QKP/QKP_S_context.csv

task_name,problem_type,instruction,instance,solution,obj,instance_variant,solution_variant,context_index,input_format,input_index_base
QKP,QKP,"Many people know that packing for a hike isn’t just about individual items—each piece has its own weight and usefulness, and some two-item combos give an extra perk when both are carried. The task here is to select a subset of those unique items so the total usefulness (sum of the usefulness values of the selected items plus any extra bonuses for specific pairs that are both in the pack) is as big as possible, while keeping the total weight inside the backpack’s limit. Items are singletons—can’t take duplicates or fractions. The exact items, weights, usefulness scores, and pair bonuses are shown below.

# total_items=5
# max_backpack_weight=4489
# gear_list=0, 1, 2, 3, 4

gear_id,base_usefulness
0,200
1,1415
2,1604
3,1141
4,2433

gear_id,item_weight
0,296
1,846
2,1941
3,399
4,1240

gear_i_id,gear_j_id,pair_bonus
0,1,501
0,4,722

When you give the final pick, just drop it into a tiny JSON snippet so it's easy to read and parse, like this:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

""solution"" is simply the list of item IDs you choose to bring in your backpack. Think of each entry in the array as the exact label for one item — that's it. This JSON is just the expected shape (a sketch of the answer format), not the actual solution itself.

Also, please make sure to use the item 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”.""","{'linear_coeffs': [200, 1415, 1604, 1141, 2433], 'quadratic_coeffs': [[731, 501, 0, 0, 722], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]], 'weights': [296, 846, 1941, 399, 1240], 'capacity': 4489, 'solution': [0, 1, 2, 4], 'obj': 6875.0, 'problem_type': 'QKP'}","[0, 1, 2, 4]",6875.0,"{'problem_type': 'QKP', 'num_items': 5, 'capacity': 4489, 'items': [0, 1, 2, 3, 4], 'linear_pairs': [{'item_id': 0, 'linear_profit': 200}, {'item_id': 1, 'linear_profit': 1415}, {'item_id': 2, 'linear_profit': 1604}, {'item_id': 3, 'linear_profit': 1141}, {'item_id': 4, 'linear_profit': 2433}], 'weight_pairs': [{'item_id': 0, 'weight': 296}, {'item_id': 1, 'weight': 846}, {'item_id': 2, 'weight': 1941}, {'item_id': 3, 'weight': 399}, {'item_id': 4, 'weight': 1240}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 731}, {'item_i_id': 0, 'item_j_id': 1, 'quadratic_profit': 501}, {'item_i_id': 0, 'item_j_id': 4, 'quadratic_profit': 722}]}","[0, 1, 2, 4]",1,csv,0
QKP,QKP,"Someone in the events crew is balancing a menu for a corporate lunch: every dish counts for its own popularity, and certain combinations add bonus appeal when they’re both present. The task is to choose which dishes to include (each dish only once, either included or not) so the combined oven load doesn’t exceed what’s available, and the final guest satisfaction — the sum of individual dish appeals plus any pair bonuses for dishes that appear together — is as large as possible. Concrete details about the dishes, their oven requirements, and the extra pair points follow below.

# num_dishes=9
# oven_capacity=441
# dish_ids=A, B, C, D, E, F, G, H, I

dish_id,dish_appeal
A,114
B,114
C,113
D,115
E,113
F,110
G,114
H,115
I,110

dish_id,oven_load
A,104
B,104
C,103
D,105
E,103
F,100
G,104
H,105
I,100

dish_i_id,dish_j_id,pair_bonus_appeal
A,B,103
A,C,103
A,D,103
A,E,101
A,F,104
A,G,100
A,H,104
A,I,102
B,C,100
B,D,100
B,E,100
B,F,100
B,G,100
B,H,100
B,I,100
C,D,105
C,E,103
C,F,100
C,G,101
C,H,103
C,I,103
D,E,105
D,F,104
D,G,100
D,H,100
D,I,100
E,F,100
E,G,100
E,H,100
E,I,100
F,G,100
F,H,101
F,I,101
G,H,102
G,I,105
H,I,100

Oh, and when you send back your choice, please stick to this simple JSON layout so it's easy to read and process:

{
  ""solution"": [<selected_dish_id>, <selected_dish_id>, ...]
}

""solution"" is just a list of the dishes you decide to include — each entry is the exact dish identifier from the instance (one per chosen dish). Think of the placeholders like little checkboxes: replace each <selected_dish_id> with the ID of a dish you want on the menu. This block is just a sketch of the shape we expect, not the actual answer.

Please 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”.""","{'linear_coeffs': [114, 114, 113, 115, 113, 110, 114, 115, 110], 'quadratic_coeffs': [[104, 103, 103, 103, 101, 104, 100, 104, 102], [0, 102, 100, 100, 100, 100, 100, 100, 100], [0, 0, 100, 105, 103, 100, 101, 103, 103], [0, 0, 0, 105, 105, 104, 100, 100, 100], [0, 0, 0, 0, 100, 100, 100, 100, 100], [0, 0, 0, 0, 0, 100, 100, 101, 101], [0, 0, 0, 0, 0, 0, 102, 102, 105], [0, 0, 0, 0, 0, 0, 0, 103, 100], [0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [104, 104, 103, 105, 103, 100, 104, 105, 100], 'capacity': 441, 'solution': [0, 2, 3, 7], 'obj': 1075.0, 'problem_type': 'QKP'}","[0, 2, 3, 7]",1075.0,"{'problem_type': 'QKP', 'num_items': 9, 'capacity': 441, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 114}, {'item_id': 'B', 'linear_profit': 114}, {'item_id': 'C', 'linear_profit': 113}, {'item_id': 'D', 'linear_profit': 115}, {'item_id': 'E', 'linear_profit': 113}, {'item_id': 'F', 'linear_profit': 110}, {'item_id': 'G', 'linear_profit': 114}, {'item_id': 'H', 'linear_profit': 115}, {'item_id': 'I', 'linear_profit': 110}], 'weight_pairs': [{'item_id': 'A', 'weight': 104}, {'item_id': 'B', 'weight': 104}, {'item_id': 'C', 'weight': 103}, {'item_id': 'D', 'weight': 105}, {'item_id': 'E', 'weight': 103}, {'item_id': 'F', 'weight': 100}, {'item_id': 'G', 'weight': 104}, {'item_id': 'H', 'weight': 105}, {'item_id': 'I', 'weight': 100}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 104}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 103}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 103}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 103}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 101}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 104}, {'item_i_id': 'A', 'item_j_id': 'G', 'quadratic_profit': 100}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 104}, {'item_i_id': 'A', 'item_j_id': 'I', 'quadratic_profit': 102}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 102}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 100}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 100}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 100}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 100}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 100}, {'item_i_id': 'B', 'item_j_id': 'H', 'quadratic_profit': 100}, {'item_i_id': 'B', 'item_j_id': 'I', 'quadratic_profit': 100}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 100}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 105}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 103}, {'item_i_id': 'C', 'item_j_id': 'F', 'quadratic_profit': 100}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 101}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 103}, {'item_i_id': 'C', 'item_j_id': 'I', 'quadratic_profit': 103}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 105}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 105}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 104}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 100}, {'item_i_id': 'D', 'item_j_id': 'H', 'quadratic_profit': 100}, {'item_i_id': 'D', 'item_j_id': 'I', 'quadratic_profit': 100}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 100}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 100}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 100}, {'item_i_id': 'E', 'item_j_id': 'H', 'quadratic_profit': 100}, {'item_i_id': 'E', 'item_j_id': 'I', 'quadratic_profit': 100}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 100}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 100}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 101}, {'item_i_id': 'F', 'item_j_id': 'I', 'quadratic_profit': 101}, {'item_i_id': 'G', 'item_j_id': 'G', 'quadratic_profit': 102}, {'item_i_id': 'G', 'item_j_id': 'H', 'quadratic_profit': 102}, {'item_i_id': 'G', 'item_j_id': 'I', 'quadratic_profit': 105}, {'item_i_id': 'H', 'item_j_id': 'H', 'quadratic_profit': 103}, {'item_i_id': 'H', 'item_j_id': 'I', 'quadratic_profit': 100}]}","['A', 'C', 'D', 'H']",2,csv,names
QKP,QKP,"Recently there was a move where the problem was choosing which boxes to cram into a single car so the trip hauled the most value. Each box brings its own value and uses up some of the trunk’s capacity, and some specific pairs of boxes add extra value when they’re together, so what’s counted is the sum of individual box values plus any bonuses from those pairs. Since the trunk can’t be overloaded and every box is unique (no duplicates), the aim is to pick the combination that yields the highest total worth without exceeding the weight allowance. The detailed list of boxes, weights, values, and pairwise bonuses follows below.

{
  ""total_boxes"": 10,
  ""trunk_weight_capacity"": 593,
  ""box_ids"": [
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    9,
    10
  ],
  ""linear"": [
    {
      ""box_id"": 1,
      ""box_value"": 101
    },
    {
      ""box_id"": 2,
      ""box_value"": 120
    },
    {
      ""box_id"": 3,
      ""box_value"": 111
    },
    {
      ""box_id"": 4,
      ""box_value"": 115
    },
    {
      ""box_id"": 5,
      ""box_value"": 98
    },
    {
      ""box_id"": 6,
      ""box_value"": 96
    },
    {
      ""box_id"": 7,
      ""box_value"": 114
    },
    {
      ""box_id"": 8,
      ""box_value"": 101
    },
    {
      ""box_id"": 9,
      ""box_value"": 102
    },
    {
      ""box_id"": 10,
      ""box_value"": 115
    }
  ],
  ""weights"": [
    {
      ""box_id"": 1,
      ""box_weight"": 100
    },
    {
      ""box_id"": 2,
      ""box_weight"": 122
    },
    {
      ""box_id"": 3,
      ""box_weight"": 114
    },
    {
      ""box_id"": 4,
      ""box_weight"": 114
    },
    {
      ""box_id"": 5,
      ""box_weight"": 100
    },
    {
      ""box_id"": 6,
      ""box_weight"": 101
    },
    {
      ""box_id"": 7,
      ""box_weight"": 109
    },
    {
      ""box_id"": 8,
      ""box_weight"": 101
    },
    {
      ""box_id"": 9,
      ""box_weight"": 106
    },
    {
      ""box_id"": 10,
      ""box_weight"": 114
    }
  ],
  ""quadratic"": [
    {
      ""box_i_id"": 1,
      ""box_j_id"": 2,
      ""pair_bonus_value"": 100
    },
    {
      ""box_i_id"": 1,
      ""box_j_id"": 3,
      ""pair_bonus_value"": 100
    },
    {
      ""box_i_id"": 1,
      ""box_j_id"": 4,
      ""pair_bonus_value"": 100
    },
    {
      ""box_i_id"": 1,
      ""box_j_id"": 5,
      ""pair_bonus_value"": 100
    },
    {
      ""box_i_id"": 1,
      ""box_j_id"": 6,
      ""pair_bonus_value"": 100
    },
    {
      ""box_i_id"": 1,
      ""box_j_id"": 7,
      ""pair_bonus_value"": 100
    },
    {
      ""box_i_id"": 1,
      ""box_j_id"": 8,
      ""pair_bonus_value"": 100
    },
    {
      ""box_i_id"": 1,
      ""box_j_id"": 9,
      ""pair_bonus_value"": 100
    },
    {
      ""box_i_id"": 1,
      ""box_j_id"": 10,
      ""pair_bonus_value"": 100
    },
    {
      ""box_i_id"": 2,
      ""box_j_id"": 3,
      ""pair_bonus_value"": 108
    },
    {
      ""box_i_id"": 2,
      ""box_j_id"": 4,
      ""pair_bonus_value"": 106
    },
    {
      ""box_i_id"": 2,
      ""box_j_id"": 5,
      ""pair_bonus_value"": 106
    },
    {
      ""box_i_id"": 2,
      ""box_j_id"": 7,
      ""pair_bonus_value"": 112
    },
    {
      ""box_i_id"": 2,
      ""box_j_id"": 9,
      ""pair_bonus_value"": 107
    },
    {
      ""box_i_id"": 2,
      ""box_j_id"": 10,
      ""pair_bonus_value"": 117
    },
    {
      ""box_i_id"": 3,
      ""box_j_id"": 4,
      ""pair_bonus_value"": 114
    },
    {
      ""box_i_id"": 3,
      ""box_j_id"": 6,
      ""pair_bonus_value"": 114
    },
    {
      ""box_i_id"": 3,
      ""box_j_id"": 7,
      ""pair_bonus_value"": 102
    },
    {
      ""box_i_id"": 3,
      ""box_j_id"": 8,
      ""pair_bonus_value"": 107
    },
    {
      ""box_i_id"": 4,
      ""box_j_id"": 5,
      ""pair_bonus_value"": 108
    },
    {
      ""box_i_id"": 4,
      ""box_j_id"": 6,
      ""pair_bonus_value"": 105
    },
    {
      ""box_i_id"": 4,
      ""box_j_id"": 7,
      ""pair_bonus_value"": 111
    },
    {
      ""box_i_id"": 4,
      ""box_j_id"": 8,
      ""pair_bonus_value"": 100
    },
    {
      ""box_i_id"": 4,
      ""box_j_id"": 10,
      ""pair_bonus_value"": 100
    },
    {
      ""box_i_id"": 5,
      ""box_j_id"": 6,
      ""pair_bonus_value"": 100
    },
    {
      ""box_i_id"": 5,
      ""box_j_id"": 7,
      ""pair_bonus_value"": 100
    },
    {
      ""box_i_id"": 5,
      ""box_j_id"": 8,
      ""pair_bonus_value"": 101
    },
    {
      ""box_i_id"": 5,
      ""box_j_id"": 9,
      ""pair_bonus_value"": 104
    },
    {
      ""box_i_id"": 5,
      ""box_j_id"": 10,
      ""pair_bonus_value"": 106
    },
    {
      ""box_i_id"": 6,
      ""box_j_id"": 7,
      ""pair_bonus_value"": 114
    },
    {
      ""box_i_id"": 6,
      ""box_j_id"": 9,
      ""pair_bonus_value"": 114
    },
    {
      ""box_i_id"": 6,
      ""box_j_id"": 10,
      ""pair_bonus_value"": 105
    },
    {
      ""box_i_id"": 7,
      ""box_j_id"": 8,
      ""pair_bonus_value"": 106
    },
    {
      ""box_i_id"": 7,
      ""box_j_id"": 9,
      ""pair_bonus_value"": 104
    },
    {
      ""box_i_id"": 7,
      ""box_j_id"": 10,
      ""pair_bonus_value"": 111
    },
    {
      ""box_i_id"": 8,
      ""box_j_id"": 9,
      ""pair_bonus_value"": 100
    },
    {
      ""box_i_id"": 8,
      ""box_j_id"": 10,
      ""pair_bonus_value"": 100
    },
    {
      ""box_i_id"": 9,
      ""box_j_id"": 10,
      ""pair_bonus_value"": 109
    }
  ]
}

Oh, and when you send back your pick, please stick to a tiny JSON snippet so it's easy to read and validate. It should follow this shape:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

Here the ""solution"" array is just a list of the box ids you decided to load into the trunk — one entry per chosen box. This is only a sketch of the expected shape, not the actual selection.

Please make sure you 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”.""","{'linear_coeffs': [101, 120, 111, 115, 98, 96, 114, 101, 102, 115], 'quadratic_coeffs': [[100, 100, 100, 100, 100, 100, 100, 100, 100, 100], [0, 0, 108, 106, 106, 0, 112, 0, 107, 117], [0, 0, 106, 114, 0, 114, 102, 107, 0, 0], [0, 0, 0, 100, 108, 105, 111, 100, 0, 100], [0, 0, 0, 0, 0, 100, 100, 101, 104, 106], [0, 0, 0, 0, 0, 0, 114, 0, 114, 105], [0, 0, 0, 0, 0, 0, 113, 106, 104, 111], [0, 0, 0, 0, 0, 0, 0, 0, 100, 100], [0, 0, 0, 0, 0, 0, 0, 0, 0, 109], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [100, 122, 114, 114, 100, 101, 109, 101, 106, 114], 'capacity': 593, 'solution': [1, 3, 4, 6, 9], 'obj': 1639.0, 'problem_type': 'QKP'}","[1, 3, 4, 6, 9]",1639.0,"{'problem_type': 'QKP', 'num_items': 10, 'capacity': 593, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 'linear_pairs': [{'item_id': 1, 'linear_profit': 101}, {'item_id': 2, 'linear_profit': 120}, {'item_id': 3, 'linear_profit': 111}, {'item_id': 4, 'linear_profit': 115}, {'item_id': 5, 'linear_profit': 98}, {'item_id': 6, 'linear_profit': 96}, {'item_id': 7, 'linear_profit': 114}, {'item_id': 8, 'linear_profit': 101}, {'item_id': 9, 'linear_profit': 102}, {'item_id': 10, 'linear_profit': 115}], 'weight_pairs': [{'item_id': 1, 'weight': 100}, {'item_id': 2, 'weight': 122}, {'item_id': 3, 'weight': 114}, {'item_id': 4, 'weight': 114}, {'item_id': 5, 'weight': 100}, {'item_id': 6, 'weight': 101}, {'item_id': 7, 'weight': 109}, {'item_id': 8, 'weight': 101}, {'item_id': 9, 'weight': 106}, {'item_id': 10, 'weight': 114}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 100}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 108}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 106}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 106}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 112}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 107}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 117}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 106}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 114}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 114}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 102}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 107}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 100}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 108}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 105}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 111}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 100}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 100}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 100}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 100}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 101}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 104}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 106}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 114}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 114}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 105}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 113}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 106}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 104}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 111}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 100}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 100}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 109}]}","[2, 4, 5, 7, 10]",3,json,1
QKP,QKP,"Out in the studio someone’s curating a compact show and balancing fun against minutes: choose which songs to play so the entire set feels as enjoyable as possible. Each track contributes a base chunk of enjoyment, and if certain tracks are both included they add a little extra because they transition nicely. The final score is the sum of every chosen song’s enjoyment plus any of those pairwise extras, and the chosen tracks must not exceed the show’s time limit or be repeated. The exact options and timings are given below.

{
  ""num_tracks"": 8,
  ""total_show_runtime"": 18419,
  ""track_ids"": [
    ""A"",
    ""B"",
    ""C"",
    ""D"",
    ""E"",
    ""F"",
    ""G"",
    ""H""
  ],
  ""linear"": [
    {
      ""track_id"": ""A"",
      ""track_enjoyment"": 41
    },
    {
      ""track_id"": ""B"",
      ""track_enjoyment"": 5069
    },
    {
      ""track_id"": ""C"",
      ""track_enjoyment"": 6389
    },
    {
      ""track_id"": ""D"",
      ""track_enjoyment"": 739
    },
    {
      ""track_id"": ""E"",
      ""track_enjoyment"": 134
    },
    {
      ""track_id"": ""F"",
      ""track_enjoyment"": 3802
    },
    {
      ""track_id"": ""G"",
      ""track_enjoyment"": 6015
    },
    {
      ""track_id"": ""H"",
      ""track_enjoyment"": 5566
    }
  ],
  ""weights"": [
    {
      ""track_id"": ""A"",
      ""track_runtime"": 265
    },
    {
      ""track_id"": ""B"",
      ""track_runtime"": 4898
    },
    {
      ""track_id"": ""C"",
      ""track_runtime"": 6356
    },
    {
      ""track_id"": ""D"",
      ""track_runtime"": 448
    },
    {
      ""track_id"": ""E"",
      ""track_runtime"": 169
    },
    {
      ""track_id"": ""F"",
      ""track_runtime"": 3651
    },
    {
      ""track_id"": ""G"",
      ""track_runtime"": 6014
    },
    {
      ""track_id"": ""H"",
      ""track_runtime"": 5527
    }
  ],
  ""quadratic"": [
    {
      ""track_i_id"": ""A"",
      ""track_j_id"": ""D"",
      ""transition_bonus"": 5300
    },
    {
      ""track_i_id"": ""A"",
      ""track_j_id"": ""E"",
      ""transition_bonus"": 4736
    },
    {
      ""track_i_id"": ""A"",
      ""track_j_id"": ""F"",
      ""transition_bonus"": 1042
    },
    {
      ""track_i_id"": ""A"",
      ""track_j_id"": ""G"",
      ""transition_bonus"": 2506
    },
    {
      ""track_i_id"": ""A"",
      ""track_j_id"": ""H"",
      ""transition_bonus"": 6239
    },
    {
      ""track_i_id"": ""B"",
      ""track_j_id"": ""C"",
      ""transition_bonus"": 2262
    },
    {
      ""track_i_id"": ""B"",
      ""track_j_id"": ""E"",
      ""transition_bonus"": 3659
    },
    {
      ""track_i_id"": ""B"",
      ""track_j_id"": ""F"",
      ""transition_bonus"": 5432
    },
    {
      ""track_i_id"": ""B"",
      ""track_j_id"": ""G"",
      ""transition_bonus"": 3206
    },
    {
      ""track_i_id"": ""C"",
      ""track_j_id"": ""D"",
      ""transition_bonus"": 4163
    },
    {
      ""track_i_id"": ""C"",
      ""track_j_id"": ""E"",
      ""transition_bonus"": 4400
    },
    {
      ""track_i_id"": ""C"",
      ""track_j_id"": ""G"",
      ""transition_bonus"": 4974
    },
    {
      ""track_i_id"": ""C"",
      ""track_j_id"": ""H"",
      ""transition_bonus"": 4676
    },
    {
      ""track_i_id"": ""D"",
      ""track_j_id"": ""E"",
      ""transition_bonus"": 332
    },
    {
      ""track_i_id"": ""D"",
      ""track_j_id"": ""G"",
      ""transition_bonus"": 4158
    },
    {
      ""track_i_id"": ""E"",
      ""track_j_id"": ""G"",
      ""transition_bonus"": 4662
    },
    {
      ""track_i_id"": ""E"",
      ""track_j_id"": ""H"",
      ""transition_bonus"": 5971
    },
    {
      ""track_i_id"": ""F"",
      ""track_j_id"": ""G"",
      ""transition_bonus"": 5149
    },
    {
      ""track_i_id"": ""F"",
      ""track_j_id"": ""H"",
      ""transition_bonus"": 884
    },
    {
      ""track_i_id"": ""G"",
      ""track_j_id"": ""H"",
      ""transition_bonus"": 817
    }
  ]
}

Also, when you hand in your final playlist, it'd be handy if you put the chosen tracks into a tiny JSON snippet so it's easy to read by whoever's cuing the set. Something like this will do:

{
  ""solution"": [<selected_track_id>, <selected_track_id>, ...]
}

Pretty simple: ""solution"" is the list of tracks you plan to play (in any order). Think of each array entry as the exact track identifier from the instance — that's the only thing the scheduler needs from you. This JSON is just a sketch of the shape I expect, not your actual answer.

Please make sure every identifier you use matches the instance exactly — no renaming, no made-up 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”.""","{'linear_coeffs': [41, 5069, 6389, 739, 134, 3802, 6015, 5566], 'quadratic_coeffs': [[0, 0, 0, 5300, 4736, 1042, 2506, 6239], [0, 2796, 2262, 0, 3659, 5432, 3206, 0], [0, 0, 4203, 4163, 4400, 0, 4974, 4676], [0, 0, 0, 2608, 332, 0, 4158, 0], [0, 0, 0, 0, 1147, 0, 4662, 5971], [0, 0, 0, 0, 0, 4706, 5149, 884], [0, 0, 0, 0, 0, 0, 0, 817], [0, 0, 0, 0, 0, 0, 0, 4593]], 'weights': [265, 4898, 6356, 448, 169, 3651, 6014, 5527], 'capacity': 18419, 'solution': [0, 1, 2, 3, 4, 6], 'obj': 62745.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 6]",62745.0,"{'problem_type': 'QKP', 'num_items': 8, 'capacity': 18419, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 41}, {'item_id': 'B', 'linear_profit': 5069}, {'item_id': 'C', 'linear_profit': 6389}, {'item_id': 'D', 'linear_profit': 739}, {'item_id': 'E', 'linear_profit': 134}, {'item_id': 'F', 'linear_profit': 3802}, {'item_id': 'G', 'linear_profit': 6015}, {'item_id': 'H', 'linear_profit': 5566}], 'weight_pairs': [{'item_id': 'A', 'weight': 265}, {'item_id': 'B', 'weight': 4898}, {'item_id': 'C', 'weight': 6356}, {'item_id': 'D', 'weight': 448}, {'item_id': 'E', 'weight': 169}, {'item_id': 'F', 'weight': 3651}, {'item_id': 'G', 'weight': 6014}, {'item_id': 'H', 'weight': 5527}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 5300}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 4736}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 1042}, {'item_i_id': 'A', 'item_j_id': 'G', 'quadratic_profit': 2506}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 6239}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 2796}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 2262}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 3659}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 5432}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 3206}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 4203}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 4163}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 4400}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 4974}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 4676}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 2608}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 332}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 4158}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 1147}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 4662}, {'item_i_id': 'E', 'item_j_id': 'H', 'quadratic_profit': 5971}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 4706}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 5149}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 884}, {'item_i_id': 'G', 'item_j_id': 'H', 'quadratic_profit': 817}, {'item_i_id': 'H', 'item_j_id': 'H', 'quadratic_profit': 4593}]}","['A', 'B', 'C', 'D', 'E', 'G']",4,json,names
QKP,QKP,"Recently a photographer had to trim a massive gear list into one bag for a fast-paced shoot, and the goal was clear: keep things light but versatile. Every lens or accessory adds a basic usefulness score, and some pairings add extra usefulness when both are included, so the final usefulness is calculated by adding each chosen item’s base score and then tacking on any pair bonuses. Weight is the limiter — the bag has a cap — and each item is either taken or not, no duplicates. The exact items, weights, and bonus pairings appear below.

{
  ""total_gear_items"": 7,
  ""bag_weight_capacity"": 2633,
  ""gear_identifiers"": [
    ""A"",
    ""B"",
    ""C"",
    ""D"",
    ""E"",
    ""F"",
    ""G""
  ],
  ""linear"": [
    {
      ""gear_id"": ""A"",
      ""base_usefulness_score"": 150
    },
    {
      ""gear_id"": ""B"",
      ""base_usefulness_score"": 814
    },
    {
      ""gear_id"": ""C"",
      ""base_usefulness_score"": 868
    },
    {
      ""gear_id"": ""D"",
      ""base_usefulness_score"": 910
    },
    {
      ""gear_id"": ""E"",
      ""base_usefulness_score"": 289
    },
    {
      ""gear_id"": ""F"",
      ""base_usefulness_score"": 650
    },
    {
      ""gear_id"": ""G"",
      ""base_usefulness_score"": 1662
    }
  ],
  ""weights"": [
    {
      ""gear_id"": ""A"",
      ""item_weight"": 1020
    },
    {
      ""gear_id"": ""B"",
      ""item_weight"": 115
    },
    {
      ""gear_id"": ""C"",
      ""item_weight"": 923
    },
    {
      ""gear_id"": ""D"",
      ""item_weight"": 1260
    },
    {
      ""gear_id"": ""E"",
      ""item_weight"": 1400
    },
    {
      ""gear_id"": ""F"",
      ""item_weight"": 1194
    },
    {
      ""gear_id"": ""G"",
      ""item_weight"": 598
    }
  ],
  ""quadratic"": [
    {
      ""gear_i_id"": ""A"",
      ""gear_j_id"": ""C"",
      ""pair_synergy_score"": 1020
    },
    {
      ""gear_i_id"": ""A"",
      ""gear_j_id"": ""D"",
      ""pair_synergy_score"": 344
    },
    {
      ""gear_i_id"": ""A"",
      ""gear_j_id"": ""E"",
      ""pair_synergy_score"": 1020
    },
    {
      ""gear_i_id"": ""A"",
      ""gear_j_id"": ""F"",
      ""pair_synergy_score"": 1020
    },
    {
      ""gear_i_id"": ""B"",
      ""gear_j_id"": ""C"",
      ""pair_synergy_score"": 115
    },
    {
      ""gear_i_id"": ""B"",
      ""gear_j_id"": ""E"",
      ""pair_synergy_score"": 115
    },
    {
      ""gear_i_id"": ""B"",
      ""gear_j_id"": ""F"",
      ""pair_synergy_score"": 115
    },
    {
      ""gear_i_id"": ""B"",
      ""gear_j_id"": ""G"",
      ""pair_synergy_score"": 115
    },
    {
      ""gear_i_id"": ""C"",
      ""gear_j_id"": ""E"",
      ""pair_synergy_score"": 923
    },
    {
      ""gear_i_id"": ""C"",
      ""gear_j_id"": ""F"",
      ""pair_synergy_score"": 598
    },
    {
      ""gear_i_id"": ""D"",
      ""gear_j_id"": ""E"",
      ""pair_synergy_score"": 344
    },
    {
      ""gear_i_id"": ""D"",
      ""gear_j_id"": ""F"",
      ""pair_synergy_score"": 344
    },
    {
      ""gear_i_id"": ""E"",
      ""gear_j_id"": ""F"",
      ""pair_synergy_score"": 499
    },
    {
      ""gear_i_id"": ""E"",
      ""gear_j_id"": ""G"",
      ""pair_synergy_score"": 624
    }
  ]
}

You can just reply with the chosen gear in a tiny JSON snippet like this — handy and simple.

{
  ""solution"": [<gear_id>, <gear_id>, ...]
}

Think of ""solution"" as the list of items you're packing for the shoot; each <gear_id> is a placeholder for one exact item identifier from the problem instance. This JSON is just a sketch of the shape I want, not the actual selection.

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”.""","{'linear_coeffs': [150, 814, 868, 910, 289, 650, 1662], 'quadratic_coeffs': [[0, 0, 1020, 344, 1020, 1020, 0], [0, 115, 115, 0, 115, 115, 115], [0, 0, 310, 0, 923, 598, 0], [0, 0, 0, 811, 344, 344, 0], [0, 0, 0, 0, 557, 499, 624], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0]], 'weights': [1020, 115, 923, 1260, 1400, 1194, 598], 'capacity': 2633, 'solution': [0, 2, 6], 'obj': 3700.0, 'problem_type': 'QKP'}","[0, 2, 6]",3700.0,"{'problem_type': 'QKP', 'num_items': 7, 'capacity': 2633, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 150}, {'item_id': 'B', 'linear_profit': 814}, {'item_id': 'C', 'linear_profit': 868}, {'item_id': 'D', 'linear_profit': 910}, {'item_id': 'E', 'linear_profit': 289}, {'item_id': 'F', 'linear_profit': 650}, {'item_id': 'G', 'linear_profit': 1662}], 'weight_pairs': [{'item_id': 'A', 'weight': 1020}, {'item_id': 'B', 'weight': 115}, {'item_id': 'C', 'weight': 923}, {'item_id': 'D', 'weight': 1260}, {'item_id': 'E', 'weight': 1400}, {'item_id': 'F', 'weight': 1194}, {'item_id': 'G', 'weight': 598}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 1020}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 344}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 1020}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 1020}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 115}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 115}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 115}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 115}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 115}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 310}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 923}, {'item_i_id': 'C', 'item_j_id': 'F', 'quadratic_profit': 598}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 811}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 344}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 344}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 557}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 499}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 624}]}","['A', 'C', 'G']",5,json,names
QKP,QKP,"I inherited a small raised bed and now have to pick which plants to put in it. Each plant takes up a bit of root space, and every choice brings a base harvest of veggies or flowers. Some combinations give a little extra — certain pairs are better buddies and boost each other’s output when they’re both planted. The trick is to choose a group of different plants whose total root space fits inside the bed, and whose overall harvest (each plant’s own yield plus any pairwise boosts) is as big as possible. The exact plant sizes, base yields and companion boosts are shown below.

{
  ""plant_count"": 5,
  ""root_space_capacity"": 297,
  ""plant_ids"": [
    ""A"",
    ""B"",
    ""C"",
    ""D"",
    ""E""
  ],
  ""linear"": [
    {
      ""plant_id"": ""A"",
      ""base_yield"": 105
    },
    {
      ""plant_id"": ""B"",
      ""base_yield"": 106
    },
    {
      ""plant_id"": ""C"",
      ""base_yield"": 113
    },
    {
      ""plant_id"": ""D"",
      ""base_yield"": 106
    },
    {
      ""plant_id"": ""E"",
      ""base_yield"": 115
    }
  ],
  ""weights"": [
    {
      ""plant_id"": ""A"",
      ""root_space"": 109
    },
    {
      ""plant_id"": ""B"",
      ""root_space"": 109
    },
    {
      ""plant_id"": ""C"",
      ""root_space"": 108
    },
    {
      ""plant_id"": ""D"",
      ""root_space"": 107
    },
    {
      ""plant_id"": ""E"",
      ""root_space"": 109
    }
  ],
  ""quadratic"": [
    {
      ""plant_i_id"": ""A"",
      ""plant_j_id"": ""B"",
      ""companion_bonus"": 109
    },
    {
      ""plant_i_id"": ""A"",
      ""plant_j_id"": ""C"",
      ""companion_bonus"": 107
    },
    {
      ""plant_i_id"": ""B"",
      ""plant_j_id"": ""C"",
      ""companion_bonus"": 102
    },
    {
      ""plant_i_id"": ""B"",
      ""plant_j_id"": ""D"",
      ""companion_bonus"": 102
    },
    {
      ""plant_i_id"": ""B"",
      ""plant_j_id"": ""E"",
      ""companion_bonus"": 100
    },
    {
      ""plant_i_id"": ""D"",
      ""plant_j_id"": ""E"",
      ""companion_bonus"": 113
    }
  ]
}

Oh, and when you're ready to tell me which plants to plant, just pop your pick into this tiny JSON layout so it's straightforward to read:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

Pretty simple: ""solution"" is just a list of the plant IDs you want in the bed. Think of it like a short shopping list — put each plant's identifier in the array, nothing else. This JSON is just a sketch of the shape I expect, 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. 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”.""","{'linear_coeffs': [105, 106, 113, 106, 115], 'quadratic_coeffs': [[109, 109, 107, 0, 0], [0, 109, 102, 102, 100], [0, 0, 123, 0, 0], [0, 0, 0, 113, 113], [0, 0, 0, 0, 0]], 'weights': [109, 109, 108, 107, 109], 'capacity': 297, 'solution': [3, 4], 'obj': 334.0, 'problem_type': 'QKP'}","[3, 4]",334.0,"{'problem_type': 'QKP', 'num_items': 5, 'capacity': 297, 'items': ['A', 'B', 'C', 'D', 'E'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 105}, {'item_id': 'B', 'linear_profit': 106}, {'item_id': 'C', 'linear_profit': 113}, {'item_id': 'D', 'linear_profit': 106}, {'item_id': 'E', 'linear_profit': 115}], 'weight_pairs': [{'item_id': 'A', 'weight': 109}, {'item_id': 'B', 'weight': 109}, {'item_id': 'C', 'weight': 108}, {'item_id': 'D', 'weight': 107}, {'item_id': 'E', 'weight': 109}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 109}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 109}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 107}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 109}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 102}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 102}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 100}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 123}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 113}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 113}]}","['D', 'E']",6,json,names
QKP,QKP,"Recently the team had to prioritize features for a tight release, and the conversation boiled down to picking a set that actually fits the sprint. Each feature takes a chunk of time and delivers a baseline user value; on top of that, certain feature pairs multiply the perceived value when they’re both delivered. The trick is to assemble features so the total time stays inside the sprint budget and the total user value equals the sum of chosen features’ base values plus any extra pair bonuses for combinations. Nothing can be split or chosen twice, and the precise numbers are shown below.

{
  ""num_features"": 8,
  ""sprint_effort_budget"": 1511,
  ""feature_ids"": [
    ""A"",
    ""B"",
    ""C"",
    ""D"",
    ""E"",
    ""F"",
    ""G"",
    ""H""
  ],
  ""linear"": [
    {
      ""feature_id"": ""A"",
      ""base_user_value"": 154
    },
    {
      ""feature_id"": ""B"",
      ""base_user_value"": 123
    },
    {
      ""feature_id"": ""C"",
      ""base_user_value"": 217
    },
    {
      ""feature_id"": ""D"",
      ""base_user_value"": 280
    },
    {
      ""feature_id"": ""E"",
      ""base_user_value"": 239
    },
    {
      ""feature_id"": ""F"",
      ""base_user_value"": 242
    },
    {
      ""feature_id"": ""G"",
      ""base_user_value"": 225
    },
    {
      ""feature_id"": ""H"",
      ""base_user_value"": 128
    }
  ],
  ""weights"": [
    {
      ""feature_id"": ""A"",
      ""development_effort"": 139
    },
    {
      ""feature_id"": ""B"",
      ""development_effort"": 111
    },
    {
      ""feature_id"": ""C"",
      ""development_effort"": 220
    },
    {
      ""feature_id"": ""D"",
      ""development_effort"": 271
    },
    {
      ""feature_id"": ""E"",
      ""development_effort"": 231
    },
    {
      ""feature_id"": ""F"",
      ""development_effort"": 254
    },
    {
      ""feature_id"": ""G"",
      ""development_effort"": 229
    },
    {
      ""feature_id"": ""H"",
      ""development_effort"": 134
    }
  ],
  ""quadratic"": [
    {
      ""feature_i_id"": ""A"",
      ""feature_j_id"": ""C"",
      ""interaction_value"": 139
    },
    {
      ""feature_i_id"": ""A"",
      ""feature_j_id"": ""D"",
      ""interaction_value"": 139
    },
    {
      ""feature_i_id"": ""A"",
      ""feature_j_id"": ""E"",
      ""interaction_value"": 139
    },
    {
      ""feature_i_id"": ""A"",
      ""feature_j_id"": ""F"",
      ""interaction_value"": 139
    },
    {
      ""feature_i_id"": ""A"",
      ""feature_j_id"": ""H"",
      ""interaction_value"": 139
    },
    {
      ""feature_i_id"": ""B"",
      ""feature_j_id"": ""C"",
      ""interaction_value"": 111
    },
    {
      ""feature_i_id"": ""B"",
      ""feature_j_id"": ""D"",
      ""interaction_value"": 111
    },
    {
      ""feature_i_id"": ""B"",
      ""feature_j_id"": ""E"",
      ""interaction_value"": 110
    },
    {
      ""feature_i_id"": ""C"",
      ""feature_j_id"": ""D"",
      ""interaction_value"": 236
    },
    {
      ""feature_i_id"": ""C"",
      ""feature_j_id"": ""E"",
      ""interaction_value"": 147
    },
    {
      ""feature_i_id"": ""C"",
      ""feature_j_id"": ""G"",
      ""interaction_value"": 107
    },
    {
      ""feature_i_id"": ""D"",
      ""feature_j_id"": ""H"",
      ""interaction_value"": 121
    },
    {
      ""feature_i_id"": ""E"",
      ""feature_j_id"": ""F"",
      ""interaction_value"": 150
    }
  ]
}

Oh, and when you send back the chosen set, a tiny JSON snippet like this works great:

{
  ""solution"": [<selected_feature_id>, <selected_feature_id>, ...]
}

Think of ""solution"" as just a list of the features you want in the sprint — put each feature's identifier in that array, that's all. This JSON is just the shape I expect, not the final picks.

Please 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”.""","{'linear_coeffs': [154, 123, 217, 280, 239, 242, 225, 128], 'quadratic_coeffs': [[139, 0, 139, 139, 139, 139, 0, 139], [0, 111, 111, 111, 110, 0, 0, 0], [0, 0, 0, 236, 147, 0, 107, 0], [0, 0, 0, 108, 0, 0, 0, 121], [0, 0, 0, 0, 0, 150, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [139, 111, 220, 271, 231, 254, 229, 134], 'capacity': 1511, 'solution': [0, 1, 2, 3, 4, 5, 7], 'obj': 3064.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 5, 7]",3064.0,"{'problem_type': 'QKP', 'num_items': 8, 'capacity': 1511, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 154}, {'item_id': 'B', 'linear_profit': 123}, {'item_id': 'C', 'linear_profit': 217}, {'item_id': 'D', 'linear_profit': 280}, {'item_id': 'E', 'linear_profit': 239}, {'item_id': 'F', 'linear_profit': 242}, {'item_id': 'G', 'linear_profit': 225}, {'item_id': 'H', 'linear_profit': 128}], 'weight_pairs': [{'item_id': 'A', 'weight': 139}, {'item_id': 'B', 'weight': 111}, {'item_id': 'C', 'weight': 220}, {'item_id': 'D', 'weight': 271}, {'item_id': 'E', 'weight': 231}, {'item_id': 'F', 'weight': 254}, {'item_id': 'G', 'weight': 229}, {'item_id': 'H', 'weight': 134}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 139}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 139}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 139}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 139}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 139}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 139}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 111}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 111}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 111}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 110}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 236}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 147}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 107}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 108}, {'item_i_id': 'D', 'item_j_id': 'H', 'quadratic_profit': 121}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 150}]}","['A', 'B', 'C', 'D', 'E', 'F', 'H']",7,json,names
QKP,QKP,"Many people don’t realize how much thought goes into a tasting menu: each plate has its own pull, certain combinations amplify that pull, and the kitchen can only handle a certain amount of prep and plating. With no dish duplicated, the challenge is to decide which courses to include so the guests’ total delight — the sum of every dish’s base delight plus any extra boost for pairs that appear together — is maximized without exceeding the kitchen’s capacity. The concrete course list, prep loads, and pairwise boosts are provided below.

{
  ""num_courses"": 9,
  ""kitchen_capacity"": 5738,
  ""course_ids"": [
    ""A"",
    ""B"",
    ""C"",
    ""D"",
    ""E"",
    ""F"",
    ""G"",
    ""H"",
    ""I""
  ],
  ""linear"": [
    {
      ""course_id"": ""A"",
      ""base_delight"": 4585
    },
    {
      ""course_id"": ""B"",
      ""base_delight"": 1986
    },
    {
      ""course_id"": ""C"",
      ""base_delight"": 6528
    },
    {
      ""course_id"": ""D"",
      ""base_delight"": 808
    },
    {
      ""course_id"": ""E"",
      ""base_delight"": 6912
    },
    {
      ""course_id"": ""F"",
      ""base_delight"": 3686
    },
    {
      ""course_id"": ""G"",
      ""base_delight"": 3378
    },
    {
      ""course_id"": ""H"",
      ""base_delight"": 4325
    },
    {
      ""course_id"": ""I"",
      ""base_delight"": 4943
    }
  ],
  ""weights"": [
    {
      ""course_id"": ""A"",
      ""prep_plating_load"": 3935
    },
    {
      ""course_id"": ""B"",
      ""prep_plating_load"": 1336
    },
    {
      ""course_id"": ""C"",
      ""prep_plating_load"": 5878
    },
    {
      ""course_id"": ""D"",
      ""prep_plating_load"": 158
    },
    {
      ""course_id"": ""E"",
      ""prep_plating_load"": 6262
    },
    {
      ""course_id"": ""F"",
      ""prep_plating_load"": 3036
    },
    {
      ""course_id"": ""G"",
      ""prep_plating_load"": 2728
    },
    {
      ""course_id"": ""H"",
      ""prep_plating_load"": 3675
    },
    {
      ""course_id"": ""I"",
      ""prep_plating_load"": 4293
    }
  ],
  ""quadratic"": [
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""B"",
      ""pair_complement_boost"": 3635
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""C"",
      ""pair_complement_boost"": 188
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""D"",
      ""pair_complement_boost"": 4986
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""E"",
      ""pair_complement_boost"": 5623
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""F"",
      ""pair_complement_boost"": 812
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""G"",
      ""pair_complement_boost"": 4713
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""H"",
      ""pair_complement_boost"": 2176
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""I"",
      ""pair_complement_boost"": 5840
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""C"",
      ""pair_complement_boost"": 5915
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""D"",
      ""pair_complement_boost"": 5299
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""E"",
      ""pair_complement_boost"": 972
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""F"",
      ""pair_complement_boost"": 3794
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""G"",
      ""pair_complement_boost"": 2217
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""H"",
      ""pair_complement_boost"": 2479
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""I"",
      ""pair_complement_boost"": 3021
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""D"",
      ""pair_complement_boost"": 1018
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""E"",
      ""pair_complement_boost"": 4788
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""F"",
      ""pair_complement_boost"": 3266
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""G"",
      ""pair_complement_boost"": 5484
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""H"",
      ""pair_complement_boost"": 909
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""I"",
      ""pair_complement_boost"": 3091
    },
    {
      ""course_i_id"": ""D"",
      ""course_j_id"": ""E"",
      ""pair_complement_boost"": 1525
    },
    {
      ""course_i_id"": ""D"",
      ""course_j_id"": ""F"",
      ""pair_complement_boost"": 4959
    },
    {
      ""course_i_id"": ""D"",
      ""course_j_id"": ""G"",
      ""pair_complement_boost"": 2900
    },
    {
      ""course_i_id"": ""D"",
      ""course_j_id"": ""H"",
      ""pair_complement_boost"": 2779
    },
    {
      ""course_i_id"": ""D"",
      ""course_j_id"": ""I"",
      ""pair_complement_boost"": 1659
    },
    {
      ""course_i_id"": ""E"",
      ""course_j_id"": ""F"",
      ""pair_complement_boost"": 4321
    },
    {
      ""course_i_id"": ""E"",
      ""course_j_id"": ""G"",
      ""pair_complement_boost"": 3748
    },
    {
      ""course_i_id"": ""E"",
      ""course_j_id"": ""H"",
      ""pair_complement_boost"": 1455
    },
    {
      ""course_i_id"": ""E"",
      ""course_j_id"": ""I"",
      ""pair_complement_boost"": 3830
    },
    {
      ""course_i_id"": ""F"",
      ""course_j_id"": ""G"",
      ""pair_complement_boost"": 2891
    },
    {
      ""course_i_id"": ""F"",
      ""course_j_id"": ""H"",
      ""pair_complement_boost"": 2692
    },
    {
      ""course_i_id"": ""F"",
      ""course_j_id"": ""I"",
      ""pair_complement_boost"": 5680
    }
  ]
}

If you want to send back a pick, a simple JSON snippet is perfect — something casual like this will do:

{
  ""solution"": [<course_id>, <course_id>, ...]
}

Here ""solution"" is just the list of course IDs you want on the tasting menu — each <course_id> is a placeholder for whatever identifier the instance uses for a dish. Think of it like filling out a tiny form: list the courses you choose in that array, and that’s your answer. This is just a sketch of the shape I expect, not the final selection 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”.""","{'linear_coeffs': [4585, 1986, 6528, 808, 6912, 3686, 3378, 4325, 4943], 'quadratic_coeffs': [[320, 3635, 188, 4986, 5623, 812, 4713, 2176, 5840], [0, 5855, 5915, 5299, 972, 3794, 2217, 2479, 3021], [0, 0, 3777, 1018, 4788, 3266, 5484, 909, 3091], [0, 0, 0, 3723, 1525, 4959, 2900, 2779, 1659], [0, 0, 0, 0, 5655, 4321, 3748, 1455, 3830], [0, 0, 0, 0, 0, 4173, 2891, 2692, 5680], [0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [3935, 1336, 5878, 158, 6262, 3036, 2728, 3675, 4293], 'capacity': 5738, 'solution': [0, 1, 3], 'obj': 21299.0, 'problem_type': 'QKP'}","[0, 1, 3]",21299.0,"{'problem_type': 'QKP', 'num_items': 9, 'capacity': 5738, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 4585}, {'item_id': 'B', 'linear_profit': 1986}, {'item_id': 'C', 'linear_profit': 6528}, {'item_id': 'D', 'linear_profit': 808}, {'item_id': 'E', 'linear_profit': 6912}, {'item_id': 'F', 'linear_profit': 3686}, {'item_id': 'G', 'linear_profit': 3378}, {'item_id': 'H', 'linear_profit': 4325}, {'item_id': 'I', 'linear_profit': 4943}], 'weight_pairs': [{'item_id': 'A', 'weight': 3935}, {'item_id': 'B', 'weight': 1336}, {'item_id': 'C', 'weight': 5878}, {'item_id': 'D', 'weight': 158}, {'item_id': 'E', 'weight': 6262}, {'item_id': 'F', 'weight': 3036}, {'item_id': 'G', 'weight': 2728}, {'item_id': 'H', 'weight': 3675}, {'item_id': 'I', 'weight': 4293}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 320}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 3635}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 188}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 4986}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 5623}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 812}, {'item_i_id': 'A', 'item_j_id': 'G', 'quadratic_profit': 4713}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 2176}, {'item_i_id': 'A', 'item_j_id': 'I', 'quadratic_profit': 5840}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 5855}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 5915}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 5299}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 972}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 3794}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 2217}, {'item_i_id': 'B', 'item_j_id': 'H', 'quadratic_profit': 2479}, {'item_i_id': 'B', 'item_j_id': 'I', 'quadratic_profit': 3021}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 3777}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 1018}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 4788}, {'item_i_id': 'C', 'item_j_id': 'F', 'quadratic_profit': 3266}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 5484}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 909}, {'item_i_id': 'C', 'item_j_id': 'I', 'quadratic_profit': 3091}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 3723}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 1525}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 4959}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 2900}, {'item_i_id': 'D', 'item_j_id': 'H', 'quadratic_profit': 2779}, {'item_i_id': 'D', 'item_j_id': 'I', 'quadratic_profit': 1659}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 5655}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 4321}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 3748}, {'item_i_id': 'E', 'item_j_id': 'H', 'quadratic_profit': 1455}, {'item_i_id': 'E', 'item_j_id': 'I', 'quadratic_profit': 3830}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 4173}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 2891}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 2692}, {'item_i_id': 'F', 'item_j_id': 'I', 'quadratic_profit': 5680}]}","['A', 'B', 'D']",8,json,names
QKP,QKP,"Recently the design team challenged themselves to build the perfect gift box that feels greater than the sum of its parts but still ships cheaply. Every piece has a weight and a basic value, and some duos create extra value when they’re both included. The task is to pick a subset of items so the total box weight stays under the shipping threshold and the overall perceived value equals adding up each chosen item’s base value plus any bonuses for included pairs; nothing can be split or doubled. The concrete details for the items and the pair bonuses are given below.

{
  ""total_items_available"": 5,
  ""shipping_capacity"": 7298,
  ""item_identifiers"": [
    0,
    1,
    2,
    3,
    4
  ],
  ""linear"": [
    {
      ""item_ref_for_weight"": 0,
      ""base_value"": 6361
    },
    {
      ""item_ref_for_weight"": 1,
      ""base_value"": 3859
    },
    {
      ""item_ref_for_weight"": 2,
      ""base_value"": 7003
    },
    {
      ""item_ref_for_weight"": 3,
      ""base_value"": 4335
    },
    {
      ""item_ref_for_weight"": 4,
      ""base_value"": 3357
    }
  ],
  ""weights"": [
    {
      ""item_ref_for_weight"": 0,
      ""item_weight"": 5711
    },
    {
      ""item_ref_for_weight"": 1,
      ""item_weight"": 3209
    },
    {
      ""item_ref_for_weight"": 2,
      ""item_weight"": 6353
    },
    {
      ""item_ref_for_weight"": 3,
      ""item_weight"": 3685
    },
    {
      ""item_ref_for_weight"": 4,
      ""item_weight"": 2707
    }
  ],
  ""quadratic"": [
    {
      ""pair_item_i"": 0,
      ""pair_item_j"": 1,
      ""pair_synergy_value"": 4018
    },
    {
      ""pair_item_i"": 0,
      ""pair_item_j"": 2,
      ""pair_synergy_value"": 4589
    },
    {
      ""pair_item_i"": 0,
      ""pair_item_j"": 3,
      ""pair_synergy_value"": 970
    },
    {
      ""pair_item_i"": 0,
      ""pair_item_j"": 4,
      ""pair_synergy_value"": 1883
    },
    {
      ""pair_item_i"": 1,
      ""pair_item_j"": 2,
      ""pair_synergy_value"": 4519
    },
    {
      ""pair_item_i"": 1,
      ""pair_item_j"": 3,
      ""pair_synergy_value"": 1242
    },
    {
      ""pair_item_i"": 1,
      ""pair_item_j"": 4,
      ""pair_synergy_value"": 381
    },
    {
      ""pair_item_i"": 2,
      ""pair_item_j"": 3,
      ""pair_synergy_value"": 5063
    },
    {
      ""pair_item_i"": 2,
      ""pair_item_j"": 4,
      ""pair_synergy_value"": 2573
    },
    {
      ""pair_item_i"": 3,
      ""pair_item_j"": 4,
      ""pair_synergy_value"": 3131
    }
  ]
}

You can just send back the chosen items in a small JSON snippet like this — it’s just the shape I expect, nothing fancy.

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

Here ""solution"" is simply a list of the items you want in the gift box. Think of each placeholder as the exact item label from the instance (the things you see in the item list). This block is a sketch of the format I want, not your final answer.

Please use the item identifiers exactly as they appear in the instance input — no renaming, no made-up 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”.""","{'linear_coeffs': [6361, 3859, 7003, 4335, 3357], 'quadratic_coeffs': [[4399, 4018, 4589, 970, 1883], [0, 2751, 4519, 1242, 381], [0, 0, 4035, 5063, 2573], [0, 0, 0, 1729, 3131], [0, 0, 0, 0, 1415]], 'weights': [5711, 3209, 6353, 3685, 2707], 'capacity': 7298, 'solution': [3, 4], 'obj': 10823.0, 'problem_type': 'QKP'}","[3, 4]",10823.0,"{'problem_type': 'QKP', 'num_items': 5, 'capacity': 7298, 'items': [0, 1, 2, 3, 4], 'linear_pairs': [{'item_id': 0, 'linear_profit': 6361}, {'item_id': 1, 'linear_profit': 3859}, {'item_id': 2, 'linear_profit': 7003}, {'item_id': 3, 'linear_profit': 4335}, {'item_id': 4, 'linear_profit': 3357}], 'weight_pairs': [{'item_id': 0, 'weight': 5711}, {'item_id': 1, 'weight': 3209}, {'item_id': 2, 'weight': 6353}, {'item_id': 3, 'weight': 3685}, {'item_id': 4, 'weight': 2707}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 4399}, {'item_i_id': 0, 'item_j_id': 1, 'quadratic_profit': 4018}, {'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 4589}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 970}, {'item_i_id': 0, 'item_j_id': 4, 'quadratic_profit': 1883}, {'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 2751}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 4519}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 1242}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 381}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 4035}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 5063}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 2573}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 1729}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 3131}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 1415}]}","[3, 4]",9,json,0
QKP,QKP,"Recently came back from visiting family with too many little gifts to fit in my luggage. Each gift has a small sentimental value and a weight, and a couple of items belong together and create extra value if both are packed. The goal is to maximize the total sentimental pay-off by adding up the chosen items’ individual values plus any pair bonuses, but the suitcase must stay under the airline’s weight allowance, and items can’t be split or duplicated. The detailed item information appears below.

{
  ""num_souvenirs"": 10,
  ""suitcase_weight_limit"": 691,
  ""souvenir_ids"": [
    0,
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    9
  ],
  ""linear"": [
    {
      ""souvenir_id"": 0,
      ""sentimental_value"": 112
    },
    {
      ""souvenir_id"": 1,
      ""sentimental_value"": 112
    },
    {
      ""souvenir_id"": 2,
      ""sentimental_value"": 128
    },
    {
      ""souvenir_id"": 3,
      ""sentimental_value"": 117
    },
    {
      ""souvenir_id"": 4,
      ""sentimental_value"": 119
    },
    {
      ""souvenir_id"": 5,
      ""sentimental_value"": 133
    },
    {
      ""souvenir_id"": 6,
      ""sentimental_value"": 124
    },
    {
      ""souvenir_id"": 7,
      ""sentimental_value"": 117
    },
    {
      ""souvenir_id"": 8,
      ""sentimental_value"": 121
    },
    {
      ""souvenir_id"": 9,
      ""sentimental_value"": 118
    }
  ],
  ""weights"": [
    {
      ""souvenir_id"": 0,
      ""souvenir_weight"": 100
    },
    {
      ""souvenir_id"": 1,
      ""souvenir_weight"": 100
    },
    {
      ""souvenir_id"": 2,
      ""souvenir_weight"": 116
    },
    {
      ""souvenir_id"": 3,
      ""souvenir_weight"": 105
    },
    {
      ""souvenir_id"": 4,
      ""souvenir_weight"": 107
    },
    {
      ""souvenir_id"": 5,
      ""souvenir_weight"": 121
    },
    {
      ""souvenir_id"": 6,
      ""souvenir_weight"": 112
    },
    {
      ""souvenir_id"": 7,
      ""souvenir_weight"": 105
    },
    {
      ""souvenir_id"": 8,
      ""souvenir_weight"": 109
    },
    {
      ""souvenir_id"": 9,
      ""souvenir_weight"": 106
    }
  ],
  ""quadratic"": [
    {
      ""souvenir_i_id"": 0,
      ""souvenir_j_id"": 4,
      ""pair_bonus_sentimental_value"": 102
    },
    {
      ""souvenir_i_id"": 0,
      ""souvenir_j_id"": 5,
      ""pair_bonus_sentimental_value"": 108
    },
    {
      ""souvenir_i_id"": 0,
      ""souvenir_j_id"": 6,
      ""pair_bonus_sentimental_value"": 121
    },
    {
      ""souvenir_i_id"": 0,
      ""souvenir_j_id"": 7,
      ""pair_bonus_sentimental_value"": 105
    },
    {
      ""souvenir_i_id"": 1,
      ""souvenir_j_id"": 5,
      ""pair_bonus_sentimental_value"": 101
    },
    {
      ""souvenir_i_id"": 1,
      ""souvenir_j_id"": 7,
      ""pair_bonus_sentimental_value"": 111
    },
    {
      ""souvenir_i_id"": 2,
      ""souvenir_j_id"": 4,
      ""pair_bonus_sentimental_value"": 105
    },
    {
      ""souvenir_i_id"": 2,
      ""souvenir_j_id"": 5,
      ""pair_bonus_sentimental_value"": 114
    },
    {
      ""souvenir_i_id"": 2,
      ""souvenir_j_id"": 9,
      ""pair_bonus_sentimental_value"": 107
    },
    {
      ""souvenir_i_id"": 3,
      ""souvenir_j_id"": 7,
      ""pair_bonus_sentimental_value"": 123
    },
    {
      ""souvenir_i_id"": 4,
      ""souvenir_j_id"": 6,
      ""pair_bonus_sentimental_value"": 107
    },
    {
      ""souvenir_i_id"": 4,
      ""souvenir_j_id"": 8,
      ""pair_bonus_sentimental_value"": 101
    },
    {
      ""souvenir_i_id"": 5,
      ""souvenir_j_id"": 6,
      ""pair_bonus_sentimental_value"": 102
    },
    {
      ""souvenir_i_id"": 6,
      ""souvenir_j_id"": 8,
      ""pair_bonus_sentimental_value"": 122
    },
    {
      ""souvenir_i_id"": 6,
      ""souvenir_j_id"": 9,
      ""pair_bonus_sentimental_value"": 110
    }
  ]
}

Also, when you send back which gifts you want in the suitcase, a tiny JSON snippet like this is perfect:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

This just lists the items you’d pack. The ""solution"" array should contain the item IDs from the instance (each <selected_item_id> is a placeholder for one chosen gift). It’s just a sketch of the shape I expect, not your final 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”.""","{'linear_coeffs': [112, 112, 128, 117, 119, 133, 124, 117, 121, 118], 'quadratic_coeffs': [[0, 0, 0, 0, 102, 108, 121, 105, 0, 0], [0, 122, 0, 0, 0, 101, 0, 111, 0, 0], [0, 0, 0, 0, 105, 114, 0, 0, 0, 107], [0, 0, 0, 0, 0, 0, 0, 123, 0, 0], [0, 0, 0, 0, 112, 0, 107, 0, 101, 0], [0, 0, 0, 0, 0, 0, 102, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 122, 110], [0, 0, 0, 0, 0, 0, 0, 116, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 124, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [100, 100, 116, 105, 107, 121, 112, 105, 109, 106], 'capacity': 691, 'solution': [0, 2, 4, 5, 6, 8], 'obj': 1719.0, 'problem_type': 'QKP'}","[0, 2, 4, 5, 6, 8]",1719.0,"{'problem_type': 'QKP', 'num_items': 10, 'capacity': 691, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 'linear_pairs': [{'item_id': 0, 'linear_profit': 112}, {'item_id': 1, 'linear_profit': 112}, {'item_id': 2, 'linear_profit': 128}, {'item_id': 3, 'linear_profit': 117}, {'item_id': 4, 'linear_profit': 119}, {'item_id': 5, 'linear_profit': 133}, {'item_id': 6, 'linear_profit': 124}, {'item_id': 7, 'linear_profit': 117}, {'item_id': 8, 'linear_profit': 121}, {'item_id': 9, 'linear_profit': 118}], 'weight_pairs': [{'item_id': 0, 'weight': 100}, {'item_id': 1, 'weight': 100}, {'item_id': 2, 'weight': 116}, {'item_id': 3, 'weight': 105}, {'item_id': 4, 'weight': 107}, {'item_id': 5, 'weight': 121}, {'item_id': 6, 'weight': 112}, {'item_id': 7, 'weight': 105}, {'item_id': 8, 'weight': 109}, {'item_id': 9, 'weight': 106}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 4, 'quadratic_profit': 102}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 108}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 121}, {'item_i_id': 0, 'item_j_id': 7, 'quadratic_profit': 105}, {'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 122}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 101}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 111}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 105}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 114}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 107}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 123}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 112}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 107}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 101}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 102}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 122}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 110}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 116}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 124}]}","[0, 2, 4, 5, 6, 8]",10,json,0
QKP,QKP,"There’s a small run of a collector’s edition to pack, and the team is deciding which deluxe components to slip into the box. Each component brings some standalone appeal, some pairs have extra synergy when included together, and every piece adds to the package’s weight. The job is to pick a selection (no duplicates, and nothing partial) that stays within the shipping capacity and gives the biggest total player value — computed as the sum of each chosen item’s base value plus any extra pair bonuses. The full breakdown of parts, weights and bonuses appears below.

Below are the 5 available deluxe components 1, 2, 3, 4, 5 and the box's shipping capacity 499.
Component 1 provides a base player value of 124.
Component 2 provides a base player value of 138.
Component 3 provides a base player value of 153.
Component 4 provides a base player value of 116.
Component 5 provides a base player value of 115.
Component 1 contributes 129 to the package weight.
Component 2 contributes 133 to the package weight.
Component 3 contributes 146 to the package weight.
Component 4 contributes 117 to the package weight.
Component 5 contributes 121 to the package weight.
Including both 1 and 3 grants an extra pair synergy of 136.
Including both 1 and 4 grants an extra pair synergy of 138.
Including both 1 and 5 grants an extra pair synergy of 138.
Including both 2 and 4 grants an extra pair synergy of 129.
Including both 3 and 4 grants an extra pair synergy of 135.
Including both 3 and 5 grants an extra pair synergy of 124.
Including both 4 and 5 grants an extra pair synergy of 110.
Selections must respect the 499 and maximize total player value.

Oh, and when you’re ready to send your pick, please use this simple JSON layout so I can read it easily:

{
  ""solution"": [<component_id>, <component_id>, ...]
}

This just means ""solution"" should be a list of the component IDs you want packed into the box. The JSON above is just a sketch of the shape I expect — not the final answer itself.

Please use the exact identifiers from the instance input — don’t rename them or make up 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”.","{'linear_coeffs': [124, 138, 153, 116, 115], 'quadratic_coeffs': [[141, 0, 136, 138, 138], [0, 130, 0, 129, 0], [0, 0, 128, 135, 124], [0, 0, 0, 112, 110], [0, 0, 0, 0, 0]], 'weights': [129, 133, 146, 117, 121], 'capacity': 499, 'solution': [0, 2, 3], 'obj': 802.0, 'problem_type': 'QKP'}","[0, 2, 3]",802.0,"{'problem_type': 'QKP', 'num_items': 5, 'capacity': 499, 'items': [1, 2, 3, 4, 5], 'linear_pairs': [{'item_id': 1, 'linear_profit': 124}, {'item_id': 2, 'linear_profit': 138}, {'item_id': 3, 'linear_profit': 153}, {'item_id': 4, 'linear_profit': 116}, {'item_id': 5, 'linear_profit': 115}], 'weight_pairs': [{'item_id': 1, 'weight': 129}, {'item_id': 2, 'weight': 133}, {'item_id': 3, 'weight': 146}, {'item_id': 4, 'weight': 117}, {'item_id': 5, 'weight': 121}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 141}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 136}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 138}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 138}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 130}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 129}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 128}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 135}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 124}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 112}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 110}]}","[1, 3, 4]",11,nl,1
QKP,QKP,"Many people underestimate how much the order and pairing of images change a printed book, so this time the plan is to pick images with both individual strength and good chemistry. Every shot has a baseline effect and a page cost, and some pairs add a special bonus if both are included. The final quality of the portfolio is the total of chosen photos’ base effects plus any added pair bonuses, and the selection can’t exceed the allowed page total or include duplicates. The specific photo sizes, base effects, and pairwise bonuses appear below.

Here are the 10 available photos (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and the page limit of 2697 pages.
Photo 0 has a baseline effect of 970.
Photo 1 has a baseline effect of 475.
Photo 2 has a baseline effect of 798.
Photo 3 has a baseline effect of 560.
Photo 4 has a baseline effect of 231.
Photo 5 has a baseline effect of 737.
Photo 6 has a baseline effect of 658.
Photo 7 has a baseline effect of 800.
Photo 8 has a baseline effect of 617.
Photo 9 has a baseline effect of 636.
Photo 0 occupies 880 pages in the layout.
Photo 1 occupies 385 pages in the layout.
Photo 2 occupies 708 pages in the layout.
Photo 3 occupies 470 pages in the layout.
Photo 4 occupies 141 pages in the layout.
Photo 5 occupies 647 pages in the layout.
Photo 6 occupies 568 pages in the layout.
Photo 7 occupies 710 pages in the layout.
Photo 8 occupies 527 pages in the layout.
Photo 9 occupies 546 pages in the layout.
If both photo 0 and photo 1 are included they yield a pairing bonus of 454.
If both photo 0 and photo 2 are included they yield a pairing bonus of 483.
If both photo 0 and photo 3 are included they yield a pairing bonus of 641.
If both photo 0 and photo 4 are included they yield a pairing bonus of 435.
If both photo 0 and photo 5 are included they yield a pairing bonus of 733.
If both photo 0 and photo 6 are included they yield a pairing bonus of 599.
If both photo 0 and photo 7 are included they yield a pairing bonus of 596.
If both photo 0 and photo 8 are included they yield a pairing bonus of 686.
If both photo 0 and photo 9 are included they yield a pairing bonus of 694.
If both photo 1 and photo 2 are included they yield a pairing bonus of 513.
If both photo 1 and photo 3 are included they yield a pairing bonus of 578.
If both photo 1 and photo 4 are included they yield a pairing bonus of 467.
If both photo 1 and photo 5 are included they yield a pairing bonus of 386.
If both photo 1 and photo 6 are included they yield a pairing bonus of 435.
If both photo 1 and photo 7 are included they yield a pairing bonus of 583.
If both photo 1 and photo 8 are included they yield a pairing bonus of 458.
If both photo 1 and photo 9 are included they yield a pairing bonus of 566.
If both photo 2 and photo 3 are included they yield a pairing bonus of 314.
If both photo 2 and photo 4 are included they yield a pairing bonus of 544.
If both photo 2 and photo 5 are included they yield a pairing bonus of 600.
If both photo 2 and photo 6 are included they yield a pairing bonus of 401.
If both photo 2 and photo 7 are included they yield a pairing bonus of 646.
If both photo 2 and photo 8 are included they yield a pairing bonus of 573.
If both photo 2 and photo 9 are included they yield a pairing bonus of 440.
If both photo 3 and photo 4 are included they yield a pairing bonus of 634.
If both photo 3 and photo 5 are included they yield a pairing bonus of 518.
If both photo 3 and photo 6 are included they yield a pairing bonus of 577.
If both photo 3 and photo 7 are included they yield a pairing bonus of 525.
If both photo 3 and photo 8 are included they yield a pairing bonus of 566.
If both photo 3 and photo 9 are included they yield a pairing bonus of 837.
If both photo 4 and photo 5 are included they yield a pairing bonus of 274.
If both photo 4 and photo 6 are included they yield a pairing bonus of 128.
If both photo 4 and photo 7 are included they yield a pairing bonus of 589.
If both photo 4 and photo 8 are included they yield a pairing bonus of 850.
If both photo 4 and photo 9 are included they yield a pairing bonus of 756.
If both photo 5 and photo 6 are included they yield a pairing bonus of 595.
If both photo 5 and photo 7 are included they yield a pairing bonus of 397.
If both photo 5 and photo 8 are included they yield a pairing bonus of 557.
If both photo 5 and photo 9 are included they yield a pairing bonus of 399.
If both photo 6 and photo 7 are included they yield a pairing bonus of 371.
If both photo 6 and photo 8 are included they yield a pairing bonus of 272.
If both photo 6 and photo 9 are included they yield a pairing bonus of 392.
If both photo 7 and photo 8 are included they yield a pairing bonus of 562.
If both photo 7 and photo 9 are included they yield a pairing bonus of 205.
If both photo 8 and photo 9 are included they yield a pairing bonus of 223.
Selections must stay within the 2697-page limit and include no duplicates so the portfolio’s total effect is maximized.

Oh, and when you send back your pick, a little JSON like the one below is perfect — nice and tidy so I can read which photos you chose.

{
  ""solution"": [<selected_photo_id>, <selected_photo_id>, ...]
}

Nothing fancy: the ""solution"" array is where you list the photo IDs you want in the final spread. This is just a sketch of the shape I expect, not the actual answer — fill it with the exact IDs from the instance when you're ready.

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”.""","{'linear_coeffs': [970, 475, 798, 560, 231, 737, 658, 800, 617, 636], 'quadratic_coeffs': [[870, 454, 483, 641, 435, 733, 599, 596, 686, 694], [0, 499, 513, 578, 467, 386, 435, 583, 458, 566], [0, 0, 576, 314, 544, 600, 401, 646, 573, 440], [0, 0, 0, 454, 634, 518, 577, 525, 566, 837], [0, 0, 0, 0, 238, 274, 128, 589, 850, 756], [0, 0, 0, 0, 0, 690, 595, 397, 557, 399], [0, 0, 0, 0, 0, 0, 491, 371, 272, 392], [0, 0, 0, 0, 0, 0, 0, 305, 562, 205], [0, 0, 0, 0, 0, 0, 0, 0, 493, 223], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [880, 385, 708, 470, 141, 647, 568, 710, 527, 546], 'capacity': 2697, 'solution': [1, 3, 4, 6, 8, 9], 'obj': 10916.0, 'problem_type': 'QKP'}","[1, 3, 4, 6, 8, 9]",10916.0,"{'problem_type': 'QKP', 'num_items': 10, 'capacity': 2697, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 'linear_pairs': [{'item_id': 0, 'linear_profit': 970}, {'item_id': 1, 'linear_profit': 475}, {'item_id': 2, 'linear_profit': 798}, {'item_id': 3, 'linear_profit': 560}, {'item_id': 4, 'linear_profit': 231}, {'item_id': 5, 'linear_profit': 737}, {'item_id': 6, 'linear_profit': 658}, {'item_id': 7, 'linear_profit': 800}, {'item_id': 8, 'linear_profit': 617}, {'item_id': 9, 'linear_profit': 636}], 'weight_pairs': [{'item_id': 0, 'weight': 880}, {'item_id': 1, 'weight': 385}, {'item_id': 2, 'weight': 708}, {'item_id': 3, 'weight': 470}, {'item_id': 4, 'weight': 141}, {'item_id': 5, 'weight': 647}, {'item_id': 6, 'weight': 568}, {'item_id': 7, 'weight': 710}, {'item_id': 8, 'weight': 527}, {'item_id': 9, 'weight': 546}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 870}, {'item_i_id': 0, 'item_j_id': 1, 'quadratic_profit': 454}, {'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 483}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 641}, {'item_i_id': 0, 'item_j_id': 4, 'quadratic_profit': 435}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 733}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 599}, {'item_i_id': 0, 'item_j_id': 7, 'quadratic_profit': 596}, {'item_i_id': 0, 'item_j_id': 8, 'quadratic_profit': 686}, {'item_i_id': 0, 'item_j_id': 9, 'quadratic_profit': 694}, {'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 499}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 513}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 578}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 467}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 386}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 435}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 583}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 458}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 566}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 576}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 314}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 544}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 600}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 401}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 646}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 573}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 440}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 454}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 634}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 518}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 577}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 525}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 566}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 837}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 238}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 274}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 128}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 589}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 850}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 756}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 690}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 595}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 397}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 557}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 399}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 491}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 371}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 272}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 392}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 305}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 562}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 205}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 493}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 223}]}","[1, 3, 4, 6, 8, 9]",12,nl,0
QKP,QKP,"Picture this: a camping menu to assemble where each food item improves meal quality a bit on its own, but certain combinations — like fresh bread with good cheese — give an extra morale kick if both make it into the pack. Every item has a weight, you either pack it or you don’t (no halves, no doubles), and the whole load needs to fit under the gear weight limit. The trip’s score is the sum of every chosen item’s base benefit plus any extra bonuses from pairs that travel together, so the job is to pick the set that yields the highest total while staying within the weight allowance. The specific items, their weights, base comforts, and pair bonuses are listed below.

You'll find 10 food options (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) and a gear weight limit of 2069 to respect.
Bring 1 — it adds 154 comfort on its own.
Bring 2 — it adds 113 comfort on its own.
Bring 3 — it adds 213 comfort on its own.
Bring 4 — it adds 207 comfort on its own.
Bring 5 — it adds 180 comfort on its own.
Bring 6 — it adds 205 comfort on its own.
Bring 7 — it adds 227 comfort on its own.
Bring 8 — it adds 146 comfort on its own.
Bring 9 — it adds 100 comfort on its own.
Bring 10 — it adds 227 comfort on its own.
1 consumes 220 of your pack's weight.
2 consumes 152 of your pack's weight.
3 consumes 294 of your pack's weight.
4 consumes 291 of your pack's weight.
5 consumes 155 of your pack's weight.
6 consumes 160 of your pack's weight.
7 consumes 292 of your pack's weight.
8 consumes 181 of your pack's weight.
9 consumes 153 of your pack's weight.
10 consumes 237 of your pack's weight.
Packing 1 with 2 grants an extra 116 morale bonus.
Packing 1 with 3 grants an extra 104 morale bonus.
Packing 1 with 4 grants an extra 227 morale bonus.
Packing 1 with 5 grants an extra 105 morale bonus.
Packing 1 with 6 grants an extra 101 morale bonus.
Packing 1 with 7 grants an extra 187 morale bonus.
Packing 1 with 8 grants an extra 167 morale bonus.
Packing 1 with 9 grants an extra 123 morale bonus.
Packing 1 with 10 grants an extra 203 morale bonus.
Packing 2 with 3 grants an extra 210 morale bonus.
Packing 2 with 4 grants an extra 205 morale bonus.
Packing 2 with 5 grants an extra 265 morale bonus.
Packing 2 with 6 grants an extra 224 morale bonus.
Packing 2 with 7 grants an extra 263 morale bonus.
Packing 2 with 8 grants an extra 251 morale bonus.
Packing 2 with 9 grants an extra 164 morale bonus.
Packing 2 with 10 grants an extra 273 morale bonus.
Packing 3 with 4 grants an extra 285 morale bonus.
Packing 3 with 5 grants an extra 242 morale bonus.
Packing 3 with 6 grants an extra 270 morale bonus.
Packing 3 with 7 grants an extra 224 morale bonus.
Packing 3 with 8 grants an extra 234 morale bonus.
Packing 3 with 9 grants an extra 230 morale bonus.
Packing 3 with 10 grants an extra 250 morale bonus.
Packing 4 with 5 grants an extra 153 morale bonus.
Packing 4 with 6 grants an extra 116 morale bonus.
Packing 4 with 7 grants an extra 292 morale bonus.
Packing 4 with 8 grants an extra 228 morale bonus.
Packing 4 with 9 grants an extra 238 morale bonus.
Packing 4 with 10 grants an extra 294 morale bonus.
Packing 5 with 6 grants an extra 122 morale bonus.
Packing 5 with 7 grants an extra 221 morale bonus.
Packing 5 with 8 grants an extra 110 morale bonus.
Packing 5 with 9 grants an extra 280 morale bonus.
Packing 5 with 10 grants an extra 210 morale bonus.
Packing 6 with 7 grants an extra 226 morale bonus.
Packing 6 with 8 grants an extra 155 morale bonus.
Packing 6 with 9 grants an extra 197 morale bonus.
Packing 6 with 10 grants an extra 294 morale bonus.
Packing 7 with 8 grants an extra 189 morale bonus.
Packing 7 with 9 grants an extra 213 morale bonus.
Packing 7 with 10 grants an extra 204 morale bonus.
Choose the set that fits the 2069 limit and maximizes overall meal comfort.

Oh, and when you send back the chosen items, please use this simple JSON layout so it's easy to read and check:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

This means: ""solution"" should be an array containing the identifiers of the items you want to pack. Think of it like a short list on a form — each entry is the ID of one selected item. The block above is just the expected shape (a sketch), not the actual answer.

Please use the item 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”.""","{'linear_coeffs': [154, 113, 213, 207, 180, 205, 227, 146, 100, 227], 'quadratic_coeffs': [[273, 116, 104, 227, 105, 101, 187, 167, 123, 203], [0, 267, 210, 205, 265, 224, 263, 251, 164, 273], [0, 0, 171, 285, 242, 270, 224, 234, 230, 250], [0, 0, 0, 147, 153, 116, 292, 228, 238, 294], [0, 0, 0, 0, 229, 122, 221, 110, 280, 210], [0, 0, 0, 0, 0, 105, 226, 155, 197, 294], [0, 0, 0, 0, 0, 0, 152, 189, 213, 204], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [220, 152, 294, 291, 155, 160, 292, 181, 153, 237], 'capacity': 2069, 'solution': [0, 1, 2, 3, 4, 5, 6, 8, 9], 'obj': 8957.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 5, 6, 8, 9]",8957.0,"{'problem_type': 'QKP', 'num_items': 10, 'capacity': 2069, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 'linear_pairs': [{'item_id': 1, 'linear_profit': 154}, {'item_id': 2, 'linear_profit': 113}, {'item_id': 3, 'linear_profit': 213}, {'item_id': 4, 'linear_profit': 207}, {'item_id': 5, 'linear_profit': 180}, {'item_id': 6, 'linear_profit': 205}, {'item_id': 7, 'linear_profit': 227}, {'item_id': 8, 'linear_profit': 146}, {'item_id': 9, 'linear_profit': 100}, {'item_id': 10, 'linear_profit': 227}], 'weight_pairs': [{'item_id': 1, 'weight': 220}, {'item_id': 2, 'weight': 152}, {'item_id': 3, 'weight': 294}, {'item_id': 4, 'weight': 291}, {'item_id': 5, 'weight': 155}, {'item_id': 6, 'weight': 160}, {'item_id': 7, 'weight': 292}, {'item_id': 8, 'weight': 181}, {'item_id': 9, 'weight': 153}, {'item_id': 10, 'weight': 237}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 273}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 116}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 104}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 227}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 105}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 101}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 187}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 167}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 123}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 203}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 267}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 210}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 205}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 265}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 224}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 263}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 251}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 164}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 273}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 171}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 285}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 242}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 270}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 224}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 234}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 230}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 250}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 147}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 153}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 116}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 292}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 228}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 238}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 294}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 229}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 122}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 221}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 110}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 280}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 210}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 105}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 226}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 155}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 197}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 294}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 152}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 189}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 213}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 204}]}","[1, 2, 3, 4, 5, 6, 7, 9, 10]",13,nl,1
QKP,QKP,"We’re trying to stock a small shelf for the office and want choices that make people happiest. Every supply has a basic usefulness score, and certain combinations create extra perks when both are on the shelf — think tape and a dispenser that together are handier than apart. The plan is to pick individual, non-duplicated items that all fit in the shelf space, then total up their individual usefulness plus any extra pair bonuses to figure out which combination gives the biggest payoff. The concrete item sizes, usefulness numbers, and pair bonuses are listed below.

# total_number_of_supplies=7
# shelf_capacity_volume=399
# supply_identifiers=0, 1, 2, 3, 4, 5, 6

supply_id,base_usefulness
0,100
1,105
2,101
3,101
4,105
5,103
6,105

supply_id,shelf_volume
0,100
1,102
2,102
3,103
4,105
5,103
6,102

supply_a_id,supply_b_id,pair_bonus_usefulness
0,1,100
0,2,100
0,5,100
0,6,103
1,6,102
2,5,102
3,4,101
3,5,103
3,6,103
4,5,102
4,6,100
5,6,100

Also, when you send back your pick, a tiny JSON snippet like the one below works great — just a single field listing the items you chose.

{
  ""solution"": [<shelf_item_id>, <shelf_item_id>, ...]
}

Pretty simple: ""solution"" is the list of items you'll put on the shelf, and each <shelf_item_id> placeholder is where you drop an exact item identifier from the instance. Think of it like filling out a short form — the JSON shows the shape I expect, but you should replace those placeholders with the real IDs.

Please use the identifiers exactly as they appear in 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”.""","{'linear_coeffs': [100, 105, 101, 101, 105, 103, 105], 'quadratic_coeffs': [[0, 100, 100, 0, 0, 100, 103], [0, 100, 0, 0, 0, 0, 102], [0, 0, 0, 0, 0, 102, 0], [0, 0, 0, 0, 101, 103, 103], [0, 0, 0, 0, 102, 102, 100], [0, 0, 0, 0, 0, 0, 100], [0, 0, 0, 0, 0, 0, 102]], 'weights': [100, 102, 102, 103, 105, 103, 102], 'capacity': 399, 'solution': [4, 5, 6], 'obj': 615.0, 'problem_type': 'QKP'}","[4, 5, 6]",615.0,"{'problem_type': 'QKP', 'num_items': 7, 'capacity': 399, 'items': [0, 1, 2, 3, 4, 5, 6], 'linear_pairs': [{'item_id': 0, 'linear_profit': 100}, {'item_id': 1, 'linear_profit': 105}, {'item_id': 2, 'linear_profit': 101}, {'item_id': 3, 'linear_profit': 101}, {'item_id': 4, 'linear_profit': 105}, {'item_id': 5, 'linear_profit': 103}, {'item_id': 6, 'linear_profit': 105}], 'weight_pairs': [{'item_id': 0, 'weight': 100}, {'item_id': 1, 'weight': 102}, {'item_id': 2, 'weight': 102}, {'item_id': 3, 'weight': 103}, {'item_id': 4, 'weight': 105}, {'item_id': 5, 'weight': 103}, {'item_id': 6, 'weight': 102}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 1, 'quadratic_profit': 100}, {'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 100}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 100}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 103}, {'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 102}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 102}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 103}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 103}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 102}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 102}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 100}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 100}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 102}]}","[4, 5, 6]",14,csv,0
QKP,QKP,"Someone in the after-school program needs to pack activity kits before Monday: every item available contributes a baseline amount of learning and has a certain weight, and a few pairs of items amplify learning when they travel together. The goal is to pick a set of distinct items that together give the highest total learning (add up each chosen item’s value and any extra value from the special pairs) without making the kit heavier than students can handle. Look below for the concrete item weights, individual values, and pair bonuses.

{
  ""num_available_items"": 7,
  ""max_kit_weight"": 8457,
  ""available_item_ids"": [
    0,
    1,
    2,
    3,
    4,
    5,
    6
  ],
  ""linear"": [
    {
      ""item_identifier"": 0,
      ""base_learning_value"": 3519
    },
    {
      ""item_identifier"": 1,
      ""base_learning_value"": 618
    },
    {
      ""item_identifier"": 2,
      ""base_learning_value"": 1481
    },
    {
      ""item_identifier"": 3,
      ""base_learning_value"": 3100
    },
    {
      ""item_identifier"": 4,
      ""base_learning_value"": 3063
    },
    {
      ""item_identifier"": 5,
      ""base_learning_value"": 1521
    },
    {
      ""item_identifier"": 6,
      ""base_learning_value"": 1585
    }
  ],
  ""weights"": [
    {
      ""item_identifier"": 0,
      ""item_weight"": 3189
    },
    {
      ""item_identifier"": 1,
      ""item_weight"": 288
    },
    {
      ""item_identifier"": 2,
      ""item_weight"": 1151
    },
    {
      ""item_identifier"": 3,
      ""item_weight"": 2770
    },
    {
      ""item_identifier"": 4,
      ""item_weight"": 2733
    },
    {
      ""item_identifier"": 5,
      ""item_weight"": 1191
    },
    {
      ""item_identifier"": 6,
      ""item_weight"": 1255
    }
  ],
  ""quadratic"": [
    {
      ""item_i_identifier"": 0,
      ""item_j_identifier"": 3,
      ""interaction_learning_bonus"": 1458
    },
    {
      ""item_i_identifier"": 0,
      ""item_j_identifier"": 6,
      ""interaction_learning_bonus"": 931
    },
    {
      ""item_i_identifier"": 1,
      ""item_j_identifier"": 5,
      ""interaction_learning_bonus"": 1533
    }
  ]
}

Also, just so we're on the same page, please give your final pick using this simple JSON layout so it's easy to read and check:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

The ""solution"" array should list the item identifiers you want packed in the kit (one identifier per chosen item). Think of it like ticking boxes on a form — put the exact ids of the items you picked inside the brackets. This JSON is just the shape I expect, not your actual answer.

Please use the item ids 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”.","{'linear_coeffs': [3519, 618, 1481, 3100, 3063, 1521, 1585], 'quadratic_coeffs': [[2067, 0, 0, 1458, 0, 0, 931], [0, 0, 0, 0, 0, 1533, 0], [0, 0, 3071, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0]], 'weights': [3189, 288, 1151, 2770, 2733, 1191, 1255], 'capacity': 8457, 'solution': [0, 3, 5, 6], 'obj': 12114.0, 'problem_type': 'QKP'}","[0, 3, 5, 6]",12114.0,"{'problem_type': 'QKP', 'num_items': 7, 'capacity': 8457, 'items': [0, 1, 2, 3, 4, 5, 6], 'linear_pairs': [{'item_id': 0, 'linear_profit': 3519}, {'item_id': 1, 'linear_profit': 618}, {'item_id': 2, 'linear_profit': 1481}, {'item_id': 3, 'linear_profit': 3100}, {'item_id': 4, 'linear_profit': 3063}, {'item_id': 5, 'linear_profit': 1521}, {'item_id': 6, 'linear_profit': 1585}], 'weight_pairs': [{'item_id': 0, 'weight': 3189}, {'item_id': 1, 'weight': 288}, {'item_id': 2, 'weight': 1151}, {'item_id': 3, 'weight': 2770}, {'item_id': 4, 'weight': 2733}, {'item_id': 5, 'weight': 1191}, {'item_id': 6, 'weight': 1255}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 2067}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 1458}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 931}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 1533}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 3071}]}","[0, 3, 5, 6]",15,json,0
QKP,QKP,"On a blank gallery wall, a curator is deciding which prints to hang to create the strongest overall impression within the space available. Each print eats into the wall allowance and offers its own visual value, and select pairs of prints give an added boost when displayed together. The decision is to pick a set of prints that fits inside the allotted area (each print only once) and to tally up the total effect by adding each chosen print’s value plus any bonuses from pairs, aiming for the highest possible total impact. The specific sizes, individual values, and pair bonuses are shown below.

There are 10 prints available — 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 — and the curator has 774 total wall area to allocate.

| print_id | base_visual_value |
|---|---|
| 0 | 140 |
| 1 | 221 |
| 2 | 237 |
| 3 | 172 |
| 4 | 281 |
| 5 | 183 |
| 6 | 278 |
| 7 | 277 |
| 8 | 223 |
| 9 | 265 |

| print_id | display_area |
|---|---|
| 0 | 126 |
| 1 | 232 |
| 2 | 226 |
| 3 | 172 |
| 4 | 292 |
| 5 | 196 |
| 6 | 274 |
| 7 | 272 |
| 8 | 221 |
| 9 | 275 |

| print_i_id | print_j_id | pair_synergy_value |
|---|---|---|
| 0 | 2 | 126 |
| 0 | 3 | 126 |
| 0 | 5 | 126 |
| 0 | 7 | 126 |
| 2 | 4 | 148 |
| 2 | 5 | 226 |
| 2 | 8 | 105 |
| 2 | 9 | 154 |
| 3 | 5 | 172 |
| 5 | 6 | 109 |

The curator will choose a combination of prints that fits the allowance and maximizes the overall visual impact.

Also, when you tell me which prints you want to hang, just send it back in this simple JSON shape so I can read it easily:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

""solution"" is just the list of prints you picked (use each print's identifier exactly as it appears in the instance). Think of it like filling out a short form: the array holds the ids of the prints you want on the wall. This JSON is only a sketch of the expected shape — not the actual answer.

Please make sure to use the identifiers exactly as given 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”.""","{'linear_coeffs': [140, 221, 237, 172, 281, 183, 278, 277, 223, 265], 'quadratic_coeffs': [[0, 0, 126, 126, 0, 126, 0, 126, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 148, 226, 0, 0, 105, 154], [0, 0, 0, 0, 0, 172, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 109, 0, 0, 0], [0, 0, 0, 0, 0, 0, 222, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [126, 232, 226, 172, 292, 196, 274, 272, 221, 275], 'capacity': 774, 'solution': [0, 2, 3, 5], 'obj': 1508.0, 'problem_type': 'QKP'}","[0, 2, 3, 5]",1508.0,"{'problem_type': 'QKP', 'num_items': 10, 'capacity': 774, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 'linear_pairs': [{'item_id': 0, 'linear_profit': 140}, {'item_id': 1, 'linear_profit': 221}, {'item_id': 2, 'linear_profit': 237}, {'item_id': 3, 'linear_profit': 172}, {'item_id': 4, 'linear_profit': 281}, {'item_id': 5, 'linear_profit': 183}, {'item_id': 6, 'linear_profit': 278}, {'item_id': 7, 'linear_profit': 277}, {'item_id': 8, 'linear_profit': 223}, {'item_id': 9, 'linear_profit': 265}], 'weight_pairs': [{'item_id': 0, 'weight': 126}, {'item_id': 1, 'weight': 232}, {'item_id': 2, 'weight': 226}, {'item_id': 3, 'weight': 172}, {'item_id': 4, 'weight': 292}, {'item_id': 5, 'weight': 196}, {'item_id': 6, 'weight': 274}, {'item_id': 7, 'weight': 272}, {'item_id': 8, 'weight': 221}, {'item_id': 9, 'weight': 275}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 126}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 126}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 126}, {'item_i_id': 0, 'item_j_id': 7, 'quadratic_profit': 126}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 148}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 226}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 105}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 154}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 172}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 109}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 222}]}","[0, 2, 3, 5]",16,markdown_table,0
QKP,QKP,"Someone on the product team thought it would be fun to treat kit design like a game: which amenities go in to get the biggest cheer from travelers? Every item adds a certain amount of satisfaction and costs a bit of kit weight, and sometimes two items together create a little extra perk that adds more satisfaction on top. The final happiness score is just the sum of the individual item scores plus any pair bonuses, while the practical limits are that the kit’s total weight can’t exceed the packing limit and each amenity is used at most once. The specific item list and pair perks are shown below.

Below are the 10 amenity options (A, B, C, D, E, F, G, H, I, J) and the kit weight capacity is 17535.
Amenity A contributes 2317 satisfaction to the kit when included.
Amenity B contributes 5016 satisfaction to the kit when included.
Amenity C contributes 5731 satisfaction to the kit when included.
Amenity D contributes 3300 satisfaction to the kit when included.
Amenity E contributes 3147 satisfaction to the kit when included.
Amenity F contributes 5522 satisfaction to the kit when included.
Amenity G contributes 4357 satisfaction to the kit when included.
Amenity H contributes 3860 satisfaction to the kit when included.
Amenity I contributes 1016 satisfaction to the kit when included.
Amenity J contributes 1756 satisfaction to the kit when included.
Amenity A has weight 2400 toward the kit's packing limit.
Amenity B has weight 5011 toward the kit's packing limit.
Amenity C has weight 5640 toward the kit's packing limit.
Amenity D has weight 3433 toward the kit's packing limit.
Amenity E has weight 2854 toward the kit's packing limit.
Amenity F has weight 5510 toward the kit's packing limit.
Amenity G has weight 4047 toward the kit's packing limit.
Amenity H has weight 3740 toward the kit's packing limit.
Amenity I has weight 1105 toward the kit's packing limit.
Amenity J has weight 1865 toward the kit's packing limit.
Amenities A and B together grant an extra 2039 satisfaction if both are packed.
Amenities A and C together grant an extra 3615 satisfaction if both are packed.
Amenities A and D together grant an extra 1310 satisfaction if both are packed.
Amenities A and E together grant an extra 3631 satisfaction if both are packed.
Amenities A and F together grant an extra 1785 satisfaction if both are packed.
Amenities A and G together grant an extra 2977 satisfaction if both are packed.
Amenities A and H together grant an extra 6094 satisfaction if both are packed.
Amenities A and I together grant an extra 2935 satisfaction if both are packed.
Amenities A and J together grant an extra 2505 satisfaction if both are packed.
Amenities B and C together grant an extra 2871 satisfaction if both are packed.
Amenities B and D together grant an extra 4087 satisfaction if both are packed.
Amenities B and E together grant an extra 1989 satisfaction if both are packed.
Amenities B and F together grant an extra 4903 satisfaction if both are packed.
Amenities B and G together grant an extra 2273 satisfaction if both are packed.
Amenities B and H together grant an extra 3466 satisfaction if both are packed.
Amenities B and I together grant an extra 3628 satisfaction if both are packed.
Amenities B and J together grant an extra 4623 satisfaction if both are packed.
Amenities C and D together grant an extra 1694 satisfaction if both are packed.
Amenities C and E together grant an extra 2565 satisfaction if both are packed.
Amenities C and F together grant an extra 3508 satisfaction if both are packed.
Amenities C and G together grant an extra 2980 satisfaction if both are packed.
Amenities C and H together grant an extra 1746 satisfaction if both are packed.
Amenities C and I together grant an extra 3350 satisfaction if both are packed.
Amenities C and J together grant an extra 1705 satisfaction if both are packed.
Amenities D and E together grant an extra 1183 satisfaction if both are packed.
Amenities D and F together grant an extra 915 satisfaction if both are packed.
Amenities D and G together grant an extra 2091 satisfaction if both are packed.
Amenities D and H together grant an extra 2161 satisfaction if both are packed.
Amenities D and I together grant an extra 2833 satisfaction if both are packed.
Amenities D and J together grant an extra 2580 satisfaction if both are packed.
Amenities E and F together grant an extra 5371 satisfaction if both are packed.
Amenities E and G together grant an extra 5533 satisfaction if both are packed.
Amenities E and H together grant an extra 3733 satisfaction if both are packed.
Amenities E and I together grant an extra 5574 satisfaction if both are packed.
Amenities E and J together grant an extra 2161 satisfaction if both are packed.
Amenities F and G together grant an extra 4016 satisfaction if both are packed.
Amenities F and H together grant an extra 2291 satisfaction if both are packed.
Amenities F and I together grant an extra 1110 satisfaction if both are packed.
Amenities F and J together grant an extra 2445 satisfaction if both are packed.
Amenities G and H together grant an extra 5098 satisfaction if both are packed.
Amenities G and I together grant an extra 5096 satisfaction if both are packed.
Amenities G and J together grant an extra 4749 satisfaction if both are packed.
Amenities H and I together grant an extra 3995 satisfaction if both are packed.
Amenities H and J together grant an extra 2753 satisfaction if both are packed.
Choose amenities to maximize total satisfaction without exceeding the 17535 kit weight.

Also, when you’re ready to tell me which items you’d pick, just send it back in a tiny JSON snippet like this:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

The ""solution"" array is just the list of amenity IDs you want in the kit — think of it like ticking boxes on a form and listing the exact IDs you picked. This JSON is just the shape I expect, 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”.""","{'linear_coeffs': [2317, 5016, 5731, 3300, 3147, 5522, 4357, 3860, 1016, 1756], 'quadratic_coeffs': [[3591, 2039, 3615, 1310, 3631, 1785, 2977, 6094, 2935, 2505], [0, 4383, 2871, 4087, 1989, 4903, 2273, 3466, 3628, 4623], [0, 0, 5724, 1694, 2565, 3508, 2980, 1746, 3350, 1705], [0, 0, 0, 3667, 1183, 915, 2091, 2161, 2833, 2580], [0, 0, 0, 0, 4690, 5371, 5533, 3733, 5574, 2161], [0, 0, 0, 0, 0, 5098, 4016, 2291, 1110, 2445], [0, 0, 0, 0, 0, 0, 4864, 5098, 5096, 4749], [0, 0, 0, 0, 0, 0, 0, 1233, 3995, 2753], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [2400, 5011, 5640, 3433, 2854, 5510, 4047, 3740, 1105, 1865], 'capacity': 17535, 'solution': [0, 4, 6, 7, 8, 9], 'obj': 73287.0, 'problem_type': 'QKP'}","[0, 4, 6, 7, 8, 9]",73287.0,"{'problem_type': 'QKP', 'num_items': 10, 'capacity': 17535, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 2317}, {'item_id': 'B', 'linear_profit': 5016}, {'item_id': 'C', 'linear_profit': 5731}, {'item_id': 'D', 'linear_profit': 3300}, {'item_id': 'E', 'linear_profit': 3147}, {'item_id': 'F', 'linear_profit': 5522}, {'item_id': 'G', 'linear_profit': 4357}, {'item_id': 'H', 'linear_profit': 3860}, {'item_id': 'I', 'linear_profit': 1016}, {'item_id': 'J', 'linear_profit': 1756}], 'weight_pairs': [{'item_id': 'A', 'weight': 2400}, {'item_id': 'B', 'weight': 5011}, {'item_id': 'C', 'weight': 5640}, {'item_id': 'D', 'weight': 3433}, {'item_id': 'E', 'weight': 2854}, {'item_id': 'F', 'weight': 5510}, {'item_id': 'G', 'weight': 4047}, {'item_id': 'H', 'weight': 3740}, {'item_id': 'I', 'weight': 1105}, {'item_id': 'J', 'weight': 1865}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 3591}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 2039}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 3615}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 1310}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 3631}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 1785}, {'item_i_id': 'A', 'item_j_id': 'G', 'quadratic_profit': 2977}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 6094}, {'item_i_id': 'A', 'item_j_id': 'I', 'quadratic_profit': 2935}, {'item_i_id': 'A', 'item_j_id': 'J', 'quadratic_profit': 2505}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 4383}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 2871}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 4087}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 1989}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 4903}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 2273}, {'item_i_id': 'B', 'item_j_id': 'H', 'quadratic_profit': 3466}, {'item_i_id': 'B', 'item_j_id': 'I', 'quadratic_profit': 3628}, {'item_i_id': 'B', 'item_j_id': 'J', 'quadratic_profit': 4623}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 5724}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 1694}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 2565}, {'item_i_id': 'C', 'item_j_id': 'F', 'quadratic_profit': 3508}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 2980}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 1746}, {'item_i_id': 'C', 'item_j_id': 'I', 'quadratic_profit': 3350}, {'item_i_id': 'C', 'item_j_id': 'J', 'quadratic_profit': 1705}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 3667}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 1183}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 915}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 2091}, {'item_i_id': 'D', 'item_j_id': 'H', 'quadratic_profit': 2161}, {'item_i_id': 'D', 'item_j_id': 'I', 'quadratic_profit': 2833}, {'item_i_id': 'D', 'item_j_id': 'J', 'quadratic_profit': 2580}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 4690}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 5371}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 5533}, {'item_i_id': 'E', 'item_j_id': 'H', 'quadratic_profit': 3733}, {'item_i_id': 'E', 'item_j_id': 'I', 'quadratic_profit': 5574}, {'item_i_id': 'E', 'item_j_id': 'J', 'quadratic_profit': 2161}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 5098}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 4016}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 2291}, {'item_i_id': 'F', 'item_j_id': 'I', 'quadratic_profit': 1110}, {'item_i_id': 'F', 'item_j_id': 'J', 'quadratic_profit': 2445}, {'item_i_id': 'G', 'item_j_id': 'G', 'quadratic_profit': 4864}, {'item_i_id': 'G', 'item_j_id': 'H', 'quadratic_profit': 5098}, {'item_i_id': 'G', 'item_j_id': 'I', 'quadratic_profit': 5096}, {'item_i_id': 'G', 'item_j_id': 'J', 'quadratic_profit': 4749}, {'item_i_id': 'H', 'item_j_id': 'H', 'quadratic_profit': 1233}, {'item_i_id': 'H', 'item_j_id': 'I', 'quadratic_profit': 3995}, {'item_i_id': 'H', 'item_j_id': 'J', 'quadratic_profit': 2753}]}","['A', 'E', 'G', 'H', 'I', 'J']",17,nl,names
QKP,QKP,"We have a limited block of bench time and a menu of possible experiments, each with its own duration and expected contribution to the project. The task is to decide which experiments to actually run so the whole schedule fits the session and the total expected scientific payoff — computed by adding every chosen experiment’s basic value and tacking on any extra value from experiments that synergize with each other — ends up as large as possible. Each experiment can be run at most once and total time can’t exceed the session. The concrete details follow below.

Specifically, there are 6 experiments labeled 1, 2, 3, 4, 5, 6 and 701 total bench time for the session.
Experiment 1 has a standalone scientific value of 150.
Experiment 2 has a standalone scientific value of 157.
Experiment 3 has a standalone scientific value of 150.
Experiment 4 has a standalone scientific value of 141.
Experiment 5 has a standalone scientific value of 140.
Experiment 6 has a standalone scientific value of 125.
Experiment 1 requires 135 units of bench time.
Experiment 2 requires 142 units of bench time.
Experiment 3 requires 135 units of bench time.
Experiment 4 requires 126 units of bench time.
Experiment 5 requires 125 units of bench time.
Experiment 6 requires 110 units of bench time.
If we run experiments 1 and 2 together, we gain an extra 126 value.
If we run experiments 1 and 3 together, we gain an extra 132 value.
If we run experiments 1 and 4 together, we gain an extra 130 value.
If we run experiments 1 and 5 together, we gain an extra 121 value.
If we run experiments 1 and 6 together, we gain an extra 124 value.
If we run experiments 2 and 3 together, we gain an extra 137 value.
If we run experiments 2 and 4 together, we gain an extra 127 value.
If we run experiments 2 and 5 together, we gain an extra 124 value.
If we run experiments 2 and 6 together, we gain an extra 118 value.
If we run experiments 3 and 4 together, we gain an extra 119 value.
If we run experiments 3 and 5 together, we gain an extra 119 value.
If we run experiments 3 and 6 together, we gain an extra 134 value.
If we run experiments 4 and 5 together, we gain an extra 116 value.
If we run experiments 4 and 6 together, we gain an extra 108 value.
We must respect the 701 session time while choosing experiments to maximize total payoff.

Also, when you send your selection back, please use this simple JSON layout so it's easy to parse:

{
  ""solution"": [<experiment_id>, <experiment_id>, ...]
}

""solution"" is just the list of experiment IDs you plan to run during the session — put each experiment's identifier inside the brackets, separated by commas. This is only a sketch of the shape I expect, not the actual answer.

Please make 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”.""","{'linear_coeffs': [150, 157, 150, 141, 140, 125], 'quadratic_coeffs': [[129, 126, 132, 130, 121, 124], [0, 136, 137, 127, 124, 118], [0, 0, 112, 119, 119, 134], [0, 0, 0, 102, 116, 108], [0, 0, 0, 0, 128, 0], [0, 0, 0, 0, 0, 0]], 'weights': [135, 142, 135, 126, 125, 110], 'capacity': 701, 'solution': [0, 1, 2, 3, 4], 'obj': 1989.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4]",1989.0,"{'problem_type': 'QKP', 'num_items': 6, 'capacity': 701, 'items': [1, 2, 3, 4, 5, 6], 'linear_pairs': [{'item_id': 1, 'linear_profit': 150}, {'item_id': 2, 'linear_profit': 157}, {'item_id': 3, 'linear_profit': 150}, {'item_id': 4, 'linear_profit': 141}, {'item_id': 5, 'linear_profit': 140}, {'item_id': 6, 'linear_profit': 125}], 'weight_pairs': [{'item_id': 1, 'weight': 135}, {'item_id': 2, 'weight': 142}, {'item_id': 3, 'weight': 135}, {'item_id': 4, 'weight': 126}, {'item_id': 5, 'weight': 125}, {'item_id': 6, 'weight': 110}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 129}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 126}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 132}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 130}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 121}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 124}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 136}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 137}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 127}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 124}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 118}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 112}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 119}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 119}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 134}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 102}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 116}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 108}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 128}]}","[1, 2, 3, 4, 5]",18,nl,1
QKP,QKP,"In a small team meeting the problem was simple: there’s one allotted airtime window and many creative options, each taking some seconds and delivering a baseline impact. On top of that, some creatives amplify each other when paired, adding extra impact only if both are scheduled. The task is to pick which creatives to run so their total time stays within the bought slot, no creative is repeated, and the campaign’s total impact — the sum of all chosen creatives’ base impacts plus any applicable pair boosts — comes out as large as possible. The specific asset data and pair boosts are provided below.

Below are the 5 creatives A, B, C, D, E and the single airtime window of 10514 seconds:

| asset_id | baseline_impact |
|---|---|
| A | 2811 |
| B | 6039 |
| C | 2025 |
| D | 4623 |
| E | 4350 |

| asset_id | airtime_seconds |
|---|---|
| A | 2161 |
| B | 5389 |
| C | 1375 |
| D | 3973 |
| E | 3700 |

| asset_i_id | asset_j_id | pair_bonus_impact |
|---|---|---|
| A | B | 2161 |
| A | C | 2161 |
| A | D | 1445 |
| A | E | 4223 |
| B | C | 2236 |
| B | D | 3291 |
| B | E | 3053 |
| C | D | 165 |
| C | E | 165 |
| D | E | 1928 |

Select creatives so total airtime stays within the 10514-second slot while maximizing total impact across the 5 assets.

Oh, and one more practical thing — when you send back the picks, please use a tiny JSON snippet so it's easy to parse. Something like this:

{
  ""solution"": [<selected_creative_id>, <selected_creative_id>, ...]
}

This just means ""solution"" should be an array listing the IDs of the creatives you chose. Think of it as filling in a short form: put each chosen creative's identifier inside the brackets, separated by commas. The JSON above is just a sketch of the shape I need, not your final answer.

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”.","{'linear_coeffs': [2811, 6039, 2025, 4623, 4350], 'quadratic_coeffs': [[2161, 2161, 2161, 1445, 4223], [0, 3053, 2236, 3291, 3053], [0, 0, 1375, 165, 165], [0, 0, 0, 262, 1928], [0, 0, 0, 0, 0]], 'weights': [2161, 5389, 1375, 3973, 3700], 'capacity': 10514, 'solution': [0, 3, 4], 'obj': 19380.0, 'problem_type': 'QKP'}","[0, 3, 4]",19380.0,"{'problem_type': 'QKP', 'num_items': 5, 'capacity': 10514, 'items': ['A', 'B', 'C', 'D', 'E'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 2811}, {'item_id': 'B', 'linear_profit': 6039}, {'item_id': 'C', 'linear_profit': 2025}, {'item_id': 'D', 'linear_profit': 4623}, {'item_id': 'E', 'linear_profit': 4350}], 'weight_pairs': [{'item_id': 'A', 'weight': 2161}, {'item_id': 'B', 'weight': 5389}, {'item_id': 'C', 'weight': 1375}, {'item_id': 'D', 'weight': 3973}, {'item_id': 'E', 'weight': 3700}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 2161}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 2161}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 2161}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 1445}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 4223}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 3053}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 2236}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 3291}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 3053}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 1375}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 165}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 165}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 262}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 1928}]}","['A', 'D', 'E']",19,markdown_table,names
QKP,QKP,"I run the catering deliveries for a small company, juggling which platters to load into the van for each event. Each platter has a weight and a base customer appeal, and certain pairs just click together and add extra appeal when both are included. The trick is to pick a set of platters whose combined weight fits in the van and that gives the highest total appeal — total appeal being the sum of each platter’s base appeal plus any extra bonuses from included platter pairs — and nobody gets duplicates of the same platter. The exact platter weights, base appeals, van capacity and pair bonuses are listed below.

Here are the 7 platters (0, 1, 2, 3, 4, 5, 6) and the van capacity 1469 that I must work with.

| platter_id | base_appeal |
|---|---|
| 0 | 314 |
| 1 | 221 |
| 2 | 317 |
| 3 | 297 |
| 4 | 322 |
| 5 | 218 |
| 6 | 244 |

| platter_id | platter_weight |
|---|---|
| 0 | 284 |
| 1 | 191 |
| 2 | 287 |
| 3 | 267 |
| 4 | 292 |
| 5 | 188 |
| 6 | 214 |

| platter_i_id | platter_j_id | pair_bonus_appeal |
|---|---|---|
| 0 | 2 | 244 |
| 0 | 3 | 111 |
| 0 | 4 | 202 |
| 0 | 5 | 112 |
| 1 | 2 | 259 |
| 1 | 3 | 108 |
| 1 | 5 | 156 |
| 1 | 6 | 102 |
| 2 | 3 | 110 |
| 3 | 4 | 282 |
| 3 | 5 | 232 |
| 3 | 6 | 170 |
| 4 | 5 | 102 |
| 4 | 6 | 112 |

I'll use these figures to pick the best set that fits the van capacity 1469.

When you tell me which platters to load, just give it in a tiny JSON like this so it's easy to read and check:

{
  ""solution"": [<selected_platter_id>, <selected_platter_id>, ...]
}

This just means ""solution"" is a list of the platters you pick for the van — one identifier per platter. Think of it like ticking boxes on a short form: put each platter's exact ID in the array in whatever order you like. This is only a sketch of the shape I expect, not the actual selection.

Please use the platter/item identifiers exactly as they appear in the problem input — don't rename them or invent new ones. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'linear_coeffs': [314, 221, 317, 297, 322, 218, 244], 'quadratic_coeffs': [[122, 0, 244, 111, 202, 112, 0], [0, 0, 259, 108, 0, 156, 102], [0, 0, 0, 110, 0, 0, 0], [0, 0, 0, 189, 282, 232, 170], [0, 0, 0, 0, 102, 102, 112], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 294]], 'weights': [284, 191, 287, 267, 292, 188, 214], 'capacity': 1469, 'solution': [0, 1, 3, 4, 5, 6], 'obj': 3305.0, 'problem_type': 'QKP'}","[0, 1, 3, 4, 5, 6]",3305.0,"{'problem_type': 'QKP', 'num_items': 7, 'capacity': 1469, 'items': [0, 1, 2, 3, 4, 5, 6], 'linear_pairs': [{'item_id': 0, 'linear_profit': 314}, {'item_id': 1, 'linear_profit': 221}, {'item_id': 2, 'linear_profit': 317}, {'item_id': 3, 'linear_profit': 297}, {'item_id': 4, 'linear_profit': 322}, {'item_id': 5, 'linear_profit': 218}, {'item_id': 6, 'linear_profit': 244}], 'weight_pairs': [{'item_id': 0, 'weight': 284}, {'item_id': 1, 'weight': 191}, {'item_id': 2, 'weight': 287}, {'item_id': 3, 'weight': 267}, {'item_id': 4, 'weight': 292}, {'item_id': 5, 'weight': 188}, {'item_id': 6, 'weight': 214}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 122}, {'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 244}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 111}, {'item_i_id': 0, 'item_j_id': 4, 'quadratic_profit': 202}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 112}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 259}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 108}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 156}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 102}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 110}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 189}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 282}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 232}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 170}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 102}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 102}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 112}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 294}]}","[0, 1, 3, 4, 5, 6]",20,markdown_table,0
QKP,QKP,"Many people working on the festival have suggested different combos of bands, but each band comes with its own cost and solo drawing power and some particular pairings give a notable extra boost when both are included. The challenge is picking a subset of bands whose combined fees fit the budget, and then measuring success by the total crowd: add each hired band’s individual draw and every extra bonus for pairs that are both in the lineup. The lineup that produces the biggest summed crowd is the one to go with, with the rule that no band is hired more than once and the total fees can’t exceed available funds. The exact cost and boost numbers are listed below.

Below are the 6 candidate bands listed by ID in A, B, C, D, E, F, with their fees and boosts; total fees must not exceed 3885.

| band_id | solo_draw |
|---|---|
| A | 2019 |
| B | 1192 |
| C | 1470 |
| D | 2466 |
| E | 2367 |
| F | 2497 |

| band_id | booking_fee |
|---|---|
| A | 3135 |
| B | 336 |
| C | 1157 |
| D | 2550 |
| E | 1201 |
| F | 383 |

| band_i_id | band_j_id | pairing_boost |
|---|---|---|
| A | B | 874 |
| A | D | 3237 |
| B | D | 2086 |
| B | F | 3242 |
| C | D | 779 |
| E | F | 2695 |

These figures can be used to evaluate lineups and identify the selection that maximizes the total crowd without exceeding the budget 3885.

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

{
  ""solution"": [<selected_band_id>, <selected_band_id>, ...]
}

This ""solution"" array should just list the IDs of the bands you want to hire (one entry per band). Think of it like filling in a form: put the band identifiers in the list, and that's your proposed lineup. The JSON above is only a sketch of the shape I need, not the actual answer.

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”. Please use the identifiers exactly as they appear in the instance input—no renaming and no new labels.","{'linear_coeffs': [2019, 1192, 1470, 2466, 2367, 2497], 'quadratic_coeffs': [[0, 874, 0, 3237, 0, 0], [0, 0, 0, 2086, 0, 3242], [0, 0, 0, 779, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2695], [0, 0, 0, 0, 0, 3106]], 'weights': [3135, 336, 1157, 2550, 1201, 383], 'capacity': 3885, 'solution': [1, 2, 4, 5], 'obj': 13463.0, 'problem_type': 'QKP'}","[1, 2, 4, 5]",13463.0,"{'problem_type': 'QKP', 'num_items': 6, 'capacity': 3885, 'items': ['A', 'B', 'C', 'D', 'E', 'F'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 2019}, {'item_id': 'B', 'linear_profit': 1192}, {'item_id': 'C', 'linear_profit': 1470}, {'item_id': 'D', 'linear_profit': 2466}, {'item_id': 'E', 'linear_profit': 2367}, {'item_id': 'F', 'linear_profit': 2497}], 'weight_pairs': [{'item_id': 'A', 'weight': 3135}, {'item_id': 'B', 'weight': 336}, {'item_id': 'C', 'weight': 1157}, {'item_id': 'D', 'weight': 2550}, {'item_id': 'E', 'weight': 1201}, {'item_id': 'F', 'weight': 383}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 874}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 3237}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 2086}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 3242}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 779}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 2695}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 3106}]}","['B', 'C', 'E', 'F']",21,markdown_table,names
QKP,QKP,"On a sunny morning the volunteers sorted through plant tags, figuring out what to put in the community bed. Each tag lists how much room a plant needs and how lovely it looks by itself, and some pairs of plants enhance one another, giving extra beauty if both are chosen. The job is to choose a combination that fits the bed’s available area and yields the greatest total beauty — add every chosen plant’s beauty and then any extra points for pairs — and each plant choice can only be used once. The exact plant items, areas, and pair bonuses are detailed below.

# num_plants=6
# bed_area_capacity=1126
# plant_ids=A, B, C, D, E, F

plant_id,base_beauty
A,252
B,351
C,394
D,128
E,246
F,211

plant_id,area_required
A,219
B,433
C,450
D,499
E,195
F,494

plant_i_id,plant_j_id,pair_bonus_beauty
A,B,146
A,C,219
A,D,219
A,E,219
A,F,177
B,C,433
B,D,177
B,E,114
B,F,146
C,D,171
C,E,171
C,F,160
D,E,120
D,F,277
E,F,468

If you want, just drop your chosen plants into a tiny JSON snippet like the one below so it's easy to read and verify.

{
  ""solution"": [<selected_plant_id>, <selected_plant_id>, ...]
}

This little block means: put the list of plant tag identifiers you picked under ""solution"". Think of each placeholder inside the brackets as the exact tag you’ll choose — replace each <selected_plant_id> with a real tag from the instance. It’s just a sketch of the shape I need, not your final answer.

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”.""","{'linear_coeffs': [252, 351, 394, 128, 246, 211], 'quadratic_coeffs': [[219, 146, 219, 219, 219, 177], [0, 317, 433, 177, 114, 146], [0, 0, 231, 171, 171, 160], [0, 0, 0, 287, 120, 277], [0, 0, 0, 0, 272, 468], [0, 0, 0, 0, 0, 344]], 'weights': [219, 433, 450, 499, 195, 494], 'capacity': 1126, 'solution': [0, 1, 2], 'obj': 1795.0, 'problem_type': 'QKP'}","[0, 1, 2]",1795.0,"{'problem_type': 'QKP', 'num_items': 6, 'capacity': 1126, 'items': ['A', 'B', 'C', 'D', 'E', 'F'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 252}, {'item_id': 'B', 'linear_profit': 351}, {'item_id': 'C', 'linear_profit': 394}, {'item_id': 'D', 'linear_profit': 128}, {'item_id': 'E', 'linear_profit': 246}, {'item_id': 'F', 'linear_profit': 211}], 'weight_pairs': [{'item_id': 'A', 'weight': 219}, {'item_id': 'B', 'weight': 433}, {'item_id': 'C', 'weight': 450}, {'item_id': 'D', 'weight': 499}, {'item_id': 'E', 'weight': 195}, {'item_id': 'F', 'weight': 494}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 219}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 146}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 219}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 219}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 219}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 177}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 317}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 433}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 177}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 114}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 146}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 231}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 171}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 171}, {'item_i_id': 'C', 'item_j_id': 'F', 'quadratic_profit': 160}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 287}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 120}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 277}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 272}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 468}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 344}]}","['A', 'B', 'C']",22,csv,names
QKP,QKP,"Many photographers know this scenario: a portrait job, a single bag, and a pile of lenses and accessories to choose from. Each piece carries weight and brings a base creative benefit, and some pairs multiply the possibilities when they’re together. The point is to decide which items to bring so the total weight remains under the bag’s capacity while the total creative score—calculated by summing each packed item’s value and adding any extra pair rewards—is as large as possible; every item is optional but indivisible. The exact kit list and stats appear below.

{
  ""total_gear_items"": 7,
  ""bag_weight_capacity"": 1513,
  ""gear_identifiers"": [
    ""A"",
    ""B"",
    ""C"",
    ""D"",
    ""E"",
    ""F"",
    ""G""
  ],
  ""linear"": [
    {
      ""gear_id"": ""A"",
      ""base_creative_value"": 911
    },
    {
      ""gear_id"": ""B"",
      ""base_creative_value"": 231
    },
    {
      ""gear_id"": ""C"",
      ""base_creative_value"": 307
    },
    {
      ""gear_id"": ""D"",
      ""base_creative_value"": 623
    },
    {
      ""gear_id"": ""E"",
      ""base_creative_value"": 742
    },
    {
      ""gear_id"": ""F"",
      ""base_creative_value"": 574
    },
    {
      ""gear_id"": ""G"",
      ""base_creative_value"": 744
    }
  ],
  ""weights"": [
    {
      ""gear_id"": ""A"",
      ""gear_weight"": 821
    },
    {
      ""gear_id"": ""B"",
      ""gear_weight"": 141
    },
    {
      ""gear_id"": ""C"",
      ""gear_weight"": 217
    },
    {
      ""gear_id"": ""D"",
      ""gear_weight"": 533
    },
    {
      ""gear_id"": ""E"",
      ""gear_weight"": 652
    },
    {
      ""gear_id"": ""F"",
      ""gear_weight"": 484
    },
    {
      ""gear_id"": ""G"",
      ""gear_weight"": 654
    }
  ],
  ""quadratic"": [
    {
      ""gear_i_id"": ""A"",
      ""gear_j_id"": ""B"",
      ""pair_synergy_bonus"": 718
    },
    {
      ""gear_i_id"": ""A"",
      ""gear_j_id"": ""C"",
      ""pair_synergy_bonus"": 806
    },
    {
      ""gear_i_id"": ""B"",
      ""gear_j_id"": ""C"",
      ""pair_synergy_bonus"": 491
    },
    {
      ""gear_i_id"": ""B"",
      ""gear_j_id"": ""D"",
      ""pair_synergy_bonus"": 826
    },
    {
      ""gear_i_id"": ""B"",
      ""gear_j_id"": ""F"",
      ""pair_synergy_bonus"": 420
    },
    {
      ""gear_i_id"": ""B"",
      ""gear_j_id"": ""G"",
      ""pair_synergy_bonus"": 368
    },
    {
      ""gear_i_id"": ""C"",
      ""gear_j_id"": ""E"",
      ""pair_synergy_bonus"": 399
    },
    {
      ""gear_i_id"": ""C"",
      ""gear_j_id"": ""F"",
      ""pair_synergy_bonus"": 317
    },
    {
      ""gear_i_id"": ""C"",
      ""gear_j_id"": ""G"",
      ""pair_synergy_bonus"": 533
    },
    {
      ""gear_i_id"": ""D"",
      ""gear_j_id"": ""E"",
      ""pair_synergy_bonus"": 795
    },
    {
      ""gear_i_id"": ""D"",
      ""gear_j_id"": ""F"",
      ""pair_synergy_bonus"": 188
    },
    {
      ""gear_i_id"": ""D"",
      ""gear_j_id"": ""G"",
      ""pair_synergy_bonus"": 514
    },
    {
      ""gear_i_id"": ""E"",
      ""gear_j_id"": ""F"",
      ""pair_synergy_bonus"": 182
    },
    {
      ""gear_i_id"": ""E"",
      ""gear_j_id"": ""G"",
      ""pair_synergy_bonus"": 536
    }
  ]
}

If you want to hand me your kit choice, just drop it in a tiny JSON snippet like this:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

Here ""solution"" is just a list of the items you’re packing — each entry should be the exact item identifier from the instance (the ID itself, not a description). Think of it like the short checklist of lenses and accessories you’ll toss into the bag. This JSON is only a sketch of the shape I expect, not the actual answer.

All identifiers must be used exactly as they appear in the instance input — no renaming and no new labels.
Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'linear_coeffs': [911, 231, 307, 623, 742, 574, 744], 'quadratic_coeffs': [[282, 718, 806, 0, 0, 0, 0], [0, 0, 491, 826, 0, 420, 368], [0, 0, 545, 0, 399, 317, 533], [0, 0, 0, 554, 795, 188, 514], [0, 0, 0, 0, 216, 182, 536], [0, 0, 0, 0, 0, 533, 0], [0, 0, 0, 0, 0, 0, 0]], 'weights': [821, 141, 217, 533, 652, 484, 654], 'capacity': 1513, 'solution': [1, 2, 5, 6], 'obj': 3985.0, 'problem_type': 'QKP'}","[1, 2, 5, 6]",3985.0,"{'problem_type': 'QKP', 'num_items': 7, 'capacity': 1513, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 911}, {'item_id': 'B', 'linear_profit': 231}, {'item_id': 'C', 'linear_profit': 307}, {'item_id': 'D', 'linear_profit': 623}, {'item_id': 'E', 'linear_profit': 742}, {'item_id': 'F', 'linear_profit': 574}, {'item_id': 'G', 'linear_profit': 744}], 'weight_pairs': [{'item_id': 'A', 'weight': 821}, {'item_id': 'B', 'weight': 141}, {'item_id': 'C', 'weight': 217}, {'item_id': 'D', 'weight': 533}, {'item_id': 'E', 'weight': 652}, {'item_id': 'F', 'weight': 484}, {'item_id': 'G', 'weight': 654}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 282}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 718}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 806}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 491}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 826}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 420}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 368}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 545}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 399}, {'item_i_id': 'C', 'item_j_id': 'F', 'quadratic_profit': 317}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 533}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 554}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 795}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 188}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 514}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 216}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 182}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 536}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 533}]}","['B', 'C', 'F', 'G']",23,json,names
QKP,QKP,"Many product teams face this: a fixed pot of money, a list of features each with a cost and its own benefit, and a few feature pairs that create extra value when shipped together. The challenge is to choose which features to fund so total spending stays within the pot, and the success metric is the total benefit — simply the sum of chosen features’ individual values plus any bonuses for chosen pairs — while never funding a feature twice or in parts. The exact numbers for costs, standalone benefits, and pair synergies are provided below.

Below are those numbers for this instance: 7 candidate features identified as 0, 1, 2, 3, 4, 5, 6 and a total development budget of 13807.

| feature_id | standalone_value |
|---|---|
| 0 | 3368 |
| 1 | 3199 |
| 2 | 3565 |
| 3 | 3406 |
| 4 | 2260 |
| 5 | 4623 |
| 6 | 5924 |

| feature_id | cost |
|---|---|
| 0 | 2718 |
| 1 | 2549 |
| 2 | 2915 |
| 3 | 2756 |
| 4 | 1610 |
| 5 | 3973 |
| 6 | 5274 |

| feature_i_id | feature_j_id | pair_synergy_value |
|---|---|---|
| 0 | 1 | 205 |
| 0 | 2 | 2718 |
| 0 | 3 | 611 |
| 0 | 4 | 596 |
| 0 | 5 | 2718 |
| 0 | 6 | 2718 |
| 1 | 2 | 734 |
| 1 | 3 | 1445 |
| 1 | 4 | 1946 |
| 1 | 5 | 2779 |
| 1 | 6 | 2915 |
| 2 | 3 | 1210 |
| 2 | 4 | 148 |
| 2 | 5 | 2261 |
| 2 | 6 | 2261 |
| 3 | 4 | 4755 |
| 3 | 5 | 2122 |
| 3 | 6 | 1928 |
| 4 | 5 | 2122 |
| 4 | 6 | 4621 |
| 5 | 6 | 1466 |

Use these figures to decide which features to fund so total spending stays within the 13807 budget while maximizing overall benefit.

When you're ready to give me the selection, just use this simple JSON shape so I can read it automatically — nothing fancy, just a list of the item IDs you want to pick:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

""solution"" is just the list of features/items you choose to fund. Think of it like filling out a form: put each item's ID in the array, and that's your selection. This block is just a sketch of the expected shape — not the real answer itself.

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”.","{'linear_coeffs': [3368, 3199, 3565, 3406, 2260, 4623, 5924], 'quadratic_coeffs': [[2718, 205, 2718, 611, 596, 2718, 2718], [0, 596, 734, 1445, 1946, 2779, 2915], [0, 0, 2915, 1210, 148, 2261, 2261], [0, 0, 0, 2407, 4755, 2122, 1928], [0, 0, 0, 0, 262, 2122, 4621], [0, 0, 0, 0, 0, 262, 1466], [0, 0, 0, 0, 0, 0, 1174]], 'weights': [2718, 2549, 2915, 2756, 1610, 3973, 5274], 'capacity': 13807, 'solution': [1, 2, 3, 4, 5], 'obj': 36575.0, 'problem_type': 'QKP'}","[1, 2, 3, 4, 5]",36575.0,"{'problem_type': 'QKP', 'num_items': 7, 'capacity': 13807, 'items': [0, 1, 2, 3, 4, 5, 6], 'linear_pairs': [{'item_id': 0, 'linear_profit': 3368}, {'item_id': 1, 'linear_profit': 3199}, {'item_id': 2, 'linear_profit': 3565}, {'item_id': 3, 'linear_profit': 3406}, {'item_id': 4, 'linear_profit': 2260}, {'item_id': 5, 'linear_profit': 4623}, {'item_id': 6, 'linear_profit': 5924}], 'weight_pairs': [{'item_id': 0, 'weight': 2718}, {'item_id': 1, 'weight': 2549}, {'item_id': 2, 'weight': 2915}, {'item_id': 3, 'weight': 2756}, {'item_id': 4, 'weight': 1610}, {'item_id': 5, 'weight': 3973}, {'item_id': 6, 'weight': 5274}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 2718}, {'item_i_id': 0, 'item_j_id': 1, 'quadratic_profit': 205}, {'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 2718}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 611}, {'item_i_id': 0, 'item_j_id': 4, 'quadratic_profit': 596}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 2718}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 2718}, {'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 596}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 734}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 1445}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 1946}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 2779}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 2915}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 2915}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 1210}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 148}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 2261}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 2261}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 2407}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 4755}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 2122}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 1928}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 262}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 2122}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 4621}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 262}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 1466}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 1174}]}","[1, 2, 3, 4, 5]",24,markdown_table,0
QKP,QKP,"We were putting together a film lineup for a friends’ marathon: every movie brings its own entertainment value and takes a chunk of the evening, and certain pairings boost the fun when both are included. The plan was to add up the happiness from each selected movie plus any extra enjoyment from pairs, while keeping the total runtime within the night’s limit and not repeating films, so the best lineup is the one with the largest total enjoyment that still fits. The specific runtimes and pairing bonuses are listed below.

# num_films=8
# available_runtime=1709
# film_ids=1, 2, 3, 4, 5, 6, 7, 8

film_id,base_enjoyment
1,259
2,318
3,259
4,260
5,263
6,143
7,251
8,308

film_id,runtime_minutes
1,229
2,288
3,229
4,230
5,233
6,113
7,221
8,278

film_i_id,film_j_id,pairing_bonus
1,2,206
1,5,110
1,8,229
2,3,283
2,5,230
3,5,229
3,6,172
3,8,230
4,5,206
4,7,104
4,8,180
6,7,106
6,8,267
7,8,172

Also, when you send back your chosen lineup, a tiny JSON snippet like the one below works great — simple and machine-friendly.

{
  ""solution"": [<selected_movie_id>, <selected_movie_id>, ...]
}

This just means: ""solution"" is the list of movies you're picking for the marathon, and each <selected_movie_id> is a placeholder for one movie's identifier from the problem instance. Think of it like filling in a short form — put each movie's ID in the array (order doesn't matter, and don't repeat a movie).

The JSON above is just a sketch of the expected shape, not the actual answer — replace those angle-bracket placeholders with the real identifiers from the instance.

Please use the identifiers exactly as they appear in the instance input — no renaming, no made-up 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”.""","{'linear_coeffs': [259, 318, 259, 260, 263, 143, 251, 308], 'quadratic_coeffs': [[0, 206, 0, 0, 110, 0, 0, 229], [0, 218, 283, 0, 230, 0, 0, 0], [0, 0, 229, 0, 229, 172, 0, 230], [0, 0, 0, 0, 206, 0, 104, 180], [0, 0, 0, 0, 105, 0, 0, 0], [0, 0, 0, 0, 0, 113, 106, 267], [0, 0, 0, 0, 0, 0, 0, 172], [0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [229, 288, 229, 230, 233, 113, 221, 278], 'capacity': 1709, 'solution': [0, 1, 2, 3, 4, 5, 7], 'obj': 4152.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 5, 7]",4152.0,"{'problem_type': 'QKP', 'num_items': 8, 'capacity': 1709, 'items': [1, 2, 3, 4, 5, 6, 7, 8], 'linear_pairs': [{'item_id': 1, 'linear_profit': 259}, {'item_id': 2, 'linear_profit': 318}, {'item_id': 3, 'linear_profit': 259}, {'item_id': 4, 'linear_profit': 260}, {'item_id': 5, 'linear_profit': 263}, {'item_id': 6, 'linear_profit': 143}, {'item_id': 7, 'linear_profit': 251}, {'item_id': 8, 'linear_profit': 308}], 'weight_pairs': [{'item_id': 1, 'weight': 229}, {'item_id': 2, 'weight': 288}, {'item_id': 3, 'weight': 229}, {'item_id': 4, 'weight': 230}, {'item_id': 5, 'weight': 233}, {'item_id': 6, 'weight': 113}, {'item_id': 7, 'weight': 221}, {'item_id': 8, 'weight': 278}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 206}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 110}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 229}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 218}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 283}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 230}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 229}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 229}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 172}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 230}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 206}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 104}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 180}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 105}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 113}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 106}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 267}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 172}]}","[1, 2, 3, 4, 5, 6, 8]",25,csv,1
QKP,QKP,"We’ve got one limited-size box to fill for a customer, and the question is which pastries to include. Every item brings its own appeal and weight, and certain combinations add extra appeal when both are present — so the overall box appeal is just adding up each chosen pastry’s appeal and any extra boosts from the pairs that happen to be included. The goal is to keep the total weight within the box limit while maximizing that summed appeal, with each pastry either in or out and none duplicated. The concrete details follow below.

We have 8 pastries (0, 1, 2, 3, 4, 5, 6, 7) and the box capacity is 1572.

| pastry_id | individual_appeal |
|---|---|
| 0 | 120 |
| 1 | 199 |
| 2 | 179 |
| 3 | 280 |
| 4 | 111 |
| 5 | 278 |
| 6 | 309 |
| 7 | 170 |

| pastry_id | pastry_weight |
|---|---|
| 0 | 112 |
| 1 | 192 |
| 2 | 184 |
| 3 | 271 |
| 4 | 121 |
| 5 | 293 |
| 6 | 295 |
| 7 | 185 |

| pastry_i_id | pastry_j_id | pair_bonus_appeal |
|---|---|---|
| 0 | 3 | 108 |
| 0 | 4 | 112 |
| 1 | 4 | 158 |
| 1 | 5 | 192 |
| 1 | 7 | 192 |
| 2 | 4 | 184 |
| 2 | 7 | 193 |
| 3 | 4 | 108 |
| 3 | 6 | 108 |
| 3 | 7 | 108 |
| 4 | 6 | 134 |
| 4 | 7 | 168 |
| 5 | 7 | 147 |
| 6 | 7 | 159 |

We'll pick the combination that fits within 1572 and maximizes the summed appeal.

Also, when you send back the final selection, just drop it in a tiny JSON snippet like this so it's easy to parse:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

""solution"" is simply the list of pastry IDs that should go into the box. Think of it as the short form of your order: just the exact IDs of the pastries you picked. This block is only a sketch of 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. 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”.","{'linear_coeffs': [120, 199, 179, 280, 111, 278, 309, 170], 'quadratic_coeffs': [[0, 0, 0, 108, 112, 0, 0, 0], [0, 154, 0, 0, 158, 192, 0, 192], [0, 0, 0, 0, 184, 0, 0, 193], [0, 0, 0, 108, 108, 0, 108, 108], [0, 0, 0, 0, 193, 0, 134, 168], [0, 0, 0, 0, 0, 193, 0, 147], [0, 0, 0, 0, 0, 0, 0, 159], [0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [112, 192, 184, 271, 121, 293, 295, 185], 'capacity': 1572, 'solution': [1, 2, 3, 4, 5, 6, 7], 'obj': 3377.0, 'problem_type': 'QKP'}","[1, 2, 3, 4, 5, 6, 7]",3377.0,"{'problem_type': 'QKP', 'num_items': 8, 'capacity': 1572, 'items': [0, 1, 2, 3, 4, 5, 6, 7], 'linear_pairs': [{'item_id': 0, 'linear_profit': 120}, {'item_id': 1, 'linear_profit': 199}, {'item_id': 2, 'linear_profit': 179}, {'item_id': 3, 'linear_profit': 280}, {'item_id': 4, 'linear_profit': 111}, {'item_id': 5, 'linear_profit': 278}, {'item_id': 6, 'linear_profit': 309}, {'item_id': 7, 'linear_profit': 170}], 'weight_pairs': [{'item_id': 0, 'weight': 112}, {'item_id': 1, 'weight': 192}, {'item_id': 2, 'weight': 184}, {'item_id': 3, 'weight': 271}, {'item_id': 4, 'weight': 121}, {'item_id': 5, 'weight': 293}, {'item_id': 6, 'weight': 295}, {'item_id': 7, 'weight': 185}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 108}, {'item_i_id': 0, 'item_j_id': 4, 'quadratic_profit': 112}, {'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 154}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 158}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 192}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 192}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 184}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 193}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 108}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 108}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 108}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 108}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 193}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 134}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 168}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 193}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 147}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 159}]}","[1, 2, 3, 4, 5, 6, 7]",26,markdown_table,0
QKP,QKP,"Someone in the group always packs like they’re moving house, so the rest of us try to be smarter: pick a set of essentials that won’t overload the pannier while getting the most convenience out of what’s carried. Each item contributes its own usefulness and some two-item combos add extra convenience when together, so the total value is the sum of chosen items’ usefulness plus any applicable pair bonuses. Items can’t be taken more than once and the total weight must stay under the pannier limit; the exact list of item weights, usefulness scores, pair bonuses, and capacity are shown below.

Below are the 7 items (1, 2, 3, 4, 5, 6, 7) and the pannier capacity 567.

| item_id | usefulness_score |
|---|---|
| 1 | 101 |
| 2 | 100 |
| 3 | 105 |
| 4 | 100 |
| 5 | 103 |
| 6 | 105 |
| 7 | 103 |

| item_id | item_weight |
|---|---|
| 1 | 102 |
| 2 | 103 |
| 3 | 105 |
| 4 | 105 |
| 5 | 102 |
| 6 | 104 |
| 7 | 103 |

| first_item_id | second_item_id | complementary_bonus |
|---|---|---|
| 1 | 4 | 105 |
| 1 | 7 | 105 |
| 2 | 3 | 103 |
| 2 | 5 | 102 |
| 2 | 7 | 101 |
| 3 | 4 | 105 |
| 3 | 6 | 102 |
| 3 | 7 | 104 |
| 5 | 7 | 100 |

Keep the total weight under 567 and pick the combo that gives us the most convenience.

Oh, and when you send back which items to take, a tiny heads-up: please use this simple JSON layout so it's easy to read and parse.

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

This just means ""solution"" should be a list containing the IDs of the items you want to pack. Think of it like ticking boxes on a checklist — each entry is the exact ID of an item you choose. The JSON above is just a sketch of the shape I expect, not the actual answer.

Please use the item identifiers exactly as they appear in the instance input — don’t rename them or invent new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'linear_coeffs': [101, 100, 105, 100, 103, 105, 103], 'quadratic_coeffs': [[102, 0, 0, 105, 0, 0, 105], [0, 105, 103, 0, 102, 0, 101], [0, 0, 0, 105, 0, 102, 104], [0, 0, 0, 105, 0, 0, 0], [0, 0, 0, 0, 0, 0, 100], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0]], 'weights': [102, 103, 105, 105, 102, 104, 103], 'capacity': 567, 'solution': [0, 1, 2, 3, 6], 'obj': 1132.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 6]",1132.0,"{'problem_type': 'QKP', 'num_items': 7, 'capacity': 567, 'items': [1, 2, 3, 4, 5, 6, 7], 'linear_pairs': [{'item_id': 1, 'linear_profit': 101}, {'item_id': 2, 'linear_profit': 100}, {'item_id': 3, 'linear_profit': 105}, {'item_id': 4, 'linear_profit': 100}, {'item_id': 5, 'linear_profit': 103}, {'item_id': 6, 'linear_profit': 105}, {'item_id': 7, 'linear_profit': 103}], 'weight_pairs': [{'item_id': 1, 'weight': 102}, {'item_id': 2, 'weight': 103}, {'item_id': 3, 'weight': 105}, {'item_id': 4, 'weight': 105}, {'item_id': 5, 'weight': 102}, {'item_id': 6, 'weight': 104}, {'item_id': 7, 'weight': 103}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 102}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 105}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 105}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 105}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 103}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 102}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 105}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 102}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 104}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 105}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 100}]}","[1, 2, 3, 4, 7]",27,markdown_table,1
QKP,QKP,"Someone on the ops team had to trim features for a constrained machine: every feature takes up memory and brings a baseline benefit, and a few feature pairings produce extra value if both are turned on. The job was to decide which features to switch on so the total memory they require doesn’t exceed what’s free, and the overall system value is as high as it can be — that value is just the sum of the selected features’ base benefits plus any bonuses from feature pairs that are active together. Features are either active or inactive (no duplicates), and the specific memory costs and benefit numbers appear below.

There were 5 optional modules listed as A, B, C, D, E, and 618 memory units were available for enabling them.

| module_id | base_benefit |
|---|---|
| A | 101 |
| B | 133 |
| C | 122 |
| D | 131 |
| E | 121 |

| module_id | memory_usage |
|---|---|
| A | 133 |
| B | 129 |
| C | 123 |
| D | 114 |
| E | 138 |

| module_i_id | module_j_id | interaction_bonus |
|---|---|---|
| A | B | 133 |
| A | C | 128 |
| A | D | 119 |
| A | E | 128 |
| B | C | 123 |
| B | E | 128 |
| C | D | 105 |
| D | E | 129 |

The ops team must choose modules so total memory stays within 618 while maximizing the system's total value.

Also, when you send your chosen set of features, please use this simple JSON layout so it's easy to read and check:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

Here ""solution"" is just the list of feature IDs you want switched on — the exact labels from the instance input, in a plain array. This JSON is only a sketch of the shape I expect, not the actual answer itself.

Make sure to 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"".","{'linear_coeffs': [101, 133, 122, 131, 121], 'quadratic_coeffs': [[115, 133, 128, 119, 128], [0, 122, 123, 0, 128], [0, 0, 127, 105, 0], [0, 0, 0, 120, 129], [0, 0, 0, 0, 117]], 'weights': [133, 129, 123, 114, 138], 'capacity': 618, 'solution': [0, 1, 3, 4], 'obj': 1123.0, 'problem_type': 'QKP'}","[0, 1, 3, 4]",1123.0,"{'problem_type': 'QKP', 'num_items': 5, 'capacity': 618, 'items': ['A', 'B', 'C', 'D', 'E'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 101}, {'item_id': 'B', 'linear_profit': 133}, {'item_id': 'C', 'linear_profit': 122}, {'item_id': 'D', 'linear_profit': 131}, {'item_id': 'E', 'linear_profit': 121}], 'weight_pairs': [{'item_id': 'A', 'weight': 133}, {'item_id': 'B', 'weight': 129}, {'item_id': 'C', 'weight': 123}, {'item_id': 'D', 'weight': 114}, {'item_id': 'E', 'weight': 138}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 115}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 133}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 128}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 119}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 128}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 122}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 123}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 128}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 127}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 105}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 120}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 129}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 117}]}","['A', 'B', 'D', 'E']",28,markdown_table,names
QKP,QKP,"Someone in the school asked for a standout display, so the librarian is choosing which books to put on the front shelf to make the learning impact as strong as possible. Every chosen title uses up some of the shelf’s length and contributes its own educational points, and a few specific pairs of titles interact to give bonus points when both are present. A great arrangement keeps the total widths within the shelf limit and scores highly by totaling each book’s points plus any pair bonuses — with each book available as one copy only, and no partial placements allowed. The full details (widths, values, and which pairs get bonuses) are listed below.

{
  ""total_books"": 7,
  ""shelf_length"": 2985,
  ""book_ids"": [
    1,
    2,
    3,
    4,
    5,
    6,
    7
  ],
  ""linear"": [
    {
      ""book_id"": 1,
      ""educational_points"": 1618
    },
    {
      ""book_id"": 2,
      ""educational_points"": 2279
    },
    {
      ""book_id"": 3,
      ""educational_points"": 1487
    },
    {
      ""book_id"": 4,
      ""educational_points"": 1512
    },
    {
      ""book_id"": 5,
      ""educational_points"": 2321
    },
    {
      ""book_id"": 6,
      ""educational_points"": 618
    },
    {
      ""book_id"": 7,
      ""educational_points"": 3214
    }
  ],
  ""weights"": [
    {
      ""book_id"": 1,
      ""width"": 3170
    },
    {
      ""book_id"": 2,
      ""width"": 2049
    },
    {
      ""book_id"": 3,
      ""width"": 1208
    },
    {
      ""book_id"": 4,
      ""width"": 646
    },
    {
      ""book_id"": 5,
      ""width"": 1984
    },
    {
      ""book_id"": 6,
      ""width"": 3024
    },
    {
      ""book_id"": 7,
      ""width"": 2839
    }
  ],
  ""quadratic"": [
    {
      ""book_i_id"": 1,
      ""book_j_id"": 2,
      ""pairing_bonus_points"": 2103
    },
    {
      ""book_i_id"": 1,
      ""book_j_id"": 6,
      ""pairing_bonus_points"": 727
    },
    {
      ""book_i_id"": 1,
      ""book_j_id"": 7,
      ""pairing_bonus_points"": 846
    },
    {
      ""book_i_id"": 2,
      ""book_j_id"": 4,
      ""pairing_bonus_points"": 1689
    },
    {
      ""book_i_id"": 2,
      ""book_j_id"": 7,
      ""pairing_bonus_points"": 1408
    },
    {
      ""book_i_id"": 3,
      ""book_j_id"": 4,
      ""pairing_bonus_points"": 1208
    },
    {
      ""book_i_id"": 3,
      ""book_j_id"": 5,
      ""pairing_bonus_points"": 629
    },
    {
      ""book_i_id"": 3,
      ""book_j_id"": 6,
      ""pairing_bonus_points"": 774
    },
    {
      ""book_i_id"": 3,
      ""book_j_id"": 7,
      ""pairing_bonus_points"": 774
    },
    {
      ""book_i_id"": 4,
      ""book_j_id"": 5,
      ""pairing_bonus_points"": 226
    },
    {
      ""book_i_id"": 4,
      ""book_j_id"": 7,
      ""pairing_bonus_points"": 2634
    },
    {
      ""book_i_id"": 5,
      ""book_j_id"": 6,
      ""pairing_bonus_points"": 1562
    },
    {
      ""book_i_id"": 6,
      ""book_j_id"": 7,
      ""pairing_bonus_points"": 124
    }
  ]
}

Oh, and when you send back your pick for the front shelf, please use this simple JSON shape so it's easy to check automatically:

{
  ""solution"": [<book_id>, <book_id>, ...]
}

Here ""solution"" is just the list of the books you want on the front shelf — each entry is the identifier for a book from the instance. Think of it like filling out a short form: put the exact book IDs you chose inside the brackets, separated by commas. This is just a sketch of the shape we expect, not the final answer itself.

Please make sure to 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”.","{'linear_coeffs': [1618, 2279, 1487, 1512, 2321, 618, 3214], 'quadratic_coeffs': [[2037, 2103, 0, 0, 0, 727, 846], [0, 0, 0, 1689, 0, 0, 1408], [0, 0, 0, 1208, 629, 774, 774], [0, 0, 0, 967, 226, 0, 2634], [0, 0, 0, 0, 1885, 1562, 0], [0, 0, 0, 0, 0, 866, 124], [0, 0, 0, 0, 0, 0, 0]], 'weights': [3170, 2049, 1208, 646, 1984, 3024, 2839], 'capacity': 2985, 'solution': [1, 3], 'obj': 5480.0, 'problem_type': 'QKP'}","[1, 3]",5480.0,"{'problem_type': 'QKP', 'num_items': 7, 'capacity': 2985, 'items': [1, 2, 3, 4, 5, 6, 7], 'linear_pairs': [{'item_id': 1, 'linear_profit': 1618}, {'item_id': 2, 'linear_profit': 2279}, {'item_id': 3, 'linear_profit': 1487}, {'item_id': 4, 'linear_profit': 1512}, {'item_id': 5, 'linear_profit': 2321}, {'item_id': 6, 'linear_profit': 618}, {'item_id': 7, 'linear_profit': 3214}], 'weight_pairs': [{'item_id': 1, 'weight': 3170}, {'item_id': 2, 'weight': 2049}, {'item_id': 3, 'weight': 1208}, {'item_id': 4, 'weight': 646}, {'item_id': 5, 'weight': 1984}, {'item_id': 6, 'weight': 3024}, {'item_id': 7, 'weight': 2839}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 2037}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 2103}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 727}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 846}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 1689}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 1408}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 1208}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 629}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 774}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 774}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 967}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 226}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 2634}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 1885}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 1562}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 866}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 124}]}","[2, 4]",29,json,1
QKP,QKP,"Many people on the scheduling desk worry about squeezing the best lineup into a fixed time slot. Every show uses up some minutes and earns a baseline audience, and there are a few beneficial pairings that add extra viewers when those two shows share the night. The job: choose a set of distinct shows that fits inside the block’s total time and maximizes the overall audience, which is just the sum of each show’s base viewers plus any applicable boosts from paired shows. The full runtimes, base ratings, and pairing bonuses appear below.

There are 9 shows available (IDs: A, B, C, D, E, F, G, H, I) and the evening block allows up to 3865 minutes.
Show A draws a baseline audience of 1044 viewers.
Show B draws a baseline audience of 411 viewers.
Show C draws a baseline audience of 86 viewers.
Show D draws a baseline audience of 1736 viewers.
Show E draws a baseline audience of 858 viewers.
Show F draws a baseline audience of 1240 viewers.
Show G draws a baseline audience of 1153 viewers.
Show H draws a baseline audience of 1139 viewers.
Show I draws a baseline audience of 1737 viewers.
Show A occupies 978 minutes of the block.
Show B occupies 377 minutes of the block.
Show C occupies 118 minutes of the block.
Show D occupies 1651 minutes of the block.
Show E occupies 927 minutes of the block.
Show F occupies 1218 minutes of the block.
Show G occupies 1148 minutes of the block.
Show H occupies 1079 minutes of the block.
Show I occupies 1660 minutes of the block.
If both A and B air, their pairing adds 695 viewers.
If both A and C air, their pairing adds 1266 viewers.
If both A and D air, their pairing adds 695 viewers.
If both A and E air, their pairing adds 417 viewers.
If both A and F air, their pairing adds 775 viewers.
If both A and G air, their pairing adds 863 viewers.
If both A and H air, their pairing adds 775 viewers.
If both A and I air, their pairing adds 633 viewers.
If both B and C air, their pairing adds 609 viewers.
If both B and D air, their pairing adds 391 viewers.
If both B and E air, their pairing adds 679 viewers.
If both B and F air, their pairing adds 897 viewers.
If both B and G air, their pairing adds 1571 viewers.
If both B and H air, their pairing adds 1130 viewers.
If both B and I air, their pairing adds 1444 viewers.
If both C and D air, their pairing adds 279 viewers.
If both C and E air, their pairing adds 434 viewers.
If both C and F air, their pairing adds 497 viewers.
If both C and G air, their pairing adds 630 viewers.
If both C and H air, their pairing adds 715 viewers.
If both C and I air, their pairing adds 604 viewers.
If both D and E air, their pairing adds 459 viewers.
If both D and F air, their pairing adds 500 viewers.
If both D and G air, their pairing adds 505 viewers.
If both D and H air, their pairing adds 583 viewers.
If both D and I air, their pairing adds 175 viewers.
If both E and F air, their pairing adds 1327 viewers.
If both E and G air, their pairing adds 1584 viewers.
If both E and H air, their pairing adds 1368 viewers.
If both E and I air, their pairing adds 1627 viewers.
If both F and G air, their pairing adds 1342 viewers.
If both F and H air, their pairing adds 1156 viewers.
If both F and I air, their pairing adds 426 viewers.
If both G and H air, their pairing adds 760 viewers.
If both G and I air, their pairing adds 582 viewers.
If both H and I air, their pairing adds 909 viewers.
Choose a lineup that fits within 3865 minutes to maximize total viewers from base audiences and any pairing uplifts.

Oh, and when you send back your pick, just stick to a tiny JSON sketch like this so it's easy to read and check:

{
  ""solution"": [<selected_show_id>, <selected_show_id>, ...]
}

This just means ""solution"" should be a list of the shows you choose (use each show's identifier from the instance). It's just a shape guide — not the actual lineup — so fill in the real show IDs when you reply.

Please use the exact identifiers from the instance input, with no renaming or invented 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”.""","{'linear_coeffs': [1044, 411, 86, 1736, 858, 1240, 1153, 1139, 1737], 'quadratic_coeffs': [[816, 695, 1266, 695, 417, 775, 863, 775, 633], [0, 639, 609, 391, 679, 897, 1571, 1130, 1444], [0, 0, 409, 279, 434, 497, 630, 715, 604], [0, 0, 0, 931, 459, 500, 505, 583, 175], [0, 0, 0, 0, 1369, 1327, 1584, 1368, 1627], [0, 0, 0, 0, 0, 1151, 1342, 1156, 426], [0, 0, 0, 0, 0, 0, 463, 760, 582], [0, 0, 0, 0, 0, 0, 0, 641, 909], [0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [978, 377, 118, 1651, 927, 1218, 1148, 1079, 1660], 'capacity': 3865, 'solution': [1, 2, 4, 5, 6], 'obj': 13318.0, 'problem_type': 'QKP'}","[1, 2, 4, 5, 6]",13318.0,"{'problem_type': 'QKP', 'num_items': 9, 'capacity': 3865, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 1044}, {'item_id': 'B', 'linear_profit': 411}, {'item_id': 'C', 'linear_profit': 86}, {'item_id': 'D', 'linear_profit': 1736}, {'item_id': 'E', 'linear_profit': 858}, {'item_id': 'F', 'linear_profit': 1240}, {'item_id': 'G', 'linear_profit': 1153}, {'item_id': 'H', 'linear_profit': 1139}, {'item_id': 'I', 'linear_profit': 1737}], 'weight_pairs': [{'item_id': 'A', 'weight': 978}, {'item_id': 'B', 'weight': 377}, {'item_id': 'C', 'weight': 118}, {'item_id': 'D', 'weight': 1651}, {'item_id': 'E', 'weight': 927}, {'item_id': 'F', 'weight': 1218}, {'item_id': 'G', 'weight': 1148}, {'item_id': 'H', 'weight': 1079}, {'item_id': 'I', 'weight': 1660}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 816}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 695}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 1266}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 695}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 417}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 775}, {'item_i_id': 'A', 'item_j_id': 'G', 'quadratic_profit': 863}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 775}, {'item_i_id': 'A', 'item_j_id': 'I', 'quadratic_profit': 633}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 639}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 609}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 391}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 679}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 897}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 1571}, {'item_i_id': 'B', 'item_j_id': 'H', 'quadratic_profit': 1130}, {'item_i_id': 'B', 'item_j_id': 'I', 'quadratic_profit': 1444}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 409}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 279}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 434}, {'item_i_id': 'C', 'item_j_id': 'F', 'quadratic_profit': 497}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 630}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 715}, {'item_i_id': 'C', 'item_j_id': 'I', 'quadratic_profit': 604}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 931}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 459}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 500}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 505}, {'item_i_id': 'D', 'item_j_id': 'H', 'quadratic_profit': 583}, {'item_i_id': 'D', 'item_j_id': 'I', 'quadratic_profit': 175}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 1369}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 1327}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 1584}, {'item_i_id': 'E', 'item_j_id': 'H', 'quadratic_profit': 1368}, {'item_i_id': 'E', 'item_j_id': 'I', 'quadratic_profit': 1627}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 1151}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 1342}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 1156}, {'item_i_id': 'F', 'item_j_id': 'I', 'quadratic_profit': 426}, {'item_i_id': 'G', 'item_j_id': 'G', 'quadratic_profit': 463}, {'item_i_id': 'G', 'item_j_id': 'H', 'quadratic_profit': 760}, {'item_i_id': 'G', 'item_j_id': 'I', 'quadratic_profit': 582}, {'item_i_id': 'H', 'item_j_id': 'H', 'quadratic_profit': 641}, {'item_i_id': 'H', 'item_j_id': 'I', 'quadratic_profit': 909}]}","['B', 'C', 'E', 'F', 'G']",30,nl,names
QKP,QKP,"Out on the river the difference between a good day and a great day can be what’s in the backpack. Every rod, lure and tool adds weight and gives a baseline chance of scoring fish, and a few specific pairings boost that chance when they travel together. The plan is to choose a combination of gear whose combined weight fits within the pack’s limit and whose total expected catch value equals the sum of all chosen items’ base values plus any pairwise bonuses, with no item repeated. The detailed item list and weight/bonus numbers are below.

Below are 10 tackle options listed as A, B, C, D, E, F, G, H, I, J, and the pack's limit is 6693.

| tackle_id | base_catch_value |
|---|---|
| A | 171 |
| B | 208 |
| C | 827 |
| D | 662 |
| E | 1458 |
| F | 195 |
| G | 1304 |
| H | 1006 |
| I | 721 |
| J | 450 |

| tackle_id | tackle_weight |
|---|---|
| A | 643 |
| B | 1353 |
| C | 941 |
| D | 250 |
| E | 245 |
| F | 1342 |
| G | 1020 |
| H | 349 |
| I | 1202 |
| J | 1212 |

| tackle_i_id | tackle_j_id | combination_bonus_value |
|---|---|---|
| A | C | 323 |
| A | D | 127 |
| A | E | 643 |
| A | F | 643 |
| A | G | 643 |
| A | H | 643 |
| B | C | 1063 |
| B | D | 699 |
| B | E | 1353 |
| B | G | 349 |
| B | H | 1353 |
| B | J | 1057 |
| C | D | 941 |
| C | G | 941 |
| C | H | 920 |
| C | I | 152 |
| C | J | 941 |
| D | E | 250 |
| D | F | 250 |
| D | G | 250 |
| D | H | 1002 |
| D | J | 671 |
| E | F | 197 |
| E | G | 1342 |
| E | I | 1137 |
| F | G | 671 |
| F | H | 240 |
| F | I | 640 |
| F | J | 477 |
| G | I | 586 |
| G | J | 430 |
| H | I | 308 |
| H | J | 761 |
| I | J | 1183 |

Pack smart: choose gear and pairings that raise total expected catch while staying within the 6693 limit.

Oh, and when you’re ready to tell me which gear you’d pack, just drop it in this simple JSON shape so I can read it cleanly:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

The ""solution"" array is just the list of the item IDs you want to take in your pack. Think of it like ticking off the gear names on a checklist — nothing fancy, just the exact labels for the items you chose. This is only a sketch of the expected shape, not the actual answer itself.

Please use the item identifiers exactly as they appear in the instance input — no renaming and no inventing 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”.""","{'linear_coeffs': [171, 208, 827, 662, 1458, 195, 1304, 1006, 721, 450], 'quadratic_coeffs': [[0, 0, 323, 127, 643, 643, 643, 643, 0, 0], [0, 1254, 1063, 699, 1353, 0, 349, 1353, 0, 1057], [0, 0, 782, 941, 0, 0, 941, 920, 152, 941], [0, 0, 0, 0, 250, 250, 250, 1002, 0, 671], [0, 0, 0, 0, 1342, 197, 1342, 0, 1137, 0], [0, 0, 0, 0, 0, 1023, 671, 240, 640, 477], [0, 0, 0, 0, 0, 0, 1183, 0, 586, 430], [0, 0, 0, 0, 0, 0, 0, 316, 308, 761], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1183], [0, 0, 0, 0, 0, 0, 0, 0, 0, 681]], 'weights': [643, 1353, 941, 250, 245, 1342, 1020, 349, 1202, 1212], 'capacity': 6693, 'solution': [1, 2, 3, 4, 6, 7, 8, 9], 'obj': 24325.0, 'problem_type': 'QKP'}","[1, 2, 3, 4, 6, 7, 8, 9]",24325.0,"{'problem_type': 'QKP', 'num_items': 10, 'capacity': 6693, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 171}, {'item_id': 'B', 'linear_profit': 208}, {'item_id': 'C', 'linear_profit': 827}, {'item_id': 'D', 'linear_profit': 662}, {'item_id': 'E', 'linear_profit': 1458}, {'item_id': 'F', 'linear_profit': 195}, {'item_id': 'G', 'linear_profit': 1304}, {'item_id': 'H', 'linear_profit': 1006}, {'item_id': 'I', 'linear_profit': 721}, {'item_id': 'J', 'linear_profit': 450}], 'weight_pairs': [{'item_id': 'A', 'weight': 643}, {'item_id': 'B', 'weight': 1353}, {'item_id': 'C', 'weight': 941}, {'item_id': 'D', 'weight': 250}, {'item_id': 'E', 'weight': 245}, {'item_id': 'F', 'weight': 1342}, {'item_id': 'G', 'weight': 1020}, {'item_id': 'H', 'weight': 349}, {'item_id': 'I', 'weight': 1202}, {'item_id': 'J', 'weight': 1212}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 323}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 127}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 643}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 643}, {'item_i_id': 'A', 'item_j_id': 'G', 'quadratic_profit': 643}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 643}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 1254}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 1063}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 699}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 1353}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 349}, {'item_i_id': 'B', 'item_j_id': 'H', 'quadratic_profit': 1353}, {'item_i_id': 'B', 'item_j_id': 'J', 'quadratic_profit': 1057}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 782}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 941}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 941}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 920}, {'item_i_id': 'C', 'item_j_id': 'I', 'quadratic_profit': 152}, {'item_i_id': 'C', 'item_j_id': 'J', 'quadratic_profit': 941}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 250}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 250}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 250}, {'item_i_id': 'D', 'item_j_id': 'H', 'quadratic_profit': 1002}, {'item_i_id': 'D', 'item_j_id': 'J', 'quadratic_profit': 671}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 1342}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 197}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 1342}, {'item_i_id': 'E', 'item_j_id': 'I', 'quadratic_profit': 1137}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 1023}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 671}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 240}, {'item_i_id': 'F', 'item_j_id': 'I', 'quadratic_profit': 640}, {'item_i_id': 'F', 'item_j_id': 'J', 'quadratic_profit': 477}, {'item_i_id': 'G', 'item_j_id': 'G', 'quadratic_profit': 1183}, {'item_i_id': 'G', 'item_j_id': 'I', 'quadratic_profit': 586}, {'item_i_id': 'G', 'item_j_id': 'J', 'quadratic_profit': 430}, {'item_i_id': 'H', 'item_j_id': 'H', 'quadratic_profit': 316}, {'item_i_id': 'H', 'item_j_id': 'I', 'quadratic_profit': 308}, {'item_i_id': 'H', 'item_j_id': 'J', 'quadratic_profit': 761}, {'item_i_id': 'I', 'item_j_id': 'J', 'quadratic_profit': 1183}, {'item_i_id': 'J', 'item_j_id': 'J', 'quadratic_profit': 681}]}","['B', 'C', 'D', 'E', 'G', 'H', 'I', 'J']",31,markdown_table,names
QKP,QKP,"We’re getting a travel-friendly portrait kit ready for a wedding day: every flash or modifier adds bulk but also contributes to the kinds of images that can be made, and some combos give bonus creative options when used together. The job is to choose which pieces to bring so the total weight fits the bag’s limit while the total benefit is the sum of each item’s standalone value plus any extra value from combos; each item can only be taken once. The concrete specs — weights, individual scores, and combo boosts — are listed below.

{
  ""total_items"": 9,
  ""carry_capacity"": 11758,
  ""item_ids"": [
    0,
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8
  ],
  ""linear"": [
    {
      ""gear_id"": 0,
      ""standalone_value"": 630
    },
    {
      ""gear_id"": 1,
      ""standalone_value"": 4384
    },
    {
      ""gear_id"": 2,
      ""standalone_value"": 4996
    },
    {
      ""gear_id"": 3,
      ""standalone_value"": 5085
    },
    {
      ""gear_id"": 4,
      ""standalone_value"": 6448
    },
    {
      ""gear_id"": 5,
      ""standalone_value"": 6189
    },
    {
      ""gear_id"": 6,
      ""standalone_value"": 1522
    },
    {
      ""gear_id"": 7,
      ""standalone_value"": 4177
    },
    {
      ""gear_id"": 8,
      ""standalone_value"": 5108
    }
  ],
  ""weights"": [
    {
      ""gear_id"": 0,
      ""item_weight"": 530
    },
    {
      ""gear_id"": 1,
      ""item_weight"": 4107
    },
    {
      ""gear_id"": 2,
      ""item_weight"": 4774
    },
    {
      ""gear_id"": 3,
      ""item_weight"": 5173
    },
    {
      ""gear_id"": 4,
      ""item_weight"": 6443
    },
    {
      ""gear_id"": 5,
      ""item_weight"": 6160
    },
    {
      ""gear_id"": 6,
      ""item_weight"": 1617
    },
    {
      ""gear_id"": 7,
      ""item_weight"": 4179
    },
    {
      ""gear_id"": 8,
      ""item_weight"": 4943
    }
  ],
  ""quadratic"": [
    {
      ""gear_i_id"": 0,
      ""gear_j_id"": 3,
      ""combo_bonus"": 1805
    },
    {
      ""gear_i_id"": 0,
      ""gear_j_id"": 5,
      ""combo_bonus"": 1663
    },
    {
      ""gear_i_id"": 0,
      ""gear_j_id"": 8,
      ""combo_bonus"": 2756
    },
    {
      ""gear_i_id"": 1,
      ""gear_j_id"": 2,
      ""combo_bonus"": 4424
    },
    {
      ""gear_i_id"": 1,
      ""gear_j_id"": 8,
      ""combo_bonus"": 4763
    },
    {
      ""gear_i_id"": 3,
      ""gear_j_id"": 4,
      ""combo_bonus"": 4853
    },
    {
      ""gear_i_id"": 3,
      ""gear_j_id"": 5,
      ""combo_bonus"": 5141
    },
    {
      ""gear_i_id"": 4,
      ""gear_j_id"": 6,
      ""combo_bonus"": 2522
    },
    {
      ""gear_i_id"": 5,
      ""gear_j_id"": 7,
      ""combo_bonus"": 4481
    }
  ]
}

Also, when you send back your pick, please use this simple JSON layout so it's easy to parse which kit bits you're bringing:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

The ""solution"" array should just list the item identifiers you want to pack — like ticking off the pieces you'll bring in your travel kit. It's just a sketch of the expected shape, 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”.""","{'linear_coeffs': [630, 4384, 4996, 5085, 6448, 6189, 1522, 4177, 5108], 'quadratic_coeffs': [[1806, 0, 0, 1805, 0, 1663, 0, 0, 2756], [0, 0, 4424, 0, 0, 0, 0, 0, 4763], [0, 0, 3695, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4853, 5141, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2522, 0, 0], [0, 0, 0, 0, 0, 4620, 0, 4481, 0], [0, 0, 0, 0, 0, 0, 1149, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4090, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [530, 4107, 4774, 5173, 6443, 6160, 1617, 4179, 4943], 'capacity': 11758, 'solution': [0, 1, 6, 8], 'obj': 19163.0, 'problem_type': 'QKP'}","[0, 1, 6, 8]",19163.0,"{'problem_type': 'QKP', 'num_items': 9, 'capacity': 11758, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8], 'linear_pairs': [{'item_id': 0, 'linear_profit': 630}, {'item_id': 1, 'linear_profit': 4384}, {'item_id': 2, 'linear_profit': 4996}, {'item_id': 3, 'linear_profit': 5085}, {'item_id': 4, 'linear_profit': 6448}, {'item_id': 5, 'linear_profit': 6189}, {'item_id': 6, 'linear_profit': 1522}, {'item_id': 7, 'linear_profit': 4177}, {'item_id': 8, 'linear_profit': 5108}], 'weight_pairs': [{'item_id': 0, 'weight': 530}, {'item_id': 1, 'weight': 4107}, {'item_id': 2, 'weight': 4774}, {'item_id': 3, 'weight': 5173}, {'item_id': 4, 'weight': 6443}, {'item_id': 5, 'weight': 6160}, {'item_id': 6, 'weight': 1617}, {'item_id': 7, 'weight': 4179}, {'item_id': 8, 'weight': 4943}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 1806}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 1805}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 1663}, {'item_i_id': 0, 'item_j_id': 8, 'quadratic_profit': 2756}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 4424}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 4763}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 3695}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 4853}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 5141}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 2522}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 4620}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 4481}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 1149}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 4090}]}","[0, 1, 6, 8]",32,json,0
QKP,QKP,"Around the neighborhood a landscaper is planning a compact front yard and sorting through which plants to use. Each shrub or tree takes up a certain area and brings a base curb-appeal score, and certain pairs have little synergy that adds to the overall look if both are planted. The choice is which plants to place so the total area stays within the plot’s limit (each plant chosen once), and the yard’s total curb appeal comes from adding each chosen plant’s own appeal plus any extra points for every planted pair that has synergy. The concrete list of plants and their stats is provided below.

They are considering 8 plants (0, 1, 2, 3, 4, 5, 6, 7) and must keep the total planted area within 11172.
Plant 0 contributes a base curb-appeal of 3236.
Plant 1 contributes a base curb-appeal of 967.
Plant 2 contributes a base curb-appeal of 984.
Plant 3 contributes a base curb-appeal of 894.
Plant 4 contributes a base curb-appeal of 1903.
Plant 5 contributes a base curb-appeal of 2323.
Plant 6 contributes a base curb-appeal of 3523.
Plant 7 contributes a base curb-appeal of 1290.
Plant 0 requires 2906 area units of the plot.
Plant 1 requires 637 area units of the plot.
Plant 2 requires 654 area units of the plot.
Plant 3 requires 564 area units of the plot.
Plant 4 requires 1573 area units of the plot.
Plant 5 requires 1993 area units of the plot.
Plant 6 requires 3193 area units of the plot.
Plant 7 requires 960 area units of the plot.
The pair 0 and 3 yields an extra 1996 curb-appeal when both are planted.
The pair 0 and 4 yields an extra 2808 curb-appeal when both are planted.
The pair 0 and 6 yields an extra 2909 curb-appeal when both are planted.
The pair 1 and 3 yields an extra 1353 curb-appeal when both are planted.
The pair 1 and 7 yields an extra 1123 curb-appeal when both are planted.
The landscaper will select from the 8 options to maximize total curb appeal while staying within 11172.

Oh, and when you're ready to give the final pick, a little JSON snippet like the one below is handy — just a relaxed, predictable shape so whoever reads it knows where to look.

{
  ""solution"": [<selected_plant_id>, <selected_plant_id>, ...]
}

Think of ""solution"" as the list of plant IDs you decide to put in the yard. The placeholders inside the brackets stand in for the exact plant identifiers from the instance input; this block is just a sketch of the expected shape, not the actual answer.

Please make sure to 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”.","{'linear_coeffs': [3236, 967, 984, 894, 1903, 2323, 3523, 1290], 'quadratic_coeffs': [[0, 0, 0, 1996, 2808, 0, 2909, 0], [0, 0, 0, 1353, 0, 0, 0, 1123], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [2906, 637, 654, 564, 1573, 1993, 3193, 960], 'capacity': 11172, 'solution': [0, 1, 2, 3, 4, 6, 7], 'obj': 22986.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 6, 7]",22986.0,"{'problem_type': 'QKP', 'num_items': 8, 'capacity': 11172, 'items': [0, 1, 2, 3, 4, 5, 6, 7], 'linear_pairs': [{'item_id': 0, 'linear_profit': 3236}, {'item_id': 1, 'linear_profit': 967}, {'item_id': 2, 'linear_profit': 984}, {'item_id': 3, 'linear_profit': 894}, {'item_id': 4, 'linear_profit': 1903}, {'item_id': 5, 'linear_profit': 2323}, {'item_id': 6, 'linear_profit': 3523}, {'item_id': 7, 'linear_profit': 1290}], 'weight_pairs': [{'item_id': 0, 'weight': 2906}, {'item_id': 1, 'weight': 637}, {'item_id': 2, 'weight': 654}, {'item_id': 3, 'weight': 564}, {'item_id': 4, 'weight': 1573}, {'item_id': 5, 'weight': 1993}, {'item_id': 6, 'weight': 3193}, {'item_id': 7, 'weight': 960}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 1996}, {'item_i_id': 0, 'item_j_id': 4, 'quadratic_profit': 2808}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 2909}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 1353}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 1123}]}","[0, 1, 2, 3, 4, 6, 7]",33,nl,0
QKP,QKP,"Recently a small dinner was planned and the host had to choose appetizers carefully. Every choice uses a bit of pantry/fridge space and gives a base amount of satisfaction, and a few particular combinations add extra joy when they’re paired. The challenge is to select which appetizers will fit into the available storage space without repeating items, and to get the highest possible total happiness, counted as the sum of the base satisfactions plus any bonuses for pairs that are both selected. The specific dishes and their space/satisfaction/pairing numbers are shown below.

There are 5 appetizer options (1, 2, 3, 4, 5), and the host has 623 units of storage space to work with.

| appetizer_id | base_satisfaction |
|---|---|
| 1 | 209 |
| 2 | 137 |
| 3 | 211 |
| 4 | 157 |
| 5 | 190 |

| appetizer_id | storage_space |
|---|---|
| 1 | 189 |
| 2 | 117 |
| 3 | 191 |
| 4 | 137 |
| 5 | 170 |

| appetizer_i_id | appetizer_j_id | pairing_bonus |
|---|---|---|
| 1 | 2 | 141 |
| 1 | 3 | 189 |
| 1 | 4 | 142 |
| 1 | 5 | 173 |
| 2 | 3 | 146 |
| 2 | 4 | 133 |
| 2 | 5 | 141 |
| 3 | 4 | 155 |
| 3 | 5 | 192 |
| 4 | 5 | 136 |

The host must pick appetizers that fit within the 623 to maximize total happiness.

If you want to send me your pick, just drop it in this simple JSON shape so it's easy to read:

{
  ""solution"": [<selected_appetizer_id>, <selected_appetizer_id>, ...]
}

Here ""solution"" is just the list of appetizer IDs you’re choosing for the spread — one entry per selected dish. Think of it like filling out a short form: put each dish’s identifier in the array, and that’s your selection. This block is only a sketch of the shape I expect, not the actual answer.

Please make sure to use the identifiers exactly as they appear in the instance input — don’t rename them or invent new labels.  
- ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'linear_coeffs': [209, 137, 211, 157, 190], 'quadratic_coeffs': [[146, 141, 189, 142, 173], [0, 124, 146, 133, 141], [0, 0, 169, 155, 192], [0, 0, 0, 142, 136], [0, 0, 0, 0, 0]], 'weights': [189, 117, 191, 137, 170], 'capacity': 623, 'solution': [1, 2, 3, 4], 'obj': 1598.0, 'problem_type': 'QKP'}","[1, 2, 3, 4]",1598.0,"{'problem_type': 'QKP', 'num_items': 5, 'capacity': 623, 'items': [1, 2, 3, 4, 5], 'linear_pairs': [{'item_id': 1, 'linear_profit': 209}, {'item_id': 2, 'linear_profit': 137}, {'item_id': 3, 'linear_profit': 211}, {'item_id': 4, 'linear_profit': 157}, {'item_id': 5, 'linear_profit': 190}], 'weight_pairs': [{'item_id': 1, 'weight': 189}, {'item_id': 2, 'weight': 117}, {'item_id': 3, 'weight': 191}, {'item_id': 4, 'weight': 137}, {'item_id': 5, 'weight': 170}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 146}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 141}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 189}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 142}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 173}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 124}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 146}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 133}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 141}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 169}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 155}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 192}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 142}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 136}]}","[2, 3, 4, 5]",34,markdown_table,1
QKP,QKP,"We’re helping a student prep for back-to-back classes and need to pick which textbooks and materials to carry in one backpack. Every item adds a base study value and has a weight, and certain combinations (like a textbook and its workbook) give an extra perk if both are brought. The trick is to get the highest possible study payoff by adding up the values of the chosen items and including any extra bonuses for compatible pairs, all without exceeding the backpack’s weight capacity and without duplicating items. The concrete details are listed below.

Here are those details for the 6 materials: ids 0, 1, 2, 3, 4, 5, backpack weight limit 457.

| material_id | base_study_value |
|---|---|
| 0 | 102 |
| 1 | 105 |
| 2 | 100 |
| 3 | 105 |
| 4 | 102 |
| 5 | 105 |

| material_id | material_weight |
|---|---|
| 0 | 100 |
| 1 | 102 |
| 2 | 100 |
| 3 | 105 |
| 4 | 100 |
| 5 | 100 |

| material_i_id | material_j_id | pair_study_bonus |
|---|---|---|
| 0 | 2 | 100 |
| 1 | 2 | 100 |
| 1 | 3 | 102 |
| 1 | 4 | 102 |
| 2 | 3 | 100 |
| 4 | 5 | 102 |

We'll pack to maximize study payoff while staying within the 457 limit.

When you send your pick, just drop it in a tiny JSON snippet so it's easy to read and check. Something casual like this works:

{
  ""solution"": [<selected_book_id>, <selected_book_id>, ...]
}

This just means: ""solution"" is the list of items (books/materials) you're planning to bring — put the identifiers for each selected item inside the array. Think of it like filling out a short form: list the exact IDs of the things you want in your backpack. The JSON above is only a sketch of the shape we expect, not the actual answer.

Please be sure to use the item 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”.","{'linear_coeffs': [102, 105, 100, 105, 102, 105], 'quadratic_coeffs': [[100, 0, 100, 0, 0, 0], [0, 102, 100, 102, 102, 0], [0, 0, 0, 100, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 101, 102], [0, 0, 0, 0, 0, 0]], 'weights': [100, 102, 100, 105, 100, 100], 'capacity': 457, 'solution': [1, 2, 3, 4], 'obj': 816.0, 'problem_type': 'QKP'}","[1, 2, 3, 4]",816.0,"{'problem_type': 'QKP', 'num_items': 6, 'capacity': 457, 'items': [0, 1, 2, 3, 4, 5], 'linear_pairs': [{'item_id': 0, 'linear_profit': 102}, {'item_id': 1, 'linear_profit': 105}, {'item_id': 2, 'linear_profit': 100}, {'item_id': 3, 'linear_profit': 105}, {'item_id': 4, 'linear_profit': 102}, {'item_id': 5, 'linear_profit': 105}], 'weight_pairs': [{'item_id': 0, 'weight': 100}, {'item_id': 1, 'weight': 102}, {'item_id': 2, 'weight': 100}, {'item_id': 3, 'weight': 105}, {'item_id': 4, 'weight': 100}, {'item_id': 5, 'weight': 100}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 100}, {'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 102}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 102}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 102}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 100}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 101}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 102}]}","[1, 2, 3, 4]",35,markdown_table,0
QKP,QKP,"We’re putting together a single promotional shelf and deciding which product lines to feature. Every item has a weight and an estimated sales figure, and certain combinations give a little bonus because they lift each other’s sales. The trick is to decide which items to put on that shelf so the combined weight doesn’t exceed what the shelf can hold, you only pick each product once (either it’s on the shelf or it isn’t), and the overall revenue — calculated by adding each chosen item’s expected sales plus any pairwise bonus sales — comes out as high as possible. The concrete details and numbers are listed below.

We have 10 product lines (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) and a shelf capacity of 376.

| product_id | estimated_sales |
|---|---|
| 1 | 215 |
| 2 | 155 |
| 3 | 191 |
| 4 | 269 |
| 5 | 102 |
| 6 | 154 |
| 7 | 141 |
| 8 | 199 |
| 9 | 191 |
| 10 | 289 |

| product_id | product_weight |
|---|---|
| 1 | 155 |
| 2 | 209 |
| 3 | 138 |
| 4 | 146 |
| 5 | 216 |
| 6 | 149 |
| 7 | 235 |
| 8 | 160 |
| 9 | 159 |
| 10 | 106 |

| product_i_id | product_j_id | cross_sale_bonus |
|---|---|---|
| 1 | 2 | 155 |
| 1 | 3 | 155 |
| 1 | 4 | 133 |
| 1 | 5 | 155 |
| 1 | 6 | 155 |
| 1 | 7 | 155 |
| 1 | 8 | 155 |
| 1 | 9 | 155 |
| 1 | 10 | 155 |
| 2 | 3 | 169 |
| 2 | 4 | 125 |
| 2 | 5 | 209 |
| 2 | 6 | 209 |
| 2 | 7 | 209 |
| 2 | 8 | 196 |
| 2 | 9 | 119 |
| 2 | 10 | 134 |
| 3 | 4 | 138 |
| 3 | 5 | 114 |
| 3 | 6 | 138 |
| 3 | 7 | 138 |
| 3 | 8 | 138 |
| 3 | 9 | 108 |
| 3 | 10 | 102 |
| 4 | 5 | 146 |
| 4 | 6 | 146 |
| 4 | 7 | 146 |
| 4 | 8 | 146 |
| 4 | 9 | 146 |
| 4 | 10 | 112 |
| 5 | 6 | 204 |
| 5 | 7 | 158 |
| 5 | 8 | 100 |
| 5 | 9 | 133 |
| 5 | 10 | 133 |
| 6 | 7 | 109 |
| 6 | 8 | 104 |
| 6 | 9 | 224 |
| 6 | 10 | 174 |
| 7 | 8 | 235 |
| 7 | 9 | 153 |
| 7 | 10 | 153 |
| 8 | 9 | 143 |
| 8 | 10 | 143 |
| 9 | 10 | 108 |

We’ll use these details to choose which products to place so we maximize total sales including any pairwise bonuses.

Also, to keep things machine-friendly, please put your chosen items into a simple JSON object like this:

{
  ""solution"": [<selected_product_id>, <selected_product_id>, ...]
}

Here ""solution"" is just the list of product IDs you want to feature on the shelf — one entry per chosen product. Think of it like a short form: the array lists which products go on the shelf, and the order doesn't matter. This JSON is only a sketch of the shape I need, not the final answer itself.

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”.""","{'linear_coeffs': [215, 155, 191, 269, 102, 154, 141, 199, 191, 289], 'quadratic_coeffs': [[155, 155, 155, 133, 155, 155, 155, 155, 155, 155], [0, 187, 169, 125, 209, 209, 209, 196, 119, 134], [0, 0, 138, 138, 114, 138, 138, 138, 108, 102], [0, 0, 0, 146, 146, 146, 146, 146, 146, 112], [0, 0, 0, 0, 216, 204, 158, 100, 133, 133], [0, 0, 0, 0, 0, 149, 109, 104, 224, 174], [0, 0, 0, 0, 0, 0, 148, 235, 153, 153], [0, 0, 0, 0, 0, 0, 0, 115, 143, 143], [0, 0, 0, 0, 0, 0, 0, 0, 248, 108], [0, 0, 0, 0, 0, 0, 0, 0, 0, 108]], 'weights': [155, 209, 138, 146, 216, 149, 235, 160, 159, 106], 'capacity': 376, 'solution': [3, 9], 'obj': 670.0, 'problem_type': 'QKP'}","[3, 9]",670.0,"{'problem_type': 'QKP', 'num_items': 10, 'capacity': 376, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 'linear_pairs': [{'item_id': 1, 'linear_profit': 215}, {'item_id': 2, 'linear_profit': 155}, {'item_id': 3, 'linear_profit': 191}, {'item_id': 4, 'linear_profit': 269}, {'item_id': 5, 'linear_profit': 102}, {'item_id': 6, 'linear_profit': 154}, {'item_id': 7, 'linear_profit': 141}, {'item_id': 8, 'linear_profit': 199}, {'item_id': 9, 'linear_profit': 191}, {'item_id': 10, 'linear_profit': 289}], 'weight_pairs': [{'item_id': 1, 'weight': 155}, {'item_id': 2, 'weight': 209}, {'item_id': 3, 'weight': 138}, {'item_id': 4, 'weight': 146}, {'item_id': 5, 'weight': 216}, {'item_id': 6, 'weight': 149}, {'item_id': 7, 'weight': 235}, {'item_id': 8, 'weight': 160}, {'item_id': 9, 'weight': 159}, {'item_id': 10, 'weight': 106}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 155}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 155}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 155}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 133}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 155}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 155}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 155}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 155}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 155}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 155}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 187}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 169}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 125}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 209}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 209}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 209}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 196}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 119}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 134}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 138}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 138}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 114}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 138}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 138}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 138}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 108}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 102}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 146}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 146}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 146}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 146}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 146}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 146}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 112}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 216}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 204}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 158}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 100}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 133}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 133}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 149}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 109}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 104}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 224}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 174}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 148}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 235}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 153}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 153}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 115}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 143}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 143}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 248}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 108}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 108}]}","[4, 10]",36,markdown_table,1
QKP,QKP,"On a tight schedule, the relief coordinator has one truck and several crates to pick from; each crate has its own weight and life‑saving worth, and a few specific pairs of crates give an extra lift to the relief effort if both are sent. The aim is straightforward in everyday terms: get the biggest total relief impact by adding each loaded crate’s value and any extra pair bonuses, while keeping the total load under the truck’s capacity — no crate can be taken more than once. The detailed list of crates, weights, values and pair bonuses follows below.

Below are the 9 crates (A, B, C, D, E, F, G, H, I) and the truck's weight capacity 874.
Crate A has base life‑saving value 142.
Crate B has base life‑saving value 147.
Crate C has base life‑saving value 143.
Crate D has base life‑saving value 144.
Crate E has base life‑saving value 118.
Crate F has base life‑saving value 136.
Crate G has base life‑saving value 134.
Crate H has base life‑saving value 125.
Crate I has base life‑saving value 147.
Crate A weighs 127.
Crate B weighs 132.
Crate C weighs 128.
Crate D weighs 129.
Crate E weighs 103.
Crate F weighs 121.
Crate G weighs 119.
Crate H weighs 110.
Crate I weighs 132.
If both crate A and crate B are loaded, they add 110 extra life‑saving value.
If both crate A and crate C are loaded, they add 120 extra life‑saving value.
If both crate A and crate D are loaded, they add 106 extra life‑saving value.
If both crate A and crate E are loaded, they add 132 extra life‑saving value.
If both crate A and crate F are loaded, they add 114 extra life‑saving value.
If both crate A and crate G are loaded, they add 104 extra life‑saving value.
If both crate A and crate H are loaded, they add 119 extra life‑saving value.
If both crate A and crate I are loaded, they add 128 extra life‑saving value.
If both crate B and crate C are loaded, they add 115 extra life‑saving value.
If both crate B and crate D are loaded, they add 119 extra life‑saving value.
If both crate B and crate E are loaded, they add 132 extra life‑saving value.
If both crate B and crate F are loaded, they add 125 extra life‑saving value.
If both crate B and crate G are loaded, they add 104 extra life‑saving value.
If both crate B and crate H are loaded, they add 138 extra life‑saving value.
If both crate B and crate I are loaded, they add 104 extra life‑saving value.
If both crate C and crate D are loaded, they add 121 extra life‑saving value.
If both crate C and crate E are loaded, they add 119 extra life‑saving value.
If both crate C and crate F are loaded, they add 104 extra life‑saving value.
If both crate C and crate G are loaded, they add 134 extra life‑saving value.
If both crate C and crate H are loaded, they add 146 extra life‑saving value.
If both crate C and crate I are loaded, they add 128 extra life‑saving value.
If both crate D and crate E are loaded, they add 116 extra life‑saving value.
If both crate D and crate F are loaded, they add 115 extra life‑saving value.
If both crate D and crate G are loaded, they add 100 extra life‑saving value.
If both crate D and crate H are loaded, they add 121 extra life‑saving value.
If both crate D and crate I are loaded, they add 139 extra life‑saving value.
If both crate E and crate F are loaded, they add 105 extra life‑saving value.
If both crate E and crate G are loaded, they add 137 extra life‑saving value.
If both crate E and crate H are loaded, they add 130 extra life‑saving value.
If both crate E and crate I are loaded, they add 144 extra life‑saving value.
If both crate F and crate G are loaded, they add 114 extra life‑saving value.
If both crate F and crate H are loaded, they add 107 extra life‑saving value.
If both crate F and crate I are loaded, they add 112 extra life‑saving value.
If both crate G and crate H are loaded, they add 112 extra life‑saving value.
If both crate G and crate I are loaded, they add 125 extra life‑saving value.
If both crate H and crate I are loaded, they add 115 extra life‑saving value.
The relief coordinator will select crates so total weight stays within 874 while maximizing life‑saving impact.

If you want to hand me the picks in a machine-friendly way, a little JSON snippet like this works nicely — just list the crate IDs you plan to load.

{
  ""solution"": [<selected_crate_id>, <selected_crate_id>, ...]
}

Think of it like a simple form: ""solution"" is the list of crates you choose to put on the truck, and each angle-bracket entry is where you drop the exact crate identifier from the instance. This JSON is just a sketch of the shape I expect — not the actual chosen crates.

Please be careful 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”.""","{'linear_coeffs': [142, 147, 143, 144, 118, 136, 134, 125, 147], 'quadratic_coeffs': [[106, 110, 120, 106, 132, 114, 104, 119, 128], [0, 140, 115, 119, 132, 125, 104, 138, 104], [0, 0, 146, 121, 119, 104, 134, 146, 128], [0, 0, 0, 111, 116, 115, 100, 121, 139], [0, 0, 0, 0, 107, 105, 137, 130, 144], [0, 0, 0, 0, 0, 148, 114, 107, 112], [0, 0, 0, 0, 0, 0, 123, 112, 125], [0, 0, 0, 0, 0, 0, 0, 126, 115], [0, 0, 0, 0, 0, 0, 0, 0, 136]], 'weights': [127, 132, 128, 129, 103, 121, 119, 110, 132], 'capacity': 874, 'solution': [0, 1, 2, 3, 4, 7, 8], 'obj': 3568.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 7, 8]",3568.0,"{'problem_type': 'QKP', 'num_items': 9, 'capacity': 874, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 142}, {'item_id': 'B', 'linear_profit': 147}, {'item_id': 'C', 'linear_profit': 143}, {'item_id': 'D', 'linear_profit': 144}, {'item_id': 'E', 'linear_profit': 118}, {'item_id': 'F', 'linear_profit': 136}, {'item_id': 'G', 'linear_profit': 134}, {'item_id': 'H', 'linear_profit': 125}, {'item_id': 'I', 'linear_profit': 147}], 'weight_pairs': [{'item_id': 'A', 'weight': 127}, {'item_id': 'B', 'weight': 132}, {'item_id': 'C', 'weight': 128}, {'item_id': 'D', 'weight': 129}, {'item_id': 'E', 'weight': 103}, {'item_id': 'F', 'weight': 121}, {'item_id': 'G', 'weight': 119}, {'item_id': 'H', 'weight': 110}, {'item_id': 'I', 'weight': 132}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 106}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 110}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 120}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 106}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 132}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 114}, {'item_i_id': 'A', 'item_j_id': 'G', 'quadratic_profit': 104}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 119}, {'item_i_id': 'A', 'item_j_id': 'I', 'quadratic_profit': 128}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 140}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 115}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 119}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 132}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 125}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 104}, {'item_i_id': 'B', 'item_j_id': 'H', 'quadratic_profit': 138}, {'item_i_id': 'B', 'item_j_id': 'I', 'quadratic_profit': 104}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 146}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 121}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 119}, {'item_i_id': 'C', 'item_j_id': 'F', 'quadratic_profit': 104}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 134}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 146}, {'item_i_id': 'C', 'item_j_id': 'I', 'quadratic_profit': 128}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 111}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 116}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 115}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 100}, {'item_i_id': 'D', 'item_j_id': 'H', 'quadratic_profit': 121}, {'item_i_id': 'D', 'item_j_id': 'I', 'quadratic_profit': 139}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 107}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 105}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 137}, {'item_i_id': 'E', 'item_j_id': 'H', 'quadratic_profit': 130}, {'item_i_id': 'E', 'item_j_id': 'I', 'quadratic_profit': 144}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 148}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 114}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 107}, {'item_i_id': 'F', 'item_j_id': 'I', 'quadratic_profit': 112}, {'item_i_id': 'G', 'item_j_id': 'G', 'quadratic_profit': 123}, {'item_i_id': 'G', 'item_j_id': 'H', 'quadratic_profit': 112}, {'item_i_id': 'G', 'item_j_id': 'I', 'quadratic_profit': 125}, {'item_i_id': 'H', 'item_j_id': 'H', 'quadratic_profit': 126}, {'item_i_id': 'H', 'item_j_id': 'I', 'quadratic_profit': 115}, {'item_i_id': 'I', 'item_j_id': 'I', 'quadratic_profit': 136}]}","['A', 'B', 'C', 'D', 'E', 'H', 'I']",37,nl,names
QKP,QKP,"At the community theater, the stage manager is figuring out which flats, props and backdrops to load into the single available truck. Every item uses up some of the truck’s capacity and brings a rental value, and a few pairings create extra payoff when both pieces arrive because they enhance the scene together. The sensible plan is to pick items that don’t overload the truck and that make the most total return — that total being each chosen piece’s rental added together plus any extra payoffs from pairs that are both taken. Each piece is taken once or left behind; the exact item weights, rental values and pair bonuses are listed below.

Listed below are the 8 pieces (A, B, C, D, E, F, G, H) and the truck's load limit 441.
Piece A brings a rental of 100.
Piece B brings a rental of 101.
Piece C brings a rental of 105.
Piece D brings a rental of 104.
Piece E brings a rental of 100.
Piece F brings a rental of 103.
Piece G brings a rental of 100.
Piece H brings a rental of 103.
Piece A occupies 104 capacity on the truck.
Piece B occupies 103 capacity on the truck.
Piece C occupies 101 capacity on the truck.
Piece D occupies 105 capacity on the truck.
Piece E occupies 104 capacity on the truck.
Piece F occupies 105 capacity on the truck.
Piece G occupies 101 capacity on the truck.
Piece H occupies 100 capacity on the truck.
Pieces A and B together add an extra 102 payoff.
Pieces A and C together add an extra 104 payoff.
Pieces A and D together add an extra 102 payoff.
Pieces A and E together add an extra 101 payoff.
Pieces A and F together add an extra 104 payoff.
Pieces A and G together add an extra 102 payoff.
Pieces A and H together add an extra 103 payoff.
Pieces B and C together add an extra 103 payoff.
Pieces B and D together add an extra 101 payoff.
Pieces B and E together add an extra 103 payoff.
Pieces B and F together add an extra 101 payoff.
Pieces B and G together add an extra 102 payoff.
Pieces B and H together add an extra 103 payoff.
Pieces C and D together add an extra 101 payoff.
Pieces C and E together add an extra 101 payoff.
Pieces C and F together add an extra 100 payoff.
Pieces C and G together add an extra 100 payoff.
Pieces C and H together add an extra 105 payoff.
Pieces D and E together add an extra 102 payoff.
Pieces D and F together add an extra 100 payoff.
Pieces D and G together add an extra 102 payoff.
Pieces D and H together add an extra 100 payoff.
Pieces E and F together add an extra 102 payoff.
Pieces E and G together add an extra 102 payoff.
Pieces E and H together add an extra 101 payoff.
Pieces F and G together add an extra 102 payoff.
Pieces F and H together add an extra 101 payoff.
Pieces G and H together add an extra 101 payoff.
The stage manager will choose pieces to maximize total rental while staying within the 441 truck limit.

If you want to return your pick in a tidy way, just use this simple JSON shape so it's easy to read and check:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

Here the ""solution"" array is where you list the ids of the flats, props and backdrops you want loaded into the truck — one id per chosen piece. Think of it like filling out a short form: list the items you’re taking, and nothing else. This JSON is just a sketch of the shape I expect, not the actual answer.

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”.""","{'linear_coeffs': [100, 101, 105, 104, 100, 103, 100, 103], 'quadratic_coeffs': [[101, 102, 104, 102, 101, 104, 102, 103], [0, 103, 103, 101, 103, 101, 102, 103], [0, 0, 101, 101, 101, 100, 100, 105], [0, 0, 0, 102, 102, 100, 102, 100], [0, 0, 0, 0, 100, 102, 102, 101], [0, 0, 0, 0, 0, 103, 102, 101], [0, 0, 0, 0, 0, 0, 102, 101], [0, 0, 0, 0, 0, 0, 0, 103]], 'weights': [104, 103, 101, 105, 104, 105, 101, 100], 'capacity': 441, 'solution': [0, 1, 2, 7], 'obj': 1029.0, 'problem_type': 'QKP'}","[0, 1, 2, 7]",1029.0,"{'problem_type': 'QKP', 'num_items': 8, 'capacity': 441, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 100}, {'item_id': 'B', 'linear_profit': 101}, {'item_id': 'C', 'linear_profit': 105}, {'item_id': 'D', 'linear_profit': 104}, {'item_id': 'E', 'linear_profit': 100}, {'item_id': 'F', 'linear_profit': 103}, {'item_id': 'G', 'linear_profit': 100}, {'item_id': 'H', 'linear_profit': 103}], 'weight_pairs': [{'item_id': 'A', 'weight': 104}, {'item_id': 'B', 'weight': 103}, {'item_id': 'C', 'weight': 101}, {'item_id': 'D', 'weight': 105}, {'item_id': 'E', 'weight': 104}, {'item_id': 'F', 'weight': 105}, {'item_id': 'G', 'weight': 101}, {'item_id': 'H', 'weight': 100}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 101}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 102}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 104}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 102}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 101}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 104}, {'item_i_id': 'A', 'item_j_id': 'G', 'quadratic_profit': 102}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 103}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 103}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 103}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 101}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 103}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 101}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 102}, {'item_i_id': 'B', 'item_j_id': 'H', 'quadratic_profit': 103}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 101}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 101}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 101}, {'item_i_id': 'C', 'item_j_id': 'F', 'quadratic_profit': 100}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 100}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 105}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 102}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 102}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 100}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 102}, {'item_i_id': 'D', 'item_j_id': 'H', 'quadratic_profit': 100}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 100}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 102}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 102}, {'item_i_id': 'E', 'item_j_id': 'H', 'quadratic_profit': 101}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 103}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 102}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 101}, {'item_i_id': 'G', 'item_j_id': 'G', 'quadratic_profit': 102}, {'item_i_id': 'G', 'item_j_id': 'H', 'quadratic_profit': 101}, {'item_i_id': 'H', 'item_j_id': 'H', 'quadratic_profit': 103}]}","['A', 'B', 'C', 'H']",38,nl,names
QKP,QKP,"We’re curating a weekend gallery and can only ship what fits in one crate. Every artwork comes with an estimated sale value and takes up some space, and certain combinations of paintings boost interest when they’re displayed together, giving extra value. The idea is to pick which pieces to include so the crate stays within its capacity, no piece is repeated, and the total expected return (individual values plus any extra for specific pairs) is as large as it can be. The concrete weights, values, and synergy numbers are shown below.

{
  ""total_artworks"": 6,
  ""crate_capacity"": 609,
  ""artwork_ids"": [
    1,
    2,
    3,
    4,
    5,
    6
  ],
  ""linear"": [
    {
      ""artwork_id"": 1,
      ""expected_sale_value"": 191
    },
    {
      ""artwork_id"": 2,
      ""expected_sale_value"": 216
    },
    {
      ""artwork_id"": 3,
      ""expected_sale_value"": 184
    },
    {
      ""artwork_id"": 4,
      ""expected_sale_value"": 191
    },
    {
      ""artwork_id"": 5,
      ""expected_sale_value"": 294
    },
    {
      ""artwork_id"": 6,
      ""expected_sale_value"": 276
    }
  ],
  ""weights"": [
    {
      ""artwork_id"": 1,
      ""space_required"": 161
    },
    {
      ""artwork_id"": 2,
      ""space_required"": 186
    },
    {
      ""artwork_id"": 3,
      ""space_required"": 154
    },
    {
      ""artwork_id"": 4,
      ""space_required"": 161
    },
    {
      ""artwork_id"": 5,
      ""space_required"": 264
    },
    {
      ""artwork_id"": 6,
      ""space_required"": 246
    }
  ],
  ""quadratic"": [
    {
      ""artwork_i_id"": 1,
      ""artwork_j_id"": 2,
      ""synergy_bonus_value"": 297
    },
    {
      ""artwork_i_id"": 1,
      ""artwork_j_id"": 3,
      ""synergy_bonus_value"": 260
    },
    {
      ""artwork_i_id"": 1,
      ""artwork_j_id"": 4,
      ""synergy_bonus_value"": 181
    },
    {
      ""artwork_i_id"": 1,
      ""artwork_j_id"": 5,
      ""synergy_bonus_value"": 125
    },
    {
      ""artwork_i_id"": 1,
      ""artwork_j_id"": 6,
      ""synergy_bonus_value"": 104
    },
    {
      ""artwork_i_id"": 2,
      ""artwork_j_id"": 3,
      ""synergy_bonus_value"": 275
    },
    {
      ""artwork_i_id"": 2,
      ""artwork_j_id"": 4,
      ""synergy_bonus_value"": 145
    },
    {
      ""artwork_i_id"": 2,
      ""artwork_j_id"": 5,
      ""synergy_bonus_value"": 144
    },
    {
      ""artwork_i_id"": 2,
      ""artwork_j_id"": 6,
      ""synergy_bonus_value"": 189
    },
    {
      ""artwork_i_id"": 3,
      ""artwork_j_id"": 4,
      ""synergy_bonus_value"": 165
    },
    {
      ""artwork_i_id"": 3,
      ""artwork_j_id"": 5,
      ""synergy_bonus_value"": 138
    },
    {
      ""artwork_i_id"": 3,
      ""artwork_j_id"": 6,
      ""synergy_bonus_value"": 250
    },
    {
      ""artwork_i_id"": 4,
      ""artwork_j_id"": 5,
      ""synergy_bonus_value"": 217
    },
    {
      ""artwork_i_id"": 4,
      ""artwork_j_id"": 6,
      ""synergy_bonus_value"": 160
    },
    {
      ""artwork_i_id"": 5,
      ""artwork_j_id"": 6,
      ""synergy_bonus_value"": 246
    }
  ]
}

If you want to give me your final pick, toss it in a tiny JSON snippet like this so it's easy to read:

{
  ""solution"": [<selected_artwork_id>, <selected_artwork_id>, ...]
}

""solution"" is just a list of the artworks you're packing into the crate — each entry should be the identifier for a piece from the instance. Think of this JSON as a little form: the array holds the chosen items, and that's it. It's only a sketch of the shape I expect, not the actual answer.

Please make 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”.""","{'linear_coeffs': [191, 216, 184, 191, 294, 276], 'quadratic_coeffs': [[160, 297, 260, 181, 125, 104], [0, 137, 275, 145, 144, 189], [0, 0, 178, 165, 138, 250], [0, 0, 0, 192, 217, 160], [0, 0, 0, 0, 142, 246], [0, 0, 0, 0, 0, 199]], 'weights': [161, 186, 154, 161, 264, 246], 'capacity': 609, 'solution': [0, 1, 2], 'obj': 1423.0, 'problem_type': 'QKP'}","[0, 1, 2]",1423.0,"{'problem_type': 'QKP', 'num_items': 6, 'capacity': 609, 'items': [1, 2, 3, 4, 5, 6], 'linear_pairs': [{'item_id': 1, 'linear_profit': 191}, {'item_id': 2, 'linear_profit': 216}, {'item_id': 3, 'linear_profit': 184}, {'item_id': 4, 'linear_profit': 191}, {'item_id': 5, 'linear_profit': 294}, {'item_id': 6, 'linear_profit': 276}], 'weight_pairs': [{'item_id': 1, 'weight': 161}, {'item_id': 2, 'weight': 186}, {'item_id': 3, 'weight': 154}, {'item_id': 4, 'weight': 161}, {'item_id': 5, 'weight': 264}, {'item_id': 6, 'weight': 246}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 160}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 297}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 260}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 181}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 125}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 104}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 137}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 275}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 145}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 144}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 189}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 178}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 165}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 138}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 250}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 192}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 217}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 160}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 142}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 246}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 199}]}","[1, 2, 3]",39,json,1
QKP,QKP,"Recently the branch got a pile of donations and only so much shelf room to display them. Each book takes up some space and has its own circulation value, and some combinations of titles actually increase traffic when they’re displayed together. The plan is to pick which books to show so the shelf doesn’t overflow and the total payoff — the sum of the individual book benefits plus any extra pairwise bonuses when those two are both chosen — is maximized, with no duplicates allowed. The specific book sizes, benefits, and pairwise boosts follow below.

# num_books=10
# shelf_capacity=37245
# book_ids=1, 2, 3, 4, 5, 6, 7, 8, 9, 10

book_id,book_benefit
1,3316
2,5357
3,3658
4,5455
5,4447
6,6253
7,5676
8,5940
9,4257
10,2227

book_id,book_size
1,2666
2,4707
3,3008
4,4805
5,3797
6,5603
7,5026
8,5290
9,3607
10,1577

book_i_id,book_j_id,pair_bonus
1,2,2666
1,3,2666
1,5,2666
1,6,870
1,7,2425
1,8,3261
1,9,575
1,10,5832
2,5,3938
2,6,3293
2,9,3261
2,10,3261
3,5,3008
3,6,3008
3,8,342
3,9,1811
4,5,4805
4,9,361
4,10,3246
5,7,1402
5,8,1402
6,7,3322
6,8,936
6,9,1465
6,10,4817
7,8,1571
7,9,917

When you send your pick, just stick to this simple JSON layout so I know which books you chose — nothing fancy, just the list of book ids.

{
  ""solution"": [<book_id>, <book_id>, ...]
}

Pretty straightforward: ""solution"" is the list of the books you want to put on the shelf, each entry being the book's identifier from the instance. Think of it like filling in a short form that names the books you’re displaying.

This JSON is only a sketch of the shape I expect you to return — not the final answer itself.

Please make sure to use the identifiers exactly as they appear in the instance input — no renaming, and don’t introduce any 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”.","{'linear_coeffs': [3316, 5357, 3658, 5455, 4447, 6253, 5676, 5940, 4257, 2227], 'quadratic_coeffs': [[0, 2666, 2666, 0, 2666, 870, 2425, 3261, 575, 5832], [0, 4150, 0, 0, 3938, 3293, 0, 0, 3261, 3261], [0, 0, 2229, 0, 3008, 3008, 0, 342, 1811, 0], [0, 0, 0, 1497, 4805, 0, 0, 0, 361, 3246], [0, 0, 0, 0, 3797, 0, 1402, 1402, 0, 0], [0, 0, 0, 0, 0, 3322, 3322, 936, 1465, 4817], [0, 0, 0, 0, 0, 0, 0, 1571, 917, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [2666, 4707, 3008, 4805, 3797, 5603, 5026, 5290, 3607, 1577], 'capacity': 37245, 'solution': [0, 1, 2, 3, 4, 5, 6, 7, 9], 'obj': 101066.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 5, 6, 7, 9]",101066.0,"{'problem_type': 'QKP', 'num_items': 10, 'capacity': 37245, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 'linear_pairs': [{'item_id': 1, 'linear_profit': 3316}, {'item_id': 2, 'linear_profit': 5357}, {'item_id': 3, 'linear_profit': 3658}, {'item_id': 4, 'linear_profit': 5455}, {'item_id': 5, 'linear_profit': 4447}, {'item_id': 6, 'linear_profit': 6253}, {'item_id': 7, 'linear_profit': 5676}, {'item_id': 8, 'linear_profit': 5940}, {'item_id': 9, 'linear_profit': 4257}, {'item_id': 10, 'linear_profit': 2227}], 'weight_pairs': [{'item_id': 1, 'weight': 2666}, {'item_id': 2, 'weight': 4707}, {'item_id': 3, 'weight': 3008}, {'item_id': 4, 'weight': 4805}, {'item_id': 5, 'weight': 3797}, {'item_id': 6, 'weight': 5603}, {'item_id': 7, 'weight': 5026}, {'item_id': 8, 'weight': 5290}, {'item_id': 9, 'weight': 3607}, {'item_id': 10, 'weight': 1577}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 2666}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 2666}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 2666}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 870}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 2425}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 3261}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 575}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 5832}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 4150}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 3938}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 3293}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 3261}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 3261}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 2229}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 3008}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 3008}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 342}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 1811}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 1497}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 4805}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 361}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 3246}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 3797}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 1402}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 1402}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 3322}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 3322}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 936}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 1465}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 4817}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 1571}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 917}]}","[1, 2, 3, 4, 5, 6, 7, 8, 10]",40,csv,1
QKP,QKP,"I’m prepping for a pop-up plant sale and trying to pack a single flatbed without going over its maximum load, so there’s a decision to make about which pots to bring. Every pot contributes a base margin, and a few specific pairings add extra margin if both make the trip. The aim is to pick the set of plants whose combined weight stays under the limit and that gives the largest possible take — by summing up each chosen pot’s margin plus any extra bonuses for those special pairs — and each plant is either taken once or left. The concrete details (plant list, weights, margins, pair bonuses) are given below.

{
  ""num_plants"": 5,
  ""flatbed_max_load"": 4961,
  ""plant_ids"": [
    ""A"",
    ""B"",
    ""C"",
    ""D"",
    ""E""
  ],
  ""linear"": [
    {
      ""plant_id"": ""A"",
      ""base_margin"": 876
    },
    {
      ""plant_id"": ""B"",
      ""base_margin"": 2080
    },
    {
      ""plant_id"": ""C"",
      ""base_margin"": 3185
    },
    {
      ""plant_id"": ""D"",
      ""base_margin"": 959
    },
    {
      ""plant_id"": ""E"",
      ""base_margin"": 2428
    }
  ],
  ""weights"": [
    {
      ""plant_id"": ""A"",
      ""pot_weight"": 1898
    },
    {
      ""plant_id"": ""B"",
      ""pot_weight"": 903
    },
    {
      ""plant_id"": ""C"",
      ""pot_weight"": 1326
    },
    {
      ""plant_id"": ""D"",
      ""pot_weight"": 508
    },
    {
      ""plant_id"": ""E"",
      ""pot_weight"": 774
    }
  ],
  ""quadratic"": [
    {
      ""plant_i_id"": ""A"",
      ""plant_j_id"": ""B"",
      ""pair_bonus_margin"": 338
    },
    {
      ""plant_i_id"": ""B"",
      ""plant_j_id"": ""E"",
      ""pair_bonus_margin"": 1144
    },
    {
      ""plant_i_id"": ""C"",
      ""plant_j_id"": ""E"",
      ""pair_bonus_margin"": 2897
    }
  ]
}

Also, when you give your pick, please put it in a tiny JSON snippet like this so it's easy to read and check:

{
  ""solution"": [""<plant_id>"", ""<plant_id>"", ...]
}

Here ""solution"" is just a list of the plant identifiers you're planning to load onto the flatbed. Think of it as the simple answer form: the names (IDs) of the pots you’ll bring, nothing else. This is just a sketch of the shape I expect, not the actual packing decision.

Please 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”.""","{'linear_coeffs': [876, 2080, 3185, 959, 2428], 'quadratic_coeffs': [[0, 338, 0, 0, 0], [0, 0, 0, 0, 1144], [0, 0, 0, 0, 2897], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]], 'weights': [1898, 903, 1326, 508, 774], 'capacity': 4961, 'solution': [0, 1, 2, 4], 'obj': 12948.0, 'problem_type': 'QKP'}","[0, 1, 2, 4]",12948.0,"{'problem_type': 'QKP', 'num_items': 5, 'capacity': 4961, 'items': ['A', 'B', 'C', 'D', 'E'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 876}, {'item_id': 'B', 'linear_profit': 2080}, {'item_id': 'C', 'linear_profit': 3185}, {'item_id': 'D', 'linear_profit': 959}, {'item_id': 'E', 'linear_profit': 2428}], 'weight_pairs': [{'item_id': 'A', 'weight': 1898}, {'item_id': 'B', 'weight': 903}, {'item_id': 'C', 'weight': 1326}, {'item_id': 'D', 'weight': 508}, {'item_id': 'E', 'weight': 774}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 338}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 1144}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 2897}]}","['A', 'B', 'C', 'E']",41,json,names
QKP,QKP,"I’m hauling a crate of vinyl to a gig and need to pick what goes in it. Each record has its own crowd-pleasing score, and certain pairs of records boost the vibe even more if both are in the set — the total payoff is the sum of each record’s score plus any extra bonuses from pairs that end up together. The catch is the crate can only hold so much weight and there’s only one copy of each record, so the aim is to get the highest total score while keeping the total weight under the transport limit. The exact records, weights, and pair bonuses are listed below.

# total_records=10
# crate_capacity=36116
# record_ids=0, 1, 2, 3, 4, 5, 6, 7, 8, 9

record_id,crowd_pleasing_score
0,6474
1,4790
2,3477
3,5662
4,5675
5,441
6,2791
7,2788
8,2557
9,6084

record_id,record_weight
0,6119
1,5983
2,1382
3,3123
4,3341
5,437
6,3582
7,3395
8,5435
9,3335

record_i_id,record_j_id,pair_synergy_bonus
0,1,5106
0,5,3582
0,6,2761
1,3,3664
1,9,4669
2,4,1382
2,6,1382
2,8,1558
2,9,3109
3,6,4914
3,7,2339
4,5,2563
5,9,755

Also, when you tell me which records make the cut, a tiny JSON snippet like this is the easiest way to send it back — nice and simple:

{
  ""solution"": [<record_id>, <record_id>, ...]
}

Here, ""solution"" is just a list of the record IDs you want in the crate — nothing fancy, just the exact identifiers of the records you picked. Think of it as the checklist: each placeholder like <record_id> stands for one record you include.

This is just the shape I need, not your final score or any calculations — just the chosen IDs. Please use the item 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”.""","{'linear_coeffs': [6474, 4790, 3477, 5662, 5675, 441, 2791, 2788, 2557, 6084], 'quadratic_coeffs': [[0, 5106, 0, 0, 0, 3582, 2761, 0, 0, 0], [0, 0, 0, 3664, 0, 0, 0, 0, 0, 4669], [0, 0, 646, 0, 1382, 0, 1382, 0, 1558, 3109], [0, 0, 0, 0, 0, 0, 4914, 2339, 0, 0], [0, 0, 0, 0, 0, 2563, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4236, 0, 0, 0, 755], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 281, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [6119, 5983, 1382, 3123, 3341, 437, 3582, 3395, 5435, 3335], 'capacity': 36116, 'solution': [0, 1, 2, 3, 4, 5, 6, 7, 9], 'obj': 74408.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 5, 6, 7, 9]",74408.0,"{'problem_type': 'QKP', 'num_items': 10, 'capacity': 36116, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 'linear_pairs': [{'item_id': 0, 'linear_profit': 6474}, {'item_id': 1, 'linear_profit': 4790}, {'item_id': 2, 'linear_profit': 3477}, {'item_id': 3, 'linear_profit': 5662}, {'item_id': 4, 'linear_profit': 5675}, {'item_id': 5, 'linear_profit': 441}, {'item_id': 6, 'linear_profit': 2791}, {'item_id': 7, 'linear_profit': 2788}, {'item_id': 8, 'linear_profit': 2557}, {'item_id': 9, 'linear_profit': 6084}], 'weight_pairs': [{'item_id': 0, 'weight': 6119}, {'item_id': 1, 'weight': 5983}, {'item_id': 2, 'weight': 1382}, {'item_id': 3, 'weight': 3123}, {'item_id': 4, 'weight': 3341}, {'item_id': 5, 'weight': 437}, {'item_id': 6, 'weight': 3582}, {'item_id': 7, 'weight': 3395}, {'item_id': 8, 'weight': 5435}, {'item_id': 9, 'weight': 3335}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 1, 'quadratic_profit': 5106}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 3582}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 2761}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 3664}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 4669}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 646}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 1382}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 1382}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 1558}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 3109}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 4914}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 2339}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 2563}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 4236}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 755}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 281}]}","[0, 1, 2, 3, 4, 5, 6, 7, 9]",42,csv,0
QKP,QKP,"We picture an aid team packing a single cargo box: there are a bunch of different aid kits, each with its own weight and a base amount of good it will do once delivered. On top of that, certain combinations of two kits create extra effectiveness when both travel together. The task is simple in idea — try to get as much total relief as possible by adding up each selected kit’s base impact and any extra pairwise bonuses — but stay within the box’s weight limit. Only whole kits are available and you can’t pick the same kit twice. The exact weights, impacts, and combo bonuses appear below.

We list 7 kits as A, B, C, D, E, F, G, and we can pack up to 344 total weight.

| kit_id | base_impact |
|---|---|
| A | 115 |
| B | 113 |
| C | 114 |
| D | 113 |
| E | 113 |
| F | 115 |
| G | 112 |

| kit_id | kit_weight |
|---|---|
| A | 105 |
| B | 103 |
| C | 104 |
| D | 103 |
| E | 103 |
| F | 105 |
| G | 102 |

| kit_i_id | kit_j_id | pairwise_boost |
|---|---|---|
| A | B | 101 |
| A | C | 102 |
| A | D | 101 |
| A | E | 100 |
| A | F | 102 |
| A | G | 102 |
| B | C | 100 |
| B | D | 103 |
| B | E | 103 |
| B | F | 100 |
| B | G | 100 |
| C | D | 102 |
| C | E | 100 |
| C | F | 100 |
| C | G | 100 |
| D | E | 105 |
| D | F | 101 |
| D | G | 100 |
| E | F | 100 |
| E | G | 103 |
| F | G | 100 |

We must pack only whole kits and keep the total weight within 344.

Oh—and before you send your pick, it'd be great if you put it in a tiny JSON note like this so it's easy to read by the rest of the team:

{
  ""solution"": [<selected_kit_id>, <selected_kit_id>, ...]
}

Here ""solution"" is just the list of which kits you want in the box — put each kit's identifier in the array. This is just a sketch of the shape I expect, not your final answer; replace the placeholders with the actual identifiers from the instance.

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”.""","{'linear_coeffs': [115, 113, 114, 113, 113, 115, 112], 'quadratic_coeffs': [[105, 101, 102, 101, 100, 102, 102], [0, 103, 100, 103, 103, 100, 100], [0, 0, 104, 102, 100, 100, 100], [0, 0, 0, 100, 105, 101, 100], [0, 0, 0, 0, 100, 100, 103], [0, 0, 0, 0, 0, 102, 100], [0, 0, 0, 0, 0, 0, 0]], 'weights': [105, 103, 104, 103, 103, 105, 102], 'capacity': 344, 'solution': [0, 2, 5], 'obj': 648.0, 'problem_type': 'QKP'}","[0, 2, 5]",648.0,"{'problem_type': 'QKP', 'num_items': 7, 'capacity': 344, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 115}, {'item_id': 'B', 'linear_profit': 113}, {'item_id': 'C', 'linear_profit': 114}, {'item_id': 'D', 'linear_profit': 113}, {'item_id': 'E', 'linear_profit': 113}, {'item_id': 'F', 'linear_profit': 115}, {'item_id': 'G', 'linear_profit': 112}], 'weight_pairs': [{'item_id': 'A', 'weight': 105}, {'item_id': 'B', 'weight': 103}, {'item_id': 'C', 'weight': 104}, {'item_id': 'D', 'weight': 103}, {'item_id': 'E', 'weight': 103}, {'item_id': 'F', 'weight': 105}, {'item_id': 'G', 'weight': 102}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 105}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 101}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 102}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 101}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 100}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 102}, {'item_i_id': 'A', 'item_j_id': 'G', 'quadratic_profit': 102}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 103}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 100}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 103}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 103}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 100}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 100}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 104}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 102}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 100}, {'item_i_id': 'C', 'item_j_id': 'F', 'quadratic_profit': 100}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 100}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 100}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 105}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 101}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 100}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 100}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 100}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 103}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 102}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 100}]}","['A', 'C', 'F']",43,markdown_table,names
QKP,QKP,"I was loading up a mobile repair van and had to pick which tools to bring so the total weight fits in the back. Each tool can help on its own with a certain service value, and some tool pairs make jobs faster or smoother — adding a little extra value if both are on board. The trick is to pick a combination that gives the highest possible total service value (add up every chosen tool’s individual value, and include any extra bonuses for pairs that are both included) while making sure the summed weight doesn’t exceed the van’s storage limit; every tool is unique, so only one of each can be taken. The concrete weights, values, and pair bonuses are shown below.

{
  ""total_tools_count"": 10,
  ""van_weight_capacity"": 11905,
  ""tool_identifiers"": [
    ""A"",
    ""B"",
    ""C"",
    ""D"",
    ""E"",
    ""F"",
    ""G"",
    ""H"",
    ""I"",
    ""J""
  ],
  ""linear"": [
    {
      ""tool_id"": ""A"",
      ""tool_service_value"": 416
    },
    {
      ""tool_id"": ""B"",
      ""tool_service_value"": 114
    },
    {
      ""tool_id"": ""C"",
      ""tool_service_value"": 1717
    },
    {
      ""tool_id"": ""D"",
      ""tool_service_value"": 1037
    },
    {
      ""tool_id"": ""E"",
      ""tool_service_value"": 1042
    },
    {
      ""tool_id"": ""F"",
      ""tool_service_value"": 1974
    },
    {
      ""tool_id"": ""G"",
      ""tool_service_value"": 2638
    },
    {
      ""tool_id"": ""H"",
      ""tool_service_value"": 1638
    },
    {
      ""tool_id"": ""I"",
      ""tool_service_value"": 854
    },
    {
      ""tool_id"": ""J"",
      ""tool_service_value"": 133
    }
  ],
  ""weights"": [
    {
      ""tool_id"": ""A"",
      ""tool_weight"": 255
    },
    {
      ""tool_id"": ""B"",
      ""tool_weight"": 887
    },
    {
      ""tool_id"": ""C"",
      ""tool_weight"": 1706
    },
    {
      ""tool_id"": ""D"",
      ""tool_weight"": 368
    },
    {
      ""tool_id"": ""E"",
      ""tool_weight"": 2101
    },
    {
      ""tool_id"": ""F"",
      ""tool_weight"": 2329
    },
    {
      ""tool_id"": ""G"",
      ""tool_weight"": 1406
    },
    {
      ""tool_id"": ""H"",
      ""tool_weight"": 1539
    },
    {
      ""tool_id"": ""I"",
      ""tool_weight"": 2180
    },
    {
      ""tool_id"": ""J"",
      ""tool_weight"": 1346
    }
  ],
  ""quadratic"": [
    {
      ""tool_i_id"": ""A"",
      ""tool_j_id"": ""B"",
      ""pair_bonus_service"": 626
    },
    {
      ""tool_i_id"": ""A"",
      ""tool_j_id"": ""C"",
      ""pair_bonus_service"": 1514
    },
    {
      ""tool_i_id"": ""A"",
      ""tool_j_id"": ""D"",
      ""pair_bonus_service"": 747
    },
    {
      ""tool_i_id"": ""A"",
      ""tool_j_id"": ""F"",
      ""pair_bonus_service"": 2661
    },
    {
      ""tool_i_id"": ""A"",
      ""tool_j_id"": ""G"",
      ""pair_bonus_service"": 835
    },
    {
      ""tool_i_id"": ""A"",
      ""tool_j_id"": ""H"",
      ""pair_bonus_service"": 2539
    },
    {
      ""tool_i_id"": ""A"",
      ""tool_j_id"": ""I"",
      ""pair_bonus_service"": 1920
    },
    {
      ""tool_i_id"": ""A"",
      ""tool_j_id"": ""J"",
      ""pair_bonus_service"": 237
    },
    {
      ""tool_i_id"": ""B"",
      ""tool_j_id"": ""C"",
      ""pair_bonus_service"": 824
    },
    {
      ""tool_i_id"": ""B"",
      ""tool_j_id"": ""D"",
      ""pair_bonus_service"": 2091
    },
    {
      ""tool_i_id"": ""B"",
      ""tool_j_id"": ""E"",
      ""pair_bonus_service"": 305
    },
    {
      ""tool_i_id"": ""B"",
      ""tool_j_id"": ""F"",
      ""pair_bonus_service"": 2132
    },
    {
      ""tool_i_id"": ""B"",
      ""tool_j_id"": ""G"",
      ""pair_bonus_service"": 1045
    },
    {
      ""tool_i_id"": ""B"",
      ""tool_j_id"": ""I"",
      ""pair_bonus_service"": 945
    },
    {
      ""tool_i_id"": ""B"",
      ""tool_j_id"": ""J"",
      ""pair_bonus_service"": 2327
    },
    {
      ""tool_i_id"": ""C"",
      ""tool_j_id"": ""D"",
      ""pair_bonus_service"": 2400
    },
    {
      ""tool_i_id"": ""C"",
      ""tool_j_id"": ""E"",
      ""pair_bonus_service"": 198
    },
    {
      ""tool_i_id"": ""C"",
      ""tool_j_id"": ""H"",
      ""pair_bonus_service"": 2544
    },
    {
      ""tool_i_id"": ""C"",
      ""tool_j_id"": ""I"",
      ""pair_bonus_service"": 727
    },
    {
      ""tool_i_id"": ""C"",
      ""tool_j_id"": ""J"",
      ""pair_bonus_service"": 389
    },
    {
      ""tool_i_id"": ""D"",
      ""tool_j_id"": ""E"",
      ""pair_bonus_service"": 2724
    },
    {
      ""tool_i_id"": ""D"",
      ""tool_j_id"": ""F"",
      ""pair_bonus_service"": 2404
    },
    {
      ""tool_i_id"": ""D"",
      ""tool_j_id"": ""G"",
      ""pair_bonus_service"": 419
    },
    {
      ""tool_i_id"": ""D"",
      ""tool_j_id"": ""H"",
      ""pair_bonus_service"": 1117
    },
    {
      ""tool_i_id"": ""D"",
      ""tool_j_id"": ""I"",
      ""pair_bonus_service"": 280
    },
    {
      ""tool_i_id"": ""E"",
      ""tool_j_id"": ""F"",
      ""pair_bonus_service"": 1047
    },
    {
      ""tool_i_id"": ""E"",
      ""tool_j_id"": ""G"",
      ""pair_bonus_service"": 327
    },
    {
      ""tool_i_id"": ""E"",
      ""tool_j_id"": ""H"",
      ""pair_bonus_service"": 2597
    },
    {
      ""tool_i_id"": ""E"",
      ""tool_j_id"": ""I"",
      ""pair_bonus_service"": 845
    },
    {
      ""tool_i_id"": ""E"",
      ""tool_j_id"": ""J"",
      ""pair_bonus_service"": 2903
    },
    {
      ""tool_i_id"": ""F"",
      ""tool_j_id"": ""G"",
      ""pair_bonus_service"": 1802
    },
    {
      ""tool_i_id"": ""F"",
      ""tool_j_id"": ""H"",
      ""pair_bonus_service"": 2287
    },
    {
      ""tool_i_id"": ""F"",
      ""tool_j_id"": ""J"",
      ""pair_bonus_service"": 2722
    },
    {
      ""tool_i_id"": ""G"",
      ""tool_j_id"": ""H"",
      ""pair_bonus_service"": 2250
    },
    {
      ""tool_i_id"": ""G"",
      ""tool_j_id"": ""I"",
      ""pair_bonus_service"": 952
    },
    {
      ""tool_i_id"": ""G"",
      ""tool_j_id"": ""J"",
      ""pair_bonus_service"": 2883
    },
    {
      ""tool_i_id"": ""H"",
      ""tool_j_id"": ""I"",
      ""pair_bonus_service"": 1184
    },
    {
      ""tool_i_id"": ""H"",
      ""tool_j_id"": ""J"",
      ""pair_bonus_service"": 2891
    }
  ]
}

You can just show your pick as a tiny JSON snippet so it's easy to read and copy back — something like this:

{
  ""solution"": [<selected_tool_id>, <selected_tool_id>, ...]
}

""solution"" is the list of tool IDs you'd bring in the van; each <selected_tool_id> is a placeholder for one of the tools from the instance. Think of it like filling in a simple checklist: put the exact tool IDs from the problem inside the brackets. This JSON is only a sketch of the shape I expect, not the final selection.

Please make sure to 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”.""","{'linear_coeffs': [416, 114, 1717, 1037, 1042, 1974, 2638, 1638, 854, 133], 'quadratic_coeffs': [[0, 626, 1514, 747, 0, 2661, 835, 2539, 1920, 237], [0, 0, 824, 2091, 305, 2132, 1045, 0, 945, 2327], [0, 0, 2673, 2400, 198, 0, 0, 2544, 727, 389], [0, 0, 0, 0, 2724, 2404, 419, 1117, 280, 0], [0, 0, 0, 0, 861, 1047, 327, 2597, 845, 2903], [0, 0, 0, 0, 0, 2894, 1802, 2287, 0, 2722], [0, 0, 0, 0, 0, 0, 1120, 2250, 952, 2883], [0, 0, 0, 0, 0, 0, 0, 697, 1184, 2891], [0, 0, 0, 0, 0, 0, 0, 0, 2433, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [255, 887, 1706, 368, 2101, 2329, 1406, 1539, 2180, 1346], 'capacity': 11905, 'solution': [0, 1, 2, 3, 4, 6, 7, 8, 9], 'obj': 53174.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 6, 7, 8, 9]",53174.0,"{'problem_type': 'QKP', 'num_items': 10, 'capacity': 11905, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 416}, {'item_id': 'B', 'linear_profit': 114}, {'item_id': 'C', 'linear_profit': 1717}, {'item_id': 'D', 'linear_profit': 1037}, {'item_id': 'E', 'linear_profit': 1042}, {'item_id': 'F', 'linear_profit': 1974}, {'item_id': 'G', 'linear_profit': 2638}, {'item_id': 'H', 'linear_profit': 1638}, {'item_id': 'I', 'linear_profit': 854}, {'item_id': 'J', 'linear_profit': 133}], 'weight_pairs': [{'item_id': 'A', 'weight': 255}, {'item_id': 'B', 'weight': 887}, {'item_id': 'C', 'weight': 1706}, {'item_id': 'D', 'weight': 368}, {'item_id': 'E', 'weight': 2101}, {'item_id': 'F', 'weight': 2329}, {'item_id': 'G', 'weight': 1406}, {'item_id': 'H', 'weight': 1539}, {'item_id': 'I', 'weight': 2180}, {'item_id': 'J', 'weight': 1346}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 626}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 1514}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 747}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 2661}, {'item_i_id': 'A', 'item_j_id': 'G', 'quadratic_profit': 835}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 2539}, {'item_i_id': 'A', 'item_j_id': 'I', 'quadratic_profit': 1920}, {'item_i_id': 'A', 'item_j_id': 'J', 'quadratic_profit': 237}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 824}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 2091}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 305}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 2132}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 1045}, {'item_i_id': 'B', 'item_j_id': 'I', 'quadratic_profit': 945}, {'item_i_id': 'B', 'item_j_id': 'J', 'quadratic_profit': 2327}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 2673}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 2400}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 198}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 2544}, {'item_i_id': 'C', 'item_j_id': 'I', 'quadratic_profit': 727}, {'item_i_id': 'C', 'item_j_id': 'J', 'quadratic_profit': 389}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 2724}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 2404}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 419}, {'item_i_id': 'D', 'item_j_id': 'H', 'quadratic_profit': 1117}, {'item_i_id': 'D', 'item_j_id': 'I', 'quadratic_profit': 280}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 861}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 1047}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 327}, {'item_i_id': 'E', 'item_j_id': 'H', 'quadratic_profit': 2597}, {'item_i_id': 'E', 'item_j_id': 'I', 'quadratic_profit': 845}, {'item_i_id': 'E', 'item_j_id': 'J', 'quadratic_profit': 2903}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 2894}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 1802}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 2287}, {'item_i_id': 'F', 'item_j_id': 'J', 'quadratic_profit': 2722}, {'item_i_id': 'G', 'item_j_id': 'G', 'quadratic_profit': 1120}, {'item_i_id': 'G', 'item_j_id': 'H', 'quadratic_profit': 2250}, {'item_i_id': 'G', 'item_j_id': 'I', 'quadratic_profit': 952}, {'item_i_id': 'G', 'item_j_id': 'J', 'quadratic_profit': 2883}, {'item_i_id': 'H', 'item_j_id': 'H', 'quadratic_profit': 697}, {'item_i_id': 'H', 'item_j_id': 'I', 'quadratic_profit': 1184}, {'item_i_id': 'H', 'item_j_id': 'J', 'quadratic_profit': 2891}, {'item_i_id': 'I', 'item_j_id': 'I', 'quadratic_profit': 2433}]}","['A', 'B', 'C', 'D', 'E', 'G', 'H', 'I', 'J']",44,json,names
QKP,QKP,"Recently the boutique owner decided to refresh a display case and now has to pick a handful of accessories to show. Each accessory takes up some room and contributes a basic sales value, and a few lucky couples of items add extra cross-selling value when they’re placed together. The task is to assemble a set that fits in the case and brings in the highest combined revenue—simply the sum of each selected item’s base value plus any pair bonuses for items that are both included—keeping total bulk under the capacity and never duplicating an item. The full breakdown of items, sizes, values and pair bonuses is shown below.

# num_accessories=6
# case_capacity=341
# accessory_ids=1, 2, 3, 4, 5, 6

accessory_id,base_sales_value
1,103
2,101
3,102
4,101
5,104
6,105

accessory_id,space_required
1,101
2,103
3,102
4,100
5,103
6,104

accessory_i_id,accessory_j_id,cross_selling_bonus
1,3,102
1,4,100
2,3,102
3,4,102
4,6,103

If you'd like the final pick in a simple, copy‑pasteable form, just send it back using this JSON layout:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

This ""solution"" list is where you put the accessories you choose for the display case — just drop in each selected item's identifier exactly as it appears in the instance. Think of it like a little checklist: the array holds the item labels you want to show. The JSON above is only a sketch of the shape I expect, not the actual answer.

Please use the exact identifiers from 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”.""","{'linear_coeffs': [103, 101, 102, 101, 104, 105], 'quadratic_coeffs': [[0, 0, 102, 100, 0, 0], [0, 0, 102, 0, 0, 0], [0, 0, 102, 102, 0, 0], [0, 0, 0, 102, 0, 103], [0, 0, 0, 0, 100, 0], [0, 0, 0, 0, 0, 0]], 'weights': [101, 103, 102, 100, 103, 104], 'capacity': 341, 'solution': [0, 2, 3], 'obj': 610.0, 'problem_type': 'QKP'}","[0, 2, 3]",610.0,"{'problem_type': 'QKP', 'num_items': 6, 'capacity': 341, 'items': [1, 2, 3, 4, 5, 6], 'linear_pairs': [{'item_id': 1, 'linear_profit': 103}, {'item_id': 2, 'linear_profit': 101}, {'item_id': 3, 'linear_profit': 102}, {'item_id': 4, 'linear_profit': 101}, {'item_id': 5, 'linear_profit': 104}, {'item_id': 6, 'linear_profit': 105}], 'weight_pairs': [{'item_id': 1, 'weight': 101}, {'item_id': 2, 'weight': 103}, {'item_id': 3, 'weight': 102}, {'item_id': 4, 'weight': 100}, {'item_id': 5, 'weight': 103}, {'item_id': 6, 'weight': 104}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 102}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 100}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 102}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 102}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 102}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 102}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 103}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 100}]}","[1, 3, 4]",45,csv,1
QKP,QKP,"Someone in the kitchen is trying to build a killer tasting platter from a set of small plates. Each option has a satisfaction score and uses some of the platter’s limited space, and a few specific pairings add extra enjoyment if both dishes make the cut. The plan is to choose a selection of different plates (no repeats) that stays within the platter’s capacity, and to maximize the total satisfaction by summing the plates’ scores plus any bonus points for chosen pairs. Exact weights, scores, and combo values are shown below.

They have 6 distinct small plates available — 1, 2, 3, 4, 5, 6 — and a platter capacity of 570; the exact plate scores, space usages, and pairing bonuses are listed below.

| plate_id | satisfaction_score |
|---|---|
| 1 | 120 |
| 2 | 124 |
| 3 | 120 |
| 4 | 133 |
| 5 | 124 |
| 6 | 113 |

| plate_id | plate_space |
|---|---|
| 1 | 108 |
| 2 | 112 |
| 3 | 108 |
| 4 | 121 |
| 5 | 112 |
| 6 | 101 |

| plate_i_id | plate_j_id | combo_bonus |
|---|---|---|
| 1 | 2 | 108 |
| 1 | 4 | 101 |
| 1 | 5 | 101 |
| 1 | 6 | 101 |
| 2 | 3 | 112 |
| 2 | 4 | 122 |
| 2 | 5 | 121 |
| 2 | 6 | 116 |
| 3 | 5 | 105 |
| 3 | 6 | 104 |
| 4 | 6 | 107 |
| 5 | 6 | 104 |

The goal is to choose plates that fit within the 570 platter space and maximize total satisfaction.

Oh, and when you send back your chosen plates, you can just drop them into a tiny JSON snippet so it's clear and machine-friendly. Something like this works:

{
  ""solution"": [<plate_id>, <plate_id>, ...]
}

Here ""solution"" is just the list of plates you want on the tasting platter — one entry per chosen plate, using the exact plate identifiers from the instance. This block is just a little template showing the shape I expect, not the final answer itself.

Please make sure to use the identifiers exactly as they appear in the instance input — no renaming, no made-up labels.
- ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'linear_coeffs': [120, 124, 120, 133, 124, 113], 'quadratic_coeffs': [[104, 108, 0, 101, 101, 101], [0, 105, 112, 122, 121, 116], [0, 0, 0, 0, 105, 104], [0, 0, 0, 103, 0, 107], [0, 0, 0, 0, 0, 104], [0, 0, 0, 0, 0, 0]], 'weights': [108, 112, 108, 121, 112, 101], 'capacity': 570, 'solution': [0, 1, 3, 4, 5], 'obj': 1595.0, 'problem_type': 'QKP'}","[0, 1, 3, 4, 5]",1595.0,"{'problem_type': 'QKP', 'num_items': 6, 'capacity': 570, 'items': [1, 2, 3, 4, 5, 6], 'linear_pairs': [{'item_id': 1, 'linear_profit': 120}, {'item_id': 2, 'linear_profit': 124}, {'item_id': 3, 'linear_profit': 120}, {'item_id': 4, 'linear_profit': 133}, {'item_id': 5, 'linear_profit': 124}, {'item_id': 6, 'linear_profit': 113}], 'weight_pairs': [{'item_id': 1, 'weight': 108}, {'item_id': 2, 'weight': 112}, {'item_id': 3, 'weight': 108}, {'item_id': 4, 'weight': 121}, {'item_id': 5, 'weight': 112}, {'item_id': 6, 'weight': 101}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 104}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 108}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 101}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 101}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 101}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 105}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 112}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 122}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 121}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 116}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 105}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 104}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 103}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 107}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 104}]}","[1, 2, 4, 5, 6]",46,markdown_table,1
QKP,QKP,"I’m loading up a rental truck with a pile of furniture and trying to decide which pieces to take so the truck doesn’t get overloaded. Each item has its own resale or reuse value, and a few specific combinations — like a matching table and set of chairs — add extra value if they travel together. The idea is to pick a bunch of items whose total weight stays under the truck’s limit while getting the biggest total payout, which is just the sum of each piece’s value plus any pair bonuses for items that were both brought along; each item can only be taken once or left behind. The exact list of pieces, weights, values and the pair bonuses is shown below.

# num_pieces=8
# truck_weight_limit=1632
# piece_ids=A, B, C, D, E, F, G, H

piece_id,resale_value
A,834
B,686
C,515
D,540
E,363
F,258
G,862
H,256

piece_id,piece_weight
A,744
B,596
C,425
D,450
E,273
F,168
G,772
H,166

piece_a_id,piece_b_id,pair_bonus_value
A,F,693
A,G,412
A,H,394
B,C,696
B,D,657
B,E,369
C,D,340
C,E,573
C,H,681
E,F,600
E,G,557
E,H,649
F,H,442
G,H,162

Oh, and when you send your pick, please use this simple JSON layout so it's easy to read and parse:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

Think of ""solution"" as just a short list of the pieces you’re loading into the truck — each entry is the identifier for one item you want to take. This JSON is just a sketch of the shape I expect, not the actual answer itself.

Please use the exact identifiers from 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”.""","{'linear_coeffs': [834, 686, 515, 540, 363, 258, 862, 256], 'quadratic_coeffs': [[790, 0, 0, 0, 0, 693, 412, 394], [0, 0, 696, 657, 369, 0, 0, 0], [0, 0, 557, 340, 573, 0, 0, 681], [0, 0, 0, 421, 0, 0, 0, 0], [0, 0, 0, 0, 322, 600, 557, 649], [0, 0, 0, 0, 0, 0, 0, 442], [0, 0, 0, 0, 0, 0, 273, 162], [0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [744, 596, 425, 450, 273, 168, 772, 166], 'capacity': 1632, 'solution': [1, 2, 4, 5, 7], 'obj': 6088.0, 'problem_type': 'QKP'}","[1, 2, 4, 5, 7]",6088.0,"{'problem_type': 'QKP', 'num_items': 8, 'capacity': 1632, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 834}, {'item_id': 'B', 'linear_profit': 686}, {'item_id': 'C', 'linear_profit': 515}, {'item_id': 'D', 'linear_profit': 540}, {'item_id': 'E', 'linear_profit': 363}, {'item_id': 'F', 'linear_profit': 258}, {'item_id': 'G', 'linear_profit': 862}, {'item_id': 'H', 'linear_profit': 256}], 'weight_pairs': [{'item_id': 'A', 'weight': 744}, {'item_id': 'B', 'weight': 596}, {'item_id': 'C', 'weight': 425}, {'item_id': 'D', 'weight': 450}, {'item_id': 'E', 'weight': 273}, {'item_id': 'F', 'weight': 168}, {'item_id': 'G', 'weight': 772}, {'item_id': 'H', 'weight': 166}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 790}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 693}, {'item_i_id': 'A', 'item_j_id': 'G', 'quadratic_profit': 412}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 394}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 696}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 657}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 369}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 557}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 340}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 573}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 681}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 421}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 322}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 600}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 557}, {'item_i_id': 'E', 'item_j_id': 'H', 'quadratic_profit': 649}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 442}, {'item_i_id': 'G', 'item_j_id': 'G', 'quadratic_profit': 273}, {'item_i_id': 'G', 'item_j_id': 'H', 'quadratic_profit': 162}]}","['B', 'C', 'E', 'F', 'H']",47,csv,names
QKP,QKP,"Back at the makerspace there’s one shelf with a strict weight capacity and a bunch of cool equipment to pick from. Each item brings its own usefulness number and adds to the total weight, and certain item pairs produce an extra boost if they’re both on the shelf. The objective is to choose which individual items to put there — no duplicates, and don’t exceed the shelf’s weight — so that the overall usefulness (individual scores plus any pairwise boosts that apply) ends up as high as possible. The full list of weights, usefulness values, and pair bonuses is given below.

Here are the 10 pieces of equipment 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and the shelf's weight capacity of 978.
1 contributes a base usefulness of 121.
2 contributes a base usefulness of 101.
3 contributes a base usefulness of 123.
4 contributes a base usefulness of 125.
5 contributes a base usefulness of 109.
6 contributes a base usefulness of 140.
7 contributes a base usefulness of 117.
8 contributes a base usefulness of 109.
9 contributes a base usefulness of 139.
10 contributes a base usefulness of 143.
1 adds 124 to the shelf's total weight.
2 adds 101 to the shelf's total weight.
3 adds 120 to the shelf's total weight.
4 adds 121 to the shelf's total weight.
5 adds 113 to the shelf's total weight.
6 adds 132 to the shelf's total weight.
7 adds 119 to the shelf's total weight.
8 adds 113 to the shelf's total weight.
9 adds 144 to the shelf's total weight.
10 adds 141 to the shelf's total weight.
If both 1 and 2 are placed, they yield an extra 124 usefulness.
If both 1 and 3 are placed, they yield an extra 124 usefulness.
If both 1 and 4 are placed, they yield an extra 124 usefulness.
If both 1 and 5 are placed, they yield an extra 102 usefulness.
If both 1 and 6 are placed, they yield an extra 118 usefulness.
If both 1 and 7 are placed, they yield an extra 113 usefulness.
If both 1 and 8 are placed, they yield an extra 124 usefulness.
If both 1 and 9 are placed, they yield an extra 124 usefulness.
If both 1 and 10 are placed, they yield an extra 105 usefulness.
If both 2 and 3 are placed, they yield an extra 101 usefulness.
If both 2 and 4 are placed, they yield an extra 100 usefulness.
If both 2 and 5 are placed, they yield an extra 101 usefulness.
If both 2 and 6 are placed, they yield an extra 101 usefulness.
If both 2 and 7 are placed, they yield an extra 101 usefulness.
If both 2 and 8 are placed, they yield an extra 101 usefulness.
If both 2 and 9 are placed, they yield an extra 101 usefulness.
If both 2 and 10 are placed, they yield an extra 101 usefulness.
If both 3 and 4 are placed, they yield an extra 120 usefulness.
If both 3 and 5 are placed, they yield an extra 120 usefulness.
If both 3 and 6 are placed, they yield an extra 120 usefulness.
If both 3 and 7 are placed, they yield an extra 120 usefulness.
If both 3 and 8 are placed, they yield an extra 117 usefulness.
If both 3 and 9 are placed, they yield an extra 114 usefulness.
If both 3 and 10 are placed, they yield an extra 108 usefulness.
If both 4 and 5 are placed, they yield an extra 121 usefulness.
If both 4 and 6 are placed, they yield an extra 121 usefulness.
If both 4 and 7 are placed, they yield an extra 121 usefulness.
If both 4 and 8 are placed, they yield an extra 121 usefulness.
If both 4 and 9 are placed, they yield an extra 104 usefulness.
If both 4 and 10 are placed, they yield an extra 140 usefulness.
If both 5 and 6 are placed, they yield an extra 113 usefulness.
If both 5 and 7 are placed, they yield an extra 113 usefulness.
If both 5 and 8 are placed, they yield an extra 101 usefulness.
If both 5 and 9 are placed, they yield an extra 102 usefulness.
If both 5 and 10 are placed, they yield an extra 102 usefulness.
If both 6 and 7 are placed, they yield an extra 112 usefulness.
If both 6 and 8 are placed, they yield an extra 112 usefulness.
If both 6 and 9 are placed, they yield an extra 100 usefulness.
If both 6 and 10 are placed, they yield an extra 100 usefulness.
If both 7 and 8 are placed, they yield an extra 140 usefulness.
If both 7 and 9 are placed, they yield an extra 127 usefulness.
If both 7 and 10 are placed, they yield an extra 136 usefulness.
If both 8 and 9 are placed, they yield an extra 112 usefulness.
If both 8 and 10 are placed, they yield an extra 104 usefulness.
If both 9 and 10 are placed, they yield an extra 136 usefulness.
Use these entries to pick items that fit within the 978 limit and maximize total usefulness.

Oh, and when you send back your pick, just use this little JSON shape so it's easy to read and machine-friendly:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

""solution"" is just the list of items you want on the shelf — put each item's identifier in that array. Think of it like filling out a short form: which items go on the shelf (no duplicates), listed by their exact IDs. This snippet is only a sketch of the expected shape, not the actual answer.

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”.","{'linear_coeffs': [121, 101, 123, 125, 109, 140, 117, 109, 139, 143], 'quadratic_coeffs': [[118, 124, 124, 124, 102, 118, 113, 124, 124, 105], [0, 101, 101, 100, 101, 101, 101, 101, 101, 101], [0, 0, 113, 120, 120, 120, 120, 117, 114, 108], [0, 0, 0, 121, 121, 121, 121, 121, 104, 140], [0, 0, 0, 0, 113, 113, 113, 101, 102, 102], [0, 0, 0, 0, 0, 112, 112, 112, 100, 100], [0, 0, 0, 0, 0, 0, 123, 140, 127, 136], [0, 0, 0, 0, 0, 0, 0, 132, 112, 104], [0, 0, 0, 0, 0, 0, 0, 0, 110, 136], [0, 0, 0, 0, 0, 0, 0, 0, 0, 113]], 'weights': [124, 101, 120, 121, 113, 132, 119, 113, 144, 141], 'capacity': 978, 'solution': [0, 1, 2, 3, 5, 6, 7, 9], 'obj': 4208.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 5, 6, 7, 9]",4208.0,"{'problem_type': 'QKP', 'num_items': 10, 'capacity': 978, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 'linear_pairs': [{'item_id': 1, 'linear_profit': 121}, {'item_id': 2, 'linear_profit': 101}, {'item_id': 3, 'linear_profit': 123}, {'item_id': 4, 'linear_profit': 125}, {'item_id': 5, 'linear_profit': 109}, {'item_id': 6, 'linear_profit': 140}, {'item_id': 7, 'linear_profit': 117}, {'item_id': 8, 'linear_profit': 109}, {'item_id': 9, 'linear_profit': 139}, {'item_id': 10, 'linear_profit': 143}], 'weight_pairs': [{'item_id': 1, 'weight': 124}, {'item_id': 2, 'weight': 101}, {'item_id': 3, 'weight': 120}, {'item_id': 4, 'weight': 121}, {'item_id': 5, 'weight': 113}, {'item_id': 6, 'weight': 132}, {'item_id': 7, 'weight': 119}, {'item_id': 8, 'weight': 113}, {'item_id': 9, 'weight': 144}, {'item_id': 10, 'weight': 141}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 118}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 124}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 124}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 124}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 102}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 118}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 113}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 124}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 124}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 105}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 101}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 101}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 100}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 101}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 101}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 101}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 101}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 101}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 113}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 120}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 120}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 120}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 120}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 117}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 114}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 108}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 121}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 121}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 121}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 121}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 121}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 104}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 140}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 113}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 113}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 113}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 101}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 102}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 102}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 112}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 112}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 112}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 100}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 100}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 123}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 140}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 127}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 136}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 132}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 112}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 104}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 110}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 136}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 113}]}","[1, 2, 3, 4, 6, 7, 8, 10]",48,nl,1
QKP,QKP,"Many people on the events team know the drill: one courier bag, a pile of flyers and kits, and a short list of choices. Every item adds some outreach value and some pairs bring extra bang when they’re combined. The goal is to end up with the highest possible outreach total — add the values of the chosen items and any extra pair bonuses — while respecting the bag’s weight limit and the rule that each piece is unique and can only be included once. The full breakdown of weights, values, and pair boosts is listed below.

{
  ""total_materials_count"": 8,
  ""bag_capacity"": 2209,
  ""material_ids"": [
    0,
    1,
    2,
    3,
    4,
    5,
    6,
    7
  ],
  ""linear"": [
    {
      ""material_id"": 0,
      ""outreach_value"": 2076
    },
    {
      ""material_id"": 1,
      ""outreach_value"": 3333
    },
    {
      ""material_id"": 2,
      ""outreach_value"": 1039
    },
    {
      ""material_id"": 3,
      ""outreach_value"": 3234
    },
    {
      ""material_id"": 4,
      ""outreach_value"": 1548
    },
    {
      ""material_id"": 5,
      ""outreach_value"": 1654
    },
    {
      ""material_id"": 6,
      ""outreach_value"": 1765
    },
    {
      ""material_id"": 7,
      ""outreach_value"": 693
    }
  ],
  ""weights"": [
    {
      ""material_id"": 0,
      ""item_weight"": 1911
    },
    {
      ""material_id"": 1,
      ""item_weight"": 3226
    },
    {
      ""material_id"": 2,
      ""item_weight"": 1047
    },
    {
      ""material_id"": 3,
      ""item_weight"": 3140
    },
    {
      ""material_id"": 4,
      ""item_weight"": 1445
    },
    {
      ""material_id"": 5,
      ""item_weight"": 1521
    },
    {
      ""material_id"": 6,
      ""item_weight"": 1650
    },
    {
      ""material_id"": 7,
      ""item_weight"": 753
    }
  ],
  ""quadratic"": [
    {
      ""material_i_id"": 0,
      ""material_j_id"": 1,
      ""combo_bonus"": 1905
    },
    {
      ""material_i_id"": 0,
      ""material_j_id"": 2,
      ""combo_bonus"": 2435
    },
    {
      ""material_i_id"": 0,
      ""material_j_id"": 3,
      ""combo_bonus"": 622
    },
    {
      ""material_i_id"": 0,
      ""material_j_id"": 5,
      ""combo_bonus"": 2127
    },
    {
      ""material_i_id"": 0,
      ""material_j_id"": 6,
      ""combo_bonus"": 2462
    },
    {
      ""material_i_id"": 0,
      ""material_j_id"": 7,
      ""combo_bonus"": 1692
    },
    {
      ""material_i_id"": 1,
      ""material_j_id"": 2,
      ""combo_bonus"": 1549
    },
    {
      ""material_i_id"": 1,
      ""material_j_id"": 3,
      ""combo_bonus"": 3198
    },
    {
      ""material_i_id"": 1,
      ""material_j_id"": 4,
      ""combo_bonus"": 1340
    },
    {
      ""material_i_id"": 1,
      ""material_j_id"": 5,
      ""combo_bonus"": 619
    },
    {
      ""material_i_id"": 1,
      ""material_j_id"": 6,
      ""combo_bonus"": 3181
    },
    {
      ""material_i_id"": 1,
      ""material_j_id"": 7,
      ""combo_bonus"": 1030
    },
    {
      ""material_i_id"": 2,
      ""material_j_id"": 5,
      ""combo_bonus"": 1378
    },
    {
      ""material_i_id"": 2,
      ""material_j_id"": 7,
      ""combo_bonus"": 510
    },
    {
      ""material_i_id"": 3,
      ""material_j_id"": 4,
      ""combo_bonus"": 834
    },
    {
      ""material_i_id"": 3,
      ""material_j_id"": 6,
      ""combo_bonus"": 416
    },
    {
      ""material_i_id"": 3,
      ""material_j_id"": 7,
      ""combo_bonus"": 166
    },
    {
      ""material_i_id"": 4,
      ""material_j_id"": 5,
      ""combo_bonus"": 2719
    },
    {
      ""material_i_id"": 5,
      ""material_j_id"": 6,
      ""combo_bonus"": 991
    },
    {
      ""material_i_id"": 6,
      ""material_j_id"": 7,
      ""combo_bonus"": 1986
    }
  ]
}

Oh, and when you tell me which items you’re packing, a tiny JSON snippet like this works great — super simple and keeps everything clear:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

Think of ""solution"" as the list of item IDs you picked to put in the bag. The angle-bracket placeholders are just a sketch of the shape I expect — when you send your actual answer, swap those placeholders for the real item IDs from the instance.

Please use the item identifiers exactly as they appear in the problem 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"".","{'linear_coeffs': [2076, 3333, 1039, 3234, 1548, 1654, 1765, 693], 'quadratic_coeffs': [[0, 1905, 2435, 622, 0, 2127, 2462, 1692], [0, 0, 1549, 3198, 1340, 619, 3181, 1030], [0, 0, 1477, 0, 0, 1378, 0, 510], [0, 0, 0, 0, 834, 0, 416, 166], [0, 0, 0, 0, 2507, 2719, 0, 0], [0, 0, 0, 0, 0, 2805, 991, 0], [0, 0, 0, 0, 0, 0, 1313, 1986], [0, 0, 0, 0, 0, 0, 0, 1987]], 'weights': [1911, 3226, 1047, 3140, 1445, 1521, 1650, 753], 'capacity': 2209, 'solution': [2, 7], 'obj': 2242.0, 'problem_type': 'QKP'}","[2, 7]",2242.0,"{'problem_type': 'QKP', 'num_items': 8, 'capacity': 2209, 'items': [0, 1, 2, 3, 4, 5, 6, 7], 'linear_pairs': [{'item_id': 0, 'linear_profit': 2076}, {'item_id': 1, 'linear_profit': 3333}, {'item_id': 2, 'linear_profit': 1039}, {'item_id': 3, 'linear_profit': 3234}, {'item_id': 4, 'linear_profit': 1548}, {'item_id': 5, 'linear_profit': 1654}, {'item_id': 6, 'linear_profit': 1765}, {'item_id': 7, 'linear_profit': 693}], 'weight_pairs': [{'item_id': 0, 'weight': 1911}, {'item_id': 1, 'weight': 3226}, {'item_id': 2, 'weight': 1047}, {'item_id': 3, 'weight': 3140}, {'item_id': 4, 'weight': 1445}, {'item_id': 5, 'weight': 1521}, {'item_id': 6, 'weight': 1650}, {'item_id': 7, 'weight': 753}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 1, 'quadratic_profit': 1905}, {'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 2435}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 622}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 2127}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 2462}, {'item_i_id': 0, 'item_j_id': 7, 'quadratic_profit': 1692}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 1549}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 3198}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 1340}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 619}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 3181}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 1030}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 1477}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 1378}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 510}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 834}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 416}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 166}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 2507}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 2719}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 2805}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 991}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 1313}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 1986}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 1987}]}","[2, 7]",49,json,0
QKP,QKP,"We’ve got a stack of product samples and one small mailer to fill for a marketing push. Every sample contributes a certain conversion value on its own, and certain combinations add a little extra lift when they’re sent together. The best choice is the kit that brings in the most total conversion value — add up each sample’s value and add any extra value for pairs that are both present — but the total weight has to fit the mailer and each sample can only appear once. The exact items, weights, and numbers are listed below.

Below are the 6 samples 1, 2, 3, 4, 5, 6 and the mailer capacity 4337.

| sample_id | individual_conversion_value |
|---|---|
| 1 | 912 |
| 2 | 764 |
| 3 | 450 |
| 4 | 2291 |
| 5 | 1949 |
| 6 | 1000 |

| sample_id | sample_weight |
|---|---|
| 1 | 1664 |
| 2 | 626 |
| 3 | 2597 |
| 4 | 1590 |
| 5 | 2867 |
| 6 | 437 |

| sample_i_id | sample_j_id | pair_bonus_conversion |
|---|---|---|
| 1 | 5 | 1399 |
| 2 | 6 | 696 |

We’ll pick the kit that maximizes total conversion while staying within the 4337 limit.

Also, when you’re ready to give your final pick, a tiny JSON snippet like this works great — simple and machine-friendly, but still easy to read:

{
  ""solution"": [<selected_item_id>, <selected_item_id>, ...]
}

The ""solution"" array is where you list the samples you want to put in the mailer — one entry per selected item. Think of it like filling in a shopping list: just put the exact item IDs you’re choosing inside the brackets. This JSON is just the shape I’m expecting, not your final answer.

Please use the item 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"".","{'linear_coeffs': [912, 764, 450, 2291, 1949, 1000], 'quadratic_coeffs': [[0, 0, 0, 0, 1399, 0], [0, 0, 0, 0, 0, 696], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 618, 0], [0, 0, 0, 0, 0, 0]], 'weights': [1664, 626, 2597, 1590, 2867, 437], 'capacity': 4337, 'solution': [0, 1, 3, 5], 'obj': 5663.0, 'problem_type': 'QKP'}","[0, 1, 3, 5]",5663.0,"{'problem_type': 'QKP', 'num_items': 6, 'capacity': 4337, 'items': [1, 2, 3, 4, 5, 6], 'linear_pairs': [{'item_id': 1, 'linear_profit': 912}, {'item_id': 2, 'linear_profit': 764}, {'item_id': 3, 'linear_profit': 450}, {'item_id': 4, 'linear_profit': 2291}, {'item_id': 5, 'linear_profit': 1949}, {'item_id': 6, 'linear_profit': 1000}], 'weight_pairs': [{'item_id': 1, 'weight': 1664}, {'item_id': 2, 'weight': 626}, {'item_id': 3, 'weight': 2597}, {'item_id': 4, 'weight': 1590}, {'item_id': 5, 'weight': 2867}, {'item_id': 6, 'weight': 437}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 1399}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 696}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 618}]}","[1, 2, 4, 6]",50,markdown_table,1