, 'nodes': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 'pairs': [{'from_id': 0, 'to_id': 1, 'weight': 30.0}, {'from_id': 0, 'to_id': 2, 'weight': 50.0}, {'from_id': 0, 'to_id': 3, 'weight': 16.0}, {'from_id': 0, 'to_id': 4, 'weight': 7.0}, {'from_id': 0, 'to_id': 5, 'weight': 18.0}, {'from_id': 0, 'to_id': 6, 'weight': 22.0}, {'from_id': 0, 'to_id': 7, 'weight': 64.0}, {'from_id': 0, 'to_id': 8, 'weight': 15.0}, {'from_id': 0, 'to_id': 9, 'weight': 28.0}, {'from_id': 0, 'to_id': 10, 'weight': 1.0}, {'from_id': 0, 'to_id': 11, 'weight': 19.0}, {'from_id': 0, 'to_id': 12, 'weight': 2.0}, {'from_id': 0, 'to_id': 13, 'weight': 1.0}, {'from_id': 0, 'to_id': 14, 'weight': 140.0}, {'from_id': 0, 'to_id': 15, 'weight': 3.0}, {'from_id': 0, 'to_id': 16, 'weight': 1.0}, {'from_id': 1, 'to_id': 0, 'weight': 15.0}, {'from_id': 1, 'to_id': 2, 'weight': 322.0}, {'from_id': 1, 'to_id': 3, 'weight': 320.0}, {'from_id': 1, 'to_id': 4, 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LOP,LOP,"Recently a commuter playlist experiment started: one continuous ride, every track played once, no detours. For any two different songs there’s a known mood effect tied to their order, and the goal is to line them up so those effects combine into the largest overall mood boost. The way to score a lineup is simple — add together the mood gains for each time a particular song comes before another. The concrete pairwise values are shown below.

There are 16 tracks in the commute playlist: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16.
Meaning: this is a directed mood_gain matrix over items in track_ids (16 total). Entry at row preceding_track_id=i and column following_track_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
mood_gain:
| preceding_track_id\following_track_id | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 404519.0 | 2175306.0 | 1317.0 | 15871.0 | 6297.0 | 4997.0 | 47822.0 | 760.0 | 43946.0 | 9705.0 | 10920.0 | 2745.0 | 7103.0 | 2563.0 | 3207.0 | 1747.0 |
| 2 | 201339.0 | 1133446.0 | 88588.0 | 355030.0 | 114783.0 | 69429.0 | 370960.0 | 23054.0 | 454271.0 | 60010.0 | 231108.0 | 48661.0 | 285830.0 | 85538.0 | 105886.0 | 40225.0 |
| 3 | 2339.0 | 34395.0 | 611347.0 | 240281.0 | 112899.0 | 1896.0 | 8286.0 | 2211.0 | 7523.0 | 2364.0 | 6501.0 | 24203.0 | 7409.0 | 21450.0 | 22335.0 | 2326.0 |
| 4 | 63694.0 | 524032.0 | 697229.0 | 4600451.0 | 595213.0 | 2719.0 | 20819.0 | 1509.0 | 17378.0 | 3774.0 | 34534.0 | 47503.0 | 69468.0 | 471979.0 | 65840.0 | 5138.0 |
| 5 | 4360.0 | 1130.0 | 273.0 | 15374.0 | 1018952.0 | 290.0 | 330.0 | 13.0 | 1530.0 | 240.0 | 1013.0 | 5355.0 | 290.0 | 343.0 | 5783.0 | 8786.0 |
| 6 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 163629.0 | 1174451.0 | 0.0 | 1516.0 | 0.0 | 0.0 | 516.0 | 448468.0 | 0.0 | 16411.0 | 33709.0 |
| 7 | 993.0 | 1120.0 | 284.0 | 10765.0 | 253.0 | 237412.0 | 8505108.0 | 4964.0 | 23299.0 | 3636.0 | 6255.0 | 2880.0 | 2155180.0 | 13.0 | 53847.0 | 57369.0 |
| 8 | 1093.0 | 3332.0 | 833.0 | 4738.0 | 40.0 | 600.0 | 3059.0 | 43725.0 | 2826.0 | 1106.0 | 2539.0 | 287.0 | 642210.0 | 67.0 | 1186.0 | 328.0 |
| 9 | 14919.0 | 21279.0 | 23619.0 | 85183.0 | 52128.0 | 4197.0 | 40543.0 | 4340.0 | 14747476.0 | 406500.0 | 745932.0 | 55659.0 | 78414.0 | 32504.0 | 83534.0 | 31513.0 |
| 10 | 2019.0 | 2460.0 | 20672.0 | 7333.0 | 2930.0 | 0.0 | 1636.0 | 0.0 | 389436.0 | 1349408.0 | 34312.0 | 6443.0 | 6011.0 | 8803.0 | 23850.0 | 7237.0 |
| 11 | 73398.0 | 37836.0 | 34805.0 | 218055.0 | 142894.0 | 19680.0 | 87632.0 | 7893.0 | 95246.0 | 43034.0 | 4381253.0 | 69032.0 | 48395.0 | 12293.0 | 13947.0 | 16308.0 |
| 12 | 1966.0 | 26911.0 | 32885.0 | 51722.0 | 23898.0 | 3040.0 | 15060.0 | 2555.0 | 41091.0 | 13484.0 | 64407.0 | 419423.0 | 11428.0 | 11607.0 | 58384.0 | 20565.0 |
| 13 | 11789.0 | 24052.0 | 20167.0 | 178130.0 | 15146.0 | 11831.0 | 92960.0 | 19537.0 | 96308.0 | 18783.0 | 73115.0 | 12409.0 | 101964.0 | 22594.0 | 72563.0 | 64023.0 |
| 14 | 30252.0 | 83181.0 | 48179.0 | 198518.0 | 43247.0 | 33011.0 | 123845.0 | 7947.0 | 196851.0 | 42712.0 | 121313.0 | 18924.0 | 181971.0 | 309001.0 | 141828.0 | 35930.0 |
| 15 | 8921.0 | 43649.0 | 13164.0 | 64424.0 | 13628.0 | 10813.0 | 63811.0 | 6161.0 | 58410.0 | 9766.0 | 26008.0 | 6290.0 | 79660.0 | 9982.0 | 460208.0 | 18736.0 |
| 16 | 7.0 | 65.0 | 151.0 | 271.0 | 67.0 | 322.0 | 972.0 | 122.0 | 878.0 | 192.0 | 290.0 | 38.0 | 3187.0 | 475.0 | 348.0 | 1248.0 |
Orderings of these 16 tracks are scored by summing the listed pairwise mood gains; the best playlist maximizes that total.

If you want to hand me a final playlist, use this simple JSON layout so it's easy to read and parse:

{
  ""solution"": [<first_track>, <second_track>, ..., <last_track>]
}

""solution"" is just the ordered playlist: the first entry is the very first song on the ride, the next entry is the second song, and so on until the last entry. The angle-bracket names are placeholders showing the shape I expect — replace them with the actual song identifiers from your instance when you give the final lineup. This JSON is just a sketch of the shape I need, not the real answer itself.

Please make sure to use the identifiers exactly as they appear in the instance input — do not rename 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”.","[[404519.0, 2175306.0, 1317.0, 15871.0, 6297.0, 4997.0, 47822.0, 760.0, 43946.0, 9705.0, 10920.0, 2745.0, 7103.0, 2563.0, 3207.0, 1747.0], [201339.0, 1133446.0, 88588.0, 355030.0, 114783.0, 69429.0, 370960.0, 23054.0, 454271.0, 60010.0, 231108.0, 48661.0, 285830.0, 85538.0, 105886.0, 40225.0], [2339.0, 34395.0, 611347.0, 240281.0, 112899.0, 1896.0, 8286.0, 2211.0, 7523.0, 2364.0, 6501.0, 24203.0, 7409.0, 21450.0, 22335.0, 2326.0], [63694.0, 524032.0, 697229.0, 4600451.0, 595213.0, 2719.0, 20819.0, 1509.0, 17378.0, 3774.0, 34534.0, 47503.0, 69468.0, 471979.0, 65840.0, 5138.0], [4360.0, 1130.0, 273.0, 15374.0, 1018952.0, 290.0, 330.0, 13.0, 1530.0, 240.0, 1013.0, 5355.0, 290.0, 343.0, 5783.0, 8786.0], [0.0, 0.0, 0.0, 0.0, 0.0, 163629.0, 1174451.0, 0.0, 1516.0, 0.0, 0.0, 516.0, 448468.0, 0.0, 16411.0, 33709.0], [993.0, 1120.0, 284.0, 10765.0, 253.0, 237412.0, 8505108.0, 4964.0, 23299.0, 3636.0, 6255.0, 2880.0, 2155180.0, 13.0, 53847.0, 57369.0], [1093.0, 3332.0, 833.0, 4738.0, 40.0, 600.0, 3059.0, 43725.0, 2826.0, 1106.0, 2539.0, 287.0, 642210.0, 67.0, 1186.0, 328.0], [14919.0, 21279.0, 23619.0, 85183.0, 52128.0, 4197.0, 40543.0, 4340.0, 14747476.0, 406500.0, 745932.0, 55659.0, 78414.0, 32504.0, 83534.0, 31513.0], [2019.0, 2460.0, 20672.0, 7333.0, 2930.0, 0.0, 1636.0, 0.0, 389436.0, 1349408.0, 34312.0, 6443.0, 6011.0, 8803.0, 23850.0, 7237.0], [73398.0, 37836.0, 34805.0, 218055.0, 142894.0, 19680.0, 87632.0, 7893.0, 95246.0, 43034.0, 4381253.0, 69032.0, 48395.0, 12293.0, 13947.0, 16308.0], [1966.0, 26911.0, 32885.0, 51722.0, 23898.0, 3040.0, 15060.0, 2555.0, 41091.0, 13484.0, 64407.0, 419423.0, 11428.0, 11607.0, 58384.0, 20565.0], [11789.0, 24052.0, 20167.0, 178130.0, 15146.0, 11831.0, 92960.0, 19537.0, 96308.0, 18783.0, 73115.0, 12409.0, 101964.0, 22594.0, 72563.0, 64023.0], [30252.0, 83181.0, 48179.0, 198518.0, 43247.0, 33011.0, 123845.0, 7947.0, 196851.0, 42712.0, 121313.0, 18924.0, 181971.0, 309001.0, 141828.0, 35930.0], [8921.0, 43649.0, 13164.0, 64424.0, 13628.0, 10813.0, 63811.0, 6161.0, 58410.0, 9766.0, 26008.0, 6290.0, 79660.0, 9982.0, 460208.0, 18736.0], [7.0, 65.0, 151.0, 271.0, 67.0, 322.0, 972.0, 122.0, 878.0, 192.0, 290.0, 38.0, 3187.0, 475.0, 348.0, 1248.0]]","[3, 13, 0, 1, 8, 10, 11, 9, 2, 5, 14, 6, 7, 12, 4, 15]",15153027.0,"{'problem_type': 'LOP', 'num_nodes': 16, 'nodes': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 'pairs': [{'from_id': 1, 'to_id': 2, 'weight': 2175306.0}, {'from_id': 1, 'to_id': 3, 'weight': 1317.0}, {'from_id': 1, 'to_id': 4, 'weight': 15871.0}, {'from_id': 1, 'to_id': 5, 'weight': 6297.0}, {'from_id': 1, 'to_id': 6, 'weight': 4997.0}, {'from_id': 1, 'to_id': 7, 'weight': 47822.0}, {'from_id': 1, 'to_id': 8, 'weight': 760.0}, {'from_id': 1, 'to_id': 9, 'weight': 43946.0}, {'from_id': 1, 'to_id': 10, 'weight': 9705.0}, {'from_id': 1, 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'to_id': 11, 'weight': 26008.0}, {'from_id': 15, 'to_id': 12, 'weight': 6290.0}, {'from_id': 15, 'to_id': 13, 'weight': 79660.0}, {'from_id': 15, 'to_id': 14, 'weight': 9982.0}, {'from_id': 15, 'to_id': 16, 'weight': 18736.0}, {'from_id': 16, 'to_id': 1, 'weight': 7.0}, {'from_id': 16, 'to_id': 2, 'weight': 65.0}, {'from_id': 16, 'to_id': 3, 'weight': 151.0}, {'from_id': 16, 'to_id': 4, 'weight': 271.0}, {'from_id': 16, 'to_id': 5, 'weight': 67.0}, {'from_id': 16, 'to_id': 6, 'weight': 322.0}, {'from_id': 16, 'to_id': 7, 'weight': 972.0}, {'from_id': 16, 'to_id': 8, 'weight': 122.0}, {'from_id': 16, 'to_id': 9, 'weight': 878.0}, {'from_id': 16, 'to_id': 10, 'weight': 192.0}, {'from_id': 16, 'to_id': 11, 'weight': 290.0}, {'from_id': 16, 'to_id': 12, 'weight': 38.0}, {'from_id': 16, 'to_id': 13, 'weight': 3187.0}, {'from_id': 16, 'to_id': 14, 'weight': 475.0}, {'from_id': 16, 'to_id': 15, 'weight': 348.0}]}","[4, 14, 1, 2, 9, 11, 12, 10, 3, 6, 15, 7, 8, 13, 5, 16]",19,nl,1
LOP,LOP,"There’s a conference schedule to sort out, and ordering actually matters: someone has measured how much better it is if each speaker precedes another. So to see how good a particular order is, just add up those little numbers for every pair where the first speaker appears earlier in the lineup than the second. The task is to pick one single sequence that includes every invited speaker one time only, and that gives the largest total sum of those pairwise perks. The specific speakers and their pairwise values are listed below.

Below are the 22 invited speakers: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V.
Meaning: this is a directed benefit_if_precedes matrix over items in speaker_ids (22 total). Entry at row preceding_speaker_id=i and column following_speaker_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
benefit_if_precedes:
| preceding_speaker_id\following_speaker_id | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 6759.0 | 901.0 | 0.0 | 4.0 | 48.0 | 146.0 | 24.0 | 19.0 | 0.0 | 57.0 | 22.0 | 123.0 | 10.0 | 1.0 | 1.0 | 290.0 | 25.0 | 221.0 | 221.0 | 97.0 | 429.0 | 85.0 |
| B | 14.0 | 1318.0 | 675.0 | 0.0 | 41.0 | 33.0 | 27.0 | 9.0 | 0.0 | 55.0 | 19.0 | 79.0 | 11.0 | 0.0 | 0.0 | 243.0 | 22.0 | 58.0 | 76.0 | 32.0 | 335.0 | 33.0 |
| C | 27.0 | 2.0 | 107.0 | 0.0 | 4.0 | 1.0 | 2.0 | 1.0 | 0.0 | 0.0 | 0.0 | 2.0 | 1.0 | 0.0 | 0.0 | 9.0 | 0.0 | 5.0 | 17.0 | 4.0 | 15.0 | 16.0 |
| D | 5.0 | 1.0 | 0.0 | 22.0 | 1.0 | 0.0 | 3.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2.0 | 0.0 | 3.0 | 12.0 | 1.0 | 3.0 | 5.0 |
| E | 133.0 | 51.0 | 0.0 | 0.0 | 230.0 | 42.0 | 1.0 | 0.0 | 3.0 | 3.0 | 0.0 | 15.0 | 4.0 | 0.0 | 0.0 | 50.0 | 0.0 | 57.0 | 112.0 | 21.0 | 188.0 | 24.0 |
| F | 475.0 | 86.0 | 0.0 | 0.0 | 75.0 | 3297.0 | 3.0 | 40.0 | 24.0 | 10.0 | 15.0 | 18.0 | 3.0 | 0.0 | 0.0 | 187.0 | 0.0 | 41.0 | 216.0 | 51.0 | 393.0 | 19.0 |
| G | 88.0 | 3.0 | 0.0 | 0.0 | 7.0 | 0.0 | 328.0 | 4.0 | 0.0 | 0.0 | 0.0 | 19.0 | 0.0 | 0.0 | 0.0 | 100.0 | 7.0 | 43.0 | 199.0 | 24.0 | 50.0 | 14.0 |
| H | 423.0 | 171.0 | 0.0 | 0.0 | 585.0 | 690.0 | 100.0 | 6015.0 | 0.0 | 441.0 | 40.0 | 47.0 | 5.0 | 2.0 | 2.0 | 1122.0 | 112.0 | 169.0 | 460.0 | 120.0 | 1154.0 | 132.0 |
| I | 38.0 | 22.0 | 0.0 | 0.0 | 59.0 | 84.0 | 16.0 | 3.0 | 110.0 | 33.0 | 1.0 | 11.0 | 1.0 | 0.0 | 0.0 | 78.0 | 6.0 | 15.0 | 30.0 | 14.0 | 72.0 | 23.0 |
| J | 61.0 | 10.0 | 0.0 | 2.0 | 2.0 | 183.0 | 18.0 | 3.0 | 0.0 | 336.0 | 3.0 | 29.0 | 8.0 | 1.0 | 1.0 | 438.0 | 3.0 | 17.0 | 56.0 | 83.0 | 242.0 | 17.0 |
| K | 68.0 | 59.0 | 0.0 | 0.0 | 7.0 | 14.0 | 11.0 | 1.0 | 0.0 | 0.0 | 16.0 | 26.0 | 5.0 | 0.0 | 0.0 | 36.0 | 2.0 | 11.0 | 54.0 | 6.0 | 42.0 | 10.0 |
| L | 120.0 | 19.0 | 0.0 | 0.0 | 2.0 | 28.0 | 2.0 | 10.0 | 0.0 | 7.0 | 0.0 | 330.0 | 0.0 | 1.0 | 1.0 | 142.0 | 8.0 | 49.0 | 166.0 | 234.0 | 708.0 | 41.0 |
| M | 80.0 | 5.0 | 0.0 | 0.0 | 34.0 | 0.0 | 0.0 | 14.0 | 0.0 | 0.0 | 2.0 | 51.0 | 89.0 | 1.0 | 1.0 | 416.0 | 28.0 | 29.0 | 73.0 | 119.0 | 655.0 | 53.0 |
| N | 32.0 | 2.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 9.0 | 0.0 | 45.0 | 11.0 | 47.0 | 2.0 | 9.0 | 31.0 | 14.0 | 62.0 | 16.0 |
| O | 26.0 | 5.0 | 0.0 | 0.0 | 0.0 | 0.0 | 4.0 | 3.0 | 0.0 | 0.0 | 0.0 | 3.0 | 0.0 | 1.0 | 622.0 | 190.0 | 5.0 | 34.0 | 98.0 | 20.0 | 114.0 | 30.0 |
| P | 912.0 | 228.0 | 0.0 | 7.0 | 333.0 | 318.0 | 787.0 | 1027.0 | 5.0 | 99.0 | 76.0 | 307.0 | 55.0 | 726.0 | 321.0 | 11654.0 | 1317.0 | 333.0 | 1100.0 | 836.0 | 3149.0 | 275.0 |
| Q | 406.0 | 38.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2.0 | 260.0 | 0.0 | 50.0 | 0.0 | 96.0 | 207.0 | 1.0 | 1.0 | 722.0 | 1402.0 | 439.0 | 1287.0 | 1656.0 | 2435.0 | 453.0 |
| R | 378.0 | 42.0 | 0.0 | 0.0 | 52.0 | 53.0 | 49.0 | 25.0 | 3.0 | 23.0 | 5.0 | 46.0 | 22.0 | 0.0 | 0.0 | 168.0 | 23.0 | 21.0 | 364.0 | 41.0 | 190.0 | 84.0 |
| S | 384.0 | 96.0 | 0.0 | 0.0 | 16.0 | 14.0 | 56.0 | 750.0 | 0.0 | 22.0 | 7.0 | 188.0 | 115.0 | 1.0 | 1.0 | 615.0 | 480.0 | 1972.0 | 7051.0 | 480.0 | 919.0 | 224.0 |
| T | 138.0 | 18.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2.0 | 35.0 | 0.0 | 0.0 | 2.0 | 193.0 | 21.0 | 0.0 | 0.0 | 76.0 | 4.0 | 289.0 | 141.0 | 0.0 | 168.0 | 11.0 |
| U | 602.0 | 156.0 | 0.0 | 0.0 | 0.0 | 32.0 | 23.0 | 206.0 | 0.0 | 187.0 | 47.0 | 2867.0 | 445.0 | 74.0 | 141.0 | 3834.0 | 1170.0 | 259.0 | 501.0 | 2075.0 | 15016.0 | 1041.0 |
| V | 1268.0 | 533.0 | 0.0 | 2.0 | 8.0 | 41.0 | 80.0 | 1019.0 | 489.0 | 564.0 | 13.0 | 1133.0 | 383.0 | 59.0 | 98.0 | 1851.0 | 2909.0 | 836.0 | 1475.0 | 1764.0 | 25418.0 | 72890.0 |
Arrange all 22 speakers into one sequence that maximizes the summed pairwise perks above.

Also, just so the final reply is easy to parse, please use this simple JSON shape for your answer — nothing fancy, just a single list showing the order.

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

This little block is just a template: ""solution"" should be the full lineup from first speaker to last, and each entry in the list should be the exact identifier used in the instance for that speaker. Think of it like filling in a form — put the speakers in order inside the array. The JSON above is only a sketch of the shape I need, not the finished ordering.

Please use the identifiers exactly as they appear in the 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”.""","[[6759.0, 901.0, 0.0, 4.0, 48.0, 146.0, 24.0, 19.0, 0.0, 57.0, 22.0, 123.0, 10.0, 1.0, 1.0, 290.0, 25.0, 221.0, 221.0, 97.0, 429.0, 85.0], [14.0, 1318.0, 675.0, 0.0, 41.0, 33.0, 27.0, 9.0, 0.0, 55.0, 19.0, 79.0, 11.0, 0.0, 0.0, 243.0, 22.0, 58.0, 76.0, 32.0, 335.0, 33.0], [27.0, 2.0, 107.0, 0.0, 4.0, 1.0, 2.0, 1.0, 0.0, 0.0, 0.0, 2.0, 1.0, 0.0, 0.0, 9.0, 0.0, 5.0, 17.0, 4.0, 15.0, 16.0], [5.0, 1.0, 0.0, 22.0, 1.0, 0.0, 3.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 3.0, 12.0, 1.0, 3.0, 5.0], [133.0, 51.0, 0.0, 0.0, 230.0, 42.0, 1.0, 0.0, 3.0, 3.0, 0.0, 15.0, 4.0, 0.0, 0.0, 50.0, 0.0, 57.0, 112.0, 21.0, 188.0, 24.0], [475.0, 86.0, 0.0, 0.0, 75.0, 3297.0, 3.0, 40.0, 24.0, 10.0, 15.0, 18.0, 3.0, 0.0, 0.0, 187.0, 0.0, 41.0, 216.0, 51.0, 393.0, 19.0], [88.0, 3.0, 0.0, 0.0, 7.0, 0.0, 328.0, 4.0, 0.0, 0.0, 0.0, 19.0, 0.0, 0.0, 0.0, 100.0, 7.0, 43.0, 199.0, 24.0, 50.0, 14.0], [423.0, 171.0, 0.0, 0.0, 585.0, 690.0, 100.0, 6015.0, 0.0, 441.0, 40.0, 47.0, 5.0, 2.0, 2.0, 1122.0, 112.0, 169.0, 460.0, 120.0, 1154.0, 132.0], [38.0, 22.0, 0.0, 0.0, 59.0, 84.0, 16.0, 3.0, 110.0, 33.0, 1.0, 11.0, 1.0, 0.0, 0.0, 78.0, 6.0, 15.0, 30.0, 14.0, 72.0, 23.0], [61.0, 10.0, 0.0, 2.0, 2.0, 183.0, 18.0, 3.0, 0.0, 336.0, 3.0, 29.0, 8.0, 1.0, 1.0, 438.0, 3.0, 17.0, 56.0, 83.0, 242.0, 17.0], [68.0, 59.0, 0.0, 0.0, 7.0, 14.0, 11.0, 1.0, 0.0, 0.0, 16.0, 26.0, 5.0, 0.0, 0.0, 36.0, 2.0, 11.0, 54.0, 6.0, 42.0, 10.0], [120.0, 19.0, 0.0, 0.0, 2.0, 28.0, 2.0, 10.0, 0.0, 7.0, 0.0, 330.0, 0.0, 1.0, 1.0, 142.0, 8.0, 49.0, 166.0, 234.0, 708.0, 41.0], [80.0, 5.0, 0.0, 0.0, 34.0, 0.0, 0.0, 14.0, 0.0, 0.0, 2.0, 51.0, 89.0, 1.0, 1.0, 416.0, 28.0, 29.0, 73.0, 119.0, 655.0, 53.0], [32.0, 2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 9.0, 0.0, 45.0, 11.0, 47.0, 2.0, 9.0, 31.0, 14.0, 62.0, 16.0], [26.0, 5.0, 0.0, 0.0, 0.0, 0.0, 4.0, 3.0, 0.0, 0.0, 0.0, 3.0, 0.0, 1.0, 622.0, 190.0, 5.0, 34.0, 98.0, 20.0, 114.0, 30.0], [912.0, 228.0, 0.0, 7.0, 333.0, 318.0, 787.0, 1027.0, 5.0, 99.0, 76.0, 307.0, 55.0, 726.0, 321.0, 11654.0, 1317.0, 333.0, 1100.0, 836.0, 3149.0, 275.0], [406.0, 38.0, 0.0, 0.0, 0.0, 0.0, 2.0, 260.0, 0.0, 50.0, 0.0, 96.0, 207.0, 1.0, 1.0, 722.0, 1402.0, 439.0, 1287.0, 1656.0, 2435.0, 453.0], [378.0, 42.0, 0.0, 0.0, 52.0, 53.0, 49.0, 25.0, 3.0, 23.0, 5.0, 46.0, 22.0, 0.0, 0.0, 168.0, 23.0, 21.0, 364.0, 41.0, 190.0, 84.0], [384.0, 96.0, 0.0, 0.0, 16.0, 14.0, 56.0, 750.0, 0.0, 22.0, 7.0, 188.0, 115.0, 1.0, 1.0, 615.0, 480.0, 1972.0, 7051.0, 480.0, 919.0, 224.0], [138.0, 18.0, 0.0, 0.0, 0.0, 0.0, 2.0, 35.0, 0.0, 0.0, 2.0, 193.0, 21.0, 0.0, 0.0, 76.0, 4.0, 289.0, 141.0, 0.0, 168.0, 11.0], [602.0, 156.0, 0.0, 0.0, 0.0, 32.0, 23.0, 206.0, 0.0, 187.0, 47.0, 2867.0, 445.0, 74.0, 141.0, 3834.0, 1170.0, 259.0, 501.0, 2075.0, 15016.0, 1041.0], [1268.0, 533.0, 0.0, 2.0, 8.0, 41.0, 80.0, 1019.0, 489.0, 564.0, 13.0, 1133.0, 383.0, 59.0, 98.0, 1851.0, 2909.0, 836.0, 1475.0, 1764.0, 25418.0, 72890.0]]","[21, 8, 12, 7, 9, 15, 13, 14, 5, 10, 3, 6, 16, 4, 18, 20, 19, 17, 0, 1, 2, 11]",81896.0,"{'problem_type': 'LOP', 'num_nodes': 22, 'nodes': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S', 'T', 'U', 'V'], 'pairs': [{'from_id': 'A', 'to_id': 'B', 'weight': 901.0}, {'from_id': 'A', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'D', 'weight': 4.0}, {'from_id': 'A', 'to_id': 'E', 'weight': 48.0}, {'from_id': 'A', 'to_id': 'F', 'weight': 146.0}, {'from_id': 'A', 'to_id': 'G', 'weight': 24.0}, {'from_id': 'A', 'to_id': 'H', 'weight': 19.0}, {'from_id': 'A', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'J', 'weight': 57.0}, {'from_id': 'A', 'to_id': 'K', 'weight': 22.0}, {'from_id': 'A', 'to_id': 'L', 'weight': 123.0}, {'from_id': 'A', 'to_id': 'M', 'weight': 10.0}, {'from_id': 'A', 'to_id': 'N', 'weight': 1.0}, {'from_id': 'A', 'to_id': 'O', 'weight': 1.0}, {'from_id': 'A', 'to_id': 'P', 'weight': 290.0}, {'from_id': 'A', 'to_id': 'Q', 'weight': 25.0}, {'from_id': 'A', 'to_id': 'R', 'weight': 221.0}, {'from_id': 'A', 'to_id': 'S', 'weight': 221.0}, {'from_id': 'A', 'to_id': 'T', 'weight': 97.0}, {'from_id': 'A', 'to_id': 'U', 'weight': 429.0}, {'from_id': 'A', 'to_id': 'V', 'weight': 85.0}, {'from_id': 'B', 'to_id': 'A', 'weight': 14.0}, {'from_id': 'B', 'to_id': 'C', 'weight': 675.0}, {'from_id': 'B', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'E', 'weight': 41.0}, {'from_id': 'B', 'to_id': 'F', 'weight': 33.0}, {'from_id': 'B', 'to_id': 'G', 'weight': 27.0}, {'from_id': 'B', 'to_id': 'H', 'weight': 9.0}, {'from_id': 'B', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'J', 'weight': 55.0}, {'from_id': 'B', 'to_id': 'K', 'weight': 19.0}, {'from_id': 'B', 'to_id': 'L', 'weight': 79.0}, {'from_id': 'B', 'to_id': 'M', 'weight': 11.0}, {'from_id': 'B', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'P', 'weight': 243.0}, {'from_id': 'B', 'to_id': 'Q', 'weight': 22.0}, {'from_id': 'B', 'to_id': 'R', 'weight': 58.0}, {'from_id': 'B', 'to_id': 'S', 'weight': 76.0}, {'from_id': 'B', 'to_id': 'T', 'weight': 32.0}, {'from_id': 'B', 'to_id': 'U', 'weight': 335.0}, {'from_id': 'B', 'to_id': 'V', 'weight': 33.0}, {'from_id': 'C', 'to_id': 'A', 'weight': 27.0}, {'from_id': 'C', 'to_id': 'B', 'weight': 2.0}, {'from_id': 'C', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'E', 'weight': 4.0}, {'from_id': 'C', 'to_id': 'F', 'weight': 1.0}, {'from_id': 'C', 'to_id': 'G', 'weight': 2.0}, {'from_id': 'C', 'to_id': 'H', 'weight': 1.0}, {'from_id': 'C', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'L', 'weight': 2.0}, {'from_id': 'C', 'to_id': 'M', 'weight': 1.0}, {'from_id': 'C', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'P', 'weight': 9.0}, {'from_id': 'C', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'R', 'weight': 5.0}, {'from_id': 'C', 'to_id': 'S', 'weight': 17.0}, {'from_id': 'C', 'to_id': 'T', 'weight': 4.0}, {'from_id': 'C', 'to_id': 'U', 'weight': 15.0}, {'from_id': 'C', 'to_id': 'V', 'weight': 16.0}, {'from_id': 'D', 'to_id': 'A', 'weight': 5.0}, {'from_id': 'D', 'to_id': 'B', 'weight': 1.0}, {'from_id': 'D', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'E', 'weight': 1.0}, {'from_id': 'D', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'G', 'weight': 3.0}, {'from_id': 'D', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'P', 'weight': 2.0}, {'from_id': 'D', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'R', 'weight': 3.0}, {'from_id': 'D', 'to_id': 'S', 'weight': 12.0}, {'from_id': 'D', 'to_id': 'T', 'weight': 1.0}, {'from_id': 'D', 'to_id': 'U', 'weight': 3.0}, {'from_id': 'D', 'to_id': 'V', 'weight': 5.0}, {'from_id': 'E', 'to_id': 'A', 'weight': 133.0}, {'from_id': 'E', 'to_id': 'B', 'weight': 51.0}, {'from_id': 'E', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'F', 'weight': 42.0}, {'from_id': 'E', 'to_id': 'G', 'weight': 1.0}, {'from_id': 'E', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'I', 'weight': 3.0}, {'from_id': 'E', 'to_id': 'J', 'weight': 3.0}, {'from_id': 'E', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'L', 'weight': 15.0}, {'from_id': 'E', 'to_id': 'M', 'weight': 4.0}, {'from_id': 'E', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'P', 'weight': 50.0}, {'from_id': 'E', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'R', 'weight': 57.0}, {'from_id': 'E', 'to_id': 'S', 'weight': 112.0}, {'from_id': 'E', 'to_id': 'T', 'weight': 21.0}, {'from_id': 'E', 'to_id': 'U', 'weight': 188.0}, {'from_id': 'E', 'to_id': 'V', 'weight': 24.0}, {'from_id': 'F', 'to_id': 'A', 'weight': 475.0}, {'from_id': 'F', 'to_id': 'B', 'weight': 86.0}, {'from_id': 'F', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'E', 'weight': 75.0}, {'from_id': 'F', 'to_id': 'G', 'weight': 3.0}, {'from_id': 'F', 'to_id': 'H', 'weight': 40.0}, {'from_id': 'F', 'to_id': 'I', 'weight': 24.0}, {'from_id': 'F', 'to_id': 'J', 'weight': 10.0}, {'from_id': 'F', 'to_id': 'K', 'weight': 15.0}, {'from_id': 'F', 'to_id': 'L', 'weight': 18.0}, {'from_id': 'F', 'to_id': 'M', 'weight': 3.0}, {'from_id': 'F', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'P', 'weight': 187.0}, {'from_id': 'F', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'R', 'weight': 41.0}, {'from_id': 'F', 'to_id': 'S', 'weight': 216.0}, {'from_id': 'F', 'to_id': 'T', 'weight': 51.0}, {'from_id': 'F', 'to_id': 'U', 'weight': 393.0}, {'from_id': 'F', 'to_id': 'V', 'weight': 19.0}, {'from_id': 'G', 'to_id': 'A', 'weight': 88.0}, {'from_id': 'G', 'to_id': 'B', 'weight': 3.0}, {'from_id': 'G', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'E', 'weight': 7.0}, {'from_id': 'G', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'H', 'weight': 4.0}, {'from_id': 'G', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'L', 'weight': 19.0}, {'from_id': 'G', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'N', 'weight': 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564.0}, {'from_id': 'V', 'to_id': 'K', 'weight': 13.0}, {'from_id': 'V', 'to_id': 'L', 'weight': 1133.0}, {'from_id': 'V', 'to_id': 'M', 'weight': 383.0}, {'from_id': 'V', 'to_id': 'N', 'weight': 59.0}, {'from_id': 'V', 'to_id': 'O', 'weight': 98.0}, {'from_id': 'V', 'to_id': 'P', 'weight': 1851.0}, {'from_id': 'V', 'to_id': 'Q', 'weight': 2909.0}, {'from_id': 'V', 'to_id': 'R', 'weight': 836.0}, {'from_id': 'V', 'to_id': 'S', 'weight': 1475.0}, {'from_id': 'V', 'to_id': 'T', 'weight': 1764.0}, {'from_id': 'V', 'to_id': 'U', 'weight': 25418.0}]}","['V', 'I', 'M', 'H', 'J', 'P', 'N', 'O', 'F', 'K', 'D', 'G', 'Q', 'E', 'S', 'U', 'T', 'R', 'A', 'B', 'C', 'L']",20,nl,names
LOP,LOP,"In a small gallery project the job is to arrange the paintings in a single straight line to get the most engaging visitor journey. For each pair of paintings there’s a small impact number showing how much better it is when one is placed before the other, and the total effect of any ordering is simply all those numbers added up for every ordered pair. The requirement is to use every painting exactly one time — nothing left out and nothing repeated — and to adopt the ordering that yields the largest combined impact. The specific list of works and their pairwise impact values are shown below.

They are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, a set of 16 paintings to order.

*Meaning: this is a directed visitor_impact_score matrix over items in painting_ids (16 total). Entry at row preceding_painting_id=i and column following_painting_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).*

**visitor_impact_score)**

| preceding_painting_id\following_painting_id | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 16193.0 | 9526.0 | 7761.0 | 3349.0 | 88.0 | 4127.0 | 557.0 | 1906.0 | 3852.0 | 596.0 | 3456.0 | 691.0 | 551.0 | 2040.0 | 291.0 | 1284.0 |
| 1 | 960.0 | 25994.0 | 1077.0 | 1600.0 | 0.0 | 208.0 | 78.0 | 1678.0 | 284.0 | 668.0 | 136265.0 | 148.0 | 25.0 | 69.0 | 67.0 | 92.0 |
| 2 | 550.0 | 979.0 | 23474.0 | 4337.0 | 89.0 | 907.0 | 1369.0 | 2866.0 | 670.0 | 3974.0 | 65772.0 | 2572.0 | 84.0 | 245.0 | 34.0 | 96.0 |
| 3 | 7.0 | 258.0 | 15.0 | 90133.0 | 0.0 | 307.0 | 237.0 | 28.0 | 84.0 | 77.0 | 279.0 | 0.0 | 3.0 | 0.0 | 132.0 | 28.0 |
| 4 | 0.0 | 0.0 | 0.0 | 0.0 | 66.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 5 | 12.0 | 321.0 | 366.0 | 1220.0 | 24.0 | 116048.0 | 3067.0 | 1812.0 | 245.0 | 1086.0 | 535.0 | 198.0 | 35.0 | 68.0 | 174.0 | 194.0 |
| 6 | 56.0 | 76.0 | 76.0 | 65.0 | 2.0 | 7476.0 | 25297.0 | 196.0 | 69.0 | 256.0 | 135.0 | 130.0 | 3.0 | 0.0 | 1.0 | 47.0 |
| 7 | 95.0 | 585.0 | 1724.0 | 1120.0 | 165.0 | 350.0 | 850.0 | 29223.0 | 96.0 | 1414.0 | 33725.0 | 212.0 | 119.0 | 53.0 | 15.0 | 166.0 |
| 8 | 715.0 | 4641.0 | 1433.0 | 6417.0 | 2327.0 | 1829.0 | 1935.0 | 567.0 | 81181.0 | 1237.0 | 3465.0 | 78.0 | 252.0 | 87.0 | 97.0 | 1199.0 |
| 9 | 46.0 | 100.0 | 433.0 | 60.0 | 7.0 | 1818.0 | 136.0 | 56.0 | 153.0 | 3286.0 | 188.0 | 10.0 | 8.0 | 50.0 | 9.0 | 926.0 |
| 10 | 610.0 | 717.0 | 540.0 | 577.0 | 41.0 | 419.0 | 101.0 | 307.0 | 122.0 | 101.0 | 962.0 | 365.0 | 578.0 | 12471.0 | 169.0 | 2575.0 |
| 11 | 222.0 | 432.0 | 4890.0 | 342.0 | 27.0 | 210.0 | 108.0 | 253.0 | 6105.0 | 76.0 | 2137.0 | 103.0 | 63.0 | 58.0 | 48.0 | 151.0 |
| 12 | 120.0 | 504.0 | 561.0 | 708.0 | 40.0 | 406.0 | 329.0 | 623.0 | 403.0 | 117.0 | 1497.0 | 114.0 | 234.0 | 56.0 | 6.0 | 106.0 |
| 13 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 175.0 | 0.0 | 0.0 |
| 14 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 15 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2684.0 | 0.0 | 453.0 | 0.0 | 0.0 | 116.0 | 225.0 | 2337.0 |

Arrange the paintings in a single straight line to maximize the combined visitor impact.

Oh, and when you send back the ordering, just stick to a tiny JSON snippet like this so it's easy to read and parse:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of ""solution"" as the single list that shows the paintings from left to right; each placeholder in the array is where you'd put the exact label for a painting in that position. This JSON is just a sketch of the shape I need — not the final answer itself.

Please use the exact identifiers from the instance input — don't rename them or invent new ones.  
- 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”.""","[[16193.0, 9526.0, 7761.0, 3349.0, 88.0, 4127.0, 557.0, 1906.0, 3852.0, 596.0, 3456.0, 691.0, 551.0, 2040.0, 291.0, 1284.0], [960.0, 25994.0, 1077.0, 1600.0, 0.0, 208.0, 78.0, 1678.0, 284.0, 668.0, 136265.0, 148.0, 25.0, 69.0, 67.0, 92.0], [550.0, 979.0, 23474.0, 4337.0, 89.0, 907.0, 1369.0, 2866.0, 670.0, 3974.0, 65772.0, 2572.0, 84.0, 245.0, 34.0, 96.0], [7.0, 258.0, 15.0, 90133.0, 0.0, 307.0, 237.0, 28.0, 84.0, 77.0, 279.0, 0.0, 3.0, 0.0, 132.0, 28.0], [0.0, 0.0, 0.0, 0.0, 66.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [12.0, 321.0, 366.0, 1220.0, 24.0, 116048.0, 3067.0, 1812.0, 245.0, 1086.0, 535.0, 198.0, 35.0, 68.0, 174.0, 194.0], [56.0, 76.0, 76.0, 65.0, 2.0, 7476.0, 25297.0, 196.0, 69.0, 256.0, 135.0, 130.0, 3.0, 0.0, 1.0, 47.0], [95.0, 585.0, 1724.0, 1120.0, 165.0, 350.0, 850.0, 29223.0, 96.0, 1414.0, 33725.0, 212.0, 119.0, 53.0, 15.0, 166.0], [715.0, 4641.0, 1433.0, 6417.0, 2327.0, 1829.0, 1935.0, 567.0, 81181.0, 1237.0, 3465.0, 78.0, 252.0, 87.0, 97.0, 1199.0], [46.0, 100.0, 433.0, 60.0, 7.0, 1818.0, 136.0, 56.0, 153.0, 3286.0, 188.0, 10.0, 8.0, 50.0, 9.0, 926.0], [610.0, 717.0, 540.0, 577.0, 41.0, 419.0, 101.0, 307.0, 122.0, 101.0, 962.0, 365.0, 578.0, 12471.0, 169.0, 2575.0], [222.0, 432.0, 4890.0, 342.0, 27.0, 210.0, 108.0, 253.0, 6105.0, 76.0, 2137.0, 103.0, 63.0, 58.0, 48.0, 151.0], [120.0, 504.0, 561.0, 708.0, 40.0, 406.0, 329.0, 623.0, 403.0, 117.0, 1497.0, 114.0, 234.0, 56.0, 6.0, 106.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 175.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2684.0, 0.0, 453.0, 0.0, 0.0, 116.0, 225.0, 2337.0]]","[0, 12, 11, 8, 2, 6, 5, 1, 7, 9, 10, 3, 15, 14, 13, 4]",374643.0,"{'problem_type': 'LOP', 'num_nodes': 16, 'nodes': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], 'pairs': [{'from_id': 0, 'to_id': 1, 'weight': 9526.0}, {'from_id': 0, 'to_id': 2, 'weight': 7761.0}, {'from_id': 0, 'to_id': 3, 'weight': 3349.0}, {'from_id': 0, 'to_id': 4, 'weight': 88.0}, {'from_id': 0, 'to_id': 5, 'weight': 4127.0}, {'from_id': 0, 'to_id': 6, 'weight': 557.0}, {'from_id': 0, 'to_id': 7, 'weight': 1906.0}, {'from_id': 0, 'to_id': 8, 'weight': 3852.0}, {'from_id': 0, 'to_id': 9, 'weight': 596.0}, {'from_id': 0, 'to_id': 10, 'weight': 3456.0}, {'from_id': 0, 'to_id': 11, 'weight': 691.0}, {'from_id': 0, 'to_id': 12, 'weight': 551.0}, {'from_id': 0, 'to_id': 13, 'weight': 2040.0}, {'from_id': 0, 'to_id': 14, 'weight': 291.0}, {'from_id': 0, 'to_id': 15, 'weight': 1284.0}, {'from_id': 1, 'to_id': 0, 'weight': 960.0}, {'from_id': 1, 'to_id': 2, 'weight': 1077.0}, {'from_id': 1, 'to_id': 3, 'weight': 1600.0}, {'from_id': 1, 'to_id': 4, 'weight': 0.0}, {'from_id': 1, 'to_id': 5, 'weight': 208.0}, {'from_id': 1, 'to_id': 6, 'weight': 78.0}, {'from_id': 1, 'to_id': 7, 'weight': 1678.0}, {'from_id': 1, 'to_id': 8, 'weight': 284.0}, {'from_id': 1, 'to_id': 9, 'weight': 668.0}, {'from_id': 1, 'to_id': 10, 'weight': 136265.0}, {'from_id': 1, 'to_id': 11, 'weight': 148.0}, {'from_id': 1, 'to_id': 12, 'weight': 25.0}, {'from_id': 1, 'to_id': 13, 'weight': 69.0}, {'from_id': 1, 'to_id': 14, 'weight': 67.0}, {'from_id': 1, 'to_id': 15, 'weight': 92.0}, {'from_id': 2, 'to_id': 0, 'weight': 550.0}, {'from_id': 2, 'to_id': 1, 'weight': 979.0}, {'from_id': 2, 'to_id': 3, 'weight': 4337.0}, {'from_id': 2, 'to_id': 4, 'weight': 89.0}, {'from_id': 2, 'to_id': 5, 'weight': 907.0}, {'from_id': 2, 'to_id': 6, 'weight': 1369.0}, {'from_id': 2, 'to_id': 7, 'weight': 2866.0}, {'from_id': 2, 'to_id': 8, 'weight': 670.0}, {'from_id': 2, 'to_id': 9, 'weight': 3974.0}, {'from_id': 2, 'to_id': 10, 'weight': 65772.0}, {'from_id': 2, 'to_id': 11, 'weight': 2572.0}, {'from_id': 2, 'to_id': 12, 'weight': 84.0}, {'from_id': 2, 'to_id': 13, 'weight': 245.0}, {'from_id': 2, 'to_id': 14, 'weight': 34.0}, {'from_id': 2, 'to_id': 15, 'weight': 96.0}, {'from_id': 3, 'to_id': 0, 'weight': 7.0}, {'from_id': 3, 'to_id': 1, 'weight': 258.0}, {'from_id': 3, 'to_id': 2, 'weight': 15.0}, {'from_id': 3, 'to_id': 4, 'weight': 0.0}, {'from_id': 3, 'to_id': 5, 'weight': 307.0}, {'from_id': 3, 'to_id': 6, 'weight': 237.0}, {'from_id': 3, 'to_id': 7, 'weight': 28.0}, {'from_id': 3, 'to_id': 8, 'weight': 84.0}, {'from_id': 3, 'to_id': 9, 'weight': 77.0}, {'from_id': 3, 'to_id': 10, 'weight': 279.0}, {'from_id': 3, 'to_id': 11, 'weight': 0.0}, {'from_id': 3, 'to_id': 12, 'weight': 3.0}, {'from_id': 3, 'to_id': 13, 'weight': 0.0}, {'from_id': 3, 'to_id': 14, 'weight': 132.0}, {'from_id': 3, 'to_id': 15, 'weight': 28.0}, {'from_id': 4, 'to_id': 0, 'weight': 0.0}, {'from_id': 4, 'to_id': 1, 'weight': 0.0}, {'from_id': 4, 'to_id': 2, 'weight': 0.0}, {'from_id': 4, 'to_id': 3, 'weight': 0.0}, {'from_id': 4, 'to_id': 5, 'weight': 0.0}, {'from_id': 4, 'to_id': 6, 'weight': 0.0}, {'from_id': 4, 'to_id': 7, 'weight': 0.0}, {'from_id': 4, 'to_id': 8, 'weight': 0.0}, {'from_id': 4, 'to_id': 9, 'weight': 0.0}, {'from_id': 4, 'to_id': 10, 'weight': 0.0}, {'from_id': 4, 'to_id': 11, 'weight': 0.0}, {'from_id': 4, 'to_id': 12, 'weight': 0.0}, {'from_id': 4, 'to_id': 13, 'weight': 0.0}, {'from_id': 4, 'to_id': 14, 'weight': 0.0}, {'from_id': 4, 'to_id': 15, 'weight': 0.0}, {'from_id': 5, 'to_id': 0, 'weight': 12.0}, {'from_id': 5, 'to_id': 1, 'weight': 321.0}, {'from_id': 5, 'to_id': 2, 'weight': 366.0}, {'from_id': 5, 'to_id': 3, 'weight': 1220.0}, {'from_id': 5, 'to_id': 4, 'weight': 24.0}, {'from_id': 5, 'to_id': 6, 'weight': 3067.0}, {'from_id': 5, 'to_id': 7, 'weight': 1812.0}, {'from_id': 5, 'to_id': 8, 'weight': 245.0}, {'from_id': 5, 'to_id': 9, 'weight': 1086.0}, {'from_id': 5, 'to_id': 10, 'weight': 535.0}, {'from_id': 5, 'to_id': 11, 'weight': 198.0}, {'from_id': 5, 'to_id': 12, 'weight': 35.0}, {'from_id': 5, 'to_id': 13, 'weight': 68.0}, {'from_id': 5, 'to_id': 14, 'weight': 174.0}, {'from_id': 5, 'to_id': 15, 'weight': 194.0}, {'from_id': 6, 'to_id': 0, 'weight': 56.0}, {'from_id': 6, 'to_id': 1, 'weight': 76.0}, {'from_id': 6, 'to_id': 2, 'weight': 76.0}, {'from_id': 6, 'to_id': 3, 'weight': 65.0}, {'from_id': 6, 'to_id': 4, 'weight': 2.0}, {'from_id': 6, 'to_id': 5, 'weight': 7476.0}, {'from_id': 6, 'to_id': 7, 'weight': 196.0}, {'from_id': 6, 'to_id': 8, 'weight': 69.0}, {'from_id': 6, 'to_id': 9, 'weight': 256.0}, {'from_id': 6, 'to_id': 10, 'weight': 135.0}, {'from_id': 6, 'to_id': 11, 'weight': 130.0}, {'from_id': 6, 'to_id': 12, 'weight': 3.0}, {'from_id': 6, 'to_id': 13, 'weight': 0.0}, {'from_id': 6, 'to_id': 14, 'weight': 1.0}, {'from_id': 6, 'to_id': 15, 'weight': 47.0}, {'from_id': 7, 'to_id': 0, 'weight': 95.0}, {'from_id': 7, 'to_id': 1, 'weight': 585.0}, {'from_id': 7, 'to_id': 2, 'weight': 1724.0}, {'from_id': 7, 'to_id': 3, 'weight': 1120.0}, {'from_id': 7, 'to_id': 4, 'weight': 165.0}, {'from_id': 7, 'to_id': 5, 'weight': 350.0}, {'from_id': 7, 'to_id': 6, 'weight': 850.0}, {'from_id': 7, 'to_id': 8, 'weight': 96.0}, {'from_id': 7, 'to_id': 9, 'weight': 1414.0}, {'from_id': 7, 'to_id': 10, 'weight': 33725.0}, {'from_id': 7, 'to_id': 11, 'weight': 212.0}, {'from_id': 7, 'to_id': 12, 'weight': 119.0}, {'from_id': 7, 'to_id': 13, 'weight': 53.0}, {'from_id': 7, 'to_id': 14, 'weight': 15.0}, {'from_id': 7, 'to_id': 15, 'weight': 166.0}, {'from_id': 8, 'to_id': 0, 'weight': 715.0}, {'from_id': 8, 'to_id': 1, 'weight': 4641.0}, {'from_id': 8, 'to_id': 2, 'weight': 1433.0}, {'from_id': 8, 'to_id': 3, 'weight': 6417.0}, {'from_id': 8, 'to_id': 4, 'weight': 2327.0}, {'from_id': 8, 'to_id': 5, 'weight': 1829.0}, {'from_id': 8, 'to_id': 6, 'weight': 1935.0}, {'from_id': 8, 'to_id': 7, 'weight': 567.0}, {'from_id': 8, 'to_id': 9, 'weight': 1237.0}, {'from_id': 8, 'to_id': 10, 'weight': 3465.0}, {'from_id': 8, 'to_id': 11, 'weight': 78.0}, {'from_id': 8, 'to_id': 12, 'weight': 252.0}, {'from_id': 8, 'to_id': 13, 'weight': 87.0}, {'from_id': 8, 'to_id': 14, 'weight': 97.0}, {'from_id': 8, 'to_id': 15, 'weight': 1199.0}, {'from_id': 9, 'to_id': 0, 'weight': 46.0}, {'from_id': 9, 'to_id': 1, 'weight': 100.0}, {'from_id': 9, 'to_id': 2, 'weight': 433.0}, {'from_id': 9, 'to_id': 3, 'weight': 60.0}, {'from_id': 9, 'to_id': 4, 'weight': 7.0}, {'from_id': 9, 'to_id': 5, 'weight': 1818.0}, {'from_id': 9, 'to_id': 6, 'weight': 136.0}, {'from_id': 9, 'to_id': 7, 'weight': 56.0}, {'from_id': 9, 'to_id': 8, 'weight': 153.0}, {'from_id': 9, 'to_id': 10, 'weight': 188.0}, {'from_id': 9, 'to_id': 11, 'weight': 10.0}, {'from_id': 9, 'to_id': 12, 'weight': 8.0}, {'from_id': 9, 'to_id': 13, 'weight': 50.0}, {'from_id': 9, 'to_id': 14, 'weight': 9.0}, {'from_id': 9, 'to_id': 15, 'weight': 926.0}, {'from_id': 10, 'to_id': 0, 'weight': 610.0}, {'from_id': 10, 'to_id': 1, 'weight': 717.0}, {'from_id': 10, 'to_id': 2, 'weight': 540.0}, {'from_id': 10, 'to_id': 3, 'weight': 577.0}, {'from_id': 10, 'to_id': 4, 'weight': 41.0}, {'from_id': 10, 'to_id': 5, 'weight': 419.0}, {'from_id': 10, 'to_id': 6, 'weight': 101.0}, {'from_id': 10, 'to_id': 7, 'weight': 307.0}, {'from_id': 10, 'to_id': 8, 'weight': 122.0}, {'from_id': 10, 'to_id': 9, 'weight': 101.0}, {'from_id': 10, 'to_id': 11, 'weight': 365.0}, {'from_id': 10, 'to_id': 12, 'weight': 578.0}, {'from_id': 10, 'to_id': 13, 'weight': 12471.0}, {'from_id': 10, 'to_id': 14, 'weight': 169.0}, {'from_id': 10, 'to_id': 15, 'weight': 2575.0}, {'from_id': 11, 'to_id': 0, 'weight': 222.0}, {'from_id': 11, 'to_id': 1, 'weight': 432.0}, {'from_id': 11, 'to_id': 2, 'weight': 4890.0}, {'from_id': 11, 'to_id': 3, 'weight': 342.0}, {'from_id': 11, 'to_id': 4, 'weight': 27.0}, {'from_id': 11, 'to_id': 5, 'weight': 210.0}, {'from_id': 11, 'to_id': 6, 'weight': 108.0}, {'from_id': 11, 'to_id': 7, 'weight': 253.0}, {'from_id': 11, 'to_id': 8, 'weight': 6105.0}, {'from_id': 11, 'to_id': 9, 'weight': 76.0}, {'from_id': 11, 'to_id': 10, 'weight': 2137.0}, {'from_id': 11, 'to_id': 12, 'weight': 63.0}, {'from_id': 11, 'to_id': 13, 'weight': 58.0}, {'from_id': 11, 'to_id': 14, 'weight': 48.0}, {'from_id': 11, 'to_id': 15, 'weight': 151.0}, {'from_id': 12, 'to_id': 0, 'weight': 120.0}, {'from_id': 12, 'to_id': 1, 'weight': 504.0}, {'from_id': 12, 'to_id': 2, 'weight': 561.0}, {'from_id': 12, 'to_id': 3, 'weight': 708.0}, {'from_id': 12, 'to_id': 4, 'weight': 40.0}, {'from_id': 12, 'to_id': 5, 'weight': 406.0}, {'from_id': 12, 'to_id': 6, 'weight': 329.0}, {'from_id': 12, 'to_id': 7, 'weight': 623.0}, {'from_id': 12, 'to_id': 8, 'weight': 403.0}, {'from_id': 12, 'to_id': 9, 'weight': 117.0}, {'from_id': 12, 'to_id': 10, 'weight': 1497.0}, {'from_id': 12, 'to_id': 11, 'weight': 114.0}, {'from_id': 12, 'to_id': 13, 'weight': 56.0}, {'from_id': 12, 'to_id': 14, 'weight': 6.0}, {'from_id': 12, 'to_id': 15, 'weight': 106.0}, {'from_id': 13, 'to_id': 0, 'weight': 0.0}, {'from_id': 13, 'to_id': 1, 'weight': 0.0}, {'from_id': 13, 'to_id': 2, 'weight': 0.0}, {'from_id': 13, 'to_id': 3, 'weight': 0.0}, {'from_id': 13, 'to_id': 4, 'weight': 0.0}, {'from_id': 13, 'to_id': 5, 'weight': 0.0}, {'from_id': 13, 'to_id': 6, 'weight': 0.0}, {'from_id': 13, 'to_id': 7, 'weight': 0.0}, {'from_id': 13, 'to_id': 8, 'weight': 0.0}, {'from_id': 13, 'to_id': 9, 'weight': 0.0}, {'from_id': 13, 'to_id': 10, 'weight': 0.0}, {'from_id': 13, 'to_id': 11, 'weight': 0.0}, {'from_id': 13, 'to_id': 12, 'weight': 0.0}, {'from_id': 13, 'to_id': 14, 'weight': 0.0}, {'from_id': 13, 'to_id': 15, 'weight': 0.0}, {'from_id': 14, 'to_id': 0, 'weight': 0.0}, {'from_id': 14, 'to_id': 1, 'weight': 0.0}, {'from_id': 14, 'to_id': 2, 'weight': 0.0}, {'from_id': 14, 'to_id': 3, 'weight': 0.0}, {'from_id': 14, 'to_id': 4, 'weight': 0.0}, {'from_id': 14, 'to_id': 5, 'weight': 0.0}, {'from_id': 14, 'to_id': 6, 'weight': 0.0}, {'from_id': 14, 'to_id': 7, 'weight': 0.0}, {'from_id': 14, 'to_id': 8, 'weight': 0.0}, {'from_id': 14, 'to_id': 9, 'weight': 0.0}, {'from_id': 14, 'to_id': 10, 'weight': 0.0}, {'from_id': 14, 'to_id': 11, 'weight': 0.0}, {'from_id': 14, 'to_id': 12, 'weight': 0.0}, {'from_id': 14, 'to_id': 13, 'weight': 0.0}, {'from_id': 14, 'to_id': 15, 'weight': 0.0}, {'from_id': 15, 'to_id': 0, 'weight': 0.0}, {'from_id': 15, 'to_id': 1, 'weight': 0.0}, {'from_id': 15, 'to_id': 2, 'weight': 0.0}, {'from_id': 15, 'to_id': 3, 'weight': 0.0}, {'from_id': 15, 'to_id': 4, 'weight': 0.0}, {'from_id': 15, 'to_id': 5, 'weight': 0.0}, {'from_id': 15, 'to_id': 6, 'weight': 0.0}, {'from_id': 15, 'to_id': 7, 'weight': 0.0}, {'from_id': 15, 'to_id': 8, 'weight': 2684.0}, {'from_id': 15, 'to_id': 9, 'weight': 0.0}, {'from_id': 15, 'to_id': 10, 'weight': 453.0}, {'from_id': 15, 'to_id': 11, 'weight': 0.0}, {'from_id': 15, 'to_id': 12, 'weight': 0.0}, {'from_id': 15, 'to_id': 13, 'weight': 116.0}, {'from_id': 15, 'to_id': 14, 'weight': 225.0}]}","[0, 12, 11, 8, 2, 6, 5, 1, 7, 9, 10, 3, 15, 14, 13, 4]",21,markdown_table,0
LOP,LOP,"We’re organizing interviews and trying to pick the sequence that makes the most of the day. Between every pair of candidates there’s a score that tells how much advantage comes from interviewing one ahead of the other; for a given schedule, simply add all those scores for the pairs that follow that schedule, and the schedule with the largest overall total is the one to aim for. Each candidate has to be scheduled once and only once — no duplicates, no omissions. The concrete pair-by-pair details are listed below.

{
  ""total_candidates"": 18,
  ""candidate_ids"": [
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14,
    15,
    16,
    17,
    18
  ]
}

# Meaning: this is a directed precedence_benefit matrix over items in candidate_ids (18 total). Entry at row preceding_candidate_id=i and column following_candidate_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# precedence_benefit 
| preceding_candidate_id\following_candidate_id | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 11831.0 | 9.0 | 0.0 | 0.0 | 1767.0 | 2073.0 | 10215.0 | 2.0 | 89.0 | 0.0 | 2.0 | 0.0 | 0.0 | 0.0 | 26.0 | 249.0 | 0.0 | 85.0 |
| 2 | 1015.0 | 2097.0 | 215.0 | 235.0 | 289.0 | 37.0 | 527.0 | 116.0 | 24.0 | 317.0 | 12.0 | 0.0 | 0.0 | 0.0 | 16.0 | 0.0 | 0.0 | 53.0 |
| 3 | 0.0 | 69.0 | 58.0 | 3.0 | 0.0 | 0.0 | 4.0 | 0.0 | 2.0 | 57.0 | 60.0 | 0.0 | 0.0 | 0.0 | 36.0 | 230.0 | 0.0 | 80.0 |
| 4 | 0.0 | 3427.0 | 0.0 | 21904.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2411.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 |
| 5 | 241.0 | 0.0 | 0.0 | 0.0 | 429.0 | 1.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 3.0 | 87.0 | 12.0 | 4.0 |
| 6 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 43.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2.0 | 0.0 | 15.0 | 0.0 |
| 7 | 188.0 | 63.0 | 2.0 | 324.0 | 0.0 | 0.0 | 31426.0 | 447.0 | 185.0 | 1331.0 | 14.0 | 0.0 | 0.0 | 0.0 | 142.0 | 81.0 | 0.0 | 60.0 |
| 8 | 123.0 | 284.0 | 33.0 | 428.0 | 52.0 | 4.0 | 336.0 | 127.0 | 71.0 | 342.0 | 11.0 | 12.0 | 48.0 | 0.0 | 56.0 | 121.0 | 285.0 | 63.0 |
| 9 | 0.0 | 229.0 | 2.0 | 12.0 | 10.0 | 0.0 | 360.0 | 0.0 | 5192.0 | 109.0 | 1.0 | 0.0 | 23.0 | 0.0 | 38.0 | 0.0 | 12.0 | 0.0 |
| 10 | 309.0 | 195.0 | 1.0 | 2.0 | 89.0 | 7.0 | 109.0 | 24.0 | 3.0 | 9.0 | 33.0 | 28.0 | 56.0 | 0.0 | 14.0 | 25.0 | 35.0 | 16.0 |
| 11 | 6.0 | 158.0 | 16.0 | 22.0 | 117.0 | 43.0 | 355.0 | 78.0 | 10.0 | 141.0 | 91.0 | 562.0 | 235.0 | 0.0 | 195.0 | 319.0 | 303.0 | 255.0 |
| 12 | 145.0 | 59.0 | 5.0 | 34.0 | 61.0 | 10.0 | 171.0 | 11.0 | 12.0 | 75.0 | 7.0 | 15896.0 | 79.0 | 500.0 | 115.0 | 25.0 | 204.0 | 25.0 |
| 13 | 8.0 | 140.0 | 28.0 | 181.0 | 20.0 | 18.0 | 53.0 | 13.0 | 5.0 | 39.0 | 10.0 | 448.0 | 5.0 | 1.0 | 6.0 | 0.0 | 36.0 | 48.0 |
| 14 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 15 | 7.0 | 212.0 | 24.0 | 163.0 | 97.0 | 60.0 | 218.0 | 38.0 | 15.0 | 108.0 | 31.0 | 114.0 | 14.0 | 4.0 | 17.0 | 0.0 | 110.0 | 0.0 |
| 16 | 555.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 23.0 | 0.0 | 0.0 | 5.0 | 0.0 | 0.0 | 0.0 |
| 17 | 51.0 | 1061.0 | 121.0 | 279.0 | 863.0 | 295.0 | 793.0 | 210.0 | 73.0 | 599.0 | 167.0 | 4459.0 | 71.0 | 20.0 | 296.0 | 665.0 | 584.0 | 500.0 |
| 18 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Also, when you send back the final schedule, please use this simple JSON shape so it’s easy to read by whatever’s collecting the answers:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of ""solution"" as the ordered list of candidates from first interview to last — just fill in the candidate identifiers in the order you want. This JSON is just a sketch of the shape I expect, not the actual answer itself.

Please make sure to use the exact identifiers as they appear in the instance input — don’t 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”.""","[[11831.0, 9.0, 0.0, 0.0, 1767.0, 2073.0, 10215.0, 2.0, 89.0, 0.0, 2.0, 0.0, 0.0, 0.0, 26.0, 249.0, 0.0, 85.0], [1015.0, 2097.0, 215.0, 235.0, 289.0, 37.0, 527.0, 116.0, 24.0, 317.0, 12.0, 0.0, 0.0, 0.0, 16.0, 0.0, 0.0, 53.0], [0.0, 69.0, 58.0, 3.0, 0.0, 0.0, 4.0, 0.0, 2.0, 57.0, 60.0, 0.0, 0.0, 0.0, 36.0, 230.0, 0.0, 80.0], [0.0, 3427.0, 0.0, 21904.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2411.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0], [241.0, 0.0, 0.0, 0.0, 429.0, 1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 3.0, 87.0, 12.0, 4.0], [0.0, 0.0, 0.0, 0.0, 0.0, 43.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 15.0, 0.0], [188.0, 63.0, 2.0, 324.0, 0.0, 0.0, 31426.0, 447.0, 185.0, 1331.0, 14.0, 0.0, 0.0, 0.0, 142.0, 81.0, 0.0, 60.0], [123.0, 284.0, 33.0, 428.0, 52.0, 4.0, 336.0, 127.0, 71.0, 342.0, 11.0, 12.0, 48.0, 0.0, 56.0, 121.0, 285.0, 63.0], [0.0, 229.0, 2.0, 12.0, 10.0, 0.0, 360.0, 0.0, 5192.0, 109.0, 1.0, 0.0, 23.0, 0.0, 38.0, 0.0, 12.0, 0.0], [309.0, 195.0, 1.0, 2.0, 89.0, 7.0, 109.0, 24.0, 3.0, 9.0, 33.0, 28.0, 56.0, 0.0, 14.0, 25.0, 35.0, 16.0], [6.0, 158.0, 16.0, 22.0, 117.0, 43.0, 355.0, 78.0, 10.0, 141.0, 91.0, 562.0, 235.0, 0.0, 195.0, 319.0, 303.0, 255.0], [145.0, 59.0, 5.0, 34.0, 61.0, 10.0, 171.0, 11.0, 12.0, 75.0, 7.0, 15896.0, 79.0, 500.0, 115.0, 25.0, 204.0, 25.0], [8.0, 140.0, 28.0, 181.0, 20.0, 18.0, 53.0, 13.0, 5.0, 39.0, 10.0, 448.0, 5.0, 1.0, 6.0, 0.0, 36.0, 48.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [7.0, 212.0, 24.0, 163.0, 97.0, 60.0, 218.0, 38.0, 15.0, 108.0, 31.0, 114.0, 14.0, 4.0, 17.0, 0.0, 110.0, 0.0], [555.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 23.0, 0.0, 0.0, 5.0, 0.0, 0.0, 0.0], [51.0, 1061.0, 121.0, 279.0, 863.0, 295.0, 793.0, 210.0, 73.0, 599.0, 167.0, 4459.0, 71.0, 20.0, 296.0, 665.0, 584.0, 500.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]]","[10, 7, 16, 8, 14, 12, 11, 3, 1, 2, 15, 0, 6, 9, 4, 17, 13, 5]",43982.0,"{'problem_type': 'LOP', 'num_nodes': 18, 'nodes': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], 'pairs': [{'from_id': 1, 'to_id': 2, 'weight': 9.0}, {'from_id': 1, 'to_id': 3, 'weight': 0.0}, {'from_id': 1, 'to_id': 4, 'weight': 0.0}, {'from_id': 1, 'to_id': 5, 'weight': 1767.0}, {'from_id': 1, 'to_id': 6, 'weight': 2073.0}, {'from_id': 1, 'to_id': 7, 'weight': 10215.0}, {'from_id': 1, 'to_id': 8, 'weight': 2.0}, {'from_id': 1, 'to_id': 9, 'weight': 89.0}, {'from_id': 1, 'to_id': 10, 'weight': 0.0}, {'from_id': 1, 'to_id': 11, 'weight': 2.0}, {'from_id': 1, 'to_id': 12, 'weight': 0.0}, {'from_id': 1, 'to_id': 13, 'weight': 0.0}, {'from_id': 1, 'to_id': 14, 'weight': 0.0}, {'from_id': 1, 'to_id': 15, 'weight': 26.0}, {'from_id': 1, 'to_id': 16, 'weight': 249.0}, {'from_id': 1, 'to_id': 17, 'weight': 0.0}, {'from_id': 1, 'to_id': 18, 'weight': 85.0}, {'from_id': 2, 'to_id': 1, 'weight': 1015.0}, {'from_id': 2, 'to_id': 3, 'weight': 215.0}, {'from_id': 2, 'to_id': 4, 'weight': 235.0}, {'from_id': 2, 'to_id': 5, 'weight': 289.0}, {'from_id': 2, 'to_id': 6, 'weight': 37.0}, {'from_id': 2, 'to_id': 7, 'weight': 527.0}, {'from_id': 2, 'to_id': 8, 'weight': 116.0}, {'from_id': 2, 'to_id': 9, 'weight': 24.0}, {'from_id': 2, 'to_id': 10, 'weight': 317.0}, {'from_id': 2, 'to_id': 11, 'weight': 12.0}, {'from_id': 2, 'to_id': 12, 'weight': 0.0}, {'from_id': 2, 'to_id': 13, 'weight': 0.0}, {'from_id': 2, 'to_id': 14, 'weight': 0.0}, {'from_id': 2, 'to_id': 15, 'weight': 16.0}, {'from_id': 2, 'to_id': 16, 'weight': 0.0}, {'from_id': 2, 'to_id': 17, 'weight': 0.0}, {'from_id': 2, 'to_id': 18, 'weight': 53.0}, {'from_id': 3, 'to_id': 1, 'weight': 0.0}, {'from_id': 3, 'to_id': 2, 'weight': 69.0}, {'from_id': 3, 'to_id': 4, 'weight': 3.0}, {'from_id': 3, 'to_id': 5, 'weight': 0.0}, {'from_id': 3, 'to_id': 6, 'weight': 0.0}, {'from_id': 3, 'to_id': 7, 'weight': 4.0}, {'from_id': 3, 'to_id': 8, 'weight': 0.0}, {'from_id': 3, 'to_id': 9, 'weight': 2.0}, {'from_id': 3, 'to_id': 10, 'weight': 57.0}, {'from_id': 3, 'to_id': 11, 'weight': 60.0}, {'from_id': 3, 'to_id': 12, 'weight': 0.0}, {'from_id': 3, 'to_id': 13, 'weight': 0.0}, {'from_id': 3, 'to_id': 14, 'weight': 0.0}, {'from_id': 3, 'to_id': 15, 'weight': 36.0}, {'from_id': 3, 'to_id': 16, 'weight': 230.0}, {'from_id': 3, 'to_id': 17, 'weight': 0.0}, {'from_id': 3, 'to_id': 18, 'weight': 80.0}, {'from_id': 4, 'to_id': 1, 'weight': 0.0}, {'from_id': 4, 'to_id': 2, 'weight': 3427.0}, {'from_id': 4, 'to_id': 3, 'weight': 0.0}, {'from_id': 4, 'to_id': 5, 'weight': 0.0}, {'from_id': 4, 'to_id': 6, 'weight': 0.0}, {'from_id': 4, 'to_id': 7, 'weight': 0.0}, {'from_id': 4, 'to_id': 8, 'weight': 0.0}, {'from_id': 4, 'to_id': 9, 'weight': 0.0}, {'from_id': 4, 'to_id': 10, 'weight': 2411.0}, {'from_id': 4, 'to_id': 11, 'weight': 0.0}, {'from_id': 4, 'to_id': 12, 'weight': 0.0}, {'from_id': 4, 'to_id': 13, 'weight': 0.0}, {'from_id': 4, 'to_id': 14, 'weight': 0.0}, {'from_id': 4, 'to_id': 15, 'weight': 1.0}, {'from_id': 4, 'to_id': 16, 'weight': 0.0}, {'from_id': 4, 'to_id': 17, 'weight': 0.0}, {'from_id': 4, 'to_id': 18, 'weight': 0.0}, {'from_id': 5, 'to_id': 1, 'weight': 241.0}, {'from_id': 5, 'to_id': 2, 'weight': 0.0}, {'from_id': 5, 'to_id': 3, 'weight': 0.0}, {'from_id': 5, 'to_id': 4, 'weight': 0.0}, {'from_id': 5, 'to_id': 6, 'weight': 1.0}, {'from_id': 5, 'to_id': 7, 'weight': 0.0}, {'from_id': 5, 'to_id': 8, 'weight': 0.0}, {'from_id': 5, 'to_id': 9, 'weight': 0.0}, {'from_id': 5, 'to_id': 10, 'weight': 1.0}, {'from_id': 5, 'to_id': 11, 'weight': 0.0}, {'from_id': 5, 'to_id': 12, 'weight': 0.0}, {'from_id': 5, 'to_id': 13, 'weight': 0.0}, {'from_id': 5, 'to_id': 14, 'weight': 0.0}, {'from_id': 5, 'to_id': 15, 'weight': 3.0}, {'from_id': 5, 'to_id': 16, 'weight': 87.0}, {'from_id': 5, 'to_id': 17, 'weight': 12.0}, {'from_id': 5, 'to_id': 18, 'weight': 4.0}, {'from_id': 6, 'to_id': 1, 'weight': 0.0}, {'from_id': 6, 'to_id': 2, 'weight': 0.0}, {'from_id': 6, 'to_id': 3, 'weight': 0.0}, {'from_id': 6, 'to_id': 4, 'weight': 0.0}, {'from_id': 6, 'to_id': 5, 'weight': 0.0}, {'from_id': 6, 'to_id': 7, 'weight': 0.0}, {'from_id': 6, 'to_id': 8, 'weight': 0.0}, {'from_id': 6, 'to_id': 9, 'weight': 0.0}, {'from_id': 6, 'to_id': 10, 'weight': 0.0}, {'from_id': 6, 'to_id': 11, 'weight': 0.0}, {'from_id': 6, 'to_id': 12, 'weight': 0.0}, {'from_id': 6, 'to_id': 13, 'weight': 0.0}, {'from_id': 6, 'to_id': 14, 'weight': 0.0}, {'from_id': 6, 'to_id': 15, 'weight': 2.0}, {'from_id': 6, 'to_id': 16, 'weight': 0.0}, {'from_id': 6, 'to_id': 17, 'weight': 15.0}, {'from_id': 6, 'to_id': 18, 'weight': 0.0}, {'from_id': 7, 'to_id': 1, 'weight': 188.0}, {'from_id': 7, 'to_id': 2, 'weight': 63.0}, {'from_id': 7, 'to_id': 3, 'weight': 2.0}, {'from_id': 7, 'to_id': 4, 'weight': 324.0}, {'from_id': 7, 'to_id': 5, 'weight': 0.0}, {'from_id': 7, 'to_id': 6, 'weight': 0.0}, {'from_id': 7, 'to_id': 8, 'weight': 447.0}, {'from_id': 7, 'to_id': 9, 'weight': 185.0}, {'from_id': 7, 'to_id': 10, 'weight': 1331.0}, {'from_id': 7, 'to_id': 11, 'weight': 14.0}, {'from_id': 7, 'to_id': 12, 'weight': 0.0}, {'from_id': 7, 'to_id': 13, 'weight': 0.0}, {'from_id': 7, 'to_id': 14, 'weight': 0.0}, {'from_id': 7, 'to_id': 15, 'weight': 142.0}, {'from_id': 7, 'to_id': 16, 'weight': 81.0}, {'from_id': 7, 'to_id': 17, 'weight': 0.0}, {'from_id': 7, 'to_id': 18, 'weight': 60.0}, {'from_id': 8, 'to_id': 1, 'weight': 123.0}, {'from_id': 8, 'to_id': 2, 'weight': 284.0}, {'from_id': 8, 'to_id': 3, 'weight': 33.0}, {'from_id': 8, 'to_id': 4, 'weight': 428.0}, {'from_id': 8, 'to_id': 5, 'weight': 52.0}, {'from_id': 8, 'to_id': 6, 'weight': 4.0}, {'from_id': 8, 'to_id': 7, 'weight': 336.0}, {'from_id': 8, 'to_id': 9, 'weight': 71.0}, {'from_id': 8, 'to_id': 10, 'weight': 342.0}, {'from_id': 8, 'to_id': 11, 'weight': 11.0}, {'from_id': 8, 'to_id': 12, 'weight': 12.0}, {'from_id': 8, 'to_id': 13, 'weight': 48.0}, {'from_id': 8, 'to_id': 14, 'weight': 0.0}, {'from_id': 8, 'to_id': 15, 'weight': 56.0}, {'from_id': 8, 'to_id': 16, 'weight': 121.0}, {'from_id': 8, 'to_id': 17, 'weight': 285.0}, {'from_id': 8, 'to_id': 18, 'weight': 63.0}, {'from_id': 9, 'to_id': 1, 'weight': 0.0}, {'from_id': 9, 'to_id': 2, 'weight': 229.0}, {'from_id': 9, 'to_id': 3, 'weight': 2.0}, {'from_id': 9, 'to_id': 4, 'weight': 12.0}, {'from_id': 9, 'to_id': 5, 'weight': 10.0}, {'from_id': 9, 'to_id': 6, 'weight': 0.0}, {'from_id': 9, 'to_id': 7, 'weight': 360.0}, {'from_id': 9, 'to_id': 8, 'weight': 0.0}, {'from_id': 9, 'to_id': 10, 'weight': 109.0}, {'from_id': 9, 'to_id': 11, 'weight': 1.0}, {'from_id': 9, 'to_id': 12, 'weight': 0.0}, {'from_id': 9, 'to_id': 13, 'weight': 23.0}, {'from_id': 9, 'to_id': 14, 'weight': 0.0}, {'from_id': 9, 'to_id': 15, 'weight': 38.0}, {'from_id': 9, 'to_id': 16, 'weight': 0.0}, {'from_id': 9, 'to_id': 17, 'weight': 12.0}, {'from_id': 9, 'to_id': 18, 'weight': 0.0}, {'from_id': 10, 'to_id': 1, 'weight': 309.0}, {'from_id': 10, 'to_id': 2, 'weight': 195.0}, {'from_id': 10, 'to_id': 3, 'weight': 1.0}, {'from_id': 10, 'to_id': 4, 'weight': 2.0}, {'from_id': 10, 'to_id': 5, 'weight': 89.0}, {'from_id': 10, 'to_id': 6, 'weight': 7.0}, {'from_id': 10, 'to_id': 7, 'weight': 109.0}, {'from_id': 10, 'to_id': 8, 'weight': 24.0}, {'from_id': 10, 'to_id': 9, 'weight': 3.0}, {'from_id': 10, 'to_id': 11, 'weight': 33.0}, {'from_id': 10, 'to_id': 12, 'weight': 28.0}, {'from_id': 10, 'to_id': 13, 'weight': 56.0}, {'from_id': 10, 'to_id': 14, 'weight': 0.0}, {'from_id': 10, 'to_id': 15, 'weight': 14.0}, {'from_id': 10, 'to_id': 16, 'weight': 25.0}, {'from_id': 10, 'to_id': 17, 'weight': 35.0}, {'from_id': 10, 'to_id': 18, 'weight': 16.0}, {'from_id': 11, 'to_id': 1, 'weight': 6.0}, {'from_id': 11, 'to_id': 2, 'weight': 158.0}, {'from_id': 11, 'to_id': 3, 'weight': 16.0}, {'from_id': 11, 'to_id': 4, 'weight': 22.0}, {'from_id': 11, 'to_id': 5, 'weight': 117.0}, {'from_id': 11, 'to_id': 6, 'weight': 43.0}, {'from_id': 11, 'to_id': 7, 'weight': 355.0}, {'from_id': 11, 'to_id': 8, 'weight': 78.0}, {'from_id': 11, 'to_id': 9, 'weight': 10.0}, {'from_id': 11, 'to_id': 10, 'weight': 141.0}, {'from_id': 11, 'to_id': 12, 'weight': 562.0}, {'from_id': 11, 'to_id': 13, 'weight': 235.0}, {'from_id': 11, 'to_id': 14, 'weight': 0.0}, {'from_id': 11, 'to_id': 15, 'weight': 195.0}, {'from_id': 11, 'to_id': 16, 'weight': 319.0}, {'from_id': 11, 'to_id': 17, 'weight': 303.0}, {'from_id': 11, 'to_id': 18, 'weight': 255.0}, {'from_id': 12, 'to_id': 1, 'weight': 145.0}, {'from_id': 12, 'to_id': 2, 'weight': 59.0}, {'from_id': 12, 'to_id': 3, 'weight': 5.0}, {'from_id': 12, 'to_id': 4, 'weight': 34.0}, {'from_id': 12, 'to_id': 5, 'weight': 61.0}, {'from_id': 12, 'to_id': 6, 'weight': 10.0}, {'from_id': 12, 'to_id': 7, 'weight': 171.0}, {'from_id': 12, 'to_id': 8, 'weight': 11.0}, {'from_id': 12, 'to_id': 9, 'weight': 12.0}, {'from_id': 12, 'to_id': 10, 'weight': 75.0}, {'from_id': 12, 'to_id': 11, 'weight': 7.0}, {'from_id': 12, 'to_id': 13, 'weight': 79.0}, {'from_id': 12, 'to_id': 14, 'weight': 500.0}, {'from_id': 12, 'to_id': 15, 'weight': 115.0}, {'from_id': 12, 'to_id': 16, 'weight': 25.0}, {'from_id': 12, 'to_id': 17, 'weight': 204.0}, {'from_id': 12, 'to_id': 18, 'weight': 25.0}, {'from_id': 13, 'to_id': 1, 'weight': 8.0}, {'from_id': 13, 'to_id': 2, 'weight': 140.0}, {'from_id': 13, 'to_id': 3, 'weight': 28.0}, {'from_id': 13, 'to_id': 4, 'weight': 181.0}, {'from_id': 13, 'to_id': 5, 'weight': 20.0}, {'from_id': 13, 'to_id': 6, 'weight': 18.0}, {'from_id': 13, 'to_id': 7, 'weight': 53.0}, {'from_id': 13, 'to_id': 8, 'weight': 13.0}, {'from_id': 13, 'to_id': 9, 'weight': 5.0}, {'from_id': 13, 'to_id': 10, 'weight': 39.0}, {'from_id': 13, 'to_id': 11, 'weight': 10.0}, {'from_id': 13, 'to_id': 12, 'weight': 448.0}, {'from_id': 13, 'to_id': 14, 'weight': 1.0}, {'from_id': 13, 'to_id': 15, 'weight': 6.0}, {'from_id': 13, 'to_id': 16, 'weight': 0.0}, {'from_id': 13, 'to_id': 17, 'weight': 36.0}, {'from_id': 13, 'to_id': 18, 'weight': 48.0}, {'from_id': 14, 'to_id': 1, 'weight': 0.0}, {'from_id': 14, 'to_id': 2, 'weight': 0.0}, {'from_id': 14, 'to_id': 3, 'weight': 0.0}, {'from_id': 14, 'to_id': 4, 'weight': 0.0}, {'from_id': 14, 'to_id': 5, 'weight': 0.0}, {'from_id': 14, 'to_id': 6, 'weight': 0.0}, {'from_id': 14, 'to_id': 7, 'weight': 0.0}, {'from_id': 14, 'to_id': 8, 'weight': 0.0}, {'from_id': 14, 'to_id': 9, 'weight': 0.0}, {'from_id': 14, 'to_id': 10, 'weight': 0.0}, {'from_id': 14, 'to_id': 11, 'weight': 0.0}, {'from_id': 14, 'to_id': 12, 'weight': 0.0}, {'from_id': 14, 'to_id': 13, 'weight': 0.0}, {'from_id': 14, 'to_id': 15, 'weight': 0.0}, {'from_id': 14, 'to_id': 16, 'weight': 0.0}, {'from_id': 14, 'to_id': 17, 'weight': 0.0}, {'from_id': 14, 'to_id': 18, 'weight': 0.0}, {'from_id': 15, 'to_id': 1, 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'weight': 167.0}, {'from_id': 17, 'to_id': 12, 'weight': 4459.0}, {'from_id': 17, 'to_id': 13, 'weight': 71.0}, {'from_id': 17, 'to_id': 14, 'weight': 20.0}, {'from_id': 17, 'to_id': 15, 'weight': 296.0}, {'from_id': 17, 'to_id': 16, 'weight': 665.0}, {'from_id': 17, 'to_id': 18, 'weight': 500.0}, {'from_id': 18, 'to_id': 1, 'weight': 0.0}, {'from_id': 18, 'to_id': 2, 'weight': 0.0}, {'from_id': 18, 'to_id': 3, 'weight': 0.0}, {'from_id': 18, 'to_id': 4, 'weight': 0.0}, {'from_id': 18, 'to_id': 5, 'weight': 0.0}, {'from_id': 18, 'to_id': 6, 'weight': 0.0}, {'from_id': 18, 'to_id': 7, 'weight': 0.0}, {'from_id': 18, 'to_id': 8, 'weight': 0.0}, {'from_id': 18, 'to_id': 9, 'weight': 0.0}, {'from_id': 18, 'to_id': 10, 'weight': 0.0}, {'from_id': 18, 'to_id': 11, 'weight': 0.0}, {'from_id': 18, 'to_id': 12, 'weight': 0.0}, {'from_id': 18, 'to_id': 13, 'weight': 0.0}, {'from_id': 18, 'to_id': 14, 'weight': 0.0}, {'from_id': 18, 'to_id': 15, 'weight': 0.0}, {'from_id': 18, 'to_id': 16, 'weight': 0.0}, {'from_id': 18, 'to_id': 17, 'weight': 0.0}]}","[11, 8, 17, 9, 15, 13, 12, 4, 2, 3, 16, 1, 7, 10, 5, 18, 14, 6]",22,json,1
LOP,LOP,"I’m putting together a course and need to pick the order of the lesson modules so the class gets the biggest overall learning boost. The idea is simple: for every pair of modules there’s a little advantage if one comes before the other, and the total usefulness of the whole course is just the sum of those pairwise boosts for every pair where the earlier one comes first. Every module has to appear exactly once — no repeats, no skipping — and the full list of module-to-module boosts and other specifics are shown below.

There are 23 modules: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23.
Meaning: this is a directed learning_benefit matrix over items in module_ids (23 total). Entry at row earlier_module=i and column later_module=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
learning_benefit:
| earlier_module\later_module | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 125900.0 | 0.0 | 0.0 | 1000.0 | 0.0 | 0.0 | 1500.0 | 0.0 | 0.0 | 0.0 | 365100.0 | 58900.0 | 111400.0 | 8700.0 | 600.0 | 0.0 | 10200.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 2 | 500.0 | 26500.0 | 107100.0 | 376600.0 | 0.0 | 33900.0 | 28700.0 | 2700.0 | 1500.0 | 1900.0 | 700.0 | 3600.0 | 0.0 | 800.0 | 500.0 | 800.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 3 | 0.0 | 200.0 | 400.0 | 6300.0 | 0.0 | 66300.0 | 3300.0 | 1300.0 | 200.0 | 300.0 | 0.0 | 100.0 | 0.0 | 100.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 4 | 30300.0 | 34200.0 | 8000.0 | 31600.0 | 0.0 | 148300.0 | 52600.0 | 41700.0 | 27100.0 | 19400.0 | 4400.0 | 7700.0 | 1200.0 | 3600.0 | 3800.0 | 27200.0 | 20300.0 | 0.0 | 8400.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 5 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 6 | 800.0 | 38700.0 | 0.0 | 17000.0 | 0.0 | 1528500.0 | 27000.0 | 529100.0 | 272200.0 | 110800.0 | 7200.0 | 2300.0 | 4800.0 | 500.0 | 17800.0 | 23300.0 | 600.0 | 0.0 | 18200.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 7 | 38600.0 | 2800.0 | 300.0 | 21700.0 | 0.0 | 70900.0 | 138300.0 | 29200.0 | 37000.0 | 5300.0 | 10300.0 | 29800.0 | 0.0 | 300.0 | 13100.0 | 6600.0 | 1700.0 | 0.0 | 1900.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 8 | 29300.0 | 48800.0 | 1600.0 | 22000.0 | 0.0 | 79300.0 | 45800.0 | 345000.0 | 81000.0 | 82700.0 | 4500.0 | 12100.0 | 3400.0 | 2000.0 | 6000.0 | 1500.0 | 2400.0 | 0.0 | 5100.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 9 | 4200.0 | 0.0 | 0.0 | 1100.0 | 0.0 | 13700.0 | 2900.0 | 81000.0 | 25500.0 | 200.0 | 300.0 | 600.0 | 0.0 | 100.0 | 0.0 | 43100.0 | 100.0 | 0.0 | 1100.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 10 | 8400.0 | 0.0 | 0.0 | 300.0 | 0.0 | 6800.0 | 0.0 | 400.0 | 2600.0 | 27300.0 | 0.0 | 0.0 | 0.0 | 100.0 | 200.0 | 90500.0 | 69500.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 11 | 3600.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 0.0 | 5500.0 | 1900.0 | 0.0 | 38500.0 | 200.0 | 0.0 | 2100.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 12 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 900.0 | 8600.0 | 0.0 | 0.0 | 0.0 | 0.0 | 3700.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 13 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 14 | 0.0 | 0.0 | 0.0 | 300.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2600.0 | 200.0 | 0.0 | 0.0 | 0.0 | 60500.0 | 500.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 15 | 0.0 | 100.0 | 0.0 | 5100.0 | 0.0 | 500.0 | 1900.0 | 7600.0 | 6700.0 | 800.0 | 0.0 | 300.0 | 100.0 | 2900.0 | 25400.0 | 1800.0 | 600.0 | 0.0 | 500.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 16 | 38200.0 | 28400.0 | 20300.0 | 47100.0 | 0.0 | 94200.0 | 162700.0 | 45600.0 | 22700.0 | 5000.0 | 24000.0 | 24100.0 | 5700.0 | 4600.0 | 4300.0 | 100.0 | 200.0 | 0.0 | 27400.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 17 | 7100.0 | 0.0 | 0.0 | 500.0 | 0.0 | 14300.0 | 1400.0 | 11700.0 | 6300.0 | 3900.0 | 2900.0 | 2400.0 | 2200.0 | 1900.0 | 1600.0 | 2200.0 | 0.0 | 0.0 | 39300.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 18 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 19 | 8200.0 | 1600.0 | 0.0 | 10400.0 | 0.0 | 4700.0 | 5600.0 | 22900.0 | 4200.0 | 4000.0 | 1300.0 | 2600.0 | 900.0 | 1500.0 | 1500.0 | 15000.0 | 6100.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 20 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 21 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 22 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 23 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
I'll use these to choose the module sequence that maximizes the total learning boost.

Also, when you give the final ordering, please use this simple JSON layout so I can pick it up automatically.

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

The ""solution"" array should list the lesson modules in the exact sequence you want them taught, from the first module to the last. Think of it like a form: one slot per position in the course, each filled with the module identifier. This JSON is just a sketch of the shape I expect — don't treat it as the actual answer unless you replace the placeholders with the real module IDs.

Please make sure to use the identifiers exactly as they appear in the instance input — no renaming and no 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”.""","[[125900.0, 0.0, 0.0, 1000.0, 0.0, 0.0, 1500.0, 0.0, 0.0, 0.0, 365100.0, 58900.0, 111400.0, 8700.0, 600.0, 0.0, 10200.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [500.0, 26500.0, 107100.0, 376600.0, 0.0, 33900.0, 28700.0, 2700.0, 1500.0, 1900.0, 700.0, 3600.0, 0.0, 800.0, 500.0, 800.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 200.0, 400.0, 6300.0, 0.0, 66300.0, 3300.0, 1300.0, 200.0, 300.0, 0.0, 100.0, 0.0, 100.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [30300.0, 34200.0, 8000.0, 31600.0, 0.0, 148300.0, 52600.0, 41700.0, 27100.0, 19400.0, 4400.0, 7700.0, 1200.0, 3600.0, 3800.0, 27200.0, 20300.0, 0.0, 8400.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [800.0, 38700.0, 0.0, 17000.0, 0.0, 1528500.0, 27000.0, 529100.0, 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{'from_id': 22, 'to_id': 5, 'weight': 0.0}, {'from_id': 22, 'to_id': 6, 'weight': 0.0}, {'from_id': 22, 'to_id': 7, 'weight': 0.0}, {'from_id': 22, 'to_id': 8, 'weight': 0.0}, {'from_id': 22, 'to_id': 9, 'weight': 0.0}, {'from_id': 22, 'to_id': 10, 'weight': 0.0}, {'from_id': 22, 'to_id': 11, 'weight': 0.0}, {'from_id': 22, 'to_id': 12, 'weight': 0.0}, {'from_id': 22, 'to_id': 13, 'weight': 0.0}, {'from_id': 22, 'to_id': 14, 'weight': 0.0}, {'from_id': 22, 'to_id': 15, 'weight': 0.0}, {'from_id': 22, 'to_id': 16, 'weight': 0.0}, {'from_id': 22, 'to_id': 17, 'weight': 0.0}, {'from_id': 22, 'to_id': 18, 'weight': 0.0}, {'from_id': 22, 'to_id': 19, 'weight': 0.0}, {'from_id': 22, 'to_id': 20, 'weight': 0.0}, {'from_id': 22, 'to_id': 21, 'weight': 0.0}, {'from_id': 22, 'to_id': 23, 'weight': 0.0}, {'from_id': 23, 'to_id': 1, 'weight': 0.0}, {'from_id': 23, 'to_id': 2, 'weight': 0.0}, {'from_id': 23, 'to_id': 3, 'weight': 0.0}, {'from_id': 23, 'to_id': 4, 'weight': 0.0}, {'from_id': 23, 'to_id': 5, 'weight': 0.0}, {'from_id': 23, 'to_id': 6, 'weight': 0.0}, {'from_id': 23, 'to_id': 7, 'weight': 0.0}, {'from_id': 23, 'to_id': 8, 'weight': 0.0}, {'from_id': 23, 'to_id': 9, 'weight': 0.0}, {'from_id': 23, 'to_id': 10, 'weight': 0.0}, {'from_id': 23, 'to_id': 11, 'weight': 0.0}, {'from_id': 23, 'to_id': 12, 'weight': 0.0}, {'from_id': 23, 'to_id': 13, 'weight': 0.0}, {'from_id': 23, 'to_id': 14, 'weight': 0.0}, {'from_id': 23, 'to_id': 15, 'weight': 0.0}, {'from_id': 23, 'to_id': 16, 'weight': 0.0}, {'from_id': 23, 'to_id': 17, 'weight': 0.0}, {'from_id': 23, 'to_id': 18, 'weight': 0.0}, {'from_id': 23, 'to_id': 19, 'weight': 0.0}, {'from_id': 23, 'to_id': 20, 'weight': 0.0}, {'from_id': 23, 'to_id': 21, 'weight': 0.0}, {'from_id': 23, 'to_id': 22, 'weight': 0.0}]}","[23, 22, 21, 20, 16, 2, 4, 3, 7, 6, 15, 8, 10, 17, 19, 18, 9, 1, 11, 14, 13, 12, 5]",23,nl,1
LOP,LOP,"I’m the email marketing manager putting together a campaign that’ll send a bunch of ad creatives in a row. The choice to make is the exact order to send them in. For any two creatives there’s a known effect if the first one appears before the second, and the campaign’s overall effectiveness is just the total you get by adding up each of those pairwise effects for every pair in the final line-up. Every creative has to be used once and only once — no repeats, no skipping anything. The concrete pairwise numbers and the list of creatives are shown below.

# num_creatives_total=22
# creative_ids_list=A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V

# Meaning: this is a directed pairwise_effect_value matrix over items in creative_ids_list (22 total). Entry at row first_creative_id=i and column second_creative_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# pairwise_effect_value
| first_creative_id\second_creative_id | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 175.0 | 121.0 | 12.0 | 3.0 | 65.0 | 77.0 | 26.0 | 518.0 | 73.0 | 9.0 | 50.0 | 8.0 | 44.0 | 12.0 | 35.0 | 66.0 | 44.0 | 0.0 | 213.0 | 588.0 | 83.0 | 221.0 |
| B | 33.0 | 204.0 | 5.0 | 21.0 | 113.0 | 188.0 | 65.0 | 87.0 | 7.0 | 7.0 | 317.0 | 2.0 | 14.0 | 8.0 | 6.0 | 6.0 | 3.0 | 1.0 | 63.0 | 164.0 | 51.0 | 38.0 |
| C | 0.0 | 1.0 | 188.0 | 0.0 | 3.0 | 17.0 | 5.0 | 76.0 | 0.0 | 0.0 | 27.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 3.0 | 28.0 | 5.0 | 15.0 |
| D | 0.0 | 1.0 | 0.0 | 4.0 | 2.0 | 3.0 | 1.0 | 3.0 | 0.0 | 0.0 | 1.0 | 0.0 | 2.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 1.0 | 1.0 | 1.0 | 2.0 |
| E | 12.0 | 226.0 | 17.0 | 5.0 | 1188.0 | 43.0 | 14.0 | 235.0 | 13.0 | 1.0 | 29.0 | 21.0 | 32.0 | 221.0 | 26.0 | 6.0 | 3.0 | 2.0 | 9.0 | 459.0 | 37.0 | 26.0 |
| F | 31.0 | 1247.0 | 12.0 | 8.0 | 381.0 | 3775.0 | 109.0 | 180.0 | 22.0 | 2.0 | 81.0 | 24.0 | 7.0 | 61.0 | 39.0 | 32.0 | 8.0 | 4.0 | 37.0 | 617.0 | 100.0 | 36.0 |
| G | 5.0 | 8.0 | 0.0 | 0.0 | 14.0 | 1369.0 | 161.0 | 7.0 | 2.0 | 1.0 | 6.0 | 0.0 | 1.0 | 2.0 | 2.0 | 0.0 | 1.0 | 1.0 | 4.0 | 124.0 | 18.0 | 11.0 |
| H | 3.0 | 54.0 | 739.0 | 0.0 | 8.0 | 20.0 | 15.0 | 2618.0 | 15.0 | 10.0 | 15.0 | 1.0 | 3.0 | 6.0 | 20.0 | 8.0 | 1.0 | 0.0 | 5.0 | 758.0 | 57.0 | 26.0 |
| I | 8.0 | 21.0 | 1.0 | 0.0 | 10.0 | 0.0 | 12.0 | 24.0 | 170.0 | 0.0 | 5.0 | 0.0 | 12.0 | 58.0 | 55.0 | 229.0 | 2.0 | 1.0 | 3.0 | 200.0 | 17.0 | 15.0 |
| J | 9.0 | 2.0 | 0.0 | 0.0 | 1.0 | 46.0 | 17.0 | 8.0 | 2.0 | 49.0 | 20.0 | 1.0 | 32.0 | 18.0 | 60.0 | 2.0 | 2.0 | 1.0 | 3.0 | 144.0 | 17.0 | 8.0 |
| K | 8.0 | 6.0 | 1.0 | 0.0 | 4.0 | 907.0 | 222.0 | 13.0 | 3.0 | 1.0 | 102.0 | 3.0 | 4.0 | 7.0 | 15.0 | 2.0 | 2.0 | 1.0 | 3.0 | 151.0 | 21.0 | 13.0 |
| L | 6.0 | 21.0 | 1.0 | 0.0 | 78.0 | 3.0 | 4.0 | 11.0 | 4.0 | 0.0 | 3.0 | 33.0 | 19.0 | 9.0 | 18.0 | 1.0 | 1.0 | 1.0 | 2.0 | 55.0 | 6.0 | 5.0 |
| M | 385.0 | 9.0 | 2.0 | 0.0 | 5.0 | 4.0 | 47.0 | 37.0 | 15.0 | 0.0 | 38.0 | 22.0 | 303.0 | 8.0 | 0.0 | 132.0 | 3.0 | 2.0 | 5.0 | 233.0 | 21.0 | 14.0 |
| N | 15.0 | 9.0 | 1.0 | 0.0 | 14.0 | 0.0 | 8.0 | 24.0 | 3.0 | 0.0 | 6.0 | 1.0 | 13.0 | 67.0 | 83.0 | 15.0 | 2.0 | 1.0 | 2.0 | 69.0 | 12.0 | 13.0 |
| O | 6.0 | 6.0 | 1.0 | 0.0 | 1.0 | 1.0 | 0.0 | 13.0 | 12.0 | 0.0 | 1.0 | 3.0 | 9.0 | 14.0 | 26.0 | 36.0 | 1.0 | 1.0 | 2.0 | 72.0 | 7.0 | 11.0 |
| P | 49.0 | 76.0 | 5.0 | 0.0 | 4.0 | 1.0 | 0.0 | 78.0 | 10.0 | 0.0 | 4.0 | 0.0 | 29.0 | 186.0 | 65.0 | 3747.0 | 9.0 | 6.0 | 19.0 | 962.0 | 82.0 | 31.0 |
| Q | 413.0 | 36.0 | 3.0 | 0.0 | 1.0 | 0.0 | 0.0 | 47.0 | 8.0 | 0.0 | 3.0 | 3.0 | 11.0 | 8.0 | 6.0 | 3.0 | 828.0 | 1.0 | 4.0 | 78.0 | 20.0 | 26.0 |
| R | 48.0 | 4.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 8.0 | 0.0 | 0.0 | 1.0 | 0.0 | 8.0 | 32.0 | 1.0 | 0.0 | 1.0 | 585.0 | 2.0 | 32.0 | 24.0 | 11.0 |
| S | 121.0 | 4.0 | 37.0 | 1.0 | 5500.0 | 1338.0 | 122.0 | 556.0 | 89.0 | 1.0 | 101.0 | 296.0 | 332.0 | 59.0 | 63.0 | 16.0 | 5.0 | 3.0 | 863.0 | 1528.0 | 185.0 | 151.0 |
| T | 168.0 | 9.0 | 22.0 | 0.0 | 96.0 | 17.0 | 3.0 | 334.0 | 46.0 | 0.0 | 9.0 | 4.0 | 56.0 | 156.0 | 17.0 | 62.0 | 15.0 | 16.0 | 45.0 | 1290.0 | 891.0 | 130.0 |
| U | 0.0 | 5.0 | 3.0 | 0.0 | 7.0 | 8.0 | 4.0 | 49.0 | 4.0 | 11.0 | 5.0 | 4.0 | 6.0 | 16.0 | 10.0 | 16.0 | 0.0 | 0.0 | 43.0 | 45.0 | 0.0 | 50.0 |
| V | 282.0 | 181.0 | 24.0 | 3.0 | 90.0 | 15.0 | 10.0 | 484.0 | 111.0 | 239.0 | 167.0 | 23.0 | 196.0 | 171.0 | 50.0 | 308.0 | 18.0 | 4.0 | 2736.0 | 1798.0 | 589.0 | 1430.0 |

Also, when you send the final ordering back, it'd be handy if you used this simple JSON layout so it's super clear what the answer is:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of that ""solution"" array as the exact campaign run — the creatives listed from the one that goes out first to the one that goes out last. Replace each placeholder with the creative identifier from the instance. This JSON is just a sketch of the shape I need, not the actual ordering itself.

Please make sure to 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”.""","[[175.0, 121.0, 12.0, 3.0, 65.0, 77.0, 26.0, 518.0, 73.0, 9.0, 50.0, 8.0, 44.0, 12.0, 35.0, 66.0, 44.0, 0.0, 213.0, 588.0, 83.0, 221.0], [33.0, 204.0, 5.0, 21.0, 113.0, 188.0, 65.0, 87.0, 7.0, 7.0, 317.0, 2.0, 14.0, 8.0, 6.0, 6.0, 3.0, 1.0, 63.0, 164.0, 51.0, 38.0], [0.0, 1.0, 188.0, 0.0, 3.0, 17.0, 5.0, 76.0, 0.0, 0.0, 27.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 3.0, 28.0, 5.0, 15.0], [0.0, 1.0, 0.0, 4.0, 2.0, 3.0, 1.0, 3.0, 0.0, 0.0, 1.0, 0.0, 2.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 1.0, 1.0, 2.0], [12.0, 226.0, 17.0, 5.0, 1188.0, 43.0, 14.0, 235.0, 13.0, 1.0, 29.0, 21.0, 32.0, 221.0, 26.0, 6.0, 3.0, 2.0, 9.0, 459.0, 37.0, 26.0], [31.0, 1247.0, 12.0, 8.0, 381.0, 3775.0, 109.0, 180.0, 22.0, 2.0, 81.0, 24.0, 7.0, 61.0, 39.0, 32.0, 8.0, 4.0, 37.0, 617.0, 100.0, 36.0], [5.0, 8.0, 0.0, 0.0, 14.0, 1369.0, 161.0, 7.0, 2.0, 1.0, 6.0, 0.0, 1.0, 2.0, 2.0, 0.0, 1.0, 1.0, 4.0, 124.0, 18.0, 11.0], [3.0, 54.0, 739.0, 0.0, 8.0, 20.0, 15.0, 2618.0, 15.0, 10.0, 15.0, 1.0, 3.0, 6.0, 20.0, 8.0, 1.0, 0.0, 5.0, 758.0, 57.0, 26.0], [8.0, 21.0, 1.0, 0.0, 10.0, 0.0, 12.0, 24.0, 170.0, 0.0, 5.0, 0.0, 12.0, 58.0, 55.0, 229.0, 2.0, 1.0, 3.0, 200.0, 17.0, 15.0], [9.0, 2.0, 0.0, 0.0, 1.0, 46.0, 17.0, 8.0, 2.0, 49.0, 20.0, 1.0, 32.0, 18.0, 60.0, 2.0, 2.0, 1.0, 3.0, 144.0, 17.0, 8.0], [8.0, 6.0, 1.0, 0.0, 4.0, 907.0, 222.0, 13.0, 3.0, 1.0, 102.0, 3.0, 4.0, 7.0, 15.0, 2.0, 2.0, 1.0, 3.0, 151.0, 21.0, 13.0], [6.0, 21.0, 1.0, 0.0, 78.0, 3.0, 4.0, 11.0, 4.0, 0.0, 3.0, 33.0, 19.0, 9.0, 18.0, 1.0, 1.0, 1.0, 2.0, 55.0, 6.0, 5.0], [385.0, 9.0, 2.0, 0.0, 5.0, 4.0, 47.0, 37.0, 15.0, 0.0, 38.0, 22.0, 303.0, 8.0, 0.0, 132.0, 3.0, 2.0, 5.0, 233.0, 21.0, 14.0], [15.0, 9.0, 1.0, 0.0, 14.0, 0.0, 8.0, 24.0, 3.0, 0.0, 6.0, 1.0, 13.0, 67.0, 83.0, 15.0, 2.0, 1.0, 2.0, 69.0, 12.0, 13.0], [6.0, 6.0, 1.0, 0.0, 1.0, 1.0, 0.0, 13.0, 12.0, 0.0, 1.0, 3.0, 9.0, 14.0, 26.0, 36.0, 1.0, 1.0, 2.0, 72.0, 7.0, 11.0], [49.0, 76.0, 5.0, 0.0, 4.0, 1.0, 0.0, 78.0, 10.0, 0.0, 4.0, 0.0, 29.0, 186.0, 65.0, 3747.0, 9.0, 6.0, 19.0, 962.0, 82.0, 31.0], [413.0, 36.0, 3.0, 0.0, 1.0, 0.0, 0.0, 47.0, 8.0, 0.0, 3.0, 3.0, 11.0, 8.0, 6.0, 3.0, 828.0, 1.0, 4.0, 78.0, 20.0, 26.0], [48.0, 4.0, 0.0, 0.0, 0.0, 0.0, 0.0, 8.0, 0.0, 0.0, 1.0, 0.0, 8.0, 32.0, 1.0, 0.0, 1.0, 585.0, 2.0, 32.0, 24.0, 11.0], [121.0, 4.0, 37.0, 1.0, 5500.0, 1338.0, 122.0, 556.0, 89.0, 1.0, 101.0, 296.0, 332.0, 59.0, 63.0, 16.0, 5.0, 3.0, 863.0, 1528.0, 185.0, 151.0], [168.0, 9.0, 22.0, 0.0, 96.0, 17.0, 3.0, 334.0, 46.0, 0.0, 9.0, 4.0, 56.0, 156.0, 17.0, 62.0, 15.0, 16.0, 45.0, 1290.0, 891.0, 130.0], [0.0, 5.0, 3.0, 0.0, 7.0, 8.0, 4.0, 49.0, 4.0, 11.0, 5.0, 4.0, 6.0, 16.0, 10.0, 16.0, 0.0, 0.0, 43.0, 45.0, 0.0, 50.0], [282.0, 181.0, 24.0, 3.0, 90.0, 15.0, 10.0, 484.0, 111.0, 239.0, 167.0, 23.0, 196.0, 171.0, 50.0, 308.0, 18.0, 4.0, 2736.0, 1798.0, 589.0, 1430.0]]","[17, 16, 21, 9, 18, 12, 0, 10, 6, 5, 11, 4, 8, 15, 1, 3, 7, 2, 19, 20, 13, 14]",34010.0,"{'problem_type': 'LOP', 'num_nodes': 22, 'nodes': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S', 'T', 'U', 'V'], 'pairs': [{'from_id': 'A', 'to_id': 'B', 'weight': 121.0}, {'from_id': 'A', 'to_id': 'C', 'weight': 12.0}, {'from_id': 'A', 'to_id': 'D', 'weight': 3.0}, {'from_id': 'A', 'to_id': 'E', 'weight': 65.0}, {'from_id': 'A', 'to_id': 'F', 'weight': 77.0}, {'from_id': 'A', 'to_id': 'G', 'weight': 26.0}, {'from_id': 'A', 'to_id': 'H', 'weight': 518.0}, {'from_id': 'A', 'to_id': 'I', 'weight': 73.0}, {'from_id': 'A', 'to_id': 'J', 'weight': 9.0}, {'from_id': 'A', 'to_id': 'K', 'weight': 50.0}, {'from_id': 'A', 'to_id': 'L', 'weight': 8.0}, {'from_id': 'A', 'to_id': 'M', 'weight': 44.0}, {'from_id': 'A', 'to_id': 'N', 'weight': 12.0}, {'from_id': 'A', 'to_id': 'O', 'weight': 35.0}, {'from_id': 'A', 'to_id': 'P', 'weight': 66.0}, {'from_id': 'A', 'to_id': 'Q', 'weight': 44.0}, {'from_id': 'A', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'S', 'weight': 213.0}, {'from_id': 'A', 'to_id': 'T', 'weight': 588.0}, {'from_id': 'A', 'to_id': 'U', 'weight': 83.0}, {'from_id': 'A', 'to_id': 'V', 'weight': 221.0}, {'from_id': 'B', 'to_id': 'A', 'weight': 33.0}, {'from_id': 'B', 'to_id': 'C', 'weight': 5.0}, {'from_id': 'B', 'to_id': 'D', 'weight': 21.0}, {'from_id': 'B', 'to_id': 'E', 'weight': 113.0}, {'from_id': 'B', 'to_id': 'F', 'weight': 188.0}, {'from_id': 'B', 'to_id': 'G', 'weight': 65.0}, {'from_id': 'B', 'to_id': 'H', 'weight': 87.0}, {'from_id': 'B', 'to_id': 'I', 'weight': 7.0}, {'from_id': 'B', 'to_id': 'J', 'weight': 7.0}, {'from_id': 'B', 'to_id': 'K', 'weight': 317.0}, {'from_id': 'B', 'to_id': 'L', 'weight': 2.0}, {'from_id': 'B', 'to_id': 'M', 'weight': 14.0}, {'from_id': 'B', 'to_id': 'N', 'weight': 8.0}, {'from_id': 'B', 'to_id': 'O', 'weight': 6.0}, {'from_id': 'B', 'to_id': 'P', 'weight': 6.0}, {'from_id': 'B', 'to_id': 'Q', 'weight': 3.0}, {'from_id': 'B', 'to_id': 'R', 'weight': 1.0}, {'from_id': 'B', 'to_id': 'S', 'weight': 63.0}, {'from_id': 'B', 'to_id': 'T', 'weight': 164.0}, {'from_id': 'B', 'to_id': 'U', 'weight': 51.0}, {'from_id': 'B', 'to_id': 'V', 'weight': 38.0}, {'from_id': 'C', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'B', 'weight': 1.0}, {'from_id': 'C', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'E', 'weight': 3.0}, {'from_id': 'C', 'to_id': 'F', 'weight': 17.0}, {'from_id': 'C', 'to_id': 'G', 'weight': 5.0}, {'from_id': 'C', 'to_id': 'H', 'weight': 76.0}, {'from_id': 'C', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'K', 'weight': 27.0}, {'from_id': 'C', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'N', 'weight': 1.0}, {'from_id': 'C', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'Q', 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'S', 'to_id': 'P', 'weight': 16.0}, {'from_id': 'S', 'to_id': 'Q', 'weight': 5.0}, {'from_id': 'S', 'to_id': 'R', 'weight': 3.0}, {'from_id': 'S', 'to_id': 'T', 'weight': 1528.0}, {'from_id': 'S', 'to_id': 'U', 'weight': 185.0}, {'from_id': 'S', 'to_id': 'V', 'weight': 151.0}, {'from_id': 'T', 'to_id': 'A', 'weight': 168.0}, {'from_id': 'T', 'to_id': 'B', 'weight': 9.0}, {'from_id': 'T', 'to_id': 'C', 'weight': 22.0}, {'from_id': 'T', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'E', 'weight': 96.0}, {'from_id': 'T', 'to_id': 'F', 'weight': 17.0}, {'from_id': 'T', 'to_id': 'G', 'weight': 3.0}, {'from_id': 'T', 'to_id': 'H', 'weight': 334.0}, {'from_id': 'T', 'to_id': 'I', 'weight': 46.0}, {'from_id': 'T', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'K', 'weight': 9.0}, {'from_id': 'T', 'to_id': 'L', 'weight': 4.0}, {'from_id': 'T', 'to_id': 'M', 'weight': 56.0}, {'from_id': 'T', 'to_id': 'N', 'weight': 156.0}, {'from_id': 'T', 'to_id': 'O', 'weight': 17.0}, {'from_id': 'T', 'to_id': 'P', 'weight': 62.0}, {'from_id': 'T', 'to_id': 'Q', 'weight': 15.0}, {'from_id': 'T', 'to_id': 'R', 'weight': 16.0}, {'from_id': 'T', 'to_id': 'S', 'weight': 45.0}, {'from_id': 'T', 'to_id': 'U', 'weight': 891.0}, {'from_id': 'T', 'to_id': 'V', 'weight': 130.0}, {'from_id': 'U', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'B', 'weight': 5.0}, {'from_id': 'U', 'to_id': 'C', 'weight': 3.0}, {'from_id': 'U', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'E', 'weight': 7.0}, {'from_id': 'U', 'to_id': 'F', 'weight': 8.0}, {'from_id': 'U', 'to_id': 'G', 'weight': 4.0}, {'from_id': 'U', 'to_id': 'H', 'weight': 49.0}, {'from_id': 'U', 'to_id': 'I', 'weight': 4.0}, {'from_id': 'U', 'to_id': 'J', 'weight': 11.0}, {'from_id': 'U', 'to_id': 'K', 'weight': 5.0}, {'from_id': 'U', 'to_id': 'L', 'weight': 4.0}, {'from_id': 'U', 'to_id': 'M', 'weight': 6.0}, {'from_id': 'U', 'to_id': 'N', 'weight': 16.0}, {'from_id': 'U', 'to_id': 'O', 'weight': 10.0}, {'from_id': 'U', 'to_id': 'P', 'weight': 16.0}, {'from_id': 'U', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'S', 'weight': 43.0}, {'from_id': 'U', 'to_id': 'T', 'weight': 45.0}, {'from_id': 'U', 'to_id': 'V', 'weight': 50.0}, {'from_id': 'V', 'to_id': 'A', 'weight': 282.0}, {'from_id': 'V', 'to_id': 'B', 'weight': 181.0}, {'from_id': 'V', 'to_id': 'C', 'weight': 24.0}, {'from_id': 'V', 'to_id': 'D', 'weight': 3.0}, {'from_id': 'V', 'to_id': 'E', 'weight': 90.0}, {'from_id': 'V', 'to_id': 'F', 'weight': 15.0}, {'from_id': 'V', 'to_id': 'G', 'weight': 10.0}, {'from_id': 'V', 'to_id': 'H', 'weight': 484.0}, {'from_id': 'V', 'to_id': 'I', 'weight': 111.0}, {'from_id': 'V', 'to_id': 'J', 'weight': 239.0}, {'from_id': 'V', 'to_id': 'K', 'weight': 167.0}, {'from_id': 'V', 'to_id': 'L', 'weight': 23.0}, {'from_id': 'V', 'to_id': 'M', 'weight': 196.0}, {'from_id': 'V', 'to_id': 'N', 'weight': 171.0}, {'from_id': 'V', 'to_id': 'O', 'weight': 50.0}, {'from_id': 'V', 'to_id': 'P', 'weight': 308.0}, {'from_id': 'V', 'to_id': 'Q', 'weight': 18.0}, {'from_id': 'V', 'to_id': 'R', 'weight': 4.0}, {'from_id': 'V', 'to_id': 'S', 'weight': 2736.0}, {'from_id': 'V', 'to_id': 'T', 'weight': 1798.0}, {'from_id': 'V', 'to_id': 'U', 'weight': 589.0}]}","['R', 'Q', 'V', 'J', 'S', 'M', 'A', 'K', 'G', 'F', 'L', 'E', 'I', 'P', 'B', 'D', 'H', 'C', 'T', 'U', 'N', 'O']",24,csv,names
LOP,LOP,"Recently the city asked for a friendly walking order through its sights — a sequence that hits every stop once. Each time one attraction precedes another there’s a known bonus to the day, and the overall enjoyment of the tour is just the sum of those bonuses across every pair in the schedule, so the guide’s aim is to arrange the stops to make that sum as big as it can be without repeating or skipping attractions. The exact attractions and the pairwise enjoyment values are listed below.

There are 23 attractions to include, identified as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22.
Meaning: this is a directed enjoyment_bonus matrix over items in attraction_ids (23 total). Entry at row attraction_before=i and column attraction_after=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
enjoyment_bonus:
| attraction_before\attraction_after | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 19.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 1.0 | 9.0 | 5.0 | 0.0 | 3.0 |
| 1 | 3.0 | 62.0 | 7.0 | 3.0 | 4.0 | 22.0 | 5.0 | 3.0 | 0.0 | 8.0 | 18.0 | 5.0 | 6.0 | 0.0 | 3.0 | 3.0 | 7.0 | 10.0 | 4.0 | 18.0 | 9.0 | 0.0 | 11.0 |
| 2 | 0.0 | 13.0 | 154.0 | 1.0 | 9.0 | 0.0 | 0.0 | 0.0 | 0.0 | 54.0 | 13.0 | 18.0 | 333.0 | 0.0 | 5.0 | 5.0 | 9.0 | 5.0 | 8.0 | 15.0 | 7.0 | 0.0 | 16.0 |
| 3 | 0.0 | 0.0 | 2.0 | 79.0 | 26.0 | 0.0 | 0.0 | 0.0 | 0.0 | 6.0 | 9.0 | 11.0 | 3.0 | 0.0 | 0.0 | 0.0 | 2.0 | 4.0 | 2.0 | 23.0 | 11.0 | 0.0 | 8.0 |
| 4 | 0.0 | 0.0 | 4.0 | 15.0 | 371.0 | 0.0 | 0.0 | 0.0 | 0.0 | 45.0 | 20.0 | 12.0 | 56.0 | 0.0 | 2.0 | 2.0 | 4.0 | 7.0 | 6.0 | 34.0 | 12.0 | 0.0 | 11.0 |
| 5 | 0.0 | 212.0 | 14.0 | 23.0 | 20.0 | 229.0 | 8.0 | 21.0 | 0.0 | 9.0 | 28.0 | 62.0 | 16.0 | 0.0 | 5.0 | 7.0 | 10.0 | 7.0 | 6.0 | 38.0 | 16.0 | 0.0 | 14.0 |
| 6 | 0.0 | 2.0 | 11.0 | 8.0 | 34.0 | 0.0 | 12.0 | 18.0 | 0.0 | 32.0 | 16.0 | 7.0 | 14.0 | 0.0 | 3.0 | 3.0 | 6.0 | 4.0 | 2.0 | 9.0 | 3.0 | 0.0 | 7.0 |
| 7 | 0.0 | 3.0 | 4.0 | 9.0 | 29.0 | 0.0 | 48.0 | 318.0 | 0.0 | 74.0 | 7.0 | 30.0 | 2.0 | 0.0 | 4.0 | 4.0 | 7.0 | 5.0 | 3.0 | 11.0 | 4.0 | 0.0 | 9.0 |
| 8 | 0.0 | 12.0 | 15.0 | 20.0 | 19.0 | 0.0 | 0.0 | 1.0 | 3.0 | 26.0 | 6.0 | 32.0 | 43.0 | 0.0 | 1.0 | 1.0 | 3.0 | 1.0 | 7.0 | 5.0 | 1.0 | 0.0 | 10.0 |
| 9 | 0.0 | 0.0 | 24.0 | 6.0 | 907.0 | 0.0 | 11.0 | 5.0 | 1.0 | 161.0 | 30.0 | 46.0 | 47.0 | 0.0 | 3.0 | 3.0 | 69.0 | 4.0 | 8.0 | 21.0 | 11.0 | 0.0 | 19.0 |
| 10 | 0.0 | 0.0 | 6.0 | 1.0 | 842.0 | 0.0 | 0.0 | 0.0 | 0.0 | 51.0 | 111.0 | 14.0 | 29.0 | 0.0 | 6.0 | 6.0 | 11.0 | 10.0 | 5.0 | 24.0 | 11.0 | 0.0 | 23.0 |
| 11 | 0.0 | 9.0 | 47.0 | 7.0 | 77.0 | 0.0 | 1.0 | 1.0 | 0.0 | 77.0 | 22.0 | 87.0 | 240.0 | 0.0 | 4.0 | 4.0 | 7.0 | 5.0 | 8.0 | 19.0 | 4.0 | 0.0 | 16.0 |
| 12 | 0.0 | 0.0 | 38.0 | 1.0 | 36.0 | 0.0 | 1.0 | 5.0 | 1.0 | 150.0 | 33.0 | 24.0 | 3083.0 | 1.0 | 12.0 | 13.0 | 25.0 | 25.0 | 33.0 | 45.0 | 20.0 | 0.0 | 45.0 |
| 13 | 0.0 | 0.0 | 0.0 | 0.0 | 9.0 | 0.0 | 0.0 | 0.0 | 0.0 | 22.0 | 6.0 | 0.0 | 2.0 | 64.0 | 1.0 | 1.0 | 39.0 | 3.0 | 4.0 | 14.0 | 5.0 | 0.0 | 12.0 |
| 14 | 0.0 | 0.0 | 17.0 | 0.0 | 19.0 | 0.0 | 6.0 | 68.0 | 0.0 | 26.0 | 19.0 | 18.0 | 13.0 | 38.0 | 631.0 | 4.0 | 122.0 | 12.0 | 9.0 | 14.0 | 3.0 | 0.0 | 19.0 |
| 15 | 0.0 | 0.0 | 0.0 | 1.0 | 67.0 | 0.0 | 0.0 | 0.0 | 0.0 | 91.0 | 3.0 | 18.0 | 2.0 | 0.0 | 1.0 | 171.0 | 10.0 | 5.0 | 4.0 | 12.0 | 2.0 | 0.0 | 18.0 |
| 16 | 0.0 | 4.0 | 25.0 | 7.0 | 120.0 | 0.0 | 37.0 | 358.0 | 2.0 | 570.0 | 163.0 | 177.0 | 28.0 | 407.0 | 18.0 | 13.0 | 4292.0 | 24.0 | 59.0 | 113.0 | 39.0 | 0.0 | 339.0 |
| 17 | 0.0 | 621.0 | 99.0 | 1061.0 | 98.0 | 1431.0 | 325.0 | 278.0 | 0.0 | 79.0 | 149.0 | 396.0 | 15.0 | 0.0 | 11.0 | 8.0 | 30.0 | 906.0 | 211.0 | 217.0 | 87.0 | 0.0 | 229.0 |
| 18 | 0.0 | 0.0 | 65.0 | 1.0 | 10.0 | 10.0 | 0.0 | 0.0 | 0.0 | 65.0 | 23.0 | 6.0 | 12.0 | 0.0 | 6.0 | 6.0 | 13.0 | 83.0 | 746.0 | 245.0 | 50.0 | 0.0 | 259.0 |
| 19 | 0.0 | 26.0 | 12.0 | 37.0 | 14.0 | 193.0 | 3.0 | 3.0 | 1.0 | 19.0 | 29.0 | 3.0 | 12.0 | 0.0 | 0.0 | 0.0 | 0.0 | 137.0 | 14.0 | 13.0 | 9.0 | 0.0 | 56.0 |
| 20 | 0.0 | 0.0 | 9.0 | 9.0 | 4.0 | 31.0 | 5.0 | 5.0 | 1.0 | 22.0 | 21.0 | 18.0 | 20.0 | 1.0 | 5.0 | 5.0 | 13.0 | 38.0 | 10.0 | 32.0 | 411.0 | 0.0 | 129.0 |
| 21 | 0.0 | 0.0 | 1.0 | 1.0 | 1.0 | 4.0 | 6.0 | 6.0 | 0.0 | 7.0 | 24.0 | 0.0 | 36.0 | 0.0 | 0.0 | 0.0 | 2.0 | 2820.0 | 6.0 | 7.0 | 1.0 | 0.0 | 1174.0 |
| 22 | 0.0 | 5.0 | 115.0 | 60.0 | 36.0 | 267.0 | 8.0 | 53.0 | 12.0 | 219.0 | 743.0 | 90.0 | 286.0 | 62.0 | 17.0 | 3.0 | 452.0 | 3502.0 | 1600.0 | 548.0 | 44.0 | 0.0 | 37480.0 |
The guide will arrange the stops to maximize the total enjoyment from these listed bonuses.

Oh, and if you could put your final tour in a tiny, predictable format, that helps a lot. Just use this JSON shape when you send back the ordering:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of ""solution"" as the single list that names the stops in the exact sequence you'd walk them — first stop first, last stop last. This JSON is just a sketch of the shape I expect, not the actual filled-in answer.

Please 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”.","[[19.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 1.0, 9.0, 5.0, 0.0, 3.0], [3.0, 62.0, 7.0, 3.0, 4.0, 22.0, 5.0, 3.0, 0.0, 8.0, 18.0, 5.0, 6.0, 0.0, 3.0, 3.0, 7.0, 10.0, 4.0, 18.0, 9.0, 0.0, 11.0], [0.0, 13.0, 154.0, 1.0, 9.0, 0.0, 0.0, 0.0, 0.0, 54.0, 13.0, 18.0, 333.0, 0.0, 5.0, 5.0, 9.0, 5.0, 8.0, 15.0, 7.0, 0.0, 16.0], [0.0, 0.0, 2.0, 79.0, 26.0, 0.0, 0.0, 0.0, 0.0, 6.0, 9.0, 11.0, 3.0, 0.0, 0.0, 0.0, 2.0, 4.0, 2.0, 23.0, 11.0, 0.0, 8.0], [0.0, 0.0, 4.0, 15.0, 371.0, 0.0, 0.0, 0.0, 0.0, 45.0, 20.0, 12.0, 56.0, 0.0, 2.0, 2.0, 4.0, 7.0, 6.0, 34.0, 12.0, 0.0, 11.0], [0.0, 212.0, 14.0, 23.0, 20.0, 229.0, 8.0, 21.0, 0.0, 9.0, 28.0, 62.0, 16.0, 0.0, 5.0, 7.0, 10.0, 7.0, 6.0, 38.0, 16.0, 0.0, 14.0], [0.0, 2.0, 11.0, 8.0, 34.0, 0.0, 12.0, 18.0, 0.0, 32.0, 16.0, 7.0, 14.0, 0.0, 3.0, 3.0, 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38.0, 631.0, 4.0, 122.0, 12.0, 9.0, 14.0, 3.0, 0.0, 19.0], [0.0, 0.0, 0.0, 1.0, 67.0, 0.0, 0.0, 0.0, 0.0, 91.0, 3.0, 18.0, 2.0, 0.0, 1.0, 171.0, 10.0, 5.0, 4.0, 12.0, 2.0, 0.0, 18.0], [0.0, 4.0, 25.0, 7.0, 120.0, 0.0, 37.0, 358.0, 2.0, 570.0, 163.0, 177.0, 28.0, 407.0, 18.0, 13.0, 4292.0, 24.0, 59.0, 113.0, 39.0, 0.0, 339.0], [0.0, 621.0, 99.0, 1061.0, 98.0, 1431.0, 325.0, 278.0, 0.0, 79.0, 149.0, 396.0, 15.0, 0.0, 11.0, 8.0, 30.0, 906.0, 211.0, 217.0, 87.0, 0.0, 229.0], [0.0, 0.0, 65.0, 1.0, 10.0, 10.0, 0.0, 0.0, 0.0, 65.0, 23.0, 6.0, 12.0, 0.0, 6.0, 6.0, 13.0, 83.0, 746.0, 245.0, 50.0, 0.0, 259.0], [0.0, 26.0, 12.0, 37.0, 14.0, 193.0, 3.0, 3.0, 1.0, 19.0, 29.0, 3.0, 12.0, 0.0, 0.0, 0.0, 0.0, 137.0, 14.0, 13.0, 9.0, 0.0, 56.0], [0.0, 0.0, 9.0, 9.0, 4.0, 31.0, 5.0, 5.0, 1.0, 22.0, 21.0, 18.0, 20.0, 1.0, 5.0, 5.0, 13.0, 38.0, 10.0, 32.0, 411.0, 0.0, 129.0], [0.0, 0.0, 1.0, 1.0, 1.0, 4.0, 6.0, 6.0, 0.0, 7.0, 24.0, 0.0, 36.0, 0.0, 0.0, 0.0, 2.0, 2820.0, 6.0, 7.0, 1.0, 0.0, 1174.0], [0.0, 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{'from_id': 21, 'to_id': 22, 'weight': 1174.0}, {'from_id': 22, 'to_id': 0, 'weight': 0.0}, {'from_id': 22, 'to_id': 1, 'weight': 5.0}, {'from_id': 22, 'to_id': 2, 'weight': 115.0}, {'from_id': 22, 'to_id': 3, 'weight': 60.0}, {'from_id': 22, 'to_id': 4, 'weight': 36.0}, {'from_id': 22, 'to_id': 5, 'weight': 267.0}, {'from_id': 22, 'to_id': 6, 'weight': 8.0}, {'from_id': 22, 'to_id': 7, 'weight': 53.0}, {'from_id': 22, 'to_id': 8, 'weight': 12.0}, {'from_id': 22, 'to_id': 9, 'weight': 219.0}, {'from_id': 22, 'to_id': 10, 'weight': 743.0}, {'from_id': 22, 'to_id': 11, 'weight': 90.0}, {'from_id': 22, 'to_id': 12, 'weight': 286.0}, {'from_id': 22, 'to_id': 13, 'weight': 62.0}, {'from_id': 22, 'to_id': 14, 'weight': 17.0}, {'from_id': 22, 'to_id': 15, 'weight': 3.0}, {'from_id': 22, 'to_id': 16, 'weight': 452.0}, {'from_id': 22, 'to_id': 17, 'weight': 3502.0}, {'from_id': 22, 'to_id': 18, 'weight': 1600.0}, {'from_id': 22, 'to_id': 19, 'weight': 548.0}, {'from_id': 22, 'to_id': 20, 'weight': 44.0}, {'from_id': 22, 'to_id': 21, 'weight': 0.0}]}","[21, 14, 0, 22, 8, 17, 16, 13, 18, 20, 19, 5, 7, 6, 3, 11, 2, 1, 12, 10, 15, 9, 4]",25,nl,0
LOP,LOP,"Picture this: a stack of reception cards, each naming one event, and between every two cards there’s a note saying how much better it is to have card A before card B. The challenge is to lay out all the cards in one neat row — every event must be used exactly once — and then tally up all those little notes for that particular order. The order with the biggest total tally gives the best guest experience. The exact events and the pairwise notes are shown below.

{
  ""num_events"": 19,
  ""event_list"": [
    ""A"",
    ""B"",
    ""C"",
    ""D"",
    ""E"",
    ""F"",
    ""G"",
    ""H"",
    ""I"",
    ""J"",
    ""K"",
    ""L"",
    ""M"",
    ""N"",
    ""O"",
    ""P"",
    ""Q"",
    ""R"",
    ""S""
  ]
}

# Meaning: this is a directed guest_experience_points matrix over items in event_list (19 total). Entry at row from_event=i and column to_event=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# guest_experience_points 
| from_event\to_event | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 92.0 | 248.0 | 0.0 | 12.0 | 13.0 | 5.0 | 0.0 | 1.0 | 6.0 | 5.0 | 0.0 | 1.0 | 7.0 | 14.0 | 1.0 | 5.0 | 0.0 | 5.0 | 12.0 |
| B | 42.0 | 38.0 | 0.0 | 47.0 | 156.0 | 31.0 | 5.0 | 17.0 | 49.0 | 40.0 | 5.0 | 6.0 | 48.0 | 100.0 | 12.0 | 18.0 | 0.0 | 19.0 | 15.0 |
| C | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| D | 8.0 | 0.0 | 0.0 | 2129.0 | 34.0 | 777.0 | 54.0 | 191.0 | 1.0 | 0.0 | 0.0 | 0.0 | 434.0 | 1.0 | 1.0 | 0.0 | 0.0 | 2.0 | -2.0 |
| E | 28.0 | 27.0 | 0.0 | 21.0 | 1598.0 | 82.0 | 24.0 | 41.0 | 91.0 | 462.0 | 33.0 | 99.0 | 359.0 | 66.0 | 2.0 | 10.0 | 0.0 | 76.0 | 90.0 |
| F | 24.0 | 6.0 | 0.0 | 26.0 | 77.0 | 181.0 | 42.0 | 97.0 | 79.0 | 9.0 | 12.0 | 4.0 | 735.0 | 58.0 | 2.0 | 4.0 | 0.0 | 4.0 | 2.0 |
| G | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 218.0 | 0.0 | 6.0 | 0.0 | 0.0 | 0.0 | 3.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | -1.0 |
| H | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 21.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| I | 0.0 | 0.0 | 0.0 | 0.0 | 106.0 | 5.0 | 0.0 | 0.0 | 1976.0 | 13.0 | 3.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 30.0 | 8.0 |
| J | 3.0 | 1.0 | 0.0 | 0.0 | 24.0 | 10.0 | 21.0 | 4.0 | 18.0 | 2184.0 | 32.0 | 36.0 | 16.0 | 30.0 | 0.0 | 2.0 | 0.0 | 12.0 | 6.0 |
| K | 1.0 | 0.0 | 0.0 | 0.0 | 2.0 | 2.0 | 1.0 | 0.0 | 0.0 | 23.0 | 126.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2.0 |
| L | 2.0 | 1.0 | 0.0 | 1.0 | 8.0 | 10.0 | 36.0 | 8.0 | 9.0 | 15.0 | 22.0 | 13.0 | 9.0 | 24.0 | 1.0 | 2.0 | 0.0 | 11.0 | 2.0 |
| M | 16.0 | 32.0 | 0.0 | 20.0 | 61.0 | 33.0 | 3.0 | 17.0 | 35.0 | 27.0 | 2.0 | 3.0 | 72.0 | 93.0 | 10.0 | 29.0 | 0.0 | 36.0 | -33.0 |
| N | 11.0 | 3.0 | 0.0 | 23.0 | 64.0 | 32.0 | 10.0 | 15.0 | 160.0 | 77.0 | 15.0 | 6.0 | 156.0 | 572.0 | 2.0 | 6.0 | 0.0 | 0.0 | 235.0 |
| O | 1.0 | 5.0 | 0.0 | 10.0 | 21.0 | 10.0 | 4.0 | 8.0 | 21.0 | 17.0 | 4.0 | 2.0 | 44.0 | 176.0 | 11.0 | 97.0 | 0.0 | 20.0 | 41.0 |
| P | 2.0 | 4.0 | 0.0 | 14.0 | 31.0 | 16.0 | 5.0 | 7.0 | 16.0 | 20.0 | 4.0 | 2.0 | 38.0 | 84.0 | 1.0 | 1014.0 | 0.0 | 3.0 | 18.0 |
| Q | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| R | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 7.0 | 0.0 | 3.0 | 0.0 |
| S | 32.0 | 27.0 | 0.0 | 34.0 | 26.0 | 39.0 | 7.0 | 42.0 | 33.0 | 77.0 | 5.0 | 6.0 | 737.0 | 60.0 | 21.0 | 548.0 | 0.0 | 23.0 | 259.0 |

When you send the final ordering back, just pop it into a tiny JSON sketch like this — keeps things neat and machine-friendly:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Here ""solution"" is simply the list that shows the cards in order from first to last — each entry should be the exact item identifier from the instance (the same labels shown in the problem). This block is just a template to show the shape I expect, not the actual answer itself; when you reply, replace the placeholders with the real identifiers in the sequence that gives the biggest total tally.

Please be sure to use the identifiers exactly as they appear in the instance input — no renaming, and don’t 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”.""","[[92.0, 248.0, 0.0, 12.0, 13.0, 5.0, 0.0, 1.0, 6.0, 5.0, 0.0, 1.0, 7.0, 14.0, 1.0, 5.0, 0.0, 5.0, 12.0], [42.0, 38.0, 0.0, 47.0, 156.0, 31.0, 5.0, 17.0, 49.0, 40.0, 5.0, 6.0, 48.0, 100.0, 12.0, 18.0, 0.0, 19.0, 15.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [8.0, 0.0, 0.0, 2129.0, 34.0, 777.0, 54.0, 191.0, 1.0, 0.0, 0.0, 0.0, 434.0, 1.0, 1.0, 0.0, 0.0, 2.0, -2.0], [28.0, 27.0, 0.0, 21.0, 1598.0, 82.0, 24.0, 41.0, 91.0, 462.0, 33.0, 99.0, 359.0, 66.0, 2.0, 10.0, 0.0, 76.0, 90.0], [24.0, 6.0, 0.0, 26.0, 77.0, 181.0, 42.0, 97.0, 79.0, 9.0, 12.0, 4.0, 735.0, 58.0, 2.0, 4.0, 0.0, 4.0, 2.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 218.0, 0.0, 6.0, 0.0, 0.0, 0.0, 3.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 21.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 106.0, 5.0, 0.0, 0.0, 1976.0, 13.0, 3.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 30.0, 8.0], [3.0, 1.0, 0.0, 0.0, 24.0, 10.0, 21.0, 4.0, 18.0, 2184.0, 32.0, 36.0, 16.0, 30.0, 0.0, 2.0, 0.0, 12.0, 6.0], [1.0, 0.0, 0.0, 0.0, 2.0, 2.0, 1.0, 0.0, 0.0, 23.0, 126.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0], [2.0, 1.0, 0.0, 1.0, 8.0, 10.0, 36.0, 8.0, 9.0, 15.0, 22.0, 13.0, 9.0, 24.0, 1.0, 2.0, 0.0, 11.0, 2.0], [16.0, 32.0, 0.0, 20.0, 61.0, 33.0, 3.0, 17.0, 35.0, 27.0, 2.0, 3.0, 72.0, 93.0, 10.0, 29.0, 0.0, 36.0, -33.0], [11.0, 3.0, 0.0, 23.0, 64.0, 32.0, 10.0, 15.0, 160.0, 77.0, 15.0, 6.0, 156.0, 572.0, 2.0, 6.0, 0.0, 0.0, 235.0], [1.0, 5.0, 0.0, 10.0, 21.0, 10.0, 4.0, 8.0, 21.0, 17.0, 4.0, 2.0, 44.0, 176.0, 11.0, 97.0, 0.0, 20.0, 41.0], [2.0, 4.0, 0.0, 14.0, 31.0, 16.0, 5.0, 7.0, 16.0, 20.0, 4.0, 2.0, 38.0, 84.0, 1.0, 1014.0, 0.0, 3.0, 18.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 7.0, 0.0, 3.0, 0.0], [32.0, 27.0, 0.0, 34.0, 26.0, 39.0, 7.0, 42.0, 33.0, 77.0, 5.0, 6.0, 737.0, 60.0, 21.0, 548.0, 0.0, 23.0, 259.0]]","[0, 1, 14, 4, 11, 13, 18, 15, 3, 5, 12, 9, 6, 8, 17, 16, 10, 7, 2]",8061.0,"{'problem_type': 'LOP', 'num_nodes': 19, 'nodes': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S'], 'pairs': [{'from_id': 'A', 'to_id': 'B', 'weight': 248.0}, {'from_id': 'A', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'D', 'weight': 12.0}, {'from_id': 'A', 'to_id': 'E', 'weight': 13.0}, {'from_id': 'A', 'to_id': 'F', 'weight': 5.0}, {'from_id': 'A', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'H', 'weight': 1.0}, {'from_id': 'A', 'to_id': 'I', 'weight': 6.0}, {'from_id': 'A', 'to_id': 'J', 'weight': 5.0}, {'from_id': 'A', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'L', 'weight': 1.0}, {'from_id': 'A', 'to_id': 'M', 'weight': 7.0}, {'from_id': 'A', 'to_id': 'N', 'weight': 14.0}, {'from_id': 'A', 'to_id': 'O', 'weight': 1.0}, {'from_id': 'A', 'to_id': 'P', 'weight': 5.0}, {'from_id': 'A', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'R', 'weight': 5.0}, {'from_id': 'A', 'to_id': 'S', 'weight': 12.0}, {'from_id': 'B', 'to_id': 'A', 'weight': 42.0}, {'from_id': 'B', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'D', 'weight': 47.0}, {'from_id': 'B', 'to_id': 'E', 'weight': 156.0}, {'from_id': 'B', 'to_id': 'F', 'weight': 31.0}, {'from_id': 'B', 'to_id': 'G', 'weight': 5.0}, {'from_id': 'B', 'to_id': 'H', 'weight': 17.0}, {'from_id': 'B', 'to_id': 'I', 'weight': 49.0}, {'from_id': 'B', 'to_id': 'J', 'weight': 40.0}, {'from_id': 'B', 'to_id': 'K', 'weight': 5.0}, {'from_id': 'B', 'to_id': 'L', 'weight': 6.0}, {'from_id': 'B', 'to_id': 'M', 'weight': 48.0}, {'from_id': 'B', 'to_id': 'N', 'weight': 100.0}, {'from_id': 'B', 'to_id': 'O', 'weight': 12.0}, {'from_id': 'B', 'to_id': 'P', 'weight': 18.0}, {'from_id': 'B', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'R', 'weight': 19.0}, {'from_id': 'B', 'to_id': 'S', 'weight': 15.0}, {'from_id': 'C', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'S', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'A', 'weight': 8.0}, {'from_id': 'D', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'E', 'weight': 34.0}, {'from_id': 'D', 'to_id': 'F', 'weight': 777.0}, {'from_id': 'D', 'to_id': 'G', 'weight': 54.0}, {'from_id': 'D', 'to_id': 'H', 'weight': 191.0}, {'from_id': 'D', 'to_id': 'I', 'weight': 1.0}, {'from_id': 'D', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'M', 'weight': 434.0}, {'from_id': 'D', 'to_id': 'N', 'weight': 1.0}, {'from_id': 'D', 'to_id': 'O', 'weight': 1.0}, {'from_id': 'D', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'R', 'weight': 2.0}, {'from_id': 'D', 'to_id': 'S', 'weight': -2.0}, {'from_id': 'E', 'to_id': 'A', 'weight': 28.0}, {'from_id': 'E', 'to_id': 'B', 'weight': 27.0}, {'from_id': 'E', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'D', 'weight': 21.0}, {'from_id': 'E', 'to_id': 'F', 'weight': 82.0}, {'from_id': 'E', 'to_id': 'G', 'weight': 24.0}, {'from_id': 'E', 'to_id': 'H', 'weight': 41.0}, {'from_id': 'E', 'to_id': 'I', 'weight': 91.0}, {'from_id': 'E', 'to_id': 'J', 'weight': 462.0}, {'from_id': 'E', 'to_id': 'K', 'weight': 33.0}, {'from_id': 'E', 'to_id': 'L', 'weight': 99.0}, {'from_id': 'E', 'to_id': 'M', 'weight': 359.0}, {'from_id': 'E', 'to_id': 'N', 'weight': 66.0}, {'from_id': 'E', 'to_id': 'O', 'weight': 2.0}, {'from_id': 'E', 'to_id': 'P', 'weight': 10.0}, {'from_id': 'E', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'R', 'weight': 76.0}, {'from_id': 'E', 'to_id': 'S', 'weight': 90.0}, {'from_id': 'F', 'to_id': 'A', 'weight': 24.0}, {'from_id': 'F', 'to_id': 'B', 'weight': 6.0}, {'from_id': 'F', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'D', 'weight': 26.0}, {'from_id': 'F', 'to_id': 'E', 'weight': 77.0}, {'from_id': 'F', 'to_id': 'G', 'weight': 42.0}, {'from_id': 'F', 'to_id': 'H', 'weight': 97.0}, {'from_id': 'F', 'to_id': 'I', 'weight': 79.0}, {'from_id': 'F', 'to_id': 'J', 'weight': 9.0}, {'from_id': 'F', 'to_id': 'K', 'weight': 12.0}, {'from_id': 'F', 'to_id': 'L', 'weight': 4.0}, {'from_id': 'F', 'to_id': 'M', 'weight': 735.0}, {'from_id': 'F', 'to_id': 'N', 'weight': 58.0}, {'from_id': 'F', 'to_id': 'O', 'weight': 2.0}, {'from_id': 'F', 'to_id': 'P', 'weight': 4.0}, {'from_id': 'F', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'R', 'weight': 4.0}, {'from_id': 'F', 'to_id': 'S', 'weight': 2.0}, {'from_id': 'G', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'I', 'weight': 6.0}, {'from_id': 'G', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'M', 'weight': 3.0}, {'from_id': 'G', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'S', 'weight': -1.0}, {'from_id': 'H', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'M', 'weight': 1.0}, {'from_id': 'H', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'S', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'E', 'weight': 106.0}, {'from_id': 'I', 'to_id': 'F', 'weight': 5.0}, {'from_id': 'I', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'J', 'weight': 13.0}, {'from_id': 'I', 'to_id': 'K', 'weight': 3.0}, {'from_id': 'I', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'N', 'weight': 1.0}, {'from_id': 'I', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'R', 'weight': 30.0}, {'from_id': 'I', 'to_id': 'S', 'weight': 8.0}, {'from_id': 'J', 'to_id': 'A', 'weight': 3.0}, {'from_id': 'J', 'to_id': 'B', 'weight': 1.0}, {'from_id': 'J', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'E', 'weight': 24.0}, {'from_id': 'J', 'to_id': 'F', 'weight': 10.0}, {'from_id': 'J', 'to_id': 'G', 'weight': 21.0}, {'from_id': 'J', 'to_id': 'H', 'weight': 4.0}, {'from_id': 'J', 'to_id': 'I', 'weight': 18.0}, {'from_id': 'J', 'to_id': 'K', 'weight': 32.0}, {'from_id': 'J', 'to_id': 'L', 'weight': 36.0}, {'from_id': 'J', 'to_id': 'M', 'weight': 16.0}, {'from_id': 'J', 'to_id': 'N', 'weight': 30.0}, {'from_id': 'J', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'P', 'weight': 2.0}, {'from_id': 'J', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'R', 'weight': 12.0}, {'from_id': 'J', 'to_id': 'S', 'weight': 6.0}, {'from_id': 'K', 'to_id': 'A', 'weight': 1.0}, {'from_id': 'K', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'E', 'weight': 2.0}, {'from_id': 'K', 'to_id': 'F', 'weight': 2.0}, {'from_id': 'K', 'to_id': 'G', 'weight': 1.0}, {'from_id': 'K', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'J', 'weight': 23.0}, {'from_id': 'K', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'M', 'weight': 1.0}, {'from_id': 'K', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'S', 'weight': 2.0}, {'from_id': 'L', 'to_id': 'A', 'weight': 2.0}, {'from_id': 'L', 'to_id': 'B', 'weight': 1.0}, {'from_id': 'L', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'D', 'weight': 1.0}, {'from_id': 'L', 'to_id': 'E', 'weight': 8.0}, {'from_id': 'L', 'to_id': 'F', 'weight': 10.0}, {'from_id': 'L', 'to_id': 'G', 'weight': 36.0}, {'from_id': 'L', 'to_id': 'H', 'weight': 8.0}, {'from_id': 'L', 'to_id': 'I', 'weight': 9.0}, {'from_id': 'L', 'to_id': 'J', 'weight': 15.0}, {'from_id': 'L', 'to_id': 'K', 'weight': 22.0}, {'from_id': 'L', 'to_id': 'M', 'weight': 9.0}, {'from_id': 'L', 'to_id': 'N', 'weight': 24.0}, {'from_id': 'L', 'to_id': 'O', 'weight': 1.0}, {'from_id': 'L', 'to_id': 'P', 'weight': 2.0}, {'from_id': 'L', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'R', 'weight': 11.0}, {'from_id': 'L', 'to_id': 'S', 'weight': 2.0}, {'from_id': 'M', 'to_id': 'A', 'weight': 16.0}, {'from_id': 'M', 'to_id': 'B', 'weight': 32.0}, {'from_id': 'M', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'M', 'to_id': 'D', 'weight': 20.0}, {'from_id': 'M', 'to_id': 'E', 'weight': 61.0}, {'from_id': 'M', 'to_id': 'F', 'weight': 33.0}, {'from_id': 'M', 'to_id': 'G', 'weight': 3.0}, {'from_id': 'M', 'to_id': 'H', 'weight': 17.0}, {'from_id': 'M', 'to_id': 'I', 'weight': 35.0}, {'from_id': 'M', 'to_id': 'J', 'weight': 27.0}, {'from_id': 'M', 'to_id': 'K', 'weight': 2.0}, {'from_id': 'M', 'to_id': 'L', 'weight': 3.0}, {'from_id': 'M', 'to_id': 'N', 'weight': 93.0}, {'from_id': 'M', 'to_id': 'O', 'weight': 10.0}, {'from_id': 'M', 'to_id': 'P', 'weight': 29.0}, {'from_id': 'M', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'M', 'to_id': 'R', 'weight': 36.0}, {'from_id': 'M', 'to_id': 'S', 'weight': -33.0}, {'from_id': 'N', 'to_id': 'A', 'weight': 11.0}, {'from_id': 'N', 'to_id': 'B', 'weight': 3.0}, {'from_id': 'N', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'D', 'weight': 23.0}, {'from_id': 'N', 'to_id': 'E', 'weight': 64.0}, {'from_id': 'N', 'to_id': 'F', 'weight': 32.0}, {'from_id': 'N', 'to_id': 'G', 'weight': 10.0}, {'from_id': 'N', 'to_id': 'H', 'weight': 15.0}, {'from_id': 'N', 'to_id': 'I', 'weight': 160.0}, {'from_id': 'N', 'to_id': 'J', 'weight': 77.0}, {'from_id': 'N', 'to_id': 'K', 'weight': 15.0}, {'from_id': 'N', 'to_id': 'L', 'weight': 6.0}, {'from_id': 'N', 'to_id': 'M', 'weight': 156.0}, {'from_id': 'N', 'to_id': 'O', 'weight': 2.0}, {'from_id': 'N', 'to_id': 'P', 'weight': 6.0}, {'from_id': 'N', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'S', 'weight': 235.0}, {'from_id': 'O', 'to_id': 'A', 'weight': 1.0}, {'from_id': 'O', 'to_id': 'B', 'weight': 5.0}, {'from_id': 'O', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'D', 'weight': 10.0}, {'from_id': 'O', 'to_id': 'E', 'weight': 21.0}, {'from_id': 'O', 'to_id': 'F', 'weight': 10.0}, {'from_id': 'O', 'to_id': 'G', 'weight': 4.0}, {'from_id': 'O', 'to_id': 'H', 'weight': 8.0}, {'from_id': 'O', 'to_id': 'I', 'weight': 21.0}, {'from_id': 'O', 'to_id': 'J', 'weight': 17.0}, {'from_id': 'O', 'to_id': 'K', 'weight': 4.0}, {'from_id': 'O', 'to_id': 'L', 'weight': 2.0}, {'from_id': 'O', 'to_id': 'M', 'weight': 44.0}, {'from_id': 'O', 'to_id': 'N', 'weight': 176.0}, {'from_id': 'O', 'to_id': 'P', 'weight': 97.0}, {'from_id': 'O', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'R', 'weight': 20.0}, {'from_id': 'O', 'to_id': 'S', 'weight': 41.0}, {'from_id': 'P', 'to_id': 'A', 'weight': 2.0}, {'from_id': 'P', 'to_id': 'B', 'weight': 4.0}, {'from_id': 'P', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'D', 'weight': 14.0}, {'from_id': 'P', 'to_id': 'E', 'weight': 31.0}, {'from_id': 'P', 'to_id': 'F', 'weight': 16.0}, {'from_id': 'P', 'to_id': 'G', 'weight': 5.0}, {'from_id': 'P', 'to_id': 'H', 'weight': 7.0}, {'from_id': 'P', 'to_id': 'I', 'weight': 16.0}, {'from_id': 'P', 'to_id': 'J', 'weight': 20.0}, {'from_id': 'P', 'to_id': 'K', 'weight': 4.0}, {'from_id': 'P', 'to_id': 'L', 'weight': 2.0}, {'from_id': 'P', 'to_id': 'M', 'weight': 38.0}, {'from_id': 'P', 'to_id': 'N', 'weight': 84.0}, {'from_id': 'P', 'to_id': 'O', 'weight': 1.0}, {'from_id': 'P', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'R', 'weight': 3.0}, {'from_id': 'P', 'to_id': 'S', 'weight': 18.0}, {'from_id': 'Q', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'S', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'P', 'weight': 7.0}, {'from_id': 'R', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'S', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'A', 'weight': 32.0}, {'from_id': 'S', 'to_id': 'B', 'weight': 27.0}, {'from_id': 'S', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'D', 'weight': 34.0}, {'from_id': 'S', 'to_id': 'E', 'weight': 26.0}, {'from_id': 'S', 'to_id': 'F', 'weight': 39.0}, {'from_id': 'S', 'to_id': 'G', 'weight': 7.0}, {'from_id': 'S', 'to_id': 'H', 'weight': 42.0}, {'from_id': 'S', 'to_id': 'I', 'weight': 33.0}, {'from_id': 'S', 'to_id': 'J', 'weight': 77.0}, {'from_id': 'S', 'to_id': 'K', 'weight': 5.0}, {'from_id': 'S', 'to_id': 'L', 'weight': 6.0}, {'from_id': 'S', 'to_id': 'M', 'weight': 737.0}, {'from_id': 'S', 'to_id': 'N', 'weight': 60.0}, {'from_id': 'S', 'to_id': 'O', 'weight': 21.0}, {'from_id': 'S', 'to_id': 'P', 'weight': 548.0}, {'from_id': 'S', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'R', 'weight': 23.0}]}","['A', 'B', 'O', 'E', 'L', 'N', 'S', 'P', 'D', 'F', 'M', 'J', 'G', 'I', 'R', 'Q', 'K', 'H', 'C']",26,json,names
LOP,LOP,"Someone in the editing room has to decide how to line up every scene so the audience feels the most from start to finish. Every ordered pair of scenes brings a certain amount of narrative punch if the first comes before the second, and the aim is to choose an ordering that yields the largest total when all those punches are added together. No scene can be repeated or skipped, and the detailed pairwise impacts follow below.

They must arrange 22 scenes, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, each appearing exactly once; the lines below give the narrative punch added when one scene precedes another.
Meaning: this is a directed impact_score matrix over items in scene_ids (22 total). Entry at row prior_scene_id=i and column subsequent_scene_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
impact_score:
| prior_scene_id\subsequent_scene_id | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1423157.0 | 34611.0 | 1416.0 | 912.0 | 0.0 | 384.0 | 660156.0 | 145432.0 | 76085.0 | 54081.0 | 0.0 | 4448.0 | 0.0 | 15019.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 36000.0 | 28600.0 | 23400.0 |
| 1 | 370172.0 | 2655275.0 | 92012.0 | 179174.0 | 53741.0 | 26856.0 | 13715.0 | 6201.0 | 34246.0 | 44778.0 | 565716.0 | 255067.0 | 10303.0 | 12547.0 | 1512.0 | 1775.0 | 0.0 | 34.0 | 0.0 | 64800.0 | 380700.0 | 14500.0 |
| 2 | 60450.0 | 42371.0 | 1222814.0 | 175494.0 | 80834.0 | 188979.0 | 5849.0 | 1985.0 | 3422.0 | 22473.0 | 16915.0 | 598200.0 | 6947.0 | 14193.0 | 31.0 | 0.0 | 0.0 | 0.0 | 0.0 | 235900.0 | 1300.0 | 7400.0 |
| 3 | 767.0 | 4638.0 | 260796.0 | 793376.0 | 213277.0 | 206956.0 | 143.0 | 83.0 | 43.0 | 6123.0 | 772.0 | 257272.0 | 62145.0 | 12111.0 | 182602.0 | 1053.0 | 0.0 | 799.0 | 0.0 | 413200.0 | 14900.0 | 6400.0 |
| 4 | 3707.0 | 500.0 | 119581.0 | 6570.0 | 1021373.0 | 67719.0 | 0.0 | 0.0 | 0.0 | 762.0 | 199.0 | 39975.0 | 191015.0 | 23392.0 | 3242.0 | 0.0 | 0.0 | 0.0 | 0.0 | 71600.0 | 8400.0 | 10900.0 |
| 5 | 14075.0 | 176.0 | 42727.0 | 26181.0 | 1159.0 | 613113.0 | 0.0 | 6.0 | 600.0 | 562.0 | 71.0 | 669.0 | 142041.0 | 235370.0 | 50.0 | 81.0 | 0.0 | 14.0 | 0.0 | 1023600.0 | 8100.0 | 5600.0 |
| 6 | 2108.0 | 15423.0 | 13.0 | 7.0 | 16.0 | 6.0 | 65946.0 | 0.0 | 97.0 | 3883.0 | 3.0 | 471.0 | 0.0 | 13451.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 26600.0 | 14800.0 | 50000.0 |
| 7 | 8966.0 | 8.0 | 5.0 | 2.0 | 9.0 | 0.0 | 1.0 | 3201.0 | 0.0 | 3.0 | 2.0 | 1.0 | 0.0 | 3377.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1100.0 | 0.0 | 0.0 |
| 8 | 660.0 | 8838.0 | 3620.0 | 1807.0 | 3993.0 | 3345.0 | 469.0 | 117.0 | 172415.0 | 2690.0 | 3656.0 | 1253.0 | 2789.0 | 218.0 | 1290.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2900.0 | 6500.0 | 700.0 |
| 9 | 20727.0 | 15353.0 | 31129.0 | 46805.0 | 22378.0 | 17005.0 | 176.0 | 767.0 | 3746.0 | 757392.0 | 10447.0 | 763132.0 | 7093.0 | 17466.0 | 3726.0 | 14734.0 | 0.0 | 11679.0 | 0.0 | 55000.0 | 17300.0 | 11100.0 |
| 10 | 15941.0 | 83433.0 | 112689.0 | 122637.0 | 264033.0 | 23041.0 | 8297.0 | 2557.0 | 35483.0 | 55536.0 | 100882.0 | 278193.0 | 176392.0 | 15502.0 | 16903.0 | 8373.0 | 0.0 | 5240.0 | 0.0 | 30200.0 | 6600.0 | 200.0 |
| 11 | 125155.0 | 1656.0 | 47216.0 | 11764.0 | 3391.0 | 8842.0 | 872.0 | 563.0 | 992.0 | 5783.0 | 2121.0 | 3027041.0 | 2800.0 | 11393.0 | 12056.0 | 48778.0 | 0.0 | 196514.0 | 0.0 | 767600.0 | 59500.0 | 44400.0 |
| 12 | 69244.0 | 95147.0 | 133428.0 | 58709.0 | 89266.0 | 19259.0 | 23657.0 | 7363.0 | 4263.0 | 30553.0 | 32962.0 | 272810.0 | 43470.0 | 19558.0 | 59561.0 | 1428.0 | 0.0 | 0.0 | 0.0 | 81800.0 | 11600.0 | 5000.0 |
| 13 | 8255.0 | 52680.0 | 39197.0 | 23108.0 | 20551.0 | 19035.0 | 1854.0 | 6133.0 | 3097.0 | 7996.0 | 7113.0 | 12452.0 | 12172.0 | 1648102.0 | 43934.0 | 13081.0 | 0.0 | 0.0 | 0.0 | 70900.0 | 4900.0 | 4900.0 |
| 14 | 17320.0 | 16164.0 | 37604.0 | 27605.0 | 19117.0 | 11547.0 | 2192.0 | 1569.0 | 1821.0 | 8838.0 | 7419.0 | 58602.0 | 69947.0 | 62156.0 | 253163.0 | 382139.0 | 0.0 | 54656.0 | 0.0 | 154500.0 | 21400.0 | 30700.0 |
| 15 | 26900.0 | 14487.0 | 22785.0 | 15799.0 | 8877.0 | 18135.0 | 914.0 | 1243.0 | 1925.0 | 12357.0 | 6682.0 | 76423.0 | 151155.0 | 63748.0 | 30373.0 | 4692408.0 | 0.0 | 9788.0 | 0.0 | 6200.0 | 4000.0 | 1100.0 |
| 16 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 17 | 39406.0 | 16047.0 | 28560.0 | 24413.0 | 10560.0 | 16348.0 | 3760.0 | 899.0 | 4029.0 | 9603.0 | 9776.0 | 12571.0 | 20988.0 | 52899.0 | 3194.0 | 97585.0 | 0.0 | 14197.0 | 0.0 | 11400.0 | 41100.0 | 2100.0 |
| 18 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 19 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 20 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 21 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
The goal is to choose the ordering that maximizes the total of these pairwise punches.

If you want to hand me the chosen lineup, just drop it in this simple JSON shape so it's easy to read and reuse:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Here ""solution"" is just the list of scenes in the order you want them to appear, from the first shown to the last. Think of the JSON as a tiny form: replace those placeholders with the actual scene identifiers from the instance — this block is just a sketch of the shape, not the final answer.

Please make sure to use the exact identifiers from the input — don't 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”.""","[[1423157.0, 34611.0, 1416.0, 912.0, 0.0, 384.0, 660156.0, 145432.0, 76085.0, 54081.0, 0.0, 4448.0, 0.0, 15019.0, 0.0, 0.0, 0.0, 0.0, 0.0, 36000.0, 28600.0, 23400.0], [370172.0, 2655275.0, 92012.0, 179174.0, 53741.0, 26856.0, 13715.0, 6201.0, 34246.0, 44778.0, 565716.0, 255067.0, 10303.0, 12547.0, 1512.0, 1775.0, 0.0, 34.0, 0.0, 64800.0, 380700.0, 14500.0], [60450.0, 42371.0, 1222814.0, 175494.0, 80834.0, 188979.0, 5849.0, 1985.0, 3422.0, 22473.0, 16915.0, 598200.0, 6947.0, 14193.0, 31.0, 0.0, 0.0, 0.0, 0.0, 235900.0, 1300.0, 7400.0], [767.0, 4638.0, 260796.0, 793376.0, 213277.0, 206956.0, 143.0, 83.0, 43.0, 6123.0, 772.0, 257272.0, 62145.0, 12111.0, 182602.0, 1053.0, 0.0, 799.0, 0.0, 413200.0, 14900.0, 6400.0], [3707.0, 500.0, 119581.0, 6570.0, 1021373.0, 67719.0, 0.0, 0.0, 0.0, 762.0, 199.0, 39975.0, 191015.0, 23392.0, 3242.0, 0.0, 0.0, 0.0, 0.0, 71600.0, 8400.0, 10900.0], [14075.0, 176.0, 42727.0, 26181.0, 1159.0, 613113.0, 0.0, 6.0, 600.0, 562.0, 71.0, 669.0, 142041.0, 235370.0, 50.0, 81.0, 0.0, 14.0, 0.0, 1023600.0, 8100.0, 5600.0], [2108.0, 15423.0, 13.0, 7.0, 16.0, 6.0, 65946.0, 0.0, 97.0, 3883.0, 3.0, 471.0, 0.0, 13451.0, 0.0, 0.0, 0.0, 0.0, 0.0, 26600.0, 14800.0, 50000.0], [8966.0, 8.0, 5.0, 2.0, 9.0, 0.0, 1.0, 3201.0, 0.0, 3.0, 2.0, 1.0, 0.0, 3377.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1100.0, 0.0, 0.0], [660.0, 8838.0, 3620.0, 1807.0, 3993.0, 3345.0, 469.0, 117.0, 172415.0, 2690.0, 3656.0, 1253.0, 2789.0, 218.0, 1290.0, 0.0, 0.0, 0.0, 0.0, 2900.0, 6500.0, 700.0], [20727.0, 15353.0, 31129.0, 46805.0, 22378.0, 17005.0, 176.0, 767.0, 3746.0, 757392.0, 10447.0, 763132.0, 7093.0, 17466.0, 3726.0, 14734.0, 0.0, 11679.0, 0.0, 55000.0, 17300.0, 11100.0], [15941.0, 83433.0, 112689.0, 122637.0, 264033.0, 23041.0, 8297.0, 2557.0, 35483.0, 55536.0, 100882.0, 278193.0, 176392.0, 15502.0, 16903.0, 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'weight': 0.0}, {'from_id': 21, 'to_id': 13, 'weight': 0.0}, {'from_id': 21, 'to_id': 14, 'weight': 0.0}, {'from_id': 21, 'to_id': 15, 'weight': 0.0}, {'from_id': 21, 'to_id': 16, 'weight': 0.0}, {'from_id': 21, 'to_id': 17, 'weight': 0.0}, {'from_id': 21, 'to_id': 18, 'weight': 0.0}, {'from_id': 21, 'to_id': 19, 'weight': 0.0}, {'from_id': 21, 'to_id': 20, 'weight': 0.0}]}","[1, 10, 9, 3, 14, 15, 4, 12, 2, 11, 17, 5, 0, 6, 13, 8, 21, 20, 7, 19, 18, 16]",27,nl,0
LOP,LOP,"In the office they’re assembling a workshop playlist and discovered that order changes outcomes — each time a particular module comes before another there’s a known benefit, and reversing that pair changes the number. The job is to put all modules into one ordered list without skipping or repeating any, then compute the list’s total by summing every pairwise benefit where the earlier item comes before the later one. The best playlist is simply the one with the highest total sum. The detailed module names and pairwise values are listed below.

# total_modules=18
# module_ids=A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R

# Meaning: this is a directed pairwise_learning_benefit matrix over items in module_ids (18 total). Entry at row preceding_module=i and column succeeding_module=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# pairwise_learning_benefit
| preceding_module\succeeding_module | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 340919.0 | 27522.0 | 0.0 | 135519.0 | 142843.0 | 28719.0 | 837.0 | 10562.0 | 1737.0 | 1543.0 | 1889.0 | 6319.0 | 1397.0 | 0.0 | 1640.0 | 3144.0 | 23348.0 | 1743.0 |
| B | 26467.0 | 8076176.0 | 0.0 | 507073.0 | 1450484.0 | 211946.0 | 39449.0 | 136392.0 | 37441.0 | 13191.0 | 131976.0 | 215045.0 | 75181.0 | 26765.0 | 24839.0 | 111757.0 | 182071.0 | 62147.0 |
| C | 0.0 | 0.0 | 37089.0 | 0.0 | 2.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 34.0 | 0.0 | 34022.0 | 0.0 |
| D | 62129.0 | 25167.0 | 900.0 | 21010120.0 | 706887.0 | 4088284.0 | 369727.0 | 3088768.0 | 844136.0 | 586234.0 | 373217.0 | 276.0 | 0.0 | 0.0 | 454.0 | 0.0 | 31992.0 | 0.0 |
| E | 62250.0 | 427733.0 | 0.0 | 543291.0 | 8969643.0 | 302874.0 | 60530.0 | 700976.0 | 91884.0 | 106331.0 | 156583.0 | 185834.0 | 48236.0 | 58245.0 | 13681.0 | 407729.0 | 171430.0 | 243414.0 |
| F | 135861.0 | 68526.0 | 0.0 | 205959.0 | 118993.0 | 3071342.0 | 89015.0 | 479963.0 | 533811.0 | 28371.0 | 353449.0 | 39680.0 | 18416.0 | 4645.0 | 276.0 | 36191.0 | 25755.0 | 546.0 |
| G | 2099.0 | 9002.0 | 0.0 | 15579.0 | 35568.0 | 162371.0 | 515404.0 | 92837.0 | 60193.0 | 23681.0 | 39877.0 | 7104.0 | 10379.0 | 180.0 | 4656.0 | 16608.0 | 46752.0 | 25594.0 |
| H | 24094.0 | 16780.0 | 0.0 | 157084.0 | 68032.0 | 908411.0 | 383559.0 | 4017141.0 | 347694.0 | 45480.0 | 464064.0 | 68794.0 | 84377.0 | 31436.0 | 10781.0 | 52241.0 | 72387.0 | 12827.0 |
| I | 4360.0 | 276.0 | 0.0 | 3591.0 | 552.0 | 5545.0 | 20.0 | 5254.0 | 903919.0 | 4861.0 | 58211.0 | 276.0 | 0.0 | 0.0 | 0.0 | 157.0 | 5054.0 | 0.0 |
| J | 1726.0 | 4404.0 | 0.0 | 5714.0 | 26538.0 | 40802.0 | 27190.0 | 37061.0 | 21864.0 | 352910.0 | 15723.0 | 11275.0 | 21725.0 | 2573.0 | 9831.0 | 57452.0 | 47364.0 | 2187.0 |
| K | 9687.0 | 46286.0 | 0.0 | 2867682.0 | 159311.0 | 185464.0 | 22935.0 | 97863.0 | 47535.0 | 8728.0 | 86024.0 | 63942.0 | 79932.0 | 139535.0 | 14774.0 | 118677.0 | 18607.0 | 23803.0 |
| L | 11782.0 | 66794.0 | 0.0 | 116360.0 | 94401.0 | 127320.0 | 19651.0 | 175392.0 | 14175.0 | 11922.0 | 73428.0 | 98844.0 | 73176.0 | 40715.0 | 9853.0 | 69417.0 | 16274.0 | 2554.0 |
| M | 12763.0 | 45207.0 | 0.0 | 80486.0 | 127576.0 | 110944.0 | 18110.0 | 114985.0 | 33594.0 | 13250.0 | 98637.0 | 52453.0 | 13039466.0 | 324974.0 | 12961.0 | 64210.0 | 56577.0 | 12841.0 |
| N | 5206.0 | 17787.0 | 0.0 | 21715.0 | 58533.0 | 53113.0 | 20526.0 | 51923.0 | 15449.0 | 11368.0 | 88865.0 | 170193.0 | 102344.0 | 180.0 | 22339.0 | 189786.0 | 71925.0 | 16483.0 |
| O | 2434.0 | 25313.0 | 900.0 | 20224.0 | 101381.0 | 6214.0 | 22140.0 | 25839.0 | 87974.0 | 802.0 | 826.0 | 5747.0 | 30084.0 | 820.0 | 6267.0 | 8164.0 | 141646.0 | 1093.0 |
| P | 8904.0 | 41987.0 | 0.0 | 44133.0 | 117111.0 | 52706.0 | 12365.0 | 61940.0 | 12931.0 | 5892.0 | 24644.0 | 65974.0 | 242716.0 | 256777.0 | 3919.0 | 425757.0 | 78899.0 | 27170.0 |
| Q | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 276.0 | 0.0 | 276.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1657.0 | 552.0 | 0.0 | 2762.0 | 0.0 |
| R | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Oh, and when you send the final ordering back, just stick to a tiny JSON sketch so it's easy to parse — something like this:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Here ""solution"" should be the full playlist in order, from the first module to the last — just list the module identifiers in the sequence you choose. Think of it like filling out a one-line form: the array is the order you want to play things.

This JSON is just the shape I expect, not the actual filled-in answer. And please use the exact identifiers from the instance input — don't rename anything 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”.""","[[340919.0, 27522.0, 0.0, 135519.0, 142843.0, 28719.0, 837.0, 10562.0, 1737.0, 1543.0, 1889.0, 6319.0, 1397.0, 0.0, 1640.0, 3144.0, 23348.0, 1743.0], [26467.0, 8076176.0, 0.0, 507073.0, 1450484.0, 211946.0, 39449.0, 136392.0, 37441.0, 13191.0, 131976.0, 215045.0, 75181.0, 26765.0, 24839.0, 111757.0, 182071.0, 62147.0], [0.0, 0.0, 37089.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 34.0, 0.0, 34022.0, 0.0], [62129.0, 25167.0, 900.0, 21010120.0, 706887.0, 4088284.0, 369727.0, 3088768.0, 844136.0, 586234.0, 373217.0, 276.0, 0.0, 0.0, 454.0, 0.0, 31992.0, 0.0], [62250.0, 427733.0, 0.0, 543291.0, 8969643.0, 302874.0, 60530.0, 700976.0, 91884.0, 106331.0, 156583.0, 185834.0, 48236.0, 58245.0, 13681.0, 407729.0, 171430.0, 243414.0], [135861.0, 68526.0, 0.0, 205959.0, 118993.0, 3071342.0, 89015.0, 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{'from_id': 'P', 'to_id': 'A', 'weight': 8904.0}, {'from_id': 'P', 'to_id': 'B', 'weight': 41987.0}, {'from_id': 'P', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'D', 'weight': 44133.0}, {'from_id': 'P', 'to_id': 'E', 'weight': 117111.0}, {'from_id': 'P', 'to_id': 'F', 'weight': 52706.0}, {'from_id': 'P', 'to_id': 'G', 'weight': 12365.0}, {'from_id': 'P', 'to_id': 'H', 'weight': 61940.0}, {'from_id': 'P', 'to_id': 'I', 'weight': 12931.0}, {'from_id': 'P', 'to_id': 'J', 'weight': 5892.0}, {'from_id': 'P', 'to_id': 'K', 'weight': 24644.0}, {'from_id': 'P', 'to_id': 'L', 'weight': 65974.0}, {'from_id': 'P', 'to_id': 'M', 'weight': 242716.0}, {'from_id': 'P', 'to_id': 'N', 'weight': 256777.0}, {'from_id': 'P', 'to_id': 'O', 'weight': 3919.0}, {'from_id': 'P', 'to_id': 'Q', 'weight': 78899.0}, {'from_id': 'P', 'to_id': 'R', 'weight': 27170.0}, {'from_id': 'Q', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'F', 'weight': 276.0}, {'from_id': 'Q', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'H', 'weight': 276.0}, {'from_id': 'Q', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'N', 'weight': 1657.0}, {'from_id': 'Q', 'to_id': 'O', 'weight': 552.0}, {'from_id': 'Q', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'Q', 'weight': 0.0}]}","['B', 'K', 'O', 'A', 'E', 'P', 'M', 'N', 'L', 'D', 'H', 'J', 'G', 'F', 'R', 'I', 'C', 'Q']",28,csv,names
LOP,LOP,"Someone in marketing lined up a handful of demos and now it’s time to decide the running order — and that order actually changes how persuasive the day will be. Between any two demos there’s a specific benefit to showing one ahead of the other, so for any full sequence you can compute a total persuasiveness by adding up all the pairwise benefits that occur in that order. The plan must use every demo once, no repeats, no skips, and the best sequence is the one with the biggest total sum. The specific demos and their pairwise values are shown below.

Here are the 21 demos: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U.
Meaning: this is a directed pairwise_persuasion matrix over items in demo_ids (21 total). Entry at row demo_before_id=i and column demo_after_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
pairwise_persuasion:
| demo_before_id\demo_after_id | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 13240.0 | 59.0 | 2.0 | 1.0 | 0.0 | 0.0 | 5.0 | 7.0 | 1.0 | 8323.0 | 11164.0 | 1297.0 | 2136.0 | 21.0 | 737.0 | 4.0 | 3.0 | 1.0 | 48.0 | 3.0 | 87.0 |
| B | 0.0 | 599.0 | 3848.0 | 34.0 | 3410.0 | 0.0 | 172.0 | 151.0 | 1.0 | 7.0 | 96.0 | 2.0 | 29.0 | 5.0 | 3.0 | 248.0 | 8.0 | 2.0 | 1.0 | 5.0 | 2.0 |
| C | 1.0 | 42.0 | 36.0 | 0.0 | 2.0 | 0.0 | 2809.0 | 96.0 | 0.0 | 0.0 | 8.0 | 1.0 | 3.0 | 3.0 | 11.0 | 21.0 | 7.0 | 9.0 | 3.0 | 7.0 | 6.0 |
| D | 956.0 | 47.0 | 51.0 | 8276.0 | 656.0 | 0.0 | 889.0 | 1047.0 | 27.0 | 78.0 | 415.0 | 29.0 | 279.0 | 790.0 | 205.0 | 2787.0 | 165.0 | 124.0 | 182.0 | 176.0 | 178.0 |
| E | 630.0 | 228.0 | 626.0 | 471.0 | 1641.0 | 0.0 | 2890.0 | 1284.0 | 67.0 | 88.0 | 423.0 | 32.0 | 422.0 | 313.0 | 263.0 | 511.0 | 153.0 | 245.0 | 98.0 | 185.0 | 236.0 |
| F | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| G | 23.0 | 154.0 | 37.0 | 58.0 | 372.0 | 0.0 | 49064.0 | 438.0 | 1206.0 | 0.0 | 0.0 | 12.0 | 644.0 | 4613.0 | 0.0 | 308.0 | 33.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| H | 204.0 | 24.0 | 10.0 | 4.0 | 97.0 | 0.0 | 1531.0 | 5621.0 | 11.0 | 40.0 | 295.0 | 0.0 | 263.0 | 15706.0 | 151.0 | 86.0 | 1.0 | 0.0 | 130.0 | 3.0 | 99.0 |
| I | 32.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 700.0 | 0.0 | 0.0 | 0.0 | 0.0 | 7.0 | 0.0 | 115.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| J | 747.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 115.0 | 1044.0 | 0.0 | 0.0 | 0.0 | 814.0 | 3.0 | 0.0 | 0.0 | 60.0 | 37.0 | 151.0 |
| K | 4883.0 | 1.0 | 0.0 | 11.0 | 4.0 | 0.0 | 8.0 | 3.0 | 0.0 | 22.0 | 10733.0 | 18.0 | 3.0 | 11.0 | 3673.0 | 18.0 | 0.0 | 6.0 | 99.0 | 130.0 | 303.0 |
| L | 6.0 | 4.0 | 0.0 | 7.0 | 12.0 | 0.0 | 24.0 | 10.0 | 0.0 | 2.0 | 11.0 | 17.0 | 9.0 | 23.0 | 2348.0 | 9.0 | 0.0 | 15.0 | 0.0 | 4.0 | 0.0 |
| M | 118.0 | 106.0 | 0.0 | 6.0 | 72.0 | 0.0 | 472.0 | 345.0 | 41.0 | 12.0 | 42.0 | 13.0 | 8181.0 | 7104.0 | 56.0 | 39.0 | 10.0 | 13.0 | 16.0 | 12.0 | 47.0 |
| N | 353.0 | 254.0 | 39.0 | 111.0 | 225.0 | 0.0 | 44.0 | 84.0 | 69.0 | 5.0 | 214.0 | 104.0 | 33.0 | 603.0 | 30.0 | 395.0 | 58.0 | 157.0 | 30.0 | 66.0 | 93.0 |
| O | 57.0 | 37.0 | 4.0 | 178.0 | 4.0 | 0.0 | 327.0 | 172.0 | 17.0 | 11.0 | 194.0 | 65.0 | 233.0 | 152.0 | 0.0 | 299.0 | 26.0 | 139.0 | 3.0 | 100.0 | 0.0 |
| P | 404.0 | 43.0 | 56.0 | 526.0 | 564.0 | 0.0 | 2075.0 | 798.0 | 70.0 | 170.0 | 581.0 | 38.0 | 260.0 | 1387.0 | 253.0 | 398.0 | 136.0 | 87.0 | 72.0 | 87.0 | 59.0 |
| Q | 159.0 | 89.0 | 7.0 | 23.0 | 69.0 | 0.0 | 126.0 | 144.0 | 34.0 | 31.0 | 117.0 | 9.0 | 279.0 | 334.0 | 275.0 | 274.0 | 106.0 | 664.0 | 116.0 | 172.0 | 60.0 |
| R | 181.0 | 26.0 | 11.0 | 59.0 | 88.0 | 0.0 | 115.0 | 110.0 | 12.0 | 16.0 | 96.0 | 32.0 | 119.0 | 322.0 | 91.0 | 421.0 | 52.0 | 17065.0 | 52.0 | 86.0 | 15.0 |
| S | 528.0 | 0.0 | 0.0 | 4.0 | 10.0 | 0.0 | 1.0 | 8.0 | 0.0 | 0.0 | 8.0 | 1.0 | 8.0 | 13.0 | 4.0 | 19.0 | 3.0 | 12.0 | 0.0 | 9.0 | 5.0 |
| T | 38.0 | 22.0 | 13.0 | 34.0 | 41.0 | 0.0 | 104.0 | 71.0 | 9.0 | 15.0 | 68.0 | 15.0 | 63.0 | 634.0 | 62.0 | 44.0 | 16.0 | 617.0 | 44.0 | 37.0 | 28.0 |
| U | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
The optimal running order uses every demo once and maximizes the total summed pairwise persuasion.

If you want to hand the running order back to me in a tidy way, just use this little JSON shape — it makes it clear which demo goes first, second, and so on.

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

This shows the expected shape: ""solution"" is an ordered list with the demos in the exact running order (first item listed is shown first, last item listed is shown last). The angle-bracket placeholders just sketch where each demo identifier should go — replace them with the actual identifiers from the instance when you answer.

This JSON is just the shape I expect, not the final answer itself. Please use the identifiers exactly as they appear in the instance input — no renaming, no 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”.""","[[13240.0, 59.0, 2.0, 1.0, 0.0, 0.0, 5.0, 7.0, 1.0, 8323.0, 11164.0, 1297.0, 2136.0, 21.0, 737.0, 4.0, 3.0, 1.0, 48.0, 3.0, 87.0], [0.0, 599.0, 3848.0, 34.0, 3410.0, 0.0, 172.0, 151.0, 1.0, 7.0, 96.0, 2.0, 29.0, 5.0, 3.0, 248.0, 8.0, 2.0, 1.0, 5.0, 2.0], [1.0, 42.0, 36.0, 0.0, 2.0, 0.0, 2809.0, 96.0, 0.0, 0.0, 8.0, 1.0, 3.0, 3.0, 11.0, 21.0, 7.0, 9.0, 3.0, 7.0, 6.0], [956.0, 47.0, 51.0, 8276.0, 656.0, 0.0, 889.0, 1047.0, 27.0, 78.0, 415.0, 29.0, 279.0, 790.0, 205.0, 2787.0, 165.0, 124.0, 182.0, 176.0, 178.0], [630.0, 228.0, 626.0, 471.0, 1641.0, 0.0, 2890.0, 1284.0, 67.0, 88.0, 423.0, 32.0, 422.0, 313.0, 263.0, 511.0, 153.0, 245.0, 98.0, 185.0, 236.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [23.0, 154.0, 37.0, 58.0, 372.0, 0.0, 49064.0, 438.0, 1206.0, 0.0, 0.0, 12.0, 644.0, 4613.0, 0.0, 308.0, 33.0, 0.0, 0.0, 0.0, 0.0], [204.0, 24.0, 10.0, 4.0, 97.0, 0.0, 1531.0, 5621.0, 11.0, 40.0, 295.0, 0.0, 263.0, 15706.0, 151.0, 86.0, 1.0, 0.0, 130.0, 3.0, 99.0], [32.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 700.0, 0.0, 0.0, 0.0, 0.0, 7.0, 0.0, 115.0, 0.0, 0.0, 0.0, 0.0, 0.0], [747.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 115.0, 1044.0, 0.0, 0.0, 0.0, 814.0, 3.0, 0.0, 0.0, 60.0, 37.0, 151.0], [4883.0, 1.0, 0.0, 11.0, 4.0, 0.0, 8.0, 3.0, 0.0, 22.0, 10733.0, 18.0, 3.0, 11.0, 3673.0, 18.0, 0.0, 6.0, 99.0, 130.0, 303.0], [6.0, 4.0, 0.0, 7.0, 12.0, 0.0, 24.0, 10.0, 0.0, 2.0, 11.0, 17.0, 9.0, 23.0, 2348.0, 9.0, 0.0, 15.0, 0.0, 4.0, 0.0], [118.0, 106.0, 0.0, 6.0, 72.0, 0.0, 472.0, 345.0, 41.0, 12.0, 42.0, 13.0, 8181.0, 7104.0, 56.0, 39.0, 10.0, 13.0, 16.0, 12.0, 47.0], [353.0, 254.0, 39.0, 111.0, 225.0, 0.0, 44.0, 84.0, 69.0, 5.0, 214.0, 104.0, 33.0, 603.0, 30.0, 395.0, 58.0, 157.0, 30.0, 66.0, 93.0], [57.0, 37.0, 4.0, 178.0, 4.0, 0.0, 327.0, 172.0, 17.0, 11.0, 194.0, 65.0, 233.0, 152.0, 0.0, 299.0, 26.0, 139.0, 3.0, 100.0, 0.0], [404.0, 43.0, 56.0, 526.0, 564.0, 0.0, 2075.0, 798.0, 70.0, 170.0, 581.0, 38.0, 260.0, 1387.0, 253.0, 398.0, 136.0, 87.0, 72.0, 87.0, 59.0], [159.0, 89.0, 7.0, 23.0, 69.0, 0.0, 126.0, 144.0, 34.0, 31.0, 117.0, 9.0, 279.0, 334.0, 275.0, 274.0, 106.0, 664.0, 116.0, 172.0, 60.0], [181.0, 26.0, 11.0, 59.0, 88.0, 0.0, 115.0, 110.0, 12.0, 16.0, 96.0, 32.0, 119.0, 322.0, 91.0, 421.0, 52.0, 17065.0, 52.0, 86.0, 15.0], [528.0, 0.0, 0.0, 4.0, 10.0, 0.0, 1.0, 8.0, 0.0, 0.0, 8.0, 1.0, 8.0, 13.0, 4.0, 19.0, 3.0, 12.0, 0.0, 9.0, 5.0], [38.0, 22.0, 13.0, 34.0, 41.0, 0.0, 104.0, 71.0, 9.0, 15.0, 68.0, 15.0, 63.0, 634.0, 62.0, 44.0, 16.0, 617.0, 44.0, 37.0, 28.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]]","[3, 1, 4, 16, 19, 17, 15, 2, 7, 18, 0, 9, 10, 11, 14, 6, 12, 13, 20, 8, 5]",106664.0,"{'problem_type': 'LOP', 'num_nodes': 21, 'nodes': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S', 'T', 'U'], 'pairs': [{'from_id': 'A', 'to_id': 'B', 'weight': 59.0}, {'from_id': 'A', 'to_id': 'C', 'weight': 2.0}, {'from_id': 'A', 'to_id': 'D', 'weight': 1.0}, {'from_id': 'A', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'G', 'weight': 5.0}, {'from_id': 'A', 'to_id': 'H', 'weight': 7.0}, {'from_id': 'A', 'to_id': 'I', 'weight': 1.0}, {'from_id': 'A', 'to_id': 'J', 'weight': 8323.0}, {'from_id': 'A', 'to_id': 'K', 'weight': 11164.0}, {'from_id': 'A', 'to_id': 'L', 'weight': 1297.0}, {'from_id': 'A', 'to_id': 'M', 'weight': 2136.0}, {'from_id': 'A', 'to_id': 'N', 'weight': 21.0}, {'from_id': 'A', 'to_id': 'O', 'weight': 737.0}, {'from_id': 'A', 'to_id': 'P', 'weight': 4.0}, {'from_id': 'A', 'to_id': 'Q', 'weight': 3.0}, {'from_id': 'A', 'to_id': 'R', 'weight': 1.0}, {'from_id': 'A', 'to_id': 'S', 'weight': 48.0}, {'from_id': 'A', 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{'from_id': 'R', 'to_id': 'L', 'weight': 32.0}, {'from_id': 'R', 'to_id': 'M', 'weight': 119.0}, {'from_id': 'R', 'to_id': 'N', 'weight': 322.0}, {'from_id': 'R', 'to_id': 'O', 'weight': 91.0}, {'from_id': 'R', 'to_id': 'P', 'weight': 421.0}, {'from_id': 'R', 'to_id': 'Q', 'weight': 52.0}, {'from_id': 'R', 'to_id': 'S', 'weight': 52.0}, {'from_id': 'R', 'to_id': 'T', 'weight': 86.0}, {'from_id': 'R', 'to_id': 'U', 'weight': 15.0}, {'from_id': 'S', 'to_id': 'A', 'weight': 528.0}, {'from_id': 'S', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'D', 'weight': 4.0}, {'from_id': 'S', 'to_id': 'E', 'weight': 10.0}, {'from_id': 'S', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'G', 'weight': 1.0}, {'from_id': 'S', 'to_id': 'H', 'weight': 8.0}, {'from_id': 'S', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'K', 'weight': 8.0}, {'from_id': 'S', 'to_id': 'L', 'weight': 1.0}, {'from_id': 'S', 'to_id': 'M', 'weight': 8.0}, {'from_id': 'S', 'to_id': 'N', 'weight': 13.0}, {'from_id': 'S', 'to_id': 'O', 'weight': 4.0}, {'from_id': 'S', 'to_id': 'P', 'weight': 19.0}, {'from_id': 'S', 'to_id': 'Q', 'weight': 3.0}, {'from_id': 'S', 'to_id': 'R', 'weight': 12.0}, {'from_id': 'S', 'to_id': 'T', 'weight': 9.0}, {'from_id': 'S', 'to_id': 'U', 'weight': 5.0}, {'from_id': 'T', 'to_id': 'A', 'weight': 38.0}, {'from_id': 'T', 'to_id': 'B', 'weight': 22.0}, {'from_id': 'T', 'to_id': 'C', 'weight': 13.0}, {'from_id': 'T', 'to_id': 'D', 'weight': 34.0}, {'from_id': 'T', 'to_id': 'E', 'weight': 41.0}, {'from_id': 'T', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'G', 'weight': 104.0}, {'from_id': 'T', 'to_id': 'H', 'weight': 71.0}, {'from_id': 'T', 'to_id': 'I', 'weight': 9.0}, {'from_id': 'T', 'to_id': 'J', 'weight': 15.0}, {'from_id': 'T', 'to_id': 'K', 'weight': 68.0}, {'from_id': 'T', 'to_id': 'L', 'weight': 15.0}, {'from_id': 'T', 'to_id': 'M', 'weight': 63.0}, {'from_id': 'T', 'to_id': 'N', 'weight': 634.0}, {'from_id': 'T', 'to_id': 'O', 'weight': 62.0}, {'from_id': 'T', 'to_id': 'P', 'weight': 44.0}, {'from_id': 'T', 'to_id': 'Q', 'weight': 16.0}, {'from_id': 'T', 'to_id': 'R', 'weight': 617.0}, {'from_id': 'T', 'to_id': 'S', 'weight': 44.0}, {'from_id': 'T', 'to_id': 'U', 'weight': 28.0}, {'from_id': 'U', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'S', 'weight': 0.0}, {'from_id': 'U', 'to_id': 'T', 'weight': 0.0}]}","['D', 'B', 'E', 'Q', 'T', 'R', 'P', 'C', 'H', 'S', 'A', 'J', 'K', 'L', 'O', 'G', 'M', 'N', 'U', 'I', 'F']",29,nl,names
LOP,LOP,"A teacher imagined writing the class outline on sticky notes and swapping them until the class flowed smoothly. For any two different notes there’s a number that measures how much better the lesson is if one comes before the other. The teacher must make a single sequence that contains each topic exactly one time, then total up those pairwise numbers for every earlier-before-later pairing — the sequence with the highest total makes the most sense. The actual topics and their pairwise values are shown below.

The teacher listed 18 topics as A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R.
Meaning: this is a directed comprehension_gain matrix over items in topic_ids (18 total). Entry at row earlier_topic_id=i and column later_topic_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
comprehension_gain:
| earlier_topic_id\later_topic_id | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 404519.0 | 1999089.0 | 216886.0 | 133424.0 | 35202.0 | 15871.0 | 16685.0 | 4997.0 | 19045.0 | 760.0 | 10920.0 | 63516.0 | 16465.0 | 7103.0 | 3130.0 | 6839.0 | 0.0 | 1629.0 |
| B | 12055.0 | 27737.0 | 1768790.0 | 64392.0 | 17023.0 | 3789.0 | 4059.0 | 183.0 | 2394.0 | 453.0 | 2779.0 | 5341.0 | 2626.0 | 8387.0 | 2651.0 | 4854.0 | 820.0 | 2260.0 |
| C | 155009.0 | 12570.0 | 24768706.0 | 390415.0 | 5417989.0 | 4192814.0 | 3297487.0 | 25009.0 | 28261.0 | 15507.0 | 268871.0 | 97277.0 | 186180.0 | 316.0 | 19333.0 | 40.0 | 2232.0 | 21615.0 |
| D | 18942.0 | 3744.0 | 812988.0 | 3146146.0 | 125262.0 | 475258.0 | 354555.0 | 42995.0 | 250405.0 | 969.0 | 148798.0 | 65485.0 | 92229.0 | 84779.0 | 120174.0 | 7553.0 | 7275.0 | 48635.0 |
| E | 253014.0 | 12823.0 | 400444.0 | 315490.0 | 3960080.0 | 732672.0 | 817318.0 | 22168.0 | 58194.0 | 13936.0 | 116297.0 | 163002.0 | 127512.0 | 40040.0 | 73935.0 | 21902.0 | 4818.0 | 3566.0 |
| F | 63694.0 | 25871.0 | 225661.0 | 44497.0 | 1161057.0 | 4600451.0 | 1115143.0 | 2719.0 | 3174.0 | 1509.0 | 34534.0 | 45914.0 | 23489.0 | 69468.0 | 428986.0 | 183781.0 | 31542.0 | 23717.0 |
| G | 293.0 | 293.0 | 42735.0 | 12978.0 | 510703.0 | 6418.0 | 5015647.0 | 2804.0 | 3526.0 | 320.0 | 2779.0 | 3326.0 | 18859.0 | 1120.0 | 220539.0 | 7396.0 | 820.0 | 3019.0 |
| H | 0.0 | 0.0 | 0.0 | 240.0 | 0.0 | 0.0 | 0.0 | 163629.0 | 7565.0 | 0.0 | 0.0 | 7126.0 | 1160.0 | 448468.0 | 21600.0 | 0.0 | 0.0 | 72994.0 |
| I | 1639.0 | 546.0 | 5498.0 | 2686.0 | 3272.0 | 5827.0 | 2252.0 | 3646.0 | 637009.0 | 1173.0 | 3529.0 | 7403.0 | 720.0 | 3069845.0 | 189365.0 | 3372.0 | 1113.0 | 15219.0 |
| J | 1093.0 | 0.0 | 6557.0 | 2772.0 | 5051.0 | 4738.0 | 2472.0 | 600.0 | 2526.0 | 43725.0 | 2539.0 | 5704.0 | 1953.0 | 642210.0 | 31698.0 | 3332.0 | 1446.0 | 40.0 |
| K | 73398.0 | 96.0 | 158755.0 | 172105.0 | 139826.0 | 218055.0 | 169649.0 | 19680.0 | 102201.0 | 7893.0 | 4381253.0 | 250984.0 | 74450.0 | 48395.0 | 151816.0 | 35710.0 | 62075.0 | 34517.0 |
| L | 27107.0 | 15351.0 | 177287.0 | 510240.0 | 349336.0 | 515265.0 | 176349.0 | 220880.0 | 262521.0 | 283960.0 | 234607.0 | 8509671.0 | 193342.0 | 145729.0 | 2101894.0 | 1203650.0 | 52064.0 | 329842.0 |
| M | 25470.0 | 1360.0 | 74843.0 | 154624.0 | 536298.0 | 856805.0 | 1822179.0 | 90288.0 | 124240.0 | 16079.0 | 475584.0 | 123872.0 | 759407.0 | 30870.0 | 240723.0 | 54940.0 | 2588.0 | 96563.0 |
| N | 11789.0 | 1242.0 | 116907.0 | 72925.0 | 130460.0 | 178130.0 | 115904.0 | 11831.0 | 59024.0 | 19537.0 | 73115.0 | 138180.0 | 45916.0 | 101964.0 | 86876.0 | 246967.0 | 43302.0 | 35718.0 |
| O | 24531.0 | 14637.0 | 763478.0 | 383903.0 | 910580.0 | 562501.0 | 335760.0 | 102848.0 | 521751.0 | 188905.0 | 311760.0 | 552625.0 | 228847.0 | 55515.0 | 17158894.0 | 118859.0 | 340268.0 | 66649.0 |
| P | 38446.0 | 25874.0 | 277050.0 | 251312.0 | 489772.0 | 616692.0 | 371306.0 | 102194.0 | 255322.0 | 73776.0 | 205034.0 | 556157.0 | 208999.0 | 281853.0 | 2725825.0 | 1536818.0 | 1638561.0 | 161398.0 |
| Q | 5206.0 | 1988.0 | 21715.0 | 38119.0 | 53113.0 | 51923.0 | 28342.0 | 6003.0 | 8590.0 | 1528.0 | 45156.0 | 45996.0 | 16189.0 | 170193.0 | 102344.0 | 390435.0 | 180.0 | 106527.0 |
| R | 380.0 | 0.0 | 273.0 | 2186.0 | 2186.0 | 3552.0 | 1093.0 | 0.0 | 1639.0 | 273.0 | 2186.0 | 2459.0 | 1093.0 | 1093.0 | 15344.0 | 4372.0 | 2186.0 | 8547.0 |
Each line is one pairwise note the teacher made.

Oh — and when you send the final ordering back, a tiny JSON snippet like the one below is perfect and keeps things tidy:

{
  ""solution"": [""<first_item>"", ""<second_item>"", ""..."", ""<last_item>""]
}

Think of ""solution"" as the single list of sticky-note topics in the exact order you want them taught — first entry is the very first topic, last entry is the very last. This block is just a sketch of the shape I expect, not the real answer itself.

Please be sure to use the exact identifiers from the instance input — don't 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”.""","[[404519.0, 1999089.0, 216886.0, 133424.0, 35202.0, 15871.0, 16685.0, 4997.0, 19045.0, 760.0, 10920.0, 63516.0, 16465.0, 7103.0, 3130.0, 6839.0, 0.0, 1629.0], [12055.0, 27737.0, 1768790.0, 64392.0, 17023.0, 3789.0, 4059.0, 183.0, 2394.0, 453.0, 2779.0, 5341.0, 2626.0, 8387.0, 2651.0, 4854.0, 820.0, 2260.0], [155009.0, 12570.0, 24768706.0, 390415.0, 5417989.0, 4192814.0, 3297487.0, 25009.0, 28261.0, 15507.0, 268871.0, 97277.0, 186180.0, 316.0, 19333.0, 40.0, 2232.0, 21615.0], [18942.0, 3744.0, 812988.0, 3146146.0, 125262.0, 475258.0, 354555.0, 42995.0, 250405.0, 969.0, 148798.0, 65485.0, 92229.0, 84779.0, 120174.0, 7553.0, 7275.0, 48635.0], [253014.0, 12823.0, 400444.0, 315490.0, 3960080.0, 732672.0, 817318.0, 22168.0, 58194.0, 13936.0, 116297.0, 163002.0, 127512.0, 40040.0, 73935.0, 21902.0, 4818.0, 3566.0], [63694.0, 25871.0, 225661.0, 44497.0, 1161057.0, 4600451.0, 1115143.0, 2719.0, 3174.0, 1509.0, 34534.0, 45914.0, 23489.0, 69468.0, 428986.0, 183781.0, 31542.0, 23717.0], [293.0, 293.0, 42735.0, 12978.0, 510703.0, 6418.0, 5015647.0, 2804.0, 3526.0, 320.0, 2779.0, 3326.0, 18859.0, 1120.0, 220539.0, 7396.0, 820.0, 3019.0], [0.0, 0.0, 0.0, 240.0, 0.0, 0.0, 0.0, 163629.0, 7565.0, 0.0, 0.0, 7126.0, 1160.0, 448468.0, 21600.0, 0.0, 0.0, 72994.0], [1639.0, 546.0, 5498.0, 2686.0, 3272.0, 5827.0, 2252.0, 3646.0, 637009.0, 1173.0, 3529.0, 7403.0, 720.0, 3069845.0, 189365.0, 3372.0, 1113.0, 15219.0], [1093.0, 0.0, 6557.0, 2772.0, 5051.0, 4738.0, 2472.0, 600.0, 2526.0, 43725.0, 2539.0, 5704.0, 1953.0, 642210.0, 31698.0, 3332.0, 1446.0, 40.0], [73398.0, 96.0, 158755.0, 172105.0, 139826.0, 218055.0, 169649.0, 19680.0, 102201.0, 7893.0, 4381253.0, 250984.0, 74450.0, 48395.0, 151816.0, 35710.0, 62075.0, 34517.0], [27107.0, 15351.0, 177287.0, 510240.0, 349336.0, 515265.0, 176349.0, 220880.0, 262521.0, 283960.0, 234607.0, 8509671.0, 193342.0, 145729.0, 2101894.0, 1203650.0, 52064.0, 329842.0], [25470.0, 1360.0, 74843.0, 154624.0, 536298.0, 856805.0, 1822179.0, 90288.0, 124240.0, 16079.0, 475584.0, 123872.0, 759407.0, 30870.0, 240723.0, 54940.0, 2588.0, 96563.0], [11789.0, 1242.0, 116907.0, 72925.0, 130460.0, 178130.0, 115904.0, 11831.0, 59024.0, 19537.0, 73115.0, 138180.0, 45916.0, 101964.0, 86876.0, 246967.0, 43302.0, 35718.0], [24531.0, 14637.0, 763478.0, 383903.0, 910580.0, 562501.0, 335760.0, 102848.0, 521751.0, 188905.0, 311760.0, 552625.0, 228847.0, 55515.0, 17158894.0, 118859.0, 340268.0, 66649.0], [38446.0, 25874.0, 277050.0, 251312.0, 489772.0, 616692.0, 371306.0, 102194.0, 255322.0, 73776.0, 205034.0, 556157.0, 208999.0, 281853.0, 2725825.0, 1536818.0, 1638561.0, 161398.0], [5206.0, 1988.0, 21715.0, 38119.0, 53113.0, 51923.0, 28342.0, 6003.0, 8590.0, 1528.0, 45156.0, 45996.0, 16189.0, 170193.0, 102344.0, 390435.0, 180.0, 106527.0], [380.0, 0.0, 273.0, 2186.0, 2186.0, 3552.0, 1093.0, 0.0, 1639.0, 273.0, 2186.0, 2459.0, 1093.0, 1093.0, 15344.0, 4372.0, 2186.0, 8547.0]]","[11, 15, 12, 14, 10, 16, 0, 1, 9, 3, 7, 8, 13, 2, 5, 4, 6, 17]",52800378.0,"{'problem_type': 'LOP', 'num_nodes': 18, 'nodes': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R'], 'pairs': [{'from_id': 'A', 'to_id': 'B', 'weight': 1999089.0}, {'from_id': 'A', 'to_id': 'C', 'weight': 216886.0}, {'from_id': 'A', 'to_id': 'D', 'weight': 133424.0}, {'from_id': 'A', 'to_id': 'E', 'weight': 35202.0}, {'from_id': 'A', 'to_id': 'F', 'weight': 15871.0}, {'from_id': 'A', 'to_id': 'G', 'weight': 16685.0}, {'from_id': 'A', 'to_id': 'H', 'weight': 4997.0}, {'from_id': 'A', 'to_id': 'I', 'weight': 19045.0}, {'from_id': 'A', 'to_id': 'J', 'weight': 760.0}, {'from_id': 'A', 'to_id': 'K', 'weight': 10920.0}, {'from_id': 'A', 'to_id': 'L', 'weight': 63516.0}, {'from_id': 'A', 'to_id': 'M', 'weight': 16465.0}, {'from_id': 'A', 'to_id': 'N', 'weight': 7103.0}, 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{'from_id': 'B', 'to_id': 'R', 'weight': 2260.0}, {'from_id': 'C', 'to_id': 'A', 'weight': 155009.0}, {'from_id': 'C', 'to_id': 'B', 'weight': 12570.0}, {'from_id': 'C', 'to_id': 'D', 'weight': 390415.0}, {'from_id': 'C', 'to_id': 'E', 'weight': 5417989.0}, {'from_id': 'C', 'to_id': 'F', 'weight': 4192814.0}, {'from_id': 'C', 'to_id': 'G', 'weight': 3297487.0}, {'from_id': 'C', 'to_id': 'H', 'weight': 25009.0}, {'from_id': 'C', 'to_id': 'I', 'weight': 28261.0}, {'from_id': 'C', 'to_id': 'J', 'weight': 15507.0}, {'from_id': 'C', 'to_id': 'K', 'weight': 268871.0}, {'from_id': 'C', 'to_id': 'L', 'weight': 97277.0}, {'from_id': 'C', 'to_id': 'M', 'weight': 186180.0}, {'from_id': 'C', 'to_id': 'N', 'weight': 316.0}, {'from_id': 'C', 'to_id': 'O', 'weight': 19333.0}, {'from_id': 'C', 'to_id': 'P', 'weight': 40.0}, {'from_id': 'C', 'to_id': 'Q', 'weight': 2232.0}, {'from_id': 'C', 'to_id': 'R', 'weight': 21615.0}, {'from_id': 'D', 'to_id': 'A', 'weight': 18942.0}, {'from_id': 'D', 'to_id': 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'C', 'F', 'E', 'G', 'R']",30,nl,names
LOP,LOP,"I’m the art director putting this issue together and I’ve got a stack of photo spreads that need to run front to back. The job is to pick one long order for them so the whole magazine reads and looks great — for every pair of spreads we already know how much better it is to have one placed before the other. To score any full ordering, just add up those pairwise benefits for every pair that ends up in that direction; the best ordering is the one with the biggest total, and every spread has to appear exactly once, no repeats or skips. The exact pairwise details are shown below.

I’ve listed 19 spreads that need a single front-to-back sequence: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S.

*Meaning: this is a directed visual_benefit_score matrix over items in spread_ids (19 total). Entry at row prior_spread_id=i and column later_spread_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).*

**visual_benefit_score)**

| prior_spread_id\later_spread_id | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 18146.0 | 0.0 | 7.0 | 46.0 | 0.0 | 726.0 | 449.0 | 88.0 | 46.0 | 4.0 | 882.0 | 64.0 | 634.0 | 172.0 | 2200.0 | 429.0 | 63.0 | 8.0 | 162.0 |
| B | 0.0 | 677.0 | 1.0 | 34.0 | 68.0 | 0.0 | 39.0 | 21.0 | 0.0 | 22.0 | 0.0 | 21.0 | 44.0 | 14.0 | 147.0 | 41.0 | 36.0 | 0.0 | 6.0 |
| C | 0.0 | 0.0 | 71.0 | 0.0 | 0.0 | 51.0 | 8.0 | 1.0 | 0.0 | 0.0 | 0.0 | 15.0 | 182.0 | 64.0 | 70.0 | 2.0 | 11.0 | 6.0 | 47.0 |
| D | 0.0 | 0.0 | 24.0 | 506.0 | 0.0 | 31.0 | 1.0 | 16.0 | 0.0 | 0.0 | 0.0 | 9.0 | 24.0 | 2.0 | 0.0 | 1.0 | 13.0 | 2.0 | 7.0 |
| E | 0.0 | 0.0 | 6.0 | 0.0 | 817.0 | 826.0 | 69.0 | 73.0 | 3.0 | 0.0 | 0.0 | 0.0 | 253.0 | 36.0 | 38.0 | 12.0 | 23.0 | 7.0 | 44.0 |
| F | 1.0 | 0.0 | 35.0 | 0.0 | 0.0 | 13372.0 | 1649.0 | 829.0 | 39.0 | 6.0 | 0.0 | 31.0 | 1387.0 | 320.0 | 447.0 | 169.0 | 259.0 | 112.0 | 355.0 |
| G | 6.0 | 0.0 | 61.0 | 0.0 | 86.0 | 2022.0 | 15553.0 | 1072.0 | 43.0 | 6.0 | 0.0 | 20.0 | 1339.0 | 163.0 | 783.0 | 201.0 | 235.0 | 62.0 | 267.0 |
| H | 2.0 | 0.0 | 16.0 | 0.0 | 0.0 | 118.0 | 120.0 | 1098.0 | 46.0 | 5.0 | 0.0 | 15.0 | 456.0 | 320.0 | 95.0 | 50.0 | 230.0 | 37.0 | 130.0 |
| I | 13.0 | 0.0 | 0.0 | 0.0 | 0.0 | 31.0 | 20.0 | 219.0 | 605.0 | 127.0 | 0.0 | 6.0 | 89.0 | 61.0 | 102.0 | 17.0 | 42.0 | 12.0 | 30.0 |
| J | 991.0 | 0.0 | 7.0 | 0.0 | 0.0 | 12.0 | 30.0 | 313.0 | 58.0 | 1796.0 | 0.0 | 1.0 | 341.0 | 148.0 | 675.0 | 19.0 | 43.0 | 27.0 | 70.0 |
| K | 10915.0 | 0.0 | 18.0 | 2.0 | 0.0 | 0.0 | 19.0 | 206.0 | 0.0 | 0.0 | 1151.0 | 4.0 | 439.0 | 253.0 | 233.0 | 12.0 | 10.0 | 2.0 | 53.0 |
| L | 2073.0 | 0.0 | 25.0 | 0.0 | 0.0 | 17.0 | 94.0 | 344.0 | 4.0 | 0.0 | 9.0 | 2669.0 | 590.0 | 163.0 | 182.0 | 50.0 | 254.0 | 10.0 | 87.0 |
| M | 240.0 | 0.0 | 81.0 | 0.0 | 0.0 | 28.0 | 305.0 | 86.0 | 0.0 | 10.0 | 0.0 | 95.0 | 154.0 | 74.0 | 305.0 | 268.0 | 2722.0 | 72.0 | 235.0 |
| N | 6.0 | 0.0 | 14.0 | 0.0 | 0.0 | 0.0 | 200.0 | 30.0 | 11.0 | 10.0 | 0.0 | 58.0 | 70.0 | 915.0 | 203.0 | 176.0 | 1226.0 | 81.0 | 75.0 |
| O | 49.0 | 0.0 | 151.0 | 0.0 | 0.0 | 245.0 | 1213.0 | 66.0 | 6.0 | 264.0 | 0.0 | 278.0 | 463.0 | 172.0 | 905.0 | 554.0 | 222.0 | 4434.0 | 299.0 |
| P | 0.0 | 0.0 | 13.0 | 0.0 | 0.0 | 22.0 | 42.0 | 61.0 | 0.0 | 21.0 | 0.0 | 15.0 | 30.0 | 7.0 | 45.0 | 213.0 | 97.0 | 19.0 | 98.0 |
| Q | 1644.0 | 614.0 | 106.0 | 0.0 | 0.0 | 25.0 | 28.0 | 78.0 | 0.0 | 49.0 | 719.0 | 4417.0 | 1048.0 | 528.0 | 1419.0 | 29.0 | 93.0 | 33.0 | 84.0 |
| R | 1014.0 | 0.0 | 14.0 | 8.0 | 0.0 | 16.0 | 37.0 | 31.0 | 0.0 | 12.0 | 0.0 | 9.0 | 72.0 | 34.0 | 155.0 | 224.0 | 92.0 | 6.0 | 1765.0 |
| S | 1390.0 | 54.0 | 380.0 | 13.0 | 856.0 | 987.0 | 3486.0 | 1563.0 | 617.0 | 643.0 | 503.0 | 435.0 | 2026.0 | 529.0 | 2676.0 | 752.0 | 1161.0 | 1681.0 | 11435.0 |

I’ll use these pairwise scores to choose the ordering that maximizes the total visual impact.

When you send the final ordering back, just drop it into a tiny JSON object so it’s easy to parse — something casual, like this:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Pretty simple: ""solution"" holds an ordered list of the spreads from front to back — the first entry is the opener and the last entry is the closer. Think of the JSON as a little form: replace the placeholders with the actual spread identifiers from the instance input. This block is just a sketch of the shape I’m expecting, not the actual answer itself.

Please use the identifiers exactly as they appear in the instance input — no renaming and no 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”.","[[18146.0, 0.0, 7.0, 46.0, 0.0, 726.0, 449.0, 88.0, 46.0, 4.0, 882.0, 64.0, 634.0, 172.0, 2200.0, 429.0, 63.0, 8.0, 162.0], [0.0, 677.0, 1.0, 34.0, 68.0, 0.0, 39.0, 21.0, 0.0, 22.0, 0.0, 21.0, 44.0, 14.0, 147.0, 41.0, 36.0, 0.0, 6.0], [0.0, 0.0, 71.0, 0.0, 0.0, 51.0, 8.0, 1.0, 0.0, 0.0, 0.0, 15.0, 182.0, 64.0, 70.0, 2.0, 11.0, 6.0, 47.0], [0.0, 0.0, 24.0, 506.0, 0.0, 31.0, 1.0, 16.0, 0.0, 0.0, 0.0, 9.0, 24.0, 2.0, 0.0, 1.0, 13.0, 2.0, 7.0], [0.0, 0.0, 6.0, 0.0, 817.0, 826.0, 69.0, 73.0, 3.0, 0.0, 0.0, 0.0, 253.0, 36.0, 38.0, 12.0, 23.0, 7.0, 44.0], [1.0, 0.0, 35.0, 0.0, 0.0, 13372.0, 1649.0, 829.0, 39.0, 6.0, 0.0, 31.0, 1387.0, 320.0, 447.0, 169.0, 259.0, 112.0, 355.0], [6.0, 0.0, 61.0, 0.0, 86.0, 2022.0, 15553.0, 1072.0, 43.0, 6.0, 0.0, 20.0, 1339.0, 163.0, 783.0, 201.0, 235.0, 62.0, 267.0], [2.0, 0.0, 16.0, 0.0, 0.0, 118.0, 120.0, 1098.0, 46.0, 5.0, 0.0, 15.0, 456.0, 320.0, 95.0, 50.0, 230.0, 37.0, 130.0], [13.0, 0.0, 0.0, 0.0, 0.0, 31.0, 20.0, 219.0, 605.0, 127.0, 0.0, 6.0, 89.0, 61.0, 102.0, 17.0, 42.0, 12.0, 30.0], [991.0, 0.0, 7.0, 0.0, 0.0, 12.0, 30.0, 313.0, 58.0, 1796.0, 0.0, 1.0, 341.0, 148.0, 675.0, 19.0, 43.0, 27.0, 70.0], [10915.0, 0.0, 18.0, 2.0, 0.0, 0.0, 19.0, 206.0, 0.0, 0.0, 1151.0, 4.0, 439.0, 253.0, 233.0, 12.0, 10.0, 2.0, 53.0], [2073.0, 0.0, 25.0, 0.0, 0.0, 17.0, 94.0, 344.0, 4.0, 0.0, 9.0, 2669.0, 590.0, 163.0, 182.0, 50.0, 254.0, 10.0, 87.0], [240.0, 0.0, 81.0, 0.0, 0.0, 28.0, 305.0, 86.0, 0.0, 10.0, 0.0, 95.0, 154.0, 74.0, 305.0, 268.0, 2722.0, 72.0, 235.0], [6.0, 0.0, 14.0, 0.0, 0.0, 0.0, 200.0, 30.0, 11.0, 10.0, 0.0, 58.0, 70.0, 915.0, 203.0, 176.0, 1226.0, 81.0, 75.0], [49.0, 0.0, 151.0, 0.0, 0.0, 245.0, 1213.0, 66.0, 6.0, 264.0, 0.0, 278.0, 463.0, 172.0, 905.0, 554.0, 222.0, 4434.0, 299.0], [0.0, 0.0, 13.0, 0.0, 0.0, 22.0, 42.0, 61.0, 0.0, 21.0, 0.0, 15.0, 30.0, 7.0, 45.0, 213.0, 97.0, 19.0, 98.0], [1644.0, 614.0, 106.0, 0.0, 0.0, 25.0, 28.0, 78.0, 0.0, 49.0, 719.0, 4417.0, 1048.0, 528.0, 1419.0, 29.0, 93.0, 33.0, 84.0], [1014.0, 0.0, 14.0, 8.0, 0.0, 16.0, 37.0, 31.0, 0.0, 12.0, 0.0, 9.0, 72.0, 34.0, 155.0, 224.0, 92.0, 6.0, 1765.0], [1390.0, 54.0, 380.0, 13.0, 856.0, 987.0, 3486.0, 1563.0, 617.0, 643.0, 503.0, 435.0, 2026.0, 529.0, 2676.0, 752.0, 1161.0, 1681.0, 11435.0]]","[18, 8, 16, 1, 9, 11, 10, 4, 0, 3, 14, 6, 2, 5, 7, 12, 13, 17, 15]",73179.0,"{'problem_type': 'LOP', 'num_nodes': 19, 'nodes': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S'], 'pairs': [{'from_id': 'A', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'C', 'weight': 7.0}, {'from_id': 'A', 'to_id': 'D', 'weight': 46.0}, {'from_id': 'A', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'F', 'weight': 726.0}, {'from_id': 'A', 'to_id': 'G', 'weight': 449.0}, {'from_id': 'A', 'to_id': 'H', 'weight': 88.0}, {'from_id': 'A', 'to_id': 'I', 'weight': 46.0}, {'from_id': 'A', 'to_id': 'J', 'weight': 4.0}, {'from_id': 'A', 'to_id': 'K', 'weight': 882.0}, {'from_id': 'A', 'to_id': 'L', 'weight': 64.0}, {'from_id': 'A', 'to_id': 'M', 'weight': 634.0}, {'from_id': 'A', 'to_id': 'N', 'weight': 172.0}, {'from_id': 'A', 'to_id': 'O', 'weight': 2200.0}, {'from_id': 'A', 'to_id': 'P', 'weight': 429.0}, {'from_id': 'A', 'to_id': 'Q', 'weight': 63.0}, {'from_id': 'A', 'to_id': 'R', 'weight': 8.0}, {'from_id': 'A', 'to_id': 'S', 'weight': 162.0}, {'from_id': 'B', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'C', 'weight': 1.0}, {'from_id': 'B', 'to_id': 'D', 'weight': 34.0}, {'from_id': 'B', 'to_id': 'E', 'weight': 68.0}, {'from_id': 'B', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'G', 'weight': 39.0}, {'from_id': 'B', 'to_id': 'H', 'weight': 21.0}, {'from_id': 'B', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'J', 'weight': 22.0}, {'from_id': 'B', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'L', 'weight': 21.0}, {'from_id': 'B', 'to_id': 'M', 'weight': 44.0}, {'from_id': 'B', 'to_id': 'N', 'weight': 14.0}, {'from_id': 'B', 'to_id': 'O', 'weight': 147.0}, {'from_id': 'B', 'to_id': 'P', 'weight': 41.0}, {'from_id': 'B', 'to_id': 'Q', 'weight': 36.0}, {'from_id': 'B', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'S', 'weight': 6.0}, {'from_id': 'C', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'F', 'weight': 51.0}, {'from_id': 'C', 'to_id': 'G', 'weight': 8.0}, {'from_id': 'C', 'to_id': 'H', 'weight': 1.0}, {'from_id': 'C', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'L', 'weight': 15.0}, {'from_id': 'C', 'to_id': 'M', 'weight': 182.0}, {'from_id': 'C', 'to_id': 'N', 'weight': 64.0}, {'from_id': 'C', 'to_id': 'O', 'weight': 70.0}, {'from_id': 'C', 'to_id': 'P', 'weight': 2.0}, {'from_id': 'C', 'to_id': 'Q', 'weight': 11.0}, {'from_id': 'C', 'to_id': 'R', 'weight': 6.0}, {'from_id': 'C', 'to_id': 'S', 'weight': 47.0}, {'from_id': 'D', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'C', 'weight': 24.0}, {'from_id': 'D', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'F', 'weight': 31.0}, {'from_id': 'D', 'to_id': 'G', 'weight': 1.0}, {'from_id': 'D', 'to_id': 'H', 'weight': 16.0}, {'from_id': 'D', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'L', 'weight': 9.0}, {'from_id': 'D', 'to_id': 'M', 'weight': 24.0}, {'from_id': 'D', 'to_id': 'N', 'weight': 2.0}, {'from_id': 'D', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'P', 'weight': 1.0}, {'from_id': 'D', 'to_id': 'Q', 'weight': 13.0}, {'from_id': 'D', 'to_id': 'R', 'weight': 2.0}, {'from_id': 'D', 'to_id': 'S', 'weight': 7.0}, {'from_id': 'E', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'C', 'weight': 6.0}, {'from_id': 'E', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'F', 'weight': 826.0}, {'from_id': 'E', 'to_id': 'G', 'weight': 69.0}, {'from_id': 'E', 'to_id': 'H', 'weight': 73.0}, {'from_id': 'E', 'to_id': 'I', 'weight': 3.0}, {'from_id': 'E', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'M', 'weight': 253.0}, {'from_id': 'E', 'to_id': 'N', 'weight': 36.0}, {'from_id': 'E', 'to_id': 'O', 'weight': 38.0}, {'from_id': 'E', 'to_id': 'P', 'weight': 12.0}, {'from_id': 'E', 'to_id': 'Q', 'weight': 23.0}, {'from_id': 'E', 'to_id': 'R', 'weight': 7.0}, {'from_id': 'E', 'to_id': 'S', 'weight': 44.0}, {'from_id': 'F', 'to_id': 'A', 'weight': 1.0}, {'from_id': 'F', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'C', 'weight': 35.0}, {'from_id': 'F', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'G', 'weight': 1649.0}, {'from_id': 'F', 'to_id': 'H', 'weight': 829.0}, {'from_id': 'F', 'to_id': 'I', 'weight': 39.0}, {'from_id': 'F', 'to_id': 'J', 'weight': 6.0}, {'from_id': 'F', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'L', 'weight': 31.0}, {'from_id': 'F', 'to_id': 'M', 'weight': 1387.0}, {'from_id': 'F', 'to_id': 'N', 'weight': 320.0}, {'from_id': 'F', 'to_id': 'O', 'weight': 447.0}, {'from_id': 'F', 'to_id': 'P', 'weight': 169.0}, {'from_id': 'F', 'to_id': 'Q', 'weight': 259.0}, {'from_id': 'F', 'to_id': 'R', 'weight': 112.0}, {'from_id': 'F', 'to_id': 'S', 'weight': 355.0}, {'from_id': 'G', 'to_id': 'A', 'weight': 6.0}, {'from_id': 'G', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'C', 'weight': 61.0}, {'from_id': 'G', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'E', 'weight': 86.0}, {'from_id': 'G', 'to_id': 'F', 'weight': 2022.0}, {'from_id': 'G', 'to_id': 'H', 'weight': 1072.0}, {'from_id': 'G', 'to_id': 'I', 'weight': 43.0}, {'from_id': 'G', 'to_id': 'J', 'weight': 6.0}, {'from_id': 'G', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'L', 'weight': 20.0}, {'from_id': 'G', 'to_id': 'M', 'weight': 1339.0}, {'from_id': 'G', 'to_id': 'N', 'weight': 163.0}, {'from_id': 'G', 'to_id': 'O', 'weight': 783.0}, {'from_id': 'G', 'to_id': 'P', 'weight': 201.0}, {'from_id': 'G', 'to_id': 'Q', 'weight': 235.0}, {'from_id': 'G', 'to_id': 'R', 'weight': 62.0}, {'from_id': 'G', 'to_id': 'S', 'weight': 267.0}, {'from_id': 'H', 'to_id': 'A', 'weight': 2.0}, {'from_id': 'H', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'C', 'weight': 16.0}, {'from_id': 'H', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'F', 'weight': 118.0}, {'from_id': 'H', 'to_id': 'G', 'weight': 120.0}, {'from_id': 'H', 'to_id': 'I', 'weight': 46.0}, {'from_id': 'H', 'to_id': 'J', 'weight': 5.0}, {'from_id': 'H', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'L', 'weight': 15.0}, {'from_id': 'H', 'to_id': 'M', 'weight': 456.0}, {'from_id': 'H', 'to_id': 'N', 'weight': 320.0}, {'from_id': 'H', 'to_id': 'O', 'weight': 95.0}, {'from_id': 'H', 'to_id': 'P', 'weight': 50.0}, {'from_id': 'H', 'to_id': 'Q', 'weight': 230.0}, {'from_id': 'H', 'to_id': 'R', 'weight': 37.0}, {'from_id': 'H', 'to_id': 'S', 'weight': 130.0}, {'from_id': 'I', 'to_id': 'A', 'weight': 13.0}, {'from_id': 'I', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'F', 'weight': 31.0}, {'from_id': 'I', 'to_id': 'G', 'weight': 20.0}, {'from_id': 'I', 'to_id': 'H', 'weight': 219.0}, {'from_id': 'I', 'to_id': 'J', 'weight': 127.0}, {'from_id': 'I', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'L', 'weight': 6.0}, {'from_id': 'I', 'to_id': 'M', 'weight': 89.0}, {'from_id': 'I', 'to_id': 'N', 'weight': 61.0}, {'from_id': 'I', 'to_id': 'O', 'weight': 102.0}, {'from_id': 'I', 'to_id': 'P', 'weight': 17.0}, {'from_id': 'I', 'to_id': 'Q', 'weight': 42.0}, {'from_id': 'I', 'to_id': 'R', 'weight': 12.0}, {'from_id': 'I', 'to_id': 'S', 'weight': 30.0}, {'from_id': 'J', 'to_id': 'A', 'weight': 991.0}, {'from_id': 'J', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'C', 'weight': 7.0}, {'from_id': 'J', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'F', 'weight': 12.0}, {'from_id': 'J', 'to_id': 'G', 'weight': 30.0}, {'from_id': 'J', 'to_id': 'H', 'weight': 313.0}, {'from_id': 'J', 'to_id': 'I', 'weight': 58.0}, {'from_id': 'J', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'L', 'weight': 1.0}, {'from_id': 'J', 'to_id': 'M', 'weight': 341.0}, {'from_id': 'J', 'to_id': 'N', 'weight': 148.0}, {'from_id': 'J', 'to_id': 'O', 'weight': 675.0}, {'from_id': 'J', 'to_id': 'P', 'weight': 19.0}, {'from_id': 'J', 'to_id': 'Q', 'weight': 43.0}, {'from_id': 'J', 'to_id': 'R', 'weight': 27.0}, {'from_id': 'J', 'to_id': 'S', 'weight': 70.0}, {'from_id': 'K', 'to_id': 'A', 'weight': 10915.0}, {'from_id': 'K', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'C', 'weight': 18.0}, {'from_id': 'K', 'to_id': 'D', 'weight': 2.0}, {'from_id': 'K', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'G', 'weight': 19.0}, {'from_id': 'K', 'to_id': 'H', 'weight': 206.0}, {'from_id': 'K', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'L', 'weight': 4.0}, {'from_id': 'K', 'to_id': 'M', 'weight': 439.0}, {'from_id': 'K', 'to_id': 'N', 'weight': 253.0}, {'from_id': 'K', 'to_id': 'O', 'weight': 233.0}, {'from_id': 'K', 'to_id': 'P', 'weight': 12.0}, {'from_id': 'K', 'to_id': 'Q', 'weight': 10.0}, {'from_id': 'K', 'to_id': 'R', 'weight': 2.0}, {'from_id': 'K', 'to_id': 'S', 'weight': 53.0}, {'from_id': 'L', 'to_id': 'A', 'weight': 2073.0}, {'from_id': 'L', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'C', 'weight': 25.0}, {'from_id': 'L', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'F', 'weight': 17.0}, {'from_id': 'L', 'to_id': 'G', 'weight': 94.0}, {'from_id': 'L', 'to_id': 'H', 'weight': 344.0}, {'from_id': 'L', 'to_id': 'I', 'weight': 4.0}, {'from_id': 'L', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'K', 'weight': 9.0}, {'from_id': 'L', 'to_id': 'M', 'weight': 590.0}, {'from_id': 'L', 'to_id': 'N', 'weight': 163.0}, {'from_id': 'L', 'to_id': 'O', 'weight': 182.0}, {'from_id': 'L', 'to_id': 'P', 'weight': 50.0}, {'from_id': 'L', 'to_id': 'Q', 'weight': 254.0}, {'from_id': 'L', 'to_id': 'R', 'weight': 10.0}, {'from_id': 'L', 'to_id': 'S', 'weight': 87.0}, {'from_id': 'M', 'to_id': 'A', 'weight': 240.0}, {'from_id': 'M', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'M', 'to_id': 'C', 'weight': 81.0}, {'from_id': 'M', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'M', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'M', 'to_id': 'F', 'weight': 28.0}, {'from_id': 'M', 'to_id': 'G', 'weight': 305.0}, {'from_id': 'M', 'to_id': 'H', 'weight': 86.0}, {'from_id': 'M', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'M', 'to_id': 'J', 'weight': 10.0}, {'from_id': 'M', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'M', 'to_id': 'L', 'weight': 95.0}, {'from_id': 'M', 'to_id': 'N', 'weight': 74.0}, {'from_id': 'M', 'to_id': 'O', 'weight': 305.0}, {'from_id': 'M', 'to_id': 'P', 'weight': 268.0}, {'from_id': 'M', 'to_id': 'Q', 'weight': 2722.0}, {'from_id': 'M', 'to_id': 'R', 'weight': 72.0}, {'from_id': 'M', 'to_id': 'S', 'weight': 235.0}, {'from_id': 'N', 'to_id': 'A', 'weight': 6.0}, {'from_id': 'N', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'C', 'weight': 14.0}, {'from_id': 'N', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'G', 'weight': 200.0}, {'from_id': 'N', 'to_id': 'H', 'weight': 30.0}, {'from_id': 'N', 'to_id': 'I', 'weight': 11.0}, {'from_id': 'N', 'to_id': 'J', 'weight': 10.0}, {'from_id': 'N', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'L', 'weight': 58.0}, {'from_id': 'N', 'to_id': 'M', 'weight': 70.0}, {'from_id': 'N', 'to_id': 'O', 'weight': 203.0}, {'from_id': 'N', 'to_id': 'P', 'weight': 176.0}, {'from_id': 'N', 'to_id': 'Q', 'weight': 1226.0}, {'from_id': 'N', 'to_id': 'R', 'weight': 81.0}, {'from_id': 'N', 'to_id': 'S', 'weight': 75.0}, {'from_id': 'O', 'to_id': 'A', 'weight': 49.0}, {'from_id': 'O', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'C', 'weight': 151.0}, {'from_id': 'O', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'F', 'weight': 245.0}, {'from_id': 'O', 'to_id': 'G', 'weight': 1213.0}, {'from_id': 'O', 'to_id': 'H', 'weight': 66.0}, {'from_id': 'O', 'to_id': 'I', 'weight': 6.0}, {'from_id': 'O', 'to_id': 'J', 'weight': 264.0}, {'from_id': 'O', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'L', 'weight': 278.0}, {'from_id': 'O', 'to_id': 'M', 'weight': 463.0}, {'from_id': 'O', 'to_id': 'N', 'weight': 172.0}, {'from_id': 'O', 'to_id': 'P', 'weight': 554.0}, {'from_id': 'O', 'to_id': 'Q', 'weight': 222.0}, {'from_id': 'O', 'to_id': 'R', 'weight': 4434.0}, {'from_id': 'O', 'to_id': 'S', 'weight': 299.0}, {'from_id': 'P', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'C', 'weight': 13.0}, {'from_id': 'P', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'F', 'weight': 22.0}, {'from_id': 'P', 'to_id': 'G', 'weight': 42.0}, {'from_id': 'P', 'to_id': 'H', 'weight': 61.0}, {'from_id': 'P', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'J', 'weight': 21.0}, {'from_id': 'P', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'L', 'weight': 15.0}, {'from_id': 'P', 'to_id': 'M', 'weight': 30.0}, {'from_id': 'P', 'to_id': 'N', 'weight': 7.0}, {'from_id': 'P', 'to_id': 'O', 'weight': 45.0}, {'from_id': 'P', 'to_id': 'Q', 'weight': 97.0}, {'from_id': 'P', 'to_id': 'R', 'weight': 19.0}, {'from_id': 'P', 'to_id': 'S', 'weight': 98.0}, {'from_id': 'Q', 'to_id': 'A', 'weight': 1644.0}, {'from_id': 'Q', 'to_id': 'B', 'weight': 614.0}, {'from_id': 'Q', 'to_id': 'C', 'weight': 106.0}, {'from_id': 'Q', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'F', 'weight': 25.0}, {'from_id': 'Q', 'to_id': 'G', 'weight': 28.0}, {'from_id': 'Q', 'to_id': 'H', 'weight': 78.0}, {'from_id': 'Q', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'J', 'weight': 49.0}, {'from_id': 'Q', 'to_id': 'K', 'weight': 719.0}, {'from_id': 'Q', 'to_id': 'L', 'weight': 4417.0}, {'from_id': 'Q', 'to_id': 'M', 'weight': 1048.0}, {'from_id': 'Q', 'to_id': 'N', 'weight': 528.0}, {'from_id': 'Q', 'to_id': 'O', 'weight': 1419.0}, {'from_id': 'Q', 'to_id': 'P', 'weight': 29.0}, {'from_id': 'Q', 'to_id': 'R', 'weight': 33.0}, {'from_id': 'Q', 'to_id': 'S', 'weight': 84.0}, {'from_id': 'R', 'to_id': 'A', 'weight': 1014.0}, {'from_id': 'R', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'C', 'weight': 14.0}, {'from_id': 'R', 'to_id': 'D', 'weight': 8.0}, {'from_id': 'R', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'F', 'weight': 16.0}, {'from_id': 'R', 'to_id': 'G', 'weight': 37.0}, {'from_id': 'R', 'to_id': 'H', 'weight': 31.0}, {'from_id': 'R', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'J', 'weight': 12.0}, {'from_id': 'R', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'L', 'weight': 9.0}, {'from_id': 'R', 'to_id': 'M', 'weight': 72.0}, {'from_id': 'R', 'to_id': 'N', 'weight': 34.0}, {'from_id': 'R', 'to_id': 'O', 'weight': 155.0}, {'from_id': 'R', 'to_id': 'P', 'weight': 224.0}, {'from_id': 'R', 'to_id': 'Q', 'weight': 92.0}, {'from_id': 'R', 'to_id': 'S', 'weight': 1765.0}, {'from_id': 'S', 'to_id': 'A', 'weight': 1390.0}, {'from_id': 'S', 'to_id': 'B', 'weight': 54.0}, {'from_id': 'S', 'to_id': 'C', 'weight': 380.0}, {'from_id': 'S', 'to_id': 'D', 'weight': 13.0}, {'from_id': 'S', 'to_id': 'E', 'weight': 856.0}, {'from_id': 'S', 'to_id': 'F', 'weight': 987.0}, {'from_id': 'S', 'to_id': 'G', 'weight': 3486.0}, {'from_id': 'S', 'to_id': 'H', 'weight': 1563.0}, {'from_id': 'S', 'to_id': 'I', 'weight': 617.0}, {'from_id': 'S', 'to_id': 'J', 'weight': 643.0}, {'from_id': 'S', 'to_id': 'K', 'weight': 503.0}, {'from_id': 'S', 'to_id': 'L', 'weight': 435.0}, {'from_id': 'S', 'to_id': 'M', 'weight': 2026.0}, {'from_id': 'S', 'to_id': 'N', 'weight': 529.0}, {'from_id': 'S', 'to_id': 'O', 'weight': 2676.0}, {'from_id': 'S', 'to_id': 'P', 'weight': 752.0}, {'from_id': 'S', 'to_id': 'Q', 'weight': 1161.0}, {'from_id': 'S', 'to_id': 'R', 'weight': 1681.0}]}","['S', 'I', 'Q', 'B', 'J', 'L', 'K', 'E', 'A', 'D', 'O', 'G', 'C', 'F', 'H', 'M', 'N', 'R', 'P']",31,markdown_table,names
LOP,LOP,"Someone on the planning committee has to decide the sequence of panels, and the aim is to make the lineup produce the most overall discussion value. There’s a tiny score for every ordered pair of topics that tells how beneficial it is to have one precede the other; after assigning each topic a single slot (no repeats, no skipping), combine the appropriate score for every pair according to who’s earlier, and add them up. The ordering that yields the largest combined score is what we want. The exact topics and their pairwise scores appear below.

Below are the 22 topics (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21) followed by each ordered pair with its listed benefit.

*Meaning: this is a directed pairwise_benefit matrix over items in topics_list (22 total). Entry at row earlier_topic_id=i and column later_topic_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).*

**pairwise_benefit)**

| earlier_topic_id\later_topic_id | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 31.0 | 198.0 | 702.0 | 247.0 | 44.0 | 1.0 | 1.0 | 21.0 | 5.0 | 5.0 | 42.0 | 4.0 | 14.0 | 40.0 | 0.0 | 9.0 | 11.0 | 5.0 | 91.0 | 12.0 | 8.0 | 5.0 |
| 1 | 21.0 | 7571.0 | 1483.0 | 713.0 | 190.0 | 155.0 | 18.0 | 491.0 | 653.0 | 121.0 | 22154.0 | 139.0 | 263.0 | 80.0 | 0.0 | 2.0 | 0.0 | 233.0 | 648.0 | 163.0 | 237.0 | 64.0 |
| 2 | 21.0 | 921.0 | 33624.0 | 1451.0 | 823.0 | 242.0 | 273.0 | 413.0 | 229.0 | 2581.0 | 430.0 | 135.0 | 237.0 | 400.0 | 29.0 | 91.0 | 282.0 | 812.0 | 6717.0 | 867.0 | 1586.0 | 63.0 |
| 3 | 56.0 | 505.0 | 3136.0 | 8428.0 | 8403.0 | 792.0 | 993.0 | 793.0 | 417.0 | 753.0 | 19086.0 | 677.0 | 170.0 | 1014.0 | 108.0 | 79.0 | 189.0 | 47.0 | 2717.0 | 170.0 | 78.0 | 27.0 |
| 4 | 18.0 | 220.0 | 383.0 | 1943.0 | 13320.0 | 292.0 | 976.0 | 81.0 | 34.0 | 280.0 | 1188.0 | 102.0 | 32.0 | 158.0 | 14.0 | 14.0 | 62.0 | 2.0 | 995.0 | 28.0 | 3.0 | 1.0 |
| 5 | 32.0 | 45.0 | 226.0 | 222.0 | 380.0 | 2298.0 | 123.0 | 30.0 | 21.0 | 215.0 | 26.0 | 75.0 | 52.0 | 57.0 | 6.0 | 58.0 | 140.0 | 228.0 | 1609.0 | 254.0 | 183.0 | 9.0 |
| 6 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 544.0 | 0.0 | 0.0 | 0.0 | 17.0 | 85.0 | 0.0 | 0.0 | 1011.0 | 0.0 | 0.0 | 0.0 | 2024.0 | 0.0 | 0.0 | 44.0 |
| 7 | 0.0 | 3.0 | 1503.0 | 14.0 | 19.0 | 1.0 | 0.0 | 16479.0 | 1138.0 | 23.0 | 16.0 | 10.0 | 3600.0 | 0.0 | 379.0 | 0.0 | 14.0 | 196.0 | 1026.0 | 137.0 | 644.0 | 229.0 |
| 8 | 8.0 | 37.0 | 659.0 | 26.0 | 34.0 | 0.0 | 0.0 | 487.0 | 2911.0 | 84.0 | 26.0 | 33.0 | 5543.0 | 41.0 | 141.0 | 0.0 | 49.0 | 9.0 | 96.0 | 291.0 | 17.0 | 886.0 |
| 9 | 23.0 | 1716.0 | 3143.0 | 931.0 | 910.0 | 278.0 | 23.0 | 1838.0 | 553.0 | 15440.0 | 722.0 | 239.0 | 296.0 | 695.0 | 102.0 | 314.0 | 813.0 | 206.0 | 3320.0 | 1071.0 | 171.0 | 139.0 |
| 10 | 25.0 | 45.0 | 144.0 | 63.0 | 50.0 | 51.0 | 41.0 | 72.0 | 48.0 | 83.0 | 1206.0 | 11.0 | 16.0 | 828.0 | 15.0 | 55.0 | 131.0 | 96.0 | 2218.0 | 699.0 | 96.0 | 2.0 |
| 11 | 11.0 | 226.0 | 198.0 | 906.0 | 152.0 | 32.0 | 45.0 | 100.0 | 78.0 | 400.0 | 199.0 | 102.0 | 78.0 | 1000.0 | 12.0 | 57.0 | 187.0 | 92.0 | 454.0 | 176.0 | 122.0 | 8.0 |
| 12 | 6.0 | 277.0 | 114.0 | 414.0 | 505.0 | 66.0 | 24.0 | 233.0 | 231.0 | 859.0 | 110.0 | 173.0 | 80.0 | 776.0 | 47.0 | 47.0 | 214.0 | 4.0 | 1309.0 | 6.0 | 11.0 | 117.0 |
| 13 | 785.0 | 1543.0 | 1883.0 | 1379.0 | 839.0 | 227.0 | 111.0 | 1169.0 | 266.0 | 1770.0 | 3077.0 | 125.0 | 443.0 | 257.0 | 81.0 | 83.0 | 157.0 | 110.0 | 895.0 | 211.0 | 142.0 | 75.0 |
| 14 | 156.0 | 232.0 | 238.0 | 76.0 | 185.0 | 59.0 | 3.0 | 100.0 | 25.0 | 102.0 | 136.0 | 45.0 | 22.0 | 42.0 | 2302.0 | 6.0 | 75.0 | 11.0 | 342.0 | 37.0 | 30.0 | 8.0 |
| 15 | 6.0 | 246.0 | 420.0 | 499.0 | 634.0 | 104.0 | 52.0 | 149.0 | 93.0 | 389.0 | 477.0 | 88.0 | 222.0 | 859.0 | 57.0 | 175.0 | 1100.0 | 220.0 | 2100.0 | 507.0 | 138.0 | 73.0 |
| 16 | 16.0 | 176.0 | 345.0 | 216.0 | 261.0 | 32.0 | 18.0 | 111.0 | 94.0 | 280.0 | 672.0 | 82.0 | 80.0 | 1589.0 | 73.0 | 135.0 | 32424.0 | 187.0 | 1259.0 | 308.0 | 65.0 | 60.0 |
| 17 | 0.0 | 11.0 | 18.0 | 20.0 | 11.0 | 8.0 | 0.0 | 8.0 | 9.0 | 15.0 | 15.0 | 5.0 | 3.0 | 45.0 | 6.0 | 7.0 | 111.0 | 63.0 | 18945.0 | 41.0 | 10.0 | 0.0 |
| 18 | 11.0 | 98.0 | 300.0 | 226.0 | 247.0 | 54.0 | 37.0 | 180.0 | 77.0 | 150.0 | 280.0 | 30.0 | 71.0 | 164.0 | 38.0 | 63.0 | 167.0 | 63.0 | 865.0 | 201.0 | 101.0 | 21.0 |
| 19 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 135.0 | 0.0 | 0.0 | 0.0 |
| 20 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 16754.0 | 0.0 | 0.0 | 0.0 |
| 21 | 0.0 | 0.0 | 13.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 102.0 | 3521.0 | 0.0 | 17.0 | 0.0 |

Someone should pick the ordering that maximizes the summed pairwise benefits.

Also, when you send back the final ordering, please use this simple JSON shape so it's easy to read and plug into whatever tool you're using.

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

""solution"" is just a list that shows the topics in the exact order you want them to appear — first slot through last slot. This JSON is only a sketch of the shape I expect, not the actual answer.

And one small but important note: use the identifiers exactly as they appear in the instance input — don't rename them and don't 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”.""","[[31.0, 198.0, 702.0, 247.0, 44.0, 1.0, 1.0, 21.0, 5.0, 5.0, 42.0, 4.0, 14.0, 40.0, 0.0, 9.0, 11.0, 5.0, 91.0, 12.0, 8.0, 5.0], [21.0, 7571.0, 1483.0, 713.0, 190.0, 155.0, 18.0, 491.0, 653.0, 121.0, 22154.0, 139.0, 263.0, 80.0, 0.0, 2.0, 0.0, 233.0, 648.0, 163.0, 237.0, 64.0], [21.0, 921.0, 33624.0, 1451.0, 823.0, 242.0, 273.0, 413.0, 229.0, 2581.0, 430.0, 135.0, 237.0, 400.0, 29.0, 91.0, 282.0, 812.0, 6717.0, 867.0, 1586.0, 63.0], [56.0, 505.0, 3136.0, 8428.0, 8403.0, 792.0, 993.0, 793.0, 417.0, 753.0, 19086.0, 677.0, 170.0, 1014.0, 108.0, 79.0, 189.0, 47.0, 2717.0, 170.0, 78.0, 27.0], [18.0, 220.0, 383.0, 1943.0, 13320.0, 292.0, 976.0, 81.0, 34.0, 280.0, 1188.0, 102.0, 32.0, 158.0, 14.0, 14.0, 62.0, 2.0, 995.0, 28.0, 3.0, 1.0], [32.0, 45.0, 226.0, 222.0, 380.0, 2298.0, 123.0, 30.0, 21.0, 215.0, 26.0, 75.0, 52.0, 57.0, 6.0, 58.0, 140.0, 228.0, 1609.0, 254.0, 183.0, 9.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 544.0, 0.0, 0.0, 0.0, 17.0, 85.0, 0.0, 0.0, 1011.0, 0.0, 0.0, 0.0, 2024.0, 0.0, 0.0, 44.0], [0.0, 3.0, 1503.0, 14.0, 19.0, 1.0, 0.0, 16479.0, 1138.0, 23.0, 16.0, 10.0, 3600.0, 0.0, 379.0, 0.0, 14.0, 196.0, 1026.0, 137.0, 644.0, 229.0], [8.0, 37.0, 659.0, 26.0, 34.0, 0.0, 0.0, 487.0, 2911.0, 84.0, 26.0, 33.0, 5543.0, 41.0, 141.0, 0.0, 49.0, 9.0, 96.0, 291.0, 17.0, 886.0], [23.0, 1716.0, 3143.0, 931.0, 910.0, 278.0, 23.0, 1838.0, 553.0, 15440.0, 722.0, 239.0, 296.0, 695.0, 102.0, 314.0, 813.0, 206.0, 3320.0, 1071.0, 171.0, 139.0], [25.0, 45.0, 144.0, 63.0, 50.0, 51.0, 41.0, 72.0, 48.0, 83.0, 1206.0, 11.0, 16.0, 828.0, 15.0, 55.0, 131.0, 96.0, 2218.0, 699.0, 96.0, 2.0], [11.0, 226.0, 198.0, 906.0, 152.0, 32.0, 45.0, 100.0, 78.0, 400.0, 199.0, 102.0, 78.0, 1000.0, 12.0, 57.0, 187.0, 92.0, 454.0, 176.0, 122.0, 8.0], [6.0, 277.0, 114.0, 414.0, 505.0, 66.0, 24.0, 233.0, 231.0, 859.0, 110.0, 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{'from_id': 21, 'to_id': 9, 'weight': 0.0}, {'from_id': 21, 'to_id': 10, 'weight': 0.0}, {'from_id': 21, 'to_id': 11, 'weight': 0.0}, {'from_id': 21, 'to_id': 12, 'weight': 0.0}, {'from_id': 21, 'to_id': 13, 'weight': 0.0}, {'from_id': 21, 'to_id': 14, 'weight': 0.0}, {'from_id': 21, 'to_id': 15, 'weight': 0.0}, {'from_id': 21, 'to_id': 16, 'weight': 0.0}, {'from_id': 21, 'to_id': 17, 'weight': 102.0}, {'from_id': 21, 'to_id': 18, 'weight': 3521.0}, {'from_id': 21, 'to_id': 19, 'weight': 0.0}, {'from_id': 21, 'to_id': 20, 'weight': 17.0}]}","[15, 11, 16, 13, 9, 0, 1, 3, 5, 7, 8, 2, 12, 4, 6, 14, 10, 21, 17, 20, 18, 19]",32,markdown_table,0
LOP,LOP,"Recently the changelog needed rewriting, and the approach was simple — take every commit summary once and order them. Between any two entries there’s a small score that tells how much better it reads if one is placed before the other, so the task is to choose a single sequence of all commits that gives the best total clarity when those pair-by-pair scores are added together. The full set of commits and their pairwise scores are listed below.

There are 19 commit summaries: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S.

*Meaning: this is a directed clarity_score_if_precedes matrix over items in commit_ids (19 total). Entry at row preceding_commit_id=i and column following_commit_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).*

**clarity_score_if_precedes)**

| preceding_commit_id\following_commit_id | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 447.0 | 558.0 | 23.0 | 1577.0 | 33.0 | 13.0 | 94.0 | 0.0 | 15.0 | 0.0 | 9.0 | 84.0 | 8.0 | 212.0 | 92.0 | 0.0 | 3093.0 | 258.0 | 1291.0 |
| B | 7.0 | 5591.0 | 40.0 | 145.0 | 21.0 | 14.0 | 15.0 | 0.0 | 0.0 | 40.0 | 18.0 | 23.0 | 8.0 | 1.0 | 1.0 | 6.0 | 21.0 | 147.0 | 134.0 |
| C | 2.0 | 110.0 | 193.0 | 59.0 | 1.0 | 7.0 | 0.0 | 0.0 | 2.0 | 2.0 | 8.0 | 4.0 | 3.0 | 0.0 | 0.0 | 2.0 | 7.0 | 74.0 | 27.0 |
| D | 276.0 | 1222.0 | 45.0 | 13063.0 | 92.0 | 461.0 | 20.0 | 0.0 | 0.0 | 27.0 | 698.0 | 218.0 | 56.0 | 308.0 | 25.0 | 17.0 | 579.0 | 670.0 | 188.0 |
| E | 806.0 | 73.0 | 6.0 | 208.0 | 288.0 | 64.0 | 3.0 | 0.0 | 0.0 | 0.0 | 9.0 | 9.0 | 0.0 | 0.0 | 0.0 | 1.0 | 5.0 | 132.0 | 14.0 |
| F | 115.0 | 113.0 | 21.0 | 432.0 | 21.0 | 356.0 | 2.0 | 0.0 | 0.0 | 5.0 | 51.0 | 62.0 | 0.0 | 0.0 | 0.0 | 2.0 | 7.0 | 139.0 | 25.0 |
| G | 22.0 | 350.0 | 491.0 | 1205.0 | 88.0 | 37.0 | 4736.0 | 0.0 | 0.0 | 311.0 | 48.0 | 484.0 | 4.0 | 2.0 | 2.0 | 17.0 | 68.0 | 305.0 | 413.0 |
| H | 2.0 | 10.0 | 0.0 | 13.0 | 3.0 | 0.0 | 0.0 | 71.0 | 0.0 | 6.0 | 0.0 | 3.0 | 0.0 | 0.0 | 0.0 | 1.0 | 5.0 | 6.0 | 7.0 |
| I | 2.0 | 31.0 | 49.0 | 76.0 | 14.0 | 0.0 | 2.0 | 0.0 | 85.0 | 23.0 | 2.0 | 17.0 | 1.0 | 0.0 | 0.0 | 2.0 | 8.0 | 20.0 | 14.0 |
| J | 5.0 | 50.0 | 2.0 | 91.0 | 16.0 | 5.0 | 2.0 | 0.0 | 0.0 | 237.0 | 33.0 | 8.0 | 6.0 | 1.0 | 1.0 | 4.0 | 16.0 | 37.0 | 27.0 |
| K | 8.0 | 81.0 | 18.0 | 467.0 | 11.0 | 1262.0 | 4.0 | 0.0 | 0.0 | 3.0 | 258.0 | 68.0 | 0.0 | 1.0 | 1.0 | 3.0 | 105.0 | 140.0 | 27.0 |
| L | 41.0 | 355.0 | 2.0 | 2671.0 | 2.0 | 38.0 | 11.0 | 0.0 | 0.0 | 3.0 | 184.0 | 3757.0 | 24.0 | 18.0 | 1.0 | 12.0 | 47.0 | 163.0 | 161.0 |
| M | 11.0 | 66.0 | 29.0 | 15.0 | 0.0 | 9.0 | 11.0 | 0.0 | 0.0 | 0.0 | 95.0 | 4064.0 | 69.0 | 1.0 | 1.0 | 9.0 | 35.0 | 48.0 | 81.0 |
| N | 69.0 | 22.0 | 0.0 | 43.0 | 4.0 | 71.0 | 2.0 | 0.0 | 0.0 | 0.0 | 39.0 | 3.0 | 0.0 | 550.0 | 0.0 | 0.0 | 340.0 | 65.0 | 14.0 |
| O | 953.0 | 7.0 | 0.0 | 11.0 | 0.0 | 10.0 | 3.0 | 0.0 | 0.0 | 0.0 | 19.0 | 2.0 | 0.0 | 1.0 | 110.0 | 0.0 | 65.0 | 38.0 | 8.0 |
| P | 41.0 | 18.0 | 0.0 | 67.0 | 1.0 | 89.0 | 8.0 | 0.0 | 0.0 | 0.0 | 85.0 | 2.0 | 2.0 | 0.0 | 0.0 | 167.0 | 16.0 | 22.0 | 28.0 |
| Q | 10144.0 | 369.0 | 26.0 | 517.0 | 10.0 | 154.0 | 28.0 | 0.0 | 0.0 | 0.0 | 602.0 | 35.0 | 5.0 | 379.0 | 411.0 | 15.0 | 6479.0 | 674.0 | 518.0 |
| R | 6.0 | 318.0 | 13.0 | 30.0 | 49.0 | 27.0 | 591.0 | 83.0 | 0.0 | 16.0 | 68.0 | 35.0 | 90.0 | 1.0 | 1.0 | 9.0 | 28.0 | 4676.0 | 451.0 |
| S | 1112.0 | 112.0 | 11.0 | 213.0 | 0.0 | 69.0 | 10.0 | 0.0 | 0.0 | 18.0 | 85.0 | 14.0 | 13.0 | 39.0 | 2.0 | 14.0 | 394.0 | 70.0 | 3671.0 |

Choose the single sequence that maximizes the total clarity from these pairwise scores.

And when you reply, just pop the ordering into a tiny JSON snippet like this — keeps things simple and easy to parse:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

The ""solution"" array is just the commits in the order you want them to appear, from first to last. Each placeholder like <first_item> stands in for the actual commit identifier from the input — think of it as a little form where you list every commit exactly once.

This JSON is only a sketch of the shape I expect, not the final answer itself. Please make sure to use the identifiers exactly as they appear in the instance input — do not rename them or add 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”.""","[[447.0, 558.0, 23.0, 1577.0, 33.0, 13.0, 94.0, 0.0, 15.0, 0.0, 9.0, 84.0, 8.0, 212.0, 92.0, 0.0, 3093.0, 258.0, 1291.0], [7.0, 5591.0, 40.0, 145.0, 21.0, 14.0, 15.0, 0.0, 0.0, 40.0, 18.0, 23.0, 8.0, 1.0, 1.0, 6.0, 21.0, 147.0, 134.0], [2.0, 110.0, 193.0, 59.0, 1.0, 7.0, 0.0, 0.0, 2.0, 2.0, 8.0, 4.0, 3.0, 0.0, 0.0, 2.0, 7.0, 74.0, 27.0], [276.0, 1222.0, 45.0, 13063.0, 92.0, 461.0, 20.0, 0.0, 0.0, 27.0, 698.0, 218.0, 56.0, 308.0, 25.0, 17.0, 579.0, 670.0, 188.0], [806.0, 73.0, 6.0, 208.0, 288.0, 64.0, 3.0, 0.0, 0.0, 0.0, 9.0, 9.0, 0.0, 0.0, 0.0, 1.0, 5.0, 132.0, 14.0], [115.0, 113.0, 21.0, 432.0, 21.0, 356.0, 2.0, 0.0, 0.0, 5.0, 51.0, 62.0, 0.0, 0.0, 0.0, 2.0, 7.0, 139.0, 25.0], [22.0, 350.0, 491.0, 1205.0, 88.0, 37.0, 4736.0, 0.0, 0.0, 311.0, 48.0, 484.0, 4.0, 2.0, 2.0, 17.0, 68.0, 305.0, 413.0], [2.0, 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'weight': 0.0}, {'from_id': 'O', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'K', 'weight': 19.0}, {'from_id': 'O', 'to_id': 'L', 'weight': 2.0}, {'from_id': 'O', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'N', 'weight': 1.0}, {'from_id': 'O', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'Q', 'weight': 65.0}, {'from_id': 'O', 'to_id': 'R', 'weight': 38.0}, {'from_id': 'O', 'to_id': 'S', 'weight': 8.0}, {'from_id': 'P', 'to_id': 'A', 'weight': 41.0}, {'from_id': 'P', 'to_id': 'B', 'weight': 18.0}, {'from_id': 'P', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'D', 'weight': 67.0}, {'from_id': 'P', 'to_id': 'E', 'weight': 1.0}, {'from_id': 'P', 'to_id': 'F', 'weight': 89.0}, {'from_id': 'P', 'to_id': 'G', 'weight': 8.0}, {'from_id': 'P', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'K', 'weight': 85.0}, {'from_id': 'P', 'to_id': 'L', 'weight': 2.0}, {'from_id': 'P', 'to_id': 'M', 'weight': 2.0}, {'from_id': 'P', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'Q', 'weight': 16.0}, {'from_id': 'P', 'to_id': 'R', 'weight': 22.0}, {'from_id': 'P', 'to_id': 'S', 'weight': 28.0}, {'from_id': 'Q', 'to_id': 'A', 'weight': 10144.0}, {'from_id': 'Q', 'to_id': 'B', 'weight': 369.0}, {'from_id': 'Q', 'to_id': 'C', 'weight': 26.0}, {'from_id': 'Q', 'to_id': 'D', 'weight': 517.0}, {'from_id': 'Q', 'to_id': 'E', 'weight': 10.0}, {'from_id': 'Q', 'to_id': 'F', 'weight': 154.0}, {'from_id': 'Q', 'to_id': 'G', 'weight': 28.0}, {'from_id': 'Q', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'K', 'weight': 602.0}, {'from_id': 'Q', 'to_id': 'L', 'weight': 35.0}, {'from_id': 'Q', 'to_id': 'M', 'weight': 5.0}, {'from_id': 'Q', 'to_id': 'N', 'weight': 379.0}, {'from_id': 'Q', 'to_id': 'O', 'weight': 411.0}, {'from_id': 'Q', 'to_id': 'P', 'weight': 15.0}, {'from_id': 'Q', 'to_id': 'R', 'weight': 674.0}, {'from_id': 'Q', 'to_id': 'S', 'weight': 518.0}, {'from_id': 'R', 'to_id': 'A', 'weight': 6.0}, {'from_id': 'R', 'to_id': 'B', 'weight': 318.0}, {'from_id': 'R', 'to_id': 'C', 'weight': 13.0}, {'from_id': 'R', 'to_id': 'D', 'weight': 30.0}, {'from_id': 'R', 'to_id': 'E', 'weight': 49.0}, {'from_id': 'R', 'to_id': 'F', 'weight': 27.0}, {'from_id': 'R', 'to_id': 'G', 'weight': 591.0}, {'from_id': 'R', 'to_id': 'H', 'weight': 83.0}, {'from_id': 'R', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'J', 'weight': 16.0}, {'from_id': 'R', 'to_id': 'K', 'weight': 68.0}, {'from_id': 'R', 'to_id': 'L', 'weight': 35.0}, {'from_id': 'R', 'to_id': 'M', 'weight': 90.0}, {'from_id': 'R', 'to_id': 'N', 'weight': 1.0}, {'from_id': 'R', 'to_id': 'O', 'weight': 1.0}, {'from_id': 'R', 'to_id': 'P', 'weight': 9.0}, {'from_id': 'R', 'to_id': 'Q', 'weight': 28.0}, {'from_id': 'R', 'to_id': 'S', 'weight': 451.0}, {'from_id': 'S', 'to_id': 'A', 'weight': 1112.0}, {'from_id': 'S', 'to_id': 'B', 'weight': 112.0}, {'from_id': 'S', 'to_id': 'C', 'weight': 11.0}, {'from_id': 'S', 'to_id': 'D', 'weight': 213.0}, {'from_id': 'S', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'F', 'weight': 69.0}, {'from_id': 'S', 'to_id': 'G', 'weight': 10.0}, {'from_id': 'S', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'J', 'weight': 18.0}, {'from_id': 'S', 'to_id': 'K', 'weight': 85.0}, {'from_id': 'S', 'to_id': 'L', 'weight': 14.0}, {'from_id': 'S', 'to_id': 'M', 'weight': 13.0}, {'from_id': 'S', 'to_id': 'N', 'weight': 39.0}, {'from_id': 'S', 'to_id': 'O', 'weight': 2.0}, {'from_id': 'S', 'to_id': 'P', 'weight': 14.0}, {'from_id': 'S', 'to_id': 'Q', 'weight': 394.0}, {'from_id': 'S', 'to_id': 'R', 'weight': 70.0}]}","['I', 'M', 'G', 'J', 'P', 'Q', 'O', 'E', 'A', 'L', 'D', 'N', 'K', 'F', 'C', 'R', 'H', 'B', 'S']",33,markdown_table,names
LOP,LOP,"Recently the crew tried to plan a day’s run using a handy list that assigns a benefit for putting one stop ahead of another. The idea was to build one continuous route that hits every stop exactly once — no repeats, no gaps — and to choose the order that yields the highest total benefit. To score any candidate route, add the benefit number for every pair of stops where the earlier one is placed before the later one; the bigger that sum, the better the route. The full details and the benefit table are given below.

They had 24 scheduled stops: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X. The table below lists the benefit gained when the earlier stop is placed before the later one.

*Meaning: this is a directed pair_benefit matrix over items in stop_ids (24 total). Entry at row from_stop_id=i and column to_stop_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).*

**pair_benefit)**

| from_stop_id\to_stop_id | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 928.0 | 0.0 | 508.0 | 32.0 | 0.0 | 5.0 | 11.0 | 7.0 | 4.0 | 131.0 | 19.0 | 4.0 | 33.0 | 3.0 | 17.0 | 298.0 | 126.0 | 0.0 | 38.0 | 15.0 | 168.0 | 116.0 | 235.0 | 0.0 |
| B | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| C | 114.0 | 0.0 | 6175.0 | 280.0 | 23.0 | 24.0 | 32.0 | 26.0 | 46.0 | 238.0 | 297.0 | 27.0 | 134.0 | 8.0 | 17916.0 | 360.0 | 21.0 | 1.0 | 1.0 | 39.0 | 43.0 | 109.0 | 42.0 | 0.0 |
| D | 960.0 | 0.0 | 449.0 | 3163.0 | 251.0 | 380.0 | 944.0 | 175.0 | 353.0 | 1041.0 | 351.0 | 125.0 | 883.0 | 32.0 | 6620.0 | 100.0 | 217.0 | 77.0 | 16.0 | 0.0 | 119.0 | 0.0 | 124.0 | 0.0 |
| E | 0.0 | 0.0 | 0.0 | 69.0 | 58.0 | 3.0 | 3.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 2.0 | 0.0 | 0.0 | 26.0 | 3.0 | 60.0 | 0.0 | 36.0 | 0.0 | 109.0 | 0.0 |
| F | 8.0 | 0.0 | 15.0 | 42.0 | 0.0 | 21904.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 30.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 25.0 | 0.0 |
| G | 0.0 | 0.0 | 13.0 | 7.0 | 1.0 | 60.0 | 776.0 | 1.0 | 0.0 | 2.0 | 0.0 | 0.0 | 0.0 | 2.0 | 0.0 | 0.0 | 381.0 | 612.0 | 0.0 | 0.0 | 0.0 | 0.0 | 90.0 | 0.0 |
| H | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 870.0 | 6.0 | 558.0 | 0.0 | 612.0 | 0.0 | 48.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 3.0 | 0.0 | 8.0 | 0.0 |
| I | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 33.0 | 72.0 | 1280.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2.0 | 0.0 | 3.0 | 0.0 |
| J | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 159.0 | 65.0 | 14530.0 | 754.0 | 19.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 23.0 | 150.0 | 9.0 | 0.0 |
| K | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 5.0 | 0.0 | 194.0 | 429.0 | 0.0 | 0.0 | 0.0 | 24.0 | 8.0 | 0.0 | 1.0 | 0.0 | 0.0 | 3.0 | 12.0 | 3.0 | 0.0 |
| L | 59.0 | 0.0 | 4.0 | 10.0 | 4.0 | 0.0 | 0.0 | 0.0 | 0.0 | 6.0 | 1.0 | 1477.0 | 32.0 | 10.0 | 0.0 | 0.0 | 2.0 | 0.0 | 3.0 | 0.0 | 4.0 | 23.0 | 3.0 | 0.0 |
| M | 118.0 | 0.0 | 405.0 | 315.0 | 8.0 | 68.0 | 71.0 | 8.0 | 3.0 | 113.0 | 20.0 | 30.0 | 4000.0 | 35.0 | 7884.0 | 46.0 | 156.0 | 0.0 | 10.0 | 35.0 | 136.0 | 373.0 | 67.0 | 0.0 |
| N | 0.0 | 0.0 | 0.0 | 29.0 | 2.0 | 12.0 | 1.0 | 0.0 | 0.0 | 12.0 | 10.0 | 1.0 | 0.0 | 5192.0 | 111.0 | 62.0 | 3.0 | 0.0 | 1.0 | 23.0 | 38.0 | 12.0 | 38.0 | 0.0 |
| O | 168.0 | 0.0 | 404.0 | 83.0 | 5.0 | 35.0 | 14.0 | 26.0 | 22.0 | 113.0 | 104.0 | 43.0 | 87.0 | 26.0 | 0.0 | 1011.0 | 25.0 | 238.0 | 51.0 | 290.0 | 791.0 | 270.0 | 825.0 | 0.0 |
| P | 402.0 | 0.0 | 1064.0 | 1259.0 | 35.0 | 583.0 | 249.0 | 1224.0 | 521.0 | 1705.0 | 470.0 | 331.0 | 1024.0 | 161.0 | 7347.0 | 0.0 | 551.0 | 520.0 | 38.0 | 95.0 | 276.0 | 643.0 | 256.0 | 0.0 |
| Q | 130.0 | 0.0 | 535.0 | 434.0 | 29.0 | 176.0 | 134.0 | 172.0 | 377.0 | 1024.0 | 267.0 | 56.0 | 591.0 | 29.0 | 3134.0 | 106.0 | 124.0 | 114.0 | 349.0 | 28.0 | 47.0 | 55.0 | 381.0 | 0.0 |
| R | 0.0 | 0.0 | 34.0 | 23.0 | 0.0 | 47.0 | 2.0 | 11.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 10.0 | 0.0 | 395.0 | 0.0 | 10.0 | 0.0 | 0.0 | 0.0 | 0.0 | 3.0 | 0.0 |
| S | 72.0 | 0.0 | 171.0 | 149.0 | 16.0 | 22.0 | 32.0 | 25.0 | 14.0 | 193.0 | 117.0 | 25.0 | 219.0 | 10.0 | 183.0 | 2041.0 | 710.0 | 187.0 | 91.0 | 235.0 | 195.0 | 303.0 | 78.0 | 0.0 |
| T | 19.0 | 0.0 | 34.0 | 53.0 | 28.0 | 181.0 | 41.0 | 30.0 | 20.0 | 91.0 | 20.0 | 5.0 | 17.0 | 5.0 | 201.0 | 50.0 | 9.0 | 31.0 | 10.0 | 5.0 | 6.0 | 36.0 | 10.0 | 0.0 |
| U | 57.0 | 0.0 | 139.0 | 261.0 | 24.0 | 163.0 | 46.0 | 30.0 | 20.0 | 180.0 | 97.0 | 8.0 | 54.0 | 15.0 | 300.0 | 120.0 | 21.0 | 268.0 | 31.0 | 14.0 | 17.0 | 110.0 | 0.0 | 0.0 |
| V | 304.0 | 0.0 | 448.0 | 760.0 | 121.0 | 279.0 | 142.0 | 192.0 | 114.0 | 1031.0 | 863.0 | 52.0 | 290.0 | 73.0 | 1525.0 | 705.0 | 120.0 | 503.0 | 167.0 | 71.0 | 296.0 | 584.0 | 570.0 | 0.0 |
| W | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| X | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

The crew sought the ordering that maximized the sum of these pairwise benefits.

When you’re ready to send the final route back, please stick to a tiny JSON sketch so it’s easy to read and parse. Something like this:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Here’s what that little block means in plain, non-technical terms: ""solution"" is the ordered list of stops — the first entry is where you start the run, the next is where you go after that, and so on until the last stop. Think of the placeholders as blanks to fill with the actual stop identifiers from the instance. This is just the shape I’m expecting, not the actual route itself.

Please make sure you use the exact identifiers from the instance input — don’t 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”.""","[[928.0, 0.0, 508.0, 32.0, 0.0, 5.0, 11.0, 7.0, 4.0, 131.0, 19.0, 4.0, 33.0, 3.0, 17.0, 298.0, 126.0, 0.0, 38.0, 15.0, 168.0, 116.0, 235.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [114.0, 0.0, 6175.0, 280.0, 23.0, 24.0, 32.0, 26.0, 46.0, 238.0, 297.0, 27.0, 134.0, 8.0, 17916.0, 360.0, 21.0, 1.0, 1.0, 39.0, 43.0, 109.0, 42.0, 0.0], [960.0, 0.0, 449.0, 3163.0, 251.0, 380.0, 944.0, 175.0, 353.0, 1041.0, 351.0, 125.0, 883.0, 32.0, 6620.0, 100.0, 217.0, 77.0, 16.0, 0.0, 119.0, 0.0, 124.0, 0.0], [0.0, 0.0, 0.0, 69.0, 58.0, 3.0, 3.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 26.0, 3.0, 60.0, 0.0, 36.0, 0.0, 109.0, 0.0], [8.0, 0.0, 15.0, 42.0, 0.0, 21904.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 30.0, 0.0, 0.0, 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0.0}, {'from_id': 'X', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'X', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'X', 'to_id': 'S', 'weight': 0.0}, {'from_id': 'X', 'to_id': 'T', 'weight': 0.0}, {'from_id': 'X', 'to_id': 'U', 'weight': 0.0}, {'from_id': 'X', 'to_id': 'V', 'weight': 0.0}, {'from_id': 'X', 'to_id': 'W', 'weight': 0.0}]}","['X', 'S', 'V', 'P', 'Q', 'D', 'M', 'A', 'C', 'N', 'O', 'T', 'E', 'U', 'G', 'R', 'I', 'H', 'J', 'L', 'K', 'F', 'W', 'B']",34,markdown_table,names
LOP,LOP,"I’m the late‑night DJ trying to stitch together a tight set from a pile of tracks. The trick is to put every song into one continuous run — no repeats, nothing left in the crate — and order them so the flow between each pair feels great. Each possible transition has a little engagement score, and the whole show’s success comes from adding up those scores for every consecutive pair; the higher that total, the better the ride. The exact tracks and their transition scores are shown below.

I’m lining up 24 tracks to stitch into one continuous late‑night set: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24.
Meaning: this is a directed transition_engagement matrix over items in track_ids (24 total). Entry at row from_track_id=i and column to_track_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
transition_engagement:
| from_track_id\to_track_id | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 8276.0 | 889.0 | 1047.0 | 2313.0 | 281.0 | 41.0 | 231.0 | 535.0 | 27.0 | 155.0 | 78.0 | 415.0 | 259.0 | 29.0 | 266.0 | 286.0 | 115.0 | 205.0 | 2787.0 | 124.0 | 3.0 | 182.0 | 1048.0 | 189.0 |
| 2 | 58.0 | 49064.0 | 438.0 | 1744.0 | 15430.0 | 815.0 | 5575.0 | 4305.0 | 1206.0 | 3.0 | 0.0 | 0.0 | 0.0 | 12.0 | 46.0 | 59.0 | 216.0 | 0.0 | 308.0 | 0.0 | 0.0 | 0.0 | 67.0 | 110.0 |
| 3 | 4.0 | 1531.0 | 5621.0 | 985.0 | 316.0 | 103.0 | 665.0 | 507.0 | 11.0 | 119.0 | 40.0 | 295.0 | 285.0 | 0.0 | 0.0 | 89.0 | 141.0 | 151.0 | 86.0 | 0.0 | 0.0 | 130.0 | 320.0 | 80.0 |
| 4 | 580.0 | 1036.0 | 580.0 | 15304.0 | 824.0 | 174.0 | 1330.0 | 1476.0 | 128.0 | 227.0 | 43.0 | 301.0 | 154.0 | 131.0 | 5190.0 | 1561.0 | 4587.0 | 414.0 | 149.0 | 131.0 | 5.0 | 638.0 | 227.0 | 326.0 |
| 5 | 195.0 | 869.0 | 285.0 | 1379.0 | 6161.0 | 522.0 | 2756.0 | 4330.0 | 471.0 | 408.0 | 122.0 | 661.0 | 233.0 | 120.0 | 345.0 | 428.0 | 450.0 | 236.0 | 297.0 | 84.0 | 4.0 | 25.0 | 1062.0 | 112.0 |
| 6 | 23.0 | 50.0 | 24.0 | 122.0 | 89.0 | 1284.0 | 195.0 | 308.0 | 74.0 | 8.0 | 2.0 | 15.0 | 13.0 | 5.0 | 20.0 | 62.0 | 14.0 | 26.0 | 22.0 | 31.0 | 2.0 | 102.0 | 288.0 | 88.0 |
| 7 | 12.0 | 174.0 | 28.0 | 89.0 | 315.0 | 591.0 | 7714.0 | 1909.0 | 213.0 | 6.0 | 0.0 | 9.0 | 4.0 | 3.0 | 8.0 | 25.0 | 15.0 | 185.0 | 226.0 | 294.0 | 6.0 | 32.0 | 274.0 | 32.0 |
| 8 | 6.0 | 29.0 | 7.0 | 19.0 | 47.0 | 1.0 | 11.0 | 8645.0 | 52.0 | 4.0 | 1.0 | 5.0 | 4.0 | 1.0 | 5.0 | 8.0 | 64.0 | 4.0 | 343.0 | 11.0 | 0.0 | 11.0 | 1294.0 | 3.0 |
| 9 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 134.0 | 700.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 115.0 | 0.0 | 0.0 | 0.0 | 1604.0 | 0.0 |
| 10 | 0.0 | 0.0 | 0.0 | 44.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 595.0 | 0.0 | 506.0 | 0.0 | 0.0 | 47.0 | 0.0 | 0.0 | 1503.0 | 4.0 | 0.0 | 0.0 | 43.0 | 146.0 | 114.0 |
| 11 | 0.0 | 0.0 | 0.0 | 63.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 115.0 | 1044.0 | 7.0 | 0.0 | 0.0 | 5.0 | 0.0 | 814.0 | 3.0 | 0.0 | 0.0 | 60.0 | 90.0 | 10.0 |
| 12 | 11.0 | 8.0 | 3.0 | 628.0 | 9.0 | 1.0 | 13.0 | 9.0 | 0.0 | 333.0 | 22.0 | 10733.0 | 934.0 | 18.0 | 5.0 | 108.0 | 35.0 | 3673.0 | 18.0 | 6.0 | 0.0 | 99.0 | 285.0 | 62.0 |
| 13 | 8.0 | 20.0 | 9.0 | 294.0 | 7.0 | 0.0 | 19.0 | 8.0 | 0.0 | 6.0 | 3.0 | 183.0 | 1958.0 | 4.0 | 3.0 | 22.0 | 2.0 | 5632.0 | 14.0 | 10.0 | 1.0 | 2.0 | 6.0 | 20.0 |
| 14 | 7.0 | 24.0 | 10.0 | 31.0 | 13.0 | 3.0 | 17.0 | 9.0 | 0.0 | 7.0 | 2.0 | 11.0 | 9.0 | 17.0 | 10.0 | 20.0 | 7.0 | 2348.0 | 9.0 | 15.0 | 0.0 | 0.0 | 5.0 | 0.0 |
| 15 | 1.0 | 11.0 | 53.0 | 164.0 | 70.0 | 14.0 | 133.0 | 263.0 | 15.0 | 14.0 | 3.0 | 31.0 | 31.0 | 1.0 | 17765.0 | 235.0 | 969.0 | 126.0 | 50.0 | 52.0 | 3.0 | 74.0 | 875.0 | 197.0 |
| 16 | 192.0 | 403.0 | 935.0 | 2208.0 | 562.0 | 195.0 | 1040.0 | 388.0 | 23.0 | 305.0 | 206.0 | 1321.0 | 441.0 | 288.0 | 650.0 | 8905.0 | 316.0 | 341.0 | 252.0 | 593.0 | 12.0 | 150.0 | 1401.0 | 400.0 |
| 17 | 142.0 | 46.0 | 220.0 | 1113.0 | 506.0 | 398.0 | 1477.0 | 2853.0 | 97.0 | 166.0 | 67.0 | 414.0 | 101.0 | 33.0 | 329.0 | 188.0 | 1801.0 | 64.0 | 189.0 | 14.0 | 3.0 | 224.0 | 356.0 | 287.0 |
| 18 | 178.0 | 327.0 | 172.0 | 91.0 | 246.0 | 51.0 | 534.0 | 314.0 | 17.0 | 72.0 | 11.0 | 194.0 | 162.0 | 65.0 | 182.0 | 388.0 | 119.0 | 0.0 | 299.0 | 139.0 | 5.0 | 3.0 | 143.0 | 0.0 |
| 19 | 526.0 | 2075.0 | 798.0 | 1008.0 | 606.0 | 52.0 | 350.0 | 354.0 | 70.0 | 257.0 | 170.0 | 581.0 | 203.0 | 38.0 | 248.0 | 432.0 | 155.0 | 253.0 | 398.0 | 87.0 | 10.0 | 72.0 | 358.0 | 93.0 |
| 20 | 59.0 | 115.0 | 110.0 | 174.0 | 117.0 | 23.0 | 188.0 | 163.0 | 12.0 | 41.0 | 16.0 | 96.0 | 59.0 | 32.0 | 95.0 | 126.0 | 50.0 | 91.0 | 421.0 | 17065.0 | 4.0 | 52.0 | 249.0 | 59.0 |
| 21 | 2.0 | 6.0 | 7.0 | 3.0 | 6.0 | 2.0 | 11.0 | 5.0 | 0.0 | 3.0 | 0.0 | 5.0 | 2.0 | 1.0 | 3.0 | 9.0 | 3.0 | 18.0 | 27.0 | 20.0 | 0.0 | 8.0 | 82.0 | 22.0 |
| 22 | 4.0 | 1.0 | 8.0 | 15.0 | 10.0 | 5.0 | 13.0 | 4.0 | 0.0 | 8.0 | 0.0 | 8.0 | 6.0 | 1.0 | 7.0 | 9.0 | 4.0 | 4.0 | 19.0 | 12.0 | 2.0 | 0.0 | 62.0 | 18.0 |
| 23 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 24 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
Time to mix them into one flawless run.

If you want to hand me the run in a simple, machine-friendly way, just drop it into a little JSON sketch like this:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of ""solution"" as the playlist order — the first element is the track that opens the set, the next is what follows it, and so on until the last track closes the run. This JSON is just a shape hint for how I expect the answer to be laid out, not the actual ordering itself.

Please use the identifiers exactly as they appear in the instance input — no renaming and no 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”.""","[[8276.0, 889.0, 1047.0, 2313.0, 281.0, 41.0, 231.0, 535.0, 27.0, 155.0, 78.0, 415.0, 259.0, 29.0, 266.0, 286.0, 115.0, 205.0, 2787.0, 124.0, 3.0, 182.0, 1048.0, 189.0], [58.0, 49064.0, 438.0, 1744.0, 15430.0, 815.0, 5575.0, 4305.0, 1206.0, 3.0, 0.0, 0.0, 0.0, 12.0, 46.0, 59.0, 216.0, 0.0, 308.0, 0.0, 0.0, 0.0, 67.0, 110.0], [4.0, 1531.0, 5621.0, 985.0, 316.0, 103.0, 665.0, 507.0, 11.0, 119.0, 40.0, 295.0, 285.0, 0.0, 0.0, 89.0, 141.0, 151.0, 86.0, 0.0, 0.0, 130.0, 320.0, 80.0], [580.0, 1036.0, 580.0, 15304.0, 824.0, 174.0, 1330.0, 1476.0, 128.0, 227.0, 43.0, 301.0, 154.0, 131.0, 5190.0, 1561.0, 4587.0, 414.0, 149.0, 131.0, 5.0, 638.0, 227.0, 326.0], [195.0, 869.0, 285.0, 1379.0, 6161.0, 522.0, 2756.0, 4330.0, 471.0, 408.0, 122.0, 661.0, 233.0, 120.0, 345.0, 428.0, 450.0, 236.0, 297.0, 84.0, 4.0, 25.0, 1062.0, 112.0], [23.0, 50.0, 24.0, 122.0, 89.0, 1284.0, 195.0, 308.0, 74.0, 8.0, 2.0, 15.0, 13.0, 5.0, 20.0, 62.0, 14.0, 26.0, 22.0, 31.0, 2.0, 102.0, 288.0, 88.0], [12.0, 174.0, 28.0, 89.0, 315.0, 591.0, 7714.0, 1909.0, 213.0, 6.0, 0.0, 9.0, 4.0, 3.0, 8.0, 25.0, 15.0, 185.0, 226.0, 294.0, 6.0, 32.0, 274.0, 32.0], [6.0, 29.0, 7.0, 19.0, 47.0, 1.0, 11.0, 8645.0, 52.0, 4.0, 1.0, 5.0, 4.0, 1.0, 5.0, 8.0, 64.0, 4.0, 343.0, 11.0, 0.0, 11.0, 1294.0, 3.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 134.0, 700.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 115.0, 0.0, 0.0, 0.0, 1604.0, 0.0], [0.0, 0.0, 0.0, 44.0, 0.0, 0.0, 0.0, 0.0, 0.0, 595.0, 0.0, 506.0, 0.0, 0.0, 47.0, 0.0, 0.0, 1503.0, 4.0, 0.0, 0.0, 43.0, 146.0, 114.0], [0.0, 0.0, 0.0, 63.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 115.0, 1044.0, 7.0, 0.0, 0.0, 5.0, 0.0, 814.0, 3.0, 0.0, 0.0, 60.0, 90.0, 10.0], [11.0, 8.0, 3.0, 628.0, 9.0, 1.0, 13.0, 9.0, 0.0, 333.0, 22.0, 10733.0, 934.0, 18.0, 5.0, 108.0, 35.0, 3673.0, 18.0, 6.0, 0.0, 99.0, 285.0, 62.0], [8.0, 20.0, 9.0, 294.0, 7.0, 0.0, 19.0, 8.0, 0.0, 6.0, 3.0, 183.0, 1958.0, 4.0, 3.0, 22.0, 2.0, 5632.0, 14.0, 10.0, 1.0, 2.0, 6.0, 20.0], [7.0, 24.0, 10.0, 31.0, 13.0, 3.0, 17.0, 9.0, 0.0, 7.0, 2.0, 11.0, 9.0, 17.0, 10.0, 20.0, 7.0, 2348.0, 9.0, 15.0, 0.0, 0.0, 5.0, 0.0], [1.0, 11.0, 53.0, 164.0, 70.0, 14.0, 133.0, 263.0, 15.0, 14.0, 3.0, 31.0, 31.0, 1.0, 17765.0, 235.0, 969.0, 126.0, 50.0, 52.0, 3.0, 74.0, 875.0, 197.0], [192.0, 403.0, 935.0, 2208.0, 562.0, 195.0, 1040.0, 388.0, 23.0, 305.0, 206.0, 1321.0, 441.0, 288.0, 650.0, 8905.0, 316.0, 341.0, 252.0, 593.0, 12.0, 150.0, 1401.0, 400.0], [142.0, 46.0, 220.0, 1113.0, 506.0, 398.0, 1477.0, 2853.0, 97.0, 166.0, 67.0, 414.0, 101.0, 33.0, 329.0, 188.0, 1801.0, 64.0, 189.0, 14.0, 3.0, 224.0, 356.0, 287.0], [178.0, 327.0, 172.0, 91.0, 246.0, 51.0, 534.0, 314.0, 17.0, 72.0, 11.0, 194.0, 162.0, 65.0, 182.0, 388.0, 119.0, 0.0, 299.0, 139.0, 5.0, 3.0, 143.0, 0.0], [526.0, 2075.0, 798.0, 1008.0, 606.0, 52.0, 350.0, 354.0, 70.0, 257.0, 170.0, 581.0, 203.0, 38.0, 248.0, 432.0, 155.0, 253.0, 398.0, 87.0, 10.0, 72.0, 358.0, 93.0], [59.0, 115.0, 110.0, 174.0, 117.0, 23.0, 188.0, 163.0, 12.0, 41.0, 16.0, 96.0, 59.0, 32.0, 95.0, 126.0, 50.0, 91.0, 421.0, 17065.0, 4.0, 52.0, 249.0, 59.0], [2.0, 6.0, 7.0, 3.0, 6.0, 2.0, 11.0, 5.0, 0.0, 3.0, 0.0, 5.0, 2.0, 1.0, 3.0, 9.0, 3.0, 18.0, 27.0, 20.0, 0.0, 8.0, 82.0, 22.0], [4.0, 1.0, 8.0, 15.0, 10.0, 5.0, 13.0, 4.0, 0.0, 8.0, 0.0, 8.0, 6.0, 1.0, 7.0, 9.0, 4.0, 4.0, 19.0, 12.0, 2.0, 0.0, 62.0, 18.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]]","[0, 15, 20, 19, 18, 2, 1, 4, 3, 14, 16, 13, 10, 9, 11, 12, 17, 6, 5, 8, 7, 21, 23, 22]",132752.0,"{'problem_type': 'LOP', 'num_nodes': 24, 'nodes': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], 'pairs': [{'from_id': 1, 'to_id': 2, 'weight': 889.0}, {'from_id': 1, 'to_id': 3, 'weight': 1047.0}, {'from_id': 1, 'to_id': 4, 'weight': 2313.0}, {'from_id': 1, 'to_id': 5, 'weight': 281.0}, {'from_id': 1, 'to_id': 6, 'weight': 41.0}, {'from_id': 1, 'to_id': 7, 'weight': 231.0}, {'from_id': 1, 'to_id': 8, 'weight': 535.0}, {'from_id': 1, 'to_id': 9, 'weight': 27.0}, {'from_id': 1, 'to_id': 10, 'weight': 155.0}, {'from_id': 1, 'to_id': 11, 'weight': 78.0}, {'from_id': 1, 'to_id': 12, 'weight': 415.0}, {'from_id': 1, 'to_id': 13, 'weight': 259.0}, {'from_id': 1, 'to_id': 14, 'weight': 29.0}, {'from_id': 1, 'to_id': 15, 'weight': 266.0}, {'from_id': 1, 'to_id': 16, 'weight': 286.0}, {'from_id': 1, 'to_id': 17, 'weight': 115.0}, {'from_id': 1, 'to_id': 18, 'weight': 205.0}, {'from_id': 1, 'to_id': 19, 'weight': 2787.0}, {'from_id': 1, 'to_id': 20, 'weight': 124.0}, {'from_id': 1, 'to_id': 21, 'weight': 3.0}, {'from_id': 1, 'to_id': 22, 'weight': 182.0}, {'from_id': 1, 'to_id': 23, 'weight': 1048.0}, {'from_id': 1, 'to_id': 24, 'weight': 189.0}, {'from_id': 2, 'to_id': 1, 'weight': 58.0}, {'from_id': 2, 'to_id': 3, 'weight': 438.0}, {'from_id': 2, 'to_id': 4, 'weight': 1744.0}, {'from_id': 2, 'to_id': 5, 'weight': 15430.0}, {'from_id': 2, 'to_id': 6, 'weight': 815.0}, {'from_id': 2, 'to_id': 7, 'weight': 5575.0}, {'from_id': 2, 'to_id': 8, 'weight': 4305.0}, {'from_id': 2, 'to_id': 9, 'weight': 1206.0}, {'from_id': 2, 'to_id': 10, 'weight': 3.0}, {'from_id': 2, 'to_id': 11, 'weight': 0.0}, {'from_id': 2, 'to_id': 12, 'weight': 0.0}, {'from_id': 2, 'to_id': 13, 'weight': 0.0}, {'from_id': 2, 'to_id': 14, 'weight': 12.0}, {'from_id': 2, 'to_id': 15, 'weight': 46.0}, {'from_id': 2, 'to_id': 16, 'weight': 59.0}, {'from_id': 2, 'to_id': 17, 'weight': 216.0}, {'from_id': 2, 'to_id': 18, 'weight': 0.0}, {'from_id': 2, 'to_id': 19, 'weight': 308.0}, {'from_id': 2, 'to_id': 20, 'weight': 0.0}, {'from_id': 2, 'to_id': 21, 'weight': 0.0}, {'from_id': 2, 'to_id': 22, 'weight': 0.0}, {'from_id': 2, 'to_id': 23, 'weight': 67.0}, {'from_id': 2, 'to_id': 24, 'weight': 110.0}, {'from_id': 3, 'to_id': 1, 'weight': 4.0}, {'from_id': 3, 'to_id': 2, 'weight': 1531.0}, {'from_id': 3, 'to_id': 4, 'weight': 985.0}, {'from_id': 3, 'to_id': 5, 'weight': 316.0}, {'from_id': 3, 'to_id': 6, 'weight': 103.0}, {'from_id': 3, 'to_id': 7, 'weight': 665.0}, {'from_id': 3, 'to_id': 8, 'weight': 507.0}, {'from_id': 3, 'to_id': 9, 'weight': 11.0}, {'from_id': 3, 'to_id': 10, 'weight': 119.0}, {'from_id': 3, 'to_id': 11, 'weight': 40.0}, {'from_id': 3, 'to_id': 12, 'weight': 295.0}, {'from_id': 3, 'to_id': 13, 'weight': 285.0}, {'from_id': 3, 'to_id': 14, 'weight': 0.0}, {'from_id': 3, 'to_id': 15, 'weight': 0.0}, {'from_id': 3, 'to_id': 16, 'weight': 89.0}, {'from_id': 3, 'to_id': 17, 'weight': 141.0}, {'from_id': 3, 'to_id': 18, 'weight': 151.0}, {'from_id': 3, 'to_id': 19, 'weight': 86.0}, {'from_id': 3, 'to_id': 20, 'weight': 0.0}, {'from_id': 3, 'to_id': 21, 'weight': 0.0}, {'from_id': 3, 'to_id': 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110.0}, {'from_id': 20, 'to_id': 4, 'weight': 174.0}, {'from_id': 20, 'to_id': 5, 'weight': 117.0}, {'from_id': 20, 'to_id': 6, 'weight': 23.0}, {'from_id': 20, 'to_id': 7, 'weight': 188.0}, {'from_id': 20, 'to_id': 8, 'weight': 163.0}, {'from_id': 20, 'to_id': 9, 'weight': 12.0}, {'from_id': 20, 'to_id': 10, 'weight': 41.0}, {'from_id': 20, 'to_id': 11, 'weight': 16.0}, {'from_id': 20, 'to_id': 12, 'weight': 96.0}, {'from_id': 20, 'to_id': 13, 'weight': 59.0}, {'from_id': 20, 'to_id': 14, 'weight': 32.0}, {'from_id': 20, 'to_id': 15, 'weight': 95.0}, {'from_id': 20, 'to_id': 16, 'weight': 126.0}, {'from_id': 20, 'to_id': 17, 'weight': 50.0}, {'from_id': 20, 'to_id': 18, 'weight': 91.0}, {'from_id': 20, 'to_id': 19, 'weight': 421.0}, {'from_id': 20, 'to_id': 21, 'weight': 4.0}, {'from_id': 20, 'to_id': 22, 'weight': 52.0}, {'from_id': 20, 'to_id': 23, 'weight': 249.0}, {'from_id': 20, 'to_id': 24, 'weight': 59.0}, {'from_id': 21, 'to_id': 1, 'weight': 2.0}, {'from_id': 21, 'to_id': 2, 'weight': 6.0}, {'from_id': 21, 'to_id': 3, 'weight': 7.0}, {'from_id': 21, 'to_id': 4, 'weight': 3.0}, {'from_id': 21, 'to_id': 5, 'weight': 6.0}, {'from_id': 21, 'to_id': 6, 'weight': 2.0}, {'from_id': 21, 'to_id': 7, 'weight': 11.0}, {'from_id': 21, 'to_id': 8, 'weight': 5.0}, {'from_id': 21, 'to_id': 9, 'weight': 0.0}, {'from_id': 21, 'to_id': 10, 'weight': 3.0}, {'from_id': 21, 'to_id': 11, 'weight': 0.0}, {'from_id': 21, 'to_id': 12, 'weight': 5.0}, {'from_id': 21, 'to_id': 13, 'weight': 2.0}, {'from_id': 21, 'to_id': 14, 'weight': 1.0}, {'from_id': 21, 'to_id': 15, 'weight': 3.0}, {'from_id': 21, 'to_id': 16, 'weight': 9.0}, {'from_id': 21, 'to_id': 17, 'weight': 3.0}, {'from_id': 21, 'to_id': 18, 'weight': 18.0}, {'from_id': 21, 'to_id': 19, 'weight': 27.0}, {'from_id': 21, 'to_id': 20, 'weight': 20.0}, {'from_id': 21, 'to_id': 22, 'weight': 8.0}, {'from_id': 21, 'to_id': 23, 'weight': 82.0}, {'from_id': 21, 'to_id': 24, 'weight': 22.0}, {'from_id': 22, 'to_id': 1, 'weight': 4.0}, {'from_id': 22, 'to_id': 2, 'weight': 1.0}, {'from_id': 22, 'to_id': 3, 'weight': 8.0}, {'from_id': 22, 'to_id': 4, 'weight': 15.0}, {'from_id': 22, 'to_id': 5, 'weight': 10.0}, {'from_id': 22, 'to_id': 6, 'weight': 5.0}, {'from_id': 22, 'to_id': 7, 'weight': 13.0}, {'from_id': 22, 'to_id': 8, 'weight': 4.0}, {'from_id': 22, 'to_id': 9, 'weight': 0.0}, {'from_id': 22, 'to_id': 10, 'weight': 8.0}, {'from_id': 22, 'to_id': 11, 'weight': 0.0}, {'from_id': 22, 'to_id': 12, 'weight': 8.0}, {'from_id': 22, 'to_id': 13, 'weight': 6.0}, {'from_id': 22, 'to_id': 14, 'weight': 1.0}, {'from_id': 22, 'to_id': 15, 'weight': 7.0}, {'from_id': 22, 'to_id': 16, 'weight': 9.0}, {'from_id': 22, 'to_id': 17, 'weight': 4.0}, {'from_id': 22, 'to_id': 18, 'weight': 4.0}, {'from_id': 22, 'to_id': 19, 'weight': 19.0}, {'from_id': 22, 'to_id': 20, 'weight': 12.0}, {'from_id': 22, 'to_id': 21, 'weight': 2.0}, {'from_id': 22, 'to_id': 23, 'weight': 62.0}, {'from_id': 22, 'to_id': 24, 'weight': 18.0}, {'from_id': 23, 'to_id': 1, 'weight': 0.0}, {'from_id': 23, 'to_id': 2, 'weight': 0.0}, {'from_id': 23, 'to_id': 3, 'weight': 0.0}, {'from_id': 23, 'to_id': 4, 'weight': 0.0}, {'from_id': 23, 'to_id': 5, 'weight': 0.0}, {'from_id': 23, 'to_id': 6, 'weight': 0.0}, {'from_id': 23, 'to_id': 7, 'weight': 0.0}, {'from_id': 23, 'to_id': 8, 'weight': 0.0}, {'from_id': 23, 'to_id': 9, 'weight': 0.0}, {'from_id': 23, 'to_id': 10, 'weight': 0.0}, {'from_id': 23, 'to_id': 11, 'weight': 0.0}, {'from_id': 23, 'to_id': 12, 'weight': 0.0}, {'from_id': 23, 'to_id': 13, 'weight': 0.0}, {'from_id': 23, 'to_id': 14, 'weight': 0.0}, {'from_id': 23, 'to_id': 15, 'weight': 0.0}, {'from_id': 23, 'to_id': 16, 'weight': 0.0}, {'from_id': 23, 'to_id': 17, 'weight': 0.0}, {'from_id': 23, 'to_id': 18, 'weight': 0.0}, {'from_id': 23, 'to_id': 19, 'weight': 0.0}, {'from_id': 23, 'to_id': 20, 'weight': 0.0}, {'from_id': 23, 'to_id': 21, 'weight': 0.0}, {'from_id': 23, 'to_id': 22, 'weight': 0.0}, {'from_id': 23, 'to_id': 24, 'weight': 0.0}, {'from_id': 24, 'to_id': 1, 'weight': 0.0}, {'from_id': 24, 'to_id': 2, 'weight': 0.0}, {'from_id': 24, 'to_id': 3, 'weight': 0.0}, {'from_id': 24, 'to_id': 4, 'weight': 0.0}, {'from_id': 24, 'to_id': 5, 'weight': 0.0}, {'from_id': 24, 'to_id': 6, 'weight': 0.0}, {'from_id': 24, 'to_id': 7, 'weight': 0.0}, {'from_id': 24, 'to_id': 8, 'weight': 0.0}, {'from_id': 24, 'to_id': 9, 'weight': 0.0}, {'from_id': 24, 'to_id': 10, 'weight': 0.0}, {'from_id': 24, 'to_id': 11, 'weight': 0.0}, {'from_id': 24, 'to_id': 12, 'weight': 0.0}, {'from_id': 24, 'to_id': 13, 'weight': 0.0}, {'from_id': 24, 'to_id': 14, 'weight': 0.0}, {'from_id': 24, 'to_id': 15, 'weight': 0.0}, {'from_id': 24, 'to_id': 16, 'weight': 0.0}, {'from_id': 24, 'to_id': 17, 'weight': 0.0}, {'from_id': 24, 'to_id': 18, 'weight': 0.0}, {'from_id': 24, 'to_id': 19, 'weight': 0.0}, {'from_id': 24, 'to_id': 20, 'weight': 0.0}, {'from_id': 24, 'to_id': 21, 'weight': 0.0}, {'from_id': 24, 'to_id': 22, 'weight': 0.0}, {'from_id': 24, 'to_id': 23, 'weight': 0.0}]}","[1, 16, 21, 20, 19, 3, 2, 5, 4, 15, 17, 14, 11, 10, 12, 13, 18, 7, 6, 9, 8, 22, 24, 23]",35,nl,1
LOP,LOP,"Many people on the team are treating this like a puzzle: arrange the whole batch of images into a single chain so every picture appears once and no image is left out or repeated. There’s a tiny benefit assigned to every possible ordered pairing — if photo X comes before photo Y you pick up the listed amount — and the story’s overall strength is the sum of all those picked-up amounts across the entire ordering. So the goal is to order the pictures to collect the biggest total reward. The exact images and their pairwise rewards are shown below.

Here are the 20 photos (A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T) that must each appear once in the final chain.

*Meaning: this is a directed pairwise_story_gain matrix over items in image_ids (20 total). Entry at row preceding_image_id=i and column following_image_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).*

**pairwise_story_gain)**

| preceding_image_id\following_image_id | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 18093.0 | 0.0 | 0.0 | 8.0 | 473.0 | 0.0 | 0.0 | 4.0 | 15030.0 | 8451.0 | 713.0 | 428.0 | 191.0 | 9.0 | 39.0 | 0.0 | 0.0 | 2.0 | 32.0 | 16.0 |
| B | 0.0 | 80.0 | 0.0 | 1605.0 | 192.0 | 64.0 | 0.0 | 10.0 | 3.0 | 7.0 | 2.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 21.0 | 5.0 |
| C | 807.0 | 0.0 | 5624.0 | 293.0 | 748.0 | 442.0 | 36.0 | 71.0 | 42.0 | 224.0 | 74.0 | 0.0 | 1500.0 | 430.0 | 750.0 | 80.0 | 188.0 | 641.0 | 615.0 | 33.0 |
| D | 416.0 | 0.0 | 122.0 | 15913.0 | 711.0 | 4950.0 | 210.0 | 795.0 | 0.0 | 12.0 | 0.0 | 3.0 | 3593.0 | 114.0 | 8.0 | 0.0 | 1.0 | 0.0 | 87.0 | 17.0 |
| E | 2764.0 | 9.0 | 383.0 | 213.0 | 5386.0 | 347.0 | 41.0 | 100.0 | 6.0 | 122.0 | 166.0 | 1.0 | 1104.0 | 214.0 | 26.0 | 5.0 | 15.0 | 210.0 | 249.0 | 23.0 |
| F | 29.0 | 4.0 | 311.0 | 228.0 | 638.0 | 1626.0 | 343.0 | 1144.0 | 36.0 | 366.0 | 90.0 | 0.0 | 4738.0 | 464.0 | 17.0 | 0.0 | 5.0 | 20.0 | 303.0 | 23.0 |
| G | 3.0 | 0.0 | 12.0 | 11.0 | 10.0 | 46.0 | 39.0 | 128.0 | 0.0 | 2.0 | 2.0 | 0.0 | 61.0 | 19.0 | 0.0 | 50.0 | 0.0 | 4.0 | 207.0 | 150.0 |
| H | 81.0 | 0.0 | 0.0 | 13.0 | 0.0 | 2.0 | 0.0 | 9.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 190.0 | 298.0 | 0.0 | 14.0 | 0.0 | 3504.0 | 1.0 |
| I | 4.0 | 0.0 | 0.0 | 0.0 | 67.0 | 0.0 | 0.0 | 0.0 | 451.0 | 82.0 | 0.0 | 0.0 | 0.0 | 11.0 | 31.0 | 1.0 | 0.0 | 0.0 | 82.0 | 0.0 |
| J | 3749.0 | 0.0 | 9.0 | 0.0 | 475.0 | 8.0 | 0.0 | 0.0 | 226.0 | 6161.0 | 225.0 | 0.0 | 0.0 | 3.0 | 4.0 | 0.0 | 0.0 | 3.0 | 52.0 | 6.0 |
| K | 0.0 | 0.0 | 0.0 | 0.0 | 2.0 | 0.0 | 0.0 | 0.0 | 0.0 | 87.0 | 196.0 | 0.0 | 0.0 | 0.0 | 38.0 | 0.0 | 0.0 | 0.0 | 3.0 | 0.0 |
| L | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| M | 723.0 | 6.0 | 211.0 | 186.0 | 340.0 | 383.0 | 83.0 | 101.0 | 28.0 | 147.0 | 63.0 | 3.0 | 0.0 | 55.0 | 21.0 | 111.0 | 58.0 | 243.0 | 3097.0 | 421.0 |
| N | 1662.0 | 0.0 | 98.0 | 1226.0 | 88.0 | 261.0 | 33.0 | 50.0 | 12.0 | 97.0 | 54.0 | 0.0 | 573.0 | 6.0 | 27.0 | 32.0 | 83.0 | 234.0 | -42.0 | 9.0 |
| O | 10.0 | 0.0 | 48.0 | 11.0 | 46.0 | 16.0 | 3.0 | 19.0 | 6.0 | 3.0 | 6.0 | 3.0 | 58.0 | 2.0 | 385.0 | 49.0 | 8.0 | 74.0 | 241.0 | 28.0 |
| P | 277.0 | 0.0 | 64.0 | 67.0 | 94.0 | 69.0 | 10.0 | 22.0 | 37.0 | 88.0 | 30.0 | 0.0 | 305.0 | 64.0 | 43.0 | 9917.0 | 13.0 | 207.0 | 230.0 | 24.0 |
| Q | 22.0 | 0.0 | 20.0 | 83.0 | 80.0 | 148.0 | 17.0 | 10.0 | 9.0 | 95.0 | 105.0 | 2.0 | 70.0 | 12.0 | 0.0 | 150.0 | 0.0 | 72.0 | 153.0 | 2.0 |
| R | 1246.0 | 18.0 | 319.0 | 478.0 | 1173.0 | 829.0 | 180.0 | 281.0 | 117.0 | 347.0 | 319.0 | 3.0 | 4739.0 | 410.0 | 141.0 | 2052.0 | 43.0 | 36.0 | 1119.0 | 57.0 |
| S | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| T | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

List each ordered pair so the total narrative gain for the full ordering can be computed.

Also, when you send the final ordering back, keep it simple — a tiny JSON object with one key, ""solution"", whose value is the ordered list of the pictures from first to last. Here’s the shape I’m expecting:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of ""solution"" as just the ordered chain: the first entry is the picture that appears first, the second entry appears next, and so on until the last entry. This JSON is just a sketch of the shape I want, not the actual ordering — fill it with the exact identifiers from the instance when you reply.

Please make sure all identifiers are used exactly as they appear in the instance input — do not rename them and do not 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”.""","[[18093.0, 0.0, 0.0, 8.0, 473.0, 0.0, 0.0, 4.0, 15030.0, 8451.0, 713.0, 428.0, 191.0, 9.0, 39.0, 0.0, 0.0, 2.0, 32.0, 16.0], [0.0, 80.0, 0.0, 1605.0, 192.0, 64.0, 0.0, 10.0, 3.0, 7.0, 2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 21.0, 5.0], [807.0, 0.0, 5624.0, 293.0, 748.0, 442.0, 36.0, 71.0, 42.0, 224.0, 74.0, 0.0, 1500.0, 430.0, 750.0, 80.0, 188.0, 641.0, 615.0, 33.0], [416.0, 0.0, 122.0, 15913.0, 711.0, 4950.0, 210.0, 795.0, 0.0, 12.0, 0.0, 3.0, 3593.0, 114.0, 8.0, 0.0, 1.0, 0.0, 87.0, 17.0], [2764.0, 9.0, 383.0, 213.0, 5386.0, 347.0, 41.0, 100.0, 6.0, 122.0, 166.0, 1.0, 1104.0, 214.0, 26.0, 5.0, 15.0, 210.0, 249.0, 23.0], [29.0, 4.0, 311.0, 228.0, 638.0, 1626.0, 343.0, 1144.0, 36.0, 366.0, 90.0, 0.0, 4738.0, 464.0, 17.0, 0.0, 5.0, 20.0, 303.0, 23.0], [3.0, 0.0, 12.0, 11.0, 10.0, 46.0, 39.0, 128.0, 0.0, 2.0, 2.0, 0.0, 61.0, 19.0, 0.0, 50.0, 0.0, 4.0, 207.0, 150.0], [81.0, 0.0, 0.0, 13.0, 0.0, 2.0, 0.0, 9.0, 0.0, 0.0, 0.0, 0.0, 0.0, 190.0, 298.0, 0.0, 14.0, 0.0, 3504.0, 1.0], [4.0, 0.0, 0.0, 0.0, 67.0, 0.0, 0.0, 0.0, 451.0, 82.0, 0.0, 0.0, 0.0, 11.0, 31.0, 1.0, 0.0, 0.0, 82.0, 0.0], [3749.0, 0.0, 9.0, 0.0, 475.0, 8.0, 0.0, 0.0, 226.0, 6161.0, 225.0, 0.0, 0.0, 3.0, 4.0, 0.0, 0.0, 3.0, 52.0, 6.0], [0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 87.0, 196.0, 0.0, 0.0, 0.0, 38.0, 0.0, 0.0, 0.0, 3.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [723.0, 6.0, 211.0, 186.0, 340.0, 383.0, 83.0, 101.0, 28.0, 147.0, 63.0, 3.0, 0.0, 55.0, 21.0, 111.0, 58.0, 243.0, 3097.0, 421.0], [1662.0, 0.0, 98.0, 1226.0, 88.0, 261.0, 33.0, 50.0, 12.0, 97.0, 54.0, 0.0, 573.0, 6.0, 27.0, 32.0, 83.0, 234.0, -42.0, 9.0], [10.0, 0.0, 48.0, 11.0, 46.0, 16.0, 3.0, 19.0, 6.0, 3.0, 6.0, 3.0, 58.0, 2.0, 385.0, 49.0, 8.0, 74.0, 241.0, 28.0], [277.0, 0.0, 64.0, 67.0, 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'weight': 70.0}, {'from_id': 'Q', 'to_id': 'N', 'weight': 12.0}, {'from_id': 'Q', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'P', 'weight': 150.0}, {'from_id': 'Q', 'to_id': 'R', 'weight': 72.0}, {'from_id': 'Q', 'to_id': 'S', 'weight': 153.0}, {'from_id': 'Q', 'to_id': 'T', 'weight': 2.0}, {'from_id': 'R', 'to_id': 'A', 'weight': 1246.0}, {'from_id': 'R', 'to_id': 'B', 'weight': 18.0}, {'from_id': 'R', 'to_id': 'C', 'weight': 319.0}, {'from_id': 'R', 'to_id': 'D', 'weight': 478.0}, {'from_id': 'R', 'to_id': 'E', 'weight': 1173.0}, {'from_id': 'R', 'to_id': 'F', 'weight': 829.0}, {'from_id': 'R', 'to_id': 'G', 'weight': 180.0}, {'from_id': 'R', 'to_id': 'H', 'weight': 281.0}, {'from_id': 'R', 'to_id': 'I', 'weight': 117.0}, {'from_id': 'R', 'to_id': 'J', 'weight': 347.0}, {'from_id': 'R', 'to_id': 'K', 'weight': 319.0}, {'from_id': 'R', 'to_id': 'L', 'weight': 3.0}, {'from_id': 'R', 'to_id': 'M', 'weight': 4739.0}, {'from_id': 'R', 'to_id': 'N', 'weight': 410.0}, {'from_id': 'R', 'to_id': 'O', 'weight': 141.0}, {'from_id': 'R', 'to_id': 'P', 'weight': 2052.0}, {'from_id': 'R', 'to_id': 'Q', 'weight': 43.0}, {'from_id': 'R', 'to_id': 'S', 'weight': 1119.0}, {'from_id': 'R', 'to_id': 'T', 'weight': 57.0}, {'from_id': 'S', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'T', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'S', 'weight': 0.0}]}","['C', 'R', 'N', 'Q', 'P', 'B', 'D', 'F', 'E', 'M', 'G', 'H', 'A', 'J', 'K', 'I', 'O', 'T', 'S', 'L']",36,markdown_table,names
LOP,LOP,"I’m putting together a meeting agenda and need to line up every topic into one continuous list — nothing left out, nothing repeated. For every pair of topics there’s a little score saying how much better it is to cover one before the other, and the overall “goodness” of the agenda is just the sum of all those scores for each time a topic appears ahead of another. The trick is to pick the order that makes that total sum as large as it can be while still including each topic exactly once. The exact topics and their pairwise scores are shown below.

# total_topics=21
# topic_ids=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21

# Meaning: this is a directed coherence_score matrix over items in topic_ids (21 total). Entry at row predecessor_topic_id=i and column successor_topic_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# coherence_score
| predecessor_topic_id\successor_topic_id | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1470.0 | 45.0 | 0.0 | 44.0 | 18.0 | 3.0 | 37.0 | 64.0 | 4.0 | 3.0 | 5.0 | 10.0 | 129.0 | 345.0 | 6.0 | 12.0 | 0.0 | 0.0 | 75.0 | 157.0 | 0.0 |
| 2 | 15.0 | 15.0 | 0.0 | 32.0 | 26.0 | 2.0 | 12.0 | 41.0 | 1.0 | 8.0 | 7.0 | 9.0 | 55.0 | 20.0 | 6.0 | 10.0 | 0.0 | 0.0 | 78.0 | 95.0 | 0.0 |
| 3 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 4 | 1.0 | 2.0 | 0.0 | 63.0 | 4.0 | 2.0 | 11.0 | 13.0 | 0.0 | 0.0 | 139.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 18.0 | 28.0 | 0.0 |
| 5 | 20.0 | 3.0 | 0.0 | 4.0 | 89.0 | 12.0 | 84.0 | 86.0 | 2.0 | 5.0 | 33.0 | 31.0 | 2.0 | 41.0 | 8.0 | 2.0 | 0.0 | 0.0 | 18.0 | 17.0 | 0.0 |
| 6 | 1.0 | 0.0 | 0.0 | 1.0 | 1.0 | 142.0 | 2.0 | 2.0 | 0.0 | 0.0 | 0.0 | 60.0 | 7.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 54.0 | 0.0 |
| 7 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 74.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 8 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 44.0 | 1103.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 4.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 5.0 | 0.0 |
| 9 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 10 | 6.0 | 2.0 | 0.0 | 1.0 | 11.0 | 30.0 | 2.0 | 4.0 | 2.0 | 7.0 | 1.0 | 8.0 | 21.0 | 17.0 | 1.0 | 2.0 | 0.0 | 0.0 | 20.0 | 8.0 | 0.0 |
| 11 | 9.0 | 2.0 | 0.0 | 5.0 | 18.0 | 4.0 | 4.0 | 12.0 | 1.0 | 4.0 | 0.0 | 6.0 | 1.0 | 1.0 | 1.0 | 1.0 | 0.0 | 0.0 | 15.0 | 20.0 | 0.0 |
| 12 | 4.0 | 0.0 | 0.0 | 6.0 | 6.0 | 1.0 | 8.0 | 8.0 | 1.0 | 0.0 | 1.0 | 26.0 | 111.0 | 2.0 | 1.0 | 5.0 | 0.0 | 0.0 | 12.0 | 5.0 | 0.0 |
| 13 | 6.0 | 1.0 | 0.0 | 6.0 | 4.0 | 2.0 | 27.0 | 10.0 | 2.0 | 1.0 | 2.0 | 15.0 | 133.0 | 23.0 | 2.0 | 9.0 | 0.0 | 0.0 | 35.0 | 69.0 | 0.0 |
| 14 | 2.0 | 0.0 | 0.0 | 0.0 | 4.0 | 0.0 | 0.0 | 7.0 | 0.0 | 0.0 | 0.0 | 8.0 | 2.0 | 0.0 | 16.0 | 7.0 | 0.0 | 0.0 | 17.0 | 14.0 | 0.0 |
| 15 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2.0 | 254.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2.0 | 0.0 | 0.0 |
| 16 | 15.0 | 3.0 | 0.0 | 7.0 | 24.0 | 4.0 | 8.0 | 20.0 | 2.0 | 2.0 | 2.0 | 7.0 | 32.0 | 149.0 | 8.0 | 14.0 | 0.0 | 11.0 | 11.0 | 40.0 | 0.0 |
| 17 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 18 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 19 | 18.0 | 17.0 | 0.0 | 41.0 | 37.0 | 6.0 | 9.0 | 31.0 | 2.0 | 5.0 | 13.0 | 12.0 | 55.0 | 154.0 | 8.0 | 79.0 | 0.0 | 0.0 | 35.0 | 456.0 | 0.0 |
| 20 | 5.0 | 3.0 | 0.0 | 3.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 13.0 | 0.0 | 0.0 | 0.0 | 4.0 | 112.0 | 0.0 | 0.0 |
| 21 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Oh, and when you send back the final ordering, please use a tiny JSON layout like this so it’s easy to read and parse — just list the topics in order under ""solution"".

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

That ""solution"" array is where you put every topic in the final sequence, from the one to cover first to the one to cover last. Think of it like filling out a simple form: one entry per slot, in order. This block is just a sketch of the shape I’m expecting, not the actual answer.

Please use the exact identifiers from the instance input — don’t 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”.""","[[1470.0, 45.0, 0.0, 44.0, 18.0, 3.0, 37.0, 64.0, 4.0, 3.0, 5.0, 10.0, 129.0, 345.0, 6.0, 12.0, 0.0, 0.0, 75.0, 157.0, 0.0], [15.0, 15.0, 0.0, 32.0, 26.0, 2.0, 12.0, 41.0, 1.0, 8.0, 7.0, 9.0, 55.0, 20.0, 6.0, 10.0, 0.0, 0.0, 78.0, 95.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [1.0, 2.0, 0.0, 63.0, 4.0, 2.0, 11.0, 13.0, 0.0, 0.0, 139.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 18.0, 28.0, 0.0], [20.0, 3.0, 0.0, 4.0, 89.0, 12.0, 84.0, 86.0, 2.0, 5.0, 33.0, 31.0, 2.0, 41.0, 8.0, 2.0, 0.0, 0.0, 18.0, 17.0, 0.0], [1.0, 0.0, 0.0, 1.0, 1.0, 142.0, 2.0, 2.0, 0.0, 0.0, 0.0, 60.0, 7.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 54.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 74.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 44.0, 1103.0, 0.0, 0.0, 0.0, 0.0, 1.0, 4.0, 0.0, 0.0, 0.0, 0.0, 0.0, 5.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [6.0, 2.0, 0.0, 1.0, 11.0, 30.0, 2.0, 4.0, 2.0, 7.0, 1.0, 8.0, 21.0, 17.0, 1.0, 2.0, 0.0, 0.0, 20.0, 8.0, 0.0], [9.0, 2.0, 0.0, 5.0, 18.0, 4.0, 4.0, 12.0, 1.0, 4.0, 0.0, 6.0, 1.0, 1.0, 1.0, 1.0, 0.0, 0.0, 15.0, 20.0, 0.0], [4.0, 0.0, 0.0, 6.0, 6.0, 1.0, 8.0, 8.0, 1.0, 0.0, 1.0, 26.0, 111.0, 2.0, 1.0, 5.0, 0.0, 0.0, 12.0, 5.0, 0.0], [6.0, 1.0, 0.0, 6.0, 4.0, 2.0, 27.0, 10.0, 2.0, 1.0, 2.0, 15.0, 133.0, 23.0, 2.0, 9.0, 0.0, 0.0, 35.0, 69.0, 0.0], [2.0, 0.0, 0.0, 0.0, 4.0, 0.0, 0.0, 7.0, 0.0, 0.0, 0.0, 8.0, 2.0, 0.0, 16.0, 7.0, 0.0, 0.0, 17.0, 14.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 254.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0], [15.0, 3.0, 0.0, 7.0, 24.0, 4.0, 8.0, 20.0, 2.0, 2.0, 2.0, 7.0, 32.0, 149.0, 8.0, 14.0, 0.0, 11.0, 11.0, 40.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [18.0, 17.0, 0.0, 41.0, 37.0, 6.0, 9.0, 31.0, 2.0, 5.0, 13.0, 12.0, 55.0, 154.0, 8.0, 79.0, 0.0, 0.0, 35.0, 456.0, 0.0], [5.0, 3.0, 0.0, 3.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 13.0, 0.0, 0.0, 0.0, 4.0, 112.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]]","[20, 0, 1, 9, 18, 15, 4, 5, 11, 12, 3, 10, 14, 6, 13, 7, 19, 17, 16, 8, 2]",4024.0,"{'problem_type': 'LOP', 'num_nodes': 21, 'nodes': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], 'pairs': [{'from_id': 1, 'to_id': 2, 'weight': 45.0}, {'from_id': 1, 'to_id': 3, 'weight': 0.0}, {'from_id': 1, 'to_id': 4, 'weight': 44.0}, {'from_id': 1, 'to_id': 5, 'weight': 18.0}, {'from_id': 1, 'to_id': 6, 'weight': 3.0}, {'from_id': 1, 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'weight': 0.0}, {'from_id': 21, 'to_id': 12, 'weight': 0.0}, {'from_id': 21, 'to_id': 13, 'weight': 0.0}, {'from_id': 21, 'to_id': 14, 'weight': 0.0}, {'from_id': 21, 'to_id': 15, 'weight': 0.0}, {'from_id': 21, 'to_id': 16, 'weight': 0.0}, {'from_id': 21, 'to_id': 17, 'weight': 0.0}, {'from_id': 21, 'to_id': 18, 'weight': 0.0}, {'from_id': 21, 'to_id': 19, 'weight': 0.0}, {'from_id': 21, 'to_id': 20, 'weight': 0.0}]}","[21, 1, 2, 10, 19, 16, 5, 6, 12, 13, 4, 11, 15, 7, 14, 8, 20, 18, 17, 9, 3]",37,csv,1
LOP,LOP,"We’ve got a stretch of gallery and a stack of exhibits that all need to be placed in a single row — each item gets exactly one spot, nobody gets left out and nothing is doubled up. Between every two different exhibits there’s a given benefit for placing one ahead of the other, and the crowd’s overall reaction to a particular arrangement is just the sum of those little benefits for every time one piece comes before another. So the plan that gives the highest combined interest is the one to aim for. The exact exhibits and the interest values follow below.

Here are the 19 exhibits we need to place: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S.

*Meaning: this is a directed interest_gain matrix over items in exhibit_ids (19 total). Entry at row preceding_exhibit=i and column succeeding_exhibit=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).*

**interest_gain)**

| preceding_exhibit\succeeding_exhibit | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 18.0 | 28.0 | 0.0 | 27106.0 | 25.0 | 21.0 | 0.0 | 84.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| B | 52.0 | 220610.0 | 6.0 | 5781.0 | 16314.0 | 980.0 | 550.0 | 4801.0 | 30.0 | 53.0 | 573.0 | 48734.0 | 32766.0 | 578.0 | 1871.0 | 139.0 | 423.0 | 656.0 | 126.0 |
| C | 0.0 | 0.0 | 48.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| D | 0.0 | 712.0 | 0.0 | 137890.0 | 3098.0 | 24405.0 | 860.0 | 3675.0 | 10.0 | 501.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 7.0 |
| E | 0.0 | 75.0 | 0.0 | 4358.0 | 25994.0 | 1468.0 | 395.0 | 1600.0 | 0.0 | 148.0 | 182.0 | 168.0 | 29.0 | 18.0 | 69.0 | 18.0 | 11.0 | 102.0 | 2.0 |
| F | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 44392.0 | 0.0 | 0.0 | 0.0 | 4394.0 | 0.0 | 8439.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| G | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 763.0 | 3886.0 | 0.0 | 0.0 | 0.0 | 0.0 | 160.0 | 0.0 | 0.0 | 77.0 | 130.0 | 572.0 | 115.0 |
| H | 0.0 | 12.0 | 0.0 | 7.0 | 258.0 | 4.0 | 2156.0 | 90133.0 | 0.0 | 0.0 | 10.0 | 16.0 | 563.0 | 0.0 | 0.0 | 281.0 | 173.0 | 1782.0 | 548.0 |
| I | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 66.0 | 0.0 | 0.0 | 0.0 | 80.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 5.0 |
| J | 7.0 | 16.0 | 0.0 | 38031.0 | 432.0 | 339.0 | 46.0 | 342.0 | 27.0 | 103.0 | 4602.0 | 34247.0 | 51.0 | 87.0 | 58.0 | 65.0 | 87.0 | 158.0 | 47.0 |
| K | 141.0 | 844.0 | 4.0 | 4049.0 | 4361.0 | 2864.0 | 847.0 | 6047.0 | 47.0 | 3418.0 | 30757.0 | 9027.0 | 2544.0 | 1218.0 | 836.0 | 444.0 | 383.0 | 2913.0 | 788.0 |
| L | 651.0 | 1401.0 | 5.0 | 6687.0 | 11086.0 | 2609.0 | 2190.0 | 11516.0 | 641.0 | 681.0 | 2982.0 | 2956.0 | 1279.0 | 768.0 | 786.0 | 537.0 | 1195.0 | 1016.0 | 446.0 |
| M | 528.0 | 679.0 | 0.0 | 2123.0 | 1873.0 | 254.0 | 33.0 | 575.0 | 92.0 | 57.0 | 732.0 | 987.0 | 765.0 | 385.0 | 38.0 | 6.0 | 22.0 | 118.0 | 28.0 |
| N | 125.0 | 931.0 | 3.0 | 1910.0 | 2636.0 | 1029.0 | 190.0 | 2164.0 | 289.0 | 385.0 | 13977.0 | 6159.0 | 3297.0 | 11178.0 | 3289.0 | 75.0 | 359.0 | 462.0 | 310.0 |
| O | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 175.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| P | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| Q | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| R | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| S | 20.0 | 103.0 | 0.0 | 265.0 | 373.0 | 185.0 | 31.0 | 241.0 | 20.0 | 96.0 | 893.0 | 356.0 | 154.0 | 335.0 | 145.0 | 62.0 | 88.0 | 0.0 | 0.0 |

That’s everything — we should pick the ordering that maximizes the total interest.

If you want to hand the final ordering back to me, just drop it into a tiny JSON snippet like this — simple and tidy:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of ""solution"" as the single row of exhibits, listed from the one at the very front to the one at the very end. Each placeholder in the array stands for one exhibit identifier — that's all the JSON is doing here, just showing the shape I expect. This is only a sketch of the shape, not the actual answer.

Please use the exact identifiers from 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”.""","[[18.0, 28.0, 0.0, 27106.0, 25.0, 21.0, 0.0, 84.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [52.0, 220610.0, 6.0, 5781.0, 16314.0, 980.0, 550.0, 4801.0, 30.0, 53.0, 573.0, 48734.0, 32766.0, 578.0, 1871.0, 139.0, 423.0, 656.0, 126.0], [0.0, 0.0, 48.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 712.0, 0.0, 137890.0, 3098.0, 24405.0, 860.0, 3675.0, 10.0, 501.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 7.0], [0.0, 75.0, 0.0, 4358.0, 25994.0, 1468.0, 395.0, 1600.0, 0.0, 148.0, 182.0, 168.0, 29.0, 18.0, 69.0, 18.0, 11.0, 102.0, 2.0], [0.0, 0.0, 0.0, 0.0, 0.0, 44392.0, 0.0, 0.0, 0.0, 4394.0, 0.0, 8439.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 763.0, 3886.0, 0.0, 0.0, 0.0, 0.0, 160.0, 0.0, 0.0, 77.0, 130.0, 572.0, 115.0], [0.0, 12.0, 0.0, 7.0, 258.0, 4.0, 2156.0, 90133.0, 0.0, 0.0, 10.0, 16.0, 563.0, 0.0, 0.0, 281.0, 173.0, 1782.0, 548.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 66.0, 0.0, 0.0, 0.0, 80.0, 0.0, 0.0, 0.0, 0.0, 0.0, 5.0], [7.0, 16.0, 0.0, 38031.0, 432.0, 339.0, 46.0, 342.0, 27.0, 103.0, 4602.0, 34247.0, 51.0, 87.0, 58.0, 65.0, 87.0, 158.0, 47.0], [141.0, 844.0, 4.0, 4049.0, 4361.0, 2864.0, 847.0, 6047.0, 47.0, 3418.0, 30757.0, 9027.0, 2544.0, 1218.0, 836.0, 444.0, 383.0, 2913.0, 788.0], [651.0, 1401.0, 5.0, 6687.0, 11086.0, 2609.0, 2190.0, 11516.0, 641.0, 681.0, 2982.0, 2956.0, 1279.0, 768.0, 786.0, 537.0, 1195.0, 1016.0, 446.0], [528.0, 679.0, 0.0, 2123.0, 1873.0, 254.0, 33.0, 575.0, 92.0, 57.0, 732.0, 987.0, 765.0, 385.0, 38.0, 6.0, 22.0, 118.0, 28.0], [125.0, 931.0, 3.0, 1910.0, 2636.0, 1029.0, 190.0, 2164.0, 289.0, 385.0, 13977.0, 6159.0, 3297.0, 11178.0, 3289.0, 75.0, 359.0, 462.0, 310.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 175.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [20.0, 103.0, 0.0, 265.0, 373.0, 185.0, 31.0, 241.0, 20.0, 96.0, 893.0, 356.0, 154.0, 335.0, 145.0, 62.0, 88.0, 0.0, 0.0]]","[18, 13, 9, 10, 1, 11, 12, 0, 4, 3, 6, 7, 17, 16, 15, 14, 8, 5, 2]",384581.0,"{'problem_type': 'LOP', 'num_nodes': 19, 'nodes': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S'], 'pairs': [{'from_id': 'A', 'to_id': 'B', 'weight': 28.0}, {'from_id': 'A', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'D', 'weight': 27106.0}, {'from_id': 'A', 'to_id': 'E', 'weight': 25.0}, {'from_id': 'A', 'to_id': 'F', 'weight': 21.0}, {'from_id': 'A', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'H', 'weight': 84.0}, {'from_id': 'A', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'S', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'A', 'weight': 52.0}, {'from_id': 'B', 'to_id': 'C', 'weight': 6.0}, {'from_id': 'B', 'to_id': 'D', 'weight': 5781.0}, {'from_id': 'B', 'to_id': 'E', 'weight': 16314.0}, {'from_id': 'B', 'to_id': 'F', 'weight': 980.0}, {'from_id': 'B', 'to_id': 'G', 'weight': 550.0}, {'from_id': 'B', 'to_id': 'H', 'weight': 4801.0}, {'from_id': 'B', 'to_id': 'I', 'weight': 30.0}, {'from_id': 'B', 'to_id': 'J', 'weight': 53.0}, {'from_id': 'B', 'to_id': 'K', 'weight': 573.0}, {'from_id': 'B', 'to_id': 'L', 'weight': 48734.0}, {'from_id': 'B', 'to_id': 'M', 'weight': 32766.0}, {'from_id': 'B', 'to_id': 'N', 'weight': 578.0}, {'from_id': 'B', 'to_id': 'O', 'weight': 1871.0}, {'from_id': 'B', 'to_id': 'P', 'weight': 139.0}, {'from_id': 'B', 'to_id': 'Q', 'weight': 423.0}, {'from_id': 'B', 'to_id': 'R', 'weight': 656.0}, {'from_id': 'B', 'to_id': 'S', 'weight': 126.0}, {'from_id': 'C', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'C', 'to_id': 'N', 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{'from_id': 'P', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'S', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'S', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'S', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'A', 'weight': 20.0}, {'from_id': 'S', 'to_id': 'B', 'weight': 103.0}, {'from_id': 'S', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'D', 'weight': 265.0}, {'from_id': 'S', 'to_id': 'E', 'weight': 373.0}, {'from_id': 'S', 'to_id': 'F', 'weight': 185.0}, {'from_id': 'S', 'to_id': 'G', 'weight': 31.0}, {'from_id': 'S', 'to_id': 'H', 'weight': 241.0}, {'from_id': 'S', 'to_id': 'I', 'weight': 20.0}, {'from_id': 'S', 'to_id': 'J', 'weight': 96.0}, {'from_id': 'S', 'to_id': 'K', 'weight': 893.0}, {'from_id': 'S', 'to_id': 'L', 'weight': 356.0}, {'from_id': 'S', 'to_id': 'M', 'weight': 154.0}, {'from_id': 'S', 'to_id': 'N', 'weight': 335.0}, {'from_id': 'S', 'to_id': 'O', 'weight': 145.0}, {'from_id': 'S', 'to_id': 'P', 'weight': 62.0}, {'from_id': 'S', 'to_id': 'Q', 'weight': 88.0}, {'from_id': 'S', 'to_id': 'R', 'weight': 0.0}]}","['S', 'N', 'J', 'K', 'B', 'L', 'M', 'A', 'E', 'D', 'G', 'H', 'R', 'Q', 'P', 'O', 'I', 'F', 'C']",38,markdown_table,names
LOP,LOP,"Picture a long buffet table where each platter must take a unique spot in the line — no duplicates, nothing skipped. For each pair of distinct platters there’s a little benefit score for putting one ahead of the other, and by summing those scores for every ordered pair in a lineup, the caterer can see which ordering makes the most overall satisfaction. The concrete platter names and the pairwise scores are listed below.

# total_platter_count=16
# platter_ids=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

# Meaning: this is a directed benefit_score matrix over items in platter_ids (16 total). Entry at row earlier_platter_id=i and column later_platter_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# benefit_score
| earlier_platter_id\later_platter_id | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 0.0 | 0.0 | 8.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 2 | 0.0 | 0.0 | 57.0 | 21.0 | 0.0 | 0.0 | 0.0 | 8.0 | 0.0 | 0.0 | 0.0 | 108.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 3 | 0.0 | 0.0 | 1284.0 | 101.0 | 2.0 | 0.0 | 21.0 | 37.0 | 41.0 | 17.0 | 123.0 | 47.0 | 55.0 | 0.0 | 0.0 | 0.0 |
| 4 | 0.0 | 0.0 | 0.0 | 212.0 | 0.0 | 0.0 | 17.0 | 6.0 | 2.0 | 2.0 | 12.0 | 5.0 | 7.0 | 0.0 | 0.0 | 0.0 |
| 5 | 0.0 | 0.0 | 72.0 | 0.0 | 25.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 13.0 | 7.0 | 5.0 | 0.0 | 0.0 | 0.0 |
| 6 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 7 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 10.0 | 0.0 | 0.0 | 0.0 | 0.0 | 8.0 | 0.0 | 0.0 | 0.0 |
| 8 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 645.0 | 138.0 | 0.0 | 0.0 | 0.0 | 18.0 | 0.0 | 0.0 | 0.0 |
| 9 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2.0 | 8.0 | 0.0 | 0.0 | 0.0 | 4.0 | 0.0 | 0.0 | 0.0 |
| 10 | 0.0 | 0.0 | 82.0 | 27.0 | 1.0 | 0.0 | 9.0 | 16.0 | 24.0 | 4.0 | 0.0 | 43.0 | 2.0 | 0.0 | 0.0 | 0.0 |
| 11 | 0.0 | 0.0 | 0.0 | 9.0 | 0.0 | 0.0 | 1.0 | 4.0 | 18.0 | 2.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 12 | 0.0 | 0.0 | 19.0 | 27.0 | 3.0 | 0.0 | 27.0 | 13.0 | 20.0 | 10.0 | 0.0 | 0.0 | 3.0 | 0.0 | 0.0 | 0.0 |
| 13 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 14 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 15 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 16 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Oh, and when you send back the final lineup, just stick to this simple JSON shape:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of ""solution"" as the ordered list of platters from the one at the very front of the table to the one at the very end. The placeholders show where each item identifier from the instance should go — one identifier per slot, in order. This JSON is just a sketch of the shape I want, not the actual answer.

Please use the exact identifiers given 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”.""","[[0.0, 0.0, 8.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 57.0, 21.0, 0.0, 0.0, 0.0, 8.0, 0.0, 0.0, 0.0, 108.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 1284.0, 101.0, 2.0, 0.0, 21.0, 37.0, 41.0, 17.0, 123.0, 47.0, 55.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 212.0, 0.0, 0.0, 17.0, 6.0, 2.0, 2.0, 12.0, 5.0, 7.0, 0.0, 0.0, 0.0], [0.0, 0.0, 72.0, 0.0, 25.0, 0.0, 0.0, 0.0, 0.0, 0.0, 13.0, 7.0, 5.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 10.0, 0.0, 0.0, 0.0, 0.0, 8.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 645.0, 138.0, 0.0, 0.0, 0.0, 18.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 8.0, 0.0, 0.0, 0.0, 4.0, 0.0, 0.0, 0.0], [0.0, 0.0, 82.0, 27.0, 1.0, 0.0, 9.0, 16.0, 24.0, 4.0, 0.0, 43.0, 2.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 9.0, 0.0, 0.0, 1.0, 4.0, 18.0, 2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 19.0, 27.0, 3.0, 0.0, 27.0, 13.0, 20.0, 10.0, 0.0, 0.0, 3.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]]","[15, 14, 13, 9, 4, 1, 0, 2, 11, 3, 10, 6, 7, 8, 12, 5]",1263.0,"{'problem_type': 'LOP', 'num_nodes': 16, 'nodes': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 'pairs': [{'from_id': 1, 'to_id': 2, 'weight': 0.0}, {'from_id': 1, 'to_id': 3, 'weight': 8.0}, {'from_id': 1, 'to_id': 4, 'weight': 0.0}, {'from_id': 1, 'to_id': 5, 'weight': 0.0}, {'from_id': 1, 'to_id': 6, 'weight': 0.0}, {'from_id': 1, 'to_id': 7, 'weight': 0.0}, {'from_id': 1, 'to_id': 8, 'weight': 0.0}, {'from_id': 1, 'to_id': 9, 'weight': 0.0}, {'from_id': 1, 'to_id': 10, 'weight': 0.0}, {'from_id': 1, 'to_id': 11, 'weight': 0.0}, {'from_id': 1, 'to_id': 12, 'weight': 0.0}, {'from_id': 1, 'to_id': 13, 'weight': 0.0}, {'from_id': 1, 'to_id': 14, 'weight': 0.0}, {'from_id': 1, 'to_id': 15, 'weight': 0.0}, {'from_id': 1, 'to_id': 16, 'weight': 0.0}, {'from_id': 2, 'to_id': 1, 'weight': 0.0}, {'from_id': 2, 'to_id': 3, 'weight': 57.0}, {'from_id': 2, 'to_id': 4, 'weight': 21.0}, {'from_id': 2, 'to_id': 5, 'weight': 0.0}, {'from_id': 2, 'to_id': 6, 'weight': 0.0}, {'from_id': 2, 'to_id': 7, 'weight': 0.0}, {'from_id': 2, 'to_id': 8, 'weight': 8.0}, {'from_id': 2, 'to_id': 9, 'weight': 0.0}, {'from_id': 2, 'to_id': 10, 'weight': 0.0}, {'from_id': 2, 'to_id': 11, 'weight': 0.0}, {'from_id': 2, 'to_id': 12, 'weight': 108.0}, {'from_id': 2, 'to_id': 13, 'weight': 0.0}, {'from_id': 2, 'to_id': 14, 'weight': 0.0}, {'from_id': 2, 'to_id': 15, 'weight': 0.0}, {'from_id': 2, 'to_id': 16, 'weight': 0.0}, {'from_id': 3, 'to_id': 1, 'weight': 0.0}, {'from_id': 3, 'to_id': 2, 'weight': 0.0}, {'from_id': 3, 'to_id': 4, 'weight': 101.0}, {'from_id': 3, 'to_id': 5, 'weight': 2.0}, {'from_id': 3, 'to_id': 6, 'weight': 0.0}, {'from_id': 3, 'to_id': 7, 'weight': 21.0}, {'from_id': 3, 'to_id': 8, 'weight': 37.0}, {'from_id': 3, 'to_id': 9, 'weight': 41.0}, {'from_id': 3, 'to_id': 10, 'weight': 17.0}, {'from_id': 3, 'to_id': 11, 'weight': 123.0}, {'from_id': 3, 'to_id': 12, 'weight': 47.0}, {'from_id': 3, 'to_id': 13, 'weight': 55.0}, {'from_id': 3, 'to_id': 14, 'weight': 0.0}, {'from_id': 3, 'to_id': 15, 'weight': 0.0}, {'from_id': 3, 'to_id': 16, 'weight': 0.0}, {'from_id': 4, 'to_id': 1, 'weight': 0.0}, {'from_id': 4, 'to_id': 2, 'weight': 0.0}, {'from_id': 4, 'to_id': 3, 'weight': 0.0}, {'from_id': 4, 'to_id': 5, 'weight': 0.0}, {'from_id': 4, 'to_id': 6, 'weight': 0.0}, {'from_id': 4, 'to_id': 7, 'weight': 17.0}, {'from_id': 4, 'to_id': 8, 'weight': 6.0}, {'from_id': 4, 'to_id': 9, 'weight': 2.0}, {'from_id': 4, 'to_id': 10, 'weight': 2.0}, {'from_id': 4, 'to_id': 11, 'weight': 12.0}, {'from_id': 4, 'to_id': 12, 'weight': 5.0}, {'from_id': 4, 'to_id': 13, 'weight': 7.0}, {'from_id': 4, 'to_id': 14, 'weight': 0.0}, {'from_id': 4, 'to_id': 15, 'weight': 0.0}, {'from_id': 4, 'to_id': 16, 'weight': 0.0}, {'from_id': 5, 'to_id': 1, 'weight': 0.0}, {'from_id': 5, 'to_id': 2, 'weight': 0.0}, {'from_id': 5, 'to_id': 3, 'weight': 72.0}, {'from_id': 5, 'to_id': 4, 'weight': 0.0}, {'from_id': 5, 'to_id': 6, 'weight': 0.0}, {'from_id': 5, 'to_id': 7, 'weight': 0.0}, {'from_id': 5, 'to_id': 8, 'weight': 0.0}, {'from_id': 5, 'to_id': 9, 'weight': 0.0}, {'from_id': 5, 'to_id': 10, 'weight': 0.0}, {'from_id': 5, 'to_id': 11, 'weight': 13.0}, {'from_id': 5, 'to_id': 12, 'weight': 7.0}, {'from_id': 5, 'to_id': 13, 'weight': 5.0}, {'from_id': 5, 'to_id': 14, 'weight': 0.0}, {'from_id': 5, 'to_id': 15, 'weight': 0.0}, {'from_id': 5, 'to_id': 16, 'weight': 0.0}, {'from_id': 6, 'to_id': 1, 'weight': 0.0}, {'from_id': 6, 'to_id': 2, 'weight': 0.0}, {'from_id': 6, 'to_id': 3, 'weight': 0.0}, {'from_id': 6, 'to_id': 4, 'weight': 0.0}, {'from_id': 6, 'to_id': 5, 'weight': 0.0}, {'from_id': 6, 'to_id': 7, 'weight': 0.0}, {'from_id': 6, 'to_id': 8, 'weight': 0.0}, {'from_id': 6, 'to_id': 9, 'weight': 0.0}, {'from_id': 6, 'to_id': 10, 'weight': 0.0}, {'from_id': 6, 'to_id': 11, 'weight': 0.0}, {'from_id': 6, 'to_id': 12, 'weight': 0.0}, {'from_id': 6, 'to_id': 13, 'weight': 0.0}, {'from_id': 6, 'to_id': 14, 'weight': 0.0}, {'from_id': 6, 'to_id': 15, 'weight': 0.0}, {'from_id': 6, 'to_id': 16, 'weight': 0.0}, {'from_id': 7, 'to_id': 1, 'weight': 0.0}, {'from_id': 7, 'to_id': 2, 'weight': 0.0}, {'from_id': 7, 'to_id': 3, 'weight': 0.0}, {'from_id': 7, 'to_id': 4, 'weight': 0.0}, {'from_id': 7, 'to_id': 5, 'weight': 0.0}, {'from_id': 7, 'to_id': 6, 'weight': 0.0}, {'from_id': 7, 'to_id': 8, 'weight': 10.0}, {'from_id': 7, 'to_id': 9, 'weight': 0.0}, {'from_id': 7, 'to_id': 10, 'weight': 0.0}, {'from_id': 7, 'to_id': 11, 'weight': 0.0}, {'from_id': 7, 'to_id': 12, 'weight': 0.0}, {'from_id': 7, 'to_id': 13, 'weight': 8.0}, {'from_id': 7, 'to_id': 14, 'weight': 0.0}, {'from_id': 7, 'to_id': 15, 'weight': 0.0}, {'from_id': 7, 'to_id': 16, 'weight': 0.0}, {'from_id': 8, 'to_id': 1, 'weight': 0.0}, {'from_id': 8, 'to_id': 2, 'weight': 0.0}, {'from_id': 8, 'to_id': 3, 'weight': 0.0}, {'from_id': 8, 'to_id': 4, 'weight': 0.0}, {'from_id': 8, 'to_id': 5, 'weight': 0.0}, {'from_id': 8, 'to_id': 6, 'weight': 0.0}, {'from_id': 8, 'to_id': 7, 'weight': 0.0}, {'from_id': 8, 'to_id': 9, 'weight': 138.0}, {'from_id': 8, 'to_id': 10, 'weight': 0.0}, {'from_id': 8, 'to_id': 11, 'weight': 0.0}, {'from_id': 8, 'to_id': 12, 'weight': 0.0}, {'from_id': 8, 'to_id': 13, 'weight': 18.0}, {'from_id': 8, 'to_id': 14, 'weight': 0.0}, {'from_id': 8, 'to_id': 15, 'weight': 0.0}, {'from_id': 8, 'to_id': 16, 'weight': 0.0}, {'from_id': 9, 'to_id': 1, 'weight': 0.0}, {'from_id': 9, 'to_id': 2, 'weight': 0.0}, {'from_id': 9, 'to_id': 3, 'weight': 0.0}, {'from_id': 9, 'to_id': 4, 'weight': 0.0}, {'from_id': 9, 'to_id': 5, 'weight': 0.0}, {'from_id': 9, 'to_id': 6, 'weight': 0.0}, {'from_id': 9, 'to_id': 7, 'weight': 0.0}, {'from_id': 9, 'to_id': 8, 'weight': 2.0}, {'from_id': 9, 'to_id': 10, 'weight': 0.0}, {'from_id': 9, 'to_id': 11, 'weight': 0.0}, {'from_id': 9, 'to_id': 12, 'weight': 0.0}, {'from_id': 9, 'to_id': 13, 'weight': 4.0}, {'from_id': 9, 'to_id': 14, 'weight': 0.0}, {'from_id': 9, 'to_id': 15, 'weight': 0.0}, {'from_id': 9, 'to_id': 16, 'weight': 0.0}, {'from_id': 10, 'to_id': 1, 'weight': 0.0}, {'from_id': 10, 'to_id': 2, 'weight': 0.0}, {'from_id': 10, 'to_id': 3, 'weight': 82.0}, {'from_id': 10, 'to_id': 4, 'weight': 27.0}, {'from_id': 10, 'to_id': 5, 'weight': 1.0}, {'from_id': 10, 'to_id': 6, 'weight': 0.0}, {'from_id': 10, 'to_id': 7, 'weight': 9.0}, {'from_id': 10, 'to_id': 8, 'weight': 16.0}, {'from_id': 10, 'to_id': 9, 'weight': 24.0}, {'from_id': 10, 'to_id': 11, 'weight': 0.0}, {'from_id': 10, 'to_id': 12, 'weight': 43.0}, {'from_id': 10, 'to_id': 13, 'weight': 2.0}, {'from_id': 10, 'to_id': 14, 'weight': 0.0}, {'from_id': 10, 'to_id': 15, 'weight': 0.0}, {'from_id': 10, 'to_id': 16, 'weight': 0.0}, {'from_id': 11, 'to_id': 1, 'weight': 0.0}, {'from_id': 11, 'to_id': 2, 'weight': 0.0}, {'from_id': 11, 'to_id': 3, 'weight': 0.0}, {'from_id': 11, 'to_id': 4, 'weight': 9.0}, {'from_id': 11, 'to_id': 5, 'weight': 0.0}, {'from_id': 11, 'to_id': 6, 'weight': 0.0}, {'from_id': 11, 'to_id': 7, 'weight': 1.0}, {'from_id': 11, 'to_id': 8, 'weight': 4.0}, {'from_id': 11, 'to_id': 9, 'weight': 18.0}, {'from_id': 11, 'to_id': 10, 'weight': 2.0}, {'from_id': 11, 'to_id': 12, 'weight': 0.0}, {'from_id': 11, 'to_id': 13, 'weight': 0.0}, {'from_id': 11, 'to_id': 14, 'weight': 0.0}, {'from_id': 11, 'to_id': 15, 'weight': 0.0}, {'from_id': 11, 'to_id': 16, 'weight': 0.0}, {'from_id': 12, 'to_id': 1, 'weight': 0.0}, {'from_id': 12, 'to_id': 2, 'weight': 0.0}, {'from_id': 12, 'to_id': 3, 'weight': 19.0}, {'from_id': 12, 'to_id': 4, 'weight': 27.0}, {'from_id': 12, 'to_id': 5, 'weight': 3.0}, {'from_id': 12, 'to_id': 6, 'weight': 0.0}, {'from_id': 12, 'to_id': 7, 'weight': 27.0}, {'from_id': 12, 'to_id': 8, 'weight': 13.0}, {'from_id': 12, 'to_id': 9, 'weight': 20.0}, {'from_id': 12, 'to_id': 10, 'weight': 10.0}, {'from_id': 12, 'to_id': 11, 'weight': 0.0}, {'from_id': 12, 'to_id': 13, 'weight': 3.0}, {'from_id': 12, 'to_id': 14, 'weight': 0.0}, {'from_id': 12, 'to_id': 15, 'weight': 0.0}, {'from_id': 12, 'to_id': 16, 'weight': 0.0}, {'from_id': 13, 'to_id': 1, 'weight': 0.0}, {'from_id': 13, 'to_id': 2, 'weight': 0.0}, {'from_id': 13, 'to_id': 3, 'weight': 0.0}, {'from_id': 13, 'to_id': 4, 'weight': 0.0}, {'from_id': 13, 'to_id': 5, 'weight': 0.0}, {'from_id': 13, 'to_id': 6, 'weight': 0.0}, {'from_id': 13, 'to_id': 7, 'weight': 0.0}, {'from_id': 13, 'to_id': 8, 'weight': 0.0}, {'from_id': 13, 'to_id': 9, 'weight': 0.0}, {'from_id': 13, 'to_id': 10, 'weight': 0.0}, {'from_id': 13, 'to_id': 11, 'weight': 0.0}, {'from_id': 13, 'to_id': 12, 'weight': 0.0}, {'from_id': 13, 'to_id': 14, 'weight': 0.0}, {'from_id': 13, 'to_id': 15, 'weight': 0.0}, {'from_id': 13, 'to_id': 16, 'weight': 0.0}, {'from_id': 14, 'to_id': 1, 'weight': 0.0}, {'from_id': 14, 'to_id': 2, 'weight': 0.0}, {'from_id': 14, 'to_id': 3, 'weight': 0.0}, {'from_id': 14, 'to_id': 4, 'weight': 0.0}, {'from_id': 14, 'to_id': 5, 'weight': 0.0}, {'from_id': 14, 'to_id': 6, 'weight': 0.0}, {'from_id': 14, 'to_id': 7, 'weight': 0.0}, {'from_id': 14, 'to_id': 8, 'weight': 0.0}, {'from_id': 14, 'to_id': 9, 'weight': 0.0}, {'from_id': 14, 'to_id': 10, 'weight': 0.0}, {'from_id': 14, 'to_id': 11, 'weight': 0.0}, {'from_id': 14, 'to_id': 12, 'weight': 0.0}, {'from_id': 14, 'to_id': 13, 'weight': 0.0}, {'from_id': 14, 'to_id': 15, 'weight': 0.0}, {'from_id': 14, 'to_id': 16, 'weight': 0.0}, {'from_id': 15, 'to_id': 1, 'weight': 0.0}, {'from_id': 15, 'to_id': 2, 'weight': 0.0}, {'from_id': 15, 'to_id': 3, 'weight': 0.0}, {'from_id': 15, 'to_id': 4, 'weight': 0.0}, {'from_id': 15, 'to_id': 5, 'weight': 0.0}, {'from_id': 15, 'to_id': 6, 'weight': 0.0}, {'from_id': 15, 'to_id': 7, 'weight': 0.0}, {'from_id': 15, 'to_id': 8, 'weight': 0.0}, {'from_id': 15, 'to_id': 9, 'weight': 0.0}, {'from_id': 15, 'to_id': 10, 'weight': 0.0}, {'from_id': 15, 'to_id': 11, 'weight': 0.0}, {'from_id': 15, 'to_id': 12, 'weight': 0.0}, {'from_id': 15, 'to_id': 13, 'weight': 0.0}, {'from_id': 15, 'to_id': 14, 'weight': 0.0}, {'from_id': 15, 'to_id': 16, 'weight': 0.0}, {'from_id': 16, 'to_id': 1, 'weight': 0.0}, {'from_id': 16, 'to_id': 2, 'weight': 0.0}, {'from_id': 16, 'to_id': 3, 'weight': 0.0}, {'from_id': 16, 'to_id': 4, 'weight': 0.0}, {'from_id': 16, 'to_id': 5, 'weight': 0.0}, {'from_id': 16, 'to_id': 6, 'weight': 0.0}, {'from_id': 16, 'to_id': 7, 'weight': 0.0}, {'from_id': 16, 'to_id': 8, 'weight': 0.0}, {'from_id': 16, 'to_id': 9, 'weight': 0.0}, {'from_id': 16, 'to_id': 10, 'weight': 0.0}, {'from_id': 16, 'to_id': 11, 'weight': 0.0}, {'from_id': 16, 'to_id': 12, 'weight': 0.0}, {'from_id': 16, 'to_id': 13, 'weight': 0.0}, {'from_id': 16, 'to_id': 14, 'weight': 0.0}, {'from_id': 16, 'to_id': 15, 'weight': 0.0}]}","[16, 15, 14, 10, 5, 2, 1, 3, 12, 4, 11, 7, 8, 9, 13, 6]",39,csv,1
LOP,LOP,"There’s a trainer deciding how to line up a handful of exercises into a single workout so nothing’s left out and nothing happens twice. Every time the trainer puts one exercise before another, there’s a measurable payoff for that specific switch, and the trick is to arrange the whole flow so the payoffs from each transition add up to the biggest number possible. So the plan is to total the benefit of each successive pair in the routine and pick the order with the greatest combined score. The concrete exercise list and the pairwise benefits are shown below.

The trainer must arrange 18 exercises: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18.
Meaning: this is a directed transition_benefit matrix over items in exercise_list (18 total). Entry at row from_exercise=i and column to_exercise=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
transition_benefit:
| from_exercise\to_exercise | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 0.0 | 0.0 | 0.0 | 3.0 | 2.0 | 0.0 | 2.0 | 0.0 | 4.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 |
| 2 | 0.0 | 1993.0 | 134.0 | 166.0 | 19.0 | 10.0 | 17.0 | 66.0 | 132.0 | 9.0 | 4.0 | 0.0 | 216.0 | 24.0 | 395.0 | 0.0 | 36.0 | 154.0 |
| 3 | 38.0 | 24.0 | 38.0 | 41.0 | 17.0 | 17.0 | 13.0 | 15.0 | 49.0 | 8.0 | 2.0 | 1.0 | 100.0 | 49.0 | 3.0 | 0.0 | 19.0 | 113.0 |
| 4 | 0.0 | 1.0 | 0.0 | 79.0 | 1.0 | 0.0 | 1.0 | 8.0 | 18.0 | 45.0 | 0.0 | 0.0 | 0.0 | 5.0 | 0.0 | 0.0 | 6.0 | 5.0 |
| 5 | 1.0 | 14.0 | 9.0 | 7.0 | 443.0 | 218.0 | 2.0 | 2.0 | 16.0 | 2.0 | 1.0 | 0.0 | 13.0 | 6.0 | 157.0 | 0.0 | 3.0 | 377.0 |
| 6 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 21.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 315.0 | 0.0 | 0.0 | 37.0 |
| 7 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 129.0 | 0.0 | 126.0 | 0.0 | 0.0 | 0.0 | 0.0 | 65.0 | 26.0 | 0.0 | 15.0 | 3.0 |
| 8 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 3.0 | 240.0 | 2.0 | 0.0 | 0.0 | 0.0 | 55.0 | 18.0 | 0.0 | 13.0 | 1.0 |
| 9 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 7.0 | 49.0 | 1976.0 | 123.0 | 0.0 | 0.0 | 1.0 | 126.0 | 25.0 | 0.0 | 30.0 | 7.0 |
| 10 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 27.0 | 14.0 | 0.0 | 0.0 | 0.0 | 214.0 | 15.0 | 0.0 | 0.0 | 1.0 |
| 11 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 62.0 | 0.0 | 0.0 | 0.0 | 23.0 | 0.0 | 0.0 | 0.0 |
| 12 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 7.0 | 0.0 | 0.0 | 5.0 | 0.0 | 0.0 | 104.0 | 0.0 | 0.0 | 40.0 | 0.0 | 2.0 | 1.0 |
| 13 | 4.0 | 4.0 | 3.0 | 48.0 | 27.0 | 15.0 | 47.0 | 25.0 | 160.0 | 11.0 | 20.0 | 2.0 | 572.0 | 29.0 | 6.0 | 0.0 | 0.0 | 0.0 |
| 14 | 0.0 | 4.0 | 2.0 | 3.0 | 5.0 | 3.0 | 1.0 | 1.0 | 5.0 | 2.0 | 1.0 | 0.0 | 43.0 | 0.0 | 0.0 | 0.0 | 1.0 | 38.0 |
| 15 | 0.0 | 2.0 | 1.0 | 1.0 | 1.0 | 1.0 | 0.0 | 0.0 | 1.0 | 0.0 | 1.0 | 0.0 | 14.0 | 0.0 | 0.0 | 0.0 | 0.0 | 8.0 |
| 16 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 17 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 3.0 | 242.0 |
| 18 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
They will total the transition benefits and pick the routine with the greatest combined score.

Oh, and when you send back the actual ordering, please use this simple JSON shape so it's easy to read and check:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Here ""solution"" is just the final workout flow — a list of the exercises in the exact sequence you want, from the opening move to the closer. Think of it like filling in the blanks with the exercise names in order. This JSON is just the shape I expect, not the real solution itself.

Please make sure every identifier you use matches the instance input exactly — don't rename items 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”.","[[0.0, 0.0, 0.0, 3.0, 2.0, 0.0, 2.0, 0.0, 4.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0], [0.0, 1993.0, 134.0, 166.0, 19.0, 10.0, 17.0, 66.0, 132.0, 9.0, 4.0, 0.0, 216.0, 24.0, 395.0, 0.0, 36.0, 154.0], [38.0, 24.0, 38.0, 41.0, 17.0, 17.0, 13.0, 15.0, 49.0, 8.0, 2.0, 1.0, 100.0, 49.0, 3.0, 0.0, 19.0, 113.0], [0.0, 1.0, 0.0, 79.0, 1.0, 0.0, 1.0, 8.0, 18.0, 45.0, 0.0, 0.0, 0.0, 5.0, 0.0, 0.0, 6.0, 5.0], [1.0, 14.0, 9.0, 7.0, 443.0, 218.0, 2.0, 2.0, 16.0, 2.0, 1.0, 0.0, 13.0, 6.0, 157.0, 0.0, 3.0, 377.0], [0.0, 0.0, 0.0, 0.0, 0.0, 21.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 315.0, 0.0, 0.0, 37.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 129.0, 0.0, 126.0, 0.0, 0.0, 0.0, 0.0, 65.0, 26.0, 0.0, 15.0, 3.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 3.0, 240.0, 2.0, 0.0, 0.0, 0.0, 55.0, 18.0, 0.0, 13.0, 1.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 7.0, 49.0, 1976.0, 123.0, 0.0, 0.0, 1.0, 126.0, 25.0, 0.0, 30.0, 7.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 27.0, 14.0, 0.0, 0.0, 0.0, 214.0, 15.0, 0.0, 0.0, 1.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 62.0, 0.0, 0.0, 0.0, 23.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 1.0, 0.0, 7.0, 0.0, 0.0, 5.0, 0.0, 0.0, 104.0, 0.0, 0.0, 40.0, 0.0, 2.0, 1.0], [4.0, 4.0, 3.0, 48.0, 27.0, 15.0, 47.0, 25.0, 160.0, 11.0, 20.0, 2.0, 572.0, 29.0, 6.0, 0.0, 0.0, 0.0], [0.0, 4.0, 2.0, 3.0, 5.0, 3.0, 1.0, 1.0, 5.0, 2.0, 1.0, 0.0, 43.0, 0.0, 0.0, 0.0, 1.0, 38.0], [0.0, 2.0, 1.0, 1.0, 1.0, 1.0, 0.0, 0.0, 1.0, 0.0, 1.0, 0.0, 14.0, 0.0, 0.0, 0.0, 0.0, 8.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 3.0, 242.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]]","[1, 2, 12, 11, 0, 4, 3, 7, 6, 8, 9, 13, 16, 10, 5, 14, 17, 15]",4981.0,"{'problem_type': 'LOP', 'num_nodes': 18, 'nodes': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], 'pairs': [{'from_id': 1, 'to_id': 2, 'weight': 0.0}, {'from_id': 1, 'to_id': 3, 'weight': 0.0}, {'from_id': 1, 'to_id': 4, 'weight': 3.0}, {'from_id': 1, 'to_id': 5, 'weight': 2.0}, {'from_id': 1, 'to_id': 6, 'weight': 0.0}, {'from_id': 1, 'to_id': 7, 'weight': 2.0}, {'from_id': 1, 'to_id': 8, 'weight': 0.0}, {'from_id': 1, 'to_id': 9, 'weight': 4.0}, {'from_id': 1, 'to_id': 10, 'weight': 0.0}, {'from_id': 1, 'to_id': 11, 'weight': 0.0}, {'from_id': 1, 'to_id': 12, 'weight': 0.0}, {'from_id': 1, 'to_id': 13, 'weight': 0.0}, {'from_id': 1, 'to_id': 14, 'weight': 0.0}, {'from_id': 1, 'to_id': 15, 'weight': 0.0}, {'from_id': 1, 'to_id': 16, 'weight': 0.0}, {'from_id': 1, 'to_id': 17, 'weight': 1.0}, {'from_id': 1, 'to_id': 18, 'weight': 0.0}, {'from_id': 2, 'to_id': 1, 'weight': 0.0}, {'from_id': 2, 'to_id': 3, 'weight': 134.0}, {'from_id': 2, 'to_id': 4, 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'to_id': 13, 'weight': 0.0}, {'from_id': 16, 'to_id': 14, 'weight': 0.0}, {'from_id': 16, 'to_id': 15, 'weight': 0.0}, {'from_id': 16, 'to_id': 17, 'weight': 0.0}, {'from_id': 16, 'to_id': 18, 'weight': 0.0}, {'from_id': 17, 'to_id': 1, 'weight': 0.0}, {'from_id': 17, 'to_id': 2, 'weight': 0.0}, {'from_id': 17, 'to_id': 3, 'weight': 0.0}, {'from_id': 17, 'to_id': 4, 'weight': 0.0}, {'from_id': 17, 'to_id': 5, 'weight': 0.0}, {'from_id': 17, 'to_id': 6, 'weight': 0.0}, {'from_id': 17, 'to_id': 7, 'weight': 0.0}, {'from_id': 17, 'to_id': 8, 'weight': 0.0}, {'from_id': 17, 'to_id': 9, 'weight': 0.0}, {'from_id': 17, 'to_id': 10, 'weight': 0.0}, {'from_id': 17, 'to_id': 11, 'weight': 0.0}, {'from_id': 17, 'to_id': 12, 'weight': 0.0}, {'from_id': 17, 'to_id': 13, 'weight': 0.0}, {'from_id': 17, 'to_id': 14, 'weight': 0.0}, {'from_id': 17, 'to_id': 15, 'weight': 0.0}, {'from_id': 17, 'to_id': 16, 'weight': 0.0}, {'from_id': 17, 'to_id': 18, 'weight': 242.0}, {'from_id': 18, 'to_id': 1, 'weight': 0.0}, {'from_id': 18, 'to_id': 2, 'weight': 0.0}, {'from_id': 18, 'to_id': 3, 'weight': 0.0}, {'from_id': 18, 'to_id': 4, 'weight': 0.0}, {'from_id': 18, 'to_id': 5, 'weight': 0.0}, {'from_id': 18, 'to_id': 6, 'weight': 0.0}, {'from_id': 18, 'to_id': 7, 'weight': 0.0}, {'from_id': 18, 'to_id': 8, 'weight': 0.0}, {'from_id': 18, 'to_id': 9, 'weight': 0.0}, {'from_id': 18, 'to_id': 10, 'weight': 0.0}, {'from_id': 18, 'to_id': 11, 'weight': 0.0}, {'from_id': 18, 'to_id': 12, 'weight': 0.0}, {'from_id': 18, 'to_id': 13, 'weight': 0.0}, {'from_id': 18, 'to_id': 14, 'weight': 0.0}, {'from_id': 18, 'to_id': 15, 'weight': 0.0}, {'from_id': 18, 'to_id': 16, 'weight': 0.0}, {'from_id': 18, 'to_id': 17, 'weight': 0.0}]}","[2, 3, 13, 12, 1, 5, 4, 8, 7, 9, 10, 14, 17, 11, 6, 15, 18, 16]",40,nl,1
LOP,LOP,"On a busy morning the team had to arrange every product tile into a single carousel rotation — no duplicates, no missing items, every product exactly once. Between every two products someone recorded how many extra clicks you get if the first is placed before the second, so to judge an ordering you go through every pair, take the corresponding number when the earlier product comes before the later one, add them all together, and aim for the ordering with the highest total clicks. The concrete product details and pairwise values are shown below.

# total_products=22
# product_ids=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22

# Meaning: this is a directed pairwise_click_value matrix over items in product_ids (22 total). Entry at row preceding_product_id=i and column succeeding_product_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# pairwise_click_value
| preceding_product_id\succeeding_product_id | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 3984.0 | 241.0 | 4.0 | 27.0 | 494.0 | 13.0 | 5.0 | 11.0 | 4.0 | 0.0 | 7.0 | 13.0 | 6.0 | 0.0 | 0.0 | 14.0 | 134.0 | 100.0 | 112.0 | 37.0 | 0.0 | 194.0 |
| 2 | 10.0 | 552.0 | 24.0 | 23.0 | 240.0 | 13.0 | 0.0 | 10.0 | 1.0 | 0.0 | 4.0 | 12.0 | 6.0 | 0.0 | 0.0 | 6.0 | 102.0 | 73.0 | 39.0 | 12.0 | 0.0 | 127.0 |
| 3 | 17.0 | 9.0 | 93.0 | 2.0 | 276.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 4.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 9.0 | 26.0 | 48.0 | 2.0 | 0.0 | 7.0 |
| 4 | 73.0 | 30.0 | 0.0 | 195.0 | 10.0 | 1.0 | 8.0 | 0.0 | 0.0 | 0.0 | 7.0 | 1.0 | 1.0 | 0.0 | 0.0 | 3.0 | 25.0 | 162.0 | 50.0 | 9.0 | 0.0 | 115.0 |
| 5 | 132.0 | 23.0 | 36.0 | 3.0 | 4127.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 9.0 | 13.0 | 0.0 | 0.0 | 0.0 | 3.0 | 34.0 | 1194.0 | 476.0 | 25.0 | 0.0 | 227.0 |
| 6 | 51.0 | 2.0 | 0.0 | 2.0 | 26.0 | 85.0 | 26.0 | 0.0 | 0.0 | 0.0 | 5.0 | 12.0 | 0.0 | 0.0 | 0.0 | 2.0 | 38.0 | 106.0 | 102.0 | 9.0 | 0.0 | 12.0 |
| 7 | 67.0 | 5.0 | 0.0 | 19.0 | 97.0 | 16.0 | 376.0 | 0.0 | 0.0 | 0.0 | 40.0 | 13.0 | 0.0 | 0.0 | 0.0 | 4.0 | 37.0 | 130.0 | 138.0 | 12.0 | 0.0 | 18.0 |
| 8 | 44.0 | 33.0 | 0.0 | 6.0 | 20.0 | 9.0 | 34.0 | 11.0 | 16.0 | 0.0 | 29.0 | 7.0 | 3.0 | 0.0 | 0.0 | 6.0 | 23.0 | 39.0 | 33.0 | 2.0 | 0.0 | 25.0 |
| 9 | 85.0 | 49.0 | 0.0 | 3.0 | 83.0 | 10.0 | 29.0 | 44.0 | 280.0 | 0.0 | 66.0 | 32.0 | 2.0 | 0.0 | 0.0 | 7.0 | 56.0 | 115.0 | 36.0 | 9.0 | 0.0 | 62.0 |
| 10 | 16.0 | 3.0 | 0.0 | 8.0 | 5.0 | 21.0 | 19.0 | 0.0 | 1.0 | 3.0 | 23.0 | 34.0 | 0.0 | 0.0 | 0.0 | 3.0 | 46.0 | 113.0 | 21.0 | 9.0 | 0.0 | 19.0 |
| 11 | 45.0 | 7.0 | 0.0 | 12.0 | 39.0 | 6.0 | 920.0 | 10.0 | 4.0 | 1.0 | 144.0 | 49.0 | 0.0 | 0.0 | 0.0 | 65.0 | 53.0 | 173.0 | 99.0 | 21.0 | 0.0 | 104.0 |
| 12 | 92.0 | 8.0 | 0.0 | 1.0 | 21.0 | 7.0 | 78.0 | 1.0 | 1.0 | 0.0 | 69.0 | 92.0 | 6.0 | 0.0 | 3.0 | 7.0 | 67.0 | 134.0 | 44.0 | 30.0 | 0.0 | 138.0 |
| 13 | 50.0 | 3.0 | 0.0 | 25.0 | 36.0 | 0.0 | 10.0 | 1.0 | 0.0 | 1.0 | 75.0 | 54.0 | 19.0 | 0.0 | 0.0 | 15.0 | 263.0 | 327.0 | 38.0 | 51.0 | 0.0 | 342.0 |
| 14 | 27.0 | 1.0 | 0.0 | 0.0 | 6.0 | 0.0 | 18.0 | 0.0 | 0.0 | 0.0 | 42.0 | 0.0 | 0.0 | 92.0 | 16.0 | 47.0 | 57.0 | 82.0 | 24.0 | 6.0 | 0.0 | 69.0 |
| 15 | 20.0 | 3.0 | 1.0 | 0.0 | 18.0 | 3.0 | 60.0 | 0.0 | 3.0 | 0.0 | 39.0 | 17.0 | 0.0 | 2.0 | 441.0 | 252.0 | 141.0 | 119.0 | 41.0 | 8.0 | 0.0 | 69.0 |
| 16 | 229.0 | 23.0 | 18.0 | 21.0 | 195.0 | 7.0 | 122.0 | 34.0 | 315.0 | 2.0 | 511.0 | 188.0 | 4.0 | 619.0 | 483.0 | 4045.0 | 647.0 | 1287.0 | 385.0 | 76.0 | 0.0 | 488.0 |
| 17 | 664.0 | 92.0 | 14.0 | 161.0 | 402.0 | 406.0 | 255.0 | 36.0 | 201.0 | 2.0 | 500.0 | 368.0 | 14.0 | 678.0 | 216.0 | 2752.0 | 5359.0 | 1716.0 | 552.0 | 354.0 | 0.0 | 1182.0 |
| 18 | 177.0 | 10.0 | 0.0 | 1.0 | 245.0 | 2.0 | 25.0 | 0.0 | 0.0 | 0.0 | 94.0 | 2.0 | 16.0 | 0.0 | 0.0 | 11.0 | 295.0 | 976.0 | 2164.0 | 927.0 | 0.0 | 541.0 |
| 19 | 274.0 | 23.0 | 0.0 | 7.0 | 1009.0 | 24.0 | 18.0 | 3.0 | 8.0 | 0.0 | 34.0 | 36.0 | 57.0 | 1.0 | 1.0 | 19.0 | 344.0 | 250.0 | 3741.0 | 178.0 | 0.0 | 208.0 |
| 20 | 28.0 | 3.0 | 1.0 | 2.0 | 66.0 | 2.0 | 12.0 | 3.0 | 4.0 | 0.0 | 29.0 | 8.0 | 8.0 | 0.0 | 0.0 | 0.0 | 38.0 | 40.0 | 46.0 | 0.0 | 0.0 | 81.0 |
| 21 | 311.0 | 998.0 | 5.0 | 0.0 | 98.0 | 1.0 | 1.0 | 6.0 | 5.0 | 0.0 | 6.0 | 0.0 | 2.0 | 0.0 | 0.0 | 2.0 | 1285.0 | 20.0 | 19.0 | 10.0 | 0.0 | 1859.0 |
| 22 | 443.0 | 29.0 | 15.0 | 0.0 | 434.0 | 20.0 | 109.0 | 38.0 | 103.0 | 9.0 | 205.0 | 168.0 | 178.0 | 104.0 | 134.0 | 2387.0 | 2241.0 | 957.0 | 284.0 | 828.0 | 26.0 | 6617.0 |

Also, when you send back the final ordering, please use this simple JSON layout so it's easy to read and machine-friendly:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

The ""solution"" array should list the product tiles in the exact order you want them in the carousel, from first to last. Think of it as the single line on the form that says ""put these items in this order"" — nothing fancy needed.

This JSON is just a sketch of the shape I expect, not the actual answer — replace the placeholders with the real item identifiers from the instance when you reply.

All identifiers must be used exactly as they appear in the instance input — no renaming and no 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"".""","[[3984.0, 241.0, 4.0, 27.0, 494.0, 13.0, 5.0, 11.0, 4.0, 0.0, 7.0, 13.0, 6.0, 0.0, 0.0, 14.0, 134.0, 100.0, 112.0, 37.0, 0.0, 194.0], [10.0, 552.0, 24.0, 23.0, 240.0, 13.0, 0.0, 10.0, 1.0, 0.0, 4.0, 12.0, 6.0, 0.0, 0.0, 6.0, 102.0, 73.0, 39.0, 12.0, 0.0, 127.0], [17.0, 9.0, 93.0, 2.0, 276.0, 1.0, 0.0, 0.0, 0.0, 0.0, 4.0, 1.0, 0.0, 0.0, 0.0, 0.0, 9.0, 26.0, 48.0, 2.0, 0.0, 7.0], [73.0, 30.0, 0.0, 195.0, 10.0, 1.0, 8.0, 0.0, 0.0, 0.0, 7.0, 1.0, 1.0, 0.0, 0.0, 3.0, 25.0, 162.0, 50.0, 9.0, 0.0, 115.0], [132.0, 23.0, 36.0, 3.0, 4127.0, 0.0, 0.0, 0.0, 0.0, 0.0, 9.0, 13.0, 0.0, 0.0, 0.0, 3.0, 34.0, 1194.0, 476.0, 25.0, 0.0, 227.0], [51.0, 2.0, 0.0, 2.0, 26.0, 85.0, 26.0, 0.0, 0.0, 0.0, 5.0, 12.0, 0.0, 0.0, 0.0, 2.0, 38.0, 106.0, 102.0, 9.0, 0.0, 12.0], [67.0, 5.0, 0.0, 19.0, 97.0, 16.0, 376.0, 0.0, 0.0, 0.0, 40.0, 13.0, 0.0, 0.0, 0.0, 4.0, 37.0, 130.0, 138.0, 12.0, 0.0, 18.0], [44.0, 33.0, 0.0, 6.0, 20.0, 9.0, 34.0, 11.0, 16.0, 0.0, 29.0, 7.0, 3.0, 0.0, 0.0, 6.0, 23.0, 39.0, 33.0, 2.0, 0.0, 25.0], [85.0, 49.0, 0.0, 3.0, 83.0, 10.0, 29.0, 44.0, 280.0, 0.0, 66.0, 32.0, 2.0, 0.0, 0.0, 7.0, 56.0, 115.0, 36.0, 9.0, 0.0, 62.0], [16.0, 3.0, 0.0, 8.0, 5.0, 21.0, 19.0, 0.0, 1.0, 3.0, 23.0, 34.0, 0.0, 0.0, 0.0, 3.0, 46.0, 113.0, 21.0, 9.0, 0.0, 19.0], [45.0, 7.0, 0.0, 12.0, 39.0, 6.0, 920.0, 10.0, 4.0, 1.0, 144.0, 49.0, 0.0, 0.0, 0.0, 65.0, 53.0, 173.0, 99.0, 21.0, 0.0, 104.0], [92.0, 8.0, 0.0, 1.0, 21.0, 7.0, 78.0, 1.0, 1.0, 0.0, 69.0, 92.0, 6.0, 0.0, 3.0, 7.0, 67.0, 134.0, 44.0, 30.0, 0.0, 138.0], [50.0, 3.0, 0.0, 25.0, 36.0, 0.0, 10.0, 1.0, 0.0, 1.0, 75.0, 54.0, 19.0, 0.0, 0.0, 15.0, 263.0, 327.0, 38.0, 51.0, 0.0, 342.0], [27.0, 1.0, 0.0, 0.0, 6.0, 0.0, 18.0, 0.0, 0.0, 0.0, 42.0, 0.0, 0.0, 92.0, 16.0, 47.0, 57.0, 82.0, 24.0, 6.0, 0.0, 69.0], [20.0, 3.0, 1.0, 0.0, 18.0, 3.0, 60.0, 0.0, 3.0, 0.0, 39.0, 17.0, 0.0, 2.0, 441.0, 252.0, 141.0, 119.0, 41.0, 8.0, 0.0, 69.0], [229.0, 23.0, 18.0, 21.0, 195.0, 7.0, 122.0, 34.0, 315.0, 2.0, 511.0, 188.0, 4.0, 619.0, 483.0, 4045.0, 647.0, 1287.0, 385.0, 76.0, 0.0, 488.0], [664.0, 92.0, 14.0, 161.0, 402.0, 406.0, 255.0, 36.0, 201.0, 2.0, 500.0, 368.0, 14.0, 678.0, 216.0, 2752.0, 5359.0, 1716.0, 552.0, 354.0, 0.0, 1182.0], [177.0, 10.0, 0.0, 1.0, 245.0, 2.0, 25.0, 0.0, 0.0, 0.0, 94.0, 2.0, 16.0, 0.0, 0.0, 11.0, 295.0, 976.0, 2164.0, 927.0, 0.0, 541.0], [274.0, 23.0, 0.0, 7.0, 1009.0, 24.0, 18.0, 3.0, 8.0, 0.0, 34.0, 36.0, 57.0, 1.0, 1.0, 19.0, 344.0, 250.0, 3741.0, 178.0, 0.0, 208.0], [28.0, 3.0, 1.0, 2.0, 66.0, 2.0, 12.0, 3.0, 4.0, 0.0, 29.0, 8.0, 8.0, 0.0, 0.0, 0.0, 38.0, 40.0, 46.0, 0.0, 0.0, 81.0], [311.0, 998.0, 5.0, 0.0, 98.0, 1.0, 1.0, 6.0, 5.0, 0.0, 6.0, 0.0, 2.0, 0.0, 0.0, 2.0, 1285.0, 20.0, 19.0, 10.0, 0.0, 1859.0], [443.0, 29.0, 15.0, 0.0, 434.0, 20.0, 109.0, 38.0, 103.0, 9.0, 205.0, 168.0, 178.0, 104.0, 134.0, 2387.0, 2241.0, 957.0, 284.0, 828.0, 26.0, 6617.0]]","[20, 12, 9, 21, 16, 15, 13, 14, 8, 7, 2, 5, 11, 10, 6, 3, 0, 1, 4, 17, 18, 19]",39644.0,"{'problem_type': 'LOP', 'num_nodes': 22, 'nodes': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], 'pairs': [{'from_id': 1, 'to_id': 2, 'weight': 241.0}, {'from_id': 1, 'to_id': 3, 'weight': 4.0}, {'from_id': 1, 'to_id': 4, 'weight': 27.0}, {'from_id': 1, 'to_id': 5, 'weight': 494.0}, {'from_id': 1, 'to_id': 6, 'weight': 13.0}, {'from_id': 1, 'to_id': 7, 'weight': 5.0}, {'from_id': 1, 'to_id': 8, 'weight': 11.0}, {'from_id': 1, 'to_id': 9, 'weight': 4.0}, {'from_id': 1, 'to_id': 10, 'weight': 0.0}, {'from_id': 1, 'to_id': 11, 'weight': 7.0}, {'from_id': 1, 'to_id': 12, 'weight': 13.0}, {'from_id': 1, 'to_id': 13, 'weight': 6.0}, {'from_id': 1, 'to_id': 14, 'weight': 0.0}, {'from_id': 1, 'to_id': 15, 'weight': 0.0}, {'from_id': 1, 'to_id': 16, 'weight': 14.0}, {'from_id': 1, 'to_id': 17, 'weight': 134.0}, {'from_id': 1, 'to_id': 18, 'weight': 100.0}, {'from_id': 1, 'to_id': 19, 'weight': 112.0}, {'from_id': 1, 'to_id': 20, 'weight': 37.0}, {'from_id': 1, 'to_id': 21, 'weight': 0.0}, {'from_id': 1, 'to_id': 22, 'weight': 194.0}, {'from_id': 2, 'to_id': 1, 'weight': 10.0}, {'from_id': 2, 'to_id': 3, 'weight': 24.0}, {'from_id': 2, 'to_id': 4, 'weight': 23.0}, {'from_id': 2, 'to_id': 5, 'weight': 240.0}, {'from_id': 2, 'to_id': 6, 'weight': 13.0}, {'from_id': 2, 'to_id': 7, 'weight': 0.0}, {'from_id': 2, 'to_id': 8, 'weight': 10.0}, {'from_id': 2, 'to_id': 9, 'weight': 1.0}, {'from_id': 2, 'to_id': 10, 'weight': 0.0}, {'from_id': 2, 'to_id': 11, 'weight': 4.0}, {'from_id': 2, 'to_id': 12, 'weight': 12.0}, {'from_id': 2, 'to_id': 13, 'weight': 6.0}, {'from_id': 2, 'to_id': 14, 'weight': 0.0}, {'from_id': 2, 'to_id': 15, 'weight': 0.0}, {'from_id': 2, 'to_id': 16, 'weight': 6.0}, {'from_id': 2, 'to_id': 17, 'weight': 102.0}, {'from_id': 2, 'to_id': 18, 'weight': 73.0}, {'from_id': 2, 'to_id': 19, 'weight': 39.0}, {'from_id': 2, 'to_id': 20, 'weight': 12.0}, {'from_id': 2, 'to_id': 21, 'weight': 0.0}, {'from_id': 2, 'to_id': 22, 'weight': 127.0}, {'from_id': 3, 'to_id': 1, 'weight': 17.0}, {'from_id': 3, 'to_id': 2, 'weight': 9.0}, {'from_id': 3, 'to_id': 4, 'weight': 2.0}, {'from_id': 3, 'to_id': 5, 'weight': 276.0}, {'from_id': 3, 'to_id': 6, 'weight': 1.0}, {'from_id': 3, 'to_id': 7, 'weight': 0.0}, {'from_id': 3, 'to_id': 8, 'weight': 0.0}, {'from_id': 3, 'to_id': 9, 'weight': 0.0}, {'from_id': 3, 'to_id': 10, 'weight': 0.0}, {'from_id': 3, 'to_id': 11, 'weight': 4.0}, {'from_id': 3, 'to_id': 12, 'weight': 1.0}, {'from_id': 3, 'to_id': 13, 'weight': 0.0}, {'from_id': 3, 'to_id': 14, 'weight': 0.0}, {'from_id': 3, 'to_id': 15, 'weight': 0.0}, {'from_id': 3, 'to_id': 16, 'weight': 0.0}, {'from_id': 3, 'to_id': 17, 'weight': 9.0}, {'from_id': 3, 'to_id': 18, 'weight': 26.0}, {'from_id': 3, 'to_id': 19, 'weight': 48.0}, {'from_id': 3, 'to_id': 20, 'weight': 2.0}, {'from_id': 3, 'to_id': 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{'from_id': 21, 'to_id': 19, 'weight': 19.0}, {'from_id': 21, 'to_id': 20, 'weight': 10.0}, {'from_id': 21, 'to_id': 22, 'weight': 1859.0}, {'from_id': 22, 'to_id': 1, 'weight': 443.0}, {'from_id': 22, 'to_id': 2, 'weight': 29.0}, {'from_id': 22, 'to_id': 3, 'weight': 15.0}, {'from_id': 22, 'to_id': 4, 'weight': 0.0}, {'from_id': 22, 'to_id': 5, 'weight': 434.0}, {'from_id': 22, 'to_id': 6, 'weight': 20.0}, {'from_id': 22, 'to_id': 7, 'weight': 109.0}, {'from_id': 22, 'to_id': 8, 'weight': 38.0}, {'from_id': 22, 'to_id': 9, 'weight': 103.0}, {'from_id': 22, 'to_id': 10, 'weight': 9.0}, {'from_id': 22, 'to_id': 11, 'weight': 205.0}, {'from_id': 22, 'to_id': 12, 'weight': 168.0}, {'from_id': 22, 'to_id': 13, 'weight': 178.0}, {'from_id': 22, 'to_id': 14, 'weight': 104.0}, {'from_id': 22, 'to_id': 15, 'weight': 134.0}, {'from_id': 22, 'to_id': 16, 'weight': 2387.0}, {'from_id': 22, 'to_id': 17, 'weight': 2241.0}, {'from_id': 22, 'to_id': 18, 'weight': 957.0}, {'from_id': 22, 'to_id': 19, 'weight': 284.0}, {'from_id': 22, 'to_id': 20, 'weight': 828.0}, {'from_id': 22, 'to_id': 21, 'weight': 26.0}]}","[21, 13, 10, 22, 17, 16, 14, 15, 9, 8, 3, 6, 12, 11, 7, 4, 1, 2, 5, 18, 19, 20]",41,csv,1
LOP,LOP,"There’s a small team planning a how-to guide, and they’re deciding the sequence of actions. For each pair of actions, there’s a number that measures how much better learners do if that specific action is taught before the other; the aim is to pick an order that makes the sum of all those pairwise numbers as large as possible. The plan must use every action one time only — nothing left out or repeated — and the concrete steps and their benefit numbers are shown below.

{
  ""total_steps"": 23,
  ""step_ids"": [
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14,
    15,
    16,
    17,
    18,
    19,
    20,
    21,
    22,
    23
  ]
}

# Meaning: this is a directed learning_benefit matrix over items in step_ids (23 total). Entry at row preceding_step_id=i and column following_step_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# learning_benefit 
| preceding_step_id\following_step_id | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 13934.0 | 4.0 | 2.0 | 0.0 | 7.0 | 548.0 | 2.0 | 9.0 | 1.0 | 7934.0 | 11253.0 | 1366.0 | 2632.0 | 49.0 | 500.0 | 43.0 | 192.0 | 1844.0 | 14.0 | 0.0 | 814.0 | 159.0 | 60.0 |
| 2 | 2.0 | 21.0 | 0.0 | 0.0 | 2843.0 | 372.0 | 30.0 | 11.0 | 1.0 | 1.0 | 14.0 | 4.0 | 8.0 | 1.0 | 2.0 | 0.0 | 28.0 | 10.0 | 0.0 | 1.0 | 62.0 | 6.0 | 4.0 |
| 3 | 1233.0 | 54.0 | 8452.0 | 0.0 | 469.0 | 2268.0 | 320.0 | 253.0 | 34.0 | 82.0 | 353.0 | 178.0 | 305.0 | 51.0 | 179.0 | 17.0 | 1178.0 | 228.0 | 799.0 | 15.0 | 1053.0 | 98.0 | 44.0 |
| 4 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 5 | 17.0 | 16.0 | 222.0 | 0.0 | 51602.0 | 1676.0 | 5316.0 | 4903.0 | 1300.0 | 0.0 | 0.0 | 0.0 | 27.0 | 19.0 | 212.0 | 465.0 | 2880.0 | 0.0 | 28.0 | 0.0 | 132.0 | 0.0 | 1.0 |
| 6 | 1607.0 | 10.0 | 647.0 | 0.0 | 882.0 | 18959.0 | 539.0 | 992.0 | 163.0 | 44.0 | 281.0 | 144.0 | 4112.0 | 116.0 | 4459.0 | 80.0 | 332.0 | 157.0 | 21.0 | 80.0 | 3747.0 | 820.0 | 40.0 |
| 7 | 749.0 | 9.0 | 265.0 | 0.0 | 261.0 | 217.0 | 9394.0 | 681.0 | 678.0 | 9.0 | 58.0 | 26.0 | 234.0 | 29.0 | 157.0 | 4.0 | 680.0 | 24.0 | 11.0 | 0.0 | 811.0 | 2.0 | 1.0 |
| 8 | 88.0 | 9.0 | 75.0 | 0.0 | 312.0 | 177.0 | 2715.0 | 10804.0 | 266.0 | 0.0 | 9.0 | 4.0 | 6.0 | 5.0 | 16.0 | 25.0 | 3079.0 | 67.0 | 36.0 | 16.0 | 633.0 | 43.0 | 19.0 |
| 9 | 44.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 878.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 9.0 | 0.0 | 633.0 | 0.0 | 1334.0 | 0.0 | 39.0 |
| 10 | 837.0 | 0.0 | 0.0 | 0.0 | 0.0 | 72.0 | 0.0 | 0.0 | 0.0 | 964.0 | 998.0 | 5.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 632.0 | 93.0 | 168.0 | 167.0 | 155.0 | 118.0 |
| 11 | 5081.0 | 0.0 | 15.0 | 0.0 | 9.0 | 714.0 | 12.0 | 16.0 | 0.0 | 25.0 | 11779.0 | 893.0 | 5.0 | 5.0 | 43.0 | 0.0 | 13.0 | 2858.0 | 313.0 | 46.0 | 787.0 | 339.0 | 171.0 |
| 12 | 93.0 | 4.0 | 20.0 | 0.0 | 45.0 | 489.0 | 27.0 | 40.0 | 0.0 | 6.0 | 276.0 | 2056.0 | 7.0 | 6.0 | 4.0 | 0.0 | 19.0 | 4103.0 | 161.0 | 0.0 | 58.0 | 10.0 | 650.0 |
| 13 | 127.0 | 5.0 | 1.0 | 0.0 | 17.0 | 183.0 | 57.0 | 125.0 | 22.0 | 4.0 | 38.0 | 34.0 | 14567.0 | 310.0 | 1063.0 | 27.0 | 68.0 | 159.0 | 56.0 | 85.0 | 1304.0 | 218.0 | 111.0 |
| 14 | 58.0 | 3.0 | 0.0 | 0.0 | 8.0 | 11.0 | 39.0 | 5.0 | 4.0 | 0.0 | 2.0 | 4.0 | 111.0 | 994.0 | 9.0 | 6.0 | 10.0 | 0.0 | 6.0 | 0.0 | 443.0 | 27.0 | 6.0 |
| 15 | 271.0 | 3.0 | 188.0 | 0.0 | 51.0 | 1059.0 | 1295.0 | 1663.0 | 104.0 | 65.0 | 355.0 | 101.0 | 265.0 | 501.0 | 1686.0 | 65.0 | 1905.0 | 44.0 | 25.0 | 4.0 | 773.0 | 148.0 | 13.0 |
| 16 | 4.0 | 3.0 | 0.0 | 0.0 | 0.0 | 0.0 | 3.0 | 17.0 | 0.0 | 0.0 | 0.0 | 0.0 | 16.0 | 1.0 | 0.0 | 335.0 | 10.0 | 8.0 | 8.0 | 0.0 | 73.0 | 4.0 | 4.0 |
| 17 | 283.0 | 7.0 | 64.0 | 0.0 | 22.0 | 51.0 | 32.0 | 43.0 | 17.0 | 0.0 | 25.0 | 28.0 | 9.0 | 3.0 | 10.0 | 0.0 | 390.0 | 10.0 | 10.0 | 0.0 | 864.0 | 39.0 | 0.0 |
| 18 | 63.0 | 3.0 | 210.0 | 0.0 | 312.0 | 76.0 | 373.0 | 531.0 | 18.0 | 10.0 | 146.0 | 134.0 | 114.0 | 33.0 | 106.0 | 14.0 | 85.0 | 60.0 | 62.0 | 63.0 | 909.0 | 7.0 | 86.0 |
| 19 | 82.0 | 59.0 | 206.0 | 0.0 | 104.0 | 147.0 | 132.0 | 105.0 | 1.0 | 14.0 | 71.0 | 18.0 | 79.0 | 16.0 | 46.0 | 10.0 | 108.0 | 14.0 | 1869.0 | 0.0 | 218.0 | 14.0 | 5.0 |
| 20 | 14.0 | 3.0 | 8.0 | 0.0 | 24.0 | 43.0 | 38.0 | 39.0 | 1.0 | 2.0 | 17.0 | 8.0 | 11.0 | 4.0 | 9.0 | 1.0 | 30.0 | 9.0 | 4.0 | 0.0 | 0.0 | 4.0 | 1.0 |
| 21 | 66.0 | 3.0 | 35.0 | 0.0 | 113.0 | 132.0 | 132.0 | 108.0 | 28.0 | 22.0 | 81.0 | 36.0 | 66.0 | 15.0 | 32.0 | 6.0 | 138.0 | 38.0 | 27.0 | 4.0 | 271.0 | 37.0 | 14.0 |
| 22 | 6.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 6965.0 | 0.0 | 0.0 |
| 23 | 16.0 | 0.0 | 0.0 | 0.0 | 0.0 | 7.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1729.0 | 8.0 | 0.0 |

Also, when you send the final sequence back, it helps to use a tiny JSON shape so it's clear and machine-friendly. Something like this will do:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Here ""solution"" is just the ordered list of actions from first to last — put each action's identifier in the array in the order you want them taught. This is only a sketch of the shape I expect, not the actual ordering.

Please use the exact identifiers from the instance input — don't 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”.""","[[13934.0, 4.0, 2.0, 0.0, 7.0, 548.0, 2.0, 9.0, 1.0, 7934.0, 11253.0, 1366.0, 2632.0, 49.0, 500.0, 43.0, 192.0, 1844.0, 14.0, 0.0, 814.0, 159.0, 60.0], [2.0, 21.0, 0.0, 0.0, 2843.0, 372.0, 30.0, 11.0, 1.0, 1.0, 14.0, 4.0, 8.0, 1.0, 2.0, 0.0, 28.0, 10.0, 0.0, 1.0, 62.0, 6.0, 4.0], [1233.0, 54.0, 8452.0, 0.0, 469.0, 2268.0, 320.0, 253.0, 34.0, 82.0, 353.0, 178.0, 305.0, 51.0, 179.0, 17.0, 1178.0, 228.0, 799.0, 15.0, 1053.0, 98.0, 44.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [17.0, 16.0, 222.0, 0.0, 51602.0, 1676.0, 5316.0, 4903.0, 1300.0, 0.0, 0.0, 0.0, 27.0, 19.0, 212.0, 465.0, 2880.0, 0.0, 28.0, 0.0, 132.0, 0.0, 1.0], [1607.0, 10.0, 647.0, 0.0, 882.0, 18959.0, 539.0, 992.0, 163.0, 44.0, 281.0, 144.0, 4112.0, 116.0, 4459.0, 80.0, 332.0, 157.0, 21.0, 80.0, 3747.0, 820.0, 40.0], [749.0, 9.0, 265.0, 0.0, 261.0, 217.0, 9394.0, 681.0, 678.0, 9.0, 58.0, 26.0, 234.0, 29.0, 157.0, 4.0, 680.0, 24.0, 11.0, 0.0, 811.0, 2.0, 1.0], [88.0, 9.0, 75.0, 0.0, 312.0, 177.0, 2715.0, 10804.0, 266.0, 0.0, 9.0, 4.0, 6.0, 5.0, 16.0, 25.0, 3079.0, 67.0, 36.0, 16.0, 633.0, 43.0, 19.0], [44.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 878.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 9.0, 0.0, 633.0, 0.0, 1334.0, 0.0, 39.0], [837.0, 0.0, 0.0, 0.0, 0.0, 72.0, 0.0, 0.0, 0.0, 964.0, 998.0, 5.0, 0.0, 0.0, 0.0, 0.0, 0.0, 632.0, 93.0, 168.0, 167.0, 155.0, 118.0], [5081.0, 0.0, 15.0, 0.0, 9.0, 714.0, 12.0, 16.0, 0.0, 25.0, 11779.0, 893.0, 5.0, 5.0, 43.0, 0.0, 13.0, 2858.0, 313.0, 46.0, 787.0, 339.0, 171.0], [93.0, 4.0, 20.0, 0.0, 45.0, 489.0, 27.0, 40.0, 0.0, 6.0, 276.0, 2056.0, 7.0, 6.0, 4.0, 0.0, 19.0, 4103.0, 161.0, 0.0, 58.0, 10.0, 650.0], [127.0, 5.0, 1.0, 0.0, 17.0, 183.0, 57.0, 125.0, 22.0, 4.0, 38.0, 34.0, 14567.0, 310.0, 1063.0, 27.0, 68.0, 159.0, 56.0, 85.0, 1304.0, 218.0, 111.0], [58.0, 3.0, 0.0, 0.0, 8.0, 11.0, 39.0, 5.0, 4.0, 0.0, 2.0, 4.0, 111.0, 994.0, 9.0, 6.0, 10.0, 0.0, 6.0, 0.0, 443.0, 27.0, 6.0], [271.0, 3.0, 188.0, 0.0, 51.0, 1059.0, 1295.0, 1663.0, 104.0, 65.0, 355.0, 101.0, 265.0, 501.0, 1686.0, 65.0, 1905.0, 44.0, 25.0, 4.0, 773.0, 148.0, 13.0], [4.0, 3.0, 0.0, 0.0, 0.0, 0.0, 3.0, 17.0, 0.0, 0.0, 0.0, 0.0, 16.0, 1.0, 0.0, 335.0, 10.0, 8.0, 8.0, 0.0, 73.0, 4.0, 4.0], [283.0, 7.0, 64.0, 0.0, 22.0, 51.0, 32.0, 43.0, 17.0, 0.0, 25.0, 28.0, 9.0, 3.0, 10.0, 0.0, 390.0, 10.0, 10.0, 0.0, 864.0, 39.0, 0.0], [63.0, 3.0, 210.0, 0.0, 312.0, 76.0, 373.0, 531.0, 18.0, 10.0, 146.0, 134.0, 114.0, 33.0, 106.0, 14.0, 85.0, 60.0, 62.0, 63.0, 909.0, 7.0, 86.0], [82.0, 59.0, 206.0, 0.0, 104.0, 147.0, 132.0, 105.0, 1.0, 14.0, 71.0, 18.0, 79.0, 16.0, 46.0, 10.0, 108.0, 14.0, 1869.0, 0.0, 218.0, 14.0, 5.0], [14.0, 3.0, 8.0, 0.0, 24.0, 43.0, 38.0, 39.0, 1.0, 2.0, 17.0, 8.0, 11.0, 4.0, 9.0, 1.0, 30.0, 9.0, 4.0, 0.0, 0.0, 4.0, 1.0], [66.0, 3.0, 35.0, 0.0, 113.0, 132.0, 132.0, 108.0, 28.0, 22.0, 81.0, 36.0, 66.0, 15.0, 32.0, 6.0, 138.0, 38.0, 27.0, 4.0, 271.0, 37.0, 14.0], [6.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 6965.0, 0.0, 0.0], [16.0, 0.0, 0.0, 0.0, 0.0, 7.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1729.0, 8.0, 0.0]]","[2, 1, 4, 5, 0, 12, 14, 9, 10, 11, 17, 19, 13, 7, 6, 8, 18, 15, 22, 16, 21, 20, 3]",121716.0,"{'problem_type': 'LOP', 'num_nodes': 23, 'nodes': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], 'pairs': [{'from_id': 1, 'to_id': 2, 'weight': 4.0}, {'from_id': 1, 'to_id': 3, 'weight': 2.0}, {'from_id': 1, 'to_id': 4, 'weight': 0.0}, {'from_id': 1, 'to_id': 5, 'weight': 7.0}, {'from_id': 1, 'to_id': 6, 'weight': 548.0}, {'from_id': 1, 'to_id': 7, 'weight': 2.0}, {'from_id': 1, 'to_id': 8, 'weight': 9.0}, {'from_id': 1, 'to_id': 9, 'weight': 1.0}, {'from_id': 1, 'to_id': 10, 'weight': 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'weight': 17.0}, {'from_id': 20, 'to_id': 12, 'weight': 8.0}, {'from_id': 20, 'to_id': 13, 'weight': 11.0}, {'from_id': 20, 'to_id': 14, 'weight': 4.0}, {'from_id': 20, 'to_id': 15, 'weight': 9.0}, {'from_id': 20, 'to_id': 16, 'weight': 1.0}, {'from_id': 20, 'to_id': 17, 'weight': 30.0}, {'from_id': 20, 'to_id': 18, 'weight': 9.0}, {'from_id': 20, 'to_id': 19, 'weight': 4.0}, {'from_id': 20, 'to_id': 21, 'weight': 0.0}, {'from_id': 20, 'to_id': 22, 'weight': 4.0}, {'from_id': 20, 'to_id': 23, 'weight': 1.0}, {'from_id': 21, 'to_id': 1, 'weight': 66.0}, {'from_id': 21, 'to_id': 2, 'weight': 3.0}, {'from_id': 21, 'to_id': 3, 'weight': 35.0}, {'from_id': 21, 'to_id': 4, 'weight': 0.0}, {'from_id': 21, 'to_id': 5, 'weight': 113.0}, {'from_id': 21, 'to_id': 6, 'weight': 132.0}, {'from_id': 21, 'to_id': 7, 'weight': 132.0}, {'from_id': 21, 'to_id': 8, 'weight': 108.0}, {'from_id': 21, 'to_id': 9, 'weight': 28.0}, {'from_id': 21, 'to_id': 10, 'weight': 22.0}, {'from_id': 21, 'to_id': 11, 'weight': 81.0}, {'from_id': 21, 'to_id': 12, 'weight': 36.0}, {'from_id': 21, 'to_id': 13, 'weight': 66.0}, {'from_id': 21, 'to_id': 14, 'weight': 15.0}, {'from_id': 21, 'to_id': 15, 'weight': 32.0}, {'from_id': 21, 'to_id': 16, 'weight': 6.0}, {'from_id': 21, 'to_id': 17, 'weight': 138.0}, {'from_id': 21, 'to_id': 18, 'weight': 38.0}, {'from_id': 21, 'to_id': 19, 'weight': 27.0}, {'from_id': 21, 'to_id': 20, 'weight': 4.0}, {'from_id': 21, 'to_id': 22, 'weight': 37.0}, {'from_id': 21, 'to_id': 23, 'weight': 14.0}, {'from_id': 22, 'to_id': 1, 'weight': 6.0}, {'from_id': 22, 'to_id': 2, 'weight': 0.0}, {'from_id': 22, 'to_id': 3, 'weight': 0.0}, {'from_id': 22, 'to_id': 4, 'weight': 0.0}, {'from_id': 22, 'to_id': 5, 'weight': 0.0}, {'from_id': 22, 'to_id': 6, 'weight': 0.0}, {'from_id': 22, 'to_id': 7, 'weight': 0.0}, {'from_id': 22, 'to_id': 8, 'weight': 0.0}, {'from_id': 22, 'to_id': 9, 'weight': 0.0}, {'from_id': 22, 'to_id': 10, 'weight': 0.0}, {'from_id': 22, 'to_id': 11, 'weight': 0.0}, {'from_id': 22, 'to_id': 12, 'weight': 0.0}, {'from_id': 22, 'to_id': 13, 'weight': 0.0}, {'from_id': 22, 'to_id': 14, 'weight': 0.0}, {'from_id': 22, 'to_id': 15, 'weight': 0.0}, {'from_id': 22, 'to_id': 16, 'weight': 0.0}, {'from_id': 22, 'to_id': 17, 'weight': 0.0}, {'from_id': 22, 'to_id': 18, 'weight': 0.0}, {'from_id': 22, 'to_id': 19, 'weight': 0.0}, {'from_id': 22, 'to_id': 20, 'weight': 0.0}, {'from_id': 22, 'to_id': 21, 'weight': 6965.0}, {'from_id': 22, 'to_id': 23, 'weight': 0.0}, {'from_id': 23, 'to_id': 1, 'weight': 16.0}, {'from_id': 23, 'to_id': 2, 'weight': 0.0}, {'from_id': 23, 'to_id': 3, 'weight': 0.0}, {'from_id': 23, 'to_id': 4, 'weight': 0.0}, {'from_id': 23, 'to_id': 5, 'weight': 0.0}, {'from_id': 23, 'to_id': 6, 'weight': 7.0}, {'from_id': 23, 'to_id': 7, 'weight': 0.0}, {'from_id': 23, 'to_id': 8, 'weight': 0.0}, {'from_id': 23, 'to_id': 9, 'weight': 0.0}, {'from_id': 23, 'to_id': 10, 'weight': 0.0}, {'from_id': 23, 'to_id': 11, 'weight': 0.0}, {'from_id': 23, 'to_id': 12, 'weight': 0.0}, {'from_id': 23, 'to_id': 13, 'weight': 0.0}, {'from_id': 23, 'to_id': 14, 'weight': 0.0}, {'from_id': 23, 'to_id': 15, 'weight': 0.0}, {'from_id': 23, 'to_id': 16, 'weight': 0.0}, {'from_id': 23, 'to_id': 17, 'weight': 0.0}, {'from_id': 23, 'to_id': 18, 'weight': 0.0}, {'from_id': 23, 'to_id': 19, 'weight': 0.0}, {'from_id': 23, 'to_id': 20, 'weight': 0.0}, {'from_id': 23, 'to_id': 21, 'weight': 1729.0}, {'from_id': 23, 'to_id': 22, 'weight': 8.0}]}","[3, 2, 5, 6, 1, 13, 15, 10, 11, 12, 18, 20, 14, 8, 7, 9, 19, 16, 23, 17, 22, 21, 4]",42,json,1
LOP,LOP,"I’m putting together a tasting flight and need to line up every bottle into one continuous progression. The trick is that for every pair of different wines there’s a little note saying how much better the palate gets if one is tasted before the other. The job is to pick an order that uses every wine exactly once—no skips, no duplicates—and that makes the whole flight as enjoyable as possible. That enjoyment is measured by adding up those pairwise improvement numbers for every time a wine appears before another, and the plan is to choose the order with the biggest total. The exact pairwise numbers and wines are shown below.

{
  ""total_wines"": 19,
  ""wine_identifiers"": [
    0,
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14,
    15,
    16,
    17,
    18
  ]
}

# Meaning: this is a directed palate_improvement_value matrix over items in wine_identifiers (19 total). Entry at row earlier_wine_id=i and column later_wine_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# palate_improvement_value 
| earlier_wine_id\later_wine_id | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 215502.0 | 6.0 | 660.0 | 4444.0 | 4.0 | 217704.0 | 18526.0 | 7569.0 | 8866.0 | 20158.0 | 667.0 | 218.0 | 265.0 | 47.0 | 74.0 | 842.0 | 375.0 | 268.0 | 5262.0 |
| 1 | 48957.0 | 220610.0 | 16314.0 | 20692.0 | 980.0 | 4801.0 | 1023.0 | 30.0 | 967.0 | 452.0 | 63.0 | 573.0 | 32766.0 | 174.0 | 78.0 | 321.0 | 3422.0 | 423.0 | 656.0 |
| 2 | 1482.0 | 75.0 | 25994.0 | 17361.0 | 1468.0 | 1600.0 | 5358.0 | 0.0 | 208.0 | 1678.0 | 668.0 | 182.0 | 29.0 | 25.0 | 5.0 | 92.0 | 494.0 | 11.0 | 102.0 |
| 3 | 50351.0 | 2566.0 | 5671.0 | 139308.0 | 2113.0 | 6149.0 | 3114.0 | 179.0 | 30170.0 | 3723.0 | 5758.0 | 1477.0 | 692.0 | 128.0 | 111.0 | 6117.0 | 1328.0 | 157.0 | 12502.0 |
| 4 | 6.0 | 0.0 | 0.0 | 0.0 | 44392.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 4217.0 | 0.0 | 0.0 |
| 5 | 100280.0 | 12.0 | 258.0 | 5306.0 | 4.0 | 90133.0 | 6057.0 | 0.0 | 307.0 | 28.0 | 77.0 | 10.0 | 563.0 | 3.0 | 0.0 | 28.0 | 22.0 | 173.0 | 1782.0 |
| 6 | 52.0 | 4.0 | 113.0 | 2447.0 | 6.0 | 237.0 | 2554.0 | 3.0 | 19.0 | 78.0 | 7.0 | 31.0 | 862.0 | 4.0 | 1.0 | 5.0 | 5.0 | 23.0 | 313.0 |
| 7 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 66.0 | 0.0 | 0.0 | 0.0 | 0.0 | 80.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 8 | 3691.0 | 24.0 | 321.0 | 1035.0 | 1974.0 | 1220.0 | 39.0 | 24.0 | 116048.0 | 1812.0 | 1086.0 | 1565.0 | 328.0 | 35.0 | 22.0 | 194.0 | 269.0 | 39.0 | 208.0 |
| 9 | 2755.0 | 20.0 | 585.0 | 608.0 | 1182.0 | 1120.0 | 1037.0 | 165.0 | 350.0 | 29223.0 | 1414.0 | 2776.0 | 63.0 | 119.0 | 13.0 | 166.0 | 456.0 | 40.0 | 116.0 |
| 10 | 61.0 | 4.0 | 100.0 | 237.0 | 280.0 | 60.0 | 20.0 | 7.0 | 1818.0 | 56.0 | 3286.0 | 654.0 | 15.0 | 8.0 | 167.0 | 926.0 | 549.0 | 53.0 | 93.0 |
| 11 | 17459.0 | 844.0 | 4361.0 | 7490.0 | 2864.0 | 6047.0 | 1422.0 | 47.0 | 20471.0 | 3384.0 | 1401.0 | 30757.0 | 2544.0 | 396.0 | 600.0 | 949.0 | 5510.0 | 383.0 | 2913.0 |
| 12 | 616.0 | 679.0 | 1873.0 | 1579.0 | 254.0 | 575.0 | 130.0 | 92.0 | 275.0 | 1229.0 | 166.0 | 732.0 | 765.0 | 30.0 | 175.0 | 245.0 | 2523.0 | 22.0 | 118.0 |
| 13 | 2700.0 | 183.0 | 504.0 | 868.0 | 149.0 | 708.0 | 99.0 | 40.0 | 406.0 | 623.0 | 117.0 | 11155.0 | 22555.0 | 234.0 | 50.0 | 106.0 | 1437.0 | 43.0 | 70.0 |
| 14 | 4822.0 | 894.0 | 2040.0 | 16044.0 | 4190.0 | 3707.0 | 5259.0 | 562.0 | 4591.0 | 1711.0 | 715.0 | 12132.0 | 3116.0 | 642.0 | 9754.0 | 745.0 | 22317.0 | 165.0 | 557.0 |
| 15 | 79.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 263.0 | 161.0 | 0.0 | 5918.0 | 2337.0 | 4966.0 | 304.0 | 1238.0 |
| 16 | 1332.0 | 133.0 | 699.0 | 695.0 | 378.0 | 698.0 | 304.0 | 63.0 | 613.0 | 471.0 | 122.0 | 3360.0 | 457.0 | 549.0 | 311.0 | 564.0 | 0.0 | 0.0 | 0.0 |
| 17 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 18 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Also, when you send back the final ordering, please put it in a simple JSON shape so it's easy to read and parse. Something like this:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Pretty straightforward: ""solution"" should be the tasting sequence from the very first pour to the very last, listed in order. Think of it like filling in the blanks on a form — the array is just the lineup of wines, one identifier per slot. This JSON is only a sketch of the expected shape, not the actual answer.

All identifiers must be used exactly as they appear in the instance input — no renaming and no 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”.""","[[215502.0, 6.0, 660.0, 4444.0, 4.0, 217704.0, 18526.0, 7569.0, 8866.0, 20158.0, 667.0, 218.0, 265.0, 47.0, 74.0, 842.0, 375.0, 268.0, 5262.0], [48957.0, 220610.0, 16314.0, 20692.0, 980.0, 4801.0, 1023.0, 30.0, 967.0, 452.0, 63.0, 573.0, 32766.0, 174.0, 78.0, 321.0, 3422.0, 423.0, 656.0], [1482.0, 75.0, 25994.0, 17361.0, 1468.0, 1600.0, 5358.0, 0.0, 208.0, 1678.0, 668.0, 182.0, 29.0, 25.0, 5.0, 92.0, 494.0, 11.0, 102.0], [50351.0, 2566.0, 5671.0, 139308.0, 2113.0, 6149.0, 3114.0, 179.0, 30170.0, 3723.0, 5758.0, 1477.0, 692.0, 128.0, 111.0, 6117.0, 1328.0, 157.0, 12502.0], [6.0, 0.0, 0.0, 0.0, 44392.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 4217.0, 0.0, 0.0], [100280.0, 12.0, 258.0, 5306.0, 4.0, 90133.0, 6057.0, 0.0, 307.0, 28.0, 77.0, 10.0, 563.0, 3.0, 0.0, 28.0, 22.0, 173.0, 1782.0], [52.0, 4.0, 113.0, 2447.0, 6.0, 237.0, 2554.0, 3.0, 19.0, 78.0, 7.0, 31.0, 862.0, 4.0, 1.0, 5.0, 5.0, 23.0, 313.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 66.0, 0.0, 0.0, 0.0, 0.0, 80.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [3691.0, 24.0, 321.0, 1035.0, 1974.0, 1220.0, 39.0, 24.0, 116048.0, 1812.0, 1086.0, 1565.0, 328.0, 35.0, 22.0, 194.0, 269.0, 39.0, 208.0], [2755.0, 20.0, 585.0, 608.0, 1182.0, 1120.0, 1037.0, 165.0, 350.0, 29223.0, 1414.0, 2776.0, 63.0, 119.0, 13.0, 166.0, 456.0, 40.0, 116.0], [61.0, 4.0, 100.0, 237.0, 280.0, 60.0, 20.0, 7.0, 1818.0, 56.0, 3286.0, 654.0, 15.0, 8.0, 167.0, 926.0, 549.0, 53.0, 93.0], [17459.0, 844.0, 4361.0, 7490.0, 2864.0, 6047.0, 1422.0, 47.0, 20471.0, 3384.0, 1401.0, 30757.0, 2544.0, 396.0, 600.0, 949.0, 5510.0, 383.0, 2913.0], [616.0, 679.0, 1873.0, 1579.0, 254.0, 575.0, 130.0, 92.0, 275.0, 1229.0, 166.0, 732.0, 765.0, 30.0, 175.0, 245.0, 2523.0, 22.0, 118.0], [2700.0, 183.0, 504.0, 868.0, 149.0, 708.0, 99.0, 40.0, 406.0, 623.0, 117.0, 11155.0, 22555.0, 234.0, 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LOP,LOP,"On a typical afternoon the assignment is simple: create one complete sequence of study topics with every topic included exactly once. There’s a reinforcement number for each directed pairing — how much benefit results if that particular topic comes before another — and the overall benefit of a sequence is the sum of those numbers for every pair that appears in that order. The task is to find the ordering with the biggest total sum, and the concrete details are provided below.

Concretely, there are 18 topics to place, listed as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18.

*Meaning: this is a directed reinforcement_score matrix over items in topic_ids (18 total). Entry at row preceding_topic_id=i and column following_topic_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).*

**reinforcement_score)**

| preceding_topic_id\following_topic_id | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 2 | 0.0 | 7329.0 | 365.0 | 3.0 | 161.0 | 868.0 | 623.0 | 28851.0 | 286.0 | 30.0 | 207.0 | 0.0 | 0.0 | 0.0 | 122.0 | 0.0 | 35.0 | 574.0 |
| 3 | 0.0 | 389.0 | 2218.0 | 165.0 | 596.0 | 2432.0 | 196.0 | 11953.0 | 570.0 | 39.0 | 225.0 | 0.0 | 0.0 | 0.0 | 208.0 | 0.0 | 4.0 | 293.0 |
| 4 | 0.0 | 0.0 | 0.0 | 2558.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 6272.0 | 51.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 587.0 |
| 5 | 0.0 | 0.0 | 0.0 | 0.0 | 3212.0 | 1996.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1633.0 | 0.0 |
| 6 | 0.0 | 18.0 | 0.0 | 0.0 | 700.0 | 42132.0 | 1802.0 | 0.0 | 0.0 | 845.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1072.0 | 124.0 |
| 7 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 388.0 | 4084.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 17.0 |
| 8 | 0.0 | 6.0 | 3.0 | 1.0 | 7.0 | 12.0 | 4.0 | 16701.0 | 5.0 | 1.0 | 492.0 | 0.0 | 0.0 | 0.0 | 132.0 | 0.0 | 1487.0 | 18914.0 |
| 9 | 0.0 | 237.0 | 129.0 | 38.0 | 196.0 | 470.0 | 158.0 | 251.0 | 0.0 | 32.0 | 1.0 | 0.0 | 0.0 | 0.0 | 100.0 | 0.0 | 0.0 | 156.0 |
| 10 | 0.0 | 191.0 | 106.0 | 30.0 | 157.0 | 382.0 | 127.0 | 209.0 | 64.0 | 6849.0 | 84.0 | 0.0 | 0.0 | 0.0 | 225.0 | 0.0 | 112.0 | 324.0 |
| 11 | 0.0 | 121.0 | 95.0 | 25.0 | 243.0 | 288.0 | 89.0 | 136.0 | 221.0 | 4702.0 | 242.0 | 0.0 | 0.0 | 0.0 | 2.0 | 0.0 | 0.0 | 210.0 |
| 12 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 13 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 14 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 15 | 0.0 | 512.0 | 447.0 | 129.0 | 3627.0 | 1586.0 | 533.0 | 5006.0 | 2530.0 | 1253.0 | 1134.0 | 0.0 | 0.0 | 0.0 | 6577.0 | 0.0 | 2943.0 | 2388.0 |
| 16 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 17 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 18 | 0.0 | 221.0 | 99.0 | 28.0 | 283.0 | 405.0 | 139.0 | 891.0 | 177.0 | 59.0 | 790.0 | 0.0 | 0.0 | 0.0 | 145.0 | 0.0 | 562.0 | 520.0 |

Select the ordering that maximizes the total sum of these reinforcement scores.

A quick note on the output format: when you send back the ordering, please stick to this simple JSON shape so it's easy to read and check.

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of ""solution"" as the single sequence of study topics, listed from the one that comes first to the one that comes last. This JSON is just a sketch of the shape I need — replace the placeholders with the actual item identifiers from the instance when you give the real answer.

Please use the identifiers exactly as they appear in the instance input — don't 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”.""","[[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 7329.0, 365.0, 3.0, 161.0, 868.0, 623.0, 28851.0, 286.0, 30.0, 207.0, 0.0, 0.0, 0.0, 122.0, 0.0, 35.0, 574.0], [0.0, 389.0, 2218.0, 165.0, 596.0, 2432.0, 196.0, 11953.0, 570.0, 39.0, 225.0, 0.0, 0.0, 0.0, 208.0, 0.0, 4.0, 293.0], [0.0, 0.0, 0.0, 2558.0, 0.0, 0.0, 0.0, 0.0, 0.0, 6272.0, 51.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 587.0], [0.0, 0.0, 0.0, 0.0, 3212.0, 1996.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1633.0, 0.0], [0.0, 18.0, 0.0, 0.0, 700.0, 42132.0, 1802.0, 0.0, 0.0, 845.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1072.0, 124.0], [0.0, 0.0, 0.0, 0.0, 0.0, 388.0, 4084.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 17.0], [0.0, 6.0, 3.0, 1.0, 7.0, 12.0, 4.0, 16701.0, 5.0, 1.0, 492.0, 0.0, 0.0, 0.0, 132.0, 0.0, 1487.0, 18914.0], [0.0, 237.0, 129.0, 38.0, 196.0, 470.0, 158.0, 251.0, 0.0, 32.0, 1.0, 0.0, 0.0, 0.0, 100.0, 0.0, 0.0, 156.0], [0.0, 191.0, 106.0, 30.0, 157.0, 382.0, 127.0, 209.0, 64.0, 6849.0, 84.0, 0.0, 0.0, 0.0, 225.0, 0.0, 112.0, 324.0], [0.0, 121.0, 95.0, 25.0, 243.0, 288.0, 89.0, 136.0, 221.0, 4702.0, 242.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 210.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 512.0, 447.0, 129.0, 3627.0, 1586.0, 533.0, 5006.0, 2530.0, 1253.0, 1134.0, 0.0, 0.0, 0.0, 6577.0, 0.0, 2943.0, 2388.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 221.0, 99.0, 28.0, 283.0, 405.0, 139.0, 891.0, 177.0, 59.0, 790.0, 0.0, 0.0, 0.0, 145.0, 0.0, 562.0, 520.0]]","[14, 2, 1, 8, 7, 3, 17, 10, 4, 5, 9, 16, 15, 13, 12, 11, 6, 0]",114865.0,"{'problem_type': 'LOP', 'num_nodes': 18, 'nodes': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], 'pairs': [{'from_id': 1, 'to_id': 2, 'weight': 0.0}, {'from_id': 1, 'to_id': 3, 'weight': 0.0}, {'from_id': 1, 'to_id': 4, 'weight': 0.0}, {'from_id': 1, 'to_id': 5, 'weight': 0.0}, {'from_id': 1, 'to_id': 6, 'weight': 0.0}, {'from_id': 1, 'to_id': 7, 'weight': 0.0}, {'from_id': 1, 'to_id': 8, 'weight': 0.0}, {'from_id': 1, 'to_id': 9, 'weight': 0.0}, {'from_id': 1, 'to_id': 10, 'weight': 0.0}, {'from_id': 1, 'to_id': 11, 'weight': 0.0}, {'from_id': 1, 'to_id': 12, 'weight': 0.0}, {'from_id': 1, 'to_id': 13, 'weight': 0.0}, {'from_id': 1, 'to_id': 14, 'weight': 0.0}, {'from_id': 1, 'to_id': 15, 'weight': 0.0}, {'from_id': 1, 'to_id': 16, 'weight': 0.0}, {'from_id': 1, 'to_id': 17, 'weight': 0.0}, {'from_id': 1, 'to_id': 18, 'weight': 0.0}, {'from_id': 2, 'to_id': 1, 'weight': 0.0}, {'from_id': 2, 'to_id': 3, 'weight': 365.0}, {'from_id': 2, 'to_id': 4, 'weight': 3.0}, {'from_id': 2, 'to_id': 5, 'weight': 161.0}, {'from_id': 2, 'to_id': 6, 'weight': 868.0}, {'from_id': 2, 'to_id': 7, 'weight': 623.0}, {'from_id': 2, 'to_id': 8, 'weight': 28851.0}, {'from_id': 2, 'to_id': 9, 'weight': 286.0}, {'from_id': 2, 'to_id': 10, 'weight': 30.0}, {'from_id': 2, 'to_id': 11, 'weight': 207.0}, {'from_id': 2, 'to_id': 12, 'weight': 0.0}, {'from_id': 2, 'to_id': 13, 'weight': 0.0}, {'from_id': 2, 'to_id': 14, 'weight': 0.0}, {'from_id': 2, 'to_id': 15, 'weight': 122.0}, {'from_id': 2, 'to_id': 16, 'weight': 0.0}, {'from_id': 2, 'to_id': 17, 'weight': 35.0}, {'from_id': 2, 'to_id': 18, 'weight': 574.0}, {'from_id': 3, 'to_id': 1, 'weight': 0.0}, {'from_id': 3, 'to_id': 2, 'weight': 389.0}, {'from_id': 3, 'to_id': 4, 'weight': 165.0}, {'from_id': 3, 'to_id': 5, 'weight': 596.0}, {'from_id': 3, 'to_id': 6, 'weight': 2432.0}, {'from_id': 3, 'to_id': 7, 'weight': 196.0}, {'from_id': 3, 'to_id': 8, 'weight': 11953.0}, {'from_id': 3, 'to_id': 9, 'weight': 570.0}, {'from_id': 3, 'to_id': 10, 'weight': 39.0}, {'from_id': 3, 'to_id': 11, 'weight': 225.0}, {'from_id': 3, 'to_id': 12, 'weight': 0.0}, {'from_id': 3, 'to_id': 13, 'weight': 0.0}, {'from_id': 3, 'to_id': 14, 'weight': 0.0}, {'from_id': 3, 'to_id': 15, 'weight': 208.0}, {'from_id': 3, 'to_id': 16, 'weight': 0.0}, {'from_id': 3, 'to_id': 17, 'weight': 4.0}, {'from_id': 3, 'to_id': 18, 'weight': 293.0}, {'from_id': 4, 'to_id': 1, 'weight': 0.0}, {'from_id': 4, 'to_id': 2, 'weight': 0.0}, {'from_id': 4, 'to_id': 3, 'weight': 0.0}, {'from_id': 4, 'to_id': 5, 'weight': 0.0}, {'from_id': 4, 'to_id': 6, 'weight': 0.0}, {'from_id': 4, 'to_id': 7, 'weight': 0.0}, {'from_id': 4, 'to_id': 8, 'weight': 0.0}, {'from_id': 4, 'to_id': 9, 'weight': 0.0}, {'from_id': 4, 'to_id': 10, 'weight': 6272.0}, {'from_id': 4, 'to_id': 11, 'weight': 51.0}, {'from_id': 4, 'to_id': 12, 'weight': 0.0}, {'from_id': 4, 'to_id': 13, 'weight': 0.0}, {'from_id': 4, 'to_id': 14, 'weight': 0.0}, {'from_id': 4, 'to_id': 15, 'weight': 0.0}, {'from_id': 4, 'to_id': 16, 'weight': 0.0}, {'from_id': 4, 'to_id': 17, 'weight': 0.0}, {'from_id': 4, 'to_id': 18, 'weight': 587.0}, {'from_id': 5, 'to_id': 1, 'weight': 0.0}, {'from_id': 5, 'to_id': 2, 'weight': 0.0}, {'from_id': 5, 'to_id': 3, 'weight': 0.0}, {'from_id': 5, 'to_id': 4, 'weight': 0.0}, {'from_id': 5, 'to_id': 6, 'weight': 1996.0}, {'from_id': 5, 'to_id': 7, 'weight': 0.0}, {'from_id': 5, 'to_id': 8, 'weight': 0.0}, {'from_id': 5, 'to_id': 9, 'weight': 0.0}, {'from_id': 5, 'to_id': 10, 'weight': 0.0}, {'from_id': 5, 'to_id': 11, 'weight': 0.0}, {'from_id': 5, 'to_id': 12, 'weight': 0.0}, {'from_id': 5, 'to_id': 13, 'weight': 0.0}, {'from_id': 5, 'to_id': 14, 'weight': 0.0}, {'from_id': 5, 'to_id': 15, 'weight': 0.0}, {'from_id': 5, 'to_id': 16, 'weight': 0.0}, {'from_id': 5, 'to_id': 17, 'weight': 1633.0}, {'from_id': 5, 'to_id': 18, 'weight': 0.0}, {'from_id': 6, 'to_id': 1, 'weight': 0.0}, {'from_id': 6, 'to_id': 2, 'weight': 18.0}, {'from_id': 6, 'to_id': 3, 'weight': 0.0}, {'from_id': 6, 'to_id': 4, 'weight': 0.0}, {'from_id': 6, 'to_id': 5, 'weight': 700.0}, {'from_id': 6, 'to_id': 7, 'weight': 1802.0}, {'from_id': 6, 'to_id': 8, 'weight': 0.0}, {'from_id': 6, 'to_id': 9, 'weight': 0.0}, {'from_id': 6, 'to_id': 10, 'weight': 845.0}, {'from_id': 6, 'to_id': 11, 'weight': 0.0}, {'from_id': 6, 'to_id': 12, 'weight': 0.0}, {'from_id': 6, 'to_id': 13, 'weight': 0.0}, {'from_id': 6, 'to_id': 14, 'weight': 0.0}, {'from_id': 6, 'to_id': 15, 'weight': 0.0}, {'from_id': 6, 'to_id': 16, 'weight': 0.0}, {'from_id': 6, 'to_id': 17, 'weight': 1072.0}, {'from_id': 6, 'to_id': 18, 'weight': 124.0}, {'from_id': 7, 'to_id': 1, 'weight': 0.0}, {'from_id': 7, 'to_id': 2, 'weight': 0.0}, {'from_id': 7, 'to_id': 3, 'weight': 0.0}, {'from_id': 7, 'to_id': 4, 'weight': 0.0}, {'from_id': 7, 'to_id': 5, 'weight': 0.0}, {'from_id': 7, 'to_id': 6, 'weight': 388.0}, {'from_id': 7, 'to_id': 8, 'weight': 0.0}, {'from_id': 7, 'to_id': 9, 'weight': 0.0}, {'from_id': 7, 'to_id': 10, 'weight': 0.0}, {'from_id': 7, 'to_id': 11, 'weight': 0.0}, {'from_id': 7, 'to_id': 12, 'weight': 0.0}, {'from_id': 7, 'to_id': 13, 'weight': 0.0}, {'from_id': 7, 'to_id': 14, 'weight': 0.0}, {'from_id': 7, 'to_id': 15, 'weight': 0.0}, {'from_id': 7, 'to_id': 16, 'weight': 0.0}, {'from_id': 7, 'to_id': 17, 'weight': 0.0}, {'from_id': 7, 'to_id': 18, 'weight': 17.0}, {'from_id': 8, 'to_id': 1, 'weight': 0.0}, {'from_id': 8, 'to_id': 2, 'weight': 6.0}, {'from_id': 8, 'to_id': 3, 'weight': 3.0}, {'from_id': 8, 'to_id': 4, 'weight': 1.0}, {'from_id': 8, 'to_id': 5, 'weight': 7.0}, {'from_id': 8, 'to_id': 6, 'weight': 12.0}, {'from_id': 8, 'to_id': 7, 'weight': 4.0}, {'from_id': 8, 'to_id': 9, 'weight': 5.0}, {'from_id': 8, 'to_id': 10, 'weight': 1.0}, {'from_id': 8, 'to_id': 11, 'weight': 492.0}, {'from_id': 8, 'to_id': 12, 'weight': 0.0}, {'from_id': 8, 'to_id': 13, 'weight': 0.0}, {'from_id': 8, 'to_id': 14, 'weight': 0.0}, {'from_id': 8, 'to_id': 15, 'weight': 132.0}, {'from_id': 8, 'to_id': 16, 'weight': 0.0}, {'from_id': 8, 'to_id': 17, 'weight': 1487.0}, {'from_id': 8, 'to_id': 18, 'weight': 18914.0}, {'from_id': 9, 'to_id': 1, 'weight': 0.0}, {'from_id': 9, 'to_id': 2, 'weight': 237.0}, {'from_id': 9, 'to_id': 3, 'weight': 129.0}, {'from_id': 9, 'to_id': 4, 'weight': 38.0}, {'from_id': 9, 'to_id': 5, 'weight': 196.0}, {'from_id': 9, 'to_id': 6, 'weight': 470.0}, {'from_id': 9, 'to_id': 7, 'weight': 158.0}, {'from_id': 9, 'to_id': 8, 'weight': 251.0}, {'from_id': 9, 'to_id': 10, 'weight': 32.0}, {'from_id': 9, 'to_id': 11, 'weight': 1.0}, {'from_id': 9, 'to_id': 12, 'weight': 0.0}, {'from_id': 9, 'to_id': 13, 'weight': 0.0}, {'from_id': 9, 'to_id': 14, 'weight': 0.0}, {'from_id': 9, 'to_id': 15, 'weight': 100.0}, {'from_id': 9, 'to_id': 16, 'weight': 0.0}, {'from_id': 9, 'to_id': 17, 'weight': 0.0}, {'from_id': 9, 'to_id': 18, 'weight': 156.0}, {'from_id': 10, 'to_id': 1, 'weight': 0.0}, {'from_id': 10, 'to_id': 2, 'weight': 191.0}, {'from_id': 10, 'to_id': 3, 'weight': 106.0}, {'from_id': 10, 'to_id': 4, 'weight': 30.0}, {'from_id': 10, 'to_id': 5, 'weight': 157.0}, {'from_id': 10, 'to_id': 6, 'weight': 382.0}, {'from_id': 10, 'to_id': 7, 'weight': 127.0}, {'from_id': 10, 'to_id': 8, 'weight': 209.0}, {'from_id': 10, 'to_id': 9, 'weight': 64.0}, {'from_id': 10, 'to_id': 11, 'weight': 84.0}, {'from_id': 10, 'to_id': 12, 'weight': 0.0}, {'from_id': 10, 'to_id': 13, 'weight': 0.0}, {'from_id': 10, 'to_id': 14, 'weight': 0.0}, {'from_id': 10, 'to_id': 15, 'weight': 225.0}, {'from_id': 10, 'to_id': 16, 'weight': 0.0}, {'from_id': 10, 'to_id': 17, 'weight': 112.0}, {'from_id': 10, 'to_id': 18, 'weight': 324.0}, {'from_id': 11, 'to_id': 1, 'weight': 0.0}, {'from_id': 11, 'to_id': 2, 'weight': 121.0}, {'from_id': 11, 'to_id': 3, 'weight': 95.0}, {'from_id': 11, 'to_id': 4, 'weight': 25.0}, {'from_id': 11, 'to_id': 5, 'weight': 243.0}, {'from_id': 11, 'to_id': 6, 'weight': 288.0}, {'from_id': 11, 'to_id': 7, 'weight': 89.0}, {'from_id': 11, 'to_id': 8, 'weight': 136.0}, {'from_id': 11, 'to_id': 9, 'weight': 221.0}, {'from_id': 11, 'to_id': 10, 'weight': 4702.0}, {'from_id': 11, 'to_id': 12, 'weight': 0.0}, {'from_id': 11, 'to_id': 13, 'weight': 0.0}, {'from_id': 11, 'to_id': 14, 'weight': 0.0}, {'from_id': 11, 'to_id': 15, 'weight': 2.0}, {'from_id': 11, 'to_id': 16, 'weight': 0.0}, {'from_id': 11, 'to_id': 17, 'weight': 0.0}, {'from_id': 11, 'to_id': 18, 'weight': 210.0}, {'from_id': 12, 'to_id': 1, 'weight': 0.0}, {'from_id': 12, 'to_id': 2, 'weight': 0.0}, {'from_id': 12, 'to_id': 3, 'weight': 0.0}, {'from_id': 12, 'to_id': 4, 'weight': 0.0}, {'from_id': 12, 'to_id': 5, 'weight': 0.0}, {'from_id': 12, 'to_id': 6, 'weight': 0.0}, {'from_id': 12, 'to_id': 7, 'weight': 0.0}, {'from_id': 12, 'to_id': 8, 'weight': 0.0}, {'from_id': 12, 'to_id': 9, 'weight': 0.0}, {'from_id': 12, 'to_id': 10, 'weight': 0.0}, {'from_id': 12, 'to_id': 11, 'weight': 0.0}, {'from_id': 12, 'to_id': 13, 'weight': 0.0}, {'from_id': 12, 'to_id': 14, 'weight': 0.0}, {'from_id': 12, 'to_id': 15, 'weight': 0.0}, {'from_id': 12, 'to_id': 16, 'weight': 0.0}, {'from_id': 12, 'to_id': 17, 'weight': 0.0}, {'from_id': 12, 'to_id': 18, 'weight': 0.0}, {'from_id': 13, 'to_id': 1, 'weight': 0.0}, {'from_id': 13, 'to_id': 2, 'weight': 0.0}, {'from_id': 13, 'to_id': 3, 'weight': 0.0}, {'from_id': 13, 'to_id': 4, 'weight': 0.0}, {'from_id': 13, 'to_id': 5, 'weight': 0.0}, {'from_id': 13, 'to_id': 6, 'weight': 0.0}, {'from_id': 13, 'to_id': 7, 'weight': 0.0}, {'from_id': 13, 'to_id': 8, 'weight': 0.0}, {'from_id': 13, 'to_id': 9, 'weight': 0.0}, {'from_id': 13, 'to_id': 10, 'weight': 0.0}, {'from_id': 13, 'to_id': 11, 'weight': 0.0}, {'from_id': 13, 'to_id': 12, 'weight': 0.0}, {'from_id': 13, 'to_id': 14, 'weight': 0.0}, {'from_id': 13, 'to_id': 15, 'weight': 0.0}, {'from_id': 13, 'to_id': 16, 'weight': 0.0}, {'from_id': 13, 'to_id': 17, 'weight': 0.0}, {'from_id': 13, 'to_id': 18, 'weight': 0.0}, {'from_id': 14, 'to_id': 1, 'weight': 0.0}, {'from_id': 14, 'to_id': 2, 'weight': 0.0}, {'from_id': 14, 'to_id': 3, 'weight': 0.0}, {'from_id': 14, 'to_id': 4, 'weight': 0.0}, {'from_id': 14, 'to_id': 5, 'weight': 0.0}, {'from_id': 14, 'to_id': 6, 'weight': 0.0}, {'from_id': 14, 'to_id': 7, 'weight': 0.0}, {'from_id': 14, 'to_id': 8, 'weight': 0.0}, {'from_id': 14, 'to_id': 9, 'weight': 0.0}, {'from_id': 14, 'to_id': 10, 'weight': 0.0}, {'from_id': 14, 'to_id': 11, 'weight': 0.0}, {'from_id': 14, 'to_id': 12, 'weight': 0.0}, {'from_id': 14, 'to_id': 13, 'weight': 0.0}, {'from_id': 14, 'to_id': 15, 'weight': 0.0}, {'from_id': 14, 'to_id': 16, 'weight': 0.0}, {'from_id': 14, 'to_id': 17, 'weight': 0.0}, {'from_id': 14, 'to_id': 18, 'weight': 0.0}, {'from_id': 15, 'to_id': 1, 'weight': 0.0}, {'from_id': 15, 'to_id': 2, 'weight': 512.0}, {'from_id': 15, 'to_id': 3, 'weight': 447.0}, {'from_id': 15, 'to_id': 4, 'weight': 129.0}, {'from_id': 15, 'to_id': 5, 'weight': 3627.0}, {'from_id': 15, 'to_id': 6, 'weight': 1586.0}, {'from_id': 15, 'to_id': 7, 'weight': 533.0}, {'from_id': 15, 'to_id': 8, 'weight': 5006.0}, {'from_id': 15, 'to_id': 9, 'weight': 2530.0}, {'from_id': 15, 'to_id': 10, 'weight': 1253.0}, {'from_id': 15, 'to_id': 11, 'weight': 1134.0}, {'from_id': 15, 'to_id': 12, 'weight': 0.0}, {'from_id': 15, 'to_id': 13, 'weight': 0.0}, {'from_id': 15, 'to_id': 14, 'weight': 0.0}, {'from_id': 15, 'to_id': 16, 'weight': 0.0}, {'from_id': 15, 'to_id': 17, 'weight': 2943.0}, {'from_id': 15, 'to_id': 18, 'weight': 2388.0}, {'from_id': 16, 'to_id': 1, 'weight': 0.0}, {'from_id': 16, 'to_id': 2, 'weight': 0.0}, {'from_id': 16, 'to_id': 3, 'weight': 0.0}, {'from_id': 16, 'to_id': 4, 'weight': 0.0}, {'from_id': 16, 'to_id': 5, 'weight': 0.0}, {'from_id': 16, 'to_id': 6, 'weight': 0.0}, {'from_id': 16, 'to_id': 7, 'weight': 0.0}, {'from_id': 16, 'to_id': 8, 'weight': 0.0}, {'from_id': 16, 'to_id': 9, 'weight': 0.0}, {'from_id': 16, 'to_id': 10, 'weight': 0.0}, {'from_id': 16, 'to_id': 11, 'weight': 0.0}, {'from_id': 16, 'to_id': 12, 'weight': 0.0}, {'from_id': 16, 'to_id': 13, 'weight': 0.0}, {'from_id': 16, 'to_id': 14, 'weight': 0.0}, {'from_id': 16, 'to_id': 15, 'weight': 0.0}, {'from_id': 16, 'to_id': 17, 'weight': 0.0}, {'from_id': 16, 'to_id': 18, 'weight': 0.0}, {'from_id': 17, 'to_id': 1, 'weight': 0.0}, {'from_id': 17, 'to_id': 2, 'weight': 0.0}, {'from_id': 17, 'to_id': 3, 'weight': 0.0}, {'from_id': 17, 'to_id': 4, 'weight': 0.0}, {'from_id': 17, 'to_id': 5, 'weight': 0.0}, {'from_id': 17, 'to_id': 6, 'weight': 0.0}, {'from_id': 17, 'to_id': 7, 'weight': 0.0}, {'from_id': 17, 'to_id': 8, 'weight': 0.0}, {'from_id': 17, 'to_id': 9, 'weight': 0.0}, {'from_id': 17, 'to_id': 10, 'weight': 0.0}, {'from_id': 17, 'to_id': 11, 'weight': 0.0}, {'from_id': 17, 'to_id': 12, 'weight': 0.0}, {'from_id': 17, 'to_id': 13, 'weight': 0.0}, {'from_id': 17, 'to_id': 14, 'weight': 0.0}, {'from_id': 17, 'to_id': 15, 'weight': 0.0}, {'from_id': 17, 'to_id': 16, 'weight': 0.0}, {'from_id': 17, 'to_id': 18, 'weight': 0.0}, {'from_id': 18, 'to_id': 1, 'weight': 0.0}, {'from_id': 18, 'to_id': 2, 'weight': 221.0}, {'from_id': 18, 'to_id': 3, 'weight': 99.0}, {'from_id': 18, 'to_id': 4, 'weight': 28.0}, {'from_id': 18, 'to_id': 5, 'weight': 283.0}, {'from_id': 18, 'to_id': 6, 'weight': 405.0}, {'from_id': 18, 'to_id': 7, 'weight': 139.0}, {'from_id': 18, 'to_id': 8, 'weight': 891.0}, {'from_id': 18, 'to_id': 9, 'weight': 177.0}, {'from_id': 18, 'to_id': 10, 'weight': 59.0}, {'from_id': 18, 'to_id': 11, 'weight': 790.0}, {'from_id': 18, 'to_id': 12, 'weight': 0.0}, {'from_id': 18, 'to_id': 13, 'weight': 0.0}, {'from_id': 18, 'to_id': 14, 'weight': 0.0}, {'from_id': 18, 'to_id': 15, 'weight': 145.0}, {'from_id': 18, 'to_id': 16, 'weight': 0.0}, {'from_id': 18, 'to_id': 17, 'weight': 562.0}]}","[15, 3, 2, 9, 8, 4, 18, 11, 5, 6, 10, 17, 16, 14, 13, 12, 7, 1]",44,markdown_table,1
LOP,LOP,"Someone in the household sketched out a postcard gallery and wants it to flow from left to right. The rule is simple: every card gets used once in one continuous line, no repeats, and each time a card appears before another there’s a given harmony number that boosts the display — the gallery’s total charm is what you get when you add up those boosts. So the job becomes arranging the cards in the order that yields the highest total charm. The exact pairwise harmony numbers follow below.

There are 24 postcards, labeled A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, and each directed pair below gives the harmony when the first appears somewhere left of the second.
Meaning: this is a directed harmony_score matrix over items in postcard_ids (24 total). Entry at row preceding_postcard_id=i and column following_postcard_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
harmony_score:
| preceding_postcard_id\following_postcard_id | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 511.0 | 0.0 | 126.0 | 3.0 | 0.0 | 9.0 | 9.0 | 9681.0 | 7037.0 | 424.0 | 349.0 | 39.0 | 43.0 | 0.0 | 0.0 | 3.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| B | 248.0 | 279.0 | 537.0 | 220.0 | 13.0 | 80.0 | 73.0 | 35.0 | 123.0 | 42.0 | 3.0 | 35.0 | 74.0 | 0.0 | 204.0 | 19.0 | 7.0 | 0.0 | 0.0 | 0.0 | 0.0 | 419.0 | 0.0 | 0.0 |
| C | 1654.0 | 21.0 | 1769.0 | 257.0 | 14.0 | 283.0 | 83.0 | 3.0 | 59.0 | 35.0 | 1.0 | 82.0 | 690.0 | 0.0 | 110.0 | 16.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 542.0 | 0.0 | 0.0 |
| D | 6.0 | 132.0 | 18.0 | 363.0 | 23.0 | 74.0 | 312.0 | 16.0 | 49.0 | 22.0 | 0.0 | 0.0 | 893.0 | 0.0 | 72.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 12.0 | 0.0 | 0.0 |
| E | 0.0 | 0.0 | 0.0 | 30.0 | 50.0 | 0.0 | 4.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2.0 | 0.0 | 19.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2.0 | 0.0 | 0.0 |
| F | 21.0 | 41.0 | 3.0 | 427.0 | 17.0 | 97.0 | 332.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2.0 | 905.0 | 0.0 | 67.0 | 20.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 57.0 | 0.0 | 0.0 |
| G | 70.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 72.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| H | 0.0 | 0.0 | 2.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 196.0 | 0.0 | 0.0 | 424.0 | 0.0 | 0.0 | 0.0 | 2.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| I | 1928.0 | 0.0 | 348.0 | 0.0 | 0.0 | 0.0 | 0.0 | 89.0 | 1730.0 | 271.0 | 0.0 | 3.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| J | 5.0 | 0.0 | 116.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 74.0 | 62.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 5.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| K | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| L | 100.0 | 0.0 | 0.0 | 22.0 | 2.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 627.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| M | 330.0 | 493.0 | 95.0 | 123.0 | 13.0 | 57.0 | 63.0 | 18.0 | 41.0 | 14.0 | 3.0 | 18.0 | 0.0 | 0.0 | 741.0 | 27.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 275.0 | 0.0 | 0.0 |
| N | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| O | 98.0 | 66.0 | 450.0 | 327.0 | 9.0 | 101.0 | 55.0 | 175.0 | 98.0 | 75.0 | 9.0 | 32.0 | 698.0 | 0.0 | 0.0 | 3.0 | 59.0 | 0.0 | 0.0 | 0.0 | 0.0 | 255.0 | 0.0 | 0.0 |
| P | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1177.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| Q | 800.0 | 79.0 | 143.0 | 181.0 | 21.0 | 97.0 | 60.0 | 44.0 | 125.0 | 58.0 | 44.0 | 44.0 | 379.0 | 0.0 | 225.0 | 69.0 | 0.0 | 0.0 | 237.0 | 0.0 | 0.0 | 297.0 | 0.0 | 0.0 |
| R | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| S | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| T | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| U | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| V | 951.0 | 76.0 | 502.0 | 388.0 | 46.0 | 172.0 | 74.0 | 85.0 | 272.0 | 204.0 | 30.0 | 56.0 | 1569.0 | 0.0 | 39.0 | 87.0 | 340.0 | 0.0 | 54.0 | 0.0 | 0.0 | 26.0 | 0.0 | 0.0 |
| W | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| X | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
Use these pairwise harmony entries to find the left-to-right sequence that yields the highest total charm.

If you want to hand me the final line-up, a little JSON sketch makes it easy to read — something like this is perfect:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of ""solution"" as the left-to-right list of cards in the gallery; each entry in the array is one card identifier from the instance input in the order they should appear. This is just a template showing the expected shape, not the actual answer.

Please use the identifiers exactly as they appear in the instance input — no renaming and no 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”.""","[[511.0, 0.0, 126.0, 3.0, 0.0, 9.0, 9.0, 9681.0, 7037.0, 424.0, 349.0, 39.0, 43.0, 0.0, 0.0, 3.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [248.0, 279.0, 537.0, 220.0, 13.0, 80.0, 73.0, 35.0, 123.0, 42.0, 3.0, 35.0, 74.0, 0.0, 204.0, 19.0, 7.0, 0.0, 0.0, 0.0, 0.0, 419.0, 0.0, 0.0], [1654.0, 21.0, 1769.0, 257.0, 14.0, 283.0, 83.0, 3.0, 59.0, 35.0, 1.0, 82.0, 690.0, 0.0, 110.0, 16.0, 0.0, 0.0, 0.0, 0.0, 0.0, 542.0, 0.0, 0.0], [6.0, 132.0, 18.0, 363.0, 23.0, 74.0, 312.0, 16.0, 49.0, 22.0, 0.0, 0.0, 893.0, 0.0, 72.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 12.0, 0.0, 0.0], [0.0, 0.0, 0.0, 30.0, 50.0, 0.0, 4.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 19.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0], [21.0, 41.0, 3.0, 427.0, 17.0, 97.0, 332.0, 0.0, 0.0, 0.0, 0.0, 2.0, 905.0, 0.0, 67.0, 20.0, 0.0, 0.0, 0.0, 0.0, 0.0, 57.0, 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LOP,LOP,"Recently I put together a lineup of fragrance strips that needed to be tasted in one continuous run — every strip had to be included exactly once, no reruns, no drops. There’s a little table telling how well each scent precedes each other scent, and the plan is to arrange the lineup so that when those pair scores are added up for every earlier-later combination present, the grand total is as high as possible. The full list of strips and their pair scores is shown below.

There are 24 strips to order — they are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23.

*Meaning: this is a directed progression_value matrix over items in strip_ids (24 total). Entry at row preceding_strip=i and column following_strip=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).*

**progression_value)**

| preceding_strip\following_strip | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 925.0 | 770.0 | 59.0 | 39.0 | 25.0 | 479.0 | 29.0 | 7.0 | 0.0 | 39.0 | 87.0 | 3.0 | 18.0 | 0.0 | 14.0 | 54.0 | 7.0 | 9.0 | 3.0 | 188.0 | 76.0 | 16.0 | 50.0 | 17.0 |
| 1 | 1.0 | 122.0 | 2.0 | 9.0 | 1.0 | 33.0 | 0.0 | 1.0 | 0.0 | 0.0 | 9.0 | 0.0 | 0.0 | 0.0 | 1.0 | 1.0 | 0.0 | 1.0 | 0.0 | 7.0 | 6.0 | 0.0 | 11.0 | 8.0 |
| 2 | 74.0 | 31.0 | 2985.0 | 201.0 | 17.0 | 749.0 | 32.0 | 31.0 | 0.0 | 4.0 | 74.0 | 13.0 | 43.0 | 0.0 | 140.0 | 29.0 | 3.0 | 17.0 | 5.0 | 444.0 | 80.0 | 29.0 | 1288.0 | 40.0 |
| 3 | 442.0 | 47.0 | 466.0 | 4932.0 | 102.0 | 350.0 | 14.0 | 35.0 | 0.0 | 8.0 | 139.0 | 8.0 | 6.0 | 0.0 | 9.0 | 32.0 | 1.0 | 4.0 | 9.0 | 223.0 | 51.0 | 10.0 | 497.0 | 47.0 |
| 4 | 60.0 | 0.0 | 10.0 | 123.0 | 2488.0 | 54.0 | 14.0 | 31.0 | 0.0 | 7.0 | 142.0 | 1.0 | 8.0 | 0.0 | 8.0 | 12.0 | 2.0 | 2.0 | 3.0 | 145.0 | 12.0 | 0.0 | 143.0 | 10.0 |
| 5 | 39.0 | 185.0 | 7.0 | 47.0 | 4.0 | 5243.0 | 18.0 | 2.0 | 0.0 | 8.0 | 71.0 | 1.0 | 2.0 | 0.0 | 21.0 | 5.0 | 1.0 | 0.0 | 1.0 | 16.0 | 12.0 | 7.0 | 631.0 | 32.0 |
| 6 | 8.0 | 0.0 | 0.0 | 25.0 | 1.0 | 55.0 | 451.0 | 6.0 | 0.0 | 9.0 | 277.0 | 0.0 | 22.0 | 0.0 | 80.0 | 18.0 | 10.0 | 0.0 | 4.0 | 151.0 | 10.0 | 9.0 | 60.0 | 15.0 |
| 7 | 120.0 | 0.0 | 12.0 | 1939.0 | 521.0 | 454.0 | 1626.0 | 4736.0 | 0.0 | 311.0 | 1293.0 | 330.0 | 39.0 | 0.0 | 48.0 | 32.0 | 28.0 | 4.0 | 17.0 | 867.0 | 182.0 | 80.0 | 305.0 | 69.0 |
| 8 | 2.0 | 0.0 | 0.0 | 19.0 | 13.0 | 11.0 | 8.0 | 0.0 | 71.0 | 6.0 | 7.0 | 2.0 | 2.0 | 0.0 | 0.0 | 1.0 | 1.0 | 0.0 | 1.0 | 16.0 | 0.0 | 0.0 | 6.0 | 3.0 |
| 9 | 7.0 | 0.0 | 5.0 | 93.0 | 138.0 | 26.0 | 11.0 | 2.0 | 0.0 | 237.0 | 55.0 | 71.0 | 13.0 | 14.0 | 33.0 | 20.0 | 17.0 | 6.0 | 4.0 | 338.0 | 12.0 | 2.0 | 37.0 | 9.0 |
| 10 | 60.0 | 0.0 | 3.0 | 1603.0 | 611.0 | 99.0 | 116.0 | 27.0 | 0.0 | 49.0 | 1312.0 | 64.0 | 100.0 | 24.0 | 133.0 | 55.0 | 9.0 | 12.0 | 11.0 | 767.0 | 22.0 | 15.0 | 319.0 | 65.0 |
| 11 | 76.0 | 0.0 | 73.0 | 3.0 | 27.0 | 111.0 | 6.0 | 1.0 | 0.0 | 0.0 | 107.0 | 466.0 | 39.0 | 0.0 | 95.0 | 10.0 | 2.0 | 5.0 | 2.0 | 97.0 | 7.0 | 3.0 | 65.0 | 10.0 |
| 12 | 6.0 | 0.0 | 34.0 | 14.0 | 4.0 | 95.0 | 50.0 | 8.0 | 0.0 | 0.0 | 637.0 | 81.0 | 799.0 | 3.0 | 58.0 | 26.0 | 47.0 | 7.0 | 6.0 | 569.0 | 14.0 | 13.0 | 228.0 | 24.0 |
| 13 | 3.0 | 0.0 | 0.0 | 15.0 | 51.0 | 9.0 | 18.0 | 5.0 | 0.0 | 23.0 | 29.0 | 1.0 | 17.0 | 3.0 | 25.0 | 7.0 | 6.0 | 0.0 | 2.0 | 66.0 | 2.0 | 6.0 | 35.0 | 7.0 |
| 14 | 11.0 | 0.0 | 6.0 | 1.0 | 20.0 | 60.0 | 45.0 | 4.0 | 0.0 | 3.0 | 85.0 | 8.0 | 17.0 | 2.0 | 258.0 | 50.0 | 8.0 | 0.0 | 3.0 | 72.0 | 5.0 | 9.0 | 140.0 | 16.0 |
| 15 | 13.0 | 0.0 | 12.0 | 0.0 | 21.0 | 36.0 | 10.0 | 8.0 | 0.0 | 5.0 | 37.0 | 0.0 | 9.0 | 0.0 | 68.0 | 225.0 | 1.0 | 0.0 | 5.0 | 110.0 | 13.0 | 6.0 | 110.0 | 22.0 |
| 16 | 8.0 | 0.0 | 0.0 | 0.0 | 2.0 | 28.0 | 89.0 | 2.0 | 0.0 | 0.0 | 103.0 | 0.0 | 18.0 | 1.0 | 31.0 | 11.0 | 600.0 | 5.0 | 3.0 | 155.0 | 6.0 | 9.0 | 23.0 | 9.0 |
| 17 | 4.0 | 0.0 | 0.0 | 7.0 | 0.0 | 38.0 | 52.0 | 11.0 | 0.0 | 0.0 | 128.0 | 0.0 | 85.0 | 1.0 | 95.0 | 35.0 | 31.0 | 69.0 | 9.0 | 321.0 | 11.0 | 20.0 | 48.0 | 28.0 |
| 18 | 3.0 | 0.0 | 0.0 | 0.0 | 0.0 | 17.0 | 0.0 | 8.0 | 0.0 | 0.0 | 39.0 | 7.0 | 14.0 | 3.0 | 85.0 | 3.0 | 0.0 | 2.0 | 167.0 | 26.0 | 4.0 | 4.0 | 22.0 | 14.0 |
| 19 | 160.0 | 0.0 | 467.0 | 794.0 | 240.0 | 593.0 | 835.0 | 809.0 | 4.0 | 70.0 | 935.0 | 300.0 | 992.0 | 10.0 | 715.0 | 209.0 | 430.0 | 43.0 | 63.0 | 9006.0 | 145.0 | 939.0 | 729.0 | 144.0 |
| 20 | 46.0 | 0.0 | 9753.0 | 1453.0 | 632.0 | 1043.0 | 168.0 | 91.0 | 0.0 | 4.0 | 1025.0 | 325.0 | 960.0 | 0.0 | 81.0 | 175.0 | 3.0 | 49.0 | 9.0 | 2580.0 | 2527.0 | 247.0 | 1298.0 | 241.0 |
| 21 | 27.0 | 0.0 | 7.0 | 0.0 | 0.0 | 485.0 | 136.0 | 205.0 | 0.0 | 35.0 | 139.0 | 0.0 | 174.0 | 1.0 | 101.0 | 65.0 | 0.0 | 162.0 | 72.0 | 558.0 | 260.0 | 1000.0 | 853.0 | 238.0 |
| 22 | 67.0 | 0.0 | 234.0 | 30.0 | 11.0 | 2051.0 | 351.0 | 591.0 | 83.0 | 16.0 | 128.0 | 10.0 | 33.0 | 0.0 | 68.0 | 128.0 | 18.0 | 90.0 | 9.0 | 475.0 | 134.0 | 342.0 | 4676.0 | 118.0 |
| 23 | 374.0 | 0.0 | 97.0 | 35.0 | 31.0 | 1025.0 | 178.0 | 802.0 | 1121.0 | 398.0 | 310.0 | 62.0 | 260.0 | 20.0 | 256.0 | 771.0 | 94.0 | 299.0 | 2.0 | 1430.0 | 3530.0 | 2075.0 | 978.0 | 26460.0 |

I'll add up these progression values for whichever order I choose to maximize the grand total.

If you want to hand me the final lineup, just drop it into this simple JSON shape so it's easy to pick up and check:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Think of that ""solution"" list like the single continuous tasting order: the first entry is the strip you sniff first, the second is next, and so on until the last. This is just a sketch of the shape I need, not the actual answer — replace those placeholders with the real strip identifiers from the instance.

Please 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”.","[[925.0, 770.0, 59.0, 39.0, 25.0, 479.0, 29.0, 7.0, 0.0, 39.0, 87.0, 3.0, 18.0, 0.0, 14.0, 54.0, 7.0, 9.0, 3.0, 188.0, 76.0, 16.0, 50.0, 17.0], [1.0, 122.0, 2.0, 9.0, 1.0, 33.0, 0.0, 1.0, 0.0, 0.0, 9.0, 0.0, 0.0, 0.0, 1.0, 1.0, 0.0, 1.0, 0.0, 7.0, 6.0, 0.0, 11.0, 8.0], [74.0, 31.0, 2985.0, 201.0, 17.0, 749.0, 32.0, 31.0, 0.0, 4.0, 74.0, 13.0, 43.0, 0.0, 140.0, 29.0, 3.0, 17.0, 5.0, 444.0, 80.0, 29.0, 1288.0, 40.0], [442.0, 47.0, 466.0, 4932.0, 102.0, 350.0, 14.0, 35.0, 0.0, 8.0, 139.0, 8.0, 6.0, 0.0, 9.0, 32.0, 1.0, 4.0, 9.0, 223.0, 51.0, 10.0, 497.0, 47.0], [60.0, 0.0, 10.0, 123.0, 2488.0, 54.0, 14.0, 31.0, 0.0, 7.0, 142.0, 1.0, 8.0, 0.0, 8.0, 12.0, 2.0, 2.0, 3.0, 145.0, 12.0, 0.0, 143.0, 10.0], [39.0, 185.0, 7.0, 47.0, 4.0, 5243.0, 18.0, 2.0, 0.0, 8.0, 71.0, 1.0, 2.0, 0.0, 21.0, 5.0, 1.0, 0.0, 1.0, 16.0, 12.0, 7.0, 631.0, 32.0], [8.0, 0.0, 0.0, 25.0, 1.0, 55.0, 451.0, 6.0, 0.0, 9.0, 277.0, 0.0, 22.0, 0.0, 80.0, 18.0, 10.0, 0.0, 4.0, 151.0, 10.0, 9.0, 60.0, 15.0], [120.0, 0.0, 12.0, 1939.0, 521.0, 454.0, 1626.0, 4736.0, 0.0, 311.0, 1293.0, 330.0, 39.0, 0.0, 48.0, 32.0, 28.0, 4.0, 17.0, 867.0, 182.0, 80.0, 305.0, 69.0], [2.0, 0.0, 0.0, 19.0, 13.0, 11.0, 8.0, 0.0, 71.0, 6.0, 7.0, 2.0, 2.0, 0.0, 0.0, 1.0, 1.0, 0.0, 1.0, 16.0, 0.0, 0.0, 6.0, 3.0], [7.0, 0.0, 5.0, 93.0, 138.0, 26.0, 11.0, 2.0, 0.0, 237.0, 55.0, 71.0, 13.0, 14.0, 33.0, 20.0, 17.0, 6.0, 4.0, 338.0, 12.0, 2.0, 37.0, 9.0], [60.0, 0.0, 3.0, 1603.0, 611.0, 99.0, 116.0, 27.0, 0.0, 49.0, 1312.0, 64.0, 100.0, 24.0, 133.0, 55.0, 9.0, 12.0, 11.0, 767.0, 22.0, 15.0, 319.0, 65.0], [76.0, 0.0, 73.0, 3.0, 27.0, 111.0, 6.0, 1.0, 0.0, 0.0, 107.0, 466.0, 39.0, 0.0, 95.0, 10.0, 2.0, 5.0, 2.0, 97.0, 7.0, 3.0, 65.0, 10.0], [6.0, 0.0, 34.0, 14.0, 4.0, 95.0, 50.0, 8.0, 0.0, 0.0, 637.0, 81.0, 799.0, 3.0, 58.0, 26.0, 47.0, 7.0, 6.0, 569.0, 14.0, 13.0, 228.0, 24.0], [3.0, 0.0, 0.0, 15.0, 51.0, 9.0, 18.0, 5.0, 0.0, 23.0, 29.0, 1.0, 17.0, 3.0, 25.0, 7.0, 6.0, 0.0, 2.0, 66.0, 2.0, 6.0, 35.0, 7.0], [11.0, 0.0, 6.0, 1.0, 20.0, 60.0, 45.0, 4.0, 0.0, 3.0, 85.0, 8.0, 17.0, 2.0, 258.0, 50.0, 8.0, 0.0, 3.0, 72.0, 5.0, 9.0, 140.0, 16.0], [13.0, 0.0, 12.0, 0.0, 21.0, 36.0, 10.0, 8.0, 0.0, 5.0, 37.0, 0.0, 9.0, 0.0, 68.0, 225.0, 1.0, 0.0, 5.0, 110.0, 13.0, 6.0, 110.0, 22.0], [8.0, 0.0, 0.0, 0.0, 2.0, 28.0, 89.0, 2.0, 0.0, 0.0, 103.0, 0.0, 18.0, 1.0, 31.0, 11.0, 600.0, 5.0, 3.0, 155.0, 6.0, 9.0, 23.0, 9.0], [4.0, 0.0, 0.0, 7.0, 0.0, 38.0, 52.0, 11.0, 0.0, 0.0, 128.0, 0.0, 85.0, 1.0, 95.0, 35.0, 31.0, 69.0, 9.0, 321.0, 11.0, 20.0, 48.0, 28.0], [3.0, 0.0, 0.0, 0.0, 0.0, 17.0, 0.0, 8.0, 0.0, 0.0, 39.0, 7.0, 14.0, 3.0, 85.0, 3.0, 0.0, 2.0, 167.0, 26.0, 4.0, 4.0, 22.0, 14.0], [160.0, 0.0, 467.0, 794.0, 240.0, 593.0, 835.0, 809.0, 4.0, 70.0, 935.0, 300.0, 992.0, 10.0, 715.0, 209.0, 430.0, 43.0, 63.0, 9006.0, 145.0, 939.0, 729.0, 144.0], [46.0, 0.0, 9753.0, 1453.0, 632.0, 1043.0, 168.0, 91.0, 0.0, 4.0, 1025.0, 325.0, 960.0, 0.0, 81.0, 175.0, 3.0, 49.0, 9.0, 2580.0, 2527.0, 247.0, 1298.0, 241.0], [27.0, 0.0, 7.0, 0.0, 0.0, 485.0, 136.0, 205.0, 0.0, 35.0, 139.0, 0.0, 174.0, 1.0, 101.0, 65.0, 0.0, 162.0, 72.0, 558.0, 260.0, 1000.0, 853.0, 238.0], [67.0, 0.0, 234.0, 30.0, 11.0, 2051.0, 351.0, 591.0, 83.0, 16.0, 128.0, 10.0, 33.0, 0.0, 68.0, 128.0, 18.0, 90.0, 9.0, 475.0, 134.0, 342.0, 4676.0, 118.0], [374.0, 0.0, 97.0, 35.0, 31.0, 1025.0, 178.0, 802.0, 1121.0, 398.0, 310.0, 62.0, 260.0, 20.0, 256.0, 771.0, 94.0, 299.0, 2.0, 1430.0, 3530.0, 2075.0, 978.0, 26460.0]]","[23, 13, 7, 9, 20, 17, 19, 21, 18, 12, 11, 16, 10, 4, 3, 2, 14, 22, 8, 0, 6, 15, 5, 1]",69345.0,"{'problem_type': 'LOP', 'num_nodes': 24, 'nodes': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], 'pairs': [{'from_id': 0, 'to_id': 1, 'weight': 770.0}, {'from_id': 0, 'to_id': 2, 'weight': 59.0}, {'from_id': 0, 'to_id': 3, 'weight': 39.0}, {'from_id': 0, 'to_id': 4, 'weight': 25.0}, {'from_id': 0, 'to_id': 5, 'weight': 479.0}, {'from_id': 0, 'to_id': 6, 'weight': 29.0}, {'from_id': 0, 'to_id': 7, 'weight': 7.0}, {'from_id': 0, 'to_id': 8, 'weight': 0.0}, {'from_id': 0, 'to_id': 9, 'weight': 39.0}, {'from_id': 0, 'to_id': 10, 'weight': 87.0}, {'from_id': 0, 'to_id': 11, 'weight': 3.0}, {'from_id': 0, 'to_id': 12, 'weight': 18.0}, {'from_id': 0, 'to_id': 13, 'weight': 0.0}, {'from_id': 0, 'to_id': 14, 'weight': 14.0}, {'from_id': 0, 'to_id': 15, 'weight': 54.0}, {'from_id': 0, 'to_id': 16, 'weight': 7.0}, {'from_id': 0, 'to_id': 17, 'weight': 9.0}, {'from_id': 0, 'to_id': 18, 'weight': 3.0}, {'from_id': 0, 'to_id': 19, 'weight': 188.0}, {'from_id': 0, 'to_id': 20, 'weight': 76.0}, {'from_id': 0, 'to_id': 21, 'weight': 16.0}, {'from_id': 0, 'to_id': 22, 'weight': 50.0}, {'from_id': 0, 'to_id': 23, 'weight': 17.0}, {'from_id': 1, 'to_id': 0, 'weight': 1.0}, {'from_id': 1, 'to_id': 2, 'weight': 2.0}, {'from_id': 1, 'to_id': 3, 'weight': 9.0}, {'from_id': 1, 'to_id': 4, 'weight': 1.0}, {'from_id': 1, 'to_id': 5, 'weight': 33.0}, {'from_id': 1, 'to_id': 6, 'weight': 0.0}, {'from_id': 1, 'to_id': 7, 'weight': 1.0}, {'from_id': 1, 'to_id': 8, 'weight': 0.0}, {'from_id': 1, 'to_id': 9, 'weight': 0.0}, {'from_id': 1, 'to_id': 10, 'weight': 9.0}, {'from_id': 1, 'to_id': 11, 'weight': 0.0}, {'from_id': 1, 'to_id': 12, 'weight': 0.0}, {'from_id': 1, 'to_id': 13, 'weight': 0.0}, {'from_id': 1, 'to_id': 14, 'weight': 1.0}, {'from_id': 1, 'to_id': 15, 'weight': 1.0}, {'from_id': 1, 'to_id': 16, 'weight': 0.0}, {'from_id': 1, 'to_id': 17, 'weight': 1.0}, {'from_id': 1, 'to_id': 18, 'weight': 0.0}, {'from_id': 1, 'to_id': 19, 'weight': 7.0}, {'from_id': 1, 'to_id': 20, 'weight': 6.0}, {'from_id': 1, 'to_id': 21, 'weight': 0.0}, {'from_id': 1, 'to_id': 22, 'weight': 11.0}, {'from_id': 1, 'to_id': 23, 'weight': 8.0}, {'from_id': 2, 'to_id': 0, 'weight': 74.0}, {'from_id': 2, 'to_id': 1, 'weight': 31.0}, {'from_id': 2, 'to_id': 3, 'weight': 201.0}, {'from_id': 2, 'to_id': 4, 'weight': 17.0}, {'from_id': 2, 'to_id': 5, 'weight': 749.0}, {'from_id': 2, 'to_id': 6, 'weight': 32.0}, {'from_id': 2, 'to_id': 7, 'weight': 31.0}, {'from_id': 2, 'to_id': 8, 'weight': 0.0}, {'from_id': 2, 'to_id': 9, 'weight': 4.0}, {'from_id': 2, 'to_id': 10, 'weight': 74.0}, {'from_id': 2, 'to_id': 11, 'weight': 13.0}, {'from_id': 2, 'to_id': 12, 'weight': 43.0}, {'from_id': 2, 'to_id': 13, 'weight': 0.0}, {'from_id': 2, 'to_id': 14, 'weight': 140.0}, {'from_id': 2, 'to_id': 15, 'weight': 29.0}, {'from_id': 2, 'to_id': 16, 'weight': 3.0}, {'from_id': 2, 'to_id': 17, 'weight': 17.0}, {'from_id': 2, 'to_id': 18, 'weight': 5.0}, {'from_id': 2, 'to_id': 19, 'weight': 444.0}, {'from_id': 2, 'to_id': 20, 'weight': 80.0}, {'from_id': 2, 'to_id': 21, 'weight': 29.0}, {'from_id': 2, 'to_id': 22, 'weight': 1288.0}, {'from_id': 2, 'to_id': 23, 'weight': 40.0}, {'from_id': 3, 'to_id': 0, 'weight': 442.0}, {'from_id': 3, 'to_id': 1, 'weight': 47.0}, 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20, 'to_id': 10, 'weight': 1025.0}, {'from_id': 20, 'to_id': 11, 'weight': 325.0}, {'from_id': 20, 'to_id': 12, 'weight': 960.0}, {'from_id': 20, 'to_id': 13, 'weight': 0.0}, {'from_id': 20, 'to_id': 14, 'weight': 81.0}, {'from_id': 20, 'to_id': 15, 'weight': 175.0}, {'from_id': 20, 'to_id': 16, 'weight': 3.0}, {'from_id': 20, 'to_id': 17, 'weight': 49.0}, {'from_id': 20, 'to_id': 18, 'weight': 9.0}, {'from_id': 20, 'to_id': 19, 'weight': 2580.0}, {'from_id': 20, 'to_id': 21, 'weight': 247.0}, {'from_id': 20, 'to_id': 22, 'weight': 1298.0}, {'from_id': 20, 'to_id': 23, 'weight': 241.0}, {'from_id': 21, 'to_id': 0, 'weight': 27.0}, {'from_id': 21, 'to_id': 1, 'weight': 0.0}, {'from_id': 21, 'to_id': 2, 'weight': 7.0}, {'from_id': 21, 'to_id': 3, 'weight': 0.0}, {'from_id': 21, 'to_id': 4, 'weight': 0.0}, {'from_id': 21, 'to_id': 5, 'weight': 485.0}, {'from_id': 21, 'to_id': 6, 'weight': 136.0}, {'from_id': 21, 'to_id': 7, 'weight': 205.0}, {'from_id': 21, 'to_id': 8, 'weight': 0.0}, 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'weight': 591.0}, {'from_id': 22, 'to_id': 8, 'weight': 83.0}, {'from_id': 22, 'to_id': 9, 'weight': 16.0}, {'from_id': 22, 'to_id': 10, 'weight': 128.0}, {'from_id': 22, 'to_id': 11, 'weight': 10.0}, {'from_id': 22, 'to_id': 12, 'weight': 33.0}, {'from_id': 22, 'to_id': 13, 'weight': 0.0}, {'from_id': 22, 'to_id': 14, 'weight': 68.0}, {'from_id': 22, 'to_id': 15, 'weight': 128.0}, {'from_id': 22, 'to_id': 16, 'weight': 18.0}, {'from_id': 22, 'to_id': 17, 'weight': 90.0}, {'from_id': 22, 'to_id': 18, 'weight': 9.0}, {'from_id': 22, 'to_id': 19, 'weight': 475.0}, {'from_id': 22, 'to_id': 20, 'weight': 134.0}, {'from_id': 22, 'to_id': 21, 'weight': 342.0}, {'from_id': 22, 'to_id': 23, 'weight': 118.0}, {'from_id': 23, 'to_id': 0, 'weight': 374.0}, {'from_id': 23, 'to_id': 1, 'weight': 0.0}, {'from_id': 23, 'to_id': 2, 'weight': 97.0}, {'from_id': 23, 'to_id': 3, 'weight': 35.0}, {'from_id': 23, 'to_id': 4, 'weight': 31.0}, {'from_id': 23, 'to_id': 5, 'weight': 1025.0}, {'from_id': 23, 'to_id': 6, 'weight': 178.0}, {'from_id': 23, 'to_id': 7, 'weight': 802.0}, {'from_id': 23, 'to_id': 8, 'weight': 1121.0}, {'from_id': 23, 'to_id': 9, 'weight': 398.0}, {'from_id': 23, 'to_id': 10, 'weight': 310.0}, {'from_id': 23, 'to_id': 11, 'weight': 62.0}, {'from_id': 23, 'to_id': 12, 'weight': 260.0}, {'from_id': 23, 'to_id': 13, 'weight': 20.0}, {'from_id': 23, 'to_id': 14, 'weight': 256.0}, {'from_id': 23, 'to_id': 15, 'weight': 771.0}, {'from_id': 23, 'to_id': 16, 'weight': 94.0}, {'from_id': 23, 'to_id': 17, 'weight': 299.0}, {'from_id': 23, 'to_id': 18, 'weight': 2.0}, {'from_id': 23, 'to_id': 19, 'weight': 1430.0}, {'from_id': 23, 'to_id': 20, 'weight': 3530.0}, {'from_id': 23, 'to_id': 21, 'weight': 2075.0}, {'from_id': 23, 'to_id': 22, 'weight': 978.0}]}","[23, 13, 7, 9, 20, 17, 19, 21, 18, 12, 11, 16, 10, 4, 3, 2, 14, 22, 8, 0, 6, 15, 5, 1]",46,markdown_table,0
LOP,LOP,"Many people treat the morning like a tiny relay: each prep task passes the baton to the next, and every pass has a smoothness score. The job here is to set a single route that uses each task once, never repeats, and never leaves anything out. The quality of a route is simply the total you get by adding each pair’s smoothness when one task follows another, so the better route is the one with the largest total. The detailed tasks and their transition scores are provided below.

There are 24 distinct morning tasks: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23.

*Meaning: this is a directed transition_smoothness matrix over items in task_ids (24 total). Entry at row preceding_task_id=i and column following_task_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).*

**transition_smoothness)**

| preceding_task_id\following_task_id | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 340919.0 | 1742050.0 | 1244428.0 | 0.0 | 62285.0 | 15882.0 | 5325.0 | 2813.0 | 30140.0 | 758.0 | 20367.0 | 7757.0 | 9206.0 | 8782.0 | 1543.0 | 22624.0 | 16068.0 | 84134.0 | 1044.0 | 6746.0 | 0.0 | 1629.0 | 23348.0 | 273.0 |
| 1 | 11575.0 | 26777.0 | 2707.0 | 0.0 | 56467.0 | 49130.0 | 216.0 | 183.0 | 5763.0 | 453.0 | 5507.0 | 26.0 | 2299.0 | 2626.0 | 1107.0 | 21183.0 | 36388.0 | 6889.0 | 0.0 | 273.0 | 820.0 | 2260.0 | 4981.0 | 820.0 |
| 2 | 119161.0 | 226215.0 | 1054016.0 | 1260.0 | 743480.0 | 437660.0 | 57512.0 | 54001.0 | 290837.0 | 19665.0 | 344716.0 | 50485.0 | 204998.0 | 227781.0 | 38690.0 | 184722.0 | 1278406.0 | 340619.0 | 31795.0 | 34203.0 | 114624.0 | 107856.0 | 246501.0 | 37893.0 |
| 3 | 0.0 | 0.0 | 52488.0 | 37089.0 | 0.0 | 2.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 11.0 | 0.0 | 0.0 | 0.0 | 0.0 | 34022.0 | 0.0 |
| 4 | 12217.0 | 3024.0 | 41027.0 | 0.0 | 2725223.0 | 201519.0 | 39308.0 | 18561.0 | 166020.0 | 949.0 | 5519.0 | 6549.0 | 102381.0 | 77099.0 | 64391.0 | 10370324.0 | 106316.0 | 25918.0 | 2403.0 | 1803.0 | 3725.0 | 48622.0 | 40203.0 | 19191.0 |
| 5 | 106643.0 | 52548.0 | 141711.0 | 0.0 | 166985.0 | 3142868.0 | 153873.0 | 77661.0 | 470328.0 | 36206.0 | 228571.0 | 110144.0 | 644778.0 | 178149.0 | 140690.0 | 4174431.0 | 615426.0 | 104769.0 | 33024.0 | 32438.0 | 124999.0 | 25150.0 | 66755.0 | 8146.0 |
| 6 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1390184.0 | 4175.0 | 258734.0 | 0.0 | 19955.0 | 464504.0 | 33336.0 | 0.0 | 9546.0 | 0.0 | 0.0 | 1093.0 | 40896.0 | 820.0 | 0.0 | 154534.0 | 42766.0 | 42612.0 |
| 7 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 16237.0 | 35012.0 | 829392.0 | 0.0 | 316.0 | 0.0 | 0.0 | 1105.0 | 276.0 | 0.0 | 0.0 | 820.0 | 26024.0 | 820.0 | 0.0 | 72994.0 | 8285.0 | 33709.0 |
| 8 | 993.0 | 0.0 | 1093.0 | 0.0 | 900.0 | 4669.0 | 108680.0 | 57514.0 | 6838673.0 | 4964.0 | 14783.0 | 3131.0 | 4762.0 | 10067.0 | 0.0 | 3834.0 | 46193.0 | 5022.0 | 56689.0 | 3825.0 | 546.0 | 136325.0 | 27775.0 | 57072.0 |
| 9 | 1093.0 | 0.0 | 3279.0 | 0.0 | 2732.0 | 3552.0 | 1913.0 | 546.0 | 3005.0 | 40111.0 | 2732.0 | 1093.0 | 2459.0 | 1913.0 | 273.0 | 6604.0 | 55511.0 | 2459.0 | 12010.0 | 546.0 | 1366.0 | 0.0 | 20.0 | 273.0 |
| 10 | 2679.0 | 1173.0 | 10332.0 | 0.0 | 25549.0 | 39558.0 | 5571.0 | 3624.0 | 32421.0 | 3607.0 | 12381367.0 | 361217.0 | 608566.0 | 423943.0 | 33091.0 | 222574.0 | 256563.0 | 52645.0 | 20218.0 | 15731.0 | 2732.0 | 100472.0 | 77152.0 | 31496.0 |
| 11 | 2019.0 | 726.0 | 1726.0 | 0.0 | 3187.0 | 20433.0 | 546.0 | 0.0 | 1623.0 | 0.0 | 321924.0 | 1172944.0 | 30263.0 | 2479.0 | 5219.0 | 6592.0 | 50273.0 | 10921.0 | 847.0 | 3934.0 | 1093.0 | 13888.0 | 20968.0 | 7237.0 |
| 12 | 47698.0 | 96.0 | 35144.0 | 0.0 | 149086.0 | 231348.0 | 63627.0 | 14626.0 | 78731.0 | 5738.0 | 65923.0 | 30184.0 | 3597011.0 | 68135.0 | 55059.0 | 4141799.0 | 356631.0 | 36796.0 | 2961.0 | 24606.0 | 62075.0 | 34517.0 | 70822.0 | 15831.0 |
| 13 | 9628.0 | 880.0 | 30003.0 | 0.0 | 122576.0 | 231582.0 | 66262.0 | 69583.0 | 267786.0 | 13702.0 | 236350.0 | 257638.0 | 428957.0 | 726966.0 | 97348.0 | 913205.0 | 578110.0 | 241162.0 | 43879.0 | 17176.0 | 2455.0 | 96377.0 | 83731.0 | 4070.0 |
| 14 | 1726.0 | 1633.0 | 14453.0 | 0.0 | 5792.0 | 35587.0 | 7914.0 | 2925.0 | 14352.0 | 2261.0 | 23285.0 | 6158.0 | 29602.0 | 7513.0 | 352910.0 | 55176.0 | 125556.0 | 20764.0 | 5187.0 | 2331.0 | 2573.0 | 29147.0 | 47364.0 | 20555.0 |
| 15 | 83384.0 | 12398.0 | 384131.0 | 0.0 | 88167.0 | 170908.0 | 29529.0 | 13446.0 | 119852.0 | 30860.0 | 83892.0 | 11464.0 | 25197.0 | 32799.0 | 5697.0 | 220447.0 | 493119.0 | 375497.0 | 19187.0 | 71051.0 | 2455768.0 | 116375.0 | 275315.0 | 71440.0 |
| 16 | 41945.0 | 23659.0 | 298284.0 | 0.0 | 574240.0 | 811151.0 | 941391.0 | 213146.0 | 1264002.0 | 87458.0 | 1221502.0 | 324267.0 | 697559.0 | 408354.0 | 88962.0 | 2056306.0 | 1759837.0 | 497321.0 | 183065.0 | 86395.0 | 76324.0 | 368080.0 | 567577.0 | 99542.0 |
| 17 | 16871.0 | 37574.0 | 241804.0 | 0.0 | 498444.0 | 371789.0 | 120861.0 | 93226.0 | 364753.0 | 20270.0 | 219218.0 | 46573.0 | 151030.0 | 78462.0 | 28714.0 | 1292720.0 | 3234719.0 | 645966.0 | 164038.0 | 391765.0 | 14165.0 | 64157.0 | 85075.0 | 34497.0 |
| 18 | 1946.0 | 1796.0 | 10967.0 | 0.0 | 28331.0 | 31419.0 | 11162.0 | 5532.0 | 31258.0 | 5406.0 | 29275.0 | 5268.0 | 14389.0 | 10042.0 | 4137.0 | 22634.0 | 127381.0 | 95685.0 | 613164.0 | 136682.0 | 2532.0 | 30408.0 | 9147.0 | 4615.0 |
| 19 | 4430.0 | 12249.0 | 16371.0 | 0.0 | 243025.0 | 90741.0 | 53727.0 | 31475.0 | 171648.0 | 18995.0 | 96311.0 | 19229.0 | 75321.0 | 35702.0 | 24382.0 | 203747.0 | 865701.0 | 837073.0 | 905176.0 | 379703.0 | 273.0 | 1157.0 | 5364.0 | 6388.0 |
| 20 | 5206.0 | 1988.0 | 12909.0 | 0.0 | 38119.0 | 28538.0 | 7464.0 | 6003.0 | 56693.0 | 1528.0 | 78684.0 | 27320.0 | 45156.0 | 16189.0 | 11368.0 | 91696.0 | 738809.0 | 42151.0 | 6669.0 | 47752.0 | 180.0 | 106527.0 | 71925.0 | 12312.0 |
| 21 | 380.0 | 0.0 | 2732.0 | 0.0 | 2186.0 | 2732.0 | 2186.0 | 0.0 | 2186.0 | 273.0 | 1913.0 | 546.0 | 2186.0 | 1093.0 | 546.0 | 3552.0 | 34426.0 | 5591.0 | 820.0 | 2732.0 | 2186.0 | 0.0 | 68400.0 | 3421.0 |
| 22 | 0.0 | 0.0 | 552.0 | 0.0 | 0.0 | 276.0 | 0.0 | 0.0 | 0.0 | 0.0 | 276.0 | 0.0 | 276.0 | 0.0 | 0.0 | 276.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1657.0 | 552.0 | 2762.0 | 0.0 |
| 23 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 7735.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 829.0 | 0.0 |

The goal is the single route that uses each task once and maximizes the total smoothness.

If you want to hand the route back in a tidy, machine-friendly way, just use this little JSON shape when you reply:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

That's just the shape I expect: ""solution"" is an ordered list from the very first task to the very last one. Think of it like a short form where you drop in the tasks in the order you picked them — one appearance each, nothing duplicated. The JSON above is just a sketch of the format, not the actual answer.

Please make sure you use the exact identifiers from the instance input — don't 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”.""","[[340919.0, 1742050.0, 1244428.0, 0.0, 62285.0, 15882.0, 5325.0, 2813.0, 30140.0, 758.0, 20367.0, 7757.0, 9206.0, 8782.0, 1543.0, 22624.0, 16068.0, 84134.0, 1044.0, 6746.0, 0.0, 1629.0, 23348.0, 273.0], [11575.0, 26777.0, 2707.0, 0.0, 56467.0, 49130.0, 216.0, 183.0, 5763.0, 453.0, 5507.0, 26.0, 2299.0, 2626.0, 1107.0, 21183.0, 36388.0, 6889.0, 0.0, 273.0, 820.0, 2260.0, 4981.0, 820.0], [119161.0, 226215.0, 1054016.0, 1260.0, 743480.0, 437660.0, 57512.0, 54001.0, 290837.0, 19665.0, 344716.0, 50485.0, 204998.0, 227781.0, 38690.0, 184722.0, 1278406.0, 340619.0, 31795.0, 34203.0, 114624.0, 107856.0, 246501.0, 37893.0], [0.0, 0.0, 52488.0, 37089.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 11.0, 0.0, 0.0, 0.0, 0.0, 34022.0, 0.0], [12217.0, 3024.0, 41027.0, 0.0, 2725223.0, 201519.0, 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'weight': 0.0}, {'from_id': 23, 'to_id': 16, 'weight': 7735.0}, {'from_id': 23, 'to_id': 17, 'weight': 0.0}, {'from_id': 23, 'to_id': 18, 'weight': 0.0}, {'from_id': 23, 'to_id': 19, 'weight': 0.0}, {'from_id': 23, 'to_id': 20, 'weight': 0.0}, {'from_id': 23, 'to_id': 21, 'weight': 0.0}, {'from_id': 23, 'to_id': 22, 'weight': 829.0}]}","[3, 0, 2, 19, 13, 17, 1, 16, 5, 10, 12, 4, 14, 15, 20, 9, 7, 6, 8, 18, 11, 21, 23, 22]",47,markdown_table,0
LOP,LOP,"We’ve got a conference schedule to build: one straight line of presenters, each taking a single slot, with no repeats and no missing people. Between every two speakers someone has worked out how much more engaging it is if A speaks before B — those are the little gains on the sheet. For any complete ordering, add up all the gains that apply (every time one presenter appears ahead of another you include that number) and the total is the event’s audience score. The goal is to arrange the speakers so that total score is as high as possible. The concrete numbers and names are listed below.

There are 20 presenters to place in that single sequence: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T.

*Meaning: this is a directed engagement_gain matrix over items in presenter_ids (20 total). Entry at row preceding_presenter_id=i and column succeeding_presenter_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).*

**engagement_gain)**

| preceding_presenter_id\succeeding_presenter_id | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 568.0 | 1431.0 | 3.0 | 898.0 | 145.0 | 267.0 | 16.0 | 2.0 | 28.0 | 13.0 | 10.0 | 3.0 | 87.0 | 6.0 | 0.0 | 46.0 | 37.0 | 1.0 | 99.0 | 6.0 |
| B | 0.0 | 80.0 | 0.0 | 1.0 | 1605.0 | 41.0 | 33.0 | 0.0 | 11.0 | 2.0 | 2.0 | 0.0 | 1.0 | 0.0 | 0.0 | 13.0 | 7.0 | 0.0 | 21.0 | 2.0 |
| C | 26.0 | 0.0 | 5624.0 | 754.0 | 293.0 | 542.0 | 543.0 | 36.0 | 89.0 | 50.0 | 74.0 | 0.0 | 237.0 | 143.0 | 430.0 | 1600.0 | 150.0 | 129.0 | 615.0 | 11.0 |
| D | 234.0 | 163.0 | 303.0 | 755.0 | 1193.0 | 408.0 | 221.0 | 20.0 | 150.0 | 81.0 | 109.0 | 5.0 | 282.0 | 126.0 | 68.0 | 666.0 | 247.0 | 42.0 | 585.0 | 30.0 |
| E | 67.0 | 0.0 | 122.0 | 45.0 | 15913.0 | 39.0 | 3619.0 | 210.0 | 1805.0 | 2.0 | 0.0 | 3.0 | 2.0 | 41.0 | 114.0 | 32.0 | 0.0 | 1.0 | 87.0 | 2.0 |
| F | 1.0 | 8.0 | 14.0 | 51.0 | 465.0 | 696.0 | 98.0 | 65.0 | 199.0 | 23.0 | 172.0 | 2.0 | 21.0 | 83.0 | 80.0 | 117.0 | 22.0 | 0.0 | 9.0 | 0.0 |
| G | 59.0 | 23.0 | 346.0 | 172.0 | 568.0 | 192.0 | 1804.0 | 154.0 | 820.0 | 24.0 | 30.0 | 2.0 | 86.0 | 22.0 | 637.0 | 75.0 | 49.0 | 51.0 | 966.0 | 3.0 |
| H | 1.0 | 0.0 | 12.0 | 5.0 | 11.0 | 2.0 | 132.0 | 39.0 | 72.0 | 1.0 | 2.0 | 0.0 | 1.0 | 0.0 | 19.0 | 0.0 | 3.0 | 0.0 | 207.0 | 4.0 |
| I | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 145.0 | 0.0 | 3139.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2684.0 | 48.0 | 5.0 | 0.0 | 563.0 | 1.0 |
| J | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 652.0 | 0.0 | 21.0 | 0.0 |
| K | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 196.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 3700.0 | 0.0 | 3.0 | 0.0 |
| L | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| M | 9.0 | 0.0 | 0.0 | 15.0 | 9.0 | 8.0 | 166.0 | 9.0 | 480.0 | 0.0 | 17.0 | 1.0 | 12481.0 | 246.0 | 782.0 | 232.0 | 109.0 | 0.0 | 20.0 | 0.0 |
| N | 0.0 | 0.0 | 0.0 | 0.0 | 5.0 | 41.0 | 156.0 | 35.0 | 28.0 | 10.0 | 44.0 | 0.0 | 9.0 | 2705.0 | 0.0 | 356.0 | 129.0 | 0.0 | 155.0 | 5.0 |
| O | 3.0 | 0.0 | 98.0 | 84.0 | 1226.0 | 200.0 | 191.0 | 33.0 | 73.0 | 68.0 | 54.0 | 0.0 | 400.0 | 90.0 | 6.0 | 384.0 | 36.0 | 32.0 | -42.0 | 3.0 |
| P | 16.0 | 6.0 | 86.0 | 9.0 | 168.0 | 167.0 | 263.0 | 120.0 | 108.0 | 25.0 | 120.0 | 0.0 | 670.0 | 431.0 | 880.0 | 906.0 | 1786.0 | 27.0 | 130.0 | 6.0 |
| Q | 3.0 | 0.0 | 61.0 | 39.0 | 71.0 | 57.0 | 105.0 | 13.0 | 43.0 | 27.0 | 53.0 | 1.0 | 111.0 | 44.0 | 42.0 | 874.0 | 0.0 | 21.0 | 313.0 | 8.0 |
| R | 17.0 | 5.0 | 162.0 | 30.0 | 279.0 | 74.0 | 164.0 | 24.0 | 92.0 | 13.0 | 50.0 | 1.0 | 126.0 | 50.0 | 18.0 | 974.0 | 13.0 | 4.0 | 193.0 | 0.0 |
| S | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| T | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

We'll arrange them to maximize the summed gains and the total audience score.

Oh, and when you return the final ordering, it'd be great if you use this simple JSON layout so it's easy to read and plug into whatever's checking the schedule:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Here the ""solution"" array is just the lineup of presenters from the very first slot to the very last slot — list each identifier once, in order. This JSON is just a sketch of the shape I expect, not the actual answer itself.

Please make sure to use the exact identifiers from the instance input — do not rename them or invent new ones.
- 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”.""","[[568.0, 1431.0, 3.0, 898.0, 145.0, 267.0, 16.0, 2.0, 28.0, 13.0, 10.0, 3.0, 87.0, 6.0, 0.0, 46.0, 37.0, 1.0, 99.0, 6.0], [0.0, 80.0, 0.0, 1.0, 1605.0, 41.0, 33.0, 0.0, 11.0, 2.0, 2.0, 0.0, 1.0, 0.0, 0.0, 13.0, 7.0, 0.0, 21.0, 2.0], [26.0, 0.0, 5624.0, 754.0, 293.0, 542.0, 543.0, 36.0, 89.0, 50.0, 74.0, 0.0, 237.0, 143.0, 430.0, 1600.0, 150.0, 129.0, 615.0, 11.0], [234.0, 163.0, 303.0, 755.0, 1193.0, 408.0, 221.0, 20.0, 150.0, 81.0, 109.0, 5.0, 282.0, 126.0, 68.0, 666.0, 247.0, 42.0, 585.0, 30.0], [67.0, 0.0, 122.0, 45.0, 15913.0, 39.0, 3619.0, 210.0, 1805.0, 2.0, 0.0, 3.0, 2.0, 41.0, 114.0, 32.0, 0.0, 1.0, 87.0, 2.0], [1.0, 8.0, 14.0, 51.0, 465.0, 696.0, 98.0, 65.0, 199.0, 23.0, 172.0, 2.0, 21.0, 83.0, 80.0, 117.0, 22.0, 0.0, 9.0, 0.0], [59.0, 23.0, 346.0, 172.0, 568.0, 192.0, 1804.0, 154.0, 820.0, 24.0, 30.0, 2.0, 86.0, 22.0, 637.0, 75.0, 49.0, 51.0, 966.0, 3.0], [1.0, 0.0, 12.0, 5.0, 11.0, 2.0, 132.0, 39.0, 72.0, 1.0, 2.0, 0.0, 1.0, 0.0, 19.0, 0.0, 3.0, 0.0, 207.0, 4.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 145.0, 0.0, 3139.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2684.0, 48.0, 5.0, 0.0, 563.0, 1.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 652.0, 0.0, 21.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 196.0, 0.0, 0.0, 0.0, 0.0, 0.0, 3700.0, 0.0, 3.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [9.0, 0.0, 0.0, 15.0, 9.0, 8.0, 166.0, 9.0, 480.0, 0.0, 17.0, 1.0, 12481.0, 246.0, 782.0, 232.0, 109.0, 0.0, 20.0, 0.0], [0.0, 0.0, 0.0, 0.0, 5.0, 41.0, 156.0, 35.0, 28.0, 10.0, 44.0, 0.0, 9.0, 2705.0, 0.0, 356.0, 129.0, 0.0, 155.0, 5.0], [3.0, 0.0, 98.0, 84.0, 1226.0, 200.0, 191.0, 33.0, 73.0, 68.0, 54.0, 0.0, 400.0, 90.0, 6.0, 384.0, 36.0, 32.0, -42.0, 3.0], [16.0, 6.0, 86.0, 9.0, 168.0, 167.0, 263.0, 120.0, 108.0, 25.0, 120.0, 0.0, 670.0, 431.0, 880.0, 906.0, 1786.0, 27.0, 130.0, 6.0], [3.0, 0.0, 61.0, 39.0, 71.0, 57.0, 105.0, 13.0, 43.0, 27.0, 53.0, 1.0, 111.0, 44.0, 42.0, 874.0, 0.0, 21.0, 313.0, 8.0], [17.0, 5.0, 162.0, 30.0, 279.0, 74.0, 164.0, 24.0, 92.0, 13.0, 50.0, 1.0, 126.0, 50.0, 18.0, 974.0, 13.0, 4.0, 193.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]]","[17, 2, 0, 3, 1, 15, 5, 12, 13, 10, 9, 16, 4, 6, 7, 8, 18, 14, 19, 11]",42531.0,"{'problem_type': 'LOP', 'num_nodes': 20, 'nodes': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S', 'T'], 'pairs': [{'from_id': 'A', 'to_id': 'B', 'weight': 1431.0}, {'from_id': 'A', 'to_id': 'C', 'weight': 3.0}, {'from_id': 'A', 'to_id': 'D', 'weight': 898.0}, {'from_id': 'A', 'to_id': 'E', 'weight': 145.0}, {'from_id': 'A', 'to_id': 'F', 'weight': 267.0}, {'from_id': 'A', 'to_id': 'G', 'weight': 16.0}, {'from_id': 'A', 'to_id': 'H', 'weight': 2.0}, {'from_id': 'A', 'to_id': 'I', 'weight': 28.0}, {'from_id': 'A', 'to_id': 'J', 'weight': 13.0}, {'from_id': 'A', 'to_id': 'K', 'weight': 10.0}, {'from_id': 'A', 'to_id': 'L', 'weight': 3.0}, {'from_id': 'A', 'to_id': 'M', 'weight': 87.0}, {'from_id': 'A', 'to_id': 'N', 'weight': 6.0}, {'from_id': 'A', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'P', 'weight': 46.0}, {'from_id': 'A', 'to_id': 'Q', 'weight': 37.0}, {'from_id': 'A', 'to_id': 'R', 'weight': 1.0}, {'from_id': 'A', 'to_id': 'S', 'weight': 99.0}, {'from_id': 'A', 'to_id': 'T', 'weight': 6.0}, {'from_id': 'B', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'D', 'weight': 1.0}, {'from_id': 'B', 'to_id': 'E', 'weight': 1605.0}, {'from_id': 'B', 'to_id': 'F', 'weight': 41.0}, {'from_id': 'B', 'to_id': 'G', 'weight': 33.0}, {'from_id': 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'F', 'to_id': 'O', 'weight': 80.0}, {'from_id': 'F', 'to_id': 'P', 'weight': 117.0}, {'from_id': 'F', 'to_id': 'Q', 'weight': 22.0}, {'from_id': 'F', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'F', 'to_id': 'S', 'weight': 9.0}, {'from_id': 'F', 'to_id': 'T', 'weight': 0.0}, {'from_id': 'G', 'to_id': 'A', 'weight': 59.0}, {'from_id': 'G', 'to_id': 'B', 'weight': 23.0}, {'from_id': 'G', 'to_id': 'C', 'weight': 346.0}, {'from_id': 'G', 'to_id': 'D', 'weight': 172.0}, {'from_id': 'G', 'to_id': 'E', 'weight': 568.0}, {'from_id': 'G', 'to_id': 'F', 'weight': 192.0}, {'from_id': 'G', 'to_id': 'H', 'weight': 154.0}, {'from_id': 'G', 'to_id': 'I', 'weight': 820.0}, {'from_id': 'G', 'to_id': 'J', 'weight': 24.0}, {'from_id': 'G', 'to_id': 'K', 'weight': 30.0}, {'from_id': 'G', 'to_id': 'L', 'weight': 2.0}, {'from_id': 'G', 'to_id': 'M', 'weight': 86.0}, {'from_id': 'G', 'to_id': 'N', 'weight': 22.0}, {'from_id': 'G', 'to_id': 'O', 'weight': 637.0}, {'from_id': 'G', 'to_id': 'P', 'weight': 75.0}, {'from_id': 'G', 'to_id': 'Q', 'weight': 49.0}, {'from_id': 'G', 'to_id': 'R', 'weight': 51.0}, {'from_id': 'G', 'to_id': 'S', 'weight': 966.0}, {'from_id': 'G', 'to_id': 'T', 'weight': 3.0}, {'from_id': 'H', 'to_id': 'A', 'weight': 1.0}, {'from_id': 'H', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'C', 'weight': 12.0}, {'from_id': 'H', 'to_id': 'D', 'weight': 5.0}, {'from_id': 'H', 'to_id': 'E', 'weight': 11.0}, {'from_id': 'H', 'to_id': 'F', 'weight': 2.0}, {'from_id': 'H', 'to_id': 'G', 'weight': 132.0}, {'from_id': 'H', 'to_id': 'I', 'weight': 72.0}, {'from_id': 'H', 'to_id': 'J', 'weight': 1.0}, {'from_id': 'H', 'to_id': 'K', 'weight': 2.0}, {'from_id': 'H', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'M', 'weight': 1.0}, {'from_id': 'H', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'O', 'weight': 19.0}, {'from_id': 'H', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'Q', 'weight': 3.0}, {'from_id': 'H', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'H', 'to_id': 'S', 'weight': 207.0}, {'from_id': 'H', 'to_id': 'T', 'weight': 4.0}, {'from_id': 'I', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'G', 'weight': 145.0}, {'from_id': 'I', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'O', 'weight': 2684.0}, {'from_id': 'I', 'to_id': 'P', 'weight': 48.0}, {'from_id': 'I', 'to_id': 'Q', 'weight': 5.0}, {'from_id': 'I', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'I', 'to_id': 'S', 'weight': 563.0}, {'from_id': 'I', 'to_id': 'T', 'weight': 1.0}, {'from_id': 'J', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'O', 'weight': 1.0}, {'from_id': 'J', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'Q', 'weight': 652.0}, {'from_id': 'J', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'J', 'to_id': 'S', 'weight': 21.0}, {'from_id': 'J', 'to_id': 'T', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'Q', 'weight': 3700.0}, {'from_id': 'K', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'K', 'to_id': 'S', 'weight': 3.0}, {'from_id': 'K', 'to_id': 'T', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'S', 'weight': 0.0}, {'from_id': 'L', 'to_id': 'T', 'weight': 0.0}, {'from_id': 'M', 'to_id': 'A', 'weight': 9.0}, {'from_id': 'M', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'M', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'M', 'to_id': 'D', 'weight': 15.0}, {'from_id': 'M', 'to_id': 'E', 'weight': 9.0}, {'from_id': 'M', 'to_id': 'F', 'weight': 8.0}, {'from_id': 'M', 'to_id': 'G', 'weight': 166.0}, {'from_id': 'M', 'to_id': 'H', 'weight': 9.0}, {'from_id': 'M', 'to_id': 'I', 'weight': 480.0}, {'from_id': 'M', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'M', 'to_id': 'K', 'weight': 17.0}, {'from_id': 'M', 'to_id': 'L', 'weight': 1.0}, {'from_id': 'M', 'to_id': 'N', 'weight': 246.0}, {'from_id': 'M', 'to_id': 'O', 'weight': 782.0}, {'from_id': 'M', 'to_id': 'P', 'weight': 232.0}, {'from_id': 'M', 'to_id': 'Q', 'weight': 109.0}, {'from_id': 'M', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'M', 'to_id': 'S', 'weight': 20.0}, {'from_id': 'M', 'to_id': 'T', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'E', 'weight': 5.0}, {'from_id': 'N', 'to_id': 'F', 'weight': 41.0}, {'from_id': 'N', 'to_id': 'G', 'weight': 156.0}, {'from_id': 'N', 'to_id': 'H', 'weight': 35.0}, {'from_id': 'N', 'to_id': 'I', 'weight': 28.0}, {'from_id': 'N', 'to_id': 'J', 'weight': 10.0}, {'from_id': 'N', 'to_id': 'K', 'weight': 44.0}, {'from_id': 'N', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'M', 'weight': 9.0}, {'from_id': 'N', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'P', 'weight': 356.0}, {'from_id': 'N', 'to_id': 'Q', 'weight': 129.0}, {'from_id': 'N', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'N', 'to_id': 'S', 'weight': 155.0}, {'from_id': 'N', 'to_id': 'T', 'weight': 5.0}, {'from_id': 'O', 'to_id': 'A', 'weight': 3.0}, {'from_id': 'O', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'C', 'weight': 98.0}, {'from_id': 'O', 'to_id': 'D', 'weight': 84.0}, {'from_id': 'O', 'to_id': 'E', 'weight': 1226.0}, {'from_id': 'O', 'to_id': 'F', 'weight': 200.0}, {'from_id': 'O', 'to_id': 'G', 'weight': 191.0}, {'from_id': 'O', 'to_id': 'H', 'weight': 33.0}, {'from_id': 'O', 'to_id': 'I', 'weight': 73.0}, {'from_id': 'O', 'to_id': 'J', 'weight': 68.0}, {'from_id': 'O', 'to_id': 'K', 'weight': 54.0}, {'from_id': 'O', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'M', 'weight': 400.0}, {'from_id': 'O', 'to_id': 'N', 'weight': 90.0}, {'from_id': 'O', 'to_id': 'P', 'weight': 384.0}, {'from_id': 'O', 'to_id': 'Q', 'weight': 36.0}, {'from_id': 'O', 'to_id': 'R', 'weight': 32.0}, {'from_id': 'O', 'to_id': 'S', 'weight': -42.0}, {'from_id': 'O', 'to_id': 'T', 'weight': 3.0}, {'from_id': 'P', 'to_id': 'A', 'weight': 16.0}, {'from_id': 'P', 'to_id': 'B', 'weight': 6.0}, {'from_id': 'P', 'to_id': 'C', 'weight': 86.0}, {'from_id': 'P', 'to_id': 'D', 'weight': 9.0}, {'from_id': 'P', 'to_id': 'E', 'weight': 168.0}, {'from_id': 'P', 'to_id': 'F', 'weight': 167.0}, {'from_id': 'P', 'to_id': 'G', 'weight': 263.0}, {'from_id': 'P', 'to_id': 'H', 'weight': 120.0}, {'from_id': 'P', 'to_id': 'I', 'weight': 108.0}, {'from_id': 'P', 'to_id': 'J', 'weight': 25.0}, {'from_id': 'P', 'to_id': 'K', 'weight': 120.0}, {'from_id': 'P', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'M', 'weight': 670.0}, {'from_id': 'P', 'to_id': 'N', 'weight': 431.0}, {'from_id': 'P', 'to_id': 'O', 'weight': 880.0}, {'from_id': 'P', 'to_id': 'Q', 'weight': 1786.0}, {'from_id': 'P', 'to_id': 'R', 'weight': 27.0}, {'from_id': 'P', 'to_id': 'S', 'weight': 130.0}, {'from_id': 'P', 'to_id': 'T', 'weight': 6.0}, {'from_id': 'Q', 'to_id': 'A', 'weight': 3.0}, {'from_id': 'Q', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'C', 'weight': 61.0}, {'from_id': 'Q', 'to_id': 'D', 'weight': 39.0}, {'from_id': 'Q', 'to_id': 'E', 'weight': 71.0}, {'from_id': 'Q', 'to_id': 'F', 'weight': 57.0}, {'from_id': 'Q', 'to_id': 'G', 'weight': 105.0}, {'from_id': 'Q', 'to_id': 'H', 'weight': 13.0}, {'from_id': 'Q', 'to_id': 'I', 'weight': 43.0}, {'from_id': 'Q', 'to_id': 'J', 'weight': 27.0}, {'from_id': 'Q', 'to_id': 'K', 'weight': 53.0}, {'from_id': 'Q', 'to_id': 'L', 'weight': 1.0}, {'from_id': 'Q', 'to_id': 'M', 'weight': 111.0}, {'from_id': 'Q', 'to_id': 'N', 'weight': 44.0}, {'from_id': 'Q', 'to_id': 'O', 'weight': 42.0}, {'from_id': 'Q', 'to_id': 'P', 'weight': 874.0}, {'from_id': 'Q', 'to_id': 'R', 'weight': 21.0}, {'from_id': 'Q', 'to_id': 'S', 'weight': 313.0}, {'from_id': 'Q', 'to_id': 'T', 'weight': 8.0}, {'from_id': 'R', 'to_id': 'A', 'weight': 17.0}, {'from_id': 'R', 'to_id': 'B', 'weight': 5.0}, {'from_id': 'R', 'to_id': 'C', 'weight': 162.0}, {'from_id': 'R', 'to_id': 'D', 'weight': 30.0}, {'from_id': 'R', 'to_id': 'E', 'weight': 279.0}, {'from_id': 'R', 'to_id': 'F', 'weight': 74.0}, {'from_id': 'R', 'to_id': 'G', 'weight': 164.0}, {'from_id': 'R', 'to_id': 'H', 'weight': 24.0}, {'from_id': 'R', 'to_id': 'I', 'weight': 92.0}, {'from_id': 'R', 'to_id': 'J', 'weight': 13.0}, {'from_id': 'R', 'to_id': 'K', 'weight': 50.0}, {'from_id': 'R', 'to_id': 'L', 'weight': 1.0}, {'from_id': 'R', 'to_id': 'M', 'weight': 126.0}, {'from_id': 'R', 'to_id': 'N', 'weight': 50.0}, {'from_id': 'R', 'to_id': 'O', 'weight': 18.0}, {'from_id': 'R', 'to_id': 'P', 'weight': 974.0}, {'from_id': 'R', 'to_id': 'Q', 'weight': 13.0}, {'from_id': 'R', 'to_id': 'S', 'weight': 193.0}, {'from_id': 'R', 'to_id': 'T', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'S', 'to_id': 'T', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'T', 'to_id': 'S', 'weight': 0.0}]}","['R', 'C', 'A', 'D', 'B', 'P', 'F', 'M', 'N', 'K', 'J', 'Q', 'E', 'G', 'H', 'I', 'S', 'O', 'T', 'L']",48,markdown_table,names
LOP,LOP,"There’s a cartoonist juggling a pile of panels, figuring out which to read first, second, third, and so on so the payoff is funniest and clearest. The rule is simple: use every panel once and only once — don’t drop any, don’t repeat any — and for every two panels there’s a little preference value that says how much better it is to put one before the other (it’s directional, so order matters). For any candidate ordering, the overall effect is just the sum of all those pairwise values where the earlier panel earns its listed bonus; the best order is the one with the largest total. The exact panels and their pairwise timing perks follow below.

# panel_count=18
# panels=A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R

# Meaning: this is a directed pairwise_benefit matrix over items in panels (18 total). Entry at row panel_before_id=i and column panel_after_id=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# pairwise_benefit
| panel_before_id\panel_after_id | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A | 47845.0 | 0.0 | 2.0 | 0.0 | 494532.0 | 274152.0 | 4011.0 | 25806.0 | 6167.0 | 5103.0 | 0.0 | 0.0 | 448.0 | 42.0 | 0.0 | 0.0 | 8352.0 | 0.0 |
| B | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| C | 5547.0 | 0.0 | 30403.0 | 25124.0 | 834.0 | 50.0 | 924.0 | 2924.0 | 620.0 | 919.0 | 0.0 | 4622.0 | 71.0 | 236.0 | 0.0 | 0.0 | 571.0 | 0.0 |
| D | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 18904.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| E | 0.0 | 0.0 | 0.0 | 0.0 | 29261.0 | 999.0 | 60034.0 | 0.0 | 1870.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| F | 4646.0 | 0.0 | 0.0 | 0.0 | 28.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| G | 277.0 | 0.0 | 0.0 | 1138.0 | 0.0 | 0.0 | 81448.0 | 0.0 | 648.0 | 0.0 | 0.0 | 25.0 | 7.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| H | 2369.0 | 0.0 | 2620.0 | 6364.0 | 119.0 | 221.0 | 7470.0 | 1456.0 | 2127.0 | 7416.0 | 0.0 | 26.0 | 158.0 | 131.0 | 0.0 | 0.0 | 4068.0 | 0.0 |
| I | 226.0 | 0.0 | 112.0 | 82.0 | 67.0 | 47.0 | 9.0 | 42.0 | 0.0 | 2708.0 | 0.0 | 102.0 | 152.0 | 175.0 | 0.0 | 0.0 | 2491.0 | 0.0 |
| J | 27664.0 | 0.0 | 2551.0 | 3860.0 | 4332.0 | 3536.0 | 10509.0 | 2644.0 | 1373.0 | 0.0 | 0.0 | 4414.0 | 1005.0 | 880.0 | 0.0 | 0.0 | 31676.0 | 0.0 |
| K | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| L | 1013.0 | 0.0 | 2373.0 | 1374.0 | 478.0 | 633.0 | 324.0 | 504.0 | 107.0 | 1449.0 | 0.0 | 0.0 | 4968.0 | 25.0 | 0.0 | 0.0 | 1526.0 | 0.0 |
| M | 546.0 | 0.0 | 3934.0 | 2857.0 | 780.0 | 1530.0 | 344.0 | 1553.0 | 191.0 | 30834.0 | 0.0 | 1670.0 | 0.0 | 14420.0 | 0.0 | 0.0 | 10515.0 | 0.0 |
| N | 61242.0 | 0.0 | 9862.0 | 6815.0 | 4896.0 | 2546.0 | 5589.0 | 3243.0 | 2294.0 | 75504.0 | 0.0 | 15354.0 | 21508.0 | 0.0 | 0.0 | 0.0 | 10217.0 | 0.0 |
| O | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| P | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| Q | 16973.0 | 0.0 | 6018.0 | 4678.0 | 730.0 | 1458.0 | 1062.0 | 1837.0 | 419.0 | 45827.0 | 0.0 | 3753.0 | 12132.0 | 11924.0 | 0.0 | 0.0 | 1292.0 | 0.0 |
| R | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

When you send back the final ordering, please use this simple JSON shape so it's easy to read and check:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

Here ""solution"" is just the ordered list of panels from the one you want read first to the one you want read last. Put each panel's exact identifier in the array in that order — one identifier per slot. This JSON is just a sketch of the shape I expect, not the real answer itself.

Please make sure to 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”.""","[[47845.0, 0.0, 2.0, 0.0, 494532.0, 274152.0, 4011.0, 25806.0, 6167.0, 5103.0, 0.0, 0.0, 448.0, 42.0, 0.0, 0.0, 8352.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [5547.0, 0.0, 30403.0, 25124.0, 834.0, 50.0, 924.0, 2924.0, 620.0, 919.0, 0.0, 4622.0, 71.0, 236.0, 0.0, 0.0, 571.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 18904.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 29261.0, 999.0, 60034.0, 0.0, 1870.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [4646.0, 0.0, 0.0, 0.0, 28.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [277.0, 0.0, 0.0, 1138.0, 0.0, 0.0, 81448.0, 0.0, 648.0, 0.0, 0.0, 25.0, 7.0, 0.0, 0.0, 0.0, 0.0, 0.0], [2369.0, 0.0, 2620.0, 6364.0, 119.0, 221.0, 7470.0, 1456.0, 2127.0, 7416.0, 0.0, 26.0, 158.0, 131.0, 0.0, 0.0, 4068.0, 0.0], [226.0, 0.0, 112.0, 82.0, 67.0, 47.0, 9.0, 42.0, 0.0, 2708.0, 0.0, 102.0, 152.0, 175.0, 0.0, 0.0, 2491.0, 0.0], [27664.0, 0.0, 2551.0, 3860.0, 4332.0, 3536.0, 10509.0, 2644.0, 1373.0, 0.0, 0.0, 4414.0, 1005.0, 880.0, 0.0, 0.0, 31676.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [1013.0, 0.0, 2373.0, 1374.0, 478.0, 633.0, 324.0, 504.0, 107.0, 1449.0, 0.0, 0.0, 4968.0, 25.0, 0.0, 0.0, 1526.0, 0.0], [546.0, 0.0, 3934.0, 2857.0, 780.0, 1530.0, 344.0, 1553.0, 191.0, 30834.0, 0.0, 1670.0, 0.0, 14420.0, 0.0, 0.0, 10515.0, 0.0], [61242.0, 0.0, 9862.0, 6815.0, 4896.0, 2546.0, 5589.0, 3243.0, 2294.0, 75504.0, 0.0, 15354.0, 21508.0, 0.0, 0.0, 0.0, 10217.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [16973.0, 0.0, 6018.0, 4678.0, 730.0, 1458.0, 1062.0, 1837.0, 419.0, 45827.0, 0.0, 3753.0, 12132.0, 11924.0, 0.0, 0.0, 1292.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]]","[17, 16, 15, 14, 13, 12, 9, 2, 0, 7, 4, 6, 8, 3, 11, 10, 5, 1]",1366908.0,"{'problem_type': 'LOP', 'num_nodes': 18, 'nodes': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R'], 'pairs': [{'from_id': 'A', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'C', 'weight': 2.0}, {'from_id': 'A', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'E', 'weight': 494532.0}, {'from_id': 'A', 'to_id': 'F', 'weight': 274152.0}, {'from_id': 'A', 'to_id': 'G', 'weight': 4011.0}, {'from_id': 'A', 'to_id': 'H', 'weight': 25806.0}, {'from_id': 'A', 'to_id': 'I', 'weight': 6167.0}, {'from_id': 'A', 'to_id': 'J', 'weight': 5103.0}, {'from_id': 'A', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'M', 'weight': 448.0}, {'from_id': 'A', 'to_id': 'N', 'weight': 42.0}, {'from_id': 'A', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'A', 'to_id': 'Q', 'weight': 8352.0}, {'from_id': 'A', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'B', 'to_id': 'Q', 'weight': 0.0}, 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'weight': 0.0}, {'from_id': 'D', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'L', 'weight': 18904.0}, {'from_id': 'D', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'D', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'E', 'to_id': 'F', 'weight': 999.0}, {'from_id': 'E', 'to_id': 'G', 'weight': 60034.0}, {'from_id': 'E', 'to_id': 'H', 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{'from_id': 'O', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'O', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'Q', 'weight': 0.0}, {'from_id': 'P', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'A', 'weight': 16973.0}, {'from_id': 'Q', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'C', 'weight': 6018.0}, {'from_id': 'Q', 'to_id': 'D', 'weight': 4678.0}, {'from_id': 'Q', 'to_id': 'E', 'weight': 730.0}, {'from_id': 'Q', 'to_id': 'F', 'weight': 1458.0}, {'from_id': 'Q', 'to_id': 'G', 'weight': 1062.0}, {'from_id': 'Q', 'to_id': 'H', 'weight': 1837.0}, {'from_id': 'Q', 'to_id': 'I', 'weight': 419.0}, {'from_id': 'Q', 'to_id': 'J', 'weight': 45827.0}, {'from_id': 'Q', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'L', 'weight': 3753.0}, {'from_id': 'Q', 'to_id': 'M', 'weight': 12132.0}, {'from_id': 'Q', 'to_id': 'N', 'weight': 11924.0}, {'from_id': 'Q', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'Q', 'to_id': 'R', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'A', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'B', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'C', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'D', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'E', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'F', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'G', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'H', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'I', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'J', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'K', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'L', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'M', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'N', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'O', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'P', 'weight': 0.0}, {'from_id': 'R', 'to_id': 'Q', 'weight': 0.0}]}","['R', 'Q', 'P', 'O', 'N', 'M', 'J', 'C', 'A', 'H', 'E', 'G', 'I', 'D', 'L', 'K', 'F', 'B']",49,csv,names
LOP,LOP,"There’s a line cook puzzle on the counter: place all the prep items in one continuous line, don’t repeat anything, and don’t leave anything out. For each pair of ingredients someone has already written a bonus number that tells how good it is to have one placed before the other; the overall success is the sum of those bonuses for every pair depending on which one comes earlier. So the job is to pick the order that produces the highest summed score. The specific ingredients and their pairwise scores are listed below.

# num_ingredients=20
# ingredient_ids=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

# Meaning: this is a directed adjacency_efficiency_score matrix over items in ingredient_ids (20 total). Entry at row preceding_ingredient=i and column following_ingredient=j is the reward gained if i is placed BEFORE j in the final ordering (diagonal can be ignored).
# adjacency_efficiency_score
| preceding_ingredient\following_ingredient | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 17.0 | 0.0 | 0.0 | 0.0 | 1.0 | 4.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 1.0 | 1.0 | 2.0 |
| 2 | 0.0 | 107.0 | 1.0 | 0.0 | 2.0 | 18.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 9.0 | 10.0 | 36.0 | 15.0 |
| 3 | 0.0 | 0.0 | 3297.0 | 15.0 | 3.0 | 178.0 | 24.0 | 10.0 | 15.0 | 1.0 | 12.0 | 0.0 | 3.0 | 0.0 | 0.0 | 3.0 | 187.0 | 20.0 | 65.0 | 393.0 |
| 4 | 0.0 | 0.0 | 1.0 | 495.0 | 2.0 | 46.0 | 0.0 | 13.0 | 0.0 | 0.0 | 31.0 | 0.0 | 0.0 | 0.0 | 0.0 | 4.0 | 195.0 | 16.0 | 42.0 | 344.0 |
| 5 | 0.0 | 0.0 | 0.0 | 8.0 | 328.0 | 49.0 | 0.0 | 0.0 | 0.0 | 0.0 | 42.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 100.0 | 13.0 | 22.0 | 50.0 |
| 6 | 0.0 | 0.0 | 899.0 | 900.0 | 172.0 | 12493.0 | 77.0 | 205.0 | 40.0 | 55.0 | 145.0 | 5.0 | 0.0 | 3.0 | 3.0 | 32.0 | 2440.0 | 85.0 | 576.0 | 2060.0 |
| 7 | 0.0 | 0.0 | 84.0 | 12.0 | 16.0 | 536.0 | 110.0 | 33.0 | 1.0 | 2.0 | 5.0 | 0.0 | 1.0 | 0.0 | 0.0 | 2.0 | 78.0 | 10.0 | 22.0 | 72.0 |
| 8 | 0.0 | 0.0 | 183.0 | 12.0 | 18.0 | 157.0 | 0.0 | 336.0 | 3.0 | 105.0 | 19.0 | 20.0 | 8.0 | 1.0 | 1.0 | 4.0 | 438.0 | 20.0 | 43.0 | 242.0 |
| 9 | 0.0 | 0.0 | 14.0 | 15.0 | 11.0 | 14.0 | 0.0 | 0.0 | 16.0 | 29.0 | 6.0 | 0.0 | 5.0 | 0.0 | 0.0 | 2.0 | 36.0 | 6.0 | 22.0 | 42.0 |
| 10 | 0.0 | 0.0 | 36.0 | 7.0 | 16.0 | 55.0 | 0.0 | 0.0 | 83.0 | 691.0 | 56.0 | 0.0 | 6.0 | 0.0 | 0.0 | 3.0 | 126.0 | 12.0 | 38.0 | 170.0 |
| 11 | 0.0 | 0.0 | 5.0 | 55.0 | 965.0 | 77.0 | 0.0 | 0.0 | 15.0 | 120.0 | 1150.0 | 5.0 | 9.0 | 1.0 | 1.0 | 6.0 | 736.0 | 23.0 | 86.0 | 297.0 |
| 12 | 0.0 | 0.0 | 67.0 | 20.0 | 28.0 | 10.0 | 0.0 | 33.0 | 0.0 | 2.0 | 24.0 | 5.0 | 0.0 | 0.0 | 0.0 | 1.0 | 85.0 | 4.0 | 17.0 | 36.0 |
| 13 | 0.0 | 0.0 | 0.0 | 57.0 | 0.0 | 41.0 | 0.0 | 0.0 | 2.0 | 0.0 | 123.0 | 2.0 | 89.0 | 1.0 | 1.0 | 9.0 | 416.0 | 18.0 | 130.0 | 655.0 |
| 14 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 45.0 | 11.0 | 0.0 | 47.0 | 2.0 | 12.0 | 62.0 |
| 15 | 0.0 | 0.0 | 0.0 | 0.0 | 4.0 | 3.0 | 0.0 | 0.0 | 0.0 | 4.0 | 0.0 | 0.0 | 0.0 | 1.0 | 622.0 | 0.0 | 190.0 | 3.0 | 22.0 | 114.0 |
| 16 | 0.0 | 0.0 | 0.0 | 27.0 | 0.0 | 43.0 | 0.0 | 0.0 | 9.0 | 146.0 | 9.0 | 0.0 | 0.0 | 1.0 | 7.0 | 1287.0 | 224.0 | 19.0 | 130.0 | 181.0 |
| 17 | 0.0 | 0.0 | 318.0 | 916.0 | 787.0 | 1310.0 | 5.0 | 99.0 | 76.0 | 444.0 | 1428.0 | 15.0 | 55.0 | 726.0 | 321.0 | 183.0 | 11654.0 | 242.0 | 1292.0 | 3149.0 |
| 18 | 0.0 | 0.0 | 838.0 | 184.0 | 1553.0 | 1050.0 | 6.0 | 6.0 | 431.0 | 482.0 | 1382.0 | 0.0 | 63.0 | 0.0 | 0.0 | 18.0 | 3339.0 | 4212.0 | 567.0 | 703.0 |
| 19 | 0.0 | 0.0 | 0.0 | 30.0 | 0.0 | 419.0 | 0.0 | 26.0 | 26.0 | 48.0 | 163.0 | 0.0 | 17.0 | 28.0 | 44.0 | 14.0 | 562.0 | 120.0 | 5914.0 | 4023.0 |
| 20 | 0.0 | 0.0 | 32.0 | 235.0 | 23.0 | 845.0 | 0.0 | 187.0 | 47.0 | 173.0 | 681.0 | 26.0 | 445.0 | 74.0 | 141.0 | 4136.0 | 3834.0 | 900.0 | 937.0 | 15016.0 |

Also, just so we’re on the same page, when you reply please use this little JSON layout so I can pick out the ordering automatically:

{
  ""solution"": [<first_item>, <second_item>, ..., <last_item>]
}

""solution"" should be a single list that goes from the ingredient you want first to the one you want last. Think of it like filling in a simple form: put each ingredient identifier in order, exactly as it appears in the instance (don’t make up new names or tweak them). This JSON is just a sketch of the shape I expect — not the final answer itself.

Please use the identifiers exactly as given in the input — no renaming, no extra 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”.""","[[17.0, 0.0, 0.0, 0.0, 1.0, 4.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 1.0, 1.0, 2.0], [0.0, 107.0, 1.0, 0.0, 2.0, 18.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 9.0, 10.0, 36.0, 15.0], [0.0, 0.0, 3297.0, 15.0, 3.0, 178.0, 24.0, 10.0, 15.0, 1.0, 12.0, 0.0, 3.0, 0.0, 0.0, 3.0, 187.0, 20.0, 65.0, 393.0], [0.0, 0.0, 1.0, 495.0, 2.0, 46.0, 0.0, 13.0, 0.0, 0.0, 31.0, 0.0, 0.0, 0.0, 0.0, 4.0, 195.0, 16.0, 42.0, 344.0], [0.0, 0.0, 0.0, 8.0, 328.0, 49.0, 0.0, 0.0, 0.0, 0.0, 42.0, 0.0, 0.0, 0.0, 0.0, 1.0, 100.0, 13.0, 22.0, 50.0], [0.0, 0.0, 899.0, 900.0, 172.0, 12493.0, 77.0, 205.0, 40.0, 55.0, 145.0, 5.0, 0.0, 3.0, 3.0, 32.0, 2440.0, 85.0, 576.0, 2060.0], [0.0, 0.0, 84.0, 12.0, 16.0, 536.0, 110.0, 33.0, 1.0, 2.0, 5.0, 0.0, 1.0, 0.0, 0.0, 2.0, 78.0, 10.0, 22.0, 72.0], [0.0, 0.0, 183.0, 12.0, 18.0, 157.0, 0.0, 336.0, 3.0, 105.0, 19.0, 20.0, 8.0, 1.0, 1.0, 4.0, 438.0, 20.0, 43.0, 242.0], [0.0, 0.0, 14.0, 15.0, 11.0, 14.0, 0.0, 0.0, 16.0, 29.0, 6.0, 0.0, 5.0, 0.0, 0.0, 2.0, 36.0, 6.0, 22.0, 42.0], [0.0, 0.0, 36.0, 7.0, 16.0, 55.0, 0.0, 0.0, 83.0, 691.0, 56.0, 0.0, 6.0, 0.0, 0.0, 3.0, 126.0, 12.0, 38.0, 170.0], [0.0, 0.0, 5.0, 55.0, 965.0, 77.0, 0.0, 0.0, 15.0, 120.0, 1150.0, 5.0, 9.0, 1.0, 1.0, 6.0, 736.0, 23.0, 86.0, 297.0], [0.0, 0.0, 67.0, 20.0, 28.0, 10.0, 0.0, 33.0, 0.0, 2.0, 24.0, 5.0, 0.0, 0.0, 0.0, 1.0, 85.0, 4.0, 17.0, 36.0], [0.0, 0.0, 0.0, 57.0, 0.0, 41.0, 0.0, 0.0, 2.0, 0.0, 123.0, 2.0, 89.0, 1.0, 1.0, 9.0, 416.0, 18.0, 130.0, 655.0], [0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 45.0, 11.0, 0.0, 47.0, 2.0, 12.0, 62.0], [0.0, 0.0, 0.0, 0.0, 4.0, 3.0, 0.0, 0.0, 0.0, 4.0, 0.0, 0.0, 0.0, 1.0, 622.0, 0.0, 190.0, 3.0, 22.0, 114.0], [0.0, 0.0, 0.0, 27.0, 0.0, 43.0, 0.0, 0.0, 9.0, 146.0, 9.0, 0.0, 0.0, 1.0, 7.0, 1287.0, 224.0, 19.0, 130.0, 181.0], [0.0, 0.0, 318.0, 916.0, 787.0, 1310.0, 5.0, 99.0, 76.0, 444.0, 1428.0, 15.0, 55.0, 726.0, 321.0, 183.0, 11654.0, 242.0, 1292.0, 3149.0], [0.0, 0.0, 838.0, 184.0, 1553.0, 1050.0, 6.0, 6.0, 431.0, 482.0, 1382.0, 0.0, 63.0, 0.0, 0.0, 18.0, 3339.0, 4212.0, 567.0, 703.0], [0.0, 0.0, 0.0, 30.0, 0.0, 419.0, 0.0, 26.0, 26.0, 48.0, 163.0, 0.0, 17.0, 28.0, 44.0, 14.0, 562.0, 120.0, 5914.0, 4023.0], [0.0, 0.0, 32.0, 235.0, 23.0, 845.0, 0.0, 187.0, 47.0, 173.0, 681.0, 26.0, 445.0, 74.0, 141.0, 4136.0, 3834.0, 900.0, 937.0, 15016.0]]","[11, 6, 1, 0, 17, 12, 5, 7, 16, 2, 3, 18, 19, 15, 13, 10, 14, 9, 8, 4]",43547.0,"{'problem_type': 'LOP', 'num_nodes': 20, 'nodes': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], 'pairs': [{'from_id': 1, 'to_id': 2, 'weight': 0.0}, {'from_id': 1, 'to_id': 3, 'weight': 0.0}, {'from_id': 1, 'to_id': 4, 'weight': 0.0}, {'from_id': 1, 'to_id': 5, 'weight': 1.0}, {'from_id': 1, 'to_id': 6, 'weight': 4.0}, {'from_id': 1, 'to_id': 7, 'weight': 0.0}, {'from_id': 1, 'to_id': 8, 'weight': 0.0}, {'from_id': 1, 'to_id': 9, 'weight': 0.0}, {'from_id': 1, 'to_id': 10, 'weight': 0.0}, {'from_id': 1, 'to_id': 11, 'weight': 0.0}, {'from_id': 1, 'to_id': 12, 'weight': 0.0}, {'from_id': 1, 'to_id': 13, 'weight': 0.0}, {'from_id': 1, 'to_id': 14, 'weight': 0.0}, {'from_id': 1, 'to_id': 15, 'weight': 0.0}, {'from_id': 1, 'to_id': 16, 'weight': 0.0}, {'from_id': 1, 'to_id': 17, 'weight': 1.0}, {'from_id': 1, 'to_id': 18, 'weight': 1.0}, {'from_id': 1, 'to_id': 19, 'weight': 1.0}, {'from_id': 1, 'to_id': 20, 'weight': 2.0}, {'from_id': 2, 'to_id': 1, 'weight': 0.0}, {'from_id': 2, 'to_id': 3, 'weight': 1.0}, {'from_id': 2, 'to_id': 4, 'weight': 0.0}, {'from_id': 2, 'to_id': 5, 'weight': 2.0}, {'from_id': 2, 'to_id': 6, 'weight': 18.0}, {'from_id': 2, 'to_id': 7, 'weight': 0.0}, {'from_id': 2, 'to_id': 8, 'weight': 0.0}, {'from_id': 2, 'to_id': 9, 'weight': 0.0}, {'from_id': 2, 'to_id': 10, 'weight': 0.0}, {'from_id': 2, 'to_id': 11, 'weight': 0.0}, {'from_id': 2, 'to_id': 12, 'weight': 0.0}, {'from_id': 2, 'to_id': 13, 'weight': 1.0}, {'from_id': 2, 'to_id': 14, 'weight': 0.0}, {'from_id': 2, 'to_id': 15, 'weight': 0.0}, {'from_id': 2, 'to_id': 16, 'weight': 0.0}, {'from_id': 2, 'to_id': 17, 'weight': 9.0}, {'from_id': 2, 'to_id': 18, 'weight': 10.0}, {'from_id': 2, 'to_id': 19, 'weight': 36.0}, {'from_id': 2, 'to_id': 20, 'weight': 15.0}, {'from_id': 3, 'to_id': 1, 'weight': 0.0}, {'from_id': 3, 'to_id': 2, 'weight': 0.0}, {'from_id': 3, 'to_id': 4, 'weight': 15.0}, {'from_id': 3, 'to_id': 5, 'weight': 3.0}, {'from_id': 3, 'to_id': 6, 'weight': 178.0}, {'from_id': 3, 'to_id': 7, 'weight': 24.0}, {'from_id': 3, 'to_id': 8, 'weight': 10.0}, {'from_id': 3, 'to_id': 9, 'weight': 15.0}, {'from_id': 3, 'to_id': 10, 'weight': 1.0}, {'from_id': 3, 'to_id': 11, 'weight': 12.0}, {'from_id': 3, 'to_id': 12, 'weight': 0.0}, {'from_id': 3, 'to_id': 13, 'weight': 3.0}, {'from_id': 3, 'to_id': 14, 'weight': 0.0}, 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