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.

Below are the 15 available items (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15) and the backpack weight limit 262.

| gear_id | base_usefulness |
|---|---|
| 1 | 103 |
| 2 | 100 |
| 3 | 103 |
| 4 | 104 |
| 5 | 101 |
| 6 | 101 |
| 7 | 105 |
| 8 | 103 |
| 9 | 105 |
| 10 | 101 |
| 11 | 102 |
| 12 | 100 |
| 13 | 105 |
| 14 | 104 |
| 15 | 103 |

| gear_id | item_weight |
|---|---|
| 1 | 105 |
| 2 | 102 |
| 3 | 101 |
| 4 | 100 |
| 5 | 105 |
| 6 | 102 |
| 7 | 101 |
| 8 | 100 |
| 9 | 103 |
| 10 | 103 |
| 11 | 105 |
| 12 | 100 |
| 13 | 101 |
| 14 | 105 |
| 15 | 105 |

| gear_i_id | gear_j_id | pair_bonus |
|---|---|---|
| 1 | 3 | 104 |
| 1 | 14 | 101 |
| 2 | 8 | 102 |
| 2 | 9 | 102 |
| 2 | 12 | 103 |
| 2 | 15 | 102 |
| 3 | 4 | 101 |
| 3 | 6 | 101 |
| 3 | 8 | 101 |
| 3 | 15 | 101 |
| 5 | 14 | 100 |
| 5 | 15 | 102 |
| 6 | 8 | 103 |
| 6 | 11 | 104 |
| 6 | 12 | 104 |
| 6 | 14 | 102 |
| 6 | 15 | 101 |
| 9 | 11 | 100 |
| 9 | 13 | 100 |
| 10 | 12 | 100 |
| 10 | 13 | 100 |
| 10 | 15 | 101 |
| 11 | 12 | 100 |
| 12 | 14 | 101 |
| 13 | 15 | 101 |
| 14 | 15 | 100 |

Choose a subset that stays within the 262 limit to maximize total usefulness.

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': [103, 100, 103, 104, 101, 101, 105, 103, 105, 101, 102, 100, 105, 104, 103], 'quadratic_coeffs': [[0, 0, 104, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 101, 0], [0, 102, 0, 0, 0, 0, 0, 102, 102, 0, 0, 103, 0, 0, 102], [0, 0, 101, 101, 0, 101, 0, 101, 0, 0, 0, 0, 0, 0, 101], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 100, 0, 0, 0, 0, 0, 0, 0, 0, 100, 102], [0, 0, 0, 0, 0, 0, 0, 103, 0, 0, 104, 104, 0, 102, 101], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 0, 100, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 100, 0, 101], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 101, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 101], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 101, 100], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [105, 102, 101, 100, 105, 102, 101, 100, 103, 103, 105, 100, 101, 105, 105], 'capacity': 262, 'solution': [0, 2], 'obj': 310.0, 'problem_type': 'QKP'}","[0, 2]",310.0,"{'problem_type': 'QKP', 'num_items': 15, 'capacity': 262, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], 'linear_pairs': [{'item_id': 1, 'linear_profit': 103}, {'item_id': 2, 'linear_profit': 100}, {'item_id': 3, 'linear_profit': 103}, {'item_id': 4, 'linear_profit': 104}, {'item_id': 5, 'linear_profit': 101}, {'item_id': 6, 'linear_profit': 101}, {'item_id': 7, 'linear_profit': 105}, {'item_id': 8, 'linear_profit': 103}, {'item_id': 9, 'linear_profit': 105}, {'item_id': 10, 'linear_profit': 101}, {'item_id': 11, 'linear_profit': 102}, {'item_id': 12, 'linear_profit': 100}, {'item_id': 13, 'linear_profit': 105}, {'item_id': 14, 'linear_profit': 104}, {'item_id': 15, 'linear_profit': 103}], 'weight_pairs': [{'item_id': 1, 'weight': 105}, {'item_id': 2, 'weight': 102}, {'item_id': 3, 'weight': 101}, {'item_id': 4, 'weight': 100}, {'item_id': 5, 'weight': 105}, {'item_id': 6, 'weight': 102}, {'item_id': 7, 'weight': 101}, {'item_id': 8, 'weight': 100}, {'item_id': 9, 'weight': 103}, {'item_id': 10, 'weight': 103}, {'item_id': 11, 'weight': 105}, {'item_id': 12, 'weight': 100}, {'item_id': 13, 'weight': 101}, {'item_id': 14, 'weight': 105}, {'item_id': 15, 'weight': 105}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 104}, {'item_i_id': 1, 'item_j_id': 14, 'quadratic_profit': 101}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 102}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 102}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 102}, {'item_i_id': 2, 'item_j_id': 12, 'quadratic_profit': 103}, {'item_i_id': 2, 'item_j_id': 15, 'quadratic_profit': 102}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 15, 'quadratic_profit': 101}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 100}, {'item_i_id': 5, 'item_j_id': 14, 'quadratic_profit': 100}, {'item_i_id': 5, 'item_j_id': 15, 'quadratic_profit': 102}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 103}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 104}, {'item_i_id': 6, 'item_j_id': 12, 'quadratic_profit': 104}, {'item_i_id': 6, 'item_j_id': 14, 'quadratic_profit': 102}, {'item_i_id': 6, 'item_j_id': 15, 'quadratic_profit': 101}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 100}, {'item_i_id': 9, 'item_j_id': 13, 'quadratic_profit': 100}, {'item_i_id': 10, 'item_j_id': 12, 'quadratic_profit': 100}, {'item_i_id': 10, 'item_j_id': 13, 'quadratic_profit': 100}, {'item_i_id': 10, 'item_j_id': 15, 'quadratic_profit': 101}, {'item_i_id': 11, 'item_j_id': 12, 'quadratic_profit': 100}, {'item_i_id': 12, 'item_j_id': 14, 'quadratic_profit': 101}, {'item_i_id': 13, 'item_j_id': 15, 'quadratic_profit': 101}, {'item_i_id': 14, 'item_j_id': 14, 'quadratic_profit': 101}, {'item_i_id': 14, 'item_j_id': 15, 'quadratic_profit': 100}]}","[1, 3]",1,markdown_table,1
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.

They are weighing 13 candidate dishes 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 against an oven capacity of 776.

| dish_id | dish_appeal |
|---|---|
| 0 | 104 |
| 1 | 113 |
| 2 | 121 |
| 3 | 107 |
| 4 | 123 |
| 5 | 122 |
| 6 | 101 |
| 7 | 114 |
| 8 | 101 |
| 9 | 115 |
| 10 | 116 |
| 11 | 108 |
| 12 | 107 |

| dish_id | oven_load |
|---|---|
| 0 | 112 |
| 1 | 105 |
| 2 | 102 |
| 3 | 114 |
| 4 | 124 |
| 5 | 114 |
| 6 | 105 |
| 7 | 103 |
| 8 | 121 |
| 9 | 104 |
| 10 | 123 |
| 11 | 108 |
| 12 | 107 |

| dish_i_id | dish_j_id | pair_bonus_appeal |
|---|---|---|
| 0 | 1 | 113 |
| 0 | 2 | 117 |
| 0 | 3 | 124 |
| 0 | 4 | 113 |
| 0 | 5 | 106 |
| 0 | 6 | 107 |
| 0 | 7 | 100 |
| 0 | 8 | 118 |
| 0 | 9 | 116 |
| 0 | 10 | 116 |
| 0 | 11 | 102 |
| 0 | 12 | 102 |
| 1 | 2 | 111 |
| 1 | 3 | 117 |
| 1 | 4 | 109 |
| 1 | 5 | 101 |
| 1 | 6 | 116 |
| 1 | 7 | 108 |
| 1 | 8 | 119 |
| 1 | 9 | 100 |
| 1 | 10 | 103 |
| 1 | 11 | 119 |
| 1 | 12 | 108 |
| 2 | 3 | 119 |
| 2 | 4 | 116 |
| 2 | 5 | 104 |
| 2 | 6 | 118 |
| 2 | 7 | 115 |
| 2 | 8 | 114 |
| 2 | 9 | 106 |
| 2 | 10 | 100 |
| 2 | 11 | 108 |
| 2 | 12 | 115 |
| 3 | 4 | 108 |
| 3 | 5 | 121 |
| 3 | 6 | 108 |
| 3 | 7 | 101 |
| 3 | 8 | 124 |
| 3 | 9 | 115 |
| 3 | 10 | 105 |
| 3 | 11 | 111 |
| 3 | 12 | 110 |
| 4 | 5 | 115 |
| 4 | 6 | 114 |
| 4 | 7 | 105 |
| 4 | 8 | 109 |
| 4 | 9 | 103 |
| 4 | 10 | 100 |
| 4 | 11 | 109 |
| 4 | 12 | 123 |
| 5 | 6 | 111 |
| 5 | 7 | 101 |
| 5 | 8 | 101 |
| 5 | 9 | 103 |
| 5 | 10 | 118 |
| 5 | 11 | 112 |
| 5 | 12 | 121 |
| 6 | 7 | 121 |
| 6 | 8 | 116 |
| 6 | 9 | 106 |
| 6 | 10 | 112 |
| 6 | 11 | 101 |
| 6 | 12 | 112 |
| 7 | 8 | 114 |
| 7 | 9 | 107 |
| 7 | 10 | 111 |
| 7 | 11 | 119 |
| 7 | 12 | 111 |
| 8 | 9 | 108 |
| 8 | 10 | 101 |
| 8 | 11 | 124 |
| 8 | 12 | 105 |
| 9 | 10 | 105 |
| 9 | 11 | 102 |
| 9 | 12 | 124 |
| 10 | 11 | 111 |
| 10 | 12 | 123 |
| 11 | 12 | 108 |

The aim is to select a subset within the 776 oven limit that maximizes total guest satisfaction across the 13 dishes.

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': [104, 113, 121, 107, 123, 122, 101, 114, 101, 115, 116, 108, 107], 'quadratic_coeffs': [[120, 113, 117, 124, 113, 106, 107, 100, 118, 116, 116, 102, 102], [0, 120, 111, 117, 109, 101, 116, 108, 119, 100, 103, 119, 108], [0, 0, 120, 119, 116, 104, 118, 115, 114, 106, 100, 108, 115], [0, 0, 0, 114, 108, 121, 108, 101, 124, 115, 105, 111, 110], [0, 0, 0, 0, 103, 115, 114, 105, 109, 103, 100, 109, 123], [0, 0, 0, 0, 0, 120, 111, 101, 101, 103, 118, 112, 121], [0, 0, 0, 0, 0, 0, 117, 121, 116, 106, 112, 101, 112], [0, 0, 0, 0, 0, 0, 0, 116, 114, 107, 111, 119, 111], [0, 0, 0, 0, 0, 0, 0, 0, 101, 108, 101, 124, 105], [0, 0, 0, 0, 0, 0, 0, 0, 0, 102, 105, 102, 124], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 115, 111, 123], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 123, 108], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 113]], 'weights': [112, 105, 102, 114, 124, 114, 105, 103, 121, 104, 123, 108, 107], 'capacity': 776, 'solution': [1, 2, 3, 4, 5, 6, 12], 'obj': 3171.0, 'problem_type': 'QKP'}","[1, 2, 3, 4, 5, 6, 12]",3171.0,"{'problem_type': 'QKP', 'num_items': 13, 'capacity': 776, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 'linear_pairs': [{'item_id': 0, 'linear_profit': 104}, {'item_id': 1, 'linear_profit': 113}, {'item_id': 2, 'linear_profit': 121}, {'item_id': 3, 'linear_profit': 107}, {'item_id': 4, 'linear_profit': 123}, {'item_id': 5, 'linear_profit': 122}, {'item_id': 6, 'linear_profit': 101}, {'item_id': 7, 'linear_profit': 114}, {'item_id': 8, 'linear_profit': 101}, {'item_id': 9, 'linear_profit': 115}, {'item_id': 10, 'linear_profit': 116}, {'item_id': 11, 'linear_profit': 108}, {'item_id': 12, 'linear_profit': 107}], 'weight_pairs': [{'item_id': 0, 'weight': 112}, {'item_id': 1, 'weight': 105}, {'item_id': 2, 'weight': 102}, {'item_id': 3, 'weight': 114}, {'item_id': 4, 'weight': 124}, {'item_id': 5, 'weight': 114}, {'item_id': 6, 'weight': 105}, {'item_id': 7, 'weight': 103}, {'item_id': 8, 'weight': 121}, {'item_id': 9, 'weight': 104}, {'item_id': 10, 'weight': 123}, {'item_id': 11, 'weight': 108}, {'item_id': 12, 'weight': 107}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 120}, {'item_i_id': 0, 'item_j_id': 1, 'quadratic_profit': 113}, {'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 117}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 124}, {'item_i_id': 0, 'item_j_id': 4, 'quadratic_profit': 113}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 106}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 107}, {'item_i_id': 0, 'item_j_id': 7, 'quadratic_profit': 100}, {'item_i_id': 0, 'item_j_id': 8, 'quadratic_profit': 118}, {'item_i_id': 0, 'item_j_id': 9, 'quadratic_profit': 116}, {'item_i_id': 0, 'item_j_id': 10, 'quadratic_profit': 116}, {'item_i_id': 0, 'item_j_id': 11, 'quadratic_profit': 102}, {'item_i_id': 0, 'item_j_id': 12, 'quadratic_profit': 102}, {'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 120}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 111}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 117}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 109}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 101}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 116}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 108}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 119}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 103}, {'item_i_id': 1, 'item_j_id': 11, 'quadratic_profit': 119}, {'item_i_id': 1, 'item_j_id': 12, 'quadratic_profit': 108}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 120}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 119}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 116}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 104}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 118}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 115}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 114}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 106}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 100}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 108}, {'item_i_id': 2, 'item_j_id': 12, 'quadratic_profit': 115}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 114}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 108}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 121}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 108}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 124}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 115}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 105}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 111}, {'item_i_id': 3, 'item_j_id': 12, 'quadratic_profit': 110}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 103}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 115}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 114}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 105}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 109}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 103}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 100}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 109}, {'item_i_id': 4, 'item_j_id': 12, 'quadratic_profit': 123}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 120}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 111}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 101}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 101}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 103}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 118}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 112}, {'item_i_id': 5, 'item_j_id': 12, 'quadratic_profit': 121}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 117}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 121}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 116}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 106}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 112}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 101}, {'item_i_id': 6, 'item_j_id': 12, 'quadratic_profit': 112}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 116}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 114}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 107}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 111}, {'item_i_id': 7, 'item_j_id': 11, 'quadratic_profit': 119}, {'item_i_id': 7, 'item_j_id': 12, 'quadratic_profit': 111}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 101}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 108}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 101}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 124}, {'item_i_id': 8, 'item_j_id': 12, 'quadratic_profit': 105}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 102}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 105}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 102}, {'item_i_id': 9, 'item_j_id': 12, 'quadratic_profit': 124}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 115}, {'item_i_id': 10, 'item_j_id': 11, 'quadratic_profit': 111}, {'item_i_id': 10, 'item_j_id': 12, 'quadratic_profit': 123}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 123}, {'item_i_id': 11, 'item_j_id': 12, 'quadratic_profit': 108}, {'item_i_id': 12, 'item_j_id': 12, 'quadratic_profit': 113}]}","[1, 2, 3, 4, 5, 6, 12]",2,markdown_table,0
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.

Below are the 13 distinct boxes (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13) and the trunk weight capacity 1437.
Box 1 provides base value 124.
Box 2 provides base value 291.
Box 3 provides base value 282.
Box 4 provides base value 259.
Box 5 provides base value 228.
Box 6 provides base value 88.
Box 7 provides base value 155.
Box 8 provides base value 179.
Box 9 provides base value 235.
Box 10 provides base value 121.
Box 11 provides base value 185.
Box 12 provides base value 193.
Box 13 provides base value 94.
Box 1 has weight 130.
Box 2 has weight 284.
Box 3 has weight 269.
Box 4 has weight 251.
Box 5 has weight 242.
Box 6 has weight 103.
Box 7 has weight 166.
Box 8 has weight 171.
Box 9 has weight 228.
Box 10 has weight 106.
Box 11 has weight 196.
Box 12 has weight 191.
Box 13 has weight 101.
Boxes 1 and 2 together add pair bonus 130.
Boxes 1 and 3 together add pair bonus 130.
Boxes 1 and 4 together add pair bonus 130.
Boxes 1 and 8 together add pair bonus 122.
Boxes 1 and 9 together add pair bonus 130.
Boxes 1 and 10 together add pair bonus 130.
Boxes 2 and 10 together add pair bonus 274.
Boxes 2 and 13 together add pair bonus 126.
Boxes 3 and 7 together add pair bonus 213.
Boxes 3 and 9 together add pair bonus 269.
Boxes 3 and 10 together add pair bonus 122.
Boxes 3 and 12 together add pair bonus 183.
Boxes 4 and 6 together add pair bonus 251.
Boxes 5 and 7 together add pair bonus 104.
Boxes 6 and 7 together add pair bonus 123.
Boxes 6 and 10 together add pair bonus 184.
Boxes 7 and 9 together add pair bonus 110.
Boxes 9 and 11 together add pair bonus 100.
Boxes 9 and 12 together add pair bonus 100.
Pick the combination that yields the highest total worth without exceeding the trunk weight capacity 1437.

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': [124, 291, 282, 259, 228, 88, 155, 179, 235, 121, 185, 193, 94], 'quadratic_coeffs': [[0, 130, 130, 130, 0, 0, 0, 122, 130, 130, 0, 0, 0], [0, 181, 0, 0, 0, 0, 0, 0, 0, 274, 0, 0, 126], [0, 0, 180, 0, 0, 0, 213, 0, 269, 122, 0, 183, 0], [0, 0, 0, 0, 0, 251, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 104, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 123, 0, 0, 184, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 110, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 100, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 128, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 184, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [130, 284, 269, 251, 242, 103, 166, 171, 228, 106, 196, 191, 101], 'capacity': 1437, 'solution': [0, 2, 3, 5, 6, 7, 8, 9], 'obj': 3357.0, 'problem_type': 'QKP'}","[0, 2, 3, 5, 6, 7, 8, 9]",3357.0,"{'problem_type': 'QKP', 'num_items': 13, 'capacity': 1437, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 'linear_pairs': [{'item_id': 1, 'linear_profit': 124}, {'item_id': 2, 'linear_profit': 291}, {'item_id': 3, 'linear_profit': 282}, {'item_id': 4, 'linear_profit': 259}, {'item_id': 5, 'linear_profit': 228}, {'item_id': 6, 'linear_profit': 88}, {'item_id': 7, 'linear_profit': 155}, {'item_id': 8, 'linear_profit': 179}, {'item_id': 9, 'linear_profit': 235}, {'item_id': 10, 'linear_profit': 121}, {'item_id': 11, 'linear_profit': 185}, {'item_id': 12, 'linear_profit': 193}, {'item_id': 13, 'linear_profit': 94}], 'weight_pairs': [{'item_id': 1, 'weight': 130}, {'item_id': 2, 'weight': 284}, {'item_id': 3, 'weight': 269}, {'item_id': 4, 'weight': 251}, {'item_id': 5, 'weight': 242}, {'item_id': 6, 'weight': 103}, {'item_id': 7, 'weight': 166}, {'item_id': 8, 'weight': 171}, {'item_id': 9, 'weight': 228}, {'item_id': 10, 'weight': 106}, {'item_id': 11, 'weight': 196}, {'item_id': 12, 'weight': 191}, {'item_id': 13, 'weight': 101}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 130}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 130}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 130}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 122}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 130}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 130}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 181}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 274}, {'item_i_id': 2, 'item_j_id': 13, 'quadratic_profit': 126}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 180}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 213}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 269}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 122}, {'item_i_id': 3, 'item_j_id': 12, 'quadratic_profit': 183}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 251}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 104}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 123}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 184}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 110}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 100}, {'item_i_id': 9, 'item_j_id': 12, 'quadratic_profit': 100}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 128}, {'item_i_id': 12, 'item_j_id': 12, 'quadratic_profit': 184}]}","[1, 3, 4, 6, 7, 8, 9, 10]",3,nl,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.

There are 11 tracks available (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) and the set must fit within 1029 minutes.

| track_id | track_enjoyment |
|---|---|
| 1 | 132 |
| 2 | 118 |
| 3 | 113 |
| 4 | 122 |
| 5 | 120 |
| 6 | 134 |
| 7 | 114 |
| 8 | 118 |
| 9 | 113 |
| 10 | 117 |
| 11 | 126 |

| track_id | track_runtime |
|---|---|
| 1 | 120 |
| 2 | 106 |
| 3 | 101 |
| 4 | 110 |
| 5 | 108 |
| 6 | 122 |
| 7 | 102 |
| 8 | 106 |
| 9 | 101 |
| 10 | 105 |
| 11 | 114 |

| track_i_id | track_j_id | transition_bonus |
|---|---|---|
| 1 | 2 | 108 |
| 1 | 3 | 109 |
| 1 | 4 | 106 |
| 1 | 5 | 111 |
| 1 | 6 | 102 |
| 1 | 8 | 120 |
| 1 | 9 | 118 |
| 1 | 10 | 106 |
| 2 | 3 | 105 |
| 2 | 5 | 106 |
| 2 | 7 | 106 |
| 2 | 8 | 102 |
| 2 | 9 | 106 |
| 2 | 11 | 106 |
| 3 | 4 | 101 |
| 3 | 7 | 101 |
| 3 | 9 | 100 |
| 3 | 10 | 101 |
| 3 | 11 | 101 |
| 4 | 5 | 110 |
| 4 | 6 | 110 |
| 4 | 7 | 110 |
| 4 | 8 | 109 |
| 4 | 9 | 108 |
| 4 | 10 | 105 |
| 4 | 11 | 106 |
| 5 | 6 | 101 |
| 5 | 7 | 108 |
| 5 | 9 | 108 |
| 5 | 10 | 107 |
| 5 | 11 | 108 |
| 6 | 7 | 114 |
| 6 | 8 | 117 |
| 6 | 10 | 104 |
| 7 | 8 | 102 |
| 7 | 10 | 103 |
| 7 | 11 | 103 |
| 8 | 9 | 101 |
| 8 | 11 | 101 |
| 9 | 10 | 112 |
| 9 | 11 | 105 |
| 10 | 11 | 103 |

Choose the combination that maximizes total enjoyment without exceeding 1029 minutes.

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': [132, 118, 113, 122, 120, 134, 114, 118, 113, 117, 126], 'quadratic_coeffs': [[108, 108, 109, 106, 111, 102, 0, 120, 118, 106, 0], [0, 0, 105, 0, 106, 0, 106, 102, 106, 0, 106], [0, 0, 101, 101, 0, 0, 101, 0, 100, 101, 101], [0, 0, 0, 102, 110, 110, 110, 109, 108, 105, 106], [0, 0, 0, 0, 108, 101, 108, 0, 108, 107, 108], [0, 0, 0, 0, 0, 122, 114, 117, 0, 104, 0], [0, 0, 0, 0, 0, 0, 101, 102, 0, 103, 103], [0, 0, 0, 0, 0, 0, 0, 0, 101, 0, 101], [0, 0, 0, 0, 0, 0, 0, 0, 106, 112, 105], [0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 103], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [120, 106, 101, 110, 108, 122, 102, 106, 101, 105, 114], 'capacity': 1029, 'solution': [0, 1, 2, 3, 4, 6, 8, 9, 10], 'obj': 4262.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 6, 8, 9, 10]",4262.0,"{'problem_type': 'QKP', 'num_items': 11, 'capacity': 1029, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 'linear_pairs': [{'item_id': 1, 'linear_profit': 132}, {'item_id': 2, 'linear_profit': 118}, {'item_id': 3, 'linear_profit': 113}, {'item_id': 4, 'linear_profit': 122}, {'item_id': 5, 'linear_profit': 120}, {'item_id': 6, 'linear_profit': 134}, {'item_id': 7, 'linear_profit': 114}, {'item_id': 8, 'linear_profit': 118}, {'item_id': 9, 'linear_profit': 113}, {'item_id': 10, 'linear_profit': 117}, {'item_id': 11, 'linear_profit': 126}], 'weight_pairs': [{'item_id': 1, 'weight': 120}, {'item_id': 2, 'weight': 106}, {'item_id': 3, 'weight': 101}, {'item_id': 4, 'weight': 110}, {'item_id': 5, 'weight': 108}, {'item_id': 6, 'weight': 122}, {'item_id': 7, 'weight': 102}, {'item_id': 8, 'weight': 106}, {'item_id': 9, 'weight': 101}, {'item_id': 10, 'weight': 105}, {'item_id': 11, 'weight': 114}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 108}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 108}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 109}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 106}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 111}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 102}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 120}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 118}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 106}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 105}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 106}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 106}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 102}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 106}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 106}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 100}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 101}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 102}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 110}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 110}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 110}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 109}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 108}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 105}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 106}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 108}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 101}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 108}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 108}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 107}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 108}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 122}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 114}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 117}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 104}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 101}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 102}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 103}, {'item_i_id': 7, 'item_j_id': 11, 'quadratic_profit': 103}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 101}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 101}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 106}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 112}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 105}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 100}, {'item_i_id': 10, 'item_j_id': 11, 'quadratic_profit': 103}]}","[1, 2, 3, 4, 5, 7, 9, 10, 11]",4,markdown_table,1
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.

There are 11 distinct lenses and accessories listed as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and the bag's weight capacity is 1015.
Item 0 provides a base usefulness score of 131.
Item 1 provides a base usefulness score of 123.
Item 2 provides a base usefulness score of 132.
Item 3 provides a base usefulness score of 114.
Item 4 provides a base usefulness score of 124.
Item 5 provides a base usefulness score of 123.
Item 6 provides a base usefulness score of 141.
Item 7 provides a base usefulness score of 101.
Item 8 provides a base usefulness score of 110.
Item 9 provides a base usefulness score of 106.
Item 10 provides a base usefulness score of 103.
Item 0 has a weight of 125.
Item 1 has a weight of 117.
Item 2 has a weight of 138.
Item 3 has a weight of 121.
Item 4 has a weight of 118.
Item 5 has a weight of 116.
Item 6 has a weight of 137.
Item 7 has a weight of 103.
Item 8 has a weight of 102.
Item 9 has a weight of 100.
Item 10 has a weight of 106.
If both 0 and 2 are packed, they add an extra usefulness of 107.
If both 0 and 3 are packed, they add an extra usefulness of 140.
If both 0 and 5 are packed, they add an extra usefulness of 112.
If both 0 and 8 are packed, they add an extra usefulness of 106.
If both 1 and 3 are packed, they add an extra usefulness of 143.
If both 1 and 6 are packed, they add an extra usefulness of 107.
If both 2 and 3 are packed, they add an extra usefulness of 113.
If both 2 and 5 are packed, they add an extra usefulness of 147.
If both 2 and 7 are packed, they add an extra usefulness of 112.
If both 2 and 8 are packed, they add an extra usefulness of 125.
If both 2 and 10 are packed, they add an extra usefulness of 119.
If both 3 and 4 are packed, they add an extra usefulness of 123.
If both 3 and 9 are packed, they add an extra usefulness of 126.
If both 3 and 10 are packed, they add an extra usefulness of 110.
If both 4 and 7 are packed, they add an extra usefulness of 107.
If both 4 and 9 are packed, they add an extra usefulness of 105.
If both 4 and 10 are packed, they add an extra usefulness of 124.
If both 5 and 7 are packed, they add an extra usefulness of 125.
If both 5 and 9 are packed, they add an extra usefulness of 138.
If both 6 and 9 are packed, they add an extra usefulness of 138.
If both 6 and 10 are packed, they add an extra usefulness of 125.
If both 7 and 10 are packed, they add an extra usefulness of 125.
If both 8 and 9 are packed, they add an extra usefulness of 141.
If both 8 and 10 are packed, they add an extra usefulness of 121.
Selections must stay within the 1015 limit to maximize overall usefulness across the 11 candidates.

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': [131, 123, 132, 114, 124, 123, 141, 101, 110, 106, 103], 'quadratic_coeffs': [[103, 0, 107, 140, 0, 112, 0, 0, 106, 0, 0], [0, 109, 0, 143, 0, 0, 107, 0, 0, 0, 0], [0, 0, 0, 113, 0, 147, 0, 112, 125, 0, 119], [0, 0, 0, 146, 123, 0, 0, 0, 0, 126, 110], [0, 0, 0, 0, 124, 0, 0, 107, 0, 105, 124], [0, 0, 0, 0, 0, 148, 0, 125, 0, 138, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 138, 125], [0, 0, 0, 0, 0, 0, 0, 144, 0, 0, 125], [0, 0, 0, 0, 0, 0, 0, 0, 116, 141, 121], [0, 0, 0, 0, 0, 0, 0, 0, 0, 149, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [125, 117, 138, 121, 118, 116, 137, 103, 102, 100, 106], 'capacity': 1015, 'solution': [0, 2, 3, 4, 5, 8, 9, 10], 'obj': 2900.0, 'problem_type': 'QKP'}","[0, 2, 3, 4, 5, 8, 9, 10]",2900.0,"{'problem_type': 'QKP', 'num_items': 11, 'capacity': 1015, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 'linear_pairs': [{'item_id': 0, 'linear_profit': 131}, {'item_id': 1, 'linear_profit': 123}, {'item_id': 2, 'linear_profit': 132}, {'item_id': 3, 'linear_profit': 114}, {'item_id': 4, 'linear_profit': 124}, {'item_id': 5, 'linear_profit': 123}, {'item_id': 6, 'linear_profit': 141}, {'item_id': 7, 'linear_profit': 101}, {'item_id': 8, 'linear_profit': 110}, {'item_id': 9, 'linear_profit': 106}, {'item_id': 10, 'linear_profit': 103}], 'weight_pairs': [{'item_id': 0, 'weight': 125}, {'item_id': 1, 'weight': 117}, {'item_id': 2, 'weight': 138}, {'item_id': 3, 'weight': 121}, {'item_id': 4, 'weight': 118}, {'item_id': 5, 'weight': 116}, {'item_id': 6, 'weight': 137}, {'item_id': 7, 'weight': 103}, {'item_id': 8, 'weight': 102}, {'item_id': 9, 'weight': 100}, {'item_id': 10, 'weight': 106}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 103}, {'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 107}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 140}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 112}, {'item_i_id': 0, 'item_j_id': 8, 'quadratic_profit': 106}, {'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 109}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 143}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 107}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 113}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 147}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 112}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 125}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 119}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 146}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 123}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 126}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 110}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 124}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 107}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 105}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 124}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 148}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 125}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 138}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 138}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 125}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 144}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 125}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 116}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 141}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 121}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 149}]}","[0, 2, 3, 4, 5, 8, 9, 10]",5,nl,0
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"": 15,
  ""root_space_capacity"": 750,
  ""plant_ids"": [
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14,
    15
  ],
  ""linear"": [
    {
      ""plant_id"": 1,
      ""base_yield"": 104
    },
    {
      ""plant_id"": 2,
      ""base_yield"": 142
    },
    {
      ""plant_id"": 3,
      ""base_yield"": 123
    },
    {
      ""plant_id"": 4,
      ""base_yield"": 107
    },
    {
      ""plant_id"": 5,
      ""base_yield"": 124
    },
    {
      ""plant_id"": 6,
      ""base_yield"": 97
    },
    {
      ""plant_id"": 7,
      ""base_yield"": 142
    },
    {
      ""plant_id"": 8,
      ""base_yield"": 108
    },
    {
      ""plant_id"": 9,
      ""base_yield"": 145
    },
    {
      ""plant_id"": 10,
      ""base_yield"": 134
    },
    {
      ""plant_id"": 11,
      ""base_yield"": 108
    },
    {
      ""plant_id"": 12,
      ""base_yield"": 149
    },
    {
      ""plant_id"": 13,
      ""base_yield"": 114
    },
    {
      ""plant_id"": 14,
      ""base_yield"": 132
    },
    {
      ""plant_id"": 15,
      ""base_yield"": 98
    }
  ],
  ""weights"": [
    {
      ""plant_id"": 1,
      ""root_space"": 111
    },
    {
      ""plant_id"": 2,
      ""root_space"": 145
    },
    {
      ""plant_id"": 3,
      ""root_space"": 129
    },
    {
      ""plant_id"": 4,
      ""root_space"": 106
    },
    {
      ""plant_id"": 5,
      ""root_space"": 130
    },
    {
      ""plant_id"": 6,
      ""root_space"": 100
    },
    {
      ""plant_id"": 7,
      ""root_space"": 147
    },
    {
      ""plant_id"": 8,
      ""root_space"": 113
    },
    {
      ""plant_id"": 9,
      ""root_space"": 143
    },
    {
      ""plant_id"": 10,
      ""root_space"": 138
    },
    {
      ""plant_id"": 11,
      ""root_space"": 110
    },
    {
      ""plant_id"": 12,
      ""root_space"": 148
    },
    {
      ""plant_id"": 13,
      ""root_space"": 110
    },
    {
      ""plant_id"": 14,
      ""root_space"": 128
    },
    {
      ""plant_id"": 15,
      ""root_space"": 102
    }
  ],
  ""quadratic"": [
    {
      ""plant_i_id"": 1,
      ""plant_j_id"": 3,
      ""companion_bonus"": 124
    },
    {
      ""plant_i_id"": 1,
      ""plant_j_id"": 4,
      ""companion_bonus"": 125
    },
    {
      ""plant_i_id"": 1,
      ""plant_j_id"": 5,
      ""companion_bonus"": 113
    },
    {
      ""plant_i_id"": 1,
      ""plant_j_id"": 8,
      ""companion_bonus"": 123
    },
    {
      ""plant_i_id"": 1,
      ""plant_j_id"": 9,
      ""companion_bonus"": 108
    },
    {
      ""plant_i_id"": 1,
      ""plant_j_id"": 11,
      ""companion_bonus"": 105
    },
    {
      ""plant_i_id"": 1,
      ""plant_j_id"": 13,
      ""companion_bonus"": 126
    },
    {
      ""plant_i_id"": 1,
      ""plant_j_id"": 14,
      ""companion_bonus"": 117
    },
    {
      ""plant_i_id"": 1,
      ""plant_j_id"": 15,
      ""companion_bonus"": 125
    },
    {
      ""plant_i_id"": 2,
      ""plant_j_id"": 4,
      ""companion_bonus"": 145
    },
    {
      ""plant_i_id"": 2,
      ""plant_j_id"": 7,
      ""companion_bonus"": 139
    },
    {
      ""plant_i_id"": 2,
      ""plant_j_id"": 8,
      ""companion_bonus"": 131
    },
    {
      ""plant_i_id"": 2,
      ""plant_j_id"": 9,
      ""companion_bonus"": 130
    },
    {
      ""plant_i_id"": 2,
      ""plant_j_id"": 10,
      ""companion_bonus"": 142
    },
    {
      ""plant_i_id"": 2,
      ""plant_j_id"": 11,
      ""companion_bonus"": 145
    },
    {
      ""plant_i_id"": 2,
      ""plant_j_id"": 12,
      ""companion_bonus"": 131
    },
    {
      ""plant_i_id"": 2,
      ""plant_j_id"": 13,
      ""companion_bonus"": 144
    },
    {
      ""plant_i_id"": 2,
      ""plant_j_id"": 14,
      ""companion_bonus"": 138
    },
    {
      ""plant_i_id"": 2,
      ""plant_j_id"": 15,
      ""companion_bonus"": 146
    },
    {
      ""plant_i_id"": 3,
      ""plant_j_id"": 4,
      ""companion_bonus"": 124
    },
    {
      ""plant_i_id"": 3,
      ""plant_j_id"": 5,
      ""companion_bonus"": 127
    },
    {
      ""plant_i_id"": 3,
      ""plant_j_id"": 6,
      ""companion_bonus"": 120
    },
    {
      ""plant_i_id"": 3,
      ""plant_j_id"": 7,
      ""companion_bonus"": 129
    },
    {
      ""plant_i_id"": 3,
      ""plant_j_id"": 9,
      ""companion_bonus"": 124
    },
    {
      ""plant_i_id"": 3,
      ""plant_j_id"": 11,
      ""companion_bonus"": 138
    },
    {
      ""plant_i_id"": 3,
      ""plant_j_id"": 12,
      ""companion_bonus"": 137
    },
    {
      ""plant_i_id"": 3,
      ""plant_j_id"": 13,
      ""companion_bonus"": 124
    },
    {
      ""plant_i_id"": 3,
      ""plant_j_id"": 14,
      ""companion_bonus"": 123
    },
    {
      ""plant_i_id"": 3,
      ""plant_j_id"": 15,
      ""companion_bonus"": 130
    },
    {
      ""plant_i_id"": 4,
      ""plant_j_id"": 5,
      ""companion_bonus"": 104
    },
    {
      ""plant_i_id"": 4,
      ""plant_j_id"": 6,
      ""companion_bonus"": 112
    },
    {
      ""plant_i_id"": 4,
      ""plant_j_id"": 7,
      ""companion_bonus"": 114
    },
    {
      ""plant_i_id"": 4,
      ""plant_j_id"": 8,
      ""companion_bonus"": 114
    },
    {
      ""plant_i_id"": 4,
      ""plant_j_id"": 9,
      ""companion_bonus"": 111
    },
    {
      ""plant_i_id"": 4,
      ""plant_j_id"": 10,
      ""companion_bonus"": 119
    },
    {
      ""plant_i_id"": 4,
      ""plant_j_id"": 11,
      ""companion_bonus"": 124
    },
    {
      ""plant_i_id"": 4,
      ""plant_j_id"": 12,
      ""companion_bonus"": 123
    },
    {
      ""plant_i_id"": 4,
      ""plant_j_id"": 15,
      ""companion_bonus"": 138
    },
    {
      ""plant_i_id"": 5,
      ""plant_j_id"": 6,
      ""companion_bonus"": 136
    },
    {
      ""plant_i_id"": 5,
      ""plant_j_id"": 8,
      ""companion_bonus"": 122
    },
    {
      ""plant_i_id"": 5,
      ""plant_j_id"": 9,
      ""companion_bonus"": 132
    },
    {
      ""plant_i_id"": 5,
      ""plant_j_id"": 10,
      ""companion_bonus"": 120
    },
    {
      ""plant_i_id"": 5,
      ""plant_j_id"": 12,
      ""companion_bonus"": 127
    },
    {
      ""plant_i_id"": 5,
      ""plant_j_id"": 14,
      ""companion_bonus"": 118
    },
    {
      ""plant_i_id"": 5,
      ""plant_j_id"": 15,
      ""companion_bonus"": 116
    },
    {
      ""plant_i_id"": 6,
      ""plant_j_id"": 7,
      ""companion_bonus"": 104
    },
    {
      ""plant_i_id"": 6,
      ""plant_j_id"": 8,
      ""companion_bonus"": 121
    },
    {
      ""plant_i_id"": 6,
      ""plant_j_id"": 9,
      ""companion_bonus"": 109
    },
    {
      ""plant_i_id"": 6,
      ""plant_j_id"": 10,
      ""companion_bonus"": 100
    },
    {
      ""plant_i_id"": 6,
      ""plant_j_id"": 11,
      ""companion_bonus"": 141
    },
    {
      ""plant_i_id"": 6,
      ""plant_j_id"": 12,
      ""companion_bonus"": 129
    },
    {
      ""plant_i_id"": 6,
      ""plant_j_id"": 13,
      ""companion_bonus"": 121
    },
    {
      ""plant_i_id"": 7,
      ""plant_j_id"": 8,
      ""companion_bonus"": 147
    },
    {
      ""plant_i_id"": 7,
      ""plant_j_id"": 10,
      ""companion_bonus"": 131
    },
    {
      ""plant_i_id"": 7,
      ""plant_j_id"": 12,
      ""companion_bonus"": 115
    },
    {
      ""plant_i_id"": 7,
      ""plant_j_id"": 13,
      ""companion_bonus"": 124
    },
    {
      ""plant_i_id"": 8,
      ""plant_j_id"": 9,
      ""companion_bonus"": 109
    },
    {
      ""plant_i_id"": 8,
      ""plant_j_id"": 10,
      ""companion_bonus"": 122
    },
    {
      ""plant_i_id"": 8,
      ""plant_j_id"": 12,
      ""companion_bonus"": 147
    },
    {
      ""plant_i_id"": 8,
      ""plant_j_id"": 13,
      ""companion_bonus"": 144
    },
    {
      ""plant_i_id"": 8,
      ""plant_j_id"": 14,
      ""companion_bonus"": 126
    },
    {
      ""plant_i_id"": 8,
      ""plant_j_id"": 15,
      ""companion_bonus"": 141
    },
    {
      ""plant_i_id"": 9,
      ""plant_j_id"": 10,
      ""companion_bonus"": 142
    },
    {
      ""plant_i_id"": 9,
      ""plant_j_id"": 11,
      ""companion_bonus"": 144
    },
    {
      ""plant_i_id"": 9,
      ""plant_j_id"": 12,
      ""companion_bonus"": 130
    },
    {
      ""plant_i_id"": 9,
      ""plant_j_id"": 13,
      ""companion_bonus"": 144
    },
    {
      ""plant_i_id"": 9,
      ""plant_j_id"": 14,
      ""companion_bonus"": 137
    },
    {
      ""plant_i_id"": 9,
      ""plant_j_id"": 15,
      ""companion_bonus"": 145
    },
    {
      ""plant_i_id"": 10,
      ""plant_j_id"": 11,
      ""companion_bonus"": 133
    },
    {
      ""plant_i_id"": 10,
      ""plant_j_id"": 13,
      ""companion_bonus"": 120
    },
    {
      ""plant_i_id"": 10,
      ""plant_j_id"": 14,
      ""companion_bonus"": 136
    },
    {
      ""plant_i_id"": 10,
      ""plant_j_id"": 15,
      ""companion_bonus"": 126
    },
    {
      ""plant_i_id"": 11,
      ""plant_j_id"": 12,
      ""companion_bonus"": 113
    },
    {
      ""plant_i_id"": 11,
      ""plant_j_id"": 13,
      ""companion_bonus"": 127
    },
    {
      ""plant_i_id"": 11,
      ""plant_j_id"": 14,
      ""companion_bonus"": 142
    },
    {
      ""plant_i_id"": 12,
      ""plant_j_id"": 13,
      ""companion_bonus"": 108
    },
    {
      ""plant_i_id"": 12,
      ""plant_j_id"": 14,
      ""companion_bonus"": 140
    },
    {
      ""plant_i_id"": 12,
      ""plant_j_id"": 15,
      ""companion_bonus"": 137
    },
    {
      ""plant_i_id"": 13,
      ""plant_j_id"": 14,
      ""companion_bonus"": 127
    },
    {
      ""plant_i_id"": 13,
      ""plant_j_id"": 15,
      ""companion_bonus"": 127
    }
  ]
}

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': [104, 142, 123, 107, 124, 97, 142, 108, 145, 134, 108, 149, 114, 132, 98], 'quadratic_coeffs': [[0, 0, 124, 125, 113, 0, 0, 123, 108, 0, 105, 0, 126, 117, 125], [0, 143, 0, 145, 0, 0, 139, 131, 130, 142, 145, 131, 144, 138, 146], [0, 0, 116, 124, 127, 120, 129, 0, 124, 0, 138, 137, 124, 123, 130], [0, 0, 0, 103, 104, 112, 114, 114, 111, 119, 124, 123, 0, 0, 138], [0, 0, 0, 0, 129, 136, 0, 122, 132, 120, 0, 127, 0, 118, 116], [0, 0, 0, 0, 0, 120, 104, 121, 109, 100, 141, 129, 121, 0, 0], [0, 0, 0, 0, 0, 0, 147, 147, 0, 131, 0, 115, 124, 0, 0], [0, 0, 0, 0, 0, 0, 0, 108, 109, 122, 0, 147, 144, 126, 141], [0, 0, 0, 0, 0, 0, 0, 0, 0, 142, 144, 130, 144, 137, 145], [0, 0, 0, 0, 0, 0, 0, 0, 0, 122, 133, 0, 120, 136, 126], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 129, 113, 127, 142, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 122, 108, 140, 137], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 144, 127, 127], [0, 0, 0, 0, 0, 0, 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': [111, 145, 129, 106, 130, 100, 147, 113, 143, 138, 110, 148, 110, 128, 102], 'capacity': 750, 'solution': [1, 3, 7, 8, 9, 14], 'obj': 2695.0, 'problem_type': 'QKP'}","[1, 3, 7, 8, 9, 14]",2695.0,"{'problem_type': 'QKP', 'num_items': 15, 'capacity': 750, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], 'linear_pairs': [{'item_id': 1, 'linear_profit': 104}, {'item_id': 2, 'linear_profit': 142}, {'item_id': 3, 'linear_profit': 123}, {'item_id': 4, 'linear_profit': 107}, {'item_id': 5, 'linear_profit': 124}, {'item_id': 6, 'linear_profit': 97}, {'item_id': 7, 'linear_profit': 142}, {'item_id': 8, 'linear_profit': 108}, {'item_id': 9, 'linear_profit': 145}, {'item_id': 10, 'linear_profit': 134}, {'item_id': 11, 'linear_profit': 108}, {'item_id': 12, 'linear_profit': 149}, {'item_id': 13, 'linear_profit': 114}, {'item_id': 14, 'linear_profit': 132}, {'item_id': 15, 'linear_profit': 98}], 'weight_pairs': [{'item_id': 1, 'weight': 111}, {'item_id': 2, 'weight': 145}, {'item_id': 3, 'weight': 129}, {'item_id': 4, 'weight': 106}, {'item_id': 5, 'weight': 130}, {'item_id': 6, 'weight': 100}, {'item_id': 7, 'weight': 147}, {'item_id': 8, 'weight': 113}, {'item_id': 9, 'weight': 143}, {'item_id': 10, 'weight': 138}, {'item_id': 11, 'weight': 110}, {'item_id': 12, 'weight': 148}, {'item_id': 13, 'weight': 110}, {'item_id': 14, 'weight': 128}, {'item_id': 15, 'weight': 102}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 124}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 125}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 113}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 123}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 108}, {'item_i_id': 1, 'item_j_id': 11, 'quadratic_profit': 105}, {'item_i_id': 1, 'item_j_id': 13, 'quadratic_profit': 126}, {'item_i_id': 1, 'item_j_id': 14, 'quadratic_profit': 117}, {'item_i_id': 1, 'item_j_id': 15, 'quadratic_profit': 125}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 143}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 145}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 139}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 131}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 130}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 142}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 145}, {'item_i_id': 2, 'item_j_id': 12, 'quadratic_profit': 131}, {'item_i_id': 2, 'item_j_id': 13, 'quadratic_profit': 144}, {'item_i_id': 2, 'item_j_id': 14, 'quadratic_profit': 138}, {'item_i_id': 2, 'item_j_id': 15, 'quadratic_profit': 146}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 116}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 124}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 127}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 120}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 129}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 124}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 138}, {'item_i_id': 3, 'item_j_id': 12, 'quadratic_profit': 137}, {'item_i_id': 3, 'item_j_id': 13, 'quadratic_profit': 124}, {'item_i_id': 3, 'item_j_id': 14, 'quadratic_profit': 123}, {'item_i_id': 3, 'item_j_id': 15, 'quadratic_profit': 130}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 103}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 104}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 112}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 114}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 114}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 111}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 119}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 124}, {'item_i_id': 4, 'item_j_id': 12, 'quadratic_profit': 123}, {'item_i_id': 4, 'item_j_id': 15, 'quadratic_profit': 138}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 129}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 136}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 122}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 132}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 120}, {'item_i_id': 5, 'item_j_id': 12, 'quadratic_profit': 127}, {'item_i_id': 5, 'item_j_id': 14, 'quadratic_profit': 118}, {'item_i_id': 5, 'item_j_id': 15, 'quadratic_profit': 116}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 120}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 104}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 121}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 109}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 100}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 141}, {'item_i_id': 6, 'item_j_id': 12, 'quadratic_profit': 129}, {'item_i_id': 6, 'item_j_id': 13, 'quadratic_profit': 121}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 147}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 147}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 131}, {'item_i_id': 7, 'item_j_id': 12, 'quadratic_profit': 115}, {'item_i_id': 7, 'item_j_id': 13, 'quadratic_profit': 124}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 108}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 109}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 122}, {'item_i_id': 8, 'item_j_id': 12, 'quadratic_profit': 147}, {'item_i_id': 8, 'item_j_id': 13, 'quadratic_profit': 144}, {'item_i_id': 8, 'item_j_id': 14, 'quadratic_profit': 126}, {'item_i_id': 8, 'item_j_id': 15, 'quadratic_profit': 141}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 142}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 144}, {'item_i_id': 9, 'item_j_id': 12, 'quadratic_profit': 130}, {'item_i_id': 9, 'item_j_id': 13, 'quadratic_profit': 144}, {'item_i_id': 9, 'item_j_id': 14, 'quadratic_profit': 137}, {'item_i_id': 9, 'item_j_id': 15, 'quadratic_profit': 145}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 122}, {'item_i_id': 10, 'item_j_id': 11, 'quadratic_profit': 133}, {'item_i_id': 10, 'item_j_id': 13, 'quadratic_profit': 120}, {'item_i_id': 10, 'item_j_id': 14, 'quadratic_profit': 136}, {'item_i_id': 10, 'item_j_id': 15, 'quadratic_profit': 126}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 129}, {'item_i_id': 11, 'item_j_id': 12, 'quadratic_profit': 113}, {'item_i_id': 11, 'item_j_id': 13, 'quadratic_profit': 127}, {'item_i_id': 11, 'item_j_id': 14, 'quadratic_profit': 142}, {'item_i_id': 12, 'item_j_id': 12, 'quadratic_profit': 122}, {'item_i_id': 12, 'item_j_id': 13, 'quadratic_profit': 108}, {'item_i_id': 12, 'item_j_id': 14, 'quadratic_profit': 140}, {'item_i_id': 12, 'item_j_id': 15, 'quadratic_profit': 137}, {'item_i_id': 13, 'item_j_id': 13, 'quadratic_profit': 144}, {'item_i_id': 13, 'item_j_id': 14, 'quadratic_profit': 127}, {'item_i_id': 13, 'item_j_id': 15, 'quadratic_profit': 127}]}","[2, 4, 8, 9, 10, 15]",6,json,1
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.

The team is considering 14 candidate features 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 with a sprint effort budget of 1890.
Feature 1 delivers a baseline user value of 361.
Feature 2 delivers a baseline user value of 823.
Feature 3 delivers a baseline user value of 302.
Feature 4 delivers a baseline user value of 201.
Feature 5 delivers a baseline user value of 580.
Feature 6 delivers a baseline user value of 737.
Feature 7 delivers a baseline user value of 821.
Feature 8 delivers a baseline user value of 174.
Feature 9 delivers a baseline user value of 565.
Feature 10 delivers a baseline user value of 845.
Feature 11 delivers a baseline user value of 828.
Feature 12 delivers a baseline user value of 593.
Feature 13 delivers a baseline user value of 792.
Feature 14 delivers a baseline user value of 795.
Feature 1 requires 397 effort.
Feature 2 requires 829 effort.
Feature 3 requires 287 effort.
Feature 4 requires 240 effort.
Feature 5 requires 590 effort.
Feature 6 requires 710 effort.
Feature 7 requires 814 effort.
Feature 8 requires 202 effort.
Feature 9 requires 567 effort.
Feature 10 requires 806 effort.
Feature 11 requires 853 effort.
Feature 12 requires 612 effort.
Feature 13 requires 769 effort.
Feature 14 requires 826 effort.
Features 1 and 2 together yield an additional interaction value of 536.
Features 1 and 3 together yield an additional interaction value of 562.
Features 1 and 4 together yield an additional interaction value of 557.
Features 1 and 5 together yield an additional interaction value of 567.
Features 1 and 6 together yield an additional interaction value of 332.
Features 1 and 7 together yield an additional interaction value of 480.
Features 1 and 8 together yield an additional interaction value of 479.
Features 1 and 9 together yield an additional interaction value of 220.
Features 1 and 10 together yield an additional interaction value of 561.
Features 1 and 11 together yield an additional interaction value of 483.
Features 1 and 12 together yield an additional interaction value of 531.
Features 1 and 13 together yield an additional interaction value of 572.
Features 1 and 14 together yield an additional interaction value of 372.
Features 2 and 3 together yield an additional interaction value of 699.
Features 2 and 4 together yield an additional interaction value of 782.
Features 2 and 5 together yield an additional interaction value of 763.
Features 2 and 6 together yield an additional interaction value of 599.
Features 2 and 7 together yield an additional interaction value of 808.
Features 2 and 8 together yield an additional interaction value of 528.
Features 2 and 9 together yield an additional interaction value of 811.
Features 2 and 10 together yield an additional interaction value of 808.
Features 2 and 11 together yield an additional interaction value of 743.
Features 2 and 12 together yield an additional interaction value of 849.
Features 2 and 13 together yield an additional interaction value of 482.
Features 2 and 14 together yield an additional interaction value of 775.
Features 3 and 4 together yield an additional interaction value of 462.
Features 3 and 5 together yield an additional interaction value of 281.
Features 3 and 6 together yield an additional interaction value of 420.
Features 3 and 7 together yield an additional interaction value of 503.
Features 3 and 8 together yield an additional interaction value of 406.
Features 3 and 9 together yield an additional interaction value of 475.
Features 3 and 10 together yield an additional interaction value of 436.
Features 3 and 11 together yield an additional interaction value of 474.
Features 3 and 12 together yield an additional interaction value of 469.
Features 3 and 13 together yield an additional interaction value of 235.
Features 3 and 14 together yield an additional interaction value of 450.
Features 4 and 5 together yield an additional interaction value of 290.
Features 4 and 6 together yield an additional interaction value of 246.
Features 4 and 7 together yield an additional interaction value of 228.
Features 4 and 8 together yield an additional interaction value of 341.
Features 4 and 9 together yield an additional interaction value of 425.
Features 4 and 10 together yield an additional interaction value of 297.
Features 4 and 11 together yield an additional interaction value of 375.
Features 4 and 12 together yield an additional interaction value of 452.
Features 4 and 13 together yield an additional interaction value of 185.
Features 4 and 14 together yield an additional interaction value of 452.
Features 5 and 6 together yield an additional interaction value of 643.
Features 5 and 7 together yield an additional interaction value of 718.
Features 5 and 8 together yield an additional interaction value of 332.
Features 5 and 9 together yield an additional interaction value of 387.
Features 5 and 10 together yield an additional interaction value of 658.
Features 5 and 11 together yield an additional interaction value of 335.
Features 5 and 12 together yield an additional interaction value of 407.
Features 5 and 13 together yield an additional interaction value of 470.
Features 5 and 14 together yield an additional interaction value of 602.
Features 6 and 7 together yield an additional interaction value of 586.
Features 6 and 8 together yield an additional interaction value of 758.
Features 6 and 9 together yield an additional interaction value of 658.
Features 6 and 10 together yield an additional interaction value of 604.
Features 6 and 11 together yield an additional interaction value of 446.
Features 6 and 12 together yield an additional interaction value of 328.
Features 6 and 13 together yield an additional interaction value of 399.
Features 6 and 14 together yield an additional interaction value of 499.
Features 7 and 8 together yield an additional interaction value of 288.
Features 7 and 9 together yield an additional interaction value of 843.
Features 7 and 10 together yield an additional interaction value of 497.
Features 7 and 11 together yield an additional interaction value of 507.
Features 7 and 12 together yield an additional interaction value of 379.
Features 7 and 13 together yield an additional interaction value of 474.
Features 7 and 14 together yield an additional interaction value of 360.
Features 8 and 9 together yield an additional interaction value of 301.
Features 8 and 10 together yield an additional interaction value of 413.
Features 8 and 11 together yield an additional interaction value of 288.
Features 8 and 12 together yield an additional interaction value of 332.
Features 8 and 13 together yield an additional interaction value of 602.
Features 8 and 14 together yield an additional interaction value of 552.
Features 9 and 10 together yield an additional interaction value of 674.
Features 9 and 11 together yield an additional interaction value of 602.
Features 9 and 12 together yield an additional interaction value of 565.
Features 9 and 13 together yield an additional interaction value of 664.
Features 9 and 14 together yield an additional interaction value of 440.
Features 10 and 11 together yield an additional interaction value of 364.
Features 10 and 12 together yield an additional interaction value of 347.
Features 10 and 13 together yield an additional interaction value of 636.
Features 10 and 14 together yield an additional interaction value of 624.
Features 11 and 12 together yield an additional interaction value of 447.
Features 11 and 13 together yield an additional interaction value of 182.
Features 11 and 14 together yield an additional interaction value of 446.
Features 12 and 13 together yield an additional interaction value of 512.
Features 12 and 14 together yield an additional interaction value of 458.
Features 13 and 14 together yield an additional interaction value of 530.
Choose a combination that stays within the 1890 budget and maximizes total user value.

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': [361, 823, 302, 201, 580, 737, 821, 174, 565, 845, 828, 593, 792, 795], 'quadratic_coeffs': [[591, 536, 562, 557, 567, 332, 480, 479, 220, 561, 483, 531, 572, 372], [0, 812, 699, 782, 763, 599, 808, 528, 811, 808, 743, 849, 482, 775], [0, 0, 402, 462, 281, 420, 503, 406, 475, 436, 474, 469, 235, 450], [0, 0, 0, 276, 290, 246, 228, 341, 425, 297, 375, 452, 185, 452], [0, 0, 0, 0, 694, 643, 718, 332, 387, 658, 335, 407, 470, 602], [0, 0, 0, 0, 0, 792, 586, 758, 658, 604, 446, 328, 399, 499], [0, 0, 0, 0, 0, 0, 363, 288, 843, 497, 507, 379, 474, 360], [0, 0, 0, 0, 0, 0, 0, 217, 301, 413, 288, 332, 602, 552], [0, 0, 0, 0, 0, 0, 0, 0, 586, 674, 602, 565, 664, 440], [0, 0, 0, 0, 0, 0, 0, 0, 0, 758, 364, 347, 636, 624], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 370, 447, 182, 446], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 383, 512, 458], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 632, 530], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 489]], 'weights': [397, 829, 287, 240, 590, 710, 814, 202, 567, 806, 853, 612, 769, 826], 'capacity': 1890, 'solution': [0, 2, 3, 5, 7], 'obj': 6338.0, 'problem_type': 'QKP'}","[0, 2, 3, 5, 7]",6338.0,"{'problem_type': 'QKP', 'num_items': 14, 'capacity': 1890, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], 'linear_pairs': [{'item_id': 1, 'linear_profit': 361}, {'item_id': 2, 'linear_profit': 823}, {'item_id': 3, 'linear_profit': 302}, {'item_id': 4, 'linear_profit': 201}, {'item_id': 5, 'linear_profit': 580}, {'item_id': 6, 'linear_profit': 737}, {'item_id': 7, 'linear_profit': 821}, {'item_id': 8, 'linear_profit': 174}, {'item_id': 9, 'linear_profit': 565}, {'item_id': 10, 'linear_profit': 845}, {'item_id': 11, 'linear_profit': 828}, {'item_id': 12, 'linear_profit': 593}, {'item_id': 13, 'linear_profit': 792}, {'item_id': 14, 'linear_profit': 795}], 'weight_pairs': [{'item_id': 1, 'weight': 397}, {'item_id': 2, 'weight': 829}, {'item_id': 3, 'weight': 287}, {'item_id': 4, 'weight': 240}, {'item_id': 5, 'weight': 590}, {'item_id': 6, 'weight': 710}, {'item_id': 7, 'weight': 814}, {'item_id': 8, 'weight': 202}, {'item_id': 9, 'weight': 567}, {'item_id': 10, 'weight': 806}, {'item_id': 11, 'weight': 853}, {'item_id': 12, 'weight': 612}, {'item_id': 13, 'weight': 769}, {'item_id': 14, 'weight': 826}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 591}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 536}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 562}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 557}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 567}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 332}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 480}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 479}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 220}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 561}, {'item_i_id': 1, 'item_j_id': 11, 'quadratic_profit': 483}, {'item_i_id': 1, 'item_j_id': 12, 'quadratic_profit': 531}, {'item_i_id': 1, 'item_j_id': 13, 'quadratic_profit': 572}, {'item_i_id': 1, 'item_j_id': 14, 'quadratic_profit': 372}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 812}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 699}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 782}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 763}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 599}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 808}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 528}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 811}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 808}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 743}, {'item_i_id': 2, 'item_j_id': 12, 'quadratic_profit': 849}, {'item_i_id': 2, 'item_j_id': 13, 'quadratic_profit': 482}, {'item_i_id': 2, 'item_j_id': 14, 'quadratic_profit': 775}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 402}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 462}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 281}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 420}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 503}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 406}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 475}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 436}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 474}, {'item_i_id': 3, 'item_j_id': 12, 'quadratic_profit': 469}, {'item_i_id': 3, 'item_j_id': 13, 'quadratic_profit': 235}, {'item_i_id': 3, 'item_j_id': 14, 'quadratic_profit': 450}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 276}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 290}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 246}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 228}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 341}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 425}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 297}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 375}, {'item_i_id': 4, 'item_j_id': 12, 'quadratic_profit': 452}, {'item_i_id': 4, 'item_j_id': 13, 'quadratic_profit': 185}, {'item_i_id': 4, 'item_j_id': 14, 'quadratic_profit': 452}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 694}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 643}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 718}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 332}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 387}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 658}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 335}, {'item_i_id': 5, 'item_j_id': 12, 'quadratic_profit': 407}, {'item_i_id': 5, 'item_j_id': 13, 'quadratic_profit': 470}, {'item_i_id': 5, 'item_j_id': 14, 'quadratic_profit': 602}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 792}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 586}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 758}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 658}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 604}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 446}, {'item_i_id': 6, 'item_j_id': 12, 'quadratic_profit': 328}, {'item_i_id': 6, 'item_j_id': 13, 'quadratic_profit': 399}, {'item_i_id': 6, 'item_j_id': 14, 'quadratic_profit': 499}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 363}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 288}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 843}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 497}, {'item_i_id': 7, 'item_j_id': 11, 'quadratic_profit': 507}, {'item_i_id': 7, 'item_j_id': 12, 'quadratic_profit': 379}, {'item_i_id': 7, 'item_j_id': 13, 'quadratic_profit': 474}, {'item_i_id': 7, 'item_j_id': 14, 'quadratic_profit': 360}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 217}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 301}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 413}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 288}, {'item_i_id': 8, 'item_j_id': 12, 'quadratic_profit': 332}, {'item_i_id': 8, 'item_j_id': 13, 'quadratic_profit': 602}, {'item_i_id': 8, 'item_j_id': 14, 'quadratic_profit': 552}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 586}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 674}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 602}, {'item_i_id': 9, 'item_j_id': 12, 'quadratic_profit': 565}, {'item_i_id': 9, 'item_j_id': 13, 'quadratic_profit': 664}, {'item_i_id': 9, 'item_j_id': 14, 'quadratic_profit': 440}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 758}, {'item_i_id': 10, 'item_j_id': 11, 'quadratic_profit': 364}, {'item_i_id': 10, 'item_j_id': 12, 'quadratic_profit': 347}, {'item_i_id': 10, 'item_j_id': 13, 'quadratic_profit': 636}, {'item_i_id': 10, 'item_j_id': 14, 'quadratic_profit': 624}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 370}, {'item_i_id': 11, 'item_j_id': 12, 'quadratic_profit': 447}, {'item_i_id': 11, 'item_j_id': 13, 'quadratic_profit': 182}, {'item_i_id': 11, 'item_j_id': 14, 'quadratic_profit': 446}, {'item_i_id': 12, 'item_j_id': 12, 'quadratic_profit': 383}, {'item_i_id': 12, 'item_j_id': 13, 'quadratic_profit': 512}, {'item_i_id': 12, 'item_j_id': 14, 'quadratic_profit': 458}, {'item_i_id': 13, 'item_j_id': 13, 'quadratic_profit': 632}, {'item_i_id': 13, 'item_j_id': 14, 'quadratic_profit': 530}, {'item_i_id': 14, 'item_j_id': 14, 'quadratic_profit': 489}]}","[1, 3, 4, 6, 8]",7,nl,1
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"": 15,
  ""kitchen_capacity"": 1120,
  ""course_ids"": [
    ""A"",
    ""B"",
    ""C"",
    ""D"",
    ""E"",
    ""F"",
    ""G"",
    ""H"",
    ""I"",
    ""J"",
    ""K"",
    ""L"",
    ""M"",
    ""N"",
    ""O""
  ],
  ""linear"": [
    {
      ""course_id"": ""A"",
      ""base_delight"": 112
    },
    {
      ""course_id"": ""B"",
      ""base_delight"": 112
    },
    {
      ""course_id"": ""C"",
      ""base_delight"": 114
    },
    {
      ""course_id"": ""D"",
      ""base_delight"": 113
    },
    {
      ""course_id"": ""E"",
      ""base_delight"": 111
    },
    {
      ""course_id"": ""F"",
      ""base_delight"": 113
    },
    {
      ""course_id"": ""G"",
      ""base_delight"": 110
    },
    {
      ""course_id"": ""H"",
      ""base_delight"": 112
    },
    {
      ""course_id"": ""I"",
      ""base_delight"": 111
    },
    {
      ""course_id"": ""J"",
      ""base_delight"": 111
    },
    {
      ""course_id"": ""K"",
      ""base_delight"": 113
    },
    {
      ""course_id"": ""L"",
      ""base_delight"": 110
    },
    {
      ""course_id"": ""M"",
      ""base_delight"": 111
    },
    {
      ""course_id"": ""N"",
      ""base_delight"": 110
    },
    {
      ""course_id"": ""O"",
      ""base_delight"": 114
    }
  ],
  ""weights"": [
    {
      ""course_id"": ""A"",
      ""prep_plating_load"": 102
    },
    {
      ""course_id"": ""B"",
      ""prep_plating_load"": 102
    },
    {
      ""course_id"": ""C"",
      ""prep_plating_load"": 104
    },
    {
      ""course_id"": ""D"",
      ""prep_plating_load"": 103
    },
    {
      ""course_id"": ""E"",
      ""prep_plating_load"": 101
    },
    {
      ""course_id"": ""F"",
      ""prep_plating_load"": 103
    },
    {
      ""course_id"": ""G"",
      ""prep_plating_load"": 100
    },
    {
      ""course_id"": ""H"",
      ""prep_plating_load"": 102
    },
    {
      ""course_id"": ""I"",
      ""prep_plating_load"": 101
    },
    {
      ""course_id"": ""J"",
      ""prep_plating_load"": 101
    },
    {
      ""course_id"": ""K"",
      ""prep_plating_load"": 103
    },
    {
      ""course_id"": ""L"",
      ""prep_plating_load"": 100
    },
    {
      ""course_id"": ""M"",
      ""prep_plating_load"": 101
    },
    {
      ""course_id"": ""N"",
      ""prep_plating_load"": 100
    },
    {
      ""course_id"": ""O"",
      ""prep_plating_load"": 104
    }
  ],
  ""quadratic"": [
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""B"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""C"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""D"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""E"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""F"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""G"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""H"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""I"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""J"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""K"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""L"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""M"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""N"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""A"",
      ""course_j_id"": ""O"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""C"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""D"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""E"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""F"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""G"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""H"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""I"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""J"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""K"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""L"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""M"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""N"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""B"",
      ""course_j_id"": ""O"",
      ""pair_complement_boost"": 103
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""D"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""E"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""F"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""G"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""H"",
      ""pair_complement_boost"": 103
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""I"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""J"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""K"",
      ""pair_complement_boost"": 104
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""L"",
      ""pair_complement_boost"": 103
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""M"",
      ""pair_complement_boost"": 104
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""N"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""C"",
      ""course_j_id"": ""O"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""D"",
      ""course_j_id"": ""E"",
      ""pair_complement_boost"": 103
    },
    {
      ""course_i_id"": ""D"",
      ""course_j_id"": ""F"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""D"",
      ""course_j_id"": ""G"",
      ""pair_complement_boost"": 103
    },
    {
      ""course_i_id"": ""D"",
      ""course_j_id"": ""H"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""D"",
      ""course_j_id"": ""I"",
      ""pair_complement_boost"": 103
    },
    {
      ""course_i_id"": ""D"",
      ""course_j_id"": ""J"",
      ""pair_complement_boost"": 103
    },
    {
      ""course_i_id"": ""D"",
      ""course_j_id"": ""K"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""D"",
      ""course_j_id"": ""L"",
      ""pair_complement_boost"": 103
    },
    {
      ""course_i_id"": ""D"",
      ""course_j_id"": ""M"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""D"",
      ""course_j_id"": ""N"",
      ""pair_complement_boost"": 103
    },
    {
      ""course_i_id"": ""D"",
      ""course_j_id"": ""O"",
      ""pair_complement_boost"": 103
    },
    {
      ""course_i_id"": ""E"",
      ""course_j_id"": ""F"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""E"",
      ""course_j_id"": ""G"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""E"",
      ""course_j_id"": ""H"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""E"",
      ""course_j_id"": ""I"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""E"",
      ""course_j_id"": ""J"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""E"",
      ""course_j_id"": ""K"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""E"",
      ""course_j_id"": ""L"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""E"",
      ""course_j_id"": ""M"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""E"",
      ""course_j_id"": ""N"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""E"",
      ""course_j_id"": ""O"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""F"",
      ""course_j_id"": ""G"",
      ""pair_complement_boost"": 103
    },
    {
      ""course_i_id"": ""F"",
      ""course_j_id"": ""H"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""F"",
      ""course_j_id"": ""I"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""F"",
      ""course_j_id"": ""J"",
      ""pair_complement_boost"": 103
    },
    {
      ""course_i_id"": ""F"",
      ""course_j_id"": ""K"",
      ""pair_complement_boost"": 103
    },
    {
      ""course_i_id"": ""F"",
      ""course_j_id"": ""L"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""F"",
      ""course_j_id"": ""M"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""F"",
      ""course_j_id"": ""N"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""F"",
      ""course_j_id"": ""O"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""G"",
      ""course_j_id"": ""H"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""G"",
      ""course_j_id"": ""I"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""G"",
      ""course_j_id"": ""J"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""G"",
      ""course_j_id"": ""K"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""G"",
      ""course_j_id"": ""L"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""G"",
      ""course_j_id"": ""M"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""G"",
      ""course_j_id"": ""N"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""G"",
      ""course_j_id"": ""O"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""H"",
      ""course_j_id"": ""I"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""H"",
      ""course_j_id"": ""J"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""H"",
      ""course_j_id"": ""K"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""H"",
      ""course_j_id"": ""L"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""H"",
      ""course_j_id"": ""M"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""H"",
      ""course_j_id"": ""N"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""H"",
      ""course_j_id"": ""O"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""I"",
      ""course_j_id"": ""J"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""I"",
      ""course_j_id"": ""K"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""I"",
      ""course_j_id"": ""L"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""I"",
      ""course_j_id"": ""M"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""I"",
      ""course_j_id"": ""N"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""I"",
      ""course_j_id"": ""O"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""J"",
      ""course_j_id"": ""K"",
      ""pair_complement_boost"": 103
    },
    {
      ""course_i_id"": ""J"",
      ""course_j_id"": ""L"",
      ""pair_complement_boost"": 103
    },
    {
      ""course_i_id"": ""J"",
      ""course_j_id"": ""M"",
      ""pair_complement_boost"": 103
    },
    {
      ""course_i_id"": ""J"",
      ""course_j_id"": ""N"",
      ""pair_complement_boost"": 102
    },
    {
      ""course_i_id"": ""J"",
      ""course_j_id"": ""O"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""K"",
      ""course_j_id"": ""L"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""K"",
      ""course_j_id"": ""M"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""K"",
      ""course_j_id"": ""N"",
      ""pair_complement_boost"": 104
    },
    {
      ""course_i_id"": ""K"",
      ""course_j_id"": ""O"",
      ""pair_complement_boost"": 100
    },
    {
      ""course_i_id"": ""L"",
      ""course_j_id"": ""M"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""L"",
      ""course_j_id"": ""N"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""L"",
      ""course_j_id"": ""O"",
      ""pair_complement_boost"": 101
    },
    {
      ""course_i_id"": ""M"",
      ""course_j_id"": ""N"",
      ""pair_complement_boost"": 103
    },
    {
      ""course_i_id"": ""M"",
      ""course_j_id"": ""O"",
      ""pair_complement_boost"": 102
    }
  ]
}

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': [112, 112, 114, 113, 111, 113, 110, 112, 111, 111, 113, 110, 111, 110, 114], 'quadratic_coeffs': [[102, 102, 102, 102, 100, 102, 102, 101, 102, 101, 100, 102, 102, 102, 100], [0, 100, 102, 102, 102, 100, 100, 100, 102, 102, 102, 102, 102, 102, 103], [0, 0, 101, 101, 102, 102, 100, 103, 101, 100, 104, 103, 104, 100, 100], [0, 0, 0, 100, 103, 101, 103, 102, 103, 103, 101, 103, 102, 103, 103], [0, 0, 0, 0, 101, 101, 101, 101, 101, 101, 101, 101, 101, 101, 101], [0, 0, 0, 0, 0, 103, 103, 101, 100, 103, 103, 102, 102, 100, 100], [0, 0, 0, 0, 0, 0, 100, 100, 100, 100, 100, 100, 100, 100, 100], [0, 0, 0, 0, 0, 0, 0, 102, 101, 102, 102, 101, 100, 101, 102], [0, 0, 0, 0, 0, 0, 0, 0, 101, 101, 100, 100, 100, 101, 101], [0, 0, 0, 0, 0, 0, 0, 0, 0, 103, 103, 103, 103, 102, 100], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 101, 101, 104, 100], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 101, 101, 101, 101], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 103, 102], [0, 0, 0, 0, 0, 0, 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': [102, 102, 104, 103, 101, 103, 100, 102, 101, 101, 103, 100, 101, 100, 104], 'capacity': 1120, 'solution': [0, 2, 3, 5, 7, 8, 9, 10, 11, 12, 13], 'obj': 6824.0, 'problem_type': 'QKP'}","[0, 2, 3, 5, 7, 8, 9, 10, 11, 12, 13]",6824.0,"{'problem_type': 'QKP', 'num_items': 15, 'capacity': 1120, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 112}, {'item_id': 'B', 'linear_profit': 112}, {'item_id': 'C', 'linear_profit': 114}, {'item_id': 'D', 'linear_profit': 113}, {'item_id': 'E', 'linear_profit': 111}, {'item_id': 'F', 'linear_profit': 113}, {'item_id': 'G', 'linear_profit': 110}, {'item_id': 'H', 'linear_profit': 112}, {'item_id': 'I', 'linear_profit': 111}, {'item_id': 'J', 'linear_profit': 111}, {'item_id': 'K', 'linear_profit': 113}, {'item_id': 'L', 'linear_profit': 110}, {'item_id': 'M', 'linear_profit': 111}, {'item_id': 'N', 'linear_profit': 110}, {'item_id': 'O', 'linear_profit': 114}], 'weight_pairs': [{'item_id': 'A', 'weight': 102}, {'item_id': 'B', 'weight': 102}, {'item_id': 'C', 'weight': 104}, {'item_id': 'D', 'weight': 103}, {'item_id': 'E', 'weight': 101}, {'item_id': 'F', 'weight': 103}, {'item_id': 'G', 'weight': 100}, {'item_id': 'H', 'weight': 102}, {'item_id': 'I', 'weight': 101}, {'item_id': 'J', 'weight': 101}, {'item_id': 'K', 'weight': 103}, {'item_id': 'L', 'weight': 100}, {'item_id': 'M', 'weight': 101}, {'item_id': 'N', 'weight': 100}, {'item_id': 'O', 'weight': 104}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 102}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 102}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 102}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 102}, {'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': 'A', 'item_j_id': 'H', 'quadratic_profit': 101}, {'item_i_id': 'A', 'item_j_id': 'I', 'quadratic_profit': 102}, {'item_i_id': 'A', 'item_j_id': 'J', 'quadratic_profit': 101}, {'item_i_id': 'A', 'item_j_id': 'K', 'quadratic_profit': 100}, {'item_i_id': 'A', 'item_j_id': 'L', 'quadratic_profit': 102}, {'item_i_id': 'A', 'item_j_id': 'M', 'quadratic_profit': 102}, {'item_i_id': 'A', 'item_j_id': 'N', 'quadratic_profit': 102}, {'item_i_id': 'A', 'item_j_id': 'O', 'quadratic_profit': 100}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 100}, {'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': 102}, {'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': 102}, {'item_i_id': 'B', 'item_j_id': 'J', 'quadratic_profit': 102}, {'item_i_id': 'B', 'item_j_id': 'K', 'quadratic_profit': 102}, {'item_i_id': 'B', 'item_j_id': 'L', 'quadratic_profit': 102}, {'item_i_id': 'B', 'item_j_id': 'M', 'quadratic_profit': 102}, {'item_i_id': 'B', 'item_j_id': 'N', 'quadratic_profit': 102}, {'item_i_id': 'B', 'item_j_id': 'O', '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': 102}, {'item_i_id': 'C', 'item_j_id': 'F', 'quadratic_profit': 102}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 100}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 103}, {'item_i_id': 'C', 'item_j_id': 'I', 'quadratic_profit': 101}, {'item_i_id': 'C', 'item_j_id': 'J', 'quadratic_profit': 100}, {'item_i_id': 'C', 'item_j_id': 'K', 'quadratic_profit': 104}, {'item_i_id': 'C', 'item_j_id': 'L', 'quadratic_profit': 103}, {'item_i_id': 'C', 'item_j_id': 'M', 'quadratic_profit': 104}, {'item_i_id': 'C', 'item_j_id': 'N', 'quadratic_profit': 100}, {'item_i_id': 'C', 'item_j_id': 'O', 'quadratic_profit': 100}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 100}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 103}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 101}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 103}, {'item_i_id': 'D', 'item_j_id': 'H', 'quadratic_profit': 102}, {'item_i_id': 'D', 'item_j_id': 'I', 'quadratic_profit': 103}, {'item_i_id': 'D', 'item_j_id': 'J', 'quadratic_profit': 103}, {'item_i_id': 'D', 'item_j_id': 'K', 'quadratic_profit': 101}, {'item_i_id': 'D', 'item_j_id': 'L', 'quadratic_profit': 103}, {'item_i_id': 'D', 'item_j_id': 'M', 'quadratic_profit': 102}, {'item_i_id': 'D', 'item_j_id': 'N', 'quadratic_profit': 103}, {'item_i_id': 'D', 'item_j_id': 'O', 'quadratic_profit': 103}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 101}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 101}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 101}, {'item_i_id': 'E', 'item_j_id': 'H', 'quadratic_profit': 101}, {'item_i_id': 'E', 'item_j_id': 'I', 'quadratic_profit': 101}, {'item_i_id': 'E', 'item_j_id': 'J', 'quadratic_profit': 101}, {'item_i_id': 'E', 'item_j_id': 'K', 'quadratic_profit': 101}, {'item_i_id': 'E', 'item_j_id': 'L', 'quadratic_profit': 101}, {'item_i_id': 'E', 'item_j_id': 'M', 'quadratic_profit': 101}, {'item_i_id': 'E', 'item_j_id': 'N', 'quadratic_profit': 101}, {'item_i_id': 'E', 'item_j_id': 'O', 'quadratic_profit': 101}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 103}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 103}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 101}, {'item_i_id': 'F', 'item_j_id': 'I', 'quadratic_profit': 100}, {'item_i_id': 'F', 'item_j_id': 'J', 'quadratic_profit': 103}, {'item_i_id': 'F', 'item_j_id': 'K', 'quadratic_profit': 103}, {'item_i_id': 'F', 'item_j_id': 'L', 'quadratic_profit': 102}, {'item_i_id': 'F', 'item_j_id': 'M', 'quadratic_profit': 102}, {'item_i_id': 'F', 'item_j_id': 'N', 'quadratic_profit': 100}, {'item_i_id': 'F', 'item_j_id': 'O', 'quadratic_profit': 100}, {'item_i_id': 'G', 'item_j_id': 'G', 'quadratic_profit': 100}, {'item_i_id': 'G', 'item_j_id': 'H', 'quadratic_profit': 100}, {'item_i_id': 'G', 'item_j_id': 'I', 'quadratic_profit': 100}, {'item_i_id': 'G', 'item_j_id': 'J', 'quadratic_profit': 100}, {'item_i_id': 'G', 'item_j_id': 'K', 'quadratic_profit': 100}, {'item_i_id': 'G', 'item_j_id': 'L', 'quadratic_profit': 100}, {'item_i_id': 'G', 'item_j_id': 'M', 'quadratic_profit': 100}, {'item_i_id': 'G', 'item_j_id': 'N', 'quadratic_profit': 100}, {'item_i_id': 'G', 'item_j_id': 'O', 'quadratic_profit': 100}, {'item_i_id': 'H', 'item_j_id': 'H', 'quadratic_profit': 102}, {'item_i_id': 'H', 'item_j_id': 'I', 'quadratic_profit': 101}, {'item_i_id': 'H', 'item_j_id': 'J', 'quadratic_profit': 102}, {'item_i_id': 'H', 'item_j_id': 'K', 'quadratic_profit': 102}, {'item_i_id': 'H', 'item_j_id': 'L', 'quadratic_profit': 101}, {'item_i_id': 'H', 'item_j_id': 'M', 'quadratic_profit': 100}, {'item_i_id': 'H', 'item_j_id': 'N', 'quadratic_profit': 101}, {'item_i_id': 'H', 'item_j_id': 'O', 'quadratic_profit': 102}, {'item_i_id': 'I', 'item_j_id': 'I', 'quadratic_profit': 101}, {'item_i_id': 'I', 'item_j_id': 'J', 'quadratic_profit': 101}, {'item_i_id': 'I', 'item_j_id': 'K', 'quadratic_profit': 100}, {'item_i_id': 'I', 'item_j_id': 'L', 'quadratic_profit': 100}, {'item_i_id': 'I', 'item_j_id': 'M', 'quadratic_profit': 100}, {'item_i_id': 'I', 'item_j_id': 'N', 'quadratic_profit': 101}, {'item_i_id': 'I', 'item_j_id': 'O', 'quadratic_profit': 101}, {'item_i_id': 'J', 'item_j_id': 'J', 'quadratic_profit': 103}, {'item_i_id': 'J', 'item_j_id': 'K', 'quadratic_profit': 103}, {'item_i_id': 'J', 'item_j_id': 'L', 'quadratic_profit': 103}, {'item_i_id': 'J', 'item_j_id': 'M', 'quadratic_profit': 103}, {'item_i_id': 'J', 'item_j_id': 'N', 'quadratic_profit': 102}, {'item_i_id': 'J', 'item_j_id': 'O', 'quadratic_profit': 100}, {'item_i_id': 'K', 'item_j_id': 'K', 'quadratic_profit': 100}, {'item_i_id': 'K', 'item_j_id': 'L', 'quadratic_profit': 101}, {'item_i_id': 'K', 'item_j_id': 'M', 'quadratic_profit': 101}, {'item_i_id': 'K', 'item_j_id': 'N', 'quadratic_profit': 104}, {'item_i_id': 'K', 'item_j_id': 'O', 'quadratic_profit': 100}, {'item_i_id': 'L', 'item_j_id': 'L', 'quadratic_profit': 101}, {'item_i_id': 'L', 'item_j_id': 'M', 'quadratic_profit': 101}, {'item_i_id': 'L', 'item_j_id': 'N', 'quadratic_profit': 101}, {'item_i_id': 'L', 'item_j_id': 'O', 'quadratic_profit': 101}, {'item_i_id': 'M', 'item_j_id': 'M', 'quadratic_profit': 100}, {'item_i_id': 'M', 'item_j_id': 'N', 'quadratic_profit': 103}, {'item_i_id': 'M', 'item_j_id': 'O', 'quadratic_profit': 102}]}","['A', 'C', 'D', 'F', 'H', 'I', 'J', 'K', 'L', 'M', 'N']",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.

There are 14 available, listed as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and the shipping threshold is 3318.

| item_ref_for_weight | base_value |
|---|---|
| 1 | 219 |
| 2 | 516 |
| 3 | 366 |
| 4 | 189 |
| 5 | 243 |
| 6 | 510 |
| 7 | 339 |
| 8 | 275 |
| 9 | 257 |
| 10 | 340 |
| 11 | 490 |
| 12 | 318 |
| 13 | 190 |
| 14 | 227 |

| item_ref_for_weight | item_weight |
|---|---|
| 1 | 169 |
| 2 | 466 |
| 3 | 316 |
| 4 | 139 |
| 5 | 193 |
| 6 | 460 |
| 7 | 289 |
| 8 | 225 |
| 9 | 207 |
| 10 | 290 |
| 11 | 440 |
| 12 | 268 |
| 13 | 140 |
| 14 | 177 |

| pair_item_i | pair_item_j | pair_synergy_value |
|---|---|---|
| 1 | 3 | 169 |
| 1 | 5 | 169 |
| 1 | 7 | 169 |
| 1 | 8 | 169 |
| 1 | 9 | 169 |
| 1 | 14 | 169 |
| 2 | 8 | 262 |
| 2 | 12 | 283 |
| 3 | 6 | 316 |
| 3 | 11 | 204 |
| 4 | 8 | 139 |
| 4 | 10 | 250 |
| 4 | 14 | 250 |
| 5 | 10 | 314 |
| 6 | 7 | 155 |
| 6 | 12 | 155 |
| 6 | 14 | 155 |
| 7 | 9 | 150 |
| 7 | 10 | 150 |
| 8 | 11 | 425 |
| 8 | 12 | 337 |
| 8 | 13 | 339 |
| 9 | 10 | 214 |
| 9 | 12 | 290 |
| 11 | 13 | 242 |

Selections must maximize the box’s perceived value while keeping total weight under the shipping threshold.

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': [219, 516, 366, 189, 243, 510, 339, 275, 257, 340, 490, 318, 190, 227], 'quadratic_coeffs': [[0, 0, 169, 0, 169, 0, 169, 169, 169, 0, 0, 0, 0, 169], [0, 0, 0, 0, 0, 0, 0, 262, 0, 0, 0, 283, 0, 0], [0, 0, 162, 0, 0, 316, 0, 0, 0, 0, 204, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 139, 0, 250, 0, 0, 0, 250], [0, 0, 0, 0, 193, 0, 0, 0, 0, 314, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 155, 0, 0, 0, 0, 155, 0, 155], [0, 0, 0, 0, 0, 0, 0, 0, 150, 150, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 425, 337, 339, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 214, 0, 290, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 214, 0, 242, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [169, 466, 316, 139, 193, 460, 289, 225, 207, 290, 440, 268, 140, 177], 'capacity': 3318, 'solution': [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 'obj': 9062.0, 'problem_type': 'QKP'}","[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]",9062.0,"{'problem_type': 'QKP', 'num_items': 14, 'capacity': 3318, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], 'linear_pairs': [{'item_id': 1, 'linear_profit': 219}, {'item_id': 2, 'linear_profit': 516}, {'item_id': 3, 'linear_profit': 366}, {'item_id': 4, 'linear_profit': 189}, {'item_id': 5, 'linear_profit': 243}, {'item_id': 6, 'linear_profit': 510}, {'item_id': 7, 'linear_profit': 339}, {'item_id': 8, 'linear_profit': 275}, {'item_id': 9, 'linear_profit': 257}, {'item_id': 10, 'linear_profit': 340}, {'item_id': 11, 'linear_profit': 490}, {'item_id': 12, 'linear_profit': 318}, {'item_id': 13, 'linear_profit': 190}, {'item_id': 14, 'linear_profit': 227}], 'weight_pairs': [{'item_id': 1, 'weight': 169}, {'item_id': 2, 'weight': 466}, {'item_id': 3, 'weight': 316}, {'item_id': 4, 'weight': 139}, {'item_id': 5, 'weight': 193}, {'item_id': 6, 'weight': 460}, {'item_id': 7, 'weight': 289}, {'item_id': 8, 'weight': 225}, {'item_id': 9, 'weight': 207}, {'item_id': 10, 'weight': 290}, {'item_id': 11, 'weight': 440}, {'item_id': 12, 'weight': 268}, {'item_id': 13, 'weight': 140}, {'item_id': 14, 'weight': 177}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 169}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 169}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 169}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 169}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 169}, {'item_i_id': 1, 'item_j_id': 14, 'quadratic_profit': 169}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 262}, {'item_i_id': 2, 'item_j_id': 12, 'quadratic_profit': 283}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 162}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 316}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 204}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 139}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 250}, {'item_i_id': 4, 'item_j_id': 14, 'quadratic_profit': 250}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 193}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 314}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 155}, {'item_i_id': 6, 'item_j_id': 12, 'quadratic_profit': 155}, {'item_i_id': 6, 'item_j_id': 14, 'quadratic_profit': 155}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 150}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 150}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 425}, {'item_i_id': 8, 'item_j_id': 12, 'quadratic_profit': 337}, {'item_i_id': 8, 'item_j_id': 13, 'quadratic_profit': 339}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 214}, {'item_i_id': 9, 'item_j_id': 12, 'quadratic_profit': 290}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 214}, {'item_i_id': 11, 'item_j_id': 13, 'quadratic_profit': 242}]}","[1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]",9,markdown_table,1
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.

I have 11 souvenirs listed as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and my suitcase weight limit is 937.
Souvenir 0 is worth 117 sentimental points to me.
Souvenir 1 is worth 116 sentimental points to me.
Souvenir 2 is worth 114 sentimental points to me.
Souvenir 3 is worth 119 sentimental points to me.
Souvenir 4 is worth 118 sentimental points to me.
Souvenir 5 is worth 117 sentimental points to me.
Souvenir 6 is worth 113 sentimental points to me.
Souvenir 7 is worth 117 sentimental points to me.
Souvenir 8 is worth 111 sentimental points to me.
Souvenir 9 is worth 119 sentimental points to me.
Souvenir 10 is worth 115 sentimental points to me.
Souvenir 0 weighs 106 units in my suitcase.
Souvenir 1 weighs 105 units in my suitcase.
Souvenir 2 weighs 103 units in my suitcase.
Souvenir 3 weighs 108 units in my suitcase.
Souvenir 4 weighs 107 units in my suitcase.
Souvenir 5 weighs 106 units in my suitcase.
Souvenir 6 weighs 102 units in my suitcase.
Souvenir 7 weighs 106 units in my suitcase.
Souvenir 8 weighs 100 units in my suitcase.
Souvenir 9 weighs 108 units in my suitcase.
Souvenir 10 weighs 104 units in my suitcase.
Packing 0 together with 3 gives me an extra 103 sentimental points.
Packing 0 together with 7 gives me an extra 105 sentimental points.
Packing 0 together with 9 gives me an extra 104 sentimental points.
Packing 1 together with 3 gives me an extra 108 sentimental points.
Packing 1 together with 6 gives me an extra 105 sentimental points.
Packing 1 together with 7 gives me an extra 108 sentimental points.
Packing 1 together with 10 gives me an extra 105 sentimental points.
Packing 2 together with 8 gives me an extra 105 sentimental points.
Packing 3 together with 7 gives me an extra 106 sentimental points.
Packing 3 together with 10 gives me an extra 102 sentimental points.
Packing 4 together with 10 gives me an extra 104 sentimental points.
Packing 6 together with 7 gives me an extra 100 sentimental points.
I'll use these details to decide what to pack so I maximize sentimental payoff while staying under the 937 limit.

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': [117, 116, 114, 119, 118, 117, 113, 117, 111, 119, 115], 'quadratic_coeffs': [[0, 0, 0, 103, 0, 0, 0, 105, 0, 104, 0], [0, 0, 0, 108, 0, 0, 105, 108, 0, 0, 105], [0, 0, 0, 0, 0, 0, 0, 0, 105, 0, 0], [0, 0, 0, 0, 0, 0, 0, 106, 0, 0, 102], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 104], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 100, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 106, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [106, 105, 103, 108, 107, 106, 102, 106, 100, 108, 104], 'capacity': 937, 'solution': [0, 1, 3, 4, 6, 7, 9, 10], 'obj': 2084.0, 'problem_type': 'QKP'}","[0, 1, 3, 4, 6, 7, 9, 10]",2084.0,"{'problem_type': 'QKP', 'num_items': 11, 'capacity': 937, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 'linear_pairs': [{'item_id': 0, 'linear_profit': 117}, {'item_id': 1, 'linear_profit': 116}, {'item_id': 2, 'linear_profit': 114}, {'item_id': 3, 'linear_profit': 119}, {'item_id': 4, 'linear_profit': 118}, {'item_id': 5, 'linear_profit': 117}, {'item_id': 6, 'linear_profit': 113}, {'item_id': 7, 'linear_profit': 117}, {'item_id': 8, 'linear_profit': 111}, {'item_id': 9, 'linear_profit': 119}, {'item_id': 10, 'linear_profit': 115}], 'weight_pairs': [{'item_id': 0, 'weight': 106}, {'item_id': 1, 'weight': 105}, {'item_id': 2, 'weight': 103}, {'item_id': 3, 'weight': 108}, {'item_id': 4, 'weight': 107}, {'item_id': 5, 'weight': 106}, {'item_id': 6, 'weight': 102}, {'item_id': 7, 'weight': 106}, {'item_id': 8, 'weight': 100}, {'item_id': 9, 'weight': 108}, {'item_id': 10, 'weight': 104}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 103}, {'item_i_id': 0, 'item_j_id': 7, 'quadratic_profit': 105}, {'item_i_id': 0, 'item_j_id': 9, 'quadratic_profit': 104}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 108}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 105}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 108}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 105}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 105}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 106}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 102}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 104}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 100}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 106}]}","[0, 1, 3, 4, 6, 7, 9, 10]",10,nl,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 11 available deluxe components 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and the box's shipping capacity 412.
Component 1 provides a base player value of 271.
Component 2 provides a base player value of 143.
Component 3 provides a base player value of 298.
Component 4 provides a base player value of 231.
Component 5 provides a base player value of 180.
Component 6 provides a base player value of 418.
Component 7 provides a base player value of 504.
Component 8 provides a base player value of 444.
Component 9 provides a base player value of 221.
Component 10 provides a base player value of 184.
Component 11 provides a base player value of 391.
Component 1 contributes 277 to the package weight.
Component 2 contributes 123 to the package weight.
Component 3 contributes 285 to the package weight.
Component 4 contributes 241 to the package weight.
Component 5 contributes 175 to the package weight.
Component 6 contributes 427 to the package weight.
Component 7 contributes 489 to the package weight.
Component 8 contributes 468 to the package weight.
Component 9 contributes 242 to the package weight.
Component 10 contributes 206 to the package weight.
Component 11 contributes 404 to the package weight.
Including both 1 and 3 grants an extra pair synergy of 328.
Including both 2 and 7 grants an extra pair synergy of 177.
Including both 3 and 11 grants an extra pair synergy of 250.
Including both 4 and 11 grants an extra pair synergy of 228.
Including both 5 and 7 grants an extra pair synergy of 268.
Including both 5 and 8 grants an extra pair synergy of 329.
Including both 5 and 11 grants an extra pair synergy of 157.
Including both 6 and 8 grants an extra pair synergy of 403.
Including both 6 and 10 grants an extra pair synergy of 318.
Including both 8 and 9 grants an extra pair synergy of 399.
Selections must respect the 412 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': [271, 143, 298, 231, 180, 418, 504, 444, 221, 184, 391], 'quadratic_coeffs': [[0, 0, 328, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 177, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 250], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 228], [0, 0, 0, 0, 0, 0, 268, 329, 0, 0, 157], [0, 0, 0, 0, 0, 388, 0, 403, 0, 318, 0], [0, 0, 0, 0, 0, 0, 364, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 430, 399, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 260, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 372, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [277, 123, 285, 241, 175, 427, 489, 468, 242, 206, 404], 'capacity': 412, 'solution': [1, 2], 'obj': 441.0, 'problem_type': 'QKP'}","[1, 2]",441.0,"{'problem_type': 'QKP', 'num_items': 11, 'capacity': 412, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 'linear_pairs': [{'item_id': 1, 'linear_profit': 271}, {'item_id': 2, 'linear_profit': 143}, {'item_id': 3, 'linear_profit': 298}, {'item_id': 4, 'linear_profit': 231}, {'item_id': 5, 'linear_profit': 180}, {'item_id': 6, 'linear_profit': 418}, {'item_id': 7, 'linear_profit': 504}, {'item_id': 8, 'linear_profit': 444}, {'item_id': 9, 'linear_profit': 221}, {'item_id': 10, 'linear_profit': 184}, {'item_id': 11, 'linear_profit': 391}], 'weight_pairs': [{'item_id': 1, 'weight': 277}, {'item_id': 2, 'weight': 123}, {'item_id': 3, 'weight': 285}, {'item_id': 4, 'weight': 241}, {'item_id': 5, 'weight': 175}, {'item_id': 6, 'weight': 427}, {'item_id': 7, 'weight': 489}, {'item_id': 8, 'weight': 468}, {'item_id': 9, 'weight': 242}, {'item_id': 10, 'weight': 206}, {'item_id': 11, 'weight': 404}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 328}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 177}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 250}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 228}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 268}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 329}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 157}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 388}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 403}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 318}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 364}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 430}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 399}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 260}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 372}]}","[2, 3]",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 15 available photos (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14) and the page limit of 1163 pages.

| photo_id | baseline_effect |
|---|---|
| 0 | 102 |
| 1 | 104 |
| 2 | 101 |
| 3 | 104 |
| 4 | 103 |
| 5 | 103 |
| 6 | 100 |
| 7 | 101 |
| 8 | 104 |
| 9 | 105 |
| 10 | 100 |
| 11 | 100 |
| 12 | 104 |
| 13 | 103 |
| 14 | 102 |

| photo_id | page_cost |
|---|---|
| 0 | 105 |
| 1 | 102 |
| 2 | 105 |
| 3 | 103 |
| 4 | 104 |
| 5 | 101 |
| 6 | 103 |
| 7 | 100 |
| 8 | 105 |
| 9 | 105 |
| 10 | 103 |
| 11 | 102 |
| 12 | 102 |
| 13 | 100 |
| 14 | 103 |

| photo_i_id | photo_j_id | pairing_bonus |
|---|---|---|
| 0 | 1 | 103 |
| 0 | 2 | 103 |
| 0 | 3 | 103 |
| 0 | 5 | 100 |
| 0 | 6 | 104 |
| 0 | 8 | 104 |
| 0 | 9 | 103 |
| 0 | 11 | 104 |
| 1 | 4 | 102 |
| 1 | 6 | 101 |
| 1 | 8 | 100 |
| 1 | 9 | 100 |
| 1 | 10 | 100 |
| 1 | 11 | 101 |
| 1 | 12 | 102 |
| 1 | 13 | 102 |
| 2 | 7 | 105 |
| 2 | 9 | 100 |
| 2 | 10 | 104 |
| 2 | 12 | 103 |
| 2 | 14 | 104 |
| 3 | 4 | 101 |
| 3 | 7 | 102 |
| 3 | 8 | 103 |
| 3 | 11 | 101 |
| 3 | 12 | 103 |
| 3 | 13 | 102 |
| 3 | 14 | 102 |
| 4 | 5 | 104 |
| 4 | 6 | 104 |
| 4 | 14 | 105 |
| 5 | 6 | 101 |
| 5 | 8 | 101 |
| 5 | 10 | 101 |
| 5 | 12 | 104 |
| 6 | 10 | 102 |
| 6 | 12 | 100 |
| 6 | 14 | 100 |
| 7 | 9 | 100 |
| 7 | 10 | 100 |
| 7 | 13 | 101 |
| 7 | 14 | 101 |
| 8 | 9 | 103 |
| 8 | 10 | 101 |
| 8 | 11 | 105 |
| 8 | 12 | 103 |
| 9 | 11 | 100 |
| 9 | 12 | 102 |
| 9 | 14 | 102 |
| 10 | 12 | 105 |
| 10 | 13 | 104 |
| 10 | 14 | 102 |
| 11 | 13 | 100 |

Selections must stay within the 1163-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': [102, 104, 101, 104, 103, 103, 100, 101, 104, 105, 100, 100, 104, 103, 102], 'quadratic_coeffs': [[100, 103, 103, 103, 0, 100, 104, 0, 104, 103, 0, 104, 0, 0, 0], [0, 0, 0, 0, 102, 0, 101, 0, 100, 100, 100, 101, 102, 102, 0], [0, 0, 0, 0, 0, 0, 0, 105, 0, 100, 104, 0, 103, 0, 104], [0, 0, 0, 0, 101, 0, 0, 102, 103, 0, 0, 101, 103, 102, 102], [0, 0, 0, 0, 0, 104, 104, 0, 0, 0, 0, 0, 0, 0, 105], [0, 0, 0, 0, 0, 100, 101, 0, 101, 0, 101, 0, 104, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 102, 0, 100, 0, 100], [0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 100, 0, 0, 101, 101], [0, 0, 0, 0, 0, 0, 0, 0, 105, 103, 101, 105, 103, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 0, 100, 102, 0, 102], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 102, 0, 105, 104, 102], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100]], 'weights': [105, 102, 105, 103, 104, 101, 103, 100, 105, 105, 103, 102, 102, 100, 103], 'capacity': 1163, 'solution': [0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 14], 'obj': 4501.0, 'problem_type': 'QKP'}","[0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 14]",4501.0,"{'problem_type': 'QKP', 'num_items': 15, 'capacity': 1163, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], 'linear_pairs': [{'item_id': 0, 'linear_profit': 102}, {'item_id': 1, 'linear_profit': 104}, {'item_id': 2, 'linear_profit': 101}, {'item_id': 3, 'linear_profit': 104}, {'item_id': 4, 'linear_profit': 103}, {'item_id': 5, 'linear_profit': 103}, {'item_id': 6, 'linear_profit': 100}, {'item_id': 7, 'linear_profit': 101}, {'item_id': 8, 'linear_profit': 104}, {'item_id': 9, 'linear_profit': 105}, {'item_id': 10, 'linear_profit': 100}, {'item_id': 11, 'linear_profit': 100}, {'item_id': 12, 'linear_profit': 104}, {'item_id': 13, 'linear_profit': 103}, {'item_id': 14, 'linear_profit': 102}], 'weight_pairs': [{'item_id': 0, 'weight': 105}, {'item_id': 1, 'weight': 102}, {'item_id': 2, 'weight': 105}, {'item_id': 3, 'weight': 103}, {'item_id': 4, 'weight': 104}, {'item_id': 5, 'weight': 101}, {'item_id': 6, 'weight': 103}, {'item_id': 7, 'weight': 100}, {'item_id': 8, 'weight': 105}, {'item_id': 9, 'weight': 105}, {'item_id': 10, 'weight': 103}, {'item_id': 11, 'weight': 102}, {'item_id': 12, 'weight': 102}, {'item_id': 13, 'weight': 100}, {'item_id': 14, 'weight': 103}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 100}, {'item_i_id': 0, 'item_j_id': 1, 'quadratic_profit': 103}, {'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 103}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 103}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 100}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 104}, {'item_i_id': 0, 'item_j_id': 8, 'quadratic_profit': 104}, {'item_i_id': 0, 'item_j_id': 9, 'quadratic_profit': 103}, {'item_i_id': 0, 'item_j_id': 11, 'quadratic_profit': 104}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 102}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 101}, {'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': 1, 'item_j_id': 11, 'quadratic_profit': 101}, {'item_i_id': 1, 'item_j_id': 12, 'quadratic_profit': 102}, {'item_i_id': 1, 'item_j_id': 13, 'quadratic_profit': 102}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 105}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 100}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 104}, {'item_i_id': 2, 'item_j_id': 12, 'quadratic_profit': 103}, {'item_i_id': 2, 'item_j_id': 14, 'quadratic_profit': 104}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 102}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 103}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 12, 'quadratic_profit': 103}, {'item_i_id': 3, 'item_j_id': 13, 'quadratic_profit': 102}, {'item_i_id': 3, 'item_j_id': 14, 'quadratic_profit': 102}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 104}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 104}, {'item_i_id': 4, 'item_j_id': 14, 'quadratic_profit': 105}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 100}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 101}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 101}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 101}, {'item_i_id': 5, 'item_j_id': 12, 'quadratic_profit': 104}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 102}, {'item_i_id': 6, 'item_j_id': 12, 'quadratic_profit': 100}, {'item_i_id': 6, 'item_j_id': 14, 'quadratic_profit': 100}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 100}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 100}, {'item_i_id': 7, 'item_j_id': 13, 'quadratic_profit': 101}, {'item_i_id': 7, 'item_j_id': 14, 'quadratic_profit': 101}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 105}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 103}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 101}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 105}, {'item_i_id': 8, 'item_j_id': 12, 'quadratic_profit': 103}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 100}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 100}, {'item_i_id': 9, 'item_j_id': 12, 'quadratic_profit': 102}, {'item_i_id': 9, 'item_j_id': 14, 'quadratic_profit': 102}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 102}, {'item_i_id': 10, 'item_j_id': 12, 'quadratic_profit': 105}, {'item_i_id': 10, 'item_j_id': 13, 'quadratic_profit': 104}, {'item_i_id': 10, 'item_j_id': 14, 'quadratic_profit': 102}, {'item_i_id': 11, 'item_j_id': 13, 'quadratic_profit': 100}, {'item_i_id': 14, 'item_j_id': 14, 'quadratic_profit': 100}]}","[0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 14]",12,markdown_table,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 13 food options (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13) and a gear weight limit of 16392 to respect.
Bring 1 — it adds 1957 comfort on its own.
Bring 2 — it adds 3259 comfort on its own.
Bring 3 — it adds 6099 comfort on its own.
Bring 4 — it adds 3962 comfort on its own.
Bring 5 — it adds 2775 comfort on its own.
Bring 6 — it adds 5589 comfort on its own.
Bring 7 — it adds 785 comfort on its own.
Bring 8 — it adds 5779 comfort on its own.
Bring 9 — it adds 6437 comfort on its own.
Bring 10 — it adds 4569 comfort on its own.
Bring 11 — it adds 1616 comfort on its own.
Bring 12 — it adds 2718 comfort on its own.
Bring 13 — it adds 1369 comfort on its own.
1 consumes 1307 of your pack's weight.
2 consumes 2609 of your pack's weight.
3 consumes 5449 of your pack's weight.
4 consumes 3312 of your pack's weight.
5 consumes 2125 of your pack's weight.
6 consumes 4939 of your pack's weight.
7 consumes 135 of your pack's weight.
8 consumes 5129 of your pack's weight.
9 consumes 5787 of your pack's weight.
10 consumes 3919 of your pack's weight.
11 consumes 966 of your pack's weight.
12 consumes 2068 of your pack's weight.
13 consumes 719 of your pack's weight.
Packing 1 with 3 grants an extra 2754 morale bonus.
Packing 1 with 5 grants an extra 2540 morale bonus.
Packing 1 with 8 grants an extra 2742 morale bonus.
Packing 1 with 11 grants an extra 2864 morale bonus.
Packing 2 with 3 grants an extra 664 morale bonus.
Packing 2 with 4 grants an extra 2297 morale bonus.
Packing 2 with 7 grants an extra 2139 morale bonus.
Packing 2 with 8 grants an extra 2935 morale bonus.
Packing 2 with 10 grants an extra 3008 morale bonus.
Packing 2 with 11 grants an extra 3657 morale bonus.
Packing 3 with 5 grants an extra 4534 morale bonus.
Packing 3 with 6 grants an extra 5603 morale bonus.
Packing 3 with 8 grants an extra 4228 morale bonus.
Packing 3 with 10 grants an extra 4895 morale bonus.
Packing 3 with 11 grants an extra 3902 morale bonus.
Packing 3 with 13 grants an extra 1168 morale bonus.
Packing 4 with 5 grants an extra 3602 morale bonus.
Packing 4 with 8 grants an extra 3060 morale bonus.
Packing 4 with 11 grants an extra 1316 morale bonus.
Packing 5 with 6 grants an extra 3057 morale bonus.
Packing 5 with 7 grants an extra 2664 morale bonus.
Packing 5 with 9 grants an extra 2107 morale bonus.
Packing 5 with 11 grants an extra 3640 morale bonus.
Packing 5 with 12 grants an extra 4782 morale bonus.
Packing 6 with 7 grants an extra 4912 morale bonus.
Packing 6 with 8 grants an extra 1459 morale bonus.
Packing 6 with 10 grants an extra 2878 morale bonus.
Packing 6 with 11 grants an extra 3051 morale bonus.
Packing 7 with 9 grants an extra 4100 morale bonus.
Packing 7 with 13 grants an extra 2882 morale bonus.
Packing 8 with 10 grants an extra 2486 morale bonus.
Packing 8 with 11 grants an extra 1615 morale bonus.
Packing 8 with 12 grants an extra 1853 morale bonus.
Packing 9 with 10 grants an extra 5123 morale bonus.
Packing 9 with 11 grants an extra 6087 morale bonus.
Packing 9 with 12 grants an extra 3791 morale bonus.
Packing 10 with 12 grants an extra 2962 morale bonus.
Packing 11 with 13 grants an extra 1020 morale bonus.
Packing 12 with 13 grants an extra 6009 morale bonus.
Choose the set that fits the 16392 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': [1957, 3259, 6099, 3962, 2775, 5589, 785, 5779, 6437, 4569, 1616, 2718, 1369], 'quadratic_coeffs': [[0, 0, 2754, 0, 2540, 0, 0, 2742, 0, 0, 2864, 0, 0], [0, 3853, 664, 2297, 0, 0, 2139, 2935, 0, 3008, 3657, 0, 0], [0, 0, 4372, 0, 4534, 5603, 0, 4228, 0, 4895, 3902, 0, 1168], [0, 0, 0, 0, 3602, 0, 0, 3060, 0, 0, 1316, 0, 0], [0, 0, 0, 0, 0, 3057, 2664, 0, 2107, 0, 3640, 4782, 0], [0, 0, 0, 0, 0, 0, 4912, 1459, 0, 2878, 3051, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4100, 0, 0, 0, 2882], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2486, 1615, 1853, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4439, 5123, 6087, 3791, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2781, 0, 2962, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2018, 0, 1020], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6009], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 954]], 'weights': [1307, 2609, 5449, 3312, 2125, 4939, 135, 5129, 5787, 3919, 966, 2068, 719], 'capacity': 16392, 'solution': [0, 1, 4, 6, 8, 10, 11, 12], 'obj': 69198.0, 'problem_type': 'QKP'}","[0, 1, 4, 6, 8, 10, 11, 12]",69198.0,"{'problem_type': 'QKP', 'num_items': 13, 'capacity': 16392, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 'linear_pairs': [{'item_id': 1, 'linear_profit': 1957}, {'item_id': 2, 'linear_profit': 3259}, {'item_id': 3, 'linear_profit': 6099}, {'item_id': 4, 'linear_profit': 3962}, {'item_id': 5, 'linear_profit': 2775}, {'item_id': 6, 'linear_profit': 5589}, {'item_id': 7, 'linear_profit': 785}, {'item_id': 8, 'linear_profit': 5779}, {'item_id': 9, 'linear_profit': 6437}, {'item_id': 10, 'linear_profit': 4569}, {'item_id': 11, 'linear_profit': 1616}, {'item_id': 12, 'linear_profit': 2718}, {'item_id': 13, 'linear_profit': 1369}], 'weight_pairs': [{'item_id': 1, 'weight': 1307}, {'item_id': 2, 'weight': 2609}, {'item_id': 3, 'weight': 5449}, {'item_id': 4, 'weight': 3312}, {'item_id': 5, 'weight': 2125}, {'item_id': 6, 'weight': 4939}, {'item_id': 7, 'weight': 135}, {'item_id': 8, 'weight': 5129}, {'item_id': 9, 'weight': 5787}, {'item_id': 10, 'weight': 3919}, {'item_id': 11, 'weight': 966}, {'item_id': 12, 'weight': 2068}, {'item_id': 13, 'weight': 719}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 2754}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 2540}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 2742}, {'item_i_id': 1, 'item_j_id': 11, 'quadratic_profit': 2864}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 3853}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 664}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 2297}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 2139}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 2935}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 3008}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 3657}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 4372}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 4534}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 5603}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 4228}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 4895}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 3902}, {'item_i_id': 3, 'item_j_id': 13, 'quadratic_profit': 1168}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 3602}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 3060}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 1316}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 3057}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 2664}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 2107}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 3640}, {'item_i_id': 5, 'item_j_id': 12, 'quadratic_profit': 4782}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 4912}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 1459}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 2878}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 3051}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 4100}, {'item_i_id': 7, 'item_j_id': 13, 'quadratic_profit': 2882}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 2486}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 1615}, {'item_i_id': 8, 'item_j_id': 12, 'quadratic_profit': 1853}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 4439}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 5123}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 6087}, {'item_i_id': 9, 'item_j_id': 12, 'quadratic_profit': 3791}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 2781}, {'item_i_id': 10, 'item_j_id': 12, 'quadratic_profit': 2962}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 2018}, {'item_i_id': 11, 'item_j_id': 13, 'quadratic_profit': 1020}, {'item_i_id': 12, 'item_j_id': 13, 'quadratic_profit': 6009}, {'item_i_id': 13, 'item_j_id': 13, 'quadratic_profit': 954}]}","[1, 2, 5, 7, 9, 11, 12, 13]",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.

There are 14 distinct supplies (listed as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14), and we have 691 total shelf volume to fill.
We get 137 usefulness if we stock 1.
We get 126 usefulness if we stock 2.
We get 146 usefulness if we stock 3.
We get 115 usefulness if we stock 4.
We get 111 usefulness if we stock 5.
We get 106 usefulness if we stock 6.
We get 106 usefulness if we stock 7.
We get 145 usefulness if we stock 8.
We get 114 usefulness if we stock 9.
We get 114 usefulness if we stock 10.
We get 131 usefulness if we stock 11.
We get 132 usefulness if we stock 12.
We get 150 usefulness if we stock 13.
We get 123 usefulness if we stock 14.
1 occupies 136 shelf volume.
2 occupies 120 shelf volume.
3 occupies 141 shelf volume.
4 occupies 108 shelf volume.
5 occupies 115 shelf volume.
6 occupies 109 shelf volume.
7 occupies 104 shelf volume.
8 occupies 143 shelf volume.
9 occupies 110 shelf volume.
10 occupies 113 shelf volume.
11 occupies 131 shelf volume.
12 occupies 125 shelf volume.
13 occupies 142 shelf volume.
14 occupies 117 shelf volume.
If we stock both 1 and 2, we gain 127 extra usefulness.
If we stock both 1 and 3, we gain 132 extra usefulness.
If we stock both 1 and 4, we gain 126 extra usefulness.
If we stock both 1 and 6, we gain 136 extra usefulness.
If we stock both 1 and 7, we gain 131 extra usefulness.
If we stock both 1 and 8, we gain 140 extra usefulness.
If we stock both 1 and 10, we gain 133 extra usefulness.
If we stock both 1 and 11, we gain 138 extra usefulness.
If we stock both 1 and 12, we gain 130 extra usefulness.
If we stock both 1 and 13, we gain 136 extra usefulness.
If we stock both 2 and 4, we gain 124 extra usefulness.
If we stock both 2 and 5, we gain 122 extra usefulness.
If we stock both 2 and 6, we gain 120 extra usefulness.
If we stock both 2 and 8, we gain 122 extra usefulness.
If we stock both 2 and 9, we gain 111 extra usefulness.
If we stock both 2 and 11, we gain 121 extra usefulness.
If we stock both 2 and 12, we gain 129 extra usefulness.
If we stock both 2 and 14, we gain 128 extra usefulness.
If we stock both 3 and 4, we gain 144 extra usefulness.
If we stock both 3 and 6, we gain 126 extra usefulness.
If we stock both 3 and 7, we gain 121 extra usefulness.
If we stock both 3 and 8, we gain 131 extra usefulness.
If we stock both 3 and 10, we gain 143 extra usefulness.
If we stock both 3 and 11, we gain 131 extra usefulness.
If we stock both 3 and 12, we gain 137 extra usefulness.
If we stock both 3 and 13, we gain 123 extra usefulness.
If we stock both 4 and 6, we gain 113 extra usefulness.
If we stock both 4 and 7, we gain 110 extra usefulness.
If we stock both 4 and 9, we gain 113 extra usefulness.
If we stock both 4 and 11, we gain 124 extra usefulness.
If we stock both 4 and 12, we gain 121 extra usefulness.
If we stock both 4 and 13, we gain 129 extra usefulness.
If we stock both 4 and 14, we gain 117 extra usefulness.
If we stock both 5 and 6, we gain 102 extra usefulness.
If we stock both 5 and 7, we gain 116 extra usefulness.
If we stock both 5 and 9, we gain 128 extra usefulness.
If we stock both 5 and 10, we gain 126 extra usefulness.
If we stock both 5 and 11, we gain 124 extra usefulness.
If we stock both 5 and 12, we gain 111 extra usefulness.
If we stock both 5 and 13, we gain 125 extra usefulness.
If we stock both 6 and 7, we gain 134 extra usefulness.
If we stock both 6 and 8, we gain 127 extra usefulness.
If we stock both 6 and 9, we gain 141 extra usefulness.
If we stock both 6 and 10, we gain 118 extra usefulness.
If we stock both 6 and 12, we gain 141 extra usefulness.
If we stock both 6 and 13, we gain 122 extra usefulness.
If we stock both 7 and 9, we gain 119 extra usefulness.
If we stock both 7 and 10, we gain 124 extra usefulness.
If we stock both 7 and 11, we gain 127 extra usefulness.
If we stock both 7 and 12, we gain 130 extra usefulness.
If we stock both 7 and 14, we gain 118 extra usefulness.
If we stock both 8 and 9, we gain 119 extra usefulness.
If we stock both 8 and 10, we gain 115 extra usefulness.
If we stock both 8 and 11, we gain 128 extra usefulness.
If we stock both 8 and 13, we gain 119 extra usefulness.
If we stock both 8 and 14, we gain 120 extra usefulness.
If we stock both 9 and 10, we gain 136 extra usefulness.
If we stock both 9 and 11, we gain 133 extra usefulness.
If we stock both 9 and 12, we gain 136 extra usefulness.
If we stock both 9 and 13, we gain 131 extra usefulness.
If we stock both 9 and 14, we gain 124 extra usefulness.
If we stock both 10 and 11, we gain 143 extra usefulness.
If we stock both 10 and 14, we gain 129 extra usefulness.
We'll pick non-duplicated supplies that fit within 691 to maximize overall usefulness.

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': [137, 126, 146, 115, 111, 106, 106, 145, 114, 114, 131, 132, 150, 123], 'quadratic_coeffs': [[136, 127, 132, 126, 0, 136, 131, 140, 0, 133, 138, 130, 136, 0], [0, 112, 0, 124, 122, 120, 0, 122, 111, 0, 121, 129, 0, 128], [0, 0, 128, 144, 0, 126, 121, 131, 0, 143, 131, 137, 123, 0], [0, 0, 0, 0, 0, 113, 110, 0, 113, 0, 124, 121, 129, 117], [0, 0, 0, 0, 111, 102, 116, 0, 128, 126, 124, 111, 125, 0], [0, 0, 0, 0, 0, 138, 134, 127, 141, 118, 0, 141, 122, 0], [0, 0, 0, 0, 0, 0, 141, 0, 119, 124, 127, 130, 0, 118], [0, 0, 0, 0, 0, 0, 0, 0, 119, 115, 128, 0, 119, 120], [0, 0, 0, 0, 0, 0, 0, 0, 132, 136, 133, 136, 131, 124], [0, 0, 0, 0, 0, 0, 0, 0, 0, 126, 143, 0, 0, 129], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 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': [136, 120, 141, 108, 115, 109, 104, 143, 110, 113, 131, 125, 142, 117], 'capacity': 691, 'solution': [4, 5, 6, 8, 9, 10], 'obj': 2453.0, 'problem_type': 'QKP'}","[4, 5, 6, 8, 9, 10]",2453.0,"{'problem_type': 'QKP', 'num_items': 14, 'capacity': 691, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], 'linear_pairs': [{'item_id': 1, 'linear_profit': 137}, {'item_id': 2, 'linear_profit': 126}, {'item_id': 3, 'linear_profit': 146}, {'item_id': 4, 'linear_profit': 115}, {'item_id': 5, 'linear_profit': 111}, {'item_id': 6, 'linear_profit': 106}, {'item_id': 7, 'linear_profit': 106}, {'item_id': 8, 'linear_profit': 145}, {'item_id': 9, 'linear_profit': 114}, {'item_id': 10, 'linear_profit': 114}, {'item_id': 11, 'linear_profit': 131}, {'item_id': 12, 'linear_profit': 132}, {'item_id': 13, 'linear_profit': 150}, {'item_id': 14, 'linear_profit': 123}], 'weight_pairs': [{'item_id': 1, 'weight': 136}, {'item_id': 2, 'weight': 120}, {'item_id': 3, 'weight': 141}, {'item_id': 4, 'weight': 108}, {'item_id': 5, 'weight': 115}, {'item_id': 6, 'weight': 109}, {'item_id': 7, 'weight': 104}, {'item_id': 8, 'weight': 143}, {'item_id': 9, 'weight': 110}, {'item_id': 10, 'weight': 113}, {'item_id': 11, 'weight': 131}, {'item_id': 12, 'weight': 125}, {'item_id': 13, 'weight': 142}, {'item_id': 14, 'weight': 117}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 136}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 127}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 132}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 126}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 136}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 131}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 140}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 133}, {'item_i_id': 1, 'item_j_id': 11, 'quadratic_profit': 138}, {'item_i_id': 1, 'item_j_id': 12, 'quadratic_profit': 130}, {'item_i_id': 1, 'item_j_id': 13, 'quadratic_profit': 136}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 112}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 124}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 122}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 120}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 122}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 111}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 121}, {'item_i_id': 2, 'item_j_id': 12, 'quadratic_profit': 129}, {'item_i_id': 2, 'item_j_id': 14, 'quadratic_profit': 128}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 128}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 144}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 126}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 121}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 131}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 143}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 131}, {'item_i_id': 3, 'item_j_id': 12, 'quadratic_profit': 137}, {'item_i_id': 3, 'item_j_id': 13, 'quadratic_profit': 123}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 113}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 110}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 113}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 124}, {'item_i_id': 4, 'item_j_id': 12, 'quadratic_profit': 121}, {'item_i_id': 4, 'item_j_id': 13, 'quadratic_profit': 129}, {'item_i_id': 4, 'item_j_id': 14, 'quadratic_profit': 117}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 111}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 102}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 116}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 128}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 126}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 124}, {'item_i_id': 5, 'item_j_id': 12, 'quadratic_profit': 111}, {'item_i_id': 5, 'item_j_id': 13, 'quadratic_profit': 125}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 138}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 134}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 127}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 141}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 118}, {'item_i_id': 6, 'item_j_id': 12, 'quadratic_profit': 141}, {'item_i_id': 6, 'item_j_id': 13, 'quadratic_profit': 122}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 141}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 119}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 124}, {'item_i_id': 7, 'item_j_id': 11, 'quadratic_profit': 127}, {'item_i_id': 7, 'item_j_id': 12, 'quadratic_profit': 130}, {'item_i_id': 7, 'item_j_id': 14, 'quadratic_profit': 118}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 119}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 115}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 128}, {'item_i_id': 8, 'item_j_id': 13, 'quadratic_profit': 119}, {'item_i_id': 8, 'item_j_id': 14, 'quadratic_profit': 120}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 132}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 136}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 133}, {'item_i_id': 9, 'item_j_id': 12, 'quadratic_profit': 136}, {'item_i_id': 9, 'item_j_id': 13, 'quadratic_profit': 131}, {'item_i_id': 9, 'item_j_id': 14, 'quadratic_profit': 124}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 126}, {'item_i_id': 10, 'item_j_id': 11, 'quadratic_profit': 143}, {'item_i_id': 10, 'item_j_id': 14, 'quadratic_profit': 129}]}","[5, 6, 7, 9, 10, 11]",14,nl,1
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=15
# max_kit_weight=319
# available_item_ids=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

item_identifier,base_learning_value
1,427
2,589
3,340
4,760
5,297
6,416
7,730
8,168
9,694
10,614
11,457
12,135
13,485
14,424
15,420

item_identifier,item_weight
1,239
2,313
3,588
4,583
5,751
6,890
7,759
8,646
9,102
10,205
11,101
12,801
13,494
14,132
15,748

item_i_identifier,item_j_identifier,interaction_learning_bonus
1,2,340
1,5,136
1,6,499
1,7,725
1,10,279
1,12,300
1,14,337
2,3,433
2,4,836
2,5,817
2,8,757
2,12,449
2,13,792
2,14,261
3,4,278
3,5,740
3,6,113
3,7,357
3,8,849
3,10,297
3,11,269
3,13,501
3,14,388
3,15,756
4,5,558
4,6,196
4,7,495
4,8,206
4,9,729
4,12,497
4,14,246
4,15,868
5,6,469
5,9,663
5,11,769
5,12,156
5,13,519
5,15,751
6,11,880
6,12,300
6,15,683
7,8,555
7,9,459
7,11,310
7,12,656
7,13,884
7,14,502
8,10,275
8,11,837
8,13,185
8,14,306
8,15,370
9,13,417
9,14,339
9,15,859
10,12,536
10,13,708
10,14,352
10,15,359
11,13,206
11,14,191
12,13,552
12,14,578
13,14,460
14,15,735

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': [427, 589, 340, 760, 297, 416, 730, 168, 694, 614, 457, 135, 485, 424, 420], 'quadratic_coeffs': [[0, 340, 0, 0, 136, 499, 725, 0, 0, 279, 0, 300, 0, 337, 0], [0, 0, 433, 836, 817, 0, 0, 757, 0, 0, 0, 449, 792, 261, 0], [0, 0, 847, 278, 740, 113, 357, 849, 0, 297, 269, 0, 501, 388, 756], [0, 0, 0, 0, 558, 196, 495, 206, 729, 0, 0, 497, 0, 246, 868], [0, 0, 0, 0, 0, 469, 0, 0, 663, 0, 769, 156, 519, 0, 751], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 880, 300, 0, 0, 683], [0, 0, 0, 0, 0, 0, 0, 555, 459, 0, 310, 656, 884, 502, 0], [0, 0, 0, 0, 0, 0, 0, 378, 0, 275, 837, 0, 185, 306, 370], [0, 0, 0, 0, 0, 0, 0, 0, 426, 0, 0, 0, 417, 339, 859], [0, 0, 0, 0, 0, 0, 0, 0, 0, 653, 0, 536, 708, 352, 359], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 206, 191, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 552, 578, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 462, 460, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 338, 735], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 360]], 'weights': [239, 313, 588, 583, 751, 890, 759, 646, 102, 205, 101, 801, 494, 132, 748], 'capacity': 319, 'solution': [8, 13], 'obj': 1457.0, 'problem_type': 'QKP'}","[8, 13]",1457.0,"{'problem_type': 'QKP', 'num_items': 15, 'capacity': 319, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], 'linear_pairs': [{'item_id': 1, 'linear_profit': 427}, {'item_id': 2, 'linear_profit': 589}, {'item_id': 3, 'linear_profit': 340}, {'item_id': 4, 'linear_profit': 760}, {'item_id': 5, 'linear_profit': 297}, {'item_id': 6, 'linear_profit': 416}, {'item_id': 7, 'linear_profit': 730}, {'item_id': 8, 'linear_profit': 168}, {'item_id': 9, 'linear_profit': 694}, {'item_id': 10, 'linear_profit': 614}, {'item_id': 11, 'linear_profit': 457}, {'item_id': 12, 'linear_profit': 135}, {'item_id': 13, 'linear_profit': 485}, {'item_id': 14, 'linear_profit': 424}, {'item_id': 15, 'linear_profit': 420}], 'weight_pairs': [{'item_id': 1, 'weight': 239}, {'item_id': 2, 'weight': 313}, {'item_id': 3, 'weight': 588}, {'item_id': 4, 'weight': 583}, {'item_id': 5, 'weight': 751}, {'item_id': 6, 'weight': 890}, {'item_id': 7, 'weight': 759}, {'item_id': 8, 'weight': 646}, {'item_id': 9, 'weight': 102}, {'item_id': 10, 'weight': 205}, {'item_id': 11, 'weight': 101}, {'item_id': 12, 'weight': 801}, {'item_id': 13, 'weight': 494}, {'item_id': 14, 'weight': 132}, {'item_id': 15, 'weight': 748}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 340}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 136}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 499}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 725}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 279}, {'item_i_id': 1, 'item_j_id': 12, 'quadratic_profit': 300}, {'item_i_id': 1, 'item_j_id': 14, 'quadratic_profit': 337}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 433}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 836}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 817}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 757}, {'item_i_id': 2, 'item_j_id': 12, 'quadratic_profit': 449}, {'item_i_id': 2, 'item_j_id': 13, 'quadratic_profit': 792}, {'item_i_id': 2, 'item_j_id': 14, 'quadratic_profit': 261}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 847}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 278}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 740}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 113}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 357}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 849}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 297}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 269}, {'item_i_id': 3, 'item_j_id': 13, 'quadratic_profit': 501}, {'item_i_id': 3, 'item_j_id': 14, 'quadratic_profit': 388}, {'item_i_id': 3, 'item_j_id': 15, 'quadratic_profit': 756}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 558}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 196}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 495}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 206}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 729}, {'item_i_id': 4, 'item_j_id': 12, 'quadratic_profit': 497}, {'item_i_id': 4, 'item_j_id': 14, 'quadratic_profit': 246}, {'item_i_id': 4, 'item_j_id': 15, 'quadratic_profit': 868}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 469}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 663}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 769}, {'item_i_id': 5, 'item_j_id': 12, 'quadratic_profit': 156}, {'item_i_id': 5, 'item_j_id': 13, 'quadratic_profit': 519}, {'item_i_id': 5, 'item_j_id': 15, 'quadratic_profit': 751}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 880}, {'item_i_id': 6, 'item_j_id': 12, 'quadratic_profit': 300}, {'item_i_id': 6, 'item_j_id': 15, 'quadratic_profit': 683}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 555}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 459}, {'item_i_id': 7, 'item_j_id': 11, 'quadratic_profit': 310}, {'item_i_id': 7, 'item_j_id': 12, 'quadratic_profit': 656}, {'item_i_id': 7, 'item_j_id': 13, 'quadratic_profit': 884}, {'item_i_id': 7, 'item_j_id': 14, 'quadratic_profit': 502}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 378}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 275}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 837}, {'item_i_id': 8, 'item_j_id': 13, 'quadratic_profit': 185}, {'item_i_id': 8, 'item_j_id': 14, 'quadratic_profit': 306}, {'item_i_id': 8, 'item_j_id': 15, 'quadratic_profit': 370}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 426}, {'item_i_id': 9, 'item_j_id': 13, 'quadratic_profit': 417}, {'item_i_id': 9, 'item_j_id': 14, 'quadratic_profit': 339}, {'item_i_id': 9, 'item_j_id': 15, 'quadratic_profit': 859}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 653}, {'item_i_id': 10, 'item_j_id': 12, 'quadratic_profit': 536}, {'item_i_id': 10, 'item_j_id': 13, 'quadratic_profit': 708}, {'item_i_id': 10, 'item_j_id': 14, 'quadratic_profit': 352}, {'item_i_id': 10, 'item_j_id': 15, 'quadratic_profit': 359}, {'item_i_id': 11, 'item_j_id': 13, 'quadratic_profit': 206}, {'item_i_id': 11, 'item_j_id': 14, 'quadratic_profit': 191}, {'item_i_id': 12, 'item_j_id': 13, 'quadratic_profit': 552}, {'item_i_id': 12, 'item_j_id': 14, 'quadratic_profit': 578}, {'item_i_id': 13, 'item_j_id': 13, 'quadratic_profit': 462}, {'item_i_id': 13, 'item_j_id': 14, 'quadratic_profit': 460}, {'item_i_id': 14, 'item_j_id': 14, 'quadratic_profit': 338}, {'item_i_id': 14, 'item_j_id': 15, 'quadratic_profit': 735}, {'item_i_id': 15, 'item_j_id': 15, 'quadratic_profit': 360}]}","[9, 14]",15,csv,1
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 14 prints available — A, B, C, D, E, F, G, H, I, J, K, L, M, N — and the curator has 1654 total wall area to allocate.
Print A provides a base visual value of 127 when hung.
Print B provides a base visual value of 116 when hung.
Print C provides a base visual value of 146 when hung.
Print D provides a base visual value of 105 when hung.
Print E provides a base visual value of 137 when hung.
Print F provides a base visual value of 123 when hung.
Print G provides a base visual value of 110 when hung.
Print H provides a base visual value of 147 when hung.
Print I provides a base visual value of 142 when hung.
Print J provides a base visual value of 146 when hung.
Print K provides a base visual value of 138 when hung.
Print L provides a base visual value of 112 when hung.
Print M provides a base visual value of 139 when hung.
Print N provides a base visual value of 133 when hung.
Print A consumes 113 of the available wall area.
Print B consumes 135 of the available wall area.
Print C consumes 134 of the available wall area.
Print D consumes 102 of the available wall area.
Print E consumes 135 of the available wall area.
Print F consumes 122 of the available wall area.
Print G consumes 127 of the available wall area.
Print H consumes 106 of the available wall area.
Print I consumes 121 of the available wall area.
Print J consumes 121 of the available wall area.
Print K consumes 111 of the available wall area.
Print L consumes 124 of the available wall area.
Print M consumes 140 of the available wall area.
Print N consumes 113 of the available wall area.
Displaying prints A and C together yields an additional synergy of 129.
Displaying prints A and D together yields an additional synergy of 111.
Displaying prints A and E together yields an additional synergy of 118.
Displaying prints A and F together yields an additional synergy of 114.
Displaying prints A and H together yields an additional synergy of 111.
Displaying prints A and I together yields an additional synergy of 127.
Displaying prints A and J together yields an additional synergy of 122.
Displaying prints A and K together yields an additional synergy of 128.
Displaying prints A and L together yields an additional synergy of 121.
Displaying prints B and C together yields an additional synergy of 131.
Displaying prints B and D together yields an additional synergy of 124.
Displaying prints B and E together yields an additional synergy of 120.
Displaying prints B and F together yields an additional synergy of 131.
Displaying prints B and G together yields an additional synergy of 119.
Displaying prints B and H together yields an additional synergy of 124.
Displaying prints B and I together yields an additional synergy of 122.
Displaying prints B and J together yields an additional synergy of 139.
Displaying prints B and K together yields an additional synergy of 104.
Displaying prints B and L together yields an additional synergy of 115.
Displaying prints B and M together yields an additional synergy of 122.
Displaying prints C and E together yields an additional synergy of 138.
Displaying prints C and G together yields an additional synergy of 121.
Displaying prints C and H together yields an additional synergy of 139.
Displaying prints C and I together yields an additional synergy of 116.
Displaying prints C and J together yields an additional synergy of 132.
Displaying prints C and L together yields an additional synergy of 113.
Displaying prints C and M together yields an additional synergy of 105.
Displaying prints C and N together yields an additional synergy of 118.
Displaying prints D and H together yields an additional synergy of 120.
Displaying prints D and I together yields an additional synergy of 112.
Displaying prints D and K together yields an additional synergy of 130.
Displaying prints D and L together yields an additional synergy of 125.
Displaying prints D and N together yields an additional synergy of 143.
Displaying prints E and F together yields an additional synergy of 139.
Displaying prints E and H together yields an additional synergy of 133.
Displaying prints E and J together yields an additional synergy of 123.
Displaying prints E and K together yields an additional synergy of 127.
Displaying prints E and L together yields an additional synergy of 127.
Displaying prints E and M together yields an additional synergy of 116.
Displaying prints E and N together yields an additional synergy of 116.
Displaying prints F and G together yields an additional synergy of 121.
Displaying prints F and I together yields an additional synergy of 107.
Displaying prints F and J together yields an additional synergy of 125.
Displaying prints F and K together yields an additional synergy of 114.
Displaying prints F and M together yields an additional synergy of 123.
Displaying prints F and N together yields an additional synergy of 114.
Displaying prints G and H together yields an additional synergy of 113.
Displaying prints G and J together yields an additional synergy of 124.
Displaying prints G and K together yields an additional synergy of 116.
Displaying prints G and L together yields an additional synergy of 114.
Displaying prints G and M together yields an additional synergy of 110.
Displaying prints H and I together yields an additional synergy of 115.
Displaying prints H and J together yields an additional synergy of 120.
Displaying prints H and L together yields an additional synergy of 121.
Displaying prints H and M together yields an additional synergy of 124.
Displaying prints H and N together yields an additional synergy of 117.
Displaying prints I and J together yields an additional synergy of 126.
Displaying prints I and K together yields an additional synergy of 134.
Displaying prints I and L together yields an additional synergy of 122.
Displaying prints I and M together yields an additional synergy of 117.
Displaying prints I and N together yields an additional synergy of 141.
Displaying prints J and K together yields an additional synergy of 143.
Displaying prints J and L together yields an additional synergy of 127.
Displaying prints J and M together yields an additional synergy of 122.
Displaying prints J and N together yields an additional synergy of 133.
Displaying prints K and L together yields an additional synergy of 119.
Displaying prints K and N together yields an additional synergy of 112.
Displaying prints L and M together yields an additional synergy of 125.
Displaying prints L and N together yields an additional synergy of 126.
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': [127, 116, 146, 105, 137, 123, 110, 147, 142, 146, 138, 112, 139, 133], 'quadratic_coeffs': [[114, 0, 129, 111, 118, 114, 0, 111, 127, 122, 128, 121, 0, 0], [0, 139, 131, 124, 120, 131, 119, 124, 122, 139, 104, 115, 122, 0], [0, 0, 0, 0, 138, 0, 121, 139, 116, 132, 0, 113, 105, 118], [0, 0, 0, 101, 0, 0, 0, 120, 112, 0, 130, 125, 0, 143], [0, 0, 0, 0, 122, 139, 0, 133, 0, 123, 127, 127, 116, 116], [0, 0, 0, 0, 0, 120, 121, 0, 107, 125, 114, 0, 123, 114], [0, 0, 0, 0, 0, 0, 135, 113, 0, 124, 116, 114, 110, 0], [0, 0, 0, 0, 0, 0, 0, 118, 115, 120, 0, 121, 124, 117], [0, 0, 0, 0, 0, 0, 0, 0, 103, 126, 134, 122, 117, 141], [0, 0, 0, 0, 0, 0, 0, 0, 0, 123, 143, 127, 122, 133], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 123, 119, 0, 112], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 145, 125, 126], [0, 0, 0, 0, 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': [113, 135, 134, 102, 135, 122, 127, 106, 121, 121, 111, 124, 140, 113], 'capacity': 1654, 'solution': [0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 'obj': 9301.0, 'problem_type': 'QKP'}","[0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]",9301.0,"{'problem_type': 'QKP', 'num_items': 14, 'capacity': 1654, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 127}, {'item_id': 'B', 'linear_profit': 116}, {'item_id': 'C', 'linear_profit': 146}, {'item_id': 'D', 'linear_profit': 105}, {'item_id': 'E', 'linear_profit': 137}, {'item_id': 'F', 'linear_profit': 123}, {'item_id': 'G', 'linear_profit': 110}, {'item_id': 'H', 'linear_profit': 147}, {'item_id': 'I', 'linear_profit': 142}, {'item_id': 'J', 'linear_profit': 146}, {'item_id': 'K', 'linear_profit': 138}, {'item_id': 'L', 'linear_profit': 112}, {'item_id': 'M', 'linear_profit': 139}, {'item_id': 'N', 'linear_profit': 133}], 'weight_pairs': [{'item_id': 'A', 'weight': 113}, {'item_id': 'B', 'weight': 135}, {'item_id': 'C', 'weight': 134}, {'item_id': 'D', 'weight': 102}, {'item_id': 'E', 'weight': 135}, {'item_id': 'F', 'weight': 122}, {'item_id': 'G', 'weight': 127}, {'item_id': 'H', 'weight': 106}, {'item_id': 'I', 'weight': 121}, {'item_id': 'J', 'weight': 121}, {'item_id': 'K', 'weight': 111}, {'item_id': 'L', 'weight': 124}, {'item_id': 'M', 'weight': 140}, {'item_id': 'N', 'weight': 113}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 114}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 129}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 111}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 118}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 114}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 111}, {'item_i_id': 'A', 'item_j_id': 'I', 'quadratic_profit': 127}, {'item_i_id': 'A', 'item_j_id': 'J', 'quadratic_profit': 122}, {'item_i_id': 'A', 'item_j_id': 'K', 'quadratic_profit': 128}, {'item_i_id': 'A', 'item_j_id': 'L', 'quadratic_profit': 121}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 139}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 131}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 124}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 120}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 131}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 119}, {'item_i_id': 'B', 'item_j_id': 'H', 'quadratic_profit': 124}, {'item_i_id': 'B', 'item_j_id': 'I', 'quadratic_profit': 122}, {'item_i_id': 'B', 'item_j_id': 'J', 'quadratic_profit': 139}, {'item_i_id': 'B', 'item_j_id': 'K', 'quadratic_profit': 104}, {'item_i_id': 'B', 'item_j_id': 'L', 'quadratic_profit': 115}, {'item_i_id': 'B', 'item_j_id': 'M', 'quadratic_profit': 122}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 138}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 121}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 139}, {'item_i_id': 'C', 'item_j_id': 'I', 'quadratic_profit': 116}, {'item_i_id': 'C', 'item_j_id': 'J', 'quadratic_profit': 132}, {'item_i_id': 'C', 'item_j_id': 'L', 'quadratic_profit': 113}, {'item_i_id': 'C', 'item_j_id': 'M', 'quadratic_profit': 105}, {'item_i_id': 'C', 'item_j_id': 'N', 'quadratic_profit': 118}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 101}, {'item_i_id': 'D', 'item_j_id': 'H', 'quadratic_profit': 120}, {'item_i_id': 'D', 'item_j_id': 'I', 'quadratic_profit': 112}, {'item_i_id': 'D', 'item_j_id': 'K', 'quadratic_profit': 130}, {'item_i_id': 'D', 'item_j_id': 'L', 'quadratic_profit': 125}, {'item_i_id': 'D', 'item_j_id': 'N', 'quadratic_profit': 143}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 122}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 139}, {'item_i_id': 'E', 'item_j_id': 'H', 'quadratic_profit': 133}, {'item_i_id': 'E', 'item_j_id': 'J', 'quadratic_profit': 123}, {'item_i_id': 'E', 'item_j_id': 'K', 'quadratic_profit': 127}, {'item_i_id': 'E', 'item_j_id': 'L', 'quadratic_profit': 127}, {'item_i_id': 'E', 'item_j_id': 'M', 'quadratic_profit': 116}, {'item_i_id': 'E', 'item_j_id': 'N', 'quadratic_profit': 116}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 120}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 121}, {'item_i_id': 'F', 'item_j_id': 'I', 'quadratic_profit': 107}, {'item_i_id': 'F', 'item_j_id': 'J', 'quadratic_profit': 125}, {'item_i_id': 'F', 'item_j_id': 'K', 'quadratic_profit': 114}, {'item_i_id': 'F', 'item_j_id': 'M', 'quadratic_profit': 123}, {'item_i_id': 'F', 'item_j_id': 'N', 'quadratic_profit': 114}, {'item_i_id': 'G', 'item_j_id': 'G', 'quadratic_profit': 135}, {'item_i_id': 'G', 'item_j_id': 'H', 'quadratic_profit': 113}, {'item_i_id': 'G', 'item_j_id': 'J', 'quadratic_profit': 124}, {'item_i_id': 'G', 'item_j_id': 'K', 'quadratic_profit': 116}, {'item_i_id': 'G', 'item_j_id': 'L', 'quadratic_profit': 114}, {'item_i_id': 'G', 'item_j_id': 'M', 'quadratic_profit': 110}, {'item_i_id': 'H', 'item_j_id': 'H', 'quadratic_profit': 118}, {'item_i_id': 'H', 'item_j_id': 'I', 'quadratic_profit': 115}, {'item_i_id': 'H', 'item_j_id': 'J', 'quadratic_profit': 120}, {'item_i_id': 'H', 'item_j_id': 'L', 'quadratic_profit': 121}, {'item_i_id': 'H', 'item_j_id': 'M', 'quadratic_profit': 124}, {'item_i_id': 'H', 'item_j_id': 'N', 'quadratic_profit': 117}, {'item_i_id': 'I', 'item_j_id': 'I', 'quadratic_profit': 103}, {'item_i_id': 'I', 'item_j_id': 'J', 'quadratic_profit': 126}, {'item_i_id': 'I', 'item_j_id': 'K', 'quadratic_profit': 134}, {'item_i_id': 'I', 'item_j_id': 'L', 'quadratic_profit': 122}, {'item_i_id': 'I', 'item_j_id': 'M', 'quadratic_profit': 117}, {'item_i_id': 'I', 'item_j_id': 'N', 'quadratic_profit': 141}, {'item_i_id': 'J', 'item_j_id': 'J', 'quadratic_profit': 123}, {'item_i_id': 'J', 'item_j_id': 'K', 'quadratic_profit': 143}, {'item_i_id': 'J', 'item_j_id': 'L', 'quadratic_profit': 127}, {'item_i_id': 'J', 'item_j_id': 'M', 'quadratic_profit': 122}, {'item_i_id': 'J', 'item_j_id': 'N', 'quadratic_profit': 133}, {'item_i_id': 'K', 'item_j_id': 'K', 'quadratic_profit': 123}, {'item_i_id': 'K', 'item_j_id': 'L', 'quadratic_profit': 119}, {'item_i_id': 'K', 'item_j_id': 'N', 'quadratic_profit': 112}, {'item_i_id': 'L', 'item_j_id': 'L', 'quadratic_profit': 145}, {'item_i_id': 'L', 'item_j_id': 'M', 'quadratic_profit': 125}, {'item_i_id': 'L', 'item_j_id': 'N', 'quadratic_profit': 126}]}","['A', 'B', 'C', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N']",16,nl,names
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.

{
  ""num_amenities"": 14,
  ""kit_weight_capacity"": 3110,
  ""amenity_ids"": [
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14
  ],
  ""linear"": [
    {
      ""amenity_id"": 1,
      ""base_satisfaction"": 451
    },
    {
      ""amenity_id"": 2,
      ""base_satisfaction"": 442
    },
    {
      ""amenity_id"": 3,
      ""base_satisfaction"": 373
    },
    {
      ""amenity_id"": 4,
      ""base_satisfaction"": 537
    },
    {
      ""amenity_id"": 5,
      ""base_satisfaction"": 270
    },
    {
      ""amenity_id"": 6,
      ""base_satisfaction"": 161
    },
    {
      ""amenity_id"": 7,
      ""base_satisfaction"": 288
    },
    {
      ""amenity_id"": 8,
      ""base_satisfaction"": 520
    },
    {
      ""amenity_id"": 9,
      ""base_satisfaction"": 274
    },
    {
      ""amenity_id"": 10,
      ""base_satisfaction"": 309
    },
    {
      ""amenity_id"": 11,
      ""base_satisfaction"": 257
    },
    {
      ""amenity_id"": 12,
      ""base_satisfaction"": 481
    },
    {
      ""amenity_id"": 13,
      ""base_satisfaction"": 356
    },
    {
      ""amenity_id"": 14,
      ""base_satisfaction"": 209
    }
  ],
  ""weights"": [
    {
      ""amenity_id"": 1,
      ""amenity_weight"": 401
    },
    {
      ""amenity_id"": 2,
      ""amenity_weight"": 392
    },
    {
      ""amenity_id"": 3,
      ""amenity_weight"": 323
    },
    {
      ""amenity_id"": 4,
      ""amenity_weight"": 487
    },
    {
      ""amenity_id"": 5,
      ""amenity_weight"": 220
    },
    {
      ""amenity_id"": 6,
      ""amenity_weight"": 111
    },
    {
      ""amenity_id"": 7,
      ""amenity_weight"": 238
    },
    {
      ""amenity_id"": 8,
      ""amenity_weight"": 470
    },
    {
      ""amenity_id"": 9,
      ""amenity_weight"": 224
    },
    {
      ""amenity_id"": 10,
      ""amenity_weight"": 259
    },
    {
      ""amenity_id"": 11,
      ""amenity_weight"": 207
    },
    {
      ""amenity_id"": 12,
      ""amenity_weight"": 431
    },
    {
      ""amenity_id"": 13,
      ""amenity_weight"": 306
    },
    {
      ""amenity_id"": 14,
      ""amenity_weight"": 159
    }
  ],
  ""quadratic"": [
    {
      ""amenity_i_id"": 1,
      ""amenity_j_id"": 2,
      ""pair_bonus_satisfaction"": 184
    },
    {
      ""amenity_i_id"": 1,
      ""amenity_j_id"": 3,
      ""pair_bonus_satisfaction"": 401
    },
    {
      ""amenity_i_id"": 1,
      ""amenity_j_id"": 4,
      ""pair_bonus_satisfaction"": 294
    },
    {
      ""amenity_i_id"": 1,
      ""amenity_j_id"": 5,
      ""pair_bonus_satisfaction"": 292
    },
    {
      ""amenity_i_id"": 1,
      ""amenity_j_id"": 6,
      ""pair_bonus_satisfaction"": 134
    },
    {
      ""amenity_i_id"": 1,
      ""amenity_j_id"": 7,
      ""pair_bonus_satisfaction"": 218
    },
    {
      ""amenity_i_id"": 1,
      ""amenity_j_id"": 8,
      ""pair_bonus_satisfaction"": 369
    },
    {
      ""amenity_i_id"": 1,
      ""amenity_j_id"": 9,
      ""pair_bonus_satisfaction"": 401
    },
    {
      ""amenity_i_id"": 1,
      ""amenity_j_id"": 10,
      ""pair_bonus_satisfaction"": 262
    },
    {
      ""amenity_i_id"": 1,
      ""amenity_j_id"": 11,
      ""pair_bonus_satisfaction"": 266
    },
    {
      ""amenity_i_id"": 1,
      ""amenity_j_id"": 12,
      ""pair_bonus_satisfaction"": 145
    },
    {
      ""amenity_i_id"": 1,
      ""amenity_j_id"": 13,
      ""pair_bonus_satisfaction"": 213
    },
    {
      ""amenity_i_id"": 1,
      ""amenity_j_id"": 14,
      ""pair_bonus_satisfaction"": 288
    },
    {
      ""amenity_i_id"": 2,
      ""amenity_j_id"": 3,
      ""pair_bonus_satisfaction"": 255
    },
    {
      ""amenity_i_id"": 2,
      ""amenity_j_id"": 4,
      ""pair_bonus_satisfaction"": 392
    },
    {
      ""amenity_i_id"": 2,
      ""amenity_j_id"": 5,
      ""pair_bonus_satisfaction"": 160
    },
    {
      ""amenity_i_id"": 2,
      ""amenity_j_id"": 6,
      ""pair_bonus_satisfaction"": 131
    },
    {
      ""amenity_i_id"": 2,
      ""amenity_j_id"": 7,
      ""pair_bonus_satisfaction"": 326
    },
    {
      ""amenity_i_id"": 2,
      ""amenity_j_id"": 8,
      ""pair_bonus_satisfaction"": 148
    },
    {
      ""amenity_i_id"": 2,
      ""amenity_j_id"": 9,
      ""pair_bonus_satisfaction"": 392
    },
    {
      ""amenity_i_id"": 2,
      ""amenity_j_id"": 10,
      ""pair_bonus_satisfaction"": 311
    },
    {
      ""amenity_i_id"": 2,
      ""amenity_j_id"": 11,
      ""pair_bonus_satisfaction"": 340
    },
    {
      ""amenity_i_id"": 2,
      ""amenity_j_id"": 12,
      ""pair_bonus_satisfaction"": 337
    },
    {
      ""amenity_i_id"": 2,
      ""amenity_j_id"": 13,
      ""pair_bonus_satisfaction"": 139
    },
    {
      ""amenity_i_id"": 2,
      ""amenity_j_id"": 14,
      ""pair_bonus_satisfaction"": 139
    },
    {
      ""amenity_i_id"": 3,
      ""amenity_j_id"": 4,
      ""pair_bonus_satisfaction"": 323
    },
    {
      ""amenity_i_id"": 3,
      ""amenity_j_id"": 5,
      ""pair_bonus_satisfaction"": 131
    },
    {
      ""amenity_i_id"": 3,
      ""amenity_j_id"": 6,
      ""pair_bonus_satisfaction"": 323
    },
    {
      ""amenity_i_id"": 3,
      ""amenity_j_id"": 7,
      ""pair_bonus_satisfaction"": 281
    },
    {
      ""amenity_i_id"": 3,
      ""amenity_j_id"": 8,
      ""pair_bonus_satisfaction"": 323
    },
    {
      ""amenity_i_id"": 3,
      ""amenity_j_id"": 9,
      ""pair_bonus_satisfaction"": 281
    },
    {
      ""amenity_i_id"": 3,
      ""amenity_j_id"": 10,
      ""pair_bonus_satisfaction"": 323
    },
    {
      ""amenity_i_id"": 3,
      ""amenity_j_id"": 11,
      ""pair_bonus_satisfaction"": 323
    },
    {
      ""amenity_i_id"": 3,
      ""amenity_j_id"": 12,
      ""pair_bonus_satisfaction"": 323
    },
    {
      ""amenity_i_id"": 3,
      ""amenity_j_id"": 13,
      ""pair_bonus_satisfaction"": 125
    },
    {
      ""amenity_i_id"": 3,
      ""amenity_j_id"": 14,
      ""pair_bonus_satisfaction"": 132
    },
    {
      ""amenity_i_id"": 4,
      ""amenity_j_id"": 5,
      ""pair_bonus_satisfaction"": 369
    },
    {
      ""amenity_i_id"": 4,
      ""amenity_j_id"": 6,
      ""pair_bonus_satisfaction"": 305
    },
    {
      ""amenity_i_id"": 4,
      ""amenity_j_id"": 7,
      ""pair_bonus_satisfaction"": 379
    },
    {
      ""amenity_i_id"": 4,
      ""amenity_j_id"": 8,
      ""pair_bonus_satisfaction"": 407
    },
    {
      ""amenity_i_id"": 4,
      ""amenity_j_id"": 9,
      ""pair_bonus_satisfaction"": 363
    },
    {
      ""amenity_i_id"": 4,
      ""amenity_j_id"": 10,
      ""pair_bonus_satisfaction"": 485
    },
    {
      ""amenity_i_id"": 4,
      ""amenity_j_id"": 11,
      ""pair_bonus_satisfaction"": 216
    },
    {
      ""amenity_i_id"": 4,
      ""amenity_j_id"": 12,
      ""pair_bonus_satisfaction"": 217
    },
    {
      ""amenity_i_id"": 4,
      ""amenity_j_id"": 13,
      ""pair_bonus_satisfaction"": 436
    },
    {
      ""amenity_i_id"": 4,
      ""amenity_j_id"": 14,
      ""pair_bonus_satisfaction"": 436
    },
    {
      ""amenity_i_id"": 5,
      ""amenity_j_id"": 6,
      ""pair_bonus_satisfaction"": 220
    },
    {
      ""amenity_i_id"": 5,
      ""amenity_j_id"": 7,
      ""pair_bonus_satisfaction"": 220
    },
    {
      ""amenity_i_id"": 5,
      ""amenity_j_id"": 8,
      ""pair_bonus_satisfaction"": 220
    },
    {
      ""amenity_i_id"": 5,
      ""amenity_j_id"": 9,
      ""pair_bonus_satisfaction"": 220
    },
    {
      ""amenity_i_id"": 5,
      ""amenity_j_id"": 10,
      ""pair_bonus_satisfaction"": 220
    },
    {
      ""amenity_i_id"": 5,
      ""amenity_j_id"": 11,
      ""pair_bonus_satisfaction"": 184
    },
    {
      ""amenity_i_id"": 5,
      ""amenity_j_id"": 12,
      ""pair_bonus_satisfaction"": 220
    },
    {
      ""amenity_i_id"": 5,
      ""amenity_j_id"": 13,
      ""pair_bonus_satisfaction"": 231
    },
    {
      ""amenity_i_id"": 5,
      ""amenity_j_id"": 14,
      ""pair_bonus_satisfaction"": 478
    },
    {
      ""amenity_i_id"": 6,
      ""amenity_j_id"": 7,
      ""pair_bonus_satisfaction"": 111
    },
    {
      ""amenity_i_id"": 6,
      ""amenity_j_id"": 8,
      ""pair_bonus_satisfaction"": 111
    },
    {
      ""amenity_i_id"": 6,
      ""amenity_j_id"": 9,
      ""pair_bonus_satisfaction"": 111
    },
    {
      ""amenity_i_id"": 6,
      ""amenity_j_id"": 10,
      ""pair_bonus_satisfaction"": 280
    },
    {
      ""amenity_i_id"": 6,
      ""amenity_j_id"": 11,
      ""pair_bonus_satisfaction"": 403
    },
    {
      ""amenity_i_id"": 6,
      ""amenity_j_id"": 12,
      ""pair_bonus_satisfaction"": 403
    },
    {
      ""amenity_i_id"": 6,
      ""amenity_j_id"": 13,
      ""pair_bonus_satisfaction"": 221
    },
    {
      ""amenity_i_id"": 6,
      ""amenity_j_id"": 14,
      ""pair_bonus_satisfaction"": 270
    },
    {
      ""amenity_i_id"": 7,
      ""amenity_j_id"": 8,
      ""pair_bonus_satisfaction"": 216
    },
    {
      ""amenity_i_id"": 7,
      ""amenity_j_id"": 9,
      ""pair_bonus_satisfaction"": 217
    },
    {
      ""amenity_i_id"": 7,
      ""amenity_j_id"": 10,
      ""pair_bonus_satisfaction"": 231
    },
    {
      ""amenity_i_id"": 7,
      ""amenity_j_id"": 11,
      ""pair_bonus_satisfaction"": 139
    },
    {
      ""amenity_i_id"": 7,
      ""amenity_j_id"": 12,
      ""pair_bonus_satisfaction"": 243
    },
    {
      ""amenity_i_id"": 7,
      ""amenity_j_id"": 13,
      ""pair_bonus_satisfaction"": 243
    },
    {
      ""amenity_i_id"": 7,
      ""amenity_j_id"": 14,
      ""pair_bonus_satisfaction"": 243
    },
    {
      ""amenity_i_id"": 8,
      ""amenity_j_id"": 9,
      ""pair_bonus_satisfaction"": 470
    },
    {
      ""amenity_i_id"": 8,
      ""amenity_j_id"": 10,
      ""pair_bonus_satisfaction"": 148
    },
    {
      ""amenity_i_id"": 8,
      ""amenity_j_id"": 11,
      ""pair_bonus_satisfaction"": 148
    },
    {
      ""amenity_i_id"": 8,
      ""amenity_j_id"": 12,
      ""pair_bonus_satisfaction"": 305
    },
    {
      ""amenity_i_id"": 8,
      ""amenity_j_id"": 13,
      ""pair_bonus_satisfaction"": 396
    },
    {
      ""amenity_i_id"": 8,
      ""amenity_j_id"": 14,
      ""pair_bonus_satisfaction"": 396
    },
    {
      ""amenity_i_id"": 9,
      ""amenity_j_id"": 10,
      ""pair_bonus_satisfaction"": 131
    },
    {
      ""amenity_i_id"": 9,
      ""amenity_j_id"": 11,
      ""pair_bonus_satisfaction"": 131
    },
    {
      ""amenity_i_id"": 9,
      ""amenity_j_id"": 12,
      ""pair_bonus_satisfaction"": 131
    },
    {
      ""amenity_i_id"": 9,
      ""amenity_j_id"": 13,
      ""pair_bonus_satisfaction"": 131
    },
    {
      ""amenity_i_id"": 9,
      ""amenity_j_id"": 14,
      ""pair_bonus_satisfaction"": 131
    },
    {
      ""amenity_i_id"": 10,
      ""amenity_j_id"": 11,
      ""pair_bonus_satisfaction"": 442
    },
    {
      ""amenity_i_id"": 10,
      ""amenity_j_id"": 12,
      ""pair_bonus_satisfaction"": 213
    },
    {
      ""amenity_i_id"": 10,
      ""amenity_j_id"": 13,
      ""pair_bonus_satisfaction"": 106
    },
    {
      ""amenity_i_id"": 10,
      ""amenity_j_id"": 14,
      ""pair_bonus_satisfaction"": 106
    },
    {
      ""amenity_i_id"": 11,
      ""amenity_j_id"": 12,
      ""pair_bonus_satisfaction"": 173
    },
    {
      ""amenity_i_id"": 11,
      ""amenity_j_id"": 13,
      ""pair_bonus_satisfaction"": 420
    },
    {
      ""amenity_i_id"": 11,
      ""amenity_j_id"": 14,
      ""pair_bonus_satisfaction"": 173
    },
    {
      ""amenity_i_id"": 12,
      ""amenity_j_id"": 13,
      ""pair_bonus_satisfaction"": 405
    },
    {
      ""amenity_i_id"": 12,
      ""amenity_j_id"": 14,
      ""pair_bonus_satisfaction"": 302
    }
  ]
}

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': [451, 442, 373, 537, 270, 161, 288, 520, 274, 309, 257, 481, 356, 209], 'quadratic_coeffs': [[401, 184, 401, 294, 292, 134, 218, 369, 401, 262, 266, 145, 213, 288], [0, 295, 255, 392, 160, 131, 326, 148, 392, 311, 340, 337, 139, 139], [0, 0, 323, 323, 131, 323, 281, 323, 281, 323, 323, 323, 125, 132], [0, 0, 0, 297, 369, 305, 379, 407, 363, 485, 216, 217, 436, 436], [0, 0, 0, 0, 220, 220, 220, 220, 220, 220, 184, 220, 231, 478], [0, 0, 0, 0, 0, 111, 111, 111, 111, 280, 403, 403, 221, 270], [0, 0, 0, 0, 0, 0, 238, 216, 217, 231, 139, 243, 243, 243], [0, 0, 0, 0, 0, 0, 0, 206, 470, 148, 148, 305, 396, 396], [0, 0, 0, 0, 0, 0, 0, 0, 224, 131, 131, 131, 131, 131], [0, 0, 0, 0, 0, 0, 0, 0, 0, 126, 442, 213, 106, 106], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 342, 173, 420, 173], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 405, 405, 302], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 214, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [401, 392, 323, 487, 220, 111, 238, 470, 224, 259, 207, 431, 306, 159], 'capacity': 3110, 'solution': [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13], 'obj': 18344.0, 'problem_type': 'QKP'}","[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13]",18344.0,"{'problem_type': 'QKP', 'num_items': 14, 'capacity': 3110, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], 'linear_pairs': [{'item_id': 1, 'linear_profit': 451}, {'item_id': 2, 'linear_profit': 442}, {'item_id': 3, 'linear_profit': 373}, {'item_id': 4, 'linear_profit': 537}, {'item_id': 5, 'linear_profit': 270}, {'item_id': 6, 'linear_profit': 161}, {'item_id': 7, 'linear_profit': 288}, {'item_id': 8, 'linear_profit': 520}, {'item_id': 9, 'linear_profit': 274}, {'item_id': 10, 'linear_profit': 309}, {'item_id': 11, 'linear_profit': 257}, {'item_id': 12, 'linear_profit': 481}, {'item_id': 13, 'linear_profit': 356}, {'item_id': 14, 'linear_profit': 209}], 'weight_pairs': [{'item_id': 1, 'weight': 401}, {'item_id': 2, 'weight': 392}, {'item_id': 3, 'weight': 323}, {'item_id': 4, 'weight': 487}, {'item_id': 5, 'weight': 220}, {'item_id': 6, 'weight': 111}, {'item_id': 7, 'weight': 238}, {'item_id': 8, 'weight': 470}, {'item_id': 9, 'weight': 224}, {'item_id': 10, 'weight': 259}, {'item_id': 11, 'weight': 207}, {'item_id': 12, 'weight': 431}, {'item_id': 13, 'weight': 306}, {'item_id': 14, 'weight': 159}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 401}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 184}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 401}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 294}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 292}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 134}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 218}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 369}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 401}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 262}, {'item_i_id': 1, 'item_j_id': 11, 'quadratic_profit': 266}, {'item_i_id': 1, 'item_j_id': 12, 'quadratic_profit': 145}, {'item_i_id': 1, 'item_j_id': 13, 'quadratic_profit': 213}, {'item_i_id': 1, 'item_j_id': 14, 'quadratic_profit': 288}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 295}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 255}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 392}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 160}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 131}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 326}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 148}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 392}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 311}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 340}, {'item_i_id': 2, 'item_j_id': 12, 'quadratic_profit': 337}, {'item_i_id': 2, 'item_j_id': 13, 'quadratic_profit': 139}, {'item_i_id': 2, 'item_j_id': 14, 'quadratic_profit': 139}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 323}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 323}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 131}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 323}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 281}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 323}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 281}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 323}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 323}, {'item_i_id': 3, 'item_j_id': 12, 'quadratic_profit': 323}, {'item_i_id': 3, 'item_j_id': 13, 'quadratic_profit': 125}, {'item_i_id': 3, 'item_j_id': 14, 'quadratic_profit': 132}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 297}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 369}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 305}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 379}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 407}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 363}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 485}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 216}, {'item_i_id': 4, 'item_j_id': 12, 'quadratic_profit': 217}, {'item_i_id': 4, 'item_j_id': 13, 'quadratic_profit': 436}, {'item_i_id': 4, 'item_j_id': 14, 'quadratic_profit': 436}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 220}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 220}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 220}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 220}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 220}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 220}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 184}, {'item_i_id': 5, 'item_j_id': 12, 'quadratic_profit': 220}, {'item_i_id': 5, 'item_j_id': 13, 'quadratic_profit': 231}, {'item_i_id': 5, 'item_j_id': 14, 'quadratic_profit': 478}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 111}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 111}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 111}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 111}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 280}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 403}, {'item_i_id': 6, 'item_j_id': 12, 'quadratic_profit': 403}, {'item_i_id': 6, 'item_j_id': 13, 'quadratic_profit': 221}, {'item_i_id': 6, 'item_j_id': 14, 'quadratic_profit': 270}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 238}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 216}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 217}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 231}, {'item_i_id': 7, 'item_j_id': 11, 'quadratic_profit': 139}, {'item_i_id': 7, 'item_j_id': 12, 'quadratic_profit': 243}, {'item_i_id': 7, 'item_j_id': 13, 'quadratic_profit': 243}, {'item_i_id': 7, 'item_j_id': 14, 'quadratic_profit': 243}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 206}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 470}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 148}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 148}, {'item_i_id': 8, 'item_j_id': 12, 'quadratic_profit': 305}, {'item_i_id': 8, 'item_j_id': 13, 'quadratic_profit': 396}, {'item_i_id': 8, 'item_j_id': 14, 'quadratic_profit': 396}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 224}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 131}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 131}, {'item_i_id': 9, 'item_j_id': 12, 'quadratic_profit': 131}, {'item_i_id': 9, 'item_j_id': 13, 'quadratic_profit': 131}, {'item_i_id': 9, 'item_j_id': 14, 'quadratic_profit': 131}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 126}, {'item_i_id': 10, 'item_j_id': 11, 'quadratic_profit': 442}, {'item_i_id': 10, 'item_j_id': 12, 'quadratic_profit': 213}, {'item_i_id': 10, 'item_j_id': 13, 'quadratic_profit': 106}, {'item_i_id': 10, 'item_j_id': 14, 'quadratic_profit': 106}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 342}, {'item_i_id': 11, 'item_j_id': 12, 'quadratic_profit': 173}, {'item_i_id': 11, 'item_j_id': 13, 'quadratic_profit': 420}, {'item_i_id': 11, 'item_j_id': 14, 'quadratic_profit': 173}, {'item_i_id': 12, 'item_j_id': 12, 'quadratic_profit': 405}, {'item_i_id': 12, 'item_j_id': 13, 'quadratic_profit': 405}, {'item_i_id': 12, 'item_j_id': 14, 'quadratic_profit': 302}, {'item_i_id': 13, 'item_j_id': 13, 'quadratic_profit': 214}]}","[1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14]",17,json,1
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 15 experiments labeled A, B, C, D, E, F, G, H, I, J, K, L, M, N, O and 2582 total bench time for the session.

| experiment_id_for_time | base_scientific_value |
|---|---|
| A | 443 |
| B | 465 |
| C | 225 |
| D | 354 |
| E | 347 |
| F | 91 |
| G | 155 |
| H | 107 |
| I | 418 |
| J | 229 |
| K | 418 |
| L | 394 |
| M | 436 |
| N | 120 |
| O | 325 |

| experiment_id_for_time | duration |
|---|---|
| A | 444 |
| B | 440 |
| C | 202 |
| D | 363 |
| E | 344 |
| F | 107 |
| G | 165 |
| H | 108 |
| I | 404 |
| J | 240 |
| K | 409 |
| L | 403 |
| M | 457 |
| N | 121 |
| O | 306 |

| experiment_i_id | experiment_j_id | pair_synergy_value |
|---|---|---|
| A | B | 248 |
| A | C | 128 |
| A | D | 423 |
| A | E | 285 |
| A | F | 444 |
| A | G | 339 |
| A | H | 211 |
| A | I | 108 |
| A | J | 379 |
| A | K | 318 |
| A | L | 444 |
| A | M | 306 |
| A | N | 284 |
| A | O | 218 |
| B | C | 196 |
| B | D | 440 |
| B | E | 205 |
| B | F | 141 |
| B | G | 211 |
| B | H | 238 |
| B | I | 212 |
| B | J | 409 |
| B | K | 147 |
| B | L | 440 |
| B | M | 440 |
| B | N | 170 |
| B | O | 170 |
| C | D | 136 |
| C | E | 202 |
| C | F | 202 |
| C | G | 202 |
| C | H | 202 |
| C | I | 124 |
| C | J | 202 |
| C | K | 122 |
| C | L | 146 |
| C | M | 262 |
| C | N | 196 |
| C | O | 101 |
| D | E | 158 |
| D | F | 219 |
| D | G | 356 |
| D | H | 363 |
| D | I | 275 |
| D | J | 363 |
| D | K | 323 |
| D | L | 383 |
| D | M | 383 |
| D | N | 178 |
| D | O | 281 |
| E | F | 330 |
| E | G | 238 |
| E | H | 344 |
| E | I | 146 |
| E | J | 344 |
| E | K | 336 |
| E | L | 178 |
| E | M | 258 |
| E | N | 175 |
| E | O | 336 |
| F | G | 107 |
| F | H | 107 |
| F | I | 107 |
| F | J | 107 |
| F | K | 357 |
| F | L | 357 |
| F | M | 223 |
| F | N | 238 |
| F | O | 344 |
| G | H | 165 |
| G | I | 165 |
| G | J | 165 |
| G | K | 260 |
| G | L | 260 |
| G | M | 260 |
| G | N | 218 |
| G | O | 260 |
| H | I | 108 |
| H | J | 108 |
| H | K | 412 |
| H | L | 291 |
| H | M | 194 |
| H | N | 409 |
| H | O | 146 |
| I | J | 330 |
| I | K | 283 |
| I | L | 382 |
| I | M | 322 |
| I | N | 164 |
| I | O | 164 |
| J | K | 110 |
| J | L | 152 |
| J | M | 216 |
| J | N | 216 |
| J | O | 216 |
| K | L | 218 |
| K | M | 128 |
| K | N | 218 |
| K | O | 146 |
| L | M | 381 |
| L | N | 381 |
| L | O | 340 |
| M | N | 191 |
| M | O | 335 |

We must respect the 2582 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': [443, 465, 225, 354, 347, 91, 155, 107, 418, 229, 418, 394, 436, 120, 325], 'quadratic_coeffs': [[262, 248, 128, 423, 285, 444, 339, 211, 108, 379, 318, 444, 306, 284, 218], [0, 363, 196, 440, 205, 141, 211, 238, 212, 409, 147, 440, 440, 170, 170], [0, 0, 202, 136, 202, 202, 202, 202, 124, 202, 122, 146, 262, 196, 101], [0, 0, 0, 359, 158, 219, 356, 363, 275, 363, 323, 383, 383, 178, 281], [0, 0, 0, 0, 216, 330, 238, 344, 146, 344, 336, 178, 258, 175, 336], [0, 0, 0, 0, 0, 107, 107, 107, 107, 107, 357, 357, 223, 238, 344], [0, 0, 0, 0, 0, 0, 146, 165, 165, 165, 260, 260, 260, 218, 260], [0, 0, 0, 0, 0, 0, 0, 108, 108, 108, 412, 291, 194, 409, 146], [0, 0, 0, 0, 0, 0, 0, 0, 356, 330, 283, 382, 322, 164, 164], [0, 0, 0, 0, 0, 0, 0, 0, 0, 152, 110, 152, 216, 216, 216], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 218, 218, 128, 218, 146], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 168, 381, 381, 340], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 382, 191, 335], [0, 0, 0, 0, 0, 0, 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': [444, 440, 202, 363, 344, 107, 165, 108, 404, 240, 409, 403, 457, 121, 306], 'capacity': 2582, 'solution': [0, 2, 3, 5, 6, 7, 9, 10, 11, 13], 'obj': 13955.0, 'problem_type': 'QKP'}","[0, 2, 3, 5, 6, 7, 9, 10, 11, 13]",13955.0,"{'problem_type': 'QKP', 'num_items': 15, 'capacity': 2582, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 443}, {'item_id': 'B', 'linear_profit': 465}, {'item_id': 'C', 'linear_profit': 225}, {'item_id': 'D', 'linear_profit': 354}, {'item_id': 'E', 'linear_profit': 347}, {'item_id': 'F', 'linear_profit': 91}, {'item_id': 'G', 'linear_profit': 155}, {'item_id': 'H', 'linear_profit': 107}, {'item_id': 'I', 'linear_profit': 418}, {'item_id': 'J', 'linear_profit': 229}, {'item_id': 'K', 'linear_profit': 418}, {'item_id': 'L', 'linear_profit': 394}, {'item_id': 'M', 'linear_profit': 436}, {'item_id': 'N', 'linear_profit': 120}, {'item_id': 'O', 'linear_profit': 325}], 'weight_pairs': [{'item_id': 'A', 'weight': 444}, {'item_id': 'B', 'weight': 440}, {'item_id': 'C', 'weight': 202}, {'item_id': 'D', 'weight': 363}, {'item_id': 'E', 'weight': 344}, {'item_id': 'F', 'weight': 107}, {'item_id': 'G', 'weight': 165}, {'item_id': 'H', 'weight': 108}, {'item_id': 'I', 'weight': 404}, {'item_id': 'J', 'weight': 240}, {'item_id': 'K', 'weight': 409}, {'item_id': 'L', 'weight': 403}, {'item_id': 'M', 'weight': 457}, {'item_id': 'N', 'weight': 121}, {'item_id': 'O', 'weight': 306}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 262}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 248}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 128}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 423}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 285}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 444}, {'item_i_id': 'A', 'item_j_id': 'G', 'quadratic_profit': 339}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 211}, {'item_i_id': 'A', 'item_j_id': 'I', 'quadratic_profit': 108}, {'item_i_id': 'A', 'item_j_id': 'J', 'quadratic_profit': 379}, {'item_i_id': 'A', 'item_j_id': 'K', 'quadratic_profit': 318}, {'item_i_id': 'A', 'item_j_id': 'L', 'quadratic_profit': 444}, {'item_i_id': 'A', 'item_j_id': 'M', 'quadratic_profit': 306}, {'item_i_id': 'A', 'item_j_id': 'N', 'quadratic_profit': 284}, {'item_i_id': 'A', 'item_j_id': 'O', 'quadratic_profit': 218}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 363}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 196}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 440}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 205}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 141}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 211}, {'item_i_id': 'B', 'item_j_id': 'H', 'quadratic_profit': 238}, {'item_i_id': 'B', 'item_j_id': 'I', 'quadratic_profit': 212}, {'item_i_id': 'B', 'item_j_id': 'J', 'quadratic_profit': 409}, {'item_i_id': 'B', 'item_j_id': 'K', 'quadratic_profit': 147}, {'item_i_id': 'B', 'item_j_id': 'L', 'quadratic_profit': 440}, {'item_i_id': 'B', 'item_j_id': 'M', 'quadratic_profit': 440}, {'item_i_id': 'B', 'item_j_id': 'N', 'quadratic_profit': 170}, {'item_i_id': 'B', 'item_j_id': 'O', 'quadratic_profit': 170}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 202}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 136}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 202}, {'item_i_id': 'C', 'item_j_id': 'F', 'quadratic_profit': 202}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 202}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 202}, {'item_i_id': 'C', 'item_j_id': 'I', 'quadratic_profit': 124}, {'item_i_id': 'C', 'item_j_id': 'J', 'quadratic_profit': 202}, {'item_i_id': 'C', 'item_j_id': 'K', 'quadratic_profit': 122}, {'item_i_id': 'C', 'item_j_id': 'L', 'quadratic_profit': 146}, {'item_i_id': 'C', 'item_j_id': 'M', 'quadratic_profit': 262}, {'item_i_id': 'C', 'item_j_id': 'N', 'quadratic_profit': 196}, {'item_i_id': 'C', 'item_j_id': 'O', 'quadratic_profit': 101}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 359}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 158}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 219}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 356}, {'item_i_id': 'D', 'item_j_id': 'H', 'quadratic_profit': 363}, {'item_i_id': 'D', 'item_j_id': 'I', 'quadratic_profit': 275}, {'item_i_id': 'D', 'item_j_id': 'J', 'quadratic_profit': 363}, {'item_i_id': 'D', 'item_j_id': 'K', 'quadratic_profit': 323}, {'item_i_id': 'D', 'item_j_id': 'L', 'quadratic_profit': 383}, {'item_i_id': 'D', 'item_j_id': 'M', 'quadratic_profit': 383}, {'item_i_id': 'D', 'item_j_id': 'N', 'quadratic_profit': 178}, {'item_i_id': 'D', 'item_j_id': 'O', 'quadratic_profit': 281}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 216}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 330}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 238}, {'item_i_id': 'E', 'item_j_id': 'H', 'quadratic_profit': 344}, {'item_i_id': 'E', 'item_j_id': 'I', 'quadratic_profit': 146}, {'item_i_id': 'E', 'item_j_id': 'J', 'quadratic_profit': 344}, {'item_i_id': 'E', 'item_j_id': 'K', 'quadratic_profit': 336}, {'item_i_id': 'E', 'item_j_id': 'L', 'quadratic_profit': 178}, {'item_i_id': 'E', 'item_j_id': 'M', 'quadratic_profit': 258}, {'item_i_id': 'E', 'item_j_id': 'N', 'quadratic_profit': 175}, {'item_i_id': 'E', 'item_j_id': 'O', 'quadratic_profit': 336}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 107}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 107}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 107}, {'item_i_id': 'F', 'item_j_id': 'I', 'quadratic_profit': 107}, {'item_i_id': 'F', 'item_j_id': 'J', 'quadratic_profit': 107}, {'item_i_id': 'F', 'item_j_id': 'K', 'quadratic_profit': 357}, {'item_i_id': 'F', 'item_j_id': 'L', 'quadratic_profit': 357}, {'item_i_id': 'F', 'item_j_id': 'M', 'quadratic_profit': 223}, {'item_i_id': 'F', 'item_j_id': 'N', 'quadratic_profit': 238}, {'item_i_id': 'F', 'item_j_id': 'O', 'quadratic_profit': 344}, {'item_i_id': 'G', 'item_j_id': 'G', 'quadratic_profit': 146}, {'item_i_id': 'G', 'item_j_id': 'H', 'quadratic_profit': 165}, {'item_i_id': 'G', 'item_j_id': 'I', 'quadratic_profit': 165}, {'item_i_id': 'G', 'item_j_id': 'J', 'quadratic_profit': 165}, {'item_i_id': 'G', 'item_j_id': 'K', 'quadratic_profit': 260}, {'item_i_id': 'G', 'item_j_id': 'L', 'quadratic_profit': 260}, {'item_i_id': 'G', 'item_j_id': 'M', 'quadratic_profit': 260}, {'item_i_id': 'G', 'item_j_id': 'N', 'quadratic_profit': 218}, {'item_i_id': 'G', 'item_j_id': 'O', 'quadratic_profit': 260}, {'item_i_id': 'H', 'item_j_id': 'H', 'quadratic_profit': 108}, {'item_i_id': 'H', 'item_j_id': 'I', 'quadratic_profit': 108}, {'item_i_id': 'H', 'item_j_id': 'J', 'quadratic_profit': 108}, {'item_i_id': 'H', 'item_j_id': 'K', 'quadratic_profit': 412}, {'item_i_id': 'H', 'item_j_id': 'L', 'quadratic_profit': 291}, {'item_i_id': 'H', 'item_j_id': 'M', 'quadratic_profit': 194}, {'item_i_id': 'H', 'item_j_id': 'N', 'quadratic_profit': 409}, {'item_i_id': 'H', 'item_j_id': 'O', 'quadratic_profit': 146}, {'item_i_id': 'I', 'item_j_id': 'I', 'quadratic_profit': 356}, {'item_i_id': 'I', 'item_j_id': 'J', 'quadratic_profit': 330}, {'item_i_id': 'I', 'item_j_id': 'K', 'quadratic_profit': 283}, {'item_i_id': 'I', 'item_j_id': 'L', 'quadratic_profit': 382}, {'item_i_id': 'I', 'item_j_id': 'M', 'quadratic_profit': 322}, {'item_i_id': 'I', 'item_j_id': 'N', 'quadratic_profit': 164}, {'item_i_id': 'I', 'item_j_id': 'O', 'quadratic_profit': 164}, {'item_i_id': 'J', 'item_j_id': 'J', 'quadratic_profit': 152}, {'item_i_id': 'J', 'item_j_id': 'K', 'quadratic_profit': 110}, {'item_i_id': 'J', 'item_j_id': 'L', 'quadratic_profit': 152}, {'item_i_id': 'J', 'item_j_id': 'M', 'quadratic_profit': 216}, {'item_i_id': 'J', 'item_j_id': 'N', 'quadratic_profit': 216}, {'item_i_id': 'J', 'item_j_id': 'O', 'quadratic_profit': 216}, {'item_i_id': 'K', 'item_j_id': 'K', 'quadratic_profit': 218}, {'item_i_id': 'K', 'item_j_id': 'L', 'quadratic_profit': 218}, {'item_i_id': 'K', 'item_j_id': 'M', 'quadratic_profit': 128}, {'item_i_id': 'K', 'item_j_id': 'N', 'quadratic_profit': 218}, {'item_i_id': 'K', 'item_j_id': 'O', 'quadratic_profit': 146}, {'item_i_id': 'L', 'item_j_id': 'L', 'quadratic_profit': 168}, {'item_i_id': 'L', 'item_j_id': 'M', 'quadratic_profit': 381}, {'item_i_id': 'L', 'item_j_id': 'N', 'quadratic_profit': 381}, {'item_i_id': 'L', 'item_j_id': 'O', 'quadratic_profit': 340}, {'item_i_id': 'M', 'item_j_id': 'M', 'quadratic_profit': 382}, {'item_i_id': 'M', 'item_j_id': 'N', 'quadratic_profit': 191}, {'item_i_id': 'M', 'item_j_id': 'O', 'quadratic_profit': 335}]}","['A', 'C', 'D', 'F', 'G', 'H', 'J', 'K', 'L', 'N']",18,markdown_table,names
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.

# total_assets_count=15
# total_airtime_seconds=239
# asset_identifiers=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

asset_id,baseline_impact
1,113
2,121
3,121
4,114
5,114
6,120
7,121
8,117
9,113
10,116
11,122
12,116
13,118
14,119
15,120

asset_id,airtime_seconds
1,102
2,110
3,110
4,103
5,103
6,109
7,110
8,106
9,102
10,105
11,111
12,105
13,107
14,108
15,109

asset_i_id,asset_j_id,pair_bonus_impact
1,3,102
1,8,100
1,9,109
1,13,107
2,3,111
2,7,110
2,9,104
3,12,109
3,13,103
3,15,103
5,7,106
5,9,111
6,9,107
7,12,108
8,11,102
9,10,109
9,13,107
10,12,109
10,15,109
11,13,109
14,15,106

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': [113, 121, 121, 114, 114, 120, 121, 117, 113, 116, 122, 116, 118, 119, 120], 'quadratic_coeffs': [[0, 0, 102, 0, 0, 0, 0, 100, 109, 0, 0, 0, 107, 0, 0], [0, 0, 111, 0, 0, 0, 110, 0, 104, 0, 0, 0, 0, 0, 0], [0, 0, 106, 0, 0, 0, 0, 0, 0, 0, 0, 109, 103, 0, 103], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 106, 0, 111, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 102, 0, 0, 107, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 103, 0, 0, 0, 0, 108, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 102, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 109, 0, 0, 107, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 109, 0, 0, 109], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 110, 0, 109, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 109, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 106], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [102, 110, 110, 103, 103, 109, 110, 106, 102, 105, 111, 105, 107, 108, 109], 'capacity': 239, 'solution': [1, 2], 'obj': 353.0, 'problem_type': 'QKP'}","[1, 2]",353.0,"{'problem_type': 'QKP', 'num_items': 15, 'capacity': 239, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], 'linear_pairs': [{'item_id': 1, 'linear_profit': 113}, {'item_id': 2, 'linear_profit': 121}, {'item_id': 3, 'linear_profit': 121}, {'item_id': 4, 'linear_profit': 114}, {'item_id': 5, 'linear_profit': 114}, {'item_id': 6, 'linear_profit': 120}, {'item_id': 7, 'linear_profit': 121}, {'item_id': 8, 'linear_profit': 117}, {'item_id': 9, 'linear_profit': 113}, {'item_id': 10, 'linear_profit': 116}, {'item_id': 11, 'linear_profit': 122}, {'item_id': 12, 'linear_profit': 116}, {'item_id': 13, 'linear_profit': 118}, {'item_id': 14, 'linear_profit': 119}, {'item_id': 15, 'linear_profit': 120}], 'weight_pairs': [{'item_id': 1, 'weight': 102}, {'item_id': 2, 'weight': 110}, {'item_id': 3, 'weight': 110}, {'item_id': 4, 'weight': 103}, {'item_id': 5, 'weight': 103}, {'item_id': 6, 'weight': 109}, {'item_id': 7, 'weight': 110}, {'item_id': 8, 'weight': 106}, {'item_id': 9, 'weight': 102}, {'item_id': 10, 'weight': 105}, {'item_id': 11, 'weight': 111}, {'item_id': 12, 'weight': 105}, {'item_id': 13, 'weight': 107}, {'item_id': 14, 'weight': 108}, {'item_id': 15, 'weight': 109}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 102}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 109}, {'item_i_id': 1, 'item_j_id': 13, 'quadratic_profit': 107}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 111}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 110}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 104}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 106}, {'item_i_id': 3, 'item_j_id': 12, 'quadratic_profit': 109}, {'item_i_id': 3, 'item_j_id': 13, 'quadratic_profit': 103}, {'item_i_id': 3, 'item_j_id': 15, 'quadratic_profit': 103}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 106}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 111}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 102}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 107}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 103}, {'item_i_id': 7, 'item_j_id': 12, 'quadratic_profit': 108}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 102}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 109}, {'item_i_id': 9, 'item_j_id': 13, 'quadratic_profit': 107}, {'item_i_id': 10, 'item_j_id': 12, 'quadratic_profit': 109}, {'item_i_id': 10, 'item_j_id': 15, 'quadratic_profit': 109}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 110}, {'item_i_id': 11, 'item_j_id': 13, 'quadratic_profit': 109}, {'item_i_id': 12, 'item_j_id': 12, 'quadratic_profit': 109}, {'item_i_id': 14, 'item_j_id': 15, 'quadratic_profit': 106}]}","[2, 3]",19,csv,1
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 11 platters (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10) and the van capacity 1082 that I must work with.
Platter 0: I count a base appeal of 124.
Platter 1: I count a base appeal of 140.
Platter 2: I count a base appeal of 97.
Platter 3: I count a base appeal of 105.
Platter 4: I count a base appeal of 134.
Platter 5: I count a base appeal of 118.
Platter 6: I count a base appeal of 139.
Platter 7: I count a base appeal of 111.
Platter 8: I count a base appeal of 148.
Platter 9: I count a base appeal of 140.
Platter 10: I count a base appeal of 111.
Platter 0: I note the weight is 120.
Platter 1: I note the weight is 136.
Platter 2: I note the weight is 101.
Platter 3: I note the weight is 102.
Platter 4: I note the weight is 127.
Platter 5: I note the weight is 123.
Platter 6: I note the weight is 140.
Platter 7: I note the weight is 111.
Platter 8: I note the weight is 148.
Platter 9: I note the weight is 146.
Platter 10: I note the weight is 104.
If I include both 0 and 1, I add a pair bonus of 120.
If I include both 0 and 2, I add a pair bonus of 113.
If I include both 0 and 3, I add a pair bonus of 102.
If I include both 0 and 4, I add a pair bonus of 120.
If I include both 0 and 5, I add a pair bonus of 120.
If I include both 0 and 6, I add a pair bonus of 120.
If I include both 0 and 7, I add a pair bonus of 100.
If I include both 0 and 8, I add a pair bonus of 117.
If I include both 0 and 9, I add a pair bonus of 120.
If I include both 0 and 10, I add a pair bonus of 111.
If I include both 1 and 2, I add a pair bonus of 119.
If I include both 1 and 3, I add a pair bonus of 136.
If I include both 1 and 4, I add a pair bonus of 135.
If I include both 1 and 5, I add a pair bonus of 136.
If I include both 1 and 6, I add a pair bonus of 134.
If I include both 1 and 7, I add a pair bonus of 124.
If I include both 1 and 8, I add a pair bonus of 106.
If I include both 1 and 9, I add a pair bonus of 128.
If I include both 1 and 10, I add a pair bonus of 110.
If I include both 2 and 3, I add a pair bonus of 101.
If I include both 2 and 4, I add a pair bonus of 101.
If I include both 2 and 5, I add a pair bonus of 111.
If I include both 2 and 6, I add a pair bonus of 111.
If I include both 2 and 7, I add a pair bonus of 111.
If I include both 2 and 8, I add a pair bonus of 111.
If I include both 2 and 9, I add a pair bonus of 111.
If I include both 2 and 10, I add a pair bonus of 111.
If I include both 3 and 4, I add a pair bonus of 102.
If I include both 3 and 5, I add a pair bonus of 122.
If I include both 3 and 6, I add a pair bonus of 122.
If I include both 3 and 7, I add a pair bonus of 114.
If I include both 3 and 8, I add a pair bonus of 122.
If I include both 3 and 9, I add a pair bonus of 122.
If I include both 3 and 10, I add a pair bonus of 122.
If I include both 4 and 5, I add a pair bonus of 103.
If I include both 4 and 6, I add a pair bonus of 103.
If I include both 4 and 7, I add a pair bonus of 103.
If I include both 4 and 8, I add a pair bonus of 103.
If I include both 4 and 9, I add a pair bonus of 103.
If I include both 4 and 10, I add a pair bonus of 103.
If I include both 5 and 6, I add a pair bonus of 106.
If I include both 5 and 7, I add a pair bonus of 131.
If I include both 5 and 8, I add a pair bonus of 104.
If I include both 5 and 9, I add a pair bonus of 131.
If I include both 5 and 10, I add a pair bonus of 131.
If I include both 6 and 7, I add a pair bonus of 122.
If I include both 6 and 8, I add a pair bonus of 133.
If I include both 6 and 9, I add a pair bonus of 108.
If I include both 6 and 10, I add a pair bonus of 101.
If I include both 7 and 8, I add a pair bonus of 115.
If I include both 7 and 9, I add a pair bonus of 122.
If I include both 7 and 10, I add a pair bonus of 100.
If I include both 8 and 9, I add a pair bonus of 108.
If I include both 8 and 10, I add a pair bonus of 108.
If I include both 9 and 10, I add a pair bonus of 101.
I'll use these figures to pick the best set that fits the van capacity 1082.

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': [124, 140, 97, 105, 134, 118, 139, 111, 148, 140, 111], 'quadratic_coeffs': [[120, 120, 113, 102, 120, 120, 120, 100, 117, 120, 111], [0, 116, 119, 136, 135, 136, 134, 124, 106, 128, 110], [0, 0, 101, 101, 101, 111, 111, 111, 111, 111, 111], [0, 0, 0, 102, 102, 122, 122, 114, 122, 122, 122], [0, 0, 0, 0, 118, 103, 103, 103, 103, 103, 103], [0, 0, 0, 0, 0, 124, 106, 131, 104, 131, 131], [0, 0, 0, 0, 0, 0, 141, 122, 133, 108, 101], [0, 0, 0, 0, 0, 0, 0, 118, 115, 122, 100], [0, 0, 0, 0, 0, 0, 0, 0, 108, 108, 108], [0, 0, 0, 0, 0, 0, 0, 0, 0, 103, 101], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 103]], 'weights': [120, 136, 101, 102, 127, 123, 140, 111, 148, 146, 104], 'capacity': 1082, 'solution': [0, 1, 2, 3, 4, 5, 7, 9, 10], 'obj': 5230.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 5, 7, 9, 10]",5230.0,"{'problem_type': 'QKP', 'num_items': 11, 'capacity': 1082, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 'linear_pairs': [{'item_id': 0, 'linear_profit': 124}, {'item_id': 1, 'linear_profit': 140}, {'item_id': 2, 'linear_profit': 97}, {'item_id': 3, 'linear_profit': 105}, {'item_id': 4, 'linear_profit': 134}, {'item_id': 5, 'linear_profit': 118}, {'item_id': 6, 'linear_profit': 139}, {'item_id': 7, 'linear_profit': 111}, {'item_id': 8, 'linear_profit': 148}, {'item_id': 9, 'linear_profit': 140}, {'item_id': 10, 'linear_profit': 111}], 'weight_pairs': [{'item_id': 0, 'weight': 120}, {'item_id': 1, 'weight': 136}, {'item_id': 2, 'weight': 101}, {'item_id': 3, 'weight': 102}, {'item_id': 4, 'weight': 127}, {'item_id': 5, 'weight': 123}, {'item_id': 6, 'weight': 140}, {'item_id': 7, 'weight': 111}, {'item_id': 8, 'weight': 148}, {'item_id': 9, 'weight': 146}, {'item_id': 10, 'weight': 104}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 120}, {'item_i_id': 0, 'item_j_id': 1, 'quadratic_profit': 120}, {'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 113}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 102}, {'item_i_id': 0, 'item_j_id': 4, 'quadratic_profit': 120}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 120}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 120}, {'item_i_id': 0, 'item_j_id': 7, 'quadratic_profit': 100}, {'item_i_id': 0, 'item_j_id': 8, 'quadratic_profit': 117}, {'item_i_id': 0, 'item_j_id': 9, 'quadratic_profit': 120}, {'item_i_id': 0, 'item_j_id': 10, 'quadratic_profit': 111}, {'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 116}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 119}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 136}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 135}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 136}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 134}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 124}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 106}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 128}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 110}, {'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': 101}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 111}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 111}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 111}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 111}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 111}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 111}, {'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': 3, 'item_j_id': 5, 'quadratic_profit': 122}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 122}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 114}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 122}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 122}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 122}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 118}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 103}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 103}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 103}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 103}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 103}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 103}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 124}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 106}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 131}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 104}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 131}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 131}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 141}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 122}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 133}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 108}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 101}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 118}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 115}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 122}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 100}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 108}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 108}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 108}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 103}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 101}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 103}]}","[0, 1, 2, 3, 4, 5, 7, 9, 10]",20,nl,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.

{
  ""num_bands"": 13,
  ""budget_limit"": 5309,
  ""band_ids"": [
    ""A"",
    ""B"",
    ""C"",
    ""D"",
    ""E"",
    ""F"",
    ""G"",
    ""H"",
    ""I"",
    ""J"",
    ""K"",
    ""L"",
    ""M""
  ],
  ""linear"": [
    {
      ""band_id"": ""A"",
      ""solo_draw"": 215
    },
    {
      ""band_id"": ""B"",
      ""solo_draw"": 1295
    },
    {
      ""band_id"": ""C"",
      ""solo_draw"": 793
    },
    {
      ""band_id"": ""D"",
      ""solo_draw"": 806
    },
    {
      ""band_id"": ""E"",
      ""solo_draw"": 346
    },
    {
      ""band_id"": ""F"",
      ""solo_draw"": 1225
    },
    {
      ""band_id"": ""G"",
      ""solo_draw"": 1161
    },
    {
      ""band_id"": ""H"",
      ""solo_draw"": 430
    },
    {
      ""band_id"": ""I"",
      ""solo_draw"": 879
    },
    {
      ""band_id"": ""J"",
      ""solo_draw"": 1328
    },
    {
      ""band_id"": ""K"",
      ""solo_draw"": 1638
    },
    {
      ""band_id"": ""L"",
      ""solo_draw"": 1526
    },
    {
      ""band_id"": ""M"",
      ""solo_draw"": 1087
    }
  ],
  ""weights"": [
    {
      ""band_id"": ""A"",
      ""booking_fee"": 207
    },
    {
      ""band_id"": ""B"",
      ""booking_fee"": 1289
    },
    {
      ""band_id"": ""C"",
      ""booking_fee"": 762
    },
    {
      ""band_id"": ""D"",
      ""booking_fee"": 788
    },
    {
      ""band_id"": ""E"",
      ""booking_fee"": 283
    },
    {
      ""band_id"": ""F"",
      ""booking_fee"": 1229
    },
    {
      ""band_id"": ""G"",
      ""booking_fee"": 1192
    },
    {
      ""band_id"": ""H"",
      ""booking_fee"": 479
    },
    {
      ""band_id"": ""I"",
      ""booking_fee"": 932
    },
    {
      ""band_id"": ""J"",
      ""booking_fee"": 1297
    },
    {
      ""band_id"": ""K"",
      ""booking_fee"": 1591
    },
    {
      ""band_id"": ""L"",
      ""booking_fee"": 1485
    },
    {
      ""band_id"": ""M"",
      ""booking_fee"": 1044
    }
  ],
  ""quadratic"": [
    {
      ""band_i_id"": ""A"",
      ""band_j_id"": ""B"",
      ""pairing_boost"": 403
    },
    {
      ""band_i_id"": ""A"",
      ""band_j_id"": ""C"",
      ""pairing_boost"": 577
    },
    {
      ""band_i_id"": ""A"",
      ""band_j_id"": ""D"",
      ""pairing_boost"": 279
    },
    {
      ""band_i_id"": ""A"",
      ""band_j_id"": ""E"",
      ""pairing_boost"": 381
    },
    {
      ""band_i_id"": ""A"",
      ""band_j_id"": ""F"",
      ""pairing_boost"": 440
    },
    {
      ""band_i_id"": ""A"",
      ""band_j_id"": ""G"",
      ""pairing_boost"": 227
    },
    {
      ""band_i_id"": ""A"",
      ""band_j_id"": ""H"",
      ""pairing_boost"": 511
    },
    {
      ""band_i_id"": ""A"",
      ""band_j_id"": ""I"",
      ""pairing_boost"": 544
    },
    {
      ""band_i_id"": ""A"",
      ""band_j_id"": ""J"",
      ""pairing_boost"": 159
    },
    {
      ""band_i_id"": ""A"",
      ""band_j_id"": ""K"",
      ""pairing_boost"": 350
    },
    {
      ""band_i_id"": ""A"",
      ""band_j_id"": ""L"",
      ""pairing_boost"": 528
    },
    {
      ""band_i_id"": ""A"",
      ""band_j_id"": ""M"",
      ""pairing_boost"": 699
    },
    {
      ""band_i_id"": ""B"",
      ""band_j_id"": ""C"",
      ""pairing_boost"": 1261
    },
    {
      ""band_i_id"": ""B"",
      ""band_j_id"": ""D"",
      ""pairing_boost"": 1424
    },
    {
      ""band_i_id"": ""B"",
      ""band_j_id"": ""E"",
      ""pairing_boost"": 855
    },
    {
      ""band_i_id"": ""B"",
      ""band_j_id"": ""F"",
      ""pairing_boost"": 1000
    },
    {
      ""band_i_id"": ""B"",
      ""band_j_id"": ""G"",
      ""pairing_boost"": 1357
    },
    {
      ""band_i_id"": ""B"",
      ""band_j_id"": ""H"",
      ""pairing_boost"": 659
    },
    {
      ""band_i_id"": ""B"",
      ""band_j_id"": ""I"",
      ""pairing_boost"": 654
    },
    {
      ""band_i_id"": ""B"",
      ""band_j_id"": ""J"",
      ""pairing_boost"": 1287
    },
    {
      ""band_i_id"": ""B"",
      ""band_j_id"": ""K"",
      ""pairing_boost"": 890
    },
    {
      ""band_i_id"": ""B"",
      ""band_j_id"": ""L"",
      ""pairing_boost"": 795
    },
    {
      ""band_i_id"": ""B"",
      ""band_j_id"": ""M"",
      ""pairing_boost"": 834
    },
    {
      ""band_i_id"": ""C"",
      ""band_j_id"": ""D"",
      ""pairing_boost"": 1080
    },
    {
      ""band_i_id"": ""C"",
      ""band_j_id"": ""E"",
      ""pairing_boost"": 613
    },
    {
      ""band_i_id"": ""C"",
      ""band_j_id"": ""F"",
      ""pairing_boost"": 980
    },
    {
      ""band_i_id"": ""C"",
      ""band_j_id"": ""G"",
      ""pairing_boost"": 1052
    },
    {
      ""band_i_id"": ""C"",
      ""band_j_id"": ""H"",
      ""pairing_boost"": 1044
    },
    {
      ""band_i_id"": ""C"",
      ""band_j_id"": ""I"",
      ""pairing_boost"": 1088
    },
    {
      ""band_i_id"": ""C"",
      ""band_j_id"": ""J"",
      ""pairing_boost"": 460
    },
    {
      ""band_i_id"": ""C"",
      ""band_j_id"": ""K"",
      ""pairing_boost"": 711
    },
    {
      ""band_i_id"": ""C"",
      ""band_j_id"": ""L"",
      ""pairing_boost"": 1034
    },
    {
      ""band_i_id"": ""C"",
      ""band_j_id"": ""M"",
      ""pairing_boost"": 984
    },
    {
      ""band_i_id"": ""D"",
      ""band_j_id"": ""E"",
      ""pairing_boost"": 1122
    },
    {
      ""band_i_id"": ""D"",
      ""band_j_id"": ""F"",
      ""pairing_boost"": 1140
    },
    {
      ""band_i_id"": ""D"",
      ""band_j_id"": ""G"",
      ""pairing_boost"": 894
    },
    {
      ""band_i_id"": ""D"",
      ""band_j_id"": ""H"",
      ""pairing_boost"": 989
    },
    {
      ""band_i_id"": ""D"",
      ""band_j_id"": ""I"",
      ""pairing_boost"": 594
    },
    {
      ""band_i_id"": ""D"",
      ""band_j_id"": ""J"",
      ""pairing_boost"": 796
    },
    {
      ""band_i_id"": ""D"",
      ""band_j_id"": ""K"",
      ""pairing_boost"": 715
    },
    {
      ""band_i_id"": ""D"",
      ""band_j_id"": ""L"",
      ""pairing_boost"": 1165
    },
    {
      ""band_i_id"": ""D"",
      ""band_j_id"": ""M"",
      ""pairing_boost"": 1156
    },
    {
      ""band_i_id"": ""E"",
      ""band_j_id"": ""F"",
      ""pairing_boost"": 553
    },
    {
      ""band_i_id"": ""E"",
      ""band_j_id"": ""G"",
      ""pairing_boost"": 217
    },
    {
      ""band_i_id"": ""E"",
      ""band_j_id"": ""H"",
      ""pairing_boost"": 518
    },
    {
      ""band_i_id"": ""E"",
      ""band_j_id"": ""I"",
      ""pairing_boost"": 513
    },
    {
      ""band_i_id"": ""E"",
      ""band_j_id"": ""J"",
      ""pairing_boost"": 521
    },
    {
      ""band_i_id"": ""E"",
      ""band_j_id"": ""K"",
      ""pairing_boost"": 741
    },
    {
      ""band_i_id"": ""E"",
      ""band_j_id"": ""L"",
      ""pairing_boost"": 1091
    },
    {
      ""band_i_id"": ""E"",
      ""band_j_id"": ""M"",
      ""pairing_boost"": 1134
    },
    {
      ""band_i_id"": ""F"",
      ""band_j_id"": ""G"",
      ""pairing_boost"": 901
    },
    {
      ""band_i_id"": ""F"",
      ""band_j_id"": ""H"",
      ""pairing_boost"": 770
    },
    {
      ""band_i_id"": ""F"",
      ""band_j_id"": ""I"",
      ""pairing_boost"": 1305
    },
    {
      ""band_i_id"": ""F"",
      ""band_j_id"": ""J"",
      ""pairing_boost"": 1416
    },
    {
      ""band_i_id"": ""F"",
      ""band_j_id"": ""K"",
      ""pairing_boost"": 823
    },
    {
      ""band_i_id"": ""F"",
      ""band_j_id"": ""L"",
      ""pairing_boost"": 713
    },
    {
      ""band_i_id"": ""F"",
      ""band_j_id"": ""M"",
      ""pairing_boost"": 568
    },
    {
      ""band_i_id"": ""G"",
      ""band_j_id"": ""H"",
      ""pairing_boost"": 916
    },
    {
      ""band_i_id"": ""G"",
      ""band_j_id"": ""I"",
      ""pairing_boost"": 665
    },
    {
      ""band_i_id"": ""G"",
      ""band_j_id"": ""J"",
      ""pairing_boost"": 1384
    },
    {
      ""band_i_id"": ""G"",
      ""band_j_id"": ""K"",
      ""pairing_boost"": 158
    },
    {
      ""band_i_id"": ""G"",
      ""band_j_id"": ""L"",
      ""pairing_boost"": 507
    },
    {
      ""band_i_id"": ""G"",
      ""band_j_id"": ""M"",
      ""pairing_boost"": 453
    },
    {
      ""band_i_id"": ""H"",
      ""band_j_id"": ""I"",
      ""pairing_boost"": 866
    },
    {
      ""band_i_id"": ""H"",
      ""band_j_id"": ""J"",
      ""pairing_boost"": 891
    },
    {
      ""band_i_id"": ""H"",
      ""band_j_id"": ""K"",
      ""pairing_boost"": 888
    },
    {
      ""band_i_id"": ""H"",
      ""band_j_id"": ""L"",
      ""pairing_boost"": 329
    },
    {
      ""band_i_id"": ""H"",
      ""band_j_id"": ""M"",
      ""pairing_boost"": 587
    },
    {
      ""band_i_id"": ""I"",
      ""band_j_id"": ""J"",
      ""pairing_boost"": 1227
    },
    {
      ""band_i_id"": ""I"",
      ""band_j_id"": ""K"",
      ""pairing_boost"": 714
    },
    {
      ""band_i_id"": ""I"",
      ""band_j_id"": ""L"",
      ""pairing_boost"": 856
    },
    {
      ""band_i_id"": ""I"",
      ""band_j_id"": ""M"",
      ""pairing_boost"": 962
    },
    {
      ""band_i_id"": ""J"",
      ""band_j_id"": ""K"",
      ""pairing_boost"": 1158
    },
    {
      ""band_i_id"": ""J"",
      ""band_j_id"": ""L"",
      ""pairing_boost"": 1216
    },
    {
      ""band_i_id"": ""J"",
      ""band_j_id"": ""M"",
      ""pairing_boost"": 475
    },
    {
      ""band_i_id"": ""K"",
      ""band_j_id"": ""L"",
      ""pairing_boost"": 602
    },
    {
      ""band_i_id"": ""K"",
      ""band_j_id"": ""M"",
      ""pairing_boost"": 728
    },
    {
      ""band_i_id"": ""L"",
      ""band_j_id"": ""M"",
      ""pairing_boost"": 482
    }
  ]
}

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': [215, 1295, 793, 806, 346, 1225, 1161, 430, 879, 1328, 1638, 1526, 1087], 'quadratic_coeffs': [[235, 403, 577, 279, 381, 440, 227, 511, 544, 159, 350, 528, 699], [0, 684, 1261, 1424, 855, 1000, 1357, 659, 654, 1287, 890, 795, 834], [0, 0, 732, 1080, 613, 980, 1052, 1044, 1088, 460, 711, 1034, 984], [0, 0, 0, 808, 1122, 1140, 894, 989, 594, 796, 715, 1165, 1156], [0, 0, 0, 0, 636, 553, 217, 518, 513, 521, 741, 1091, 1134], [0, 0, 0, 0, 0, 1264, 901, 770, 1305, 1416, 823, 713, 568], [0, 0, 0, 0, 0, 0, 1075, 916, 665, 1384, 158, 507, 453], [0, 0, 0, 0, 0, 0, 0, 888, 866, 891, 888, 329, 587], [0, 0, 0, 0, 0, 0, 0, 0, 353, 1227, 714, 856, 962], [0, 0, 0, 0, 0, 0, 0, 0, 0, 870, 1158, 1216, 475], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 730, 602, 728], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1308, 482], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [207, 1289, 762, 788, 283, 1229, 1192, 479, 932, 1297, 1591, 1485, 1044], 'capacity': 5309, 'solution': [0, 1, 2, 3, 4, 8, 12], 'obj': 22578.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 8, 12]",22578.0,"{'problem_type': 'QKP', 'num_items': 13, 'capacity': 5309, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 215}, {'item_id': 'B', 'linear_profit': 1295}, {'item_id': 'C', 'linear_profit': 793}, {'item_id': 'D', 'linear_profit': 806}, {'item_id': 'E', 'linear_profit': 346}, {'item_id': 'F', 'linear_profit': 1225}, {'item_id': 'G', 'linear_profit': 1161}, {'item_id': 'H', 'linear_profit': 430}, {'item_id': 'I', 'linear_profit': 879}, {'item_id': 'J', 'linear_profit': 1328}, {'item_id': 'K', 'linear_profit': 1638}, {'item_id': 'L', 'linear_profit': 1526}, {'item_id': 'M', 'linear_profit': 1087}], 'weight_pairs': [{'item_id': 'A', 'weight': 207}, {'item_id': 'B', 'weight': 1289}, {'item_id': 'C', 'weight': 762}, {'item_id': 'D', 'weight': 788}, {'item_id': 'E', 'weight': 283}, {'item_id': 'F', 'weight': 1229}, {'item_id': 'G', 'weight': 1192}, {'item_id': 'H', 'weight': 479}, {'item_id': 'I', 'weight': 932}, {'item_id': 'J', 'weight': 1297}, {'item_id': 'K', 'weight': 1591}, {'item_id': 'L', 'weight': 1485}, {'item_id': 'M', 'weight': 1044}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 235}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 403}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 577}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 279}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 381}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 440}, {'item_i_id': 'A', 'item_j_id': 'G', 'quadratic_profit': 227}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 511}, {'item_i_id': 'A', 'item_j_id': 'I', 'quadratic_profit': 544}, {'item_i_id': 'A', 'item_j_id': 'J', 'quadratic_profit': 159}, {'item_i_id': 'A', 'item_j_id': 'K', 'quadratic_profit': 350}, {'item_i_id': 'A', 'item_j_id': 'L', 'quadratic_profit': 528}, {'item_i_id': 'A', 'item_j_id': 'M', 'quadratic_profit': 699}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 684}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 1261}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 1424}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 855}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 1000}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 1357}, {'item_i_id': 'B', 'item_j_id': 'H', 'quadratic_profit': 659}, {'item_i_id': 'B', 'item_j_id': 'I', 'quadratic_profit': 654}, {'item_i_id': 'B', 'item_j_id': 'J', 'quadratic_profit': 1287}, {'item_i_id': 'B', 'item_j_id': 'K', 'quadratic_profit': 890}, {'item_i_id': 'B', 'item_j_id': 'L', 'quadratic_profit': 795}, {'item_i_id': 'B', 'item_j_id': 'M', 'quadratic_profit': 834}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 732}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 1080}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 613}, {'item_i_id': 'C', 'item_j_id': 'F', 'quadratic_profit': 980}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 1052}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 1044}, {'item_i_id': 'C', 'item_j_id': 'I', 'quadratic_profit': 1088}, {'item_i_id': 'C', 'item_j_id': 'J', 'quadratic_profit': 460}, {'item_i_id': 'C', 'item_j_id': 'K', 'quadratic_profit': 711}, {'item_i_id': 'C', 'item_j_id': 'L', 'quadratic_profit': 1034}, {'item_i_id': 'C', 'item_j_id': 'M', 'quadratic_profit': 984}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 808}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 1122}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 1140}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 894}, {'item_i_id': 'D', 'item_j_id': 'H', 'quadratic_profit': 989}, {'item_i_id': 'D', 'item_j_id': 'I', 'quadratic_profit': 594}, {'item_i_id': 'D', 'item_j_id': 'J', 'quadratic_profit': 796}, {'item_i_id': 'D', 'item_j_id': 'K', 'quadratic_profit': 715}, {'item_i_id': 'D', 'item_j_id': 'L', 'quadratic_profit': 1165}, {'item_i_id': 'D', 'item_j_id': 'M', 'quadratic_profit': 1156}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 636}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 553}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 217}, {'item_i_id': 'E', 'item_j_id': 'H', 'quadratic_profit': 518}, {'item_i_id': 'E', 'item_j_id': 'I', 'quadratic_profit': 513}, {'item_i_id': 'E', 'item_j_id': 'J', 'quadratic_profit': 521}, {'item_i_id': 'E', 'item_j_id': 'K', 'quadratic_profit': 741}, {'item_i_id': 'E', 'item_j_id': 'L', 'quadratic_profit': 1091}, {'item_i_id': 'E', 'item_j_id': 'M', 'quadratic_profit': 1134}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 1264}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 901}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 770}, {'item_i_id': 'F', 'item_j_id': 'I', 'quadratic_profit': 1305}, {'item_i_id': 'F', 'item_j_id': 'J', 'quadratic_profit': 1416}, {'item_i_id': 'F', 'item_j_id': 'K', 'quadratic_profit': 823}, {'item_i_id': 'F', 'item_j_id': 'L', 'quadratic_profit': 713}, {'item_i_id': 'F', 'item_j_id': 'M', 'quadratic_profit': 568}, {'item_i_id': 'G', 'item_j_id': 'G', 'quadratic_profit': 1075}, {'item_i_id': 'G', 'item_j_id': 'H', 'quadratic_profit': 916}, {'item_i_id': 'G', 'item_j_id': 'I', 'quadratic_profit': 665}, {'item_i_id': 'G', 'item_j_id': 'J', 'quadratic_profit': 1384}, {'item_i_id': 'G', 'item_j_id': 'K', 'quadratic_profit': 158}, {'item_i_id': 'G', 'item_j_id': 'L', 'quadratic_profit': 507}, {'item_i_id': 'G', 'item_j_id': 'M', 'quadratic_profit': 453}, {'item_i_id': 'H', 'item_j_id': 'H', 'quadratic_profit': 888}, {'item_i_id': 'H', 'item_j_id': 'I', 'quadratic_profit': 866}, {'item_i_id': 'H', 'item_j_id': 'J', 'quadratic_profit': 891}, {'item_i_id': 'H', 'item_j_id': 'K', 'quadratic_profit': 888}, {'item_i_id': 'H', 'item_j_id': 'L', 'quadratic_profit': 329}, {'item_i_id': 'H', 'item_j_id': 'M', 'quadratic_profit': 587}, {'item_i_id': 'I', 'item_j_id': 'I', 'quadratic_profit': 353}, {'item_i_id': 'I', 'item_j_id': 'J', 'quadratic_profit': 1227}, {'item_i_id': 'I', 'item_j_id': 'K', 'quadratic_profit': 714}, {'item_i_id': 'I', 'item_j_id': 'L', 'quadratic_profit': 856}, {'item_i_id': 'I', 'item_j_id': 'M', 'quadratic_profit': 962}, {'item_i_id': 'J', 'item_j_id': 'J', 'quadratic_profit': 870}, {'item_i_id': 'J', 'item_j_id': 'K', 'quadratic_profit': 1158}, {'item_i_id': 'J', 'item_j_id': 'L', 'quadratic_profit': 1216}, {'item_i_id': 'J', 'item_j_id': 'M', 'quadratic_profit': 475}, {'item_i_id': 'K', 'item_j_id': 'K', 'quadratic_profit': 730}, {'item_i_id': 'K', 'item_j_id': 'L', 'quadratic_profit': 602}, {'item_i_id': 'K', 'item_j_id': 'M', 'quadratic_profit': 728}, {'item_i_id': 'L', 'item_j_id': 'L', 'quadratic_profit': 1308}, {'item_i_id': 'L', 'item_j_id': 'M', 'quadratic_profit': 482}]}","['A', 'B', 'C', 'D', 'E', 'I', 'M']",21,json,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"": 12,
  ""bed_area_capacity"": 1188,
  ""plant_ids"": [
    0,
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11
  ],
  ""linear"": [
    {
      ""plant_id"": 0,
      ""base_beauty"": 108
    },
    {
      ""plant_id"": 1,
      ""base_beauty"": 111
    },
    {
      ""plant_id"": 2,
      ""base_beauty"": 102
    },
    {
      ""plant_id"": 3,
      ""base_beauty"": 100
    },
    {
      ""plant_id"": 4,
      ""base_beauty"": 105
    },
    {
      ""plant_id"": 5,
      ""base_beauty"": 110
    },
    {
      ""plant_id"": 6,
      ""base_beauty"": 104
    },
    {
      ""plant_id"": 7,
      ""base_beauty"": 111
    },
    {
      ""plant_id"": 8,
      ""base_beauty"": 100
    },
    {
      ""plant_id"": 9,
      ""base_beauty"": 110
    },
    {
      ""plant_id"": 10,
      ""base_beauty"": 102
    },
    {
      ""plant_id"": 11,
      ""base_beauty"": 102
    }
  ],
  ""weights"": [
    {
      ""plant_id"": 0,
      ""area_required"": 105
    },
    {
      ""plant_id"": 1,
      ""area_required"": 103
    },
    {
      ""plant_id"": 2,
      ""area_required"": 107
    },
    {
      ""plant_id"": 3,
      ""area_required"": 108
    },
    {
      ""plant_id"": 4,
      ""area_required"": 100
    },
    {
      ""plant_id"": 5,
      ""area_required"": 103
    },
    {
      ""plant_id"": 6,
      ""area_required"": 108
    },
    {
      ""plant_id"": 7,
      ""area_required"": 104
    },
    {
      ""plant_id"": 8,
      ""area_required"": 110
    },
    {
      ""plant_id"": 9,
      ""area_required"": 104
    },
    {
      ""plant_id"": 10,
      ""area_required"": 106
    },
    {
      ""plant_id"": 11,
      ""area_required"": 107
    }
  ],
  ""quadratic"": [
    {
      ""plant_i_id"": 0,
      ""plant_j_id"": 1,
      ""pair_bonus_beauty"": 102
    },
    {
      ""plant_i_id"": 0,
      ""plant_j_id"": 2,
      ""pair_bonus_beauty"": 102
    },
    {
      ""plant_i_id"": 0,
      ""plant_j_id"": 5,
      ""pair_bonus_beauty"": 105
    },
    {
      ""plant_i_id"": 0,
      ""plant_j_id"": 8,
      ""pair_bonus_beauty"": 105
    },
    {
      ""plant_i_id"": 0,
      ""plant_j_id"": 9,
      ""pair_bonus_beauty"": 102
    },
    {
      ""plant_i_id"": 0,
      ""plant_j_id"": 11,
      ""pair_bonus_beauty"": 106
    },
    {
      ""plant_i_id"": 1,
      ""plant_j_id"": 2,
      ""pair_bonus_beauty"": 106
    },
    {
      ""plant_i_id"": 1,
      ""plant_j_id"": 3,
      ""pair_bonus_beauty"": 105
    },
    {
      ""plant_i_id"": 1,
      ""plant_j_id"": 4,
      ""pair_bonus_beauty"": 105
    },
    {
      ""plant_i_id"": 1,
      ""plant_j_id"": 5,
      ""pair_bonus_beauty"": 105
    },
    {
      ""plant_i_id"": 1,
      ""plant_j_id"": 6,
      ""pair_bonus_beauty"": 101
    },
    {
      ""plant_i_id"": 1,
      ""plant_j_id"": 9,
      ""pair_bonus_beauty"": 101
    },
    {
      ""plant_i_id"": 1,
      ""plant_j_id"": 10,
      ""pair_bonus_beauty"": 101
    },
    {
      ""plant_i_id"": 1,
      ""plant_j_id"": 11,
      ""pair_bonus_beauty"": 104
    },
    {
      ""plant_i_id"": 2,
      ""plant_j_id"": 3,
      ""pair_bonus_beauty"": 103
    },
    {
      ""plant_i_id"": 2,
      ""plant_j_id"": 4,
      ""pair_bonus_beauty"": 108
    },
    {
      ""plant_i_id"": 2,
      ""plant_j_id"": 6,
      ""pair_bonus_beauty"": 107
    },
    {
      ""plant_i_id"": 2,
      ""plant_j_id"": 8,
      ""pair_bonus_beauty"": 103
    },
    {
      ""plant_i_id"": 2,
      ""plant_j_id"": 11,
      ""pair_bonus_beauty"": 103
    },
    {
      ""plant_i_id"": 3,
      ""plant_j_id"": 4,
      ""pair_bonus_beauty"": 105
    },
    {
      ""plant_i_id"": 3,
      ""plant_j_id"": 6,
      ""pair_bonus_beauty"": 103
    },
    {
      ""plant_i_id"": 3,
      ""plant_j_id"": 10,
      ""pair_bonus_beauty"": 110
    },
    {
      ""plant_i_id"": 3,
      ""plant_j_id"": 11,
      ""pair_bonus_beauty"": 107
    },
    {
      ""plant_i_id"": 4,
      ""plant_j_id"": 5,
      ""pair_bonus_beauty"": 103
    },
    {
      ""plant_i_id"": 4,
      ""plant_j_id"": 6,
      ""pair_bonus_beauty"": 100
    },
    {
      ""plant_i_id"": 4,
      ""plant_j_id"": 7,
      ""pair_bonus_beauty"": 103
    },
    {
      ""plant_i_id"": 4,
      ""plant_j_id"": 8,
      ""pair_bonus_beauty"": 106
    },
    {
      ""plant_i_id"": 4,
      ""plant_j_id"": 9,
      ""pair_bonus_beauty"": 104
    },
    {
      ""plant_i_id"": 4,
      ""plant_j_id"": 10,
      ""pair_bonus_beauty"": 106
    },
    {
      ""plant_i_id"": 4,
      ""plant_j_id"": 11,
      ""pair_bonus_beauty"": 102
    },
    {
      ""plant_i_id"": 5,
      ""plant_j_id"": 7,
      ""pair_bonus_beauty"": 103
    },
    {
      ""plant_i_id"": 5,
      ""plant_j_id"": 8,
      ""pair_bonus_beauty"": 103
    },
    {
      ""plant_i_id"": 5,
      ""plant_j_id"": 9,
      ""pair_bonus_beauty"": 104
    },
    {
      ""plant_i_id"": 5,
      ""plant_j_id"": 11,
      ""pair_bonus_beauty"": 104
    },
    {
      ""plant_i_id"": 6,
      ""plant_j_id"": 7,
      ""pair_bonus_beauty"": 108
    },
    {
      ""plant_i_id"": 6,
      ""plant_j_id"": 8,
      ""pair_bonus_beauty"": 103
    },
    {
      ""plant_i_id"": 6,
      ""plant_j_id"": 9,
      ""pair_bonus_beauty"": 104
    },
    {
      ""plant_i_id"": 6,
      ""plant_j_id"": 11,
      ""pair_bonus_beauty"": 105
    },
    {
      ""plant_i_id"": 7,
      ""plant_j_id"": 8,
      ""pair_bonus_beauty"": 106
    },
    {
      ""plant_i_id"": 7,
      ""plant_j_id"": 10,
      ""pair_bonus_beauty"": 103
    },
    {
      ""plant_i_id"": 7,
      ""plant_j_id"": 11,
      ""pair_bonus_beauty"": 103
    },
    {
      ""plant_i_id"": 8,
      ""plant_j_id"": 9,
      ""pair_bonus_beauty"": 103
    },
    {
      ""plant_i_id"": 8,
      ""plant_j_id"": 11,
      ""pair_bonus_beauty"": 108
    },
    {
      ""plant_i_id"": 9,
      ""plant_j_id"": 10,
      ""pair_bonus_beauty"": 101
    },
    {
      ""plant_i_id"": 9,
      ""plant_j_id"": 11,
      ""pair_bonus_beauty"": 105
    }
  ]
}

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': [108, 111, 102, 100, 105, 110, 104, 111, 100, 110, 102, 102], 'quadratic_coeffs': [[102, 102, 102, 0, 0, 105, 0, 0, 105, 102, 0, 106], [0, 105, 106, 105, 105, 105, 101, 0, 0, 101, 101, 104], [0, 0, 107, 103, 108, 0, 107, 0, 103, 0, 0, 103], [0, 0, 0, 106, 105, 0, 103, 0, 0, 0, 110, 107], [0, 0, 0, 0, 103, 103, 100, 103, 106, 104, 106, 102], [0, 0, 0, 0, 0, 105, 0, 103, 103, 104, 0, 104], [0, 0, 0, 0, 0, 0, 0, 108, 103, 104, 0, 105], [0, 0, 0, 0, 0, 0, 0, 105, 106, 0, 103, 103], [0, 0, 0, 0, 0, 0, 0, 0, 109, 103, 0, 108], [0, 0, 0, 0, 0, 0, 0, 0, 0, 107, 101, 105], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 101, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 108]], 'weights': [105, 103, 107, 108, 100, 103, 108, 104, 110, 104, 106, 107], 'capacity': 1188, 'solution': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11], 'obj': 5328.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11]",5328.0,"{'problem_type': 'QKP', 'num_items': 12, 'capacity': 1188, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 'linear_pairs': [{'item_id': 0, 'linear_profit': 108}, {'item_id': 1, 'linear_profit': 111}, {'item_id': 2, 'linear_profit': 102}, {'item_id': 3, 'linear_profit': 100}, {'item_id': 4, 'linear_profit': 105}, {'item_id': 5, 'linear_profit': 110}, {'item_id': 6, 'linear_profit': 104}, {'item_id': 7, 'linear_profit': 111}, {'item_id': 8, 'linear_profit': 100}, {'item_id': 9, 'linear_profit': 110}, {'item_id': 10, 'linear_profit': 102}, {'item_id': 11, 'linear_profit': 102}], 'weight_pairs': [{'item_id': 0, 'weight': 105}, {'item_id': 1, 'weight': 103}, {'item_id': 2, 'weight': 107}, {'item_id': 3, 'weight': 108}, {'item_id': 4, 'weight': 100}, {'item_id': 5, 'weight': 103}, {'item_id': 6, 'weight': 108}, {'item_id': 7, 'weight': 104}, {'item_id': 8, 'weight': 110}, {'item_id': 9, 'weight': 104}, {'item_id': 10, 'weight': 106}, {'item_id': 11, 'weight': 107}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 102}, {'item_i_id': 0, 'item_j_id': 1, 'quadratic_profit': 102}, {'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 102}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 105}, {'item_i_id': 0, 'item_j_id': 8, 'quadratic_profit': 105}, {'item_i_id': 0, 'item_j_id': 9, 'quadratic_profit': 102}, {'item_i_id': 0, 'item_j_id': 11, 'quadratic_profit': 106}, {'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 105}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 106}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 105}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 105}, {'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': 9, 'quadratic_profit': 101}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 101}, {'item_i_id': 1, 'item_j_id': 11, 'quadratic_profit': 104}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 107}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 103}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 108}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 107}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 103}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 103}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 106}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 105}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 103}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 110}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 107}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 103}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 103}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 100}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 103}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 106}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 104}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 106}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 102}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 105}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 103}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 103}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 104}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 104}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 108}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 103}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 104}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 105}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 105}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 106}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 103}, {'item_i_id': 7, 'item_j_id': 11, 'quadratic_profit': 103}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 109}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 103}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 108}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 107}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 101}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 105}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 101}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 108}]}","[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11]",22,json,0
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.

There are 13 distinct pieces listed as A, B, C, D, E, F, G, H, I, J, K, L, M, and the bag's weight limit is 1147 — choose which to pack so the kit fits and the creative score is maximized.

| gear_id | base_creative_value |
|---|---|
| A | 139 |
| B | 145 |
| C | 149 |
| D | 135 |
| E | 137 |
| F | 120 |
| G | 127 |
| H | 103 |
| I | 141 |
| J | 124 |
| K | 118 |
| L | 136 |
| M | 126 |

| gear_id | gear_weight |
|---|---|
| A | 135 |
| B | 139 |
| C | 144 |
| D | 140 |
| E | 136 |
| F | 115 |
| G | 131 |
| H | 105 |
| I | 140 |
| J | 119 |
| K | 113 |
| L | 142 |
| M | 128 |

| gear_i_id | gear_j_id | pair_synergy_bonus |
|---|---|---|
| A | B | 135 |
| A | C | 100 |
| A | I | 120 |
| A | J | 100 |
| A | K | 110 |
| A | L | 104 |
| A | M | 107 |
| B | C | 129 |
| B | D | 119 |
| B | E | 139 |
| B | F | 125 |
| B | G | 139 |
| B | H | 130 |
| B | I | 130 |
| B | J | 136 |
| B | K | 112 |
| B | L | 130 |
| B | M | 130 |
| C | D | 112 |
| C | E | 111 |
| C | G | 122 |
| C | I | 121 |
| C | J | 121 |
| C | K | 121 |
| C | M | 121 |
| D | E | 140 |
| D | F | 114 |
| D | G | 128 |
| D | J | 118 |
| D | K | 113 |
| D | L | 118 |
| E | F | 131 |
| E | G | 136 |
| E | J | 132 |
| E | K | 110 |
| E | M | 129 |
| F | G | 115 |
| F | J | 105 |
| F | K | 105 |
| G | H | 103 |
| G | I | 133 |
| G | J | 105 |
| G | K | 133 |
| G | L | 110 |
| G | M | 131 |
| H | J | 101 |
| H | K | 101 |
| I | J | 119 |
| I | K | 113 |
| J | K | 120 |
| J | L | 120 |
| J | M | 120 |
| K | M | 109 |
| L | M | 107 |

Pack wisely to maximize creative value without exceeding the 1147 limit.

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': [139, 145, 149, 135, 137, 120, 127, 103, 141, 124, 118, 136, 126], 'quadratic_coeffs': [[0, 135, 100, 0, 0, 0, 0, 0, 120, 100, 110, 104, 107], [0, 105, 129, 119, 139, 125, 139, 130, 130, 136, 112, 130, 130], [0, 0, 124, 112, 111, 0, 122, 0, 121, 121, 121, 0, 121], [0, 0, 0, 118, 140, 114, 128, 0, 0, 118, 113, 118, 0], [0, 0, 0, 0, 110, 131, 136, 0, 0, 132, 110, 0, 129], [0, 0, 0, 0, 0, 0, 115, 0, 0, 105, 105, 0, 0], [0, 0, 0, 0, 0, 0, 0, 103, 133, 105, 133, 110, 131], [0, 0, 0, 0, 0, 0, 0, 101, 0, 101, 101, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 119, 113, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 120, 120, 120, 120], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 104, 0, 109], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 109, 107], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 104]], 'weights': [135, 139, 144, 140, 136, 115, 131, 105, 140, 119, 113, 142, 128], 'capacity': 1147, 'solution': [1, 2, 3, 4, 5, 6, 7, 9, 10], 'obj': 4884.0, 'problem_type': 'QKP'}","[1, 2, 3, 4, 5, 6, 7, 9, 10]",4884.0,"{'problem_type': 'QKP', 'num_items': 13, 'capacity': 1147, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 139}, {'item_id': 'B', 'linear_profit': 145}, {'item_id': 'C', 'linear_profit': 149}, {'item_id': 'D', 'linear_profit': 135}, {'item_id': 'E', 'linear_profit': 137}, {'item_id': 'F', 'linear_profit': 120}, {'item_id': 'G', 'linear_profit': 127}, {'item_id': 'H', 'linear_profit': 103}, {'item_id': 'I', 'linear_profit': 141}, {'item_id': 'J', 'linear_profit': 124}, {'item_id': 'K', 'linear_profit': 118}, {'item_id': 'L', 'linear_profit': 136}, {'item_id': 'M', 'linear_profit': 126}], 'weight_pairs': [{'item_id': 'A', 'weight': 135}, {'item_id': 'B', 'weight': 139}, {'item_id': 'C', 'weight': 144}, {'item_id': 'D', 'weight': 140}, {'item_id': 'E', 'weight': 136}, {'item_id': 'F', 'weight': 115}, {'item_id': 'G', 'weight': 131}, {'item_id': 'H', 'weight': 105}, {'item_id': 'I', 'weight': 140}, {'item_id': 'J', 'weight': 119}, {'item_id': 'K', 'weight': 113}, {'item_id': 'L', 'weight': 142}, {'item_id': 'M', 'weight': 128}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 135}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 100}, {'item_i_id': 'A', 'item_j_id': 'I', 'quadratic_profit': 120}, {'item_i_id': 'A', 'item_j_id': 'J', 'quadratic_profit': 100}, {'item_i_id': 'A', 'item_j_id': 'K', 'quadratic_profit': 110}, {'item_i_id': 'A', 'item_j_id': 'L', 'quadratic_profit': 104}, {'item_i_id': 'A', 'item_j_id': 'M', 'quadratic_profit': 107}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 105}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 129}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 119}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 139}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 125}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 139}, {'item_i_id': 'B', 'item_j_id': 'H', 'quadratic_profit': 130}, {'item_i_id': 'B', 'item_j_id': 'I', 'quadratic_profit': 130}, {'item_i_id': 'B', 'item_j_id': 'J', 'quadratic_profit': 136}, {'item_i_id': 'B', 'item_j_id': 'K', 'quadratic_profit': 112}, {'item_i_id': 'B', 'item_j_id': 'L', 'quadratic_profit': 130}, {'item_i_id': 'B', 'item_j_id': 'M', 'quadratic_profit': 130}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 124}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 112}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 111}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 122}, {'item_i_id': 'C', 'item_j_id': 'I', 'quadratic_profit': 121}, {'item_i_id': 'C', 'item_j_id': 'J', 'quadratic_profit': 121}, {'item_i_id': 'C', 'item_j_id': 'K', 'quadratic_profit': 121}, {'item_i_id': 'C', 'item_j_id': 'M', 'quadratic_profit': 121}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 118}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 140}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 114}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 128}, {'item_i_id': 'D', 'item_j_id': 'J', 'quadratic_profit': 118}, {'item_i_id': 'D', 'item_j_id': 'K', 'quadratic_profit': 113}, {'item_i_id': 'D', 'item_j_id': 'L', 'quadratic_profit': 118}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 110}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 131}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 136}, {'item_i_id': 'E', 'item_j_id': 'J', 'quadratic_profit': 132}, {'item_i_id': 'E', 'item_j_id': 'K', 'quadratic_profit': 110}, {'item_i_id': 'E', 'item_j_id': 'M', 'quadratic_profit': 129}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 115}, {'item_i_id': 'F', 'item_j_id': 'J', 'quadratic_profit': 105}, {'item_i_id': 'F', 'item_j_id': 'K', 'quadratic_profit': 105}, {'item_i_id': 'G', 'item_j_id': 'H', 'quadratic_profit': 103}, {'item_i_id': 'G', 'item_j_id': 'I', 'quadratic_profit': 133}, {'item_i_id': 'G', 'item_j_id': 'J', 'quadratic_profit': 105}, {'item_i_id': 'G', 'item_j_id': 'K', 'quadratic_profit': 133}, {'item_i_id': 'G', 'item_j_id': 'L', 'quadratic_profit': 110}, {'item_i_id': 'G', 'item_j_id': 'M', 'quadratic_profit': 131}, {'item_i_id': 'H', 'item_j_id': 'H', 'quadratic_profit': 101}, {'item_i_id': 'H', 'item_j_id': 'J', 'quadratic_profit': 101}, {'item_i_id': 'H', 'item_j_id': 'K', 'quadratic_profit': 101}, {'item_i_id': 'I', 'item_j_id': 'J', 'quadratic_profit': 119}, {'item_i_id': 'I', 'item_j_id': 'K', 'quadratic_profit': 113}, {'item_i_id': 'J', 'item_j_id': 'J', 'quadratic_profit': 120}, {'item_i_id': 'J', 'item_j_id': 'K', 'quadratic_profit': 120}, {'item_i_id': 'J', 'item_j_id': 'L', 'quadratic_profit': 120}, {'item_i_id': 'J', 'item_j_id': 'M', 'quadratic_profit': 120}, {'item_i_id': 'K', 'item_j_id': 'K', 'quadratic_profit': 104}, {'item_i_id': 'K', 'item_j_id': 'M', 'quadratic_profit': 109}, {'item_i_id': 'L', 'item_j_id': 'L', 'quadratic_profit': 109}, {'item_i_id': 'L', 'item_j_id': 'M', 'quadratic_profit': 107}, {'item_i_id': 'M', 'item_j_id': 'M', 'quadratic_profit': 104}]}","['B', 'C', 'D', 'E', 'F', 'G', 'H', 'J', 'K']",23,markdown_table,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.

{
  ""num_features"": 14,
  ""total_budget"": 451,
  ""feature_ids"": [
    ""A"",
    ""B"",
    ""C"",
    ""D"",
    ""E"",
    ""F"",
    ""G"",
    ""H"",
    ""I"",
    ""J"",
    ""K"",
    ""L"",
    ""M"",
    ""N""
  ],
  ""linear"": [
    {
      ""feature_id"": ""A"",
      ""standalone_value"": 103
    },
    {
      ""feature_id"": ""B"",
      ""standalone_value"": 108
    },
    {
      ""feature_id"": ""C"",
      ""standalone_value"": 101
    },
    {
      ""feature_id"": ""D"",
      ""standalone_value"": 136
    },
    {
      ""feature_id"": ""E"",
      ""standalone_value"": 117
    },
    {
      ""feature_id"": ""F"",
      ""standalone_value"": 139
    },
    {
      ""feature_id"": ""G"",
      ""standalone_value"": 137
    },
    {
      ""feature_id"": ""H"",
      ""standalone_value"": 135
    },
    {
      ""feature_id"": ""I"",
      ""standalone_value"": 141
    },
    {
      ""feature_id"": ""J"",
      ""standalone_value"": 138
    },
    {
      ""feature_id"": ""K"",
      ""standalone_value"": 110
    },
    {
      ""feature_id"": ""L"",
      ""standalone_value"": 103
    },
    {
      ""feature_id"": ""M"",
      ""standalone_value"": 103
    },
    {
      ""feature_id"": ""N"",
      ""standalone_value"": 103
    }
  ],
  ""weights"": [
    {
      ""feature_id"": ""A"",
      ""cost"": 147
    },
    {
      ""feature_id"": ""B"",
      ""cost"": 111
    },
    {
      ""feature_id"": ""C"",
      ""cost"": 126
    },
    {
      ""feature_id"": ""D"",
      ""cost"": 133
    },
    {
      ""feature_id"": ""E"",
      ""cost"": 149
    },
    {
      ""feature_id"": ""F"",
      ""cost"": 122
    },
    {
      ""feature_id"": ""G"",
      ""cost"": 123
    },
    {
      ""feature_id"": ""H"",
      ""cost"": 117
    },
    {
      ""feature_id"": ""I"",
      ""cost"": 101
    },
    {
      ""feature_id"": ""J"",
      ""cost"": 140
    },
    {
      ""feature_id"": ""K"",
      ""cost"": 112
    },
    {
      ""feature_id"": ""L"",
      ""cost"": 111
    },
    {
      ""feature_id"": ""M"",
      ""cost"": 104
    },
    {
      ""feature_id"": ""N"",
      ""cost"": 110
    }
  ],
  ""quadratic"": [
    {
      ""feature_i_id"": ""A"",
      ""feature_j_id"": ""B"",
      ""pair_synergy_value"": 122
    },
    {
      ""feature_i_id"": ""A"",
      ""feature_j_id"": ""C"",
      ""pair_synergy_value"": 129
    },
    {
      ""feature_i_id"": ""A"",
      ""feature_j_id"": ""D"",
      ""pair_synergy_value"": 102
    },
    {
      ""feature_i_id"": ""A"",
      ""feature_j_id"": ""J"",
      ""pair_synergy_value"": 123
    },
    {
      ""feature_i_id"": ""A"",
      ""feature_j_id"": ""K"",
      ""pair_synergy_value"": 124
    },
    {
      ""feature_i_id"": ""A"",
      ""feature_j_id"": ""N"",
      ""pair_synergy_value"": 135
    },
    {
      ""feature_i_id"": ""B"",
      ""feature_j_id"": ""E"",
      ""pair_synergy_value"": 144
    },
    {
      ""feature_i_id"": ""B"",
      ""feature_j_id"": ""F"",
      ""pair_synergy_value"": 102
    },
    {
      ""feature_i_id"": ""B"",
      ""feature_j_id"": ""G"",
      ""pair_synergy_value"": 120
    },
    {
      ""feature_i_id"": ""B"",
      ""feature_j_id"": ""H"",
      ""pair_synergy_value"": 117
    },
    {
      ""feature_i_id"": ""B"",
      ""feature_j_id"": ""I"",
      ""pair_synergy_value"": 119
    },
    {
      ""feature_i_id"": ""B"",
      ""feature_j_id"": ""J"",
      ""pair_synergy_value"": 120
    },
    {
      ""feature_i_id"": ""B"",
      ""feature_j_id"": ""M"",
      ""pair_synergy_value"": 111
    },
    {
      ""feature_i_id"": ""B"",
      ""feature_j_id"": ""N"",
      ""pair_synergy_value"": 119
    },
    {
      ""feature_i_id"": ""C"",
      ""feature_j_id"": ""D"",
      ""pair_synergy_value"": 108
    },
    {
      ""feature_i_id"": ""C"",
      ""feature_j_id"": ""E"",
      ""pair_synergy_value"": 131
    },
    {
      ""feature_i_id"": ""C"",
      ""feature_j_id"": ""G"",
      ""pair_synergy_value"": 140
    },
    {
      ""feature_i_id"": ""C"",
      ""feature_j_id"": ""H"",
      ""pair_synergy_value"": 124
    },
    {
      ""feature_i_id"": ""C"",
      ""feature_j_id"": ""I"",
      ""pair_synergy_value"": 149
    },
    {
      ""feature_i_id"": ""C"",
      ""feature_j_id"": ""J"",
      ""pair_synergy_value"": 128
    },
    {
      ""feature_i_id"": ""C"",
      ""feature_j_id"": ""K"",
      ""pair_synergy_value"": 113
    },
    {
      ""feature_i_id"": ""C"",
      ""feature_j_id"": ""L"",
      ""pair_synergy_value"": 141
    },
    {
      ""feature_i_id"": ""C"",
      ""feature_j_id"": ""M"",
      ""pair_synergy_value"": 132
    },
    {
      ""feature_i_id"": ""C"",
      ""feature_j_id"": ""N"",
      ""pair_synergy_value"": 130
    },
    {
      ""feature_i_id"": ""D"",
      ""feature_j_id"": ""E"",
      ""pair_synergy_value"": 143
    },
    {
      ""feature_i_id"": ""D"",
      ""feature_j_id"": ""F"",
      ""pair_synergy_value"": 101
    },
    {
      ""feature_i_id"": ""D"",
      ""feature_j_id"": ""G"",
      ""pair_synergy_value"": 131
    },
    {
      ""feature_i_id"": ""D"",
      ""feature_j_id"": ""L"",
      ""pair_synergy_value"": 130
    },
    {
      ""feature_i_id"": ""D"",
      ""feature_j_id"": ""M"",
      ""pair_synergy_value"": 112
    },
    {
      ""feature_i_id"": ""D"",
      ""feature_j_id"": ""N"",
      ""pair_synergy_value"": 108
    },
    {
      ""feature_i_id"": ""E"",
      ""feature_j_id"": ""F"",
      ""pair_synergy_value"": 112
    },
    {
      ""feature_i_id"": ""E"",
      ""feature_j_id"": ""G"",
      ""pair_synergy_value"": 125
    },
    {
      ""feature_i_id"": ""E"",
      ""feature_j_id"": ""I"",
      ""pair_synergy_value"": 131
    },
    {
      ""feature_i_id"": ""E"",
      ""feature_j_id"": ""K"",
      ""pair_synergy_value"": 107
    },
    {
      ""feature_i_id"": ""E"",
      ""feature_j_id"": ""L"",
      ""pair_synergy_value"": 133
    },
    {
      ""feature_i_id"": ""E"",
      ""feature_j_id"": ""N"",
      ""pair_synergy_value"": 134
    },
    {
      ""feature_i_id"": ""F"",
      ""feature_j_id"": ""H"",
      ""pair_synergy_value"": 133
    },
    {
      ""feature_i_id"": ""F"",
      ""feature_j_id"": ""I"",
      ""pair_synergy_value"": 147
    },
    {
      ""feature_i_id"": ""F"",
      ""feature_j_id"": ""J"",
      ""pair_synergy_value"": 148
    },
    {
      ""feature_i_id"": ""F"",
      ""feature_j_id"": ""K"",
      ""pair_synergy_value"": 142
    },
    {
      ""feature_i_id"": ""F"",
      ""feature_j_id"": ""M"",
      ""pair_synergy_value"": 100
    },
    {
      ""feature_i_id"": ""F"",
      ""feature_j_id"": ""N"",
      ""pair_synergy_value"": 111
    },
    {
      ""feature_i_id"": ""G"",
      ""feature_j_id"": ""I"",
      ""pair_synergy_value"": 105
    },
    {
      ""feature_i_id"": ""G"",
      ""feature_j_id"": ""J"",
      ""pair_synergy_value"": 104
    },
    {
      ""feature_i_id"": ""G"",
      ""feature_j_id"": ""K"",
      ""pair_synergy_value"": 124
    },
    {
      ""feature_i_id"": ""G"",
      ""feature_j_id"": ""L"",
      ""pair_synergy_value"": 143
    },
    {
      ""feature_i_id"": ""G"",
      ""feature_j_id"": ""M"",
      ""pair_synergy_value"": 144
    },
    {
      ""feature_i_id"": ""G"",
      ""feature_j_id"": ""N"",
      ""pair_synergy_value"": 144
    },
    {
      ""feature_i_id"": ""H"",
      ""feature_j_id"": ""I"",
      ""pair_synergy_value"": 139
    },
    {
      ""feature_i_id"": ""H"",
      ""feature_j_id"": ""J"",
      ""pair_synergy_value"": 149
    },
    {
      ""feature_i_id"": ""H"",
      ""feature_j_id"": ""K"",
      ""pair_synergy_value"": 145
    },
    {
      ""feature_i_id"": ""H"",
      ""feature_j_id"": ""L"",
      ""pair_synergy_value"": 135
    },
    {
      ""feature_i_id"": ""H"",
      ""feature_j_id"": ""M"",
      ""pair_synergy_value"": 125
    },
    {
      ""feature_i_id"": ""H"",
      ""feature_j_id"": ""N"",
      ""pair_synergy_value"": 129
    },
    {
      ""feature_i_id"": ""I"",
      ""feature_j_id"": ""J"",
      ""pair_synergy_value"": 129
    },
    {
      ""feature_i_id"": ""I"",
      ""feature_j_id"": ""M"",
      ""pair_synergy_value"": 108
    },
    {
      ""feature_i_id"": ""I"",
      ""feature_j_id"": ""N"",
      ""pair_synergy_value"": 108
    },
    {
      ""feature_i_id"": ""J"",
      ""feature_j_id"": ""K"",
      ""pair_synergy_value"": 134
    },
    {
      ""feature_i_id"": ""J"",
      ""feature_j_id"": ""L"",
      ""pair_synergy_value"": 112
    },
    {
      ""feature_i_id"": ""J"",
      ""feature_j_id"": ""N"",
      ""pair_synergy_value"": 114
    },
    {
      ""feature_i_id"": ""K"",
      ""feature_j_id"": ""M"",
      ""pair_synergy_value"": 109
    },
    {
      ""feature_i_id"": ""K"",
      ""feature_j_id"": ""N"",
      ""pair_synergy_value"": 116
    },
    {
      ""feature_i_id"": ""L"",
      ""feature_j_id"": ""M"",
      ""pair_synergy_value"": 116
    },
    {
      ""feature_i_id"": ""L"",
      ""feature_j_id"": ""N"",
      ""pair_synergy_value"": 132
    },
    {
      ""feature_i_id"": ""M"",
      ""feature_j_id"": ""N"",
      ""pair_synergy_value"": 104
    }
  ]
}

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': [103, 108, 101, 136, 117, 139, 137, 135, 141, 138, 110, 103, 103, 103], 'quadratic_coeffs': [[114, 122, 129, 102, 0, 0, 0, 0, 0, 123, 124, 0, 0, 135], [0, 0, 0, 0, 144, 102, 120, 117, 119, 120, 0, 0, 111, 119], [0, 0, 0, 108, 131, 0, 140, 124, 149, 128, 113, 141, 132, 130], [0, 0, 0, 0, 143, 101, 131, 0, 0, 0, 0, 130, 112, 108], [0, 0, 0, 0, 109, 112, 125, 0, 131, 0, 107, 133, 0, 134], [0, 0, 0, 0, 0, 109, 0, 133, 147, 148, 142, 0, 100, 111], [0, 0, 0, 0, 0, 0, 0, 0, 105, 104, 124, 143, 144, 144], [0, 0, 0, 0, 0, 0, 0, 0, 139, 149, 145, 135, 125, 129], [0, 0, 0, 0, 0, 0, 0, 0, 109, 129, 0, 0, 108, 108], [0, 0, 0, 0, 0, 0, 0, 0, 0, 110, 134, 112, 0, 114], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 111, 0, 109, 116], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 116, 132], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 111, 104], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [147, 111, 126, 133, 149, 122, 123, 117, 101, 140, 112, 111, 104, 110], 'capacity': 451, 'solution': [5, 7, 8, 13], 'obj': 1285.0, 'problem_type': 'QKP'}","[5, 7, 8, 13]",1285.0,"{'problem_type': 'QKP', 'num_items': 14, 'capacity': 451, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 103}, {'item_id': 'B', 'linear_profit': 108}, {'item_id': 'C', 'linear_profit': 101}, {'item_id': 'D', 'linear_profit': 136}, {'item_id': 'E', 'linear_profit': 117}, {'item_id': 'F', 'linear_profit': 139}, {'item_id': 'G', 'linear_profit': 137}, {'item_id': 'H', 'linear_profit': 135}, {'item_id': 'I', 'linear_profit': 141}, {'item_id': 'J', 'linear_profit': 138}, {'item_id': 'K', 'linear_profit': 110}, {'item_id': 'L', 'linear_profit': 103}, {'item_id': 'M', 'linear_profit': 103}, {'item_id': 'N', 'linear_profit': 103}], 'weight_pairs': [{'item_id': 'A', 'weight': 147}, {'item_id': 'B', 'weight': 111}, {'item_id': 'C', 'weight': 126}, {'item_id': 'D', 'weight': 133}, {'item_id': 'E', 'weight': 149}, {'item_id': 'F', 'weight': 122}, {'item_id': 'G', 'weight': 123}, {'item_id': 'H', 'weight': 117}, {'item_id': 'I', 'weight': 101}, {'item_id': 'J', 'weight': 140}, {'item_id': 'K', 'weight': 112}, {'item_id': 'L', 'weight': 111}, {'item_id': 'M', 'weight': 104}, {'item_id': 'N', 'weight': 110}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 114}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 122}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 129}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 102}, {'item_i_id': 'A', 'item_j_id': 'J', 'quadratic_profit': 123}, {'item_i_id': 'A', 'item_j_id': 'K', 'quadratic_profit': 124}, {'item_i_id': 'A', 'item_j_id': 'N', 'quadratic_profit': 135}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 144}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 102}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 120}, {'item_i_id': 'B', 'item_j_id': 'H', 'quadratic_profit': 117}, {'item_i_id': 'B', 'item_j_id': 'I', 'quadratic_profit': 119}, {'item_i_id': 'B', 'item_j_id': 'J', 'quadratic_profit': 120}, {'item_i_id': 'B', 'item_j_id': 'M', 'quadratic_profit': 111}, {'item_i_id': 'B', 'item_j_id': 'N', 'quadratic_profit': 119}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 108}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 131}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 140}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 124}, {'item_i_id': 'C', 'item_j_id': 'I', 'quadratic_profit': 149}, {'item_i_id': 'C', 'item_j_id': 'J', 'quadratic_profit': 128}, {'item_i_id': 'C', 'item_j_id': 'K', 'quadratic_profit': 113}, {'item_i_id': 'C', 'item_j_id': 'L', 'quadratic_profit': 141}, {'item_i_id': 'C', 'item_j_id': 'M', 'quadratic_profit': 132}, {'item_i_id': 'C', 'item_j_id': 'N', 'quadratic_profit': 130}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 143}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 101}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 131}, {'item_i_id': 'D', 'item_j_id': 'L', 'quadratic_profit': 130}, {'item_i_id': 'D', 'item_j_id': 'M', 'quadratic_profit': 112}, {'item_i_id': 'D', 'item_j_id': 'N', 'quadratic_profit': 108}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 109}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 112}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 125}, {'item_i_id': 'E', 'item_j_id': 'I', 'quadratic_profit': 131}, {'item_i_id': 'E', 'item_j_id': 'K', 'quadratic_profit': 107}, {'item_i_id': 'E', 'item_j_id': 'L', 'quadratic_profit': 133}, {'item_i_id': 'E', 'item_j_id': 'N', 'quadratic_profit': 134}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 109}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 133}, {'item_i_id': 'F', 'item_j_id': 'I', 'quadratic_profit': 147}, {'item_i_id': 'F', 'item_j_id': 'J', 'quadratic_profit': 148}, {'item_i_id': 'F', 'item_j_id': 'K', 'quadratic_profit': 142}, {'item_i_id': 'F', 'item_j_id': 'M', 'quadratic_profit': 100}, {'item_i_id': 'F', 'item_j_id': 'N', 'quadratic_profit': 111}, {'item_i_id': 'G', 'item_j_id': 'I', 'quadratic_profit': 105}, {'item_i_id': 'G', 'item_j_id': 'J', 'quadratic_profit': 104}, {'item_i_id': 'G', 'item_j_id': 'K', 'quadratic_profit': 124}, {'item_i_id': 'G', 'item_j_id': 'L', 'quadratic_profit': 143}, {'item_i_id': 'G', 'item_j_id': 'M', 'quadratic_profit': 144}, {'item_i_id': 'G', 'item_j_id': 'N', 'quadratic_profit': 144}, {'item_i_id': 'H', 'item_j_id': 'I', 'quadratic_profit': 139}, {'item_i_id': 'H', 'item_j_id': 'J', 'quadratic_profit': 149}, {'item_i_id': 'H', 'item_j_id': 'K', 'quadratic_profit': 145}, {'item_i_id': 'H', 'item_j_id': 'L', 'quadratic_profit': 135}, {'item_i_id': 'H', 'item_j_id': 'M', 'quadratic_profit': 125}, {'item_i_id': 'H', 'item_j_id': 'N', 'quadratic_profit': 129}, {'item_i_id': 'I', 'item_j_id': 'I', 'quadratic_profit': 109}, {'item_i_id': 'I', 'item_j_id': 'J', 'quadratic_profit': 129}, {'item_i_id': 'I', 'item_j_id': 'M', 'quadratic_profit': 108}, {'item_i_id': 'I', 'item_j_id': 'N', 'quadratic_profit': 108}, {'item_i_id': 'J', 'item_j_id': 'J', 'quadratic_profit': 110}, {'item_i_id': 'J', 'item_j_id': 'K', 'quadratic_profit': 134}, {'item_i_id': 'J', 'item_j_id': 'L', 'quadratic_profit': 112}, {'item_i_id': 'J', 'item_j_id': 'N', 'quadratic_profit': 114}, {'item_i_id': 'K', 'item_j_id': 'K', 'quadratic_profit': 111}, {'item_i_id': 'K', 'item_j_id': 'M', 'quadratic_profit': 109}, {'item_i_id': 'K', 'item_j_id': 'N', 'quadratic_profit': 116}, {'item_i_id': 'L', 'item_j_id': 'M', 'quadratic_profit': 116}, {'item_i_id': 'L', 'item_j_id': 'N', 'quadratic_profit': 132}, {'item_i_id': 'M', 'item_j_id': 'M', 'quadratic_profit': 111}, {'item_i_id': 'M', 'item_j_id': 'N', 'quadratic_profit': 104}]}","['F', 'H', 'I', 'N']",24,json,names
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.

We had 11 films to pick from (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) and 27117 minutes for the marathon.
For film 1 we get 5643 base enjoyment on its own.
For film 2 we get 3368 base enjoyment on its own.
For film 3 we get 1342 base enjoyment on its own.
For film 4 we get 6952 base enjoyment on its own.
For film 5 we get 3429 base enjoyment on its own.
For film 6 we get 2886 base enjoyment on its own.
For film 7 we get 5149 base enjoyment on its own.
For film 8 we get 3241 base enjoyment on its own.
For film 9 we get 5271 base enjoyment on its own.
For film 10 we get 5924 base enjoyment on its own.
For film 11 we get 6750 base enjoyment on its own.
Film 1 uses up 4993 minutes of the evening.
Film 2 uses up 2718 minutes of the evening.
Film 3 uses up 692 minutes of the evening.
Film 4 uses up 6302 minutes of the evening.
Film 5 uses up 2779 minutes of the evening.
Film 6 uses up 2236 minutes of the evening.
Film 7 uses up 4499 minutes of the evening.
Film 8 uses up 2591 minutes of the evening.
Film 9 uses up 4621 minutes of the evening.
Film 10 uses up 5274 minutes of the evening.
Film 11 uses up 6100 minutes of the evening.
If we include 1 and 2 together, they add 1847 extra enjoyment.
If we include 1 and 3 together, they add 1616 extra enjoyment.
If we include 1 and 4 together, they add 4585 extra enjoyment.
If we include 1 and 5 together, they add 4993 extra enjoyment.
If we include 1 and 6 together, they add 4499 extra enjoyment.
If we include 1 and 7 together, they add 4993 extra enjoyment.
If we include 1 and 8 together, they add 2122 extra enjoyment.
If we include 1 and 9 together, they add 2448 extra enjoyment.
If we include 1 and 10 together, they add 2854 extra enjoyment.
If we include 1 and 11 together, they add 4993 extra enjoyment.
If we include 2 and 3 together, they add 1876 extra enjoyment.
If we include 2 and 4 together, they add 912 extra enjoyment.
If we include 2 and 5 together, they add 482 extra enjoyment.
If we include 2 and 6 together, they add 2718 extra enjoyment.
If we include 2 and 7 together, they add 1770 extra enjoyment.
If we include 2 and 8 together, they add 734 extra enjoyment.
If we include 2 and 9 together, they add 1167 extra enjoyment.
If we include 2 and 10 together, they add 2718 extra enjoyment.
If we include 2 and 11 together, they add 2407 extra enjoyment.
If we include 3 and 4 together, they add 692 extra enjoyment.
If we include 3 and 5 together, they add 692 extra enjoyment.
If we include 3 and 6 together, they add 692 extra enjoyment.
If we include 3 and 7 together, they add 692 extra enjoyment.
If we include 3 and 8 together, they add 692 extra enjoyment.
If we include 3 and 9 together, they add 692 extra enjoyment.
If we include 3 and 10 together, they add 692 extra enjoyment.
If we include 3 and 11 together, they add 1616 extra enjoyment.
If we include 4 and 5 together, they add 3668 extra enjoyment.
If we include 4 and 6 together, they add 2854 extra enjoyment.
If we include 4 and 7 together, they add 6100 extra enjoyment.
If we include 4 and 8 together, they add 1174 extra enjoyment.
If we include 4 and 9 together, they add 1466 extra enjoyment.
If we include 4 and 10 together, they add 1876 extra enjoyment.
If we include 4 and 11 together, they add 1205 extra enjoyment.
If we include 5 and 6 together, they add 6100 extra enjoyment.
If we include 5 and 7 together, they add 5644 extra enjoyment.
If we include 5 and 8 together, they add 6274 extra enjoyment.
If we include 5 and 9 together, they add 3700 extra enjoyment.
If we include 5 and 10 together, they add 2289 extra enjoyment.
If we include 5 and 11 together, they add 2289 extra enjoyment.
If we include 6 and 7 together, they add 3045 extra enjoyment.
If we include 6 and 8 together, they add 341 extra enjoyment.
If we include 6 and 9 together, they add 1770 extra enjoyment.
If we include 6 and 10 together, they add 596 extra enjoyment.
If we include 6 and 11 together, they add 5274 extra enjoyment.
If we include 7 and 8 together, they add 2122 extra enjoyment.
If we include 7 and 9 together, they add 2448 extra enjoyment.
If we include 7 and 10 together, they add 2854 extra enjoyment.
If we include 7 and 11 together, they add 4953 extra enjoyment.
If we include 8 and 9 together, they add 3876 extra enjoyment.
If we include 8 and 10 together, they add 596 extra enjoyment.
If we include 8 and 11 together, they add 1770 extra enjoyment.
If we include 9 and 10 together, they add 596 extra enjoyment.
If we include 9 and 11 together, they add 596 extra enjoyment.
If we include 10 and 11 together, they add 1121 extra enjoyment.
Now we can list each film's enjoyment, runtime, and pair bonuses so we pick the lineup with the largest total enjoyment that still fits into 27117 minutes.

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': [5643, 3368, 1342, 6952, 3429, 2886, 5149, 3241, 5271, 5924, 6750], 'quadratic_coeffs': [[1958, 1847, 1616, 4585, 4993, 4499, 4993, 2122, 2448, 2854, 4993], [0, 2718, 1876, 912, 482, 2718, 1770, 734, 1167, 2718, 2407], [0, 0, 692, 692, 692, 692, 692, 692, 692, 692, 1616], [0, 0, 0, 5248, 3668, 2854, 6100, 1174, 1466, 1876, 1205], [0, 0, 0, 0, 1928, 6100, 5644, 6274, 3700, 2289, 2289], [0, 0, 0, 0, 0, 2407, 3045, 341, 1770, 596, 5274], [0, 0, 0, 0, 0, 0, 3700, 2122, 2448, 2854, 4953], [0, 0, 0, 0, 0, 0, 0, 1121, 3876, 596, 1770], [0, 0, 0, 0, 0, 0, 0, 0, 1928, 596, 596], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1174, 1121], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2407]], 'weights': [4993, 2718, 692, 6302, 2779, 2236, 4499, 2591, 4621, 5274, 6100], 'capacity': 27117, 'solution': [0, 1, 2, 4, 5, 6, 7, 10], 'obj': 109054.0, 'problem_type': 'QKP'}","[0, 1, 2, 4, 5, 6, 7, 10]",109054.0,"{'problem_type': 'QKP', 'num_items': 11, 'capacity': 27117, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 'linear_pairs': [{'item_id': 1, 'linear_profit': 5643}, {'item_id': 2, 'linear_profit': 3368}, {'item_id': 3, 'linear_profit': 1342}, {'item_id': 4, 'linear_profit': 6952}, {'item_id': 5, 'linear_profit': 3429}, {'item_id': 6, 'linear_profit': 2886}, {'item_id': 7, 'linear_profit': 5149}, {'item_id': 8, 'linear_profit': 3241}, {'item_id': 9, 'linear_profit': 5271}, {'item_id': 10, 'linear_profit': 5924}, {'item_id': 11, 'linear_profit': 6750}], 'weight_pairs': [{'item_id': 1, 'weight': 4993}, {'item_id': 2, 'weight': 2718}, {'item_id': 3, 'weight': 692}, {'item_id': 4, 'weight': 6302}, {'item_id': 5, 'weight': 2779}, {'item_id': 6, 'weight': 2236}, {'item_id': 7, 'weight': 4499}, {'item_id': 8, 'weight': 2591}, {'item_id': 9, 'weight': 4621}, {'item_id': 10, 'weight': 5274}, {'item_id': 11, 'weight': 6100}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 1958}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 1847}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 1616}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 4585}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 4993}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 4499}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 4993}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 2122}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 2448}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 2854}, {'item_i_id': 1, 'item_j_id': 11, 'quadratic_profit': 4993}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 2718}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 1876}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 912}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 482}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 2718}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 1770}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 734}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 1167}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 2718}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 2407}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 692}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 692}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 692}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 692}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 692}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 692}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 692}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 692}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 1616}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 5248}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 3668}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 2854}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 6100}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 1174}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 1466}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 1876}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 1205}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 1928}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 6100}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 5644}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 6274}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 3700}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 2289}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 2289}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 2407}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 3045}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 341}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 1770}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 596}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 5274}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 3700}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 2122}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 2448}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 2854}, {'item_i_id': 7, 'item_j_id': 11, 'quadratic_profit': 4953}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 1121}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 3876}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 596}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 1770}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 1928}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 596}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 596}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 1174}, {'item_i_id': 10, 'item_j_id': 11, 'quadratic_profit': 1121}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 2407}]}","[1, 2, 3, 5, 6, 7, 8, 11]",25,nl,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 15 pastries (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15) and the box capacity is 387.
We can include pastry 1 for an individual appeal of 103.
We can include pastry 2 for an individual appeal of 101.
We can include pastry 3 for an individual appeal of 118.
We can include pastry 4 for an individual appeal of 105.
We can include pastry 5 for an individual appeal of 111.
We can include pastry 6 for an individual appeal of 119.
We can include pastry 7 for an individual appeal of 107.
We can include pastry 8 for an individual appeal of 120.
We can include pastry 9 for an individual appeal of 103.
We can include pastry 10 for an individual appeal of 123.
We can include pastry 11 for an individual appeal of 116.
We can include pastry 12 for an individual appeal of 108.
We can include pastry 13 for an individual appeal of 100.
We can include pastry 14 for an individual appeal of 120.
We can include pastry 15 for an individual appeal of 106.
We note pastry 1 weighs 102.
We note pastry 2 weighs 120.
We note pastry 3 weighs 124.
We note pastry 4 weighs 117.
We note pastry 5 weighs 106.
We note pastry 6 weighs 111.
We note pastry 7 weighs 101.
We note pastry 8 weighs 102.
We note pastry 9 weighs 124.
We note pastry 10 weighs 118.
We note pastry 11 weighs 105.
We note pastry 12 weighs 112.
We note pastry 13 weighs 117.
We note pastry 14 weighs 121.
We note pastry 15 weighs 100.
If we include pastries 1 and 2 together, they add a pair bonus appeal of 107.
If we include pastries 1 and 3 together, they add a pair bonus appeal of 103.
If we include pastries 1 and 4 together, they add a pair bonus appeal of 107.
If we include pastries 1 and 5 together, they add a pair bonus appeal of 112.
If we include pastries 1 and 6 together, they add a pair bonus appeal of 111.
If we include pastries 1 and 7 together, they add a pair bonus appeal of 105.
If we include pastries 1 and 8 together, they add a pair bonus appeal of 108.
If we include pastries 1 and 9 together, they add a pair bonus appeal of 111.
If we include pastries 1 and 10 together, they add a pair bonus appeal of 109.
If we include pastries 1 and 11 together, they add a pair bonus appeal of 107.
If we include pastries 1 and 12 together, they add a pair bonus appeal of 102.
If we include pastries 1 and 13 together, they add a pair bonus appeal of 116.
If we include pastries 1 and 14 together, they add a pair bonus appeal of 109.
If we include pastries 1 and 15 together, they add a pair bonus appeal of 114.
If we include pastries 2 and 3 together, they add a pair bonus appeal of 120.
If we include pastries 2 and 4 together, they add a pair bonus appeal of 120.
If we include pastries 2 and 5 together, they add a pair bonus appeal of 117.
If we include pastries 2 and 6 together, they add a pair bonus appeal of 115.
If we include pastries 2 and 7 together, they add a pair bonus appeal of 112.
If we include pastries 2 and 8 together, they add a pair bonus appeal of 110.
If we include pastries 2 and 9 together, they add a pair bonus appeal of 109.
If we include pastries 2 and 10 together, they add a pair bonus appeal of 111.
If we include pastries 2 and 11 together, they add a pair bonus appeal of 119.
If we include pastries 2 and 12 together, they add a pair bonus appeal of 116.
If we include pastries 2 and 13 together, they add a pair bonus appeal of 114.
If we include pastries 2 and 14 together, they add a pair bonus appeal of 118.
If we include pastries 2 and 15 together, they add a pair bonus appeal of 113.
If we include pastries 3 and 4 together, they add a pair bonus appeal of 122.
If we include pastries 3 and 5 together, they add a pair bonus appeal of 121.
If we include pastries 3 and 6 together, they add a pair bonus appeal of 111.
If we include pastries 3 and 7 together, they add a pair bonus appeal of 121.
If we include pastries 3 and 8 together, they add a pair bonus appeal of 115.
If we include pastries 3 and 9 together, they add a pair bonus appeal of 122.
If we include pastries 3 and 10 together, they add a pair bonus appeal of 112.
If we include pastries 3 and 11 together, they add a pair bonus appeal of 111.
If we include pastries 3 and 12 together, they add a pair bonus appeal of 106.
If we include pastries 3 and 13 together, they add a pair bonus appeal of 116.
If we include pastries 3 and 14 together, they add a pair bonus appeal of 111.
If we include pastries 3 and 15 together, they add a pair bonus appeal of 113.
If we include pastries 4 and 5 together, they add a pair bonus appeal of 109.
If we include pastries 4 and 6 together, they add a pair bonus appeal of 114.
If we include pastries 4 and 7 together, they add a pair bonus appeal of 117.
If we include pastries 4 and 8 together, they add a pair bonus appeal of 118.
If we include pastries 4 and 9 together, they add a pair bonus appeal of 119.
If we include pastries 4 and 10 together, they add a pair bonus appeal of 116.
If we include pastries 4 and 11 together, they add a pair bonus appeal of 110.
If we include pastries 4 and 12 together, they add a pair bonus appeal of 120.
If we include pastries 4 and 13 together, they add a pair bonus appeal of 120.
If we include pastries 4 and 14 together, they add a pair bonus appeal of 120.
If we include pastries 4 and 15 together, they add a pair bonus appeal of 117.
If we include pastries 5 and 6 together, they add a pair bonus appeal of 113.
If we include pastries 5 and 7 together, they add a pair bonus appeal of 113.
If we include pastries 5 and 8 together, they add a pair bonus appeal of 118.
If we include pastries 5 and 9 together, they add a pair bonus appeal of 111.
If we include pastries 5 and 10 together, they add a pair bonus appeal of 115.
If we include pastries 5 and 11 together, they add a pair bonus appeal of 117.
If we include pastries 5 and 12 together, they add a pair bonus appeal of 113.
If we include pastries 5 and 13 together, they add a pair bonus appeal of 107.
If we include pastries 5 and 14 together, they add a pair bonus appeal of 117.
If we include pastries 5 and 15 together, they add a pair bonus appeal of 117.
If we include pastries 6 and 7 together, they add a pair bonus appeal of 105.
If we include pastries 6 and 8 together, they add a pair bonus appeal of 115.
If we include pastries 6 and 9 together, they add a pair bonus appeal of 114.
If we include pastries 6 and 10 together, they add a pair bonus appeal of 110.
If we include pastries 6 and 11 together, they add a pair bonus appeal of 105.
If we include pastries 6 and 12 together, they add a pair bonus appeal of 112.
If we include pastries 6 and 13 together, they add a pair bonus appeal of 114.
If we include pastries 6 and 14 together, they add a pair bonus appeal of 115.
If we include pastries 6 and 15 together, they add a pair bonus appeal of 114.
If we include pastries 7 and 8 together, they add a pair bonus appeal of 105.
If we include pastries 7 and 9 together, they add a pair bonus appeal of 111.
If we include pastries 7 and 10 together, they add a pair bonus appeal of 106.
If we include pastries 7 and 11 together, they add a pair bonus appeal of 111.
If we include pastries 7 and 12 together, they add a pair bonus appeal of 112.
If we include pastries 7 and 13 together, they add a pair bonus appeal of 121.
If we include pastries 7 and 14 together, they add a pair bonus appeal of 121.
If we include pastries 7 and 15 together, they add a pair bonus appeal of 115.
If we include pastries 8 and 9 together, they add a pair bonus appeal of 122.
If we include pastries 8 and 10 together, they add a pair bonus appeal of 117.
If we include pastries 8 and 11 together, they add a pair bonus appeal of 122.
If we include pastries 8 and 12 together, they add a pair bonus appeal of 110.
If we include pastries 8 and 13 together, they add a pair bonus appeal of 122.
If we include pastries 8 and 14 together, they add a pair bonus appeal of 116.
If we include pastries 8 and 15 together, they add a pair bonus appeal of 122.
If we include pastries 9 and 10 together, they add a pair bonus appeal of 121.
If we include pastries 9 and 11 together, they add a pair bonus appeal of 111.
If we include pastries 9 and 12 together, they add a pair bonus appeal of 119.
If we include pastries 9 and 13 together, they add a pair bonus appeal of 118.
If we include pastries 9 and 14 together, they add a pair bonus appeal of 111.
If we include pastries 9 and 15 together, they add a pair bonus appeal of 116.
If we include pastries 10 and 11 together, they add a pair bonus appeal of 117.
If we include pastries 10 and 12 together, they add a pair bonus appeal of 119.
If we include pastries 10 and 13 together, they add a pair bonus appeal of 120.
If we include pastries 10 and 14 together, they add a pair bonus appeal of 108.
If we include pastries 10 and 15 together, they add a pair bonus appeal of 106.
If we include pastries 11 and 12 together, they add a pair bonus appeal of 114.
If we include pastries 11 and 13 together, they add a pair bonus appeal of 116.
If we include pastries 11 and 14 together, they add a pair bonus appeal of 117.
If we include pastries 11 and 15 together, they add a pair bonus appeal of 115.
If we include pastries 12 and 13 together, they add a pair bonus appeal of 121.
If we include pastries 12 and 14 together, they add a pair bonus appeal of 121.
If we include pastries 12 and 15 together, they add a pair bonus appeal of 120.
We'll pick the combination that fits within 387 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': [103, 101, 118, 105, 111, 119, 107, 120, 103, 123, 116, 108, 100, 120, 106], 'quadratic_coeffs': [[108, 107, 103, 107, 112, 111, 105, 108, 111, 109, 107, 102, 116, 109, 114], [0, 116, 120, 120, 117, 115, 112, 110, 109, 111, 119, 116, 114, 118, 113], [0, 0, 119, 122, 121, 111, 121, 115, 122, 112, 111, 106, 116, 111, 113], [0, 0, 0, 115, 109, 114, 117, 118, 119, 116, 110, 120, 120, 120, 117], [0, 0, 0, 0, 113, 113, 113, 118, 111, 115, 117, 113, 107, 117, 117], [0, 0, 0, 0, 0, 115, 105, 115, 114, 110, 105, 112, 114, 115, 114], [0, 0, 0, 0, 0, 0, 103, 105, 111, 106, 111, 112, 121, 121, 115], [0, 0, 0, 0, 0, 0, 0, 121, 122, 117, 122, 110, 122, 116, 122], [0, 0, 0, 0, 0, 0, 0, 0, 119, 121, 111, 119, 118, 111, 116], [0, 0, 0, 0, 0, 0, 0, 0, 0, 111, 117, 119, 120, 108, 106], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 117, 114, 116, 117, 115], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 121, 121, 121, 120], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 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': [102, 120, 124, 117, 106, 111, 101, 102, 124, 118, 105, 112, 117, 121, 100], 'capacity': 387, 'solution': [7, 9, 10], 'obj': 715.0, 'problem_type': 'QKP'}","[7, 9, 10]",715.0,"{'problem_type': 'QKP', 'num_items': 15, 'capacity': 387, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], 'linear_pairs': [{'item_id': 1, 'linear_profit': 103}, {'item_id': 2, 'linear_profit': 101}, {'item_id': 3, 'linear_profit': 118}, {'item_id': 4, 'linear_profit': 105}, {'item_id': 5, 'linear_profit': 111}, {'item_id': 6, 'linear_profit': 119}, {'item_id': 7, 'linear_profit': 107}, {'item_id': 8, 'linear_profit': 120}, {'item_id': 9, 'linear_profit': 103}, {'item_id': 10, 'linear_profit': 123}, {'item_id': 11, 'linear_profit': 116}, {'item_id': 12, 'linear_profit': 108}, {'item_id': 13, 'linear_profit': 100}, {'item_id': 14, 'linear_profit': 120}, {'item_id': 15, 'linear_profit': 106}], 'weight_pairs': [{'item_id': 1, 'weight': 102}, {'item_id': 2, 'weight': 120}, {'item_id': 3, 'weight': 124}, {'item_id': 4, 'weight': 117}, {'item_id': 5, 'weight': 106}, {'item_id': 6, 'weight': 111}, {'item_id': 7, 'weight': 101}, {'item_id': 8, 'weight': 102}, {'item_id': 9, 'weight': 124}, {'item_id': 10, 'weight': 118}, {'item_id': 11, 'weight': 105}, {'item_id': 12, 'weight': 112}, {'item_id': 13, 'weight': 117}, {'item_id': 14, 'weight': 121}, {'item_id': 15, 'weight': 100}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 108}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 107}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 103}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 107}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 112}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 111}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 105}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 108}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 111}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 109}, {'item_i_id': 1, 'item_j_id': 11, 'quadratic_profit': 107}, {'item_i_id': 1, 'item_j_id': 12, 'quadratic_profit': 102}, {'item_i_id': 1, 'item_j_id': 13, 'quadratic_profit': 116}, {'item_i_id': 1, 'item_j_id': 14, 'quadratic_profit': 109}, {'item_i_id': 1, 'item_j_id': 15, 'quadratic_profit': 114}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 116}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 120}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 120}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 117}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 115}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 112}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 110}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 109}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 111}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 119}, {'item_i_id': 2, 'item_j_id': 12, 'quadratic_profit': 116}, {'item_i_id': 2, 'item_j_id': 13, 'quadratic_profit': 114}, {'item_i_id': 2, 'item_j_id': 14, 'quadratic_profit': 118}, {'item_i_id': 2, 'item_j_id': 15, 'quadratic_profit': 113}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 119}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 122}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 121}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 111}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 121}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 115}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 122}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 112}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 111}, {'item_i_id': 3, 'item_j_id': 12, 'quadratic_profit': 106}, {'item_i_id': 3, 'item_j_id': 13, 'quadratic_profit': 116}, {'item_i_id': 3, 'item_j_id': 14, 'quadratic_profit': 111}, {'item_i_id': 3, 'item_j_id': 15, 'quadratic_profit': 113}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 115}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 109}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 114}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 117}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 118}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 119}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 116}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 110}, {'item_i_id': 4, 'item_j_id': 12, 'quadratic_profit': 120}, {'item_i_id': 4, 'item_j_id': 13, 'quadratic_profit': 120}, {'item_i_id': 4, 'item_j_id': 14, 'quadratic_profit': 120}, {'item_i_id': 4, 'item_j_id': 15, 'quadratic_profit': 117}, {'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': 118}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 111}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 115}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 117}, {'item_i_id': 5, 'item_j_id': 12, 'quadratic_profit': 113}, {'item_i_id': 5, 'item_j_id': 13, 'quadratic_profit': 107}, {'item_i_id': 5, 'item_j_id': 14, 'quadratic_profit': 117}, {'item_i_id': 5, 'item_j_id': 15, 'quadratic_profit': 117}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 115}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 105}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 115}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 114}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 110}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 105}, {'item_i_id': 6, 'item_j_id': 12, 'quadratic_profit': 112}, {'item_i_id': 6, 'item_j_id': 13, 'quadratic_profit': 114}, {'item_i_id': 6, 'item_j_id': 14, 'quadratic_profit': 115}, {'item_i_id': 6, 'item_j_id': 15, 'quadratic_profit': 114}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 103}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 105}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 111}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 106}, {'item_i_id': 7, 'item_j_id': 11, 'quadratic_profit': 111}, {'item_i_id': 7, 'item_j_id': 12, 'quadratic_profit': 112}, {'item_i_id': 7, 'item_j_id': 13, 'quadratic_profit': 121}, {'item_i_id': 7, 'item_j_id': 14, 'quadratic_profit': 121}, {'item_i_id': 7, 'item_j_id': 15, 'quadratic_profit': 115}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 121}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 122}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 117}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 122}, {'item_i_id': 8, 'item_j_id': 12, 'quadratic_profit': 110}, {'item_i_id': 8, 'item_j_id': 13, 'quadratic_profit': 122}, {'item_i_id': 8, 'item_j_id': 14, 'quadratic_profit': 116}, {'item_i_id': 8, 'item_j_id': 15, 'quadratic_profit': 122}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 119}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 121}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 111}, {'item_i_id': 9, 'item_j_id': 12, 'quadratic_profit': 119}, {'item_i_id': 9, 'item_j_id': 13, 'quadratic_profit': 118}, {'item_i_id': 9, 'item_j_id': 14, 'quadratic_profit': 111}, {'item_i_id': 9, 'item_j_id': 15, 'quadratic_profit': 116}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 111}, {'item_i_id': 10, 'item_j_id': 11, 'quadratic_profit': 117}, {'item_i_id': 10, 'item_j_id': 12, 'quadratic_profit': 119}, {'item_i_id': 10, 'item_j_id': 13, 'quadratic_profit': 120}, {'item_i_id': 10, 'item_j_id': 14, 'quadratic_profit': 108}, {'item_i_id': 10, 'item_j_id': 15, 'quadratic_profit': 106}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 117}, {'item_i_id': 11, 'item_j_id': 12, 'quadratic_profit': 114}, {'item_i_id': 11, 'item_j_id': 13, 'quadratic_profit': 116}, {'item_i_id': 11, 'item_j_id': 14, 'quadratic_profit': 117}, {'item_i_id': 11, 'item_j_id': 15, 'quadratic_profit': 115}, {'item_i_id': 12, 'item_j_id': 12, 'quadratic_profit': 121}, {'item_i_id': 12, 'item_j_id': 13, 'quadratic_profit': 121}, {'item_i_id': 12, 'item_j_id': 14, 'quadratic_profit': 121}, {'item_i_id': 12, 'item_j_id': 15, 'quadratic_profit': 120}]}","[8, 10, 11]",26,nl,1
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.

{
  ""total_items_count"": 14,
  ""pannier_capacity"": 21309,
  ""item_ids_list"": [
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14
  ],
  ""linear"": [
    {
      ""item_id"": 1,
      ""usefulness_score"": 1263
    },
    {
      ""item_id"": 2,
      ""usefulness_score"": 1784
    },
    {
      ""item_id"": 3,
      ""usefulness_score"": 1307
    },
    {
      ""item_id"": 4,
      ""usefulness_score"": 3364
    },
    {
      ""item_id"": 5,
      ""usefulness_score"": 3375
    },
    {
      ""item_id"": 6,
      ""usefulness_score"": 1347
    },
    {
      ""item_id"": 7,
      ""usefulness_score"": 823
    },
    {
      ""item_id"": 8,
      ""usefulness_score"": 2875
    },
    {
      ""item_id"": 9,
      ""usefulness_score"": 518
    },
    {
      ""item_id"": 10,
      ""usefulness_score"": 3260
    },
    {
      ""item_id"": 11,
      ""usefulness_score"": 1117
    },
    {
      ""item_id"": 12,
      ""usefulness_score"": 1941
    },
    {
      ""item_id"": 13,
      ""usefulness_score"": 3249
    },
    {
      ""item_id"": 14,
      ""usefulness_score"": 2201
    }
  ],
  ""weights"": [
    {
      ""item_id"": 1,
      ""item_weight"": 933
    },
    {
      ""item_id"": 2,
      ""item_weight"": 1454
    },
    {
      ""item_id"": 3,
      ""item_weight"": 977
    },
    {
      ""item_id"": 4,
      ""item_weight"": 3034
    },
    {
      ""item_id"": 5,
      ""item_weight"": 3045
    },
    {
      ""item_id"": 6,
      ""item_weight"": 1017
    },
    {
      ""item_id"": 7,
      ""item_weight"": 493
    },
    {
      ""item_id"": 8,
      ""item_weight"": 2545
    },
    {
      ""item_id"": 9,
      ""item_weight"": 188
    },
    {
      ""item_id"": 10,
      ""item_weight"": 2930
    },
    {
      ""item_id"": 11,
      ""item_weight"": 787
    },
    {
      ""item_id"": 12,
      ""item_weight"": 1611
    },
    {
      ""item_id"": 13,
      ""item_weight"": 2919
    },
    {
      ""item_id"": 14,
      ""item_weight"": 1871
    }
  ],
  ""quadratic"": [
    {
      ""first_item_id"": 1,
      ""second_item_id"": 2,
      ""complementary_bonus"": 897
    },
    {
      ""first_item_id"": 1,
      ""second_item_id"": 6,
      ""complementary_bonus"": 1211
    },
    {
      ""first_item_id"": 3,
      ""second_item_id"": 9,
      ""complementary_bonus"": 1341
    },
    {
      ""first_item_id"": 3,
      ""second_item_id"": 10,
      ""complementary_bonus"": 1762
    },
    {
      ""first_item_id"": 4,
      ""second_item_id"": 8,
      ""complementary_bonus"": 2608
    },
    {
      ""first_item_id"": 5,
      ""second_item_id"": 12,
      ""complementary_bonus"": 454
    },
    {
      ""first_item_id"": 5,
      ""second_item_id"": 13,
      ""complementary_bonus"": 1193
    },
    {
      ""first_item_id"": 5,
      ""second_item_id"": 14,
      ""complementary_bonus"": 406
    },
    {
      ""first_item_id"": 6,
      ""second_item_id"": 7,
      ""complementary_bonus"": 1948
    },
    {
      ""first_item_id"": 6,
      ""second_item_id"": 10,
      ""complementary_bonus"": 2003
    },
    {
      ""first_item_id"": 6,
      ""second_item_id"": 13,
      ""complementary_bonus"": 1916
    },
    {
      ""first_item_id"": 7,
      ""second_item_id"": 8,
      ""complementary_bonus"": 753
    },
    {
      ""first_item_id"": 7,
      ""second_item_id"": 9,
      ""complementary_bonus"": 1477
    },
    {
      ""first_item_id"": 8,
      ""second_item_id"": 10,
      ""complementary_bonus"": 2692
    },
    {
      ""first_item_id"": 10,
      ""second_item_id"": 12,
      ""complementary_bonus"": 1769
    }
  ]
}

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': [1263, 1784, 1307, 3364, 3375, 1347, 823, 2875, 518, 3260, 1117, 1941, 3249, 2201], 'quadratic_coeffs': [[0, 897, 0, 0, 0, 1211, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1341, 1762, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2608, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1868, 0, 0, 0, 0, 0, 0, 454, 1193, 406], [0, 0, 0, 0, 0, 0, 1948, 0, 0, 2003, 0, 0, 1916, 0], [0, 0, 0, 0, 0, 0, 0, 753, 1477, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2692, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1006, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3067, 0, 1769, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 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': [933, 1454, 977, 3034, 3045, 1017, 493, 2545, 188, 2930, 787, 1611, 2919, 1871], 'capacity': 21309, 'solution': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12], 'obj': 47130.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12]",47130.0,"{'problem_type': 'QKP', 'num_items': 14, 'capacity': 21309, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], 'linear_pairs': [{'item_id': 1, 'linear_profit': 1263}, {'item_id': 2, 'linear_profit': 1784}, {'item_id': 3, 'linear_profit': 1307}, {'item_id': 4, 'linear_profit': 3364}, {'item_id': 5, 'linear_profit': 3375}, {'item_id': 6, 'linear_profit': 1347}, {'item_id': 7, 'linear_profit': 823}, {'item_id': 8, 'linear_profit': 2875}, {'item_id': 9, 'linear_profit': 518}, {'item_id': 10, 'linear_profit': 3260}, {'item_id': 11, 'linear_profit': 1117}, {'item_id': 12, 'linear_profit': 1941}, {'item_id': 13, 'linear_profit': 3249}, {'item_id': 14, 'linear_profit': 2201}], 'weight_pairs': [{'item_id': 1, 'weight': 933}, {'item_id': 2, 'weight': 1454}, {'item_id': 3, 'weight': 977}, {'item_id': 4, 'weight': 3034}, {'item_id': 5, 'weight': 3045}, {'item_id': 6, 'weight': 1017}, {'item_id': 7, 'weight': 493}, {'item_id': 8, 'weight': 2545}, {'item_id': 9, 'weight': 188}, {'item_id': 10, 'weight': 2930}, {'item_id': 11, 'weight': 787}, {'item_id': 12, 'weight': 1611}, {'item_id': 13, 'weight': 2919}, {'item_id': 14, 'weight': 1871}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 897}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 1211}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 1341}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 1762}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 2608}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 1868}, {'item_i_id': 5, 'item_j_id': 12, 'quadratic_profit': 454}, {'item_i_id': 5, 'item_j_id': 13, 'quadratic_profit': 1193}, {'item_i_id': 5, 'item_j_id': 14, 'quadratic_profit': 406}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 1948}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 2003}, {'item_i_id': 6, 'item_j_id': 13, 'quadratic_profit': 1916}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 753}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 1477}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 2692}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 1006}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 3067}, {'item_i_id': 10, 'item_j_id': 12, 'quadratic_profit': 1769}]}","[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13]",27,json,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 15 optional modules listed as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 5695 memory units were available for enabling them.

| module_id | base_benefit |
|---|---|
| 1 | 1259 |
| 2 | 330 |
| 3 | 381 |
| 4 | 503 |
| 5 | 1333 |
| 6 | 1057 |
| 7 | 740 |
| 8 | 518 |
| 9 | 1465 |
| 10 | 915 |
| 11 | 585 |
| 12 | 441 |
| 13 | 1077 |
| 14 | 1304 |
| 15 | 1766 |

| module_id | memory_usage |
|---|---|
| 1 | 1089 |
| 2 | 160 |
| 3 | 211 |
| 4 | 333 |
| 5 | 1163 |
| 6 | 887 |
| 7 | 570 |
| 8 | 348 |
| 9 | 1295 |
| 10 | 745 |
| 11 | 415 |
| 12 | 271 |
| 13 | 907 |
| 14 | 1134 |
| 15 | 1596 |

| module_i_id | module_j_id | interaction_bonus |
|---|---|---|
| 1 | 3 | 1089 |
| 1 | 5 | 994 |
| 1 | 6 | 847 |
| 1 | 9 | 1089 |
| 1 | 10 | 878 |
| 1 | 11 | 1089 |
| 1 | 13 | 235 |
| 1 | 14 | 1089 |
| 2 | 5 | 160 |
| 2 | 6 | 160 |
| 2 | 7 | 160 |
| 2 | 8 | 160 |
| 2 | 10 | 160 |
| 2 | 11 | 160 |
| 2 | 12 | 160 |
| 2 | 13 | 160 |
| 3 | 4 | 211 |
| 3 | 6 | 211 |
| 3 | 7 | 211 |
| 3 | 9 | 211 |
| 3 | 10 | 153 |
| 3 | 11 | 211 |
| 3 | 12 | 211 |
| 3 | 13 | 211 |
| 3 | 15 | 333 |
| 4 | 6 | 333 |
| 4 | 8 | 333 |
| 4 | 9 | 333 |
| 4 | 10 | 333 |
| 4 | 11 | 333 |
| 4 | 13 | 333 |
| 4 | 15 | 308 |
| 5 | 6 | 737 |
| 5 | 7 | 414 |
| 5 | 9 | 838 |
| 5 | 11 | 1163 |
| 5 | 12 | 766 |
| 5 | 15 | 122 |
| 6 | 8 | 887 |
| 6 | 9 | 173 |
| 6 | 10 | 746 |
| 6 | 11 | 887 |
| 6 | 13 | 439 |
| 6 | 15 | 171 |
| 7 | 8 | 570 |
| 7 | 9 | 378 |
| 7 | 10 | 570 |
| 7 | 11 | 439 |
| 7 | 13 | 570 |
| 7 | 14 | 1295 |
| 7 | 15 | 1233 |
| 8 | 10 | 348 |
| 8 | 11 | 348 |
| 8 | 12 | 348 |
| 8 | 13 | 348 |
| 8 | 15 | 281 |
| 9 | 12 | 696 |
| 9 | 13 | 329 |
| 9 | 14 | 604 |
| 9 | 15 | 1067 |
| 10 | 11 | 1128 |
| 10 | 12 | 794 |
| 10 | 13 | 228 |
| 11 | 13 | 434 |
| 11 | 14 | 434 |
| 11 | 15 | 296 |
| 12 | 13 | 216 |
| 12 | 14 | 109 |
| 13 | 14 | 235 |
| 13 | 15 | 520 |
| 14 | 15 | 528 |

The ops team must choose modules so total memory stays within 5695 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': [1259, 330, 381, 503, 1333, 1057, 740, 518, 1465, 915, 585, 441, 1077, 1304, 1766], 'quadratic_coeffs': [[1089, 0, 1089, 0, 994, 847, 0, 0, 1089, 878, 1089, 0, 235, 1089, 0], [0, 160, 0, 0, 160, 160, 160, 160, 0, 160, 160, 160, 160, 0, 0], [0, 0, 211, 211, 0, 211, 211, 0, 211, 153, 211, 211, 211, 0, 333], [0, 0, 0, 0, 0, 333, 0, 333, 333, 333, 333, 0, 333, 0, 308], [0, 0, 0, 0, 1163, 737, 414, 0, 838, 0, 1163, 766, 0, 0, 122], [0, 0, 0, 0, 0, 745, 0, 887, 173, 746, 887, 0, 439, 0, 171], [0, 0, 0, 0, 0, 0, 570, 570, 378, 570, 439, 0, 570, 1295, 1233], [0, 0, 0, 0, 0, 0, 0, 348, 0, 348, 348, 348, 348, 0, 281], [0, 0, 0, 0, 0, 0, 0, 0, 1066, 0, 0, 696, 329, 604, 1067], [0, 0, 0, 0, 0, 0, 0, 0, 0, 183, 1128, 794, 228, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 153, 0, 434, 434, 296], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 743, 216, 109, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 918, 235, 520], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 528, 528], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 815]], 'weights': [1089, 160, 211, 333, 1163, 887, 570, 348, 1295, 745, 415, 271, 907, 1134, 1596], 'capacity': 5695, 'solution': [0, 1, 2, 3, 4, 5, 7, 9, 10, 11], 'obj': 23660.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 5, 7, 9, 10, 11]",23660.0,"{'problem_type': 'QKP', 'num_items': 15, 'capacity': 5695, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], 'linear_pairs': [{'item_id': 1, 'linear_profit': 1259}, {'item_id': 2, 'linear_profit': 330}, {'item_id': 3, 'linear_profit': 381}, {'item_id': 4, 'linear_profit': 503}, {'item_id': 5, 'linear_profit': 1333}, {'item_id': 6, 'linear_profit': 1057}, {'item_id': 7, 'linear_profit': 740}, {'item_id': 8, 'linear_profit': 518}, {'item_id': 9, 'linear_profit': 1465}, {'item_id': 10, 'linear_profit': 915}, {'item_id': 11, 'linear_profit': 585}, {'item_id': 12, 'linear_profit': 441}, {'item_id': 13, 'linear_profit': 1077}, {'item_id': 14, 'linear_profit': 1304}, {'item_id': 15, 'linear_profit': 1766}], 'weight_pairs': [{'item_id': 1, 'weight': 1089}, {'item_id': 2, 'weight': 160}, {'item_id': 3, 'weight': 211}, {'item_id': 4, 'weight': 333}, {'item_id': 5, 'weight': 1163}, {'item_id': 6, 'weight': 887}, {'item_id': 7, 'weight': 570}, {'item_id': 8, 'weight': 348}, {'item_id': 9, 'weight': 1295}, {'item_id': 10, 'weight': 745}, {'item_id': 11, 'weight': 415}, {'item_id': 12, 'weight': 271}, {'item_id': 13, 'weight': 907}, {'item_id': 14, 'weight': 1134}, {'item_id': 15, 'weight': 1596}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 1089}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 1089}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 994}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 847}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 1089}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 878}, {'item_i_id': 1, 'item_j_id': 11, 'quadratic_profit': 1089}, {'item_i_id': 1, 'item_j_id': 13, 'quadratic_profit': 235}, {'item_i_id': 1, 'item_j_id': 14, 'quadratic_profit': 1089}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 160}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 160}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 160}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 160}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 160}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 160}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 160}, {'item_i_id': 2, 'item_j_id': 12, 'quadratic_profit': 160}, {'item_i_id': 2, 'item_j_id': 13, 'quadratic_profit': 160}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 211}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 211}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 211}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 211}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 211}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 153}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 211}, {'item_i_id': 3, 'item_j_id': 12, 'quadratic_profit': 211}, {'item_i_id': 3, 'item_j_id': 13, 'quadratic_profit': 211}, {'item_i_id': 3, 'item_j_id': 15, 'quadratic_profit': 333}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 333}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 333}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 333}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 333}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 333}, {'item_i_id': 4, 'item_j_id': 13, 'quadratic_profit': 333}, {'item_i_id': 4, 'item_j_id': 15, 'quadratic_profit': 308}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 1163}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 737}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 414}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 838}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 1163}, {'item_i_id': 5, 'item_j_id': 12, 'quadratic_profit': 766}, {'item_i_id': 5, 'item_j_id': 15, 'quadratic_profit': 122}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 745}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 887}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 173}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 746}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 887}, {'item_i_id': 6, 'item_j_id': 13, 'quadratic_profit': 439}, {'item_i_id': 6, 'item_j_id': 15, 'quadratic_profit': 171}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 570}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 570}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 378}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 570}, {'item_i_id': 7, 'item_j_id': 11, 'quadratic_profit': 439}, {'item_i_id': 7, 'item_j_id': 13, 'quadratic_profit': 570}, {'item_i_id': 7, 'item_j_id': 14, 'quadratic_profit': 1295}, {'item_i_id': 7, 'item_j_id': 15, 'quadratic_profit': 1233}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 348}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 348}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 348}, {'item_i_id': 8, 'item_j_id': 12, 'quadratic_profit': 348}, {'item_i_id': 8, 'item_j_id': 13, 'quadratic_profit': 348}, {'item_i_id': 8, 'item_j_id': 15, 'quadratic_profit': 281}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 1066}, {'item_i_id': 9, 'item_j_id': 12, 'quadratic_profit': 696}, {'item_i_id': 9, 'item_j_id': 13, 'quadratic_profit': 329}, {'item_i_id': 9, 'item_j_id': 14, 'quadratic_profit': 604}, {'item_i_id': 9, 'item_j_id': 15, 'quadratic_profit': 1067}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 183}, {'item_i_id': 10, 'item_j_id': 11, 'quadratic_profit': 1128}, {'item_i_id': 10, 'item_j_id': 12, 'quadratic_profit': 794}, {'item_i_id': 10, 'item_j_id': 13, 'quadratic_profit': 228}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 153}, {'item_i_id': 11, 'item_j_id': 13, 'quadratic_profit': 434}, {'item_i_id': 11, 'item_j_id': 14, 'quadratic_profit': 434}, {'item_i_id': 11, 'item_j_id': 15, 'quadratic_profit': 296}, {'item_i_id': 12, 'item_j_id': 12, 'quadratic_profit': 743}, {'item_i_id': 12, 'item_j_id': 13, 'quadratic_profit': 216}, {'item_i_id': 12, 'item_j_id': 14, 'quadratic_profit': 109}, {'item_i_id': 13, 'item_j_id': 13, 'quadratic_profit': 918}, {'item_i_id': 13, 'item_j_id': 14, 'quadratic_profit': 235}, {'item_i_id': 13, 'item_j_id': 15, 'quadratic_profit': 520}, {'item_i_id': 14, 'item_j_id': 14, 'quadratic_profit': 528}, {'item_i_id': 14, 'item_j_id': 15, 'quadratic_profit': 528}, {'item_i_id': 15, 'item_j_id': 15, 'quadratic_profit': 815}]}","[1, 2, 3, 4, 5, 6, 8, 10, 11, 12]",28,markdown_table,1
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=15
# shelf_length=1853
# book_ids=A, B, C, D, E, F, G, H, I, J, K, L, M, N, O

book_id,educational_points
A,105
B,116
C,137
D,105
E,141
F,108
G,149
H,126
I,115
J,136
K,122
L,146
M,138
N,112
O,137

book_id,width
A,145
B,135
C,129
D,102
E,138
F,128
G,145
H,139
I,142
J,106
K,112
L,121
M,142
N,118
O,107

book_i_id,book_j_id,pairing_bonus_points
A,B,142
A,C,134
A,E,123
A,F,135
A,H,133
A,I,132
A,J,133
A,K,143
A,L,139
A,M,133
A,N,134
B,D,131
B,E,116
B,F,128
B,G,138
B,H,128
B,I,135
B,J,125
B,K,139
B,L,126
B,N,139
C,D,122
C,E,120
C,G,129
C,H,127
C,K,125
C,L,108
C,O,110
D,E,108
D,G,113
D,H,112
D,I,121
D,J,121
D,K,123
D,O,143
E,F,110
E,G,109
E,I,111
E,J,115
E,K,119
E,L,105
E,N,118
E,O,113
F,G,120
F,H,121
F,I,129
F,J,121
F,K,128
F,M,132
F,N,144
F,O,143
G,H,109
G,I,105
G,J,110
G,K,125
G,L,128
G,N,114
H,I,123
H,K,117
H,L,108
H,M,117
H,O,101
I,J,126
I,K,133
I,L,132
I,M,128
I,N,134
I,O,132
J,K,120
J,L,123
J,M,126
J,O,126
K,L,121
L,M,128
M,N,112
M,O,118

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': [105, 116, 137, 105, 141, 108, 149, 126, 115, 136, 122, 146, 138, 112, 137], 'quadratic_coeffs': [[130, 142, 134, 0, 123, 135, 0, 133, 132, 133, 143, 139, 133, 134, 0], [0, 0, 0, 131, 116, 128, 138, 128, 135, 125, 139, 126, 0, 139, 0], [0, 0, 0, 122, 120, 0, 129, 127, 0, 0, 125, 108, 0, 0, 110], [0, 0, 0, 101, 108, 0, 113, 112, 121, 121, 123, 0, 0, 0, 143], [0, 0, 0, 0, 0, 110, 109, 0, 111, 115, 119, 105, 0, 118, 113], [0, 0, 0, 0, 0, 0, 120, 121, 129, 121, 128, 0, 132, 144, 143], [0, 0, 0, 0, 0, 0, 110, 109, 105, 110, 125, 128, 0, 114, 0], [0, 0, 0, 0, 0, 0, 0, 118, 123, 0, 117, 108, 117, 0, 101], [0, 0, 0, 0, 0, 0, 0, 0, 0, 126, 133, 132, 128, 134, 132], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 120, 123, 126, 0, 126], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 117, 121, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 123, 128, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 122, 112, 118], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 118]], 'weights': [145, 135, 129, 102, 138, 128, 145, 139, 142, 106, 112, 121, 142, 118, 107], 'capacity': 1853, 'solution': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14], 'obj': 10308.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14]",10308.0,"{'problem_type': 'QKP', 'num_items': 15, 'capacity': 1853, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 105}, {'item_id': 'B', 'linear_profit': 116}, {'item_id': 'C', 'linear_profit': 137}, {'item_id': 'D', 'linear_profit': 105}, {'item_id': 'E', 'linear_profit': 141}, {'item_id': 'F', 'linear_profit': 108}, {'item_id': 'G', 'linear_profit': 149}, {'item_id': 'H', 'linear_profit': 126}, {'item_id': 'I', 'linear_profit': 115}, {'item_id': 'J', 'linear_profit': 136}, {'item_id': 'K', 'linear_profit': 122}, {'item_id': 'L', 'linear_profit': 146}, {'item_id': 'M', 'linear_profit': 138}, {'item_id': 'N', 'linear_profit': 112}, {'item_id': 'O', 'linear_profit': 137}], 'weight_pairs': [{'item_id': 'A', 'weight': 145}, {'item_id': 'B', 'weight': 135}, {'item_id': 'C', 'weight': 129}, {'item_id': 'D', 'weight': 102}, {'item_id': 'E', 'weight': 138}, {'item_id': 'F', 'weight': 128}, {'item_id': 'G', 'weight': 145}, {'item_id': 'H', 'weight': 139}, {'item_id': 'I', 'weight': 142}, {'item_id': 'J', 'weight': 106}, {'item_id': 'K', 'weight': 112}, {'item_id': 'L', 'weight': 121}, {'item_id': 'M', 'weight': 142}, {'item_id': 'N', 'weight': 118}, {'item_id': 'O', 'weight': 107}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 130}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 142}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 134}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 123}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 135}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 133}, {'item_i_id': 'A', 'item_j_id': 'I', 'quadratic_profit': 132}, {'item_i_id': 'A', 'item_j_id': 'J', 'quadratic_profit': 133}, {'item_i_id': 'A', 'item_j_id': 'K', 'quadratic_profit': 143}, {'item_i_id': 'A', 'item_j_id': 'L', 'quadratic_profit': 139}, {'item_i_id': 'A', 'item_j_id': 'M', 'quadratic_profit': 133}, {'item_i_id': 'A', 'item_j_id': 'N', 'quadratic_profit': 134}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 131}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 116}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 128}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 138}, {'item_i_id': 'B', 'item_j_id': 'H', 'quadratic_profit': 128}, {'item_i_id': 'B', 'item_j_id': 'I', 'quadratic_profit': 135}, {'item_i_id': 'B', 'item_j_id': 'J', 'quadratic_profit': 125}, {'item_i_id': 'B', 'item_j_id': 'K', 'quadratic_profit': 139}, {'item_i_id': 'B', 'item_j_id': 'L', 'quadratic_profit': 126}, {'item_i_id': 'B', 'item_j_id': 'N', 'quadratic_profit': 139}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 122}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 120}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 129}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 127}, {'item_i_id': 'C', 'item_j_id': 'K', 'quadratic_profit': 125}, {'item_i_id': 'C', 'item_j_id': 'L', 'quadratic_profit': 108}, {'item_i_id': 'C', 'item_j_id': 'O', 'quadratic_profit': 110}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 101}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 108}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 113}, {'item_i_id': 'D', 'item_j_id': 'H', 'quadratic_profit': 112}, {'item_i_id': 'D', 'item_j_id': 'I', 'quadratic_profit': 121}, {'item_i_id': 'D', 'item_j_id': 'J', 'quadratic_profit': 121}, {'item_i_id': 'D', 'item_j_id': 'K', 'quadratic_profit': 123}, {'item_i_id': 'D', 'item_j_id': 'O', 'quadratic_profit': 143}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 110}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 109}, {'item_i_id': 'E', 'item_j_id': 'I', 'quadratic_profit': 111}, {'item_i_id': 'E', 'item_j_id': 'J', 'quadratic_profit': 115}, {'item_i_id': 'E', 'item_j_id': 'K', 'quadratic_profit': 119}, {'item_i_id': 'E', 'item_j_id': 'L', 'quadratic_profit': 105}, {'item_i_id': 'E', 'item_j_id': 'N', 'quadratic_profit': 118}, {'item_i_id': 'E', 'item_j_id': 'O', 'quadratic_profit': 113}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 120}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 121}, {'item_i_id': 'F', 'item_j_id': 'I', 'quadratic_profit': 129}, {'item_i_id': 'F', 'item_j_id': 'J', 'quadratic_profit': 121}, {'item_i_id': 'F', 'item_j_id': 'K', 'quadratic_profit': 128}, {'item_i_id': 'F', 'item_j_id': 'M', 'quadratic_profit': 132}, {'item_i_id': 'F', 'item_j_id': 'N', 'quadratic_profit': 144}, {'item_i_id': 'F', 'item_j_id': 'O', 'quadratic_profit': 143}, {'item_i_id': 'G', 'item_j_id': 'G', 'quadratic_profit': 110}, {'item_i_id': 'G', 'item_j_id': 'H', 'quadratic_profit': 109}, {'item_i_id': 'G', 'item_j_id': 'I', 'quadratic_profit': 105}, {'item_i_id': 'G', 'item_j_id': 'J', 'quadratic_profit': 110}, {'item_i_id': 'G', 'item_j_id': 'K', 'quadratic_profit': 125}, {'item_i_id': 'G', 'item_j_id': 'L', 'quadratic_profit': 128}, {'item_i_id': 'G', 'item_j_id': 'N', 'quadratic_profit': 114}, {'item_i_id': 'H', 'item_j_id': 'H', 'quadratic_profit': 118}, {'item_i_id': 'H', 'item_j_id': 'I', 'quadratic_profit': 123}, {'item_i_id': 'H', 'item_j_id': 'K', 'quadratic_profit': 117}, {'item_i_id': 'H', 'item_j_id': 'L', 'quadratic_profit': 108}, {'item_i_id': 'H', 'item_j_id': 'M', 'quadratic_profit': 117}, {'item_i_id': 'H', 'item_j_id': 'O', 'quadratic_profit': 101}, {'item_i_id': 'I', 'item_j_id': 'J', 'quadratic_profit': 126}, {'item_i_id': 'I', 'item_j_id': 'K', 'quadratic_profit': 133}, {'item_i_id': 'I', 'item_j_id': 'L', 'quadratic_profit': 132}, {'item_i_id': 'I', 'item_j_id': 'M', 'quadratic_profit': 128}, {'item_i_id': 'I', 'item_j_id': 'N', 'quadratic_profit': 134}, {'item_i_id': 'I', 'item_j_id': 'O', 'quadratic_profit': 132}, {'item_i_id': 'J', 'item_j_id': 'K', 'quadratic_profit': 120}, {'item_i_id': 'J', 'item_j_id': 'L', 'quadratic_profit': 123}, {'item_i_id': 'J', 'item_j_id': 'M', 'quadratic_profit': 126}, {'item_i_id': 'J', 'item_j_id': 'O', 'quadratic_profit': 126}, {'item_i_id': 'K', 'item_j_id': 'K', 'quadratic_profit': 117}, {'item_i_id': 'K', 'item_j_id': 'L', 'quadratic_profit': 121}, {'item_i_id': 'L', 'item_j_id': 'L', 'quadratic_profit': 123}, {'item_i_id': 'L', 'item_j_id': 'M', 'quadratic_profit': 128}, {'item_i_id': 'M', 'item_j_id': 'M', 'quadratic_profit': 122}, {'item_i_id': 'M', 'item_j_id': 'N', 'quadratic_profit': 112}, {'item_i_id': 'M', 'item_j_id': 'O', 'quadratic_profit': 118}, {'item_i_id': 'O', 'item_j_id': 'O', 'quadratic_profit': 118}]}","['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'O']",29,csv,names
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.

# num_shows=11
# block_runtime_minutes=15795
# show_ids=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

show_id,base_viewers
1,6137
2,1590
3,6087
4,1616
5,6052
6,4542
7,6279
8,4928
9,4219
10,4289
11,5492

show_id,runtime_minutes
1,6334
2,1656
3,5927
4,1323
5,5733
6,4671
7,6395
8,5182
9,3930
10,4544
11,5254

show_i_id,show_j_id,pairing_uplift_viewers
1,2,2358
1,3,4112
1,8,5656
1,9,6033
2,3,2482
2,5,544
4,6,2154
4,7,3049
4,8,1732
4,9,2833
5,9,4306
7,9,3005
7,10,3662
8,11,3585

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': [6137, 1590, 6087, 1616, 6052, 4542, 6279, 4928, 4219, 4289, 5492], 'quadratic_coeffs': [[0, 2358, 4112, 0, 0, 0, 0, 5656, 6033, 0, 0], [0, 1306, 2482, 0, 544, 0, 0, 0, 0, 0, 0], [0, 0, 5327, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2648, 0, 2154, 3049, 1732, 2833, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4306, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2639, 0, 3005, 3662, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3585], [0, 0, 0, 0, 0, 0, 0, 0, 0, 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': [6334, 1656, 5927, 1323, 5733, 4671, 6395, 5182, 3930, 4544, 5254], 'capacity': 15795, 'solution': [0, 7, 8], 'obj': 26973.0, 'problem_type': 'QKP'}","[0, 7, 8]",26973.0,"{'problem_type': 'QKP', 'num_items': 11, 'capacity': 15795, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 'linear_pairs': [{'item_id': 1, 'linear_profit': 6137}, {'item_id': 2, 'linear_profit': 1590}, {'item_id': 3, 'linear_profit': 6087}, {'item_id': 4, 'linear_profit': 1616}, {'item_id': 5, 'linear_profit': 6052}, {'item_id': 6, 'linear_profit': 4542}, {'item_id': 7, 'linear_profit': 6279}, {'item_id': 8, 'linear_profit': 4928}, {'item_id': 9, 'linear_profit': 4219}, {'item_id': 10, 'linear_profit': 4289}, {'item_id': 11, 'linear_profit': 5492}], 'weight_pairs': [{'item_id': 1, 'weight': 6334}, {'item_id': 2, 'weight': 1656}, {'item_id': 3, 'weight': 5927}, {'item_id': 4, 'weight': 1323}, {'item_id': 5, 'weight': 5733}, {'item_id': 6, 'weight': 4671}, {'item_id': 7, 'weight': 6395}, {'item_id': 8, 'weight': 5182}, {'item_id': 9, 'weight': 3930}, {'item_id': 10, 'weight': 4544}, {'item_id': 11, 'weight': 5254}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 2358}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 4112}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 5656}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 6033}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 1306}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 2482}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 544}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 5327}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 2648}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 2154}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 3049}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 1732}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 2833}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 4306}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 2639}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 3005}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 3662}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 3585}]}","[1, 8, 9]",30,csv,1
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 12 tackle options listed as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and the pack's limit is 19422.

| tackle_id | base_catch_value |
|---|---|
| 0 | 1284 |
| 1 | 3565 |
| 2 | 2004 |
| 3 | 2843 |
| 4 | 1472 |
| 5 | 1398 |
| 6 | 6409 |
| 7 | 6104 |
| 8 | 4347 |
| 9 | 1182 |
| 10 | 6274 |
| 11 | 3856 |

| tackle_id | tackle_weight |
|---|---|
| 0 | 4862 |
| 1 | 3458 |
| 2 | 5011 |
| 3 | 3219 |
| 4 | 2813 |
| 5 | 3193 |
| 6 | 1854 |
| 7 | 5233 |
| 8 | 4443 |
| 9 | 1616 |
| 10 | 2036 |
| 11 | 4699 |

| tackle_i_id | tackle_j_id | combination_bonus_value |
|---|---|---|
| 0 | 1 | 2477 |
| 0 | 2 | 1748 |
| 0 | 3 | 3209 |
| 0 | 4 | 5010 |
| 0 | 5 | 3032 |
| 0 | 6 | 5483 |
| 0 | 7 | 3989 |
| 0 | 8 | 3441 |
| 0 | 9 | 3763 |
| 0 | 10 | 4327 |
| 0 | 11 | 783 |
| 1 | 2 | 4697 |
| 1 | 3 | 3168 |
| 1 | 4 | 1527 |
| 1 | 5 | 2943 |
| 1 | 6 | 1138 |
| 1 | 7 | 3723 |
| 1 | 8 | 2487 |
| 1 | 9 | 2168 |
| 1 | 10 | 824 |
| 1 | 11 | 1163 |
| 2 | 3 | 4815 |
| 2 | 4 | 3305 |
| 2 | 5 | 4977 |
| 2 | 6 | 4636 |
| 2 | 7 | 4839 |
| 2 | 8 | 2610 |
| 2 | 9 | 1955 |
| 2 | 10 | 699 |
| 2 | 11 | 1827 |
| 3 | 4 | 3544 |
| 3 | 5 | 2560 |
| 3 | 6 | 2092 |
| 3 | 7 | 4488 |
| 3 | 8 | 3827 |
| 3 | 9 | 2107 |
| 3 | 10 | 1262 |
| 3 | 11 | 1234 |
| 4 | 5 | 3453 |
| 4 | 6 | 1935 |
| 4 | 7 | 2570 |
| 4 | 8 | 2746 |
| 4 | 9 | 825 |
| 4 | 10 | 2041 |
| 4 | 11 | 3549 |
| 5 | 6 | 2811 |
| 5 | 7 | 4747 |
| 5 | 8 | 5221 |
| 5 | 9 | 5531 |
| 5 | 10 | 5027 |
| 5 | 11 | 4470 |
| 6 | 7 | 3498 |
| 6 | 8 | 3260 |
| 6 | 9 | 2580 |
| 6 | 10 | 2347 |
| 6 | 11 | 3063 |
| 7 | 8 | 4474 |
| 7 | 9 | 4830 |
| 7 | 10 | 5252 |
| 7 | 11 | 3682 |
| 8 | 9 | 3867 |
| 8 | 10 | 3349 |
| 8 | 11 | 2360 |
| 9 | 10 | 2446 |
| 9 | 11 | 3127 |
| 10 | 11 | 5776 |

Pack smart: choose gear and pairings that raise total expected catch while staying within the 19422 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': [1284, 3565, 2004, 2843, 1472, 1398, 6409, 6104, 4347, 1182, 6274, 3856], 'quadratic_coeffs': [[3297, 2477, 1748, 3209, 5010, 3032, 5483, 3989, 3441, 3763, 4327, 783], [0, 3681, 4697, 3168, 1527, 2943, 1138, 3723, 2487, 2168, 824, 1163], [0, 0, 3796, 4815, 3305, 4977, 4636, 4839, 2610, 1955, 699, 1827], [0, 0, 0, 1074, 3544, 2560, 2092, 4488, 3827, 2107, 1262, 1234], [0, 0, 0, 0, 2634, 3453, 1935, 2570, 2746, 825, 2041, 3549], [0, 0, 0, 0, 0, 5434, 2811, 4747, 5221, 5531, 5027, 4470], [0, 0, 0, 0, 0, 0, 2706, 3498, 3260, 2580, 2347, 3063], [0, 0, 0, 0, 0, 0, 0, 1368, 4474, 4830, 5252, 3682], [0, 0, 0, 0, 0, 0, 0, 0, 3644, 3867, 3349, 2360], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2249, 2446, 3127], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5011, 5776], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2129]], 'weights': [4862, 3458, 5011, 3219, 2813, 3193, 1854, 5233, 4443, 1616, 2036, 4699], 'capacity': 19422, 'solution': [3, 4, 5, 6, 8, 9, 10], 'obj': 86756.0, 'problem_type': 'QKP'}","[3, 4, 5, 6, 8, 9, 10]",86756.0,"{'problem_type': 'QKP', 'num_items': 12, 'capacity': 19422, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 'linear_pairs': [{'item_id': 0, 'linear_profit': 1284}, {'item_id': 1, 'linear_profit': 3565}, {'item_id': 2, 'linear_profit': 2004}, {'item_id': 3, 'linear_profit': 2843}, {'item_id': 4, 'linear_profit': 1472}, {'item_id': 5, 'linear_profit': 1398}, {'item_id': 6, 'linear_profit': 6409}, {'item_id': 7, 'linear_profit': 6104}, {'item_id': 8, 'linear_profit': 4347}, {'item_id': 9, 'linear_profit': 1182}, {'item_id': 10, 'linear_profit': 6274}, {'item_id': 11, 'linear_profit': 3856}], 'weight_pairs': [{'item_id': 0, 'weight': 4862}, {'item_id': 1, 'weight': 3458}, {'item_id': 2, 'weight': 5011}, {'item_id': 3, 'weight': 3219}, {'item_id': 4, 'weight': 2813}, {'item_id': 5, 'weight': 3193}, {'item_id': 6, 'weight': 1854}, {'item_id': 7, 'weight': 5233}, {'item_id': 8, 'weight': 4443}, {'item_id': 9, 'weight': 1616}, {'item_id': 10, 'weight': 2036}, {'item_id': 11, 'weight': 4699}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 3297}, {'item_i_id': 0, 'item_j_id': 1, 'quadratic_profit': 2477}, {'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 1748}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 3209}, {'item_i_id': 0, 'item_j_id': 4, 'quadratic_profit': 5010}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 3032}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 5483}, {'item_i_id': 0, 'item_j_id': 7, 'quadratic_profit': 3989}, {'item_i_id': 0, 'item_j_id': 8, 'quadratic_profit': 3441}, {'item_i_id': 0, 'item_j_id': 9, 'quadratic_profit': 3763}, {'item_i_id': 0, 'item_j_id': 10, 'quadratic_profit': 4327}, {'item_i_id': 0, 'item_j_id': 11, 'quadratic_profit': 783}, {'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 3681}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 4697}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 3168}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 1527}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 2943}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 1138}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 3723}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 2487}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 2168}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 824}, {'item_i_id': 1, 'item_j_id': 11, 'quadratic_profit': 1163}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 3796}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 4815}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 3305}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 4977}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 4636}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 4839}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 2610}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 1955}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 699}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 1827}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 1074}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 3544}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 2560}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 2092}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 4488}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 3827}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 2107}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 1262}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 1234}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 2634}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 3453}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 1935}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 2570}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 2746}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 825}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 2041}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 3549}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 5434}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 2811}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 4747}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 5221}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 5531}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 5027}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 4470}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 2706}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 3498}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 3260}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 2580}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 2347}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 3063}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 1368}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 4474}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 4830}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 5252}, {'item_i_id': 7, 'item_j_id': 11, 'quadratic_profit': 3682}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 3644}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 3867}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 3349}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 2360}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 2249}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 2446}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 3127}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 5011}, {'item_i_id': 10, 'item_j_id': 11, 'quadratic_profit': 5776}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 2129}]}","[3, 4, 5, 6, 8, 9, 10]",31,markdown_table,0
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"": 15,
  ""carry_capacity"": 26697,
  ""item_ids"": [
    0,
    1,
    2,
    3,
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14
  ],
  ""linear"": [
    {
      ""gear_id"": 0,
      ""standalone_value"": 3317
    },
    {
      ""gear_id"": 1,
      ""standalone_value"": 3289
    },
    {
      ""gear_id"": 2,
      ""standalone_value"": 226
    },
    {
      ""gear_id"": 3,
      ""standalone_value"": 3229
    },
    {
      ""gear_id"": 4,
      ""standalone_value"": 2997
    },
    {
      ""gear_id"": 5,
      ""standalone_value"": 724
    },
    {
      ""gear_id"": 6,
      ""standalone_value"": 424
    },
    {
      ""gear_id"": 7,
      ""standalone_value"": 267
    },
    {
      ""gear_id"": 8,
      ""standalone_value"": 3033
    },
    {
      ""gear_id"": 9,
      ""standalone_value"": 758
    },
    {
      ""gear_id"": 10,
      ""standalone_value"": 3659
    },
    {
      ""gear_id"": 11,
      ""standalone_value"": 2079
    },
    {
      ""gear_id"": 12,
      ""standalone_value"": 3930
    },
    {
      ""gear_id"": 13,
      ""standalone_value"": 2258
    },
    {
      ""gear_id"": 14,
      ""standalone_value"": 3216
    }
  ],
  ""weights"": [
    {
      ""gear_id"": 0,
      ""item_weight"": 3282
    },
    {
      ""gear_id"": 1,
      ""item_weight"": 3040
    },
    {
      ""gear_id"": 2,
      ""item_weight"": 253
    },
    {
      ""gear_id"": 3,
      ""item_weight"": 3255
    },
    {
      ""gear_id"": 4,
      ""item_weight"": 2760
    },
    {
      ""gear_id"": 5,
      ""item_weight"": 1033
    },
    {
      ""gear_id"": 6,
      ""item_weight"": 698
    },
    {
      ""gear_id"": 7,
      ""item_weight"": 294
    },
    {
      ""gear_id"": 8,
      ""item_weight"": 2961
    },
    {
      ""gear_id"": 9,
      ""item_weight"": 1045
    },
    {
      ""gear_id"": 10,
      ""item_weight"": 3358
    },
    {
      ""gear_id"": 11,
      ""item_weight"": 2135
    },
    {
      ""gear_id"": 12,
      ""item_weight"": 3613
    },
    {
      ""gear_id"": 13,
      ""item_weight"": 2201
    },
    {
      ""gear_id"": 14,
      ""item_weight"": 3534
    }
  ],
  ""quadratic"": [
    {
      ""gear_i_id"": 0,
      ""gear_j_id"": 1,
      ""combo_bonus"": 583
    },
    {
      ""gear_i_id"": 0,
      ""gear_j_id"": 5,
      ""combo_bonus"": 3975
    },
    {
      ""gear_i_id"": 0,
      ""gear_j_id"": 6,
      ""combo_bonus"": 3953
    },
    {
      ""gear_i_id"": 0,
      ""gear_j_id"": 7,
      ""combo_bonus"": 1730
    },
    {
      ""gear_i_id"": 0,
      ""gear_j_id"": 9,
      ""combo_bonus"": 3719
    },
    {
      ""gear_i_id"": 0,
      ""gear_j_id"": 14,
      ""combo_bonus"": 5059
    },
    {
      ""gear_i_id"": 1,
      ""gear_j_id"": 3,
      ""combo_bonus"": 2396
    },
    {
      ""gear_i_id"": 1,
      ""gear_j_id"": 6,
      ""combo_bonus"": 6100
    },
    {
      ""gear_i_id"": 2,
      ""gear_j_id"": 7,
      ""combo_bonus"": 1398
    },
    {
      ""gear_i_id"": 2,
      ""gear_j_id"": 9,
      ""combo_bonus"": 4649
    },
    {
      ""gear_i_id"": 2,
      ""gear_j_id"": 12,
      ""combo_bonus"": 2635
    },
    {
      ""gear_i_id"": 3,
      ""gear_j_id"": 6,
      ""combo_bonus"": 1163
    },
    {
      ""gear_i_id"": 3,
      ""gear_j_id"": 10,
      ""combo_bonus"": 3594
    },
    {
      ""gear_i_id"": 4,
      ""gear_j_id"": 6,
      ""combo_bonus"": 3874
    },
    {
      ""gear_i_id"": 4,
      ""gear_j_id"": 12,
      ""combo_bonus"": 3606
    },
    {
      ""gear_i_id"": 4,
      ""gear_j_id"": 14,
      ""combo_bonus"": 1470
    },
    {
      ""gear_i_id"": 5,
      ""gear_j_id"": 7,
      ""combo_bonus"": 1396
    },
    {
      ""gear_i_id"": 5,
      ""gear_j_id"": 8,
      ""combo_bonus"": 509
    },
    {
      ""gear_i_id"": 5,
      ""gear_j_id"": 9,
      ""combo_bonus"": 2822
    },
    {
      ""gear_i_id"": 5,
      ""gear_j_id"": 14,
      ""combo_bonus"": 3316
    },
    {
      ""gear_i_id"": 6,
      ""gear_j_id"": 13,
      ""combo_bonus"": 550
    },
    {
      ""gear_i_id"": 7,
      ""gear_j_id"": 9,
      ""combo_bonus"": 2158
    },
    {
      ""gear_i_id"": 8,
      ""gear_j_id"": 13,
      ""combo_bonus"": 1987
    },
    {
      ""gear_i_id"": 9,
      ""gear_j_id"": 13,
      ""combo_bonus"": 1499
    },
    {
      ""gear_i_id"": 11,
      ""gear_j_id"": 13,
      ""combo_bonus"": 3575
    }
  ]
}

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': [3317, 3289, 226, 3229, 2997, 724, 424, 267, 3033, 758, 3659, 2079, 3930, 2258, 3216], 'quadratic_coeffs': [[0, 583, 0, 0, 0, 3975, 3953, 1730, 0, 3719, 0, 0, 0, 0, 5059], [0, 0, 0, 2396, 0, 0, 6100, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1398, 0, 4649, 0, 0, 2635, 0, 0], [0, 0, 0, 3580, 0, 0, 1163, 0, 0, 0, 3594, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3874, 0, 0, 0, 0, 0, 3606, 0, 1470], [0, 0, 0, 0, 0, 0, 0, 1396, 509, 2822, 0, 0, 0, 0, 3316], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 550, 0], [0, 0, 0, 0, 0, 0, 0, 4903, 0, 2158, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1987, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1499, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3575, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 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': [3282, 3040, 253, 3255, 2760, 1033, 698, 294, 2961, 1045, 3358, 2135, 3613, 2201, 3534], 'capacity': 26697, 'solution': [0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14], 'obj': 85632.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14]",85632.0,"{'problem_type': 'QKP', 'num_items': 15, 'capacity': 26697, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], 'linear_pairs': [{'item_id': 0, 'linear_profit': 3317}, {'item_id': 1, 'linear_profit': 3289}, {'item_id': 2, 'linear_profit': 226}, {'item_id': 3, 'linear_profit': 3229}, {'item_id': 4, 'linear_profit': 2997}, {'item_id': 5, 'linear_profit': 724}, {'item_id': 6, 'linear_profit': 424}, {'item_id': 7, 'linear_profit': 267}, {'item_id': 8, 'linear_profit': 3033}, {'item_id': 9, 'linear_profit': 758}, {'item_id': 10, 'linear_profit': 3659}, {'item_id': 11, 'linear_profit': 2079}, {'item_id': 12, 'linear_profit': 3930}, {'item_id': 13, 'linear_profit': 2258}, {'item_id': 14, 'linear_profit': 3216}], 'weight_pairs': [{'item_id': 0, 'weight': 3282}, {'item_id': 1, 'weight': 3040}, {'item_id': 2, 'weight': 253}, {'item_id': 3, 'weight': 3255}, {'item_id': 4, 'weight': 2760}, {'item_id': 5, 'weight': 1033}, {'item_id': 6, 'weight': 698}, {'item_id': 7, 'weight': 294}, {'item_id': 8, 'weight': 2961}, {'item_id': 9, 'weight': 1045}, {'item_id': 10, 'weight': 3358}, {'item_id': 11, 'weight': 2135}, {'item_id': 12, 'weight': 3613}, {'item_id': 13, 'weight': 2201}, {'item_id': 14, 'weight': 3534}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 1, 'quadratic_profit': 583}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 3975}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 3953}, {'item_i_id': 0, 'item_j_id': 7, 'quadratic_profit': 1730}, {'item_i_id': 0, 'item_j_id': 9, 'quadratic_profit': 3719}, {'item_i_id': 0, 'item_j_id': 14, 'quadratic_profit': 5059}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 2396}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 6100}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 1398}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 4649}, {'item_i_id': 2, 'item_j_id': 12, 'quadratic_profit': 2635}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 3580}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 1163}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 3594}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 3874}, {'item_i_id': 4, 'item_j_id': 12, 'quadratic_profit': 3606}, {'item_i_id': 4, 'item_j_id': 14, 'quadratic_profit': 1470}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 1396}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 509}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 2822}, {'item_i_id': 5, 'item_j_id': 14, 'quadratic_profit': 3316}, {'item_i_id': 6, 'item_j_id': 13, 'quadratic_profit': 550}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 4903}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 2158}, {'item_i_id': 8, 'item_j_id': 13, 'quadratic_profit': 1987}, {'item_i_id': 9, 'item_j_id': 13, 'quadratic_profit': 1499}, {'item_i_id': 11, 'item_j_id': 13, 'quadratic_profit': 3575}]}","[0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14]",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.

# num_plants=11
# plot_area_capacity=2317
# plant_list=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

plant_id,base_appeal
0,2095
1,3195
2,1356
3,901
4,2536
5,325
6,1609
7,1695
8,3075
9,789
10,864

plant_id,area_required
0,243
1,1236
2,839
3,1014
4,568
5,841
6,1242
7,3093
8,2910
9,1671
10,3246

plant_i_id,plant_j_id,pair_synergy_appeal
0,1,174
0,3,292
0,5,742
0,6,641
0,10,835
1,2,1092
1,4,1079
1,5,1808
1,7,1697
1,8,1923
1,9,1198
2,4,1650
2,5,1214
2,7,430
2,9,1039
3,5,1583
3,7,1100
3,8,843
3,9,864
3,10,371
4,7,1593
4,8,623
4,9,2552
5,6,2460
5,9,1445
5,10,1125
6,7,713
6,8,950
6,9,636
7,10,1045

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': [2095, 3195, 1356, 901, 2536, 325, 1609, 1695, 3075, 789, 864], 'quadratic_coeffs': [[790, 174, 0, 292, 0, 742, 641, 0, 0, 0, 835], [0, 1346, 1092, 0, 1079, 1808, 0, 1697, 1923, 1198, 0], [0, 0, 0, 0, 1650, 1214, 0, 430, 0, 1039, 0], [0, 0, 0, 0, 0, 1583, 0, 1100, 843, 864, 371], [0, 0, 0, 0, 0, 0, 0, 1593, 623, 2552, 0], [0, 0, 0, 0, 0, 0, 2460, 0, 0, 1445, 1125], [0, 0, 0, 0, 0, 0, 0, 713, 950, 636, 0], [0, 0, 0, 0, 0, 0, 0, 1353, 0, 0, 1045], [0, 0, 0, 0, 0, 0, 0, 0, 2878, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2548, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [243, 1236, 839, 1014, 568, 841, 1242, 3093, 2910, 1671, 3246], 'capacity': 2317, 'solution': [0, 1, 4], 'obj': 9079.0, 'problem_type': 'QKP'}","[0, 1, 4]",9079.0,"{'problem_type': 'QKP', 'num_items': 11, 'capacity': 2317, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 'linear_pairs': [{'item_id': 0, 'linear_profit': 2095}, {'item_id': 1, 'linear_profit': 3195}, {'item_id': 2, 'linear_profit': 1356}, {'item_id': 3, 'linear_profit': 901}, {'item_id': 4, 'linear_profit': 2536}, {'item_id': 5, 'linear_profit': 325}, {'item_id': 6, 'linear_profit': 1609}, {'item_id': 7, 'linear_profit': 1695}, {'item_id': 8, 'linear_profit': 3075}, {'item_id': 9, 'linear_profit': 789}, {'item_id': 10, 'linear_profit': 864}], 'weight_pairs': [{'item_id': 0, 'weight': 243}, {'item_id': 1, 'weight': 1236}, {'item_id': 2, 'weight': 839}, {'item_id': 3, 'weight': 1014}, {'item_id': 4, 'weight': 568}, {'item_id': 5, 'weight': 841}, {'item_id': 6, 'weight': 1242}, {'item_id': 7, 'weight': 3093}, {'item_id': 8, 'weight': 2910}, {'item_id': 9, 'weight': 1671}, {'item_id': 10, 'weight': 3246}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 790}, {'item_i_id': 0, 'item_j_id': 1, 'quadratic_profit': 174}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 292}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 742}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 641}, {'item_i_id': 0, 'item_j_id': 10, 'quadratic_profit': 835}, {'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 1346}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 1092}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 1079}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 1808}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 1697}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 1923}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 1198}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 1650}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 1214}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 430}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 1039}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 1583}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 1100}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 843}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 864}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 371}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 1593}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 623}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 2552}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 2460}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 1445}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 1125}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 713}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 950}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 636}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 1353}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 1045}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 2878}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 2548}]}","[0, 1, 4]",33,csv,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 12 appetizer options (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12), and the host has 2230 units of storage space to work with.
1 gives a base satisfaction of 142 points when prepared.
2 gives a base satisfaction of 317 points when prepared.
3 gives a base satisfaction of 156 points when prepared.
4 gives a base satisfaction of 279 points when prepared.
5 gives a base satisfaction of 231 points when prepared.
6 gives a base satisfaction of 277 points when prepared.
7 gives a base satisfaction of 288 points when prepared.
8 gives a base satisfaction of 199 points when prepared.
9 gives a base satisfaction of 176 points when prepared.
10 gives a base satisfaction of 231 points when prepared.
11 gives a base satisfaction of 250 points when prepared.
12 gives a base satisfaction of 329 points when prepared.
Preparing 1 consumes 112 units of storage space.
Preparing 2 consumes 287 units of storage space.
Preparing 3 consumes 126 units of storage space.
Preparing 4 consumes 249 units of storage space.
Preparing 5 consumes 201 units of storage space.
Preparing 6 consumes 247 units of storage space.
Preparing 7 consumes 258 units of storage space.
Preparing 8 consumes 169 units of storage space.
Preparing 9 consumes 146 units of storage space.
Preparing 10 consumes 201 units of storage space.
Preparing 11 consumes 220 units of storage space.
Preparing 12 consumes 299 units of storage space.
If both 1 and 2 are prepared, they add 284 bonus satisfaction points.
If both 1 and 3 are prepared, they add 168 bonus satisfaction points.
If both 1 and 6 are prepared, they add 196 bonus satisfaction points.
If both 1 and 8 are prepared, they add 127 bonus satisfaction points.
If both 1 and 10 are prepared, they add 202 bonus satisfaction points.
If both 2 and 3 are prepared, they add 161 bonus satisfaction points.
If both 2 and 7 are prepared, they add 203 bonus satisfaction points.
If both 2 and 8 are prepared, they add 240 bonus satisfaction points.
If both 3 and 6 are prepared, they add 231 bonus satisfaction points.
If both 3 and 7 are prepared, they add 189 bonus satisfaction points.
If both 4 and 11 are prepared, they add 181 bonus satisfaction points.
If both 5 and 11 are prepared, they add 171 bonus satisfaction points.
If both 6 and 11 are prepared, they add 193 bonus satisfaction points.
If both 6 and 12 are prepared, they add 128 bonus satisfaction points.
If both 8 and 9 are prepared, they add 238 bonus satisfaction points.
If both 8 and 12 are prepared, they add 270 bonus satisfaction points.
If both 9 and 12 are prepared, they add 279 bonus satisfaction points.
If both 10 and 11 are prepared, they add 156 bonus satisfaction points.
If both 11 and 12 are prepared, they add 124 bonus satisfaction points.
The host must pick appetizers that fit within the 2230 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': [142, 317, 156, 279, 231, 277, 288, 199, 176, 231, 250, 329], 'quadratic_coeffs': [[0, 284, 168, 0, 0, 196, 0, 127, 0, 202, 0, 0], [0, 234, 161, 0, 0, 0, 203, 240, 0, 0, 0, 0], [0, 0, 0, 0, 0, 231, 189, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 181, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 171, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 193, 128], [0, 0, 0, 0, 0, 0, 228, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 238, 0, 0, 270], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 279], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 156, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 124], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [112, 287, 126, 249, 201, 247, 258, 169, 146, 201, 220, 299], 'capacity': 2230, 'solution': [0, 1, 2, 5, 6, 7, 8, 9, 10, 11], 'obj': 5754.0, 'problem_type': 'QKP'}","[0, 1, 2, 5, 6, 7, 8, 9, 10, 11]",5754.0,"{'problem_type': 'QKP', 'num_items': 12, 'capacity': 2230, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 'linear_pairs': [{'item_id': 1, 'linear_profit': 142}, {'item_id': 2, 'linear_profit': 317}, {'item_id': 3, 'linear_profit': 156}, {'item_id': 4, 'linear_profit': 279}, {'item_id': 5, 'linear_profit': 231}, {'item_id': 6, 'linear_profit': 277}, {'item_id': 7, 'linear_profit': 288}, {'item_id': 8, 'linear_profit': 199}, {'item_id': 9, 'linear_profit': 176}, {'item_id': 10, 'linear_profit': 231}, {'item_id': 11, 'linear_profit': 250}, {'item_id': 12, 'linear_profit': 329}], 'weight_pairs': [{'item_id': 1, 'weight': 112}, {'item_id': 2, 'weight': 287}, {'item_id': 3, 'weight': 126}, {'item_id': 4, 'weight': 249}, {'item_id': 5, 'weight': 201}, {'item_id': 6, 'weight': 247}, {'item_id': 7, 'weight': 258}, {'item_id': 8, 'weight': 169}, {'item_id': 9, 'weight': 146}, {'item_id': 10, 'weight': 201}, {'item_id': 11, 'weight': 220}, {'item_id': 12, 'weight': 299}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 284}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 168}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 196}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 127}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 202}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 234}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 161}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 203}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 240}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 231}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 189}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 181}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 171}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 193}, {'item_i_id': 6, 'item_j_id': 12, 'quadratic_profit': 128}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 228}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 238}, {'item_i_id': 8, 'item_j_id': 12, 'quadratic_profit': 270}, {'item_i_id': 9, 'item_j_id': 12, 'quadratic_profit': 279}, {'item_i_id': 10, 'item_j_id': 11, 'quadratic_profit': 156}, {'item_i_id': 11, 'item_j_id': 12, 'quadratic_profit': 124}]}","[1, 2, 3, 6, 7, 8, 9, 10, 11, 12]",34,nl,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 13 materials: ids 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, backpack weight limit 406.
We get a base study value of 100 from 0.
We get a base study value of 101 from 1.
We get a base study value of 105 from 2.
We get a base study value of 103 from 3.
We get a base study value of 100 from 4.
We get a base study value of 102 from 5.
We get a base study value of 102 from 6.
We get a base study value of 105 from 7.
We get a base study value of 105 from 8.
We get a base study value of 105 from 9.
We get a base study value of 105 from 10.
We get a base study value of 104 from 11.
We get a base study value of 101 from 12.
We note 0 weighs 101.
We note 1 weighs 105.
We note 2 weighs 103.
We note 3 weighs 101.
We note 4 weighs 103.
We note 5 weighs 103.
We note 6 weighs 103.
We note 7 weighs 104.
We note 8 weighs 100.
We note 9 weighs 104.
We note 10 weighs 102.
We note 11 weighs 101.
We note 12 weighs 105.
If we pack 0 with 2, we gain an extra 101 study bonus.
If we pack 0 with 3, we gain an extra 101 study bonus.
If we pack 0 with 4, we gain an extra 100 study bonus.
If we pack 0 with 6, we gain an extra 100 study bonus.
If we pack 0 with 7, we gain an extra 101 study bonus.
If we pack 0 with 8, we gain an extra 100 study bonus.
If we pack 0 with 9, we gain an extra 102 study bonus.
If we pack 0 with 10, we gain an extra 100 study bonus.
If we pack 0 with 12, we gain an extra 102 study bonus.
If we pack 1 with 2, we gain an extra 103 study bonus.
If we pack 1 with 4, we gain an extra 104 study bonus.
If we pack 1 with 5, we gain an extra 103 study bonus.
If we pack 1 with 6, we gain an extra 100 study bonus.
If we pack 1 with 7, we gain an extra 102 study bonus.
If we pack 1 with 10, we gain an extra 104 study bonus.
If we pack 1 with 11, we gain an extra 102 study bonus.
If we pack 1 with 12, we gain an extra 104 study bonus.
If we pack 2 with 3, we gain an extra 102 study bonus.
If we pack 2 with 4, we gain an extra 103 study bonus.
If we pack 2 with 5, we gain an extra 101 study bonus.
If we pack 2 with 7, we gain an extra 100 study bonus.
If we pack 2 with 9, we gain an extra 100 study bonus.
If we pack 2 with 10, we gain an extra 100 study bonus.
If we pack 2 with 11, we gain an extra 100 study bonus.
If we pack 3 with 4, we gain an extra 101 study bonus.
If we pack 3 with 5, we gain an extra 101 study bonus.
If we pack 3 with 6, we gain an extra 102 study bonus.
If we pack 3 with 7, we gain an extra 100 study bonus.
If we pack 3 with 8, we gain an extra 102 study bonus.
If we pack 3 with 9, we gain an extra 100 study bonus.
If we pack 3 with 10, we gain an extra 102 study bonus.
If we pack 3 with 11, we gain an extra 101 study bonus.
If we pack 4 with 5, we gain an extra 100 study bonus.
If we pack 4 with 6, we gain an extra 100 study bonus.
If we pack 4 with 7, we gain an extra 100 study bonus.
If we pack 4 with 8, we gain an extra 100 study bonus.
If we pack 4 with 9, we gain an extra 100 study bonus.
If we pack 4 with 11, we gain an extra 100 study bonus.
If we pack 4 with 12, we gain an extra 100 study bonus.
If we pack 5 with 6, we gain an extra 100 study bonus.
If we pack 5 with 7, we gain an extra 100 study bonus.
If we pack 5 with 8, we gain an extra 100 study bonus.
If we pack 5 with 9, we gain an extra 100 study bonus.
If we pack 5 with 10, we gain an extra 104 study bonus.
If we pack 5 with 11, we gain an extra 102 study bonus.
If we pack 5 with 12, we gain an extra 103 study bonus.
If we pack 6 with 7, we gain an extra 101 study bonus.
If we pack 6 with 8, we gain an extra 101 study bonus.
If we pack 6 with 9, we gain an extra 101 study bonus.
If we pack 6 with 10, we gain an extra 101 study bonus.
If we pack 6 with 11, we gain an extra 101 study bonus.
If we pack 6 with 12, we gain an extra 101 study bonus.
If we pack 7 with 8, we gain an extra 104 study bonus.
If we pack 7 with 9, we gain an extra 102 study bonus.
If we pack 7 with 11, we gain an extra 100 study bonus.
If we pack 7 with 12, we gain an extra 100 study bonus.
If we pack 8 with 9, we gain an extra 100 study bonus.
If we pack 8 with 10, we gain an extra 100 study bonus.
If we pack 9 with 11, we gain an extra 101 study bonus.
If we pack 9 with 12, we gain an extra 101 study bonus.
If we pack 10 with 12, we gain an extra 101 study bonus.
If we pack 11 with 12, we gain an extra 100 study bonus.
We'll pack to maximize study payoff while staying within the 406 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': [100, 101, 105, 103, 100, 102, 102, 105, 105, 105, 105, 104, 101], 'quadratic_coeffs': [[101, 0, 101, 101, 100, 0, 100, 101, 100, 102, 100, 0, 102], [0, 104, 103, 0, 104, 103, 100, 102, 0, 0, 104, 102, 104], [0, 0, 103, 102, 103, 101, 0, 100, 0, 100, 100, 100, 0], [0, 0, 0, 101, 101, 101, 102, 100, 102, 100, 102, 101, 0], [0, 0, 0, 0, 100, 100, 100, 100, 100, 100, 0, 100, 100], [0, 0, 0, 0, 0, 100, 100, 100, 100, 100, 104, 102, 103], [0, 0, 0, 0, 0, 0, 101, 101, 101, 101, 101, 101, 101], [0, 0, 0, 0, 0, 0, 0, 103, 104, 102, 0, 100, 100], [0, 0, 0, 0, 0, 0, 0, 0, 101, 100, 100, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 103, 0, 101, 101], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 103, 0, 101], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 101, 100], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [101, 105, 103, 101, 103, 103, 103, 104, 100, 104, 102, 101, 105], 'capacity': 406, 'solution': [3, 5, 8, 10], 'obj': 1024.0, 'problem_type': 'QKP'}","[3, 5, 8, 10]",1024.0,"{'problem_type': 'QKP', 'num_items': 13, 'capacity': 406, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 'linear_pairs': [{'item_id': 0, 'linear_profit': 100}, {'item_id': 1, 'linear_profit': 101}, {'item_id': 2, 'linear_profit': 105}, {'item_id': 3, 'linear_profit': 103}, {'item_id': 4, 'linear_profit': 100}, {'item_id': 5, 'linear_profit': 102}, {'item_id': 6, 'linear_profit': 102}, {'item_id': 7, 'linear_profit': 105}, {'item_id': 8, 'linear_profit': 105}, {'item_id': 9, 'linear_profit': 105}, {'item_id': 10, 'linear_profit': 105}, {'item_id': 11, 'linear_profit': 104}, {'item_id': 12, 'linear_profit': 101}], 'weight_pairs': [{'item_id': 0, 'weight': 101}, {'item_id': 1, 'weight': 105}, {'item_id': 2, 'weight': 103}, {'item_id': 3, 'weight': 101}, {'item_id': 4, 'weight': 103}, {'item_id': 5, 'weight': 103}, {'item_id': 6, 'weight': 103}, {'item_id': 7, 'weight': 104}, {'item_id': 8, 'weight': 100}, {'item_id': 9, 'weight': 104}, {'item_id': 10, 'weight': 102}, {'item_id': 11, 'weight': 101}, {'item_id': 12, 'weight': 105}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 101}, {'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 101}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 101}, {'item_i_id': 0, 'item_j_id': 4, 'quadratic_profit': 100}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 100}, {'item_i_id': 0, 'item_j_id': 7, 'quadratic_profit': 101}, {'item_i_id': 0, 'item_j_id': 8, 'quadratic_profit': 100}, {'item_i_id': 0, 'item_j_id': 9, 'quadratic_profit': 102}, {'item_i_id': 0, 'item_j_id': 10, 'quadratic_profit': 100}, {'item_i_id': 0, 'item_j_id': 12, 'quadratic_profit': 102}, {'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 104}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 103}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 104}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 103}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 102}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 104}, {'item_i_id': 1, 'item_j_id': 11, 'quadratic_profit': 102}, {'item_i_id': 1, 'item_j_id': 12, 'quadratic_profit': 104}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 103}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 102}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 103}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 101}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 100}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 100}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 100}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 100}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 102}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 100}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 102}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 100}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 102}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 101}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 100}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 100}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 100}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 100}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 100}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 100}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 100}, {'item_i_id': 4, 'item_j_id': 12, 'quadratic_profit': 100}, {'item_i_id': 5, 'item_j_id': 5, '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': 100}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 100}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 104}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 102}, {'item_i_id': 5, 'item_j_id': 12, 'quadratic_profit': 103}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 101}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 101}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 101}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 101}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 101}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 101}, {'item_i_id': 6, 'item_j_id': 12, 'quadratic_profit': 101}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 103}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 104}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 102}, {'item_i_id': 7, 'item_j_id': 11, 'quadratic_profit': 100}, {'item_i_id': 7, 'item_j_id': 12, 'quadratic_profit': 100}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 101}, {'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': 9, 'quadratic_profit': 103}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 101}, {'item_i_id': 9, 'item_j_id': 12, 'quadratic_profit': 101}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 103}, {'item_i_id': 10, 'item_j_id': 12, 'quadratic_profit': 101}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 101}, {'item_i_id': 11, 'item_j_id': 12, 'quadratic_profit': 100}]}","[3, 5, 8, 10]",35,nl,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.

# total_products=14
# shelf_capacity_weight=10074
# product_ids=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13

product_id,estimated_sales
0,105
1,450
2,1696
3,499
4,552
5,2377
6,548
7,2711
8,1887
9,1182
10,1071
11,403
12,761
13,3116

product_id,product_weight
0,2553
1,2597
2,567
3,2125
4,2383
5,808
6,1339
7,1000
8,1458
9,1671
10,1379
11,1143
12,1475
13,2222

product_i_id,product_j_id,cross_sale_bonus
0,3,304
0,11,3117
1,3,989
1,4,1866
1,7,980
2,4,1880
2,8,962
3,4,1066
3,5,233
3,11,1063
4,8,1555
4,10,2869
5,11,747
5,13,2621
6,7,3212
6,8,2564
7,9,1447
7,13,2263
8,11,2718

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': [105, 450, 1696, 499, 552, 2377, 548, 2711, 1887, 1182, 1071, 403, 761, 3116], 'quadratic_coeffs': [[2557, 0, 0, 304, 0, 0, 0, 0, 0, 0, 0, 3117, 0, 0], [0, 0, 0, 989, 1866, 0, 0, 980, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1880, 0, 0, 0, 962, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1066, 233, 0, 0, 0, 0, 0, 1063, 0, 0], [0, 0, 0, 0, 2358, 0, 0, 0, 1555, 0, 2869, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 747, 0, 2621], [0, 0, 0, 0, 0, 0, 1744, 3212, 2564, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1447, 0, 0, 0, 2263], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2718, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2007, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 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': [2553, 2597, 567, 2125, 2383, 808, 1339, 1000, 1458, 1671, 1379, 1143, 1475, 2222], 'capacity': 10074, 'solution': [2, 5, 6, 7, 8, 10, 11, 13], 'obj': 28896.0, 'problem_type': 'QKP'}","[2, 5, 6, 7, 8, 10, 11, 13]",28896.0,"{'problem_type': 'QKP', 'num_items': 14, 'capacity': 10074, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 'linear_pairs': [{'item_id': 0, 'linear_profit': 105}, {'item_id': 1, 'linear_profit': 450}, {'item_id': 2, 'linear_profit': 1696}, {'item_id': 3, 'linear_profit': 499}, {'item_id': 4, 'linear_profit': 552}, {'item_id': 5, 'linear_profit': 2377}, {'item_id': 6, 'linear_profit': 548}, {'item_id': 7, 'linear_profit': 2711}, {'item_id': 8, 'linear_profit': 1887}, {'item_id': 9, 'linear_profit': 1182}, {'item_id': 10, 'linear_profit': 1071}, {'item_id': 11, 'linear_profit': 403}, {'item_id': 12, 'linear_profit': 761}, {'item_id': 13, 'linear_profit': 3116}], 'weight_pairs': [{'item_id': 0, 'weight': 2553}, {'item_id': 1, 'weight': 2597}, {'item_id': 2, 'weight': 567}, {'item_id': 3, 'weight': 2125}, {'item_id': 4, 'weight': 2383}, {'item_id': 5, 'weight': 808}, {'item_id': 6, 'weight': 1339}, {'item_id': 7, 'weight': 1000}, {'item_id': 8, 'weight': 1458}, {'item_id': 9, 'weight': 1671}, {'item_id': 10, 'weight': 1379}, {'item_id': 11, 'weight': 1143}, {'item_id': 12, 'weight': 1475}, {'item_id': 13, 'weight': 2222}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 2557}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 304}, {'item_i_id': 0, 'item_j_id': 11, 'quadratic_profit': 3117}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 989}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 1866}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 980}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 1880}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 962}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 1066}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 233}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 1063}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 2358}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 1555}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 2869}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 747}, {'item_i_id': 5, 'item_j_id': 13, 'quadratic_profit': 2621}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 1744}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 3212}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 2564}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 1447}, {'item_i_id': 7, 'item_j_id': 13, 'quadratic_profit': 2263}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 2718}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 2007}]}","[2, 5, 6, 7, 8, 10, 11, 13]",36,csv,0
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 13 crates (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12) and the truck's weight capacity 2149.

| crate_id | base_life_saving_value |
|---|---|
| 0 | 190 |
| 1 | 174 |
| 2 | 312 |
| 3 | 139 |
| 4 | 261 |
| 5 | 123 |
| 6 | 308 |
| 7 | 111 |
| 8 | 105 |
| 9 | 183 |
| 10 | 120 |
| 11 | 113 |
| 12 | 179 |

| crate_id | crate_weight |
|---|---|
| 0 | 193 |
| 1 | 171 |
| 2 | 298 |
| 3 | 143 |
| 4 | 247 |
| 5 | 111 |
| 6 | 299 |
| 7 | 121 |
| 8 | 109 |
| 9 | 177 |
| 10 | 107 |
| 11 | 117 |
| 12 | 166 |

| crate_a_id | crate_b_id | pair_synergy_bonus |
|---|---|---|
| 0 | 2 | 138 |
| 0 | 3 | 193 |
| 0 | 5 | 193 |
| 0 | 6 | 193 |
| 0 | 8 | 256 |
| 1 | 2 | 171 |
| 1 | 3 | 171 |
| 1 | 4 | 171 |
| 1 | 10 | 227 |
| 2 | 4 | 108 |
| 2 | 5 | 121 |
| 2 | 8 | 143 |
| 2 | 9 | 143 |
| 2 | 11 | 143 |
| 3 | 4 | 121 |
| 3 | 7 | 143 |
| 3 | 9 | 204 |
| 4 | 6 | 236 |
| 4 | 11 | 111 |
| 4 | 12 | 111 |
| 5 | 7 | 111 |
| 5 | 9 | 203 |
| 6 | 7 | 177 |
| 6 | 8 | 142 |
| 6 | 9 | 142 |
| 6 | 10 | 142 |
| 6 | 11 | 142 |
| 7 | 9 | 142 |
| 7 | 10 | 231 |
| 7 | 12 | 150 |

The relief coordinator will select crates so total weight stays within 2149 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': [190, 174, 312, 139, 261, 123, 308, 111, 105, 183, 120, 113, 179], 'quadratic_coeffs': [[0, 0, 138, 193, 0, 193, 193, 0, 256, 0, 0, 0, 0], [0, 0, 171, 171, 171, 0, 0, 0, 0, 0, 227, 0, 0], [0, 0, 165, 0, 108, 121, 0, 0, 143, 143, 0, 143, 0], [0, 0, 0, 0, 121, 0, 0, 143, 0, 204, 0, 0, 0], [0, 0, 0, 0, 223, 0, 236, 0, 0, 0, 0, 111, 111], [0, 0, 0, 0, 0, 111, 0, 111, 0, 203, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 177, 142, 142, 142, 142, 0], [0, 0, 0, 0, 0, 0, 0, 193, 0, 142, 231, 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], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 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': [193, 171, 298, 143, 247, 111, 299, 121, 109, 177, 107, 117, 166], 'capacity': 2149, 'solution': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 'obj': 6757.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]",6757.0,"{'problem_type': 'QKP', 'num_items': 13, 'capacity': 2149, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 'linear_pairs': [{'item_id': 0, 'linear_profit': 190}, {'item_id': 1, 'linear_profit': 174}, {'item_id': 2, 'linear_profit': 312}, {'item_id': 3, 'linear_profit': 139}, {'item_id': 4, 'linear_profit': 261}, {'item_id': 5, 'linear_profit': 123}, {'item_id': 6, 'linear_profit': 308}, {'item_id': 7, 'linear_profit': 111}, {'item_id': 8, 'linear_profit': 105}, {'item_id': 9, 'linear_profit': 183}, {'item_id': 10, 'linear_profit': 120}, {'item_id': 11, 'linear_profit': 113}, {'item_id': 12, 'linear_profit': 179}], 'weight_pairs': [{'item_id': 0, 'weight': 193}, {'item_id': 1, 'weight': 171}, {'item_id': 2, 'weight': 298}, {'item_id': 3, 'weight': 143}, {'item_id': 4, 'weight': 247}, {'item_id': 5, 'weight': 111}, {'item_id': 6, 'weight': 299}, {'item_id': 7, 'weight': 121}, {'item_id': 8, 'weight': 109}, {'item_id': 9, 'weight': 177}, {'item_id': 10, 'weight': 107}, {'item_id': 11, 'weight': 117}, {'item_id': 12, 'weight': 166}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 138}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 193}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 193}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 193}, {'item_i_id': 0, 'item_j_id': 8, 'quadratic_profit': 256}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 171}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 171}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 171}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 227}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 165}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 108}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 121}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 143}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 143}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 143}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 121}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 143}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 204}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 223}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 236}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 111}, {'item_i_id': 4, 'item_j_id': 12, 'quadratic_profit': 111}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 111}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 111}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 203}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 177}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 142}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 142}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 142}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 142}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 193}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 142}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 231}, {'item_i_id': 7, 'item_j_id': 12, 'quadratic_profit': 150}]}","[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]",37,markdown_table,0
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 13 pieces (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13) and the truck's load limit 684.
Piece 1 brings a rental of 125.
Piece 2 brings a rental of 136.
Piece 3 brings a rental of 135.
Piece 4 brings a rental of 123.
Piece 5 brings a rental of 119.
Piece 6 brings a rental of 113.
Piece 7 brings a rental of 116.
Piece 8 brings a rental of 132.
Piece 9 brings a rental of 125.
Piece 10 brings a rental of 119.
Piece 11 brings a rental of 130.
Piece 12 brings a rental of 117.
Piece 13 brings a rental of 134.
Piece 1 occupies 113 capacity on the truck.
Piece 2 occupies 124 capacity on the truck.
Piece 3 occupies 123 capacity on the truck.
Piece 4 occupies 111 capacity on the truck.
Piece 5 occupies 107 capacity on the truck.
Piece 6 occupies 101 capacity on the truck.
Piece 7 occupies 104 capacity on the truck.
Piece 8 occupies 120 capacity on the truck.
Piece 9 occupies 113 capacity on the truck.
Piece 10 occupies 107 capacity on the truck.
Piece 11 occupies 118 capacity on the truck.
Piece 12 occupies 105 capacity on the truck.
Piece 13 occupies 122 capacity on the truck.
Pieces 1 and 2 together add an extra 113 payoff.
Pieces 1 and 3 together add an extra 110 payoff.
Pieces 1 and 4 together add an extra 111 payoff.
Pieces 1 and 6 together add an extra 115 payoff.
Pieces 1 and 8 together add an extra 112 payoff.
Pieces 1 and 9 together add an extra 107 payoff.
Pieces 1 and 12 together add an extra 109 payoff.
Pieces 2 and 3 together add an extra 120 payoff.
Pieces 2 and 4 together add an extra 115 payoff.
Pieces 2 and 5 together add an extra 114 payoff.
Pieces 2 and 6 together add an extra 117 payoff.
Pieces 2 and 7 together add an extra 111 payoff.
Pieces 2 and 8 together add an extra 113 payoff.
Pieces 2 and 12 together add an extra 104 payoff.
Pieces 2 and 13 together add an extra 101 payoff.
Pieces 3 and 4 together add an extra 120 payoff.
Pieces 3 and 5 together add an extra 114 payoff.
Pieces 3 and 6 together add an extra 113 payoff.
Pieces 3 and 7 together add an extra 116 payoff.
Pieces 3 and 8 together add an extra 120 payoff.
Pieces 3 and 9 together add an extra 123 payoff.
Pieces 3 and 10 together add an extra 120 payoff.
Pieces 3 and 13 together add an extra 115 payoff.
Pieces 4 and 5 together add an extra 115 payoff.
Pieces 4 and 7 together add an extra 115 payoff.
Pieces 4 and 8 together add an extra 113 payoff.
Pieces 4 and 9 together add an extra 108 payoff.
Pieces 4 and 11 together add an extra 117 payoff.
Pieces 4 and 12 together add an extra 101 payoff.
Pieces 4 and 13 together add an extra 107 payoff.
Pieces 5 and 6 together add an extra 111 payoff.
Pieces 5 and 8 together add an extra 111 payoff.
Pieces 5 and 9 together add an extra 106 payoff.
Pieces 5 and 12 together add an extra 103 payoff.
Pieces 5 and 13 together add an extra 111 payoff.
Pieces 6 and 7 together add an extra 101 payoff.
Pieces 6 and 8 together add an extra 104 payoff.
Pieces 6 and 9 together add an extra 101 payoff.
Pieces 6 and 11 together add an extra 109 payoff.
Pieces 6 and 13 together add an extra 107 payoff.
Pieces 7 and 8 together add an extra 104 payoff.
Pieces 7 and 9 together add an extra 112 payoff.
Pieces 7 and 10 together add an extra 110 payoff.
Pieces 7 and 11 together add an extra 106 payoff.
Pieces 7 and 12 together add an extra 117 payoff.
Pieces 7 and 13 together add an extra 119 payoff.
Pieces 8 and 9 together add an extra 114 payoff.
Pieces 8 and 10 together add an extra 113 payoff.
Pieces 8 and 11 together add an extra 107 payoff.
Pieces 8 and 12 together add an extra 114 payoff.
Pieces 8 and 13 together add an extra 117 payoff.
Pieces 9 and 11 together add an extra 112 payoff.
Pieces 9 and 12 together add an extra 119 payoff.
Pieces 9 and 13 together add an extra 114 payoff.
Pieces 10 and 12 together add an extra 102 payoff.
Pieces 11 and 12 together add an extra 116 payoff.
Pieces 11 and 13 together add an extra 113 payoff.
The stage manager will choose pieces to maximize total rental while staying within the 684 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': [125, 136, 135, 123, 119, 113, 116, 132, 125, 119, 130, 117, 134], 'quadratic_coeffs': [[110, 113, 110, 111, 0, 115, 0, 112, 107, 0, 0, 109, 0], [0, 117, 120, 115, 114, 117, 111, 113, 0, 0, 0, 104, 101], [0, 0, 117, 120, 114, 113, 116, 120, 123, 120, 0, 0, 115], [0, 0, 0, 111, 115, 0, 115, 113, 108, 0, 117, 101, 107], [0, 0, 0, 0, 109, 111, 0, 111, 106, 0, 0, 103, 111], [0, 0, 0, 0, 0, 101, 101, 104, 101, 0, 109, 0, 107], [0, 0, 0, 0, 0, 0, 0, 104, 112, 110, 106, 117, 119], [0, 0, 0, 0, 0, 0, 0, 0, 114, 113, 107, 114, 117], [0, 0, 0, 0, 0, 0, 0, 0, 111, 0, 112, 119, 114], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 102, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 113, 116, 113], [0, 0, 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': [113, 124, 123, 111, 107, 101, 104, 120, 113, 107, 118, 105, 122], 'capacity': 684, 'solution': [2, 5, 6, 7, 8, 12], 'obj': 2435.0, 'problem_type': 'QKP'}","[2, 5, 6, 7, 8, 12]",2435.0,"{'problem_type': 'QKP', 'num_items': 13, 'capacity': 684, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 'linear_pairs': [{'item_id': 1, 'linear_profit': 125}, {'item_id': 2, 'linear_profit': 136}, {'item_id': 3, 'linear_profit': 135}, {'item_id': 4, 'linear_profit': 123}, {'item_id': 5, 'linear_profit': 119}, {'item_id': 6, 'linear_profit': 113}, {'item_id': 7, 'linear_profit': 116}, {'item_id': 8, 'linear_profit': 132}, {'item_id': 9, 'linear_profit': 125}, {'item_id': 10, 'linear_profit': 119}, {'item_id': 11, 'linear_profit': 130}, {'item_id': 12, 'linear_profit': 117}, {'item_id': 13, 'linear_profit': 134}], 'weight_pairs': [{'item_id': 1, 'weight': 113}, {'item_id': 2, 'weight': 124}, {'item_id': 3, 'weight': 123}, {'item_id': 4, 'weight': 111}, {'item_id': 5, 'weight': 107}, {'item_id': 6, 'weight': 101}, {'item_id': 7, 'weight': 104}, {'item_id': 8, 'weight': 120}, {'item_id': 9, 'weight': 113}, {'item_id': 10, 'weight': 107}, {'item_id': 11, 'weight': 118}, {'item_id': 12, 'weight': 105}, {'item_id': 13, 'weight': 122}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 110}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 113}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 110}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 111}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 115}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 112}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 107}, {'item_i_id': 1, 'item_j_id': 12, 'quadratic_profit': 109}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 117}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 120}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 115}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 114}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 117}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 111}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 113}, {'item_i_id': 2, 'item_j_id': 12, 'quadratic_profit': 104}, {'item_i_id': 2, 'item_j_id': 13, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 117}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 120}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 114}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 113}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 116}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 120}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 123}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 120}, {'item_i_id': 3, 'item_j_id': 13, 'quadratic_profit': 115}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 111}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 115}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 115}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 113}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 108}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 117}, {'item_i_id': 4, 'item_j_id': 12, 'quadratic_profit': 101}, {'item_i_id': 4, 'item_j_id': 13, 'quadratic_profit': 107}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 109}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 111}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 111}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 106}, {'item_i_id': 5, 'item_j_id': 12, 'quadratic_profit': 103}, {'item_i_id': 5, 'item_j_id': 13, 'quadratic_profit': 111}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 101}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 101}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 104}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 101}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 109}, {'item_i_id': 6, 'item_j_id': 13, 'quadratic_profit': 107}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 104}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 112}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 110}, {'item_i_id': 7, 'item_j_id': 11, 'quadratic_profit': 106}, {'item_i_id': 7, 'item_j_id': 12, 'quadratic_profit': 117}, {'item_i_id': 7, 'item_j_id': 13, 'quadratic_profit': 119}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 114}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 113}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 107}, {'item_i_id': 8, 'item_j_id': 12, 'quadratic_profit': 114}, {'item_i_id': 8, 'item_j_id': 13, 'quadratic_profit': 117}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 111}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 112}, {'item_i_id': 9, 'item_j_id': 12, 'quadratic_profit': 119}, {'item_i_id': 9, 'item_j_id': 13, 'quadratic_profit': 114}, {'item_i_id': 10, 'item_j_id': 12, 'quadratic_profit': 102}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 113}, {'item_i_id': 11, 'item_j_id': 12, 'quadratic_profit': 116}, {'item_i_id': 11, 'item_j_id': 13, 'quadratic_profit': 113}]}","[3, 6, 7, 8, 9, 13]",38,nl,1
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.

Below are the 11 artworks (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) with their space and value figures; we must select pieces that together do not exceed the crate capacity of 550.
Artwork 1 has an expected sale value of 114.
Artwork 2 has an expected sale value of 121.
Artwork 3 has an expected sale value of 124.
Artwork 4 has an expected sale value of 104.
Artwork 5 has an expected sale value of 106.
Artwork 6 has an expected sale value of 114.
Artwork 7 has an expected sale value of 124.
Artwork 8 has an expected sale value of 132.
Artwork 9 has an expected sale value of 124.
Artwork 10 has an expected sale value of 141.
Artwork 11 has an expected sale value of 145.
Artwork 1 occupies 119 units of crate space.
Artwork 2 occupies 121 units of crate space.
Artwork 3 occupies 130 units of crate space.
Artwork 4 occupies 106 units of crate space.
Artwork 5 occupies 102 units of crate space.
Artwork 6 occupies 119 units of crate space.
Artwork 7 occupies 128 units of crate space.
Artwork 8 occupies 136 units of crate space.
Artwork 9 occupies 122 units of crate space.
Artwork 10 occupies 144 units of crate space.
Artwork 11 occupies 137 units of crate space.
Including both 1 and 4 yields an extra synergy value of 124.
Including both 1 and 5 yields an extra synergy value of 116.
Including both 1 and 6 yields an extra synergy value of 130.
Including both 1 and 7 yields an extra synergy value of 128.
Including both 1 and 8 yields an extra synergy value of 113.
Including both 1 and 9 yields an extra synergy value of 131.
Including both 1 and 11 yields an extra synergy value of 114.
Including both 2 and 4 yields an extra synergy value of 124.
Including both 2 and 5 yields an extra synergy value of 116.
Including both 2 and 6 yields an extra synergy value of 119.
Including both 2 and 7 yields an extra synergy value of 121.
Including both 2 and 10 yields an extra synergy value of 112.
Including both 2 and 11 yields an extra synergy value of 113.
Including both 3 and 4 yields an extra synergy value of 124.
Including both 3 and 6 yields an extra synergy value of 116.
Including both 3 and 7 yields an extra synergy value of 115.
Including both 3 and 8 yields an extra synergy value of 123.
Including both 3 and 10 yields an extra synergy value of 116.
Including both 3 and 11 yields an extra synergy value of 128.
Including both 4 and 5 yields an extra synergy value of 120.
Including both 4 and 6 yields an extra synergy value of 113.
Including both 4 and 7 yields an extra synergy value of 120.
Including both 4 and 9 yields an extra synergy value of 120.
Including both 4 and 10 yields an extra synergy value of 112.
Including both 4 and 11 yields an extra synergy value of 123.
Including both 5 and 6 yields an extra synergy value of 122.
Including both 5 and 8 yields an extra synergy value of 121.
Including both 5 and 9 yields an extra synergy value of 113.
Including both 5 and 10 yields an extra synergy value of 103.
Including both 5 and 11 yields an extra synergy value of 127.
Including both 6 and 7 yields an extra synergy value of 111.
Including both 6 and 8 yields an extra synergy value of 131.
Including both 6 and 10 yields an extra synergy value of 124.
Including both 6 and 11 yields an extra synergy value of 120.
Including both 7 and 8 yields an extra synergy value of 119.
Including both 7 and 10 yields an extra synergy value of 140.
Including both 8 and 9 yields an extra synergy value of 133.
Including both 8 and 10 yields an extra synergy value of 124.
Including both 8 and 11 yields an extra synergy value of 117.
Including both 9 and 10 yields an extra synergy value of 122.
Including both 9 and 11 yields an extra synergy value of 131.
Including both 10 and 11 yields an extra synergy value of 124.
We’ll use these figures to choose the combination that fits within 550 and maximizes expected return.

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': [114, 121, 124, 104, 106, 114, 124, 132, 124, 141, 145], 'quadratic_coeffs': [[126, 0, 0, 124, 116, 130, 128, 113, 131, 0, 114], [0, 119, 0, 124, 116, 119, 121, 0, 0, 112, 113], [0, 0, 129, 124, 0, 116, 115, 123, 0, 116, 128], [0, 0, 0, 125, 120, 113, 120, 0, 120, 112, 123], [0, 0, 0, 0, 119, 122, 0, 121, 113, 103, 127], [0, 0, 0, 0, 0, 114, 111, 131, 0, 124, 120], [0, 0, 0, 0, 0, 0, 125, 119, 0, 140, 0], [0, 0, 0, 0, 0, 0, 0, 0, 133, 124, 117], [0, 0, 0, 0, 0, 0, 0, 0, 109, 122, 131], [0, 0, 0, 0, 0, 0, 0, 0, 0, 112, 124], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 140]], 'weights': [119, 121, 130, 106, 102, 119, 128, 136, 122, 144, 137], 'capacity': 550, 'solution': [7, 8, 9, 10], 'obj': 1293.0, 'problem_type': 'QKP'}","[7, 8, 9, 10]",1293.0,"{'problem_type': 'QKP', 'num_items': 11, 'capacity': 550, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 'linear_pairs': [{'item_id': 1, 'linear_profit': 114}, {'item_id': 2, 'linear_profit': 121}, {'item_id': 3, 'linear_profit': 124}, {'item_id': 4, 'linear_profit': 104}, {'item_id': 5, 'linear_profit': 106}, {'item_id': 6, 'linear_profit': 114}, {'item_id': 7, 'linear_profit': 124}, {'item_id': 8, 'linear_profit': 132}, {'item_id': 9, 'linear_profit': 124}, {'item_id': 10, 'linear_profit': 141}, {'item_id': 11, 'linear_profit': 145}], 'weight_pairs': [{'item_id': 1, 'weight': 119}, {'item_id': 2, 'weight': 121}, {'item_id': 3, 'weight': 130}, {'item_id': 4, 'weight': 106}, {'item_id': 5, 'weight': 102}, {'item_id': 6, 'weight': 119}, {'item_id': 7, 'weight': 128}, {'item_id': 8, 'weight': 136}, {'item_id': 9, 'weight': 122}, {'item_id': 10, 'weight': 144}, {'item_id': 11, 'weight': 137}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 126}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 124}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 116}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 130}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 128}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 113}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 131}, {'item_i_id': 1, 'item_j_id': 11, 'quadratic_profit': 114}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 119}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 124}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 116}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 119}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 121}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 112}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 113}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 129}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 124}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 116}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 115}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 123}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 116}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 128}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 125}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 120}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 113}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 120}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 120}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 112}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 123}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 119}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 122}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 121}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 113}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 103}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 127}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 114}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 111}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 131}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 124}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 120}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 125}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 119}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 140}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 133}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 124}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 117}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 109}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 122}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 131}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 112}, {'item_i_id': 10, 'item_j_id': 11, 'quadratic_profit': 124}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 140}]}","[8, 9, 10, 11]",39,nl,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.

There are 15 donated books (A, B, C, D, E, F, G, H, I, J, K, L, M, N, O) and a shelf capacity of 12086.

| book_id | book_benefit |
|---|---|
| A | 843 |
| B | 1009 |
| C | 1731 |
| D | 1147 |
| E | 324 |
| F | 1217 |
| G | 1849 |
| H | 1166 |
| I | 1136 |
| J | 1865 |
| K | 1196 |
| L | 290 |
| M | 506 |
| N | 652 |
| O | 1002 |

| book_id | book_size |
|---|---|
| A | 673 |
| B | 839 |
| C | 1561 |
| D | 977 |
| E | 154 |
| F | 1047 |
| G | 1679 |
| H | 996 |
| I | 966 |
| J | 1695 |
| K | 1026 |
| L | 120 |
| M | 336 |
| N | 482 |
| O | 832 |

| book_i_id | book_j_id | pair_bonus |
|---|---|---|
| A | B | 914 |
| A | C | 695 |
| A | F | 1062 |
| A | J | 711 |
| A | K | 284 |
| A | L | 664 |
| A | N | 581 |
| B | C | 975 |
| B | F | 365 |
| B | G | 758 |
| B | H | 741 |
| B | K | 1012 |
| B | L | 814 |
| B | M | 1144 |
| C | D | 707 |
| C | E | 1029 |
| C | F | 613 |
| C | M | 666 |
| C | N | 1333 |
| C | O | 1563 |
| D | G | 1102 |
| D | L | 801 |
| D | M | 695 |
| D | N | 418 |
| E | F | 364 |
| E | K | 372 |
| E | N | 719 |
| E | O | 1012 |
| F | G | 1261 |
| F | H | 1190 |
| F | I | 1141 |
| F | L | 665 |
| F | M | 1131 |
| F | O | 380 |
| G | I | 977 |
| G | J | 1420 |
| G | L | 1143 |
| G | M | 1663 |
| G | N | 1292 |
| G | O | 502 |
| H | J | 707 |
| H | K | 1064 |
| H | L | 860 |
| H | N | 1183 |
| I | J | 931 |
| I | L | 790 |
| I | N | 776 |
| J | M | 397 |
| J | N | 961 |
| J | O | 540 |
| K | L | 897 |
| K | N | 1155 |
| K | O | 995 |
| L | N | 1553 |
| M | N | 777 |
| M | O | 610 |
| N | O | 1219 |

Selections must fit within the 12086 shelf limit and are chosen to maximize total payoff across the 15 donated books listed above.

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': [843, 1009, 1731, 1147, 324, 1217, 1849, 1166, 1136, 1865, 1196, 290, 506, 652, 1002], 'quadratic_coeffs': [[0, 914, 695, 0, 0, 1062, 0, 0, 0, 711, 284, 664, 0, 581, 0], [0, 0, 975, 0, 0, 365, 758, 741, 0, 0, 1012, 814, 1144, 0, 0], [0, 0, 1529, 707, 1029, 613, 0, 0, 0, 0, 0, 0, 666, 1333, 1563], [0, 0, 0, 1096, 0, 0, 1102, 0, 0, 0, 0, 801, 695, 418, 0], [0, 0, 0, 0, 0, 364, 0, 0, 0, 0, 372, 0, 0, 719, 1012], [0, 0, 0, 0, 0, 941, 1261, 1190, 1141, 0, 0, 665, 1131, 0, 380], [0, 0, 0, 0, 0, 0, 0, 0, 977, 1420, 0, 1143, 1663, 1292, 502], [0, 0, 0, 0, 0, 0, 0, 744, 0, 707, 1064, 860, 0, 1183, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1298, 931, 0, 790, 0, 776, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 397, 961, 540], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 897, 0, 1155, 995], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 930, 0, 1553, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1029, 777, 610], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1219], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1051]], 'weights': [673, 839, 1561, 977, 154, 1047, 1679, 996, 966, 1695, 1026, 120, 336, 482, 832], 'capacity': 12086, 'solution': [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14], 'obj': 58695.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14]",58695.0,"{'problem_type': 'QKP', 'num_items': 15, 'capacity': 12086, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 843}, {'item_id': 'B', 'linear_profit': 1009}, {'item_id': 'C', 'linear_profit': 1731}, {'item_id': 'D', 'linear_profit': 1147}, {'item_id': 'E', 'linear_profit': 324}, {'item_id': 'F', 'linear_profit': 1217}, {'item_id': 'G', 'linear_profit': 1849}, {'item_id': 'H', 'linear_profit': 1166}, {'item_id': 'I', 'linear_profit': 1136}, {'item_id': 'J', 'linear_profit': 1865}, {'item_id': 'K', 'linear_profit': 1196}, {'item_id': 'L', 'linear_profit': 290}, {'item_id': 'M', 'linear_profit': 506}, {'item_id': 'N', 'linear_profit': 652}, {'item_id': 'O', 'linear_profit': 1002}], 'weight_pairs': [{'item_id': 'A', 'weight': 673}, {'item_id': 'B', 'weight': 839}, {'item_id': 'C', 'weight': 1561}, {'item_id': 'D', 'weight': 977}, {'item_id': 'E', 'weight': 154}, {'item_id': 'F', 'weight': 1047}, {'item_id': 'G', 'weight': 1679}, {'item_id': 'H', 'weight': 996}, {'item_id': 'I', 'weight': 966}, {'item_id': 'J', 'weight': 1695}, {'item_id': 'K', 'weight': 1026}, {'item_id': 'L', 'weight': 120}, {'item_id': 'M', 'weight': 336}, {'item_id': 'N', 'weight': 482}, {'item_id': 'O', 'weight': 832}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 914}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 695}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 1062}, {'item_i_id': 'A', 'item_j_id': 'J', 'quadratic_profit': 711}, {'item_i_id': 'A', 'item_j_id': 'K', 'quadratic_profit': 284}, {'item_i_id': 'A', 'item_j_id': 'L', 'quadratic_profit': 664}, {'item_i_id': 'A', 'item_j_id': 'N', 'quadratic_profit': 581}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 975}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 365}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 758}, {'item_i_id': 'B', 'item_j_id': 'H', 'quadratic_profit': 741}, {'item_i_id': 'B', 'item_j_id': 'K', 'quadratic_profit': 1012}, {'item_i_id': 'B', 'item_j_id': 'L', 'quadratic_profit': 814}, {'item_i_id': 'B', 'item_j_id': 'M', 'quadratic_profit': 1144}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 1529}, {'item_i_id': 'C', 'item_j_id': 'D', 'quadratic_profit': 707}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 1029}, {'item_i_id': 'C', 'item_j_id': 'F', 'quadratic_profit': 613}, {'item_i_id': 'C', 'item_j_id': 'M', 'quadratic_profit': 666}, {'item_i_id': 'C', 'item_j_id': 'N', 'quadratic_profit': 1333}, {'item_i_id': 'C', 'item_j_id': 'O', 'quadratic_profit': 1563}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 1096}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 1102}, {'item_i_id': 'D', 'item_j_id': 'L', 'quadratic_profit': 801}, {'item_i_id': 'D', 'item_j_id': 'M', 'quadratic_profit': 695}, {'item_i_id': 'D', 'item_j_id': 'N', 'quadratic_profit': 418}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 364}, {'item_i_id': 'E', 'item_j_id': 'K', 'quadratic_profit': 372}, {'item_i_id': 'E', 'item_j_id': 'N', 'quadratic_profit': 719}, {'item_i_id': 'E', 'item_j_id': 'O', 'quadratic_profit': 1012}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 941}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 1261}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 1190}, {'item_i_id': 'F', 'item_j_id': 'I', 'quadratic_profit': 1141}, {'item_i_id': 'F', 'item_j_id': 'L', 'quadratic_profit': 665}, {'item_i_id': 'F', 'item_j_id': 'M', 'quadratic_profit': 1131}, {'item_i_id': 'F', 'item_j_id': 'O', 'quadratic_profit': 380}, {'item_i_id': 'G', 'item_j_id': 'I', 'quadratic_profit': 977}, {'item_i_id': 'G', 'item_j_id': 'J', 'quadratic_profit': 1420}, {'item_i_id': 'G', 'item_j_id': 'L', 'quadratic_profit': 1143}, {'item_i_id': 'G', 'item_j_id': 'M', 'quadratic_profit': 1663}, {'item_i_id': 'G', 'item_j_id': 'N', 'quadratic_profit': 1292}, {'item_i_id': 'G', 'item_j_id': 'O', 'quadratic_profit': 502}, {'item_i_id': 'H', 'item_j_id': 'H', 'quadratic_profit': 744}, {'item_i_id': 'H', 'item_j_id': 'J', 'quadratic_profit': 707}, {'item_i_id': 'H', 'item_j_id': 'K', 'quadratic_profit': 1064}, {'item_i_id': 'H', 'item_j_id': 'L', 'quadratic_profit': 860}, {'item_i_id': 'H', 'item_j_id': 'N', 'quadratic_profit': 1183}, {'item_i_id': 'I', 'item_j_id': 'I', 'quadratic_profit': 1298}, {'item_i_id': 'I', 'item_j_id': 'J', 'quadratic_profit': 931}, {'item_i_id': 'I', 'item_j_id': 'L', 'quadratic_profit': 790}, {'item_i_id': 'I', 'item_j_id': 'N', 'quadratic_profit': 776}, {'item_i_id': 'J', 'item_j_id': 'M', 'quadratic_profit': 397}, {'item_i_id': 'J', 'item_j_id': 'N', 'quadratic_profit': 961}, {'item_i_id': 'J', 'item_j_id': 'O', 'quadratic_profit': 540}, {'item_i_id': 'K', 'item_j_id': 'L', 'quadratic_profit': 897}, {'item_i_id': 'K', 'item_j_id': 'N', 'quadratic_profit': 1155}, {'item_i_id': 'K', 'item_j_id': 'O', 'quadratic_profit': 995}, {'item_i_id': 'L', 'item_j_id': 'L', 'quadratic_profit': 930}, {'item_i_id': 'L', 'item_j_id': 'N', 'quadratic_profit': 1553}, {'item_i_id': 'M', 'item_j_id': 'M', 'quadratic_profit': 1029}, {'item_i_id': 'M', 'item_j_id': 'N', 'quadratic_profit': 777}, {'item_i_id': 'M', 'item_j_id': 'O', 'quadratic_profit': 610}, {'item_i_id': 'N', 'item_j_id': 'O', 'quadratic_profit': 1219}, {'item_i_id': 'O', 'item_j_id': 'O', 'quadratic_profit': 1051}]}","['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'K', 'L', 'M', 'N', 'O']",40,markdown_table,names
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=13
# flatbed_max_load=5459
# plant_ids=A, B, C, D, E, F, G, H, I, J, K, L, M

plant_id,base_margin
A,1057
B,3010
C,1008
D,1248
E,1099
F,2074
G,1719
H,2421
I,513
J,1268
K,1947
L,1809
M,3105

plant_id,pot_weight
A,1057
B,3077
C,848
D,1368
E,982
F,1974
G,1777
H,2469
I,417
J,1306
K,2069
L,1728
M,2994

plant_i_id,plant_j_id,pair_bonus_margin
A,B,848
A,E,1057
A,F,101
A,H,1057
A,K,1057
A,M,437
B,D,2848
B,E,1496
B,F,2278
B,G,1929
B,H,2037
B,L,1757
B,M,1757
C,F,848
C,G,799
C,I,848
C,L,1297
D,G,1368
D,J,444
D,L,444
E,F,982
E,J,180
E,K,557
E,L,779
F,J,454
F,M,454
G,I,101
G,J,101
G,L,101
H,I,121
H,J,963
H,K,2516
H,M,1467
I,M,1411
J,L,2725
J,M,2446

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': [1057, 3010, 1008, 1248, 1099, 2074, 1719, 2421, 513, 1268, 1947, 1809, 3105], 'quadratic_coeffs': [[0, 848, 0, 0, 1057, 101, 0, 1057, 0, 0, 1057, 0, 437], [0, 0, 0, 2848, 1496, 2278, 1929, 2037, 0, 0, 0, 1757, 1757], [0, 0, 0, 0, 0, 848, 799, 0, 848, 0, 0, 1297, 0], [0, 0, 0, 1368, 0, 0, 1368, 0, 0, 444, 0, 444, 0], [0, 0, 0, 0, 982, 982, 0, 0, 0, 180, 557, 779, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 454, 0, 0, 454], [0, 0, 0, 0, 0, 0, 0, 0, 101, 101, 0, 101, 0], [0, 0, 0, 0, 0, 0, 0, 0, 121, 963, 2516, 0, 1467], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1411], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2725, 2446], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 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': [1057, 3077, 848, 1368, 982, 1974, 1777, 2469, 417, 1306, 2069, 1728, 2994], 'capacity': 5459, 'solution': [2, 4, 8, 9, 11], 'obj': 11526.0, 'problem_type': 'QKP'}","[2, 4, 8, 9, 11]",11526.0,"{'problem_type': 'QKP', 'num_items': 13, 'capacity': 5459, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 1057}, {'item_id': 'B', 'linear_profit': 3010}, {'item_id': 'C', 'linear_profit': 1008}, {'item_id': 'D', 'linear_profit': 1248}, {'item_id': 'E', 'linear_profit': 1099}, {'item_id': 'F', 'linear_profit': 2074}, {'item_id': 'G', 'linear_profit': 1719}, {'item_id': 'H', 'linear_profit': 2421}, {'item_id': 'I', 'linear_profit': 513}, {'item_id': 'J', 'linear_profit': 1268}, {'item_id': 'K', 'linear_profit': 1947}, {'item_id': 'L', 'linear_profit': 1809}, {'item_id': 'M', 'linear_profit': 3105}], 'weight_pairs': [{'item_id': 'A', 'weight': 1057}, {'item_id': 'B', 'weight': 3077}, {'item_id': 'C', 'weight': 848}, {'item_id': 'D', 'weight': 1368}, {'item_id': 'E', 'weight': 982}, {'item_id': 'F', 'weight': 1974}, {'item_id': 'G', 'weight': 1777}, {'item_id': 'H', 'weight': 2469}, {'item_id': 'I', 'weight': 417}, {'item_id': 'J', 'weight': 1306}, {'item_id': 'K', 'weight': 2069}, {'item_id': 'L', 'weight': 1728}, {'item_id': 'M', 'weight': 2994}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 848}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 1057}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 101}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 1057}, {'item_i_id': 'A', 'item_j_id': 'K', 'quadratic_profit': 1057}, {'item_i_id': 'A', 'item_j_id': 'M', 'quadratic_profit': 437}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 2848}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 1496}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 2278}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 1929}, {'item_i_id': 'B', 'item_j_id': 'H', 'quadratic_profit': 2037}, {'item_i_id': 'B', 'item_j_id': 'L', 'quadratic_profit': 1757}, {'item_i_id': 'B', 'item_j_id': 'M', 'quadratic_profit': 1757}, {'item_i_id': 'C', 'item_j_id': 'F', 'quadratic_profit': 848}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 799}, {'item_i_id': 'C', 'item_j_id': 'I', 'quadratic_profit': 848}, {'item_i_id': 'C', 'item_j_id': 'L', 'quadratic_profit': 1297}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 1368}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 1368}, {'item_i_id': 'D', 'item_j_id': 'J', 'quadratic_profit': 444}, {'item_i_id': 'D', 'item_j_id': 'L', 'quadratic_profit': 444}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 982}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 982}, {'item_i_id': 'E', 'item_j_id': 'J', 'quadratic_profit': 180}, {'item_i_id': 'E', 'item_j_id': 'K', 'quadratic_profit': 557}, {'item_i_id': 'E', 'item_j_id': 'L', 'quadratic_profit': 779}, {'item_i_id': 'F', 'item_j_id': 'J', 'quadratic_profit': 454}, {'item_i_id': 'F', 'item_j_id': 'M', 'quadratic_profit': 454}, {'item_i_id': 'G', 'item_j_id': 'I', 'quadratic_profit': 101}, {'item_i_id': 'G', 'item_j_id': 'J', 'quadratic_profit': 101}, {'item_i_id': 'G', 'item_j_id': 'L', 'quadratic_profit': 101}, {'item_i_id': 'H', 'item_j_id': 'I', 'quadratic_profit': 121}, {'item_i_id': 'H', 'item_j_id': 'J', 'quadratic_profit': 963}, {'item_i_id': 'H', 'item_j_id': 'K', 'quadratic_profit': 2516}, {'item_i_id': 'H', 'item_j_id': 'M', 'quadratic_profit': 1467}, {'item_i_id': 'I', 'item_j_id': 'M', 'quadratic_profit': 1411}, {'item_i_id': 'J', 'item_j_id': 'L', 'quadratic_profit': 2725}, {'item_i_id': 'J', 'item_j_id': 'M', 'quadratic_profit': 2446}]}","['C', 'E', 'I', 'J', 'L']",41,csv,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.

I'm choosing from 14 records: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and my crate holds up to 4880 weight — details below.

| record_id | crowd_pleasing_score |
|---|---|
| 0 | 293 |
| 1 | 768 |
| 2 | 460 |
| 3 | 171 |
| 4 | 411 |
| 5 | 768 |
| 6 | 185 |
| 7 | 503 |
| 8 | 316 |
| 9 | 829 |
| 10 | 632 |
| 11 | 137 |
| 12 | 289 |
| 13 | 743 |

| record_id | record_weight |
|---|---|
| 0 | 297 |
| 1 | 737 |
| 2 | 458 |
| 3 | 189 |
| 4 | 453 |
| 5 | 780 |
| 6 | 230 |
| 7 | 501 |
| 8 | 312 |
| 9 | 793 |
| 10 | 658 |
| 11 | 143 |
| 12 | 245 |
| 13 | 770 |

| record_i_id | record_j_id | pair_synergy_bonus |
|---|---|---|
| 0 | 2 | 110 |
| 0 | 3 | 297 |
| 0 | 4 | 223 |
| 0 | 7 | 297 |
| 0 | 8 | 297 |
| 0 | 9 | 297 |
| 0 | 10 | 297 |
| 0 | 11 | 238 |
| 0 | 12 | 297 |
| 0 | 13 | 297 |
| 1 | 2 | 105 |
| 1 | 3 | 737 |
| 1 | 4 | 349 |
| 1 | 5 | 737 |
| 1 | 7 | 316 |
| 1 | 8 | 658 |
| 1 | 9 | 689 |
| 1 | 10 | 676 |
| 1 | 11 | 310 |
| 1 | 12 | 493 |
| 1 | 13 | 152 |
| 2 | 3 | 376 |
| 2 | 5 | 458 |
| 2 | 6 | 310 |
| 2 | 7 | 458 |
| 2 | 8 | 258 |
| 2 | 10 | 393 |
| 2 | 11 | 431 |
| 2 | 12 | 319 |
| 2 | 13 | 267 |
| 3 | 5 | 189 |
| 3 | 6 | 189 |
| 3 | 7 | 105 |
| 3 | 9 | 189 |
| 3 | 13 | 263 |
| 4 | 5 | 453 |
| 4 | 6 | 453 |
| 4 | 8 | 453 |
| 4 | 11 | 312 |
| 4 | 12 | 336 |
| 4 | 13 | 336 |
| 5 | 6 | 152 |
| 5 | 8 | 140 |
| 5 | 9 | 152 |
| 5 | 10 | 152 |
| 5 | 11 | 152 |
| 5 | 12 | 152 |
| 5 | 13 | 152 |
| 6 | 7 | 790 |
| 6 | 8 | 143 |
| 6 | 9 | 246 |
| 6 | 10 | 310 |
| 6 | 12 | 152 |
| 6 | 13 | 279 |
| 7 | 8 | 121 |
| 7 | 9 | 872 |
| 7 | 10 | 105 |
| 7 | 11 | 812 |
| 7 | 12 | 312 |
| 7 | 13 | 704 |
| 8 | 9 | 431 |
| 8 | 10 | 480 |
| 8 | 12 | 246 |
| 8 | 13 | 105 |
| 9 | 10 | 225 |
| 9 | 11 | 587 |
| 9 | 12 | 172 |
| 9 | 13 | 630 |
| 10 | 12 | 258 |
| 10 | 13 | 833 |
| 12 | 13 | 258 |

I'll aim to pack the best-scoring mix without exceeding 4880.

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': [293, 768, 460, 171, 411, 768, 185, 503, 316, 829, 632, 137, 289, 743], 'quadratic_coeffs': [[297, 0, 110, 297, 223, 0, 0, 297, 297, 297, 297, 238, 297, 297], [0, 0, 105, 737, 349, 737, 0, 316, 658, 689, 676, 310, 493, 152], [0, 0, 306, 376, 0, 458, 310, 458, 258, 0, 393, 431, 319, 267], [0, 0, 0, 0, 0, 189, 189, 105, 0, 189, 0, 0, 0, 263], [0, 0, 0, 0, 143, 453, 453, 0, 453, 0, 0, 312, 336, 336], [0, 0, 0, 0, 0, 0, 152, 0, 140, 152, 152, 152, 152, 152], [0, 0, 0, 0, 0, 0, 171, 790, 143, 246, 310, 0, 152, 279], [0, 0, 0, 0, 0, 0, 0, 285, 121, 872, 105, 812, 312, 704], [0, 0, 0, 0, 0, 0, 0, 0, 152, 431, 480, 0, 246, 105], [0, 0, 0, 0, 0, 0, 0, 0, 0, 225, 225, 587, 172, 630], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 689, 0, 258, 833], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 238, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 258], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], 'weights': [297, 737, 458, 189, 453, 780, 230, 501, 312, 793, 658, 143, 245, 770], 'capacity': 4880, 'solution': [0, 1, 3, 6, 7, 8, 9, 10, 11, 12, 13], 'obj': 21031.0, 'problem_type': 'QKP'}","[0, 1, 3, 6, 7, 8, 9, 10, 11, 12, 13]",21031.0,"{'problem_type': 'QKP', 'num_items': 14, 'capacity': 4880, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 'linear_pairs': [{'item_id': 0, 'linear_profit': 293}, {'item_id': 1, 'linear_profit': 768}, {'item_id': 2, 'linear_profit': 460}, {'item_id': 3, 'linear_profit': 171}, {'item_id': 4, 'linear_profit': 411}, {'item_id': 5, 'linear_profit': 768}, {'item_id': 6, 'linear_profit': 185}, {'item_id': 7, 'linear_profit': 503}, {'item_id': 8, 'linear_profit': 316}, {'item_id': 9, 'linear_profit': 829}, {'item_id': 10, 'linear_profit': 632}, {'item_id': 11, 'linear_profit': 137}, {'item_id': 12, 'linear_profit': 289}, {'item_id': 13, 'linear_profit': 743}], 'weight_pairs': [{'item_id': 0, 'weight': 297}, {'item_id': 1, 'weight': 737}, {'item_id': 2, 'weight': 458}, {'item_id': 3, 'weight': 189}, {'item_id': 4, 'weight': 453}, {'item_id': 5, 'weight': 780}, {'item_id': 6, 'weight': 230}, {'item_id': 7, 'weight': 501}, {'item_id': 8, 'weight': 312}, {'item_id': 9, 'weight': 793}, {'item_id': 10, 'weight': 658}, {'item_id': 11, 'weight': 143}, {'item_id': 12, 'weight': 245}, {'item_id': 13, 'weight': 770}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 297}, {'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 110}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 297}, {'item_i_id': 0, 'item_j_id': 4, 'quadratic_profit': 223}, {'item_i_id': 0, 'item_j_id': 7, 'quadratic_profit': 297}, {'item_i_id': 0, 'item_j_id': 8, 'quadratic_profit': 297}, {'item_i_id': 0, 'item_j_id': 9, 'quadratic_profit': 297}, {'item_i_id': 0, 'item_j_id': 10, 'quadratic_profit': 297}, {'item_i_id': 0, 'item_j_id': 11, 'quadratic_profit': 238}, {'item_i_id': 0, 'item_j_id': 12, 'quadratic_profit': 297}, {'item_i_id': 0, 'item_j_id': 13, 'quadratic_profit': 297}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 105}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 737}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 349}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 737}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 316}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 658}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 689}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 676}, {'item_i_id': 1, 'item_j_id': 11, 'quadratic_profit': 310}, {'item_i_id': 1, 'item_j_id': 12, 'quadratic_profit': 493}, {'item_i_id': 1, 'item_j_id': 13, 'quadratic_profit': 152}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 306}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 376}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 458}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 310}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 458}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 258}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 393}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 431}, {'item_i_id': 2, 'item_j_id': 12, 'quadratic_profit': 319}, {'item_i_id': 2, 'item_j_id': 13, 'quadratic_profit': 267}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 189}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 189}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 105}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 189}, {'item_i_id': 3, 'item_j_id': 13, 'quadratic_profit': 263}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 143}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 453}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 453}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 453}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 312}, {'item_i_id': 4, 'item_j_id': 12, 'quadratic_profit': 336}, {'item_i_id': 4, 'item_j_id': 13, 'quadratic_profit': 336}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 152}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 140}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 152}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 152}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 152}, {'item_i_id': 5, 'item_j_id': 12, 'quadratic_profit': 152}, {'item_i_id': 5, 'item_j_id': 13, 'quadratic_profit': 152}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 171}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 790}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 143}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 246}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 310}, {'item_i_id': 6, 'item_j_id': 12, 'quadratic_profit': 152}, {'item_i_id': 6, 'item_j_id': 13, 'quadratic_profit': 279}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 285}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 121}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 872}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 105}, {'item_i_id': 7, 'item_j_id': 11, 'quadratic_profit': 812}, {'item_i_id': 7, 'item_j_id': 12, 'quadratic_profit': 312}, {'item_i_id': 7, 'item_j_id': 13, 'quadratic_profit': 704}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 152}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 431}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 480}, {'item_i_id': 8, 'item_j_id': 12, 'quadratic_profit': 246}, {'item_i_id': 8, 'item_j_id': 13, 'quadratic_profit': 105}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 225}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 225}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 587}, {'item_i_id': 9, 'item_j_id': 12, 'quadratic_profit': 172}, {'item_i_id': 9, 'item_j_id': 13, 'quadratic_profit': 630}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 689}, {'item_i_id': 10, 'item_j_id': 12, 'quadratic_profit': 258}, {'item_i_id': 10, 'item_j_id': 13, 'quadratic_profit': 833}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 238}, {'item_i_id': 12, 'item_j_id': 13, 'quadratic_profit': 258}]}","[0, 1, 3, 6, 7, 8, 9, 10, 11, 12, 13]",42,markdown_table,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 15 kits as A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, and we can pack up to 24675 total weight.
Kit A delivers a base relief impact of 6600 when included.
Kit B delivers a base relief impact of 6569 when included.
Kit C delivers a base relief impact of 5065 when included.
Kit D delivers a base relief impact of 3198 when included.
Kit E delivers a base relief impact of 2270 when included.
Kit F delivers a base relief impact of 6437 when included.
Kit G delivers a base relief impact of 3752 when included.
Kit H delivers a base relief impact of 2770 when included.
Kit I delivers a base relief impact of 6229 when included.
Kit J delivers a base relief impact of 3839 when included.
Kit K delivers a base relief impact of 4665 when included.
Kit L delivers a base relief impact of 6191 when included.
Kit M delivers a base relief impact of 4568 when included.
Kit N delivers a base relief impact of 4177 when included.
Kit O delivers a base relief impact of 1320 when included.
Kit A weighs 5950.
Kit B weighs 5919.
Kit C weighs 4415.
Kit D weighs 2548.
Kit E weighs 1620.
Kit F weighs 5787.
Kit G weighs 3102.
Kit H weighs 2120.
Kit I weighs 5579.
Kit J weighs 3189.
Kit K weighs 4015.
Kit L weighs 5541.
Kit M weighs 3918.
Kit N weighs 3527.
Kit O weighs 670.
Kits A and B together provide an extra 4510 impact.
Kits A and C together provide an extra 4670 impact.
Kits A and E together provide an extra 5980 impact.
Kits A and F together provide an extra 4887 impact.
Kits A and H together provide an extra 2982 impact.
Kits A and I together provide an extra 3742 impact.
Kits A and J together provide an extra 1360 impact.
Kits A and O together provide an extra 3960 impact.
Kits B and D together provide an extra 5781 impact.
Kits B and E together provide an extra 6020 impact.
Kits B and F together provide an extra 2232 impact.
Kits B and H together provide an extra 2062 impact.
Kits B and I together provide an extra 5508 impact.
Kits B and J together provide an extra 5302 impact.
Kits B and K together provide an extra 6124 impact.
Kits B and L together provide an extra 4877 impact.
Kits B and O together provide an extra 3484 impact.
Kits C and E together provide an extra 5037 impact.
Kits C and F together provide an extra 783 impact.
Kits C and G together provide an extra 3662 impact.
Kits C and H together provide an extra 3613 impact.
Kits C and J together provide an extra 860 impact.
Kits C and K together provide an extra 378 impact.
Kits C and L together provide an extra 957 impact.
Kits C and M together provide an extra 852 impact.
Kits C and N together provide an extra 620 impact.
Kits C and O together provide an extra 946 impact.
Kits D and E together provide an extra 1856 impact.
Kits D and G together provide an extra 2416 impact.
Kits D and H together provide an extra 2000 impact.
Kits D and I together provide an extra 1742 impact.
Kits D and J together provide an extra 3104 impact.
Kits D and L together provide an extra 2632 impact.
Kits D and N together provide an extra 2636 impact.
Kits E and M together provide an extra 2892 impact.
Kits E and N together provide an extra 5372 impact.
Kits E and O together provide an extra 3775 impact.
Kits F and H together provide an extra 2358 impact.
Kits F and I together provide an extra 5937 impact.
Kits F and J together provide an extra 3680 impact.
Kits F and M together provide an extra 2405 impact.
Kits F and N together provide an extra 5653 impact.
Kits F and O together provide an extra 4171 impact.
Kits G and H together provide an extra 5224 impact.
Kits G and I together provide an extra 3878 impact.
Kits G and N together provide an extra 1782 impact.
Kits G and O together provide an extra 2120 impact.
Kits H and I together provide an extra 4350 impact.
Kits H and J together provide an extra 5006 impact.
Kits H and K together provide an extra 4061 impact.
Kits H and L together provide an extra 4140 impact.
Kits H and N together provide an extra 2692 impact.
Kits H and O together provide an extra 3159 impact.
Kits I and M together provide an extra 4211 impact.
Kits I and N together provide an extra 2683 impact.
Kits J and K together provide an extra 1862 impact.
Kits J and M together provide an extra 2495 impact.
Kits J and N together provide an extra 4635 impact.
Kits J and O together provide an extra 3257 impact.
Kits K and L together provide an extra 4547 impact.
Kits K and N together provide an extra 2649 impact.
Kits M and N together provide an extra 1478 impact.
Kits M and O together provide an extra 2958 impact.
We must pack only whole kits and keep the total weight within 24675.

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': [6600, 6569, 5065, 3198, 2270, 6437, 3752, 2770, 6229, 3839, 4665, 6191, 4568, 4177, 1320], 'quadratic_coeffs': [[5789, 4510, 4670, 0, 5980, 4887, 0, 2982, 3742, 1360, 0, 0, 0, 0, 3960], [0, 0, 0, 5781, 6020, 2232, 0, 2062, 5508, 5302, 6124, 4877, 0, 0, 3484], [0, 0, 0, 0, 5037, 783, 3662, 3613, 0, 860, 378, 957, 852, 620, 946], [0, 0, 0, 0, 1856, 0, 2416, 2000, 1742, 3104, 0, 2632, 0, 2636, 0], [0, 0, 0, 0, 2257, 0, 0, 0, 0, 0, 0, 0, 2892, 5372, 3775], [0, 0, 0, 0, 0, 4439, 0, 2358, 5937, 3680, 0, 0, 2405, 5653, 4171], [0, 0, 0, 0, 0, 0, 0, 5224, 3878, 0, 0, 0, 0, 1782, 2120], [0, 0, 0, 0, 0, 0, 0, 1107, 4350, 5006, 4061, 4140, 0, 2692, 3159], [0, 0, 0, 0, 0, 0, 0, 0, 1466, 0, 0, 0, 4211, 2683, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1862, 0, 2495, 4635, 3257], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3646, 4547, 0, 2649, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1046, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5410, 1478, 2958], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2297, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1651]], 'weights': [5950, 5919, 4415, 2548, 1620, 5787, 3102, 2120, 5579, 3189, 4015, 5541, 3918, 3527, 670], 'capacity': 24675, 'solution': [1, 3, 4, 7, 9, 10, 13, 14], 'obj': 103645.0, 'problem_type': 'QKP'}","[1, 3, 4, 7, 9, 10, 13, 14]",103645.0,"{'problem_type': 'QKP', 'num_items': 15, 'capacity': 24675, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 6600}, {'item_id': 'B', 'linear_profit': 6569}, {'item_id': 'C', 'linear_profit': 5065}, {'item_id': 'D', 'linear_profit': 3198}, {'item_id': 'E', 'linear_profit': 2270}, {'item_id': 'F', 'linear_profit': 6437}, {'item_id': 'G', 'linear_profit': 3752}, {'item_id': 'H', 'linear_profit': 2770}, {'item_id': 'I', 'linear_profit': 6229}, {'item_id': 'J', 'linear_profit': 3839}, {'item_id': 'K', 'linear_profit': 4665}, {'item_id': 'L', 'linear_profit': 6191}, {'item_id': 'M', 'linear_profit': 4568}, {'item_id': 'N', 'linear_profit': 4177}, {'item_id': 'O', 'linear_profit': 1320}], 'weight_pairs': [{'item_id': 'A', 'weight': 5950}, {'item_id': 'B', 'weight': 5919}, {'item_id': 'C', 'weight': 4415}, {'item_id': 'D', 'weight': 2548}, {'item_id': 'E', 'weight': 1620}, {'item_id': 'F', 'weight': 5787}, {'item_id': 'G', 'weight': 3102}, {'item_id': 'H', 'weight': 2120}, {'item_id': 'I', 'weight': 5579}, {'item_id': 'J', 'weight': 3189}, {'item_id': 'K', 'weight': 4015}, {'item_id': 'L', 'weight': 5541}, {'item_id': 'M', 'weight': 3918}, {'item_id': 'N', 'weight': 3527}, {'item_id': 'O', 'weight': 670}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 5789}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 4510}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 4670}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 5980}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 4887}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 2982}, {'item_i_id': 'A', 'item_j_id': 'I', 'quadratic_profit': 3742}, {'item_i_id': 'A', 'item_j_id': 'J', 'quadratic_profit': 1360}, {'item_i_id': 'A', 'item_j_id': 'O', 'quadratic_profit': 3960}, {'item_i_id': 'B', 'item_j_id': 'D', 'quadratic_profit': 5781}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 6020}, {'item_i_id': 'B', 'item_j_id': 'F', 'quadratic_profit': 2232}, {'item_i_id': 'B', 'item_j_id': 'H', 'quadratic_profit': 2062}, {'item_i_id': 'B', 'item_j_id': 'I', 'quadratic_profit': 5508}, {'item_i_id': 'B', 'item_j_id': 'J', 'quadratic_profit': 5302}, {'item_i_id': 'B', 'item_j_id': 'K', 'quadratic_profit': 6124}, {'item_i_id': 'B', 'item_j_id': 'L', 'quadratic_profit': 4877}, {'item_i_id': 'B', 'item_j_id': 'O', 'quadratic_profit': 3484}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 5037}, {'item_i_id': 'C', 'item_j_id': 'F', 'quadratic_profit': 783}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 3662}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 3613}, {'item_i_id': 'C', 'item_j_id': 'J', 'quadratic_profit': 860}, {'item_i_id': 'C', 'item_j_id': 'K', 'quadratic_profit': 378}, {'item_i_id': 'C', 'item_j_id': 'L', 'quadratic_profit': 957}, {'item_i_id': 'C', 'item_j_id': 'M', 'quadratic_profit': 852}, {'item_i_id': 'C', 'item_j_id': 'N', 'quadratic_profit': 620}, {'item_i_id': 'C', 'item_j_id': 'O', 'quadratic_profit': 946}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 1856}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 2416}, {'item_i_id': 'D', 'item_j_id': 'H', 'quadratic_profit': 2000}, {'item_i_id': 'D', 'item_j_id': 'I', 'quadratic_profit': 1742}, {'item_i_id': 'D', 'item_j_id': 'J', 'quadratic_profit': 3104}, {'item_i_id': 'D', 'item_j_id': 'L', 'quadratic_profit': 2632}, {'item_i_id': 'D', 'item_j_id': 'N', 'quadratic_profit': 2636}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 2257}, {'item_i_id': 'E', 'item_j_id': 'M', 'quadratic_profit': 2892}, {'item_i_id': 'E', 'item_j_id': 'N', 'quadratic_profit': 5372}, {'item_i_id': 'E', 'item_j_id': 'O', 'quadratic_profit': 3775}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 4439}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 2358}, {'item_i_id': 'F', 'item_j_id': 'I', 'quadratic_profit': 5937}, {'item_i_id': 'F', 'item_j_id': 'J', 'quadratic_profit': 3680}, {'item_i_id': 'F', 'item_j_id': 'M', 'quadratic_profit': 2405}, {'item_i_id': 'F', 'item_j_id': 'N', 'quadratic_profit': 5653}, {'item_i_id': 'F', 'item_j_id': 'O', 'quadratic_profit': 4171}, {'item_i_id': 'G', 'item_j_id': 'H', 'quadratic_profit': 5224}, {'item_i_id': 'G', 'item_j_id': 'I', 'quadratic_profit': 3878}, {'item_i_id': 'G', 'item_j_id': 'N', 'quadratic_profit': 1782}, {'item_i_id': 'G', 'item_j_id': 'O', 'quadratic_profit': 2120}, {'item_i_id': 'H', 'item_j_id': 'H', 'quadratic_profit': 1107}, {'item_i_id': 'H', 'item_j_id': 'I', 'quadratic_profit': 4350}, {'item_i_id': 'H', 'item_j_id': 'J', 'quadratic_profit': 5006}, {'item_i_id': 'H', 'item_j_id': 'K', 'quadratic_profit': 4061}, {'item_i_id': 'H', 'item_j_id': 'L', 'quadratic_profit': 4140}, {'item_i_id': 'H', 'item_j_id': 'N', 'quadratic_profit': 2692}, {'item_i_id': 'H', 'item_j_id': 'O', 'quadratic_profit': 3159}, {'item_i_id': 'I', 'item_j_id': 'I', 'quadratic_profit': 1466}, {'item_i_id': 'I', 'item_j_id': 'M', 'quadratic_profit': 4211}, {'item_i_id': 'I', 'item_j_id': 'N', 'quadratic_profit': 2683}, {'item_i_id': 'J', 'item_j_id': 'K', 'quadratic_profit': 1862}, {'item_i_id': 'J', 'item_j_id': 'M', 'quadratic_profit': 2495}, {'item_i_id': 'J', 'item_j_id': 'N', 'quadratic_profit': 4635}, {'item_i_id': 'J', 'item_j_id': 'O', 'quadratic_profit': 3257}, {'item_i_id': 'K', 'item_j_id': 'K', 'quadratic_profit': 3646}, {'item_i_id': 'K', 'item_j_id': 'L', 'quadratic_profit': 4547}, {'item_i_id': 'K', 'item_j_id': 'N', 'quadratic_profit': 2649}, {'item_i_id': 'L', 'item_j_id': 'L', 'quadratic_profit': 1046}, {'item_i_id': 'M', 'item_j_id': 'M', 'quadratic_profit': 5410}, {'item_i_id': 'M', 'item_j_id': 'N', 'quadratic_profit': 1478}, {'item_i_id': 'M', 'item_j_id': 'O', 'quadratic_profit': 2958}, {'item_i_id': 'N', 'item_j_id': 'N', 'quadratic_profit': 2297}, {'item_i_id': 'O', 'item_j_id': 'O', 'quadratic_profit': 1651}]}","['B', 'D', 'E', 'H', 'J', 'K', 'N', 'O']",43,nl,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.

Below I list the 12 tools (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12) and the van's weight capacity of 15449.
I counted tool 1 as contributing 3559 service value on its own.
I counted tool 2 as contributing 726 service value on its own.
I counted tool 3 as contributing 858 service value on its own.
I counted tool 4 as contributing 2144 service value on its own.
I counted tool 5 as contributing 1969 service value on its own.
I counted tool 6 as contributing 1538 service value on its own.
I counted tool 7 as contributing 699 service value on its own.
I counted tool 8 as contributing 460 service value on its own.
I counted tool 9 as contributing 1470 service value on its own.
I counted tool 10 as contributing 1209 service value on its own.
I counted tool 11 as contributing 3437 service value on its own.
I counted tool 12 as contributing 1546 service value on its own.
Tool 1 weighed 3229, which I had to include in the van's total load.
Tool 2 weighed 396, which I had to include in the van's total load.
Tool 3 weighed 528, which I had to include in the van's total load.
Tool 4 weighed 1814, which I had to include in the van's total load.
Tool 5 weighed 1639, which I had to include in the van's total load.
Tool 6 weighed 1208, which I had to include in the van's total load.
Tool 7 weighed 369, which I had to include in the van's total load.
Tool 8 weighed 130, which I had to include in the van's total load.
Tool 9 weighed 1140, which I had to include in the van's total load.
Tool 10 weighed 879, which I had to include in the van's total load.
Tool 11 weighed 3107, which I had to include in the van's total load.
Tool 12 weighed 1216, which I had to include in the van's total load.
If I took both 1 and 3, they'd add an extra 2188 service value together.
If I took both 1 and 10, they'd add an extra 2220 service value together.
If I took both 1 and 12, they'd add an extra 1180 service value together.
If I took both 2 and 5, they'd add an extra 1758 service value together.
If I took both 2 and 6, they'd add an extra 1358 service value together.
If I took both 2 and 7, they'd add an extra 1540 service value together.
If I took both 2 and 9, they'd add an extra 2823 service value together.
If I took both 3 and 7, they'd add an extra 2564 service value together.
If I took both 3 and 12, they'd add an extra 450 service value together.
If I took both 4 and 5, they'd add an extra 2822 service value together.
If I took both 4 and 8, they'd add an extra 2846 service value together.
If I took both 4 and 10, they'd add an extra 2234 service value together.
If I took both 5 and 6, they'd add an extra 791 service value together.
If I took both 5 and 7, they'd add an extra 2434 service value together.
If I took both 6 and 8, they'd add an extra 2459 service value together.
If I took both 7 and 12, they'd add an extra 842 service value together.
I'll use these figures to pick the combination that fits within 15449 and yields the highest total service value.

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': [3559, 726, 858, 2144, 1969, 1538, 699, 460, 1470, 1209, 3437, 1546], 'quadratic_coeffs': [[0, 0, 2188, 0, 0, 0, 0, 0, 0, 2220, 0, 1180], [0, 0, 0, 0, 1758, 1358, 1540, 0, 2823, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2564, 0, 0, 0, 0, 450], [0, 0, 0, 0, 2822, 0, 0, 2846, 0, 2234, 0, 0], [0, 0, 0, 0, 0, 791, 2434, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2459, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 842], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1128, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 710]], 'weights': [3229, 396, 528, 1814, 1639, 1208, 369, 130, 1140, 879, 3107, 1216], 'capacity': 15449, 'solution': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11], 'obj': 46687.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11]",46687.0,"{'problem_type': 'QKP', 'num_items': 12, 'capacity': 15449, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 'linear_pairs': [{'item_id': 1, 'linear_profit': 3559}, {'item_id': 2, 'linear_profit': 726}, {'item_id': 3, 'linear_profit': 858}, {'item_id': 4, 'linear_profit': 2144}, {'item_id': 5, 'linear_profit': 1969}, {'item_id': 6, 'linear_profit': 1538}, {'item_id': 7, 'linear_profit': 699}, {'item_id': 8, 'linear_profit': 460}, {'item_id': 9, 'linear_profit': 1470}, {'item_id': 10, 'linear_profit': 1209}, {'item_id': 11, 'linear_profit': 3437}, {'item_id': 12, 'linear_profit': 1546}], 'weight_pairs': [{'item_id': 1, 'weight': 3229}, {'item_id': 2, 'weight': 396}, {'item_id': 3, 'weight': 528}, {'item_id': 4, 'weight': 1814}, {'item_id': 5, 'weight': 1639}, {'item_id': 6, 'weight': 1208}, {'item_id': 7, 'weight': 369}, {'item_id': 8, 'weight': 130}, {'item_id': 9, 'weight': 1140}, {'item_id': 10, 'weight': 879}, {'item_id': 11, 'weight': 3107}, {'item_id': 12, 'weight': 1216}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 2188}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 2220}, {'item_i_id': 1, 'item_j_id': 12, 'quadratic_profit': 1180}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 1758}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 1358}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 1540}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 2823}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 2564}, {'item_i_id': 3, 'item_j_id': 12, 'quadratic_profit': 450}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 2822}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 2846}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 2234}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 791}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 2434}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 2459}, {'item_i_id': 7, 'item_j_id': 12, 'quadratic_profit': 842}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 1128}, {'item_i_id': 12, 'item_j_id': 12, 'quadratic_profit': 710}]}","[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12]",44,nl,1
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.

There are 11 distinct accessories (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10) and the display case can hold up to 618 units of space.

| accessory_id | base_sales_value |
|---|---|
| 0 | 428 |
| 1 | 164 |
| 2 | 88 |
| 3 | 666 |
| 4 | 588 |
| 5 | 491 |
| 6 | 183 |
| 7 | 653 |
| 8 | 161 |
| 9 | 819 |
| 10 | 170 |

| accessory_id | space_required |
|---|---|
| 0 | 453 |
| 1 | 119 |
| 2 | 100 |
| 3 | 687 |
| 4 | 571 |
| 5 | 512 |
| 6 | 197 |
| 7 | 654 |
| 8 | 193 |
| 9 | 788 |
| 10 | 210 |

| accessory_i_id | accessory_j_id | cross_selling_bonus |
|---|---|---|
| 0 | 1 | 453 |
| 0 | 2 | 321 |
| 0 | 3 | 253 |
| 0 | 4 | 453 |
| 0 | 5 | 421 |
| 0 | 6 | 453 |
| 0 | 7 | 169 |
| 0 | 8 | 453 |
| 0 | 9 | 453 |
| 0 | 10 | 303 |
| 1 | 2 | 119 |
| 1 | 3 | 119 |
| 1 | 4 | 119 |
| 1 | 5 | 119 |
| 1 | 6 | 119 |
| 1 | 7 | 119 |
| 1 | 8 | 119 |
| 1 | 9 | 119 |
| 1 | 10 | 119 |
| 2 | 3 | 100 |
| 2 | 4 | 100 |
| 2 | 5 | 100 |
| 2 | 6 | 100 |
| 2 | 7 | 100 |
| 2 | 8 | 100 |
| 2 | 9 | 100 |
| 2 | 10 | 100 |
| 3 | 4 | 562 |
| 3 | 5 | 687 |
| 3 | 6 | 421 |
| 3 | 7 | 687 |
| 3 | 8 | 449 |
| 3 | 9 | 110 |
| 3 | 10 | 110 |
| 4 | 5 | 533 |
| 4 | 6 | 425 |
| 4 | 7 | 113 |
| 4 | 8 | 571 |
| 4 | 9 | 183 |
| 4 | 10 | 417 |
| 5 | 6 | 512 |
| 5 | 7 | 143 |
| 5 | 8 | 146 |
| 5 | 9 | 513 |
| 5 | 10 | 513 |
| 6 | 7 | 534 |
| 6 | 8 | 562 |
| 6 | 9 | 562 |
| 6 | 10 | 562 |
| 7 | 8 | 393 |
| 7 | 9 | 267 |
| 7 | 10 | 247 |
| 8 | 9 | 196 |
| 8 | 10 | 196 |
| 9 | 10 | 143 |

A selection must respect the 618 limit and maximize the sum of base sales values plus any cross-selling bonuses.

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': [428, 164, 88, 666, 588, 491, 183, 653, 161, 819, 170], 'quadratic_coeffs': [[453, 453, 321, 253, 453, 421, 453, 169, 453, 453, 303], [0, 119, 119, 119, 119, 119, 119, 119, 119, 119, 119], [0, 0, 100, 100, 100, 100, 100, 100, 100, 100, 100], [0, 0, 0, 687, 562, 687, 421, 687, 449, 110, 110], [0, 0, 0, 0, 571, 533, 425, 113, 571, 183, 417], [0, 0, 0, 0, 0, 512, 512, 143, 146, 513, 513], [0, 0, 0, 0, 0, 0, 197, 534, 562, 562, 562], [0, 0, 0, 0, 0, 0, 0, 193, 393, 267, 247], [0, 0, 0, 0, 0, 0, 0, 0, 499, 196, 196], [0, 0, 0, 0, 0, 0, 0, 0, 0, 256, 143], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 499]], 'weights': [453, 119, 100, 687, 571, 512, 197, 654, 193, 788, 210], 'capacity': 618, 'solution': [6, 8, 10], 'obj': 1834.0, 'problem_type': 'QKP'}","[6, 8, 10]",1834.0,"{'problem_type': 'QKP', 'num_items': 11, 'capacity': 618, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 'linear_pairs': [{'item_id': 0, 'linear_profit': 428}, {'item_id': 1, 'linear_profit': 164}, {'item_id': 2, 'linear_profit': 88}, {'item_id': 3, 'linear_profit': 666}, {'item_id': 4, 'linear_profit': 588}, {'item_id': 5, 'linear_profit': 491}, {'item_id': 6, 'linear_profit': 183}, {'item_id': 7, 'linear_profit': 653}, {'item_id': 8, 'linear_profit': 161}, {'item_id': 9, 'linear_profit': 819}, {'item_id': 10, 'linear_profit': 170}], 'weight_pairs': [{'item_id': 0, 'weight': 453}, {'item_id': 1, 'weight': 119}, {'item_id': 2, 'weight': 100}, {'item_id': 3, 'weight': 687}, {'item_id': 4, 'weight': 571}, {'item_id': 5, 'weight': 512}, {'item_id': 6, 'weight': 197}, {'item_id': 7, 'weight': 654}, {'item_id': 8, 'weight': 193}, {'item_id': 9, 'weight': 788}, {'item_id': 10, 'weight': 210}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 453}, {'item_i_id': 0, 'item_j_id': 1, 'quadratic_profit': 453}, {'item_i_id': 0, 'item_j_id': 2, 'quadratic_profit': 321}, {'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 253}, {'item_i_id': 0, 'item_j_id': 4, 'quadratic_profit': 453}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 421}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 453}, {'item_i_id': 0, 'item_j_id': 7, 'quadratic_profit': 169}, {'item_i_id': 0, 'item_j_id': 8, 'quadratic_profit': 453}, {'item_i_id': 0, 'item_j_id': 9, 'quadratic_profit': 453}, {'item_i_id': 0, 'item_j_id': 10, 'quadratic_profit': 303}, {'item_i_id': 1, 'item_j_id': 1, 'quadratic_profit': 119}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 119}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 119}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 119}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 119}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 119}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 119}, {'item_i_id': 1, 'item_j_id': 8, 'quadratic_profit': 119}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 119}, {'item_i_id': 1, 'item_j_id': 10, 'quadratic_profit': 119}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 100}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 100}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 100}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 100}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 100}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 100}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 100}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 100}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 100}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 687}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 562}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 687}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 421}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 687}, {'item_i_id': 3, 'item_j_id': 8, 'quadratic_profit': 449}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 110}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 110}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 571}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 533}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 425}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 113}, {'item_i_id': 4, 'item_j_id': 8, 'quadratic_profit': 571}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 183}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 417}, {'item_i_id': 5, 'item_j_id': 5, 'quadratic_profit': 512}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 512}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 143}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 146}, {'item_i_id': 5, 'item_j_id': 9, 'quadratic_profit': 513}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 513}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 197}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 534}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 562}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 562}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 562}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 193}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 393}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 267}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 247}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 499}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 196}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 196}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 256}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 143}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 499}]}","[6, 8, 10]",45,markdown_table,0
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 13 distinct small plates available — 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 — and a platter capacity of 228; the exact plate scores, space usages, and pairing bonuses are listed below.
Plate 0 yields 105 satisfaction points when included on the tasting platter.
Plate 1 yields 102 satisfaction points when included on the tasting platter.
Plate 2 yields 101 satisfaction points when included on the tasting platter.
Plate 3 yields 105 satisfaction points when included on the tasting platter.
Plate 4 yields 104 satisfaction points when included on the tasting platter.
Plate 5 yields 102 satisfaction points when included on the tasting platter.
Plate 6 yields 104 satisfaction points when included on the tasting platter.
Plate 7 yields 101 satisfaction points when included on the tasting platter.
Plate 8 yields 104 satisfaction points when included on the tasting platter.
Plate 9 yields 105 satisfaction points when included on the tasting platter.
Plate 10 yields 103 satisfaction points when included on the tasting platter.
Plate 11 yields 105 satisfaction points when included on the tasting platter.
Plate 12 yields 103 satisfaction points when included on the tasting platter.
Plate 0 occupies 103 units of platter space.
Plate 1 occupies 102 units of platter space.
Plate 2 occupies 103 units of platter space.
Plate 3 occupies 105 units of platter space.
Plate 4 occupies 105 units of platter space.
Plate 5 occupies 103 units of platter space.
Plate 6 occupies 102 units of platter space.
Plate 7 occupies 102 units of platter space.
Plate 8 occupies 102 units of platter space.
Plate 9 occupies 102 units of platter space.
Plate 10 occupies 100 units of platter space.
Plate 11 occupies 105 units of platter space.
Plate 12 occupies 105 units of platter space.
Serving 0 together with 8 adds 100 extra satisfaction points.
Serving 1 together with 6 adds 100 extra satisfaction points.
Serving 1 together with 9 adds 100 extra satisfaction points.
Serving 2 together with 6 adds 102 extra satisfaction points.
Serving 2 together with 7 adds 101 extra satisfaction points.
Serving 2 together with 10 adds 101 extra satisfaction points.
Serving 2 together with 11 adds 103 extra satisfaction points.
Serving 3 together with 4 adds 104 extra satisfaction points.
Serving 3 together with 12 adds 103 extra satisfaction points.
Serving 4 together with 7 adds 104 extra satisfaction points.
Serving 4 together with 10 adds 102 extra satisfaction points.
Serving 5 together with 11 adds 102 extra satisfaction points.
Serving 5 together with 12 adds 101 extra satisfaction points.
Serving 6 together with 7 adds 100 extra satisfaction points.
Serving 6 together with 9 adds 102 extra satisfaction points.
Serving 6 together with 10 adds 102 extra satisfaction points.
Serving 7 together with 9 adds 100 extra satisfaction points.
Serving 8 together with 10 adds 100 extra satisfaction points.
Serving 9 together with 10 adds 103 extra satisfaction points.
Serving 9 together with 11 adds 101 extra satisfaction points.
Serving 10 together with 12 adds 102 extra satisfaction points.
The goal is to choose plates that fit within the 228 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': [105, 102, 101, 105, 104, 102, 104, 101, 104, 105, 103, 105, 103], 'quadratic_coeffs': [[103, 0, 0, 0, 0, 0, 0, 0, 100, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 100, 0, 0, 100, 0, 0, 0], [0, 0, 0, 0, 0, 0, 102, 101, 0, 0, 101, 103, 0], [0, 0, 0, 0, 104, 0, 0, 0, 0, 0, 0, 0, 103], [0, 0, 0, 0, 101, 0, 0, 104, 0, 0, 102, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 102, 101], [0, 0, 0, 0, 0, 0, 0, 100, 0, 102, 102, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 103, 101, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 102], [0, 0, 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': [103, 102, 103, 105, 105, 103, 102, 102, 102, 102, 100, 105, 105], 'capacity': 228, 'solution': [3, 4], 'obj': 313.0, 'problem_type': 'QKP'}","[3, 4]",313.0,"{'problem_type': 'QKP', 'num_items': 13, 'capacity': 228, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 'linear_pairs': [{'item_id': 0, 'linear_profit': 105}, {'item_id': 1, 'linear_profit': 102}, {'item_id': 2, 'linear_profit': 101}, {'item_id': 3, 'linear_profit': 105}, {'item_id': 4, 'linear_profit': 104}, {'item_id': 5, 'linear_profit': 102}, {'item_id': 6, 'linear_profit': 104}, {'item_id': 7, 'linear_profit': 101}, {'item_id': 8, 'linear_profit': 104}, {'item_id': 9, 'linear_profit': 105}, {'item_id': 10, 'linear_profit': 103}, {'item_id': 11, 'linear_profit': 105}, {'item_id': 12, 'linear_profit': 103}], 'weight_pairs': [{'item_id': 0, 'weight': 103}, {'item_id': 1, 'weight': 102}, {'item_id': 2, 'weight': 103}, {'item_id': 3, 'weight': 105}, {'item_id': 4, 'weight': 105}, {'item_id': 5, 'weight': 103}, {'item_id': 6, 'weight': 102}, {'item_id': 7, 'weight': 102}, {'item_id': 8, 'weight': 102}, {'item_id': 9, 'weight': 102}, {'item_id': 10, 'weight': 100}, {'item_id': 11, 'weight': 105}, {'item_id': 12, 'weight': 105}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 0, 'quadratic_profit': 103}, {'item_i_id': 0, 'item_j_id': 8, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 100}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 102}, {'item_i_id': 2, 'item_j_id': 7, 'quadratic_profit': 101}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 101}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 103}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 104}, {'item_i_id': 3, 'item_j_id': 12, 'quadratic_profit': 103}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 101}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 104}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 102}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 102}, {'item_i_id': 5, 'item_j_id': 12, 'quadratic_profit': 101}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 100}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 102}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 102}, {'item_i_id': 7, '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': 103}, {'item_i_id': 9, 'item_j_id': 11, 'quadratic_profit': 101}, {'item_i_id': 10, 'item_j_id': 12, 'quadratic_profit': 102}]}","[3, 4]",46,nl,0
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.

Below are the 15 pieces (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14) and the truck's weight limit is 598.

| piece_id | resale_value |
|---|---|
| 0 | 120 |
| 1 | 108 |
| 2 | 139 |
| 3 | 124 |
| 4 | 133 |
| 5 | 133 |
| 6 | 105 |
| 7 | 102 |
| 8 | 135 |
| 9 | 121 |
| 10 | 136 |
| 11 | 125 |
| 12 | 114 |
| 13 | 134 |
| 14 | 109 |

| piece_id | piece_weight |
|---|---|
| 0 | 127 |
| 1 | 108 |
| 2 | 145 |
| 3 | 131 |
| 4 | 138 |
| 5 | 127 |
| 6 | 109 |
| 7 | 100 |
| 8 | 127 |
| 9 | 117 |
| 10 | 133 |
| 11 | 124 |
| 12 | 118 |
| 13 | 140 |
| 14 | 102 |

| piece_a_id | piece_b_id | pair_bonus_value |
|---|---|---|
| 0 | 3 | 129 |
| 0 | 4 | 106 |
| 0 | 5 | 141 |
| 0 | 6 | 107 |
| 0 | 7 | 109 |
| 0 | 9 | 118 |
| 0 | 10 | 137 |
| 0 | 11 | 123 |
| 0 | 12 | 142 |
| 0 | 14 | 112 |
| 1 | 2 | 110 |
| 1 | 4 | 140 |
| 1 | 7 | 101 |
| 1 | 9 | 103 |
| 1 | 11 | 127 |
| 1 | 14 | 108 |
| 2 | 3 | 130 |
| 2 | 4 | 111 |
| 2 | 8 | 130 |
| 2 | 14 | 125 |
| 3 | 4 | 115 |
| 3 | 5 | 137 |
| 3 | 6 | 111 |
| 3 | 7 | 100 |
| 3 | 9 | 121 |
| 3 | 11 | 121 |
| 3 | 13 | 149 |
| 3 | 14 | 148 |
| 4 | 5 | 139 |
| 4 | 6 | 141 |
| 4 | 11 | 118 |
| 4 | 12 | 140 |
| 5 | 10 | 126 |
| 5 | 11 | 146 |
| 5 | 12 | 111 |
| 5 | 14 | 134 |
| 6 | 10 | 130 |
| 6 | 11 | 145 |
| 6 | 14 | 123 |
| 7 | 9 | 134 |
| 7 | 10 | 125 |
| 7 | 11 | 109 |
| 7 | 13 | 118 |
| 8 | 10 | 118 |
| 8 | 11 | 132 |
| 9 | 10 | 113 |
| 10 | 14 | 139 |
| 11 | 13 | 122 |
| 11 | 14 | 139 |
| 12 | 14 | 148 |

I'll choose pieces so the total weight stays within 598 while maximizing the total payout (base values plus any pair bonuses).

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': [120, 108, 139, 124, 133, 133, 105, 102, 135, 121, 136, 125, 114, 134, 109], 'quadratic_coeffs': [[0, 0, 0, 129, 106, 141, 107, 109, 0, 118, 137, 123, 142, 0, 112], [0, 0, 110, 0, 140, 0, 0, 101, 0, 103, 0, 127, 0, 0, 108], [0, 0, 105, 130, 111, 0, 0, 0, 130, 0, 0, 0, 0, 0, 125], [0, 0, 0, 100, 115, 137, 111, 100, 0, 121, 0, 121, 0, 149, 148], [0, 0, 0, 0, 0, 139, 141, 0, 0, 0, 0, 118, 140, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 126, 146, 111, 0, 134], [0, 0, 0, 0, 0, 0, 138, 0, 0, 0, 130, 145, 0, 0, 123], [0, 0, 0, 0, 0, 0, 0, 148, 0, 134, 125, 109, 0, 118, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 118, 132, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 142, 113, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 139], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 147, 0, 122, 139], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 148], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 139]], 'weights': [127, 108, 145, 131, 138, 127, 109, 100, 127, 117, 133, 124, 118, 140, 102], 'capacity': 598, 'solution': [0, 3, 6, 11, 14], 'obj': 1841.0, 'problem_type': 'QKP'}","[0, 3, 6, 11, 14]",1841.0,"{'problem_type': 'QKP', 'num_items': 15, 'capacity': 598, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], 'linear_pairs': [{'item_id': 0, 'linear_profit': 120}, {'item_id': 1, 'linear_profit': 108}, {'item_id': 2, 'linear_profit': 139}, {'item_id': 3, 'linear_profit': 124}, {'item_id': 4, 'linear_profit': 133}, {'item_id': 5, 'linear_profit': 133}, {'item_id': 6, 'linear_profit': 105}, {'item_id': 7, 'linear_profit': 102}, {'item_id': 8, 'linear_profit': 135}, {'item_id': 9, 'linear_profit': 121}, {'item_id': 10, 'linear_profit': 136}, {'item_id': 11, 'linear_profit': 125}, {'item_id': 12, 'linear_profit': 114}, {'item_id': 13, 'linear_profit': 134}, {'item_id': 14, 'linear_profit': 109}], 'weight_pairs': [{'item_id': 0, 'weight': 127}, {'item_id': 1, 'weight': 108}, {'item_id': 2, 'weight': 145}, {'item_id': 3, 'weight': 131}, {'item_id': 4, 'weight': 138}, {'item_id': 5, 'weight': 127}, {'item_id': 6, 'weight': 109}, {'item_id': 7, 'weight': 100}, {'item_id': 8, 'weight': 127}, {'item_id': 9, 'weight': 117}, {'item_id': 10, 'weight': 133}, {'item_id': 11, 'weight': 124}, {'item_id': 12, 'weight': 118}, {'item_id': 13, 'weight': 140}, {'item_id': 14, 'weight': 102}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 129}, {'item_i_id': 0, 'item_j_id': 4, 'quadratic_profit': 106}, {'item_i_id': 0, 'item_j_id': 5, 'quadratic_profit': 141}, {'item_i_id': 0, 'item_j_id': 6, 'quadratic_profit': 107}, {'item_i_id': 0, 'item_j_id': 7, 'quadratic_profit': 109}, {'item_i_id': 0, 'item_j_id': 9, 'quadratic_profit': 118}, {'item_i_id': 0, 'item_j_id': 10, 'quadratic_profit': 137}, {'item_i_id': 0, 'item_j_id': 11, 'quadratic_profit': 123}, {'item_i_id': 0, 'item_j_id': 12, 'quadratic_profit': 142}, {'item_i_id': 0, 'item_j_id': 14, 'quadratic_profit': 112}, {'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 110}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 140}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 101}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 103}, {'item_i_id': 1, 'item_j_id': 11, 'quadratic_profit': 127}, {'item_i_id': 1, 'item_j_id': 14, 'quadratic_profit': 108}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 105}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 130}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 111}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 130}, {'item_i_id': 2, 'item_j_id': 14, 'quadratic_profit': 125}, {'item_i_id': 3, 'item_j_id': 3, 'quadratic_profit': 100}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 115}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 137}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 111}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 100}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 121}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 121}, {'item_i_id': 3, 'item_j_id': 13, 'quadratic_profit': 149}, {'item_i_id': 3, 'item_j_id': 14, 'quadratic_profit': 148}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 139}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 141}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 118}, {'item_i_id': 4, 'item_j_id': 12, 'quadratic_profit': 140}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 126}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 146}, {'item_i_id': 5, 'item_j_id': 12, 'quadratic_profit': 111}, {'item_i_id': 5, 'item_j_id': 14, 'quadratic_profit': 134}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 138}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 130}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 145}, {'item_i_id': 6, 'item_j_id': 14, 'quadratic_profit': 123}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 148}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 134}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 125}, {'item_i_id': 7, 'item_j_id': 11, 'quadratic_profit': 109}, {'item_i_id': 7, 'item_j_id': 13, 'quadratic_profit': 118}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 118}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 132}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 142}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 113}, {'item_i_id': 10, 'item_j_id': 14, 'quadratic_profit': 139}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 147}, {'item_i_id': 11, 'item_j_id': 13, 'quadratic_profit': 122}, {'item_i_id': 11, 'item_j_id': 14, 'quadratic_profit': 139}, {'item_i_id': 12, 'item_j_id': 14, 'quadratic_profit': 148}, {'item_i_id': 14, 'item_j_id': 14, 'quadratic_profit': 139}]}","[0, 3, 6, 11, 14]",47,markdown_table,0
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 15 pieces of equipment A, B, C, D, E, F, G, H, I, J, K, L, M, N, O and the shelf's weight capacity of 17015.
A contributes a base usefulness of 1717.
B contributes a base usefulness of 795.
C contributes a base usefulness of 1645.
D contributes a base usefulness of 1652.
E contributes a base usefulness of 1329.
F contributes a base usefulness of 1565.
G contributes a base usefulness of 2429.
H contributes a base usefulness of 1526.
I contributes a base usefulness of 1400.
J contributes a base usefulness of 2460.
K contributes a base usefulness of 3205.
L contributes a base usefulness of 1687.
M contributes a base usefulness of 498.
N contributes a base usefulness of 2686.
O contributes a base usefulness of 1803.
A adds 1706 to the shelf's total weight.
B adds 359 to the shelf's total weight.
C adds 909 to the shelf's total weight.
D adds 1179 to the shelf's total weight.
E adds 197 to the shelf's total weight.
F adds 1859 to the shelf's total weight.
G adds 2224 to the shelf's total weight.
H adds 588 to the shelf's total weight.
I adds 2125 to the shelf's total weight.
J adds 264 to the shelf's total weight.
K adds 2638 to the shelf's total weight.
L adds 338 to the shelf's total weight.
M adds 1811 to the shelf's total weight.
N adds 1739 to the shelf's total weight.
O adds 2240 to the shelf's total weight.
If both A and B are placed, they yield an extra 3006 usefulness.
If both A and C are placed, they yield an extra 368 usefulness.
If both A and D are placed, they yield an extra 1316 usefulness.
If both A and E are placed, they yield an extra 2821 usefulness.
If both A and F are placed, they yield an extra 1175 usefulness.
If both A and H are placed, they yield an extra 2788 usefulness.
If both A and J are placed, they yield an extra 205 usefulness.
If both A and K are placed, they yield an extra 1256 usefulness.
If both A and N are placed, they yield an extra 3035 usefulness.
If both A and O are placed, they yield an extra 2226 usefulness.
If both B and C are placed, they yield an extra 1917 usefulness.
If both B and E are placed, they yield an extra 283 usefulness.
If both B and G are placed, they yield an extra 407 usefulness.
If both B and I are placed, they yield an extra 2520 usefulness.
If both B and K are placed, they yield an extra 475 usefulness.
If both B and M are placed, they yield an extra 1076 usefulness.
If both B and N are placed, they yield an extra 2288 usefulness.
If both B and O are placed, they yield an extra 2805 usefulness.
If both C and E are placed, they yield an extra 2743 usefulness.
If both C and F are placed, they yield an extra 3286 usefulness.
If both C and G are placed, they yield an extra 873 usefulness.
If both C and H are placed, they yield an extra 383 usefulness.
If both C and I are placed, they yield an extra 3241 usefulness.
If both C and J are placed, they yield an extra 3170 usefulness.
If both C and K are placed, they yield an extra 719 usefulness.
If both C and L are placed, they yield an extra 352 usefulness.
If both C and M are placed, they yield an extra 1205 usefulness.
If both C and N are placed, they yield an extra 3253 usefulness.
If both C and O are placed, they yield an extra 2091 usefulness.
If both D and E are placed, they yield an extra 2529 usefulness.
If both D and F are placed, they yield an extra 867 usefulness.
If both D and G are placed, they yield an extra 1950 usefulness.
If both D and I are placed, they yield an extra 1688 usefulness.
If both D and J are placed, they yield an extra 2695 usefulness.
If both D and K are placed, they yield an extra 2036 usefulness.
If both D and L are placed, they yield an extra 3296 usefulness.
If both D and M are placed, they yield an extra 1265 usefulness.
If both D and O are placed, they yield an extra 2257 usefulness.
If both E and F are placed, they yield an extra 2166 usefulness.
If both E and G are placed, they yield an extra 1834 usefulness.
If both E and H are placed, they yield an extra 1235 usefulness.
If both E and J are placed, they yield an extra 1122 usefulness.
If both E and K are placed, they yield an extra 1689 usefulness.
If both E and L are placed, they yield an extra 1149 usefulness.
If both E and M are placed, they yield an extra 842 usefulness.
If both E and N are placed, they yield an extra 2554 usefulness.
If both F and G are placed, they yield an extra 551 usefulness.
If both F and H are placed, they yield an extra 1318 usefulness.
If both F and J are placed, they yield an extra 1752 usefulness.
If both F and K are placed, they yield an extra 1935 usefulness.
If both F and L are placed, they yield an extra 1048 usefulness.
If both F and M are placed, they yield an extra 2821 usefulness.
If both F and N are placed, they yield an extra 1484 usefulness.
If both F and O are placed, they yield an extra 1220 usefulness.
If both G and H are placed, they yield an extra 1245 usefulness.
If both G and I are placed, they yield an extra 2117 usefulness.
If both G and J are placed, they yield an extra 1225 usefulness.
If both G and K are placed, they yield an extra 176 usefulness.
If both G and L are placed, they yield an extra 1897 usefulness.
If both G and M are placed, they yield an extra 2483 usefulness.
If both G and N are placed, they yield an extra 1196 usefulness.
If both H and I are placed, they yield an extra 1666 usefulness.
If both H and K are placed, they yield an extra 1296 usefulness.
If both H and L are placed, they yield an extra 2342 usefulness.
If both H and M are placed, they yield an extra 2890 usefulness.
If both H and N are placed, they yield an extra 2057 usefulness.
If both I and J are placed, they yield an extra 2158 usefulness.
If both I and K are placed, they yield an extra 636 usefulness.
If both I and N are placed, they yield an extra 2754 usefulness.
If both I and O are placed, they yield an extra 575 usefulness.
If both J and K are placed, they yield an extra 1409 usefulness.
If both J and M are placed, they yield an extra 252 usefulness.
If both J and N are placed, they yield an extra 1280 usefulness.
If both J and O are placed, they yield an extra 2520 usefulness.
If both K and L are placed, they yield an extra 207 usefulness.
If both K and N are placed, they yield an extra 2864 usefulness.
If both K and O are placed, they yield an extra 1988 usefulness.
If both L and M are placed, they yield an extra 2088 usefulness.
If both L and N are placed, they yield an extra 752 usefulness.
If both L and O are placed, they yield an extra 3102 usefulness.
If both M and N are placed, they yield an extra 2364 usefulness.
Use these entries to pick items that fit within the 17015 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': [1717, 795, 1645, 1652, 1329, 1565, 2429, 1526, 1400, 2460, 3205, 1687, 498, 2686, 1803], 'quadratic_coeffs': [[2673, 3006, 368, 1316, 2821, 1175, 0, 2788, 0, 205, 1256, 0, 0, 3035, 2226], [0, 2389, 1917, 0, 283, 0, 407, 0, 2520, 0, 475, 0, 1076, 2288, 2805], [0, 0, 2826, 0, 2743, 3286, 873, 383, 3241, 3170, 719, 352, 1205, 3253, 2091], [0, 0, 0, 3269, 2529, 867, 1950, 0, 1688, 2695, 2036, 3296, 1265, 0, 2257], [0, 0, 0, 0, 3215, 2166, 1834, 1235, 0, 1122, 1689, 1149, 842, 2554, 0], [0, 0, 0, 0, 0, 2497, 551, 1318, 0, 1752, 1935, 1048, 2821, 1484, 1220], [0, 0, 0, 0, 0, 0, 2373, 1245, 2117, 1225, 176, 1897, 2483, 1196, 0], [0, 0, 0, 0, 0, 0, 0, 1745, 1666, 0, 1296, 2342, 2890, 2057, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2595, 2158, 636, 0, 0, 2754, 575], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1409, 0, 252, 1280, 2520], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1279, 207, 0, 2864, 1988], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 343, 2088, 752, 3102], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2721, 2364, 0], [0, 0, 0, 0, 0, 0, 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': [1706, 359, 909, 1179, 197, 1859, 2224, 588, 2125, 264, 2638, 338, 1811, 1739, 2240], 'capacity': 17015, 'solution': [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14], 'obj': 132838.0, 'problem_type': 'QKP'}","[0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14]",132838.0,"{'problem_type': 'QKP', 'num_items': 15, 'capacity': 17015, 'items': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O'], 'linear_pairs': [{'item_id': 'A', 'linear_profit': 1717}, {'item_id': 'B', 'linear_profit': 795}, {'item_id': 'C', 'linear_profit': 1645}, {'item_id': 'D', 'linear_profit': 1652}, {'item_id': 'E', 'linear_profit': 1329}, {'item_id': 'F', 'linear_profit': 1565}, {'item_id': 'G', 'linear_profit': 2429}, {'item_id': 'H', 'linear_profit': 1526}, {'item_id': 'I', 'linear_profit': 1400}, {'item_id': 'J', 'linear_profit': 2460}, {'item_id': 'K', 'linear_profit': 3205}, {'item_id': 'L', 'linear_profit': 1687}, {'item_id': 'M', 'linear_profit': 498}, {'item_id': 'N', 'linear_profit': 2686}, {'item_id': 'O', 'linear_profit': 1803}], 'weight_pairs': [{'item_id': 'A', 'weight': 1706}, {'item_id': 'B', 'weight': 359}, {'item_id': 'C', 'weight': 909}, {'item_id': 'D', 'weight': 1179}, {'item_id': 'E', 'weight': 197}, {'item_id': 'F', 'weight': 1859}, {'item_id': 'G', 'weight': 2224}, {'item_id': 'H', 'weight': 588}, {'item_id': 'I', 'weight': 2125}, {'item_id': 'J', 'weight': 264}, {'item_id': 'K', 'weight': 2638}, {'item_id': 'L', 'weight': 338}, {'item_id': 'M', 'weight': 1811}, {'item_id': 'N', 'weight': 1739}, {'item_id': 'O', 'weight': 2240}], 'quadratic_pairs': [{'item_i_id': 'A', 'item_j_id': 'A', 'quadratic_profit': 2673}, {'item_i_id': 'A', 'item_j_id': 'B', 'quadratic_profit': 3006}, {'item_i_id': 'A', 'item_j_id': 'C', 'quadratic_profit': 368}, {'item_i_id': 'A', 'item_j_id': 'D', 'quadratic_profit': 1316}, {'item_i_id': 'A', 'item_j_id': 'E', 'quadratic_profit': 2821}, {'item_i_id': 'A', 'item_j_id': 'F', 'quadratic_profit': 1175}, {'item_i_id': 'A', 'item_j_id': 'H', 'quadratic_profit': 2788}, {'item_i_id': 'A', 'item_j_id': 'J', 'quadratic_profit': 205}, {'item_i_id': 'A', 'item_j_id': 'K', 'quadratic_profit': 1256}, {'item_i_id': 'A', 'item_j_id': 'N', 'quadratic_profit': 3035}, {'item_i_id': 'A', 'item_j_id': 'O', 'quadratic_profit': 2226}, {'item_i_id': 'B', 'item_j_id': 'B', 'quadratic_profit': 2389}, {'item_i_id': 'B', 'item_j_id': 'C', 'quadratic_profit': 1917}, {'item_i_id': 'B', 'item_j_id': 'E', 'quadratic_profit': 283}, {'item_i_id': 'B', 'item_j_id': 'G', 'quadratic_profit': 407}, {'item_i_id': 'B', 'item_j_id': 'I', 'quadratic_profit': 2520}, {'item_i_id': 'B', 'item_j_id': 'K', 'quadratic_profit': 475}, {'item_i_id': 'B', 'item_j_id': 'M', 'quadratic_profit': 1076}, {'item_i_id': 'B', 'item_j_id': 'N', 'quadratic_profit': 2288}, {'item_i_id': 'B', 'item_j_id': 'O', 'quadratic_profit': 2805}, {'item_i_id': 'C', 'item_j_id': 'C', 'quadratic_profit': 2826}, {'item_i_id': 'C', 'item_j_id': 'E', 'quadratic_profit': 2743}, {'item_i_id': 'C', 'item_j_id': 'F', 'quadratic_profit': 3286}, {'item_i_id': 'C', 'item_j_id': 'G', 'quadratic_profit': 873}, {'item_i_id': 'C', 'item_j_id': 'H', 'quadratic_profit': 383}, {'item_i_id': 'C', 'item_j_id': 'I', 'quadratic_profit': 3241}, {'item_i_id': 'C', 'item_j_id': 'J', 'quadratic_profit': 3170}, {'item_i_id': 'C', 'item_j_id': 'K', 'quadratic_profit': 719}, {'item_i_id': 'C', 'item_j_id': 'L', 'quadratic_profit': 352}, {'item_i_id': 'C', 'item_j_id': 'M', 'quadratic_profit': 1205}, {'item_i_id': 'C', 'item_j_id': 'N', 'quadratic_profit': 3253}, {'item_i_id': 'C', 'item_j_id': 'O', 'quadratic_profit': 2091}, {'item_i_id': 'D', 'item_j_id': 'D', 'quadratic_profit': 3269}, {'item_i_id': 'D', 'item_j_id': 'E', 'quadratic_profit': 2529}, {'item_i_id': 'D', 'item_j_id': 'F', 'quadratic_profit': 867}, {'item_i_id': 'D', 'item_j_id': 'G', 'quadratic_profit': 1950}, {'item_i_id': 'D', 'item_j_id': 'I', 'quadratic_profit': 1688}, {'item_i_id': 'D', 'item_j_id': 'J', 'quadratic_profit': 2695}, {'item_i_id': 'D', 'item_j_id': 'K', 'quadratic_profit': 2036}, {'item_i_id': 'D', 'item_j_id': 'L', 'quadratic_profit': 3296}, {'item_i_id': 'D', 'item_j_id': 'M', 'quadratic_profit': 1265}, {'item_i_id': 'D', 'item_j_id': 'O', 'quadratic_profit': 2257}, {'item_i_id': 'E', 'item_j_id': 'E', 'quadratic_profit': 3215}, {'item_i_id': 'E', 'item_j_id': 'F', 'quadratic_profit': 2166}, {'item_i_id': 'E', 'item_j_id': 'G', 'quadratic_profit': 1834}, {'item_i_id': 'E', 'item_j_id': 'H', 'quadratic_profit': 1235}, {'item_i_id': 'E', 'item_j_id': 'J', 'quadratic_profit': 1122}, {'item_i_id': 'E', 'item_j_id': 'K', 'quadratic_profit': 1689}, {'item_i_id': 'E', 'item_j_id': 'L', 'quadratic_profit': 1149}, {'item_i_id': 'E', 'item_j_id': 'M', 'quadratic_profit': 842}, {'item_i_id': 'E', 'item_j_id': 'N', 'quadratic_profit': 2554}, {'item_i_id': 'F', 'item_j_id': 'F', 'quadratic_profit': 2497}, {'item_i_id': 'F', 'item_j_id': 'G', 'quadratic_profit': 551}, {'item_i_id': 'F', 'item_j_id': 'H', 'quadratic_profit': 1318}, {'item_i_id': 'F', 'item_j_id': 'J', 'quadratic_profit': 1752}, {'item_i_id': 'F', 'item_j_id': 'K', 'quadratic_profit': 1935}, {'item_i_id': 'F', 'item_j_id': 'L', 'quadratic_profit': 1048}, {'item_i_id': 'F', 'item_j_id': 'M', 'quadratic_profit': 2821}, {'item_i_id': 'F', 'item_j_id': 'N', 'quadratic_profit': 1484}, {'item_i_id': 'F', 'item_j_id': 'O', 'quadratic_profit': 1220}, {'item_i_id': 'G', 'item_j_id': 'G', 'quadratic_profit': 2373}, {'item_i_id': 'G', 'item_j_id': 'H', 'quadratic_profit': 1245}, {'item_i_id': 'G', 'item_j_id': 'I', 'quadratic_profit': 2117}, {'item_i_id': 'G', 'item_j_id': 'J', 'quadratic_profit': 1225}, {'item_i_id': 'G', 'item_j_id': 'K', 'quadratic_profit': 176}, {'item_i_id': 'G', 'item_j_id': 'L', 'quadratic_profit': 1897}, {'item_i_id': 'G', 'item_j_id': 'M', 'quadratic_profit': 2483}, {'item_i_id': 'G', 'item_j_id': 'N', 'quadratic_profit': 1196}, {'item_i_id': 'H', 'item_j_id': 'H', 'quadratic_profit': 1745}, {'item_i_id': 'H', 'item_j_id': 'I', 'quadratic_profit': 1666}, {'item_i_id': 'H', 'item_j_id': 'K', 'quadratic_profit': 1296}, {'item_i_id': 'H', 'item_j_id': 'L', 'quadratic_profit': 2342}, {'item_i_id': 'H', 'item_j_id': 'M', 'quadratic_profit': 2890}, {'item_i_id': 'H', 'item_j_id': 'N', 'quadratic_profit': 2057}, {'item_i_id': 'I', 'item_j_id': 'I', 'quadratic_profit': 2595}, {'item_i_id': 'I', 'item_j_id': 'J', 'quadratic_profit': 2158}, {'item_i_id': 'I', 'item_j_id': 'K', 'quadratic_profit': 636}, {'item_i_id': 'I', 'item_j_id': 'N', 'quadratic_profit': 2754}, {'item_i_id': 'I', 'item_j_id': 'O', 'quadratic_profit': 575}, {'item_i_id': 'J', 'item_j_id': 'K', 'quadratic_profit': 1409}, {'item_i_id': 'J', 'item_j_id': 'M', 'quadratic_profit': 252}, {'item_i_id': 'J', 'item_j_id': 'N', 'quadratic_profit': 1280}, {'item_i_id': 'J', 'item_j_id': 'O', 'quadratic_profit': 2520}, {'item_i_id': 'K', 'item_j_id': 'K', 'quadratic_profit': 1279}, {'item_i_id': 'K', 'item_j_id': 'L', 'quadratic_profit': 207}, {'item_i_id': 'K', 'item_j_id': 'N', 'quadratic_profit': 2864}, {'item_i_id': 'K', 'item_j_id': 'O', 'quadratic_profit': 1988}, {'item_i_id': 'L', 'item_j_id': 'L', 'quadratic_profit': 343}, {'item_i_id': 'L', 'item_j_id': 'M', 'quadratic_profit': 2088}, {'item_i_id': 'L', 'item_j_id': 'N', 'quadratic_profit': 752}, {'item_i_id': 'L', 'item_j_id': 'O', 'quadratic_profit': 3102}, {'item_i_id': 'M', 'item_j_id': 'M', 'quadratic_profit': 2721}, {'item_i_id': 'M', 'item_j_id': 'N', 'quadratic_profit': 2364}]}","['A', 'B', 'C', 'D', 'E', 'F', 'H', 'I', 'J', 'K', 'L', 'N', 'O']",48,nl,names
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.

There are 12 distinct pieces listed as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and the courier bag holds up to 19637 weight units.
Material 0 adds 4586 outreach when included.
Material 1 adds 6105 outreach when included.
Material 2 adds 1708 outreach when included.
Material 3 adds 4748 outreach when included.
Material 4 adds 3919 outreach when included.
Material 5 adds 2332 outreach when included.
Material 6 adds 6200 outreach when included.
Material 7 adds 7002 outreach when included.
Material 8 adds 5839 outreach when included.
Material 9 adds 3783 outreach when included.
Material 10 adds 784 outreach when included.
Material 11 adds 4338 outreach when included.
Material 0 weighs 3936 weight units.
Material 1 weighs 5455 weight units.
Material 2 weighs 1058 weight units.
Material 3 weighs 4098 weight units.
Material 4 weighs 3269 weight units.
Material 5 weighs 1682 weight units.
Material 6 weighs 5550 weight units.
Material 7 weighs 6352 weight units.
Material 8 weighs 5189 weight units.
Material 9 weighs 3133 weight units.
Material 10 weighs 134 weight units.
Material 11 weighs 3688 weight units.
Including 0 together with 3 yields an extra 3936 outreach.
Including 1 together with 3 yields an extra 5189 outreach.
Including 1 together with 6 yields an extra 3642 outreach.
Including 2 together with 4 yields an extra 1058 outreach.
Including 2 together with 10 yields an extra 376 outreach.
Including 3 together with 5 yields an extra 2136 outreach.
Including 3 together with 11 yields an extra 376 outreach.
Including 4 together with 6 yields an extra 134 outreach.
Including 4 together with 9 yields an extra 2738 outreach.
Including 4 together with 10 yields an extra 259 outreach.
Including 5 together with 7 yields an extra 1983 outreach.
Including 5 together with 11 yields an extra 3289 outreach.
Including 6 together with 8 yields an extra 2682 outreach.
Including 6 together with 10 yields an extra 962 outreach.
Including 6 together with 11 yields an extra 3894 outreach.
Including 7 together with 9 yields an extra 3038 outreach.
Including 7 together with 11 yields an extra 1829 outreach.
Including 8 together with 11 yields an extra 5394 outreach.
Stick to the bag limit and unique-piece rule — aim for the highest outreach.

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': [4586, 6105, 1708, 4748, 3919, 2332, 6200, 7002, 5839, 3783, 784, 4338], 'quadratic_coeffs': [[0, 0, 0, 3936, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 5189, 0, 0, 3642, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1058, 0, 0, 0, 0, 0, 376, 0], [0, 0, 0, 0, 0, 2136, 0, 0, 0, 0, 0, 376], [0, 0, 0, 0, 0, 0, 134, 0, 0, 2738, 259, 0], [0, 0, 0, 0, 0, 0, 0, 1983, 0, 0, 0, 3289], [0, 0, 0, 0, 0, 0, 1673, 0, 2682, 0, 962, 3894], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3038, 0, 1829], [0, 0, 0, 0, 0, 0, 0, 0, 2682, 0, 0, 5394], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1963, 0, 0], [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': [3936, 5455, 1058, 4098, 3269, 1682, 5550, 6352, 5189, 3133, 134, 3688], 'capacity': 19637, 'solution': [4, 5, 6, 8, 10, 11], 'obj': 40026.0, 'problem_type': 'QKP'}","[4, 5, 6, 8, 10, 11]",40026.0,"{'problem_type': 'QKP', 'num_items': 12, 'capacity': 19637, 'items': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 'linear_pairs': [{'item_id': 0, 'linear_profit': 4586}, {'item_id': 1, 'linear_profit': 6105}, {'item_id': 2, 'linear_profit': 1708}, {'item_id': 3, 'linear_profit': 4748}, {'item_id': 4, 'linear_profit': 3919}, {'item_id': 5, 'linear_profit': 2332}, {'item_id': 6, 'linear_profit': 6200}, {'item_id': 7, 'linear_profit': 7002}, {'item_id': 8, 'linear_profit': 5839}, {'item_id': 9, 'linear_profit': 3783}, {'item_id': 10, 'linear_profit': 784}, {'item_id': 11, 'linear_profit': 4338}], 'weight_pairs': [{'item_id': 0, 'weight': 3936}, {'item_id': 1, 'weight': 5455}, {'item_id': 2, 'weight': 1058}, {'item_id': 3, 'weight': 4098}, {'item_id': 4, 'weight': 3269}, {'item_id': 5, 'weight': 1682}, {'item_id': 6, 'weight': 5550}, {'item_id': 7, 'weight': 6352}, {'item_id': 8, 'weight': 5189}, {'item_id': 9, 'weight': 3133}, {'item_id': 10, 'weight': 134}, {'item_id': 11, 'weight': 3688}], 'quadratic_pairs': [{'item_i_id': 0, 'item_j_id': 3, 'quadratic_profit': 3936}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 5189}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 3642}, {'item_i_id': 2, 'item_j_id': 4, 'quadratic_profit': 1058}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 376}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 2136}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 376}, {'item_i_id': 4, 'item_j_id': 6, 'quadratic_profit': 134}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 2738}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 259}, {'item_i_id': 5, 'item_j_id': 7, 'quadratic_profit': 1983}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 3289}, {'item_i_id': 6, 'item_j_id': 6, 'quadratic_profit': 1673}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 2682}, {'item_i_id': 6, 'item_j_id': 10, 'quadratic_profit': 962}, {'item_i_id': 6, 'item_j_id': 11, 'quadratic_profit': 3894}, {'item_i_id': 7, 'item_j_id': 9, 'quadratic_profit': 3038}, {'item_i_id': 7, 'item_j_id': 11, 'quadratic_profit': 1829}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 2682}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 5394}, {'item_i_id': 9, 'item_j_id': 9, 'quadratic_profit': 1963}]}","[4, 5, 6, 8, 10, 11]",49,nl,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 15 samples 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and the mailer capacity 408.

| sample_id | individual_conversion_value |
|---|---|
| 1 | 124 |
| 2 | 120 |
| 3 | 134 |
| 4 | 131 |
| 5 | 117 |
| 6 | 119 |
| 7 | 117 |
| 8 | 113 |
| 9 | 133 |
| 10 | 112 |
| 11 | 135 |
| 12 | 119 |
| 13 | 116 |
| 14 | 132 |
| 15 | 126 |

| sample_id | sample_weight |
|---|---|
| 1 | 112 |
| 2 | 108 |
| 3 | 122 |
| 4 | 119 |
| 5 | 105 |
| 6 | 107 |
| 7 | 105 |
| 8 | 101 |
| 9 | 121 |
| 10 | 100 |
| 11 | 123 |
| 12 | 107 |
| 13 | 104 |
| 14 | 120 |
| 15 | 114 |

| sample_i_id | sample_j_id | pair_bonus_conversion |
|---|---|---|
| 1 | 2 | 121 |
| 1 | 3 | 123 |
| 1 | 4 | 104 |
| 1 | 5 | 112 |
| 1 | 6 | 100 |
| 1 | 7 | 104 |
| 1 | 9 | 117 |
| 1 | 11 | 118 |
| 1 | 12 | 108 |
| 1 | 13 | 117 |
| 1 | 14 | 123 |
| 2 | 3 | 113 |
| 2 | 5 | 122 |
| 2 | 6 | 106 |
| 2 | 8 | 108 |
| 2 | 9 | 123 |
| 2 | 10 | 110 |
| 2 | 11 | 107 |
| 2 | 13 | 119 |
| 2 | 15 | 102 |
| 3 | 4 | 111 |
| 3 | 5 | 115 |
| 3 | 6 | 120 |
| 3 | 7 | 102 |
| 3 | 9 | 107 |
| 3 | 10 | 104 |
| 3 | 11 | 106 |
| 3 | 12 | 106 |
| 3 | 13 | 101 |
| 3 | 15 | 103 |
| 4 | 5 | 115 |
| 4 | 7 | 102 |
| 4 | 9 | 115 |
| 4 | 10 | 114 |
| 4 | 11 | 122 |
| 4 | 13 | 124 |
| 4 | 14 | 118 |
| 4 | 15 | 118 |
| 5 | 6 | 102 |
| 5 | 8 | 106 |
| 5 | 10 | 116 |
| 5 | 11 | 101 |
| 5 | 12 | 108 |
| 5 | 13 | 107 |
| 5 | 14 | 116 |
| 5 | 15 | 100 |
| 6 | 7 | 107 |
| 6 | 8 | 111 |
| 6 | 9 | 107 |
| 6 | 13 | 100 |
| 6 | 14 | 102 |
| 6 | 15 | 109 |
| 7 | 8 | 122 |
| 7 | 10 | 101 |
| 7 | 11 | 109 |
| 7 | 14 | 110 |
| 7 | 15 | 107 |
| 8 | 9 | 104 |
| 8 | 10 | 115 |
| 8 | 11 | 104 |
| 8 | 13 | 105 |
| 8 | 14 | 123 |
| 8 | 15 | 107 |
| 9 | 10 | 122 |
| 9 | 12 | 116 |
| 9 | 13 | 100 |
| 9 | 15 | 113 |
| 10 | 11 | 124 |
| 10 | 12 | 120 |
| 10 | 13 | 112 |
| 10 | 14 | 122 |
| 10 | 15 | 102 |
| 11 | 13 | 119 |
| 11 | 14 | 111 |
| 12 | 13 | 115 |
| 12 | 14 | 121 |
| 12 | 15 | 114 |
| 13 | 15 | 121 |
| 14 | 15 | 107 |

We’ll pick the kit that maximizes total conversion while staying within the 408 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': [124, 120, 134, 131, 117, 119, 117, 113, 133, 112, 135, 119, 116, 132, 126], 'quadratic_coeffs': [[0, 121, 123, 104, 112, 100, 104, 0, 117, 0, 118, 108, 117, 123, 0], [0, 110, 113, 0, 122, 106, 0, 108, 123, 110, 107, 0, 119, 0, 102], [0, 0, 0, 111, 115, 120, 102, 0, 107, 104, 106, 106, 101, 0, 103], [0, 0, 0, 103, 115, 0, 102, 0, 115, 114, 122, 0, 124, 118, 118], [0, 0, 0, 0, 0, 102, 0, 106, 0, 116, 101, 108, 107, 116, 100], [0, 0, 0, 0, 0, 0, 107, 111, 107, 0, 0, 0, 100, 102, 109], [0, 0, 0, 0, 0, 0, 104, 122, 0, 101, 109, 0, 0, 110, 107], [0, 0, 0, 0, 0, 0, 0, 122, 104, 115, 104, 0, 105, 123, 107], [0, 0, 0, 0, 0, 0, 0, 0, 0, 122, 0, 116, 100, 0, 113], [0, 0, 0, 0, 0, 0, 0, 0, 0, 117, 124, 120, 112, 122, 102], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 0, 119, 111, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 110, 115, 121, 114], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 118, 0, 121], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 107], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 109]], 'weights': [112, 108, 122, 119, 105, 107, 105, 101, 121, 100, 123, 107, 104, 120, 114], 'capacity': 408, 'solution': [3, 10, 13], 'obj': 749.0, 'problem_type': 'QKP'}","[3, 10, 13]",749.0,"{'problem_type': 'QKP', 'num_items': 15, 'capacity': 408, 'items': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], 'linear_pairs': [{'item_id': 1, 'linear_profit': 124}, {'item_id': 2, 'linear_profit': 120}, {'item_id': 3, 'linear_profit': 134}, {'item_id': 4, 'linear_profit': 131}, {'item_id': 5, 'linear_profit': 117}, {'item_id': 6, 'linear_profit': 119}, {'item_id': 7, 'linear_profit': 117}, {'item_id': 8, 'linear_profit': 113}, {'item_id': 9, 'linear_profit': 133}, {'item_id': 10, 'linear_profit': 112}, {'item_id': 11, 'linear_profit': 135}, {'item_id': 12, 'linear_profit': 119}, {'item_id': 13, 'linear_profit': 116}, {'item_id': 14, 'linear_profit': 132}, {'item_id': 15, 'linear_profit': 126}], 'weight_pairs': [{'item_id': 1, 'weight': 112}, {'item_id': 2, 'weight': 108}, {'item_id': 3, 'weight': 122}, {'item_id': 4, 'weight': 119}, {'item_id': 5, 'weight': 105}, {'item_id': 6, 'weight': 107}, {'item_id': 7, 'weight': 105}, {'item_id': 8, 'weight': 101}, {'item_id': 9, 'weight': 121}, {'item_id': 10, 'weight': 100}, {'item_id': 11, 'weight': 123}, {'item_id': 12, 'weight': 107}, {'item_id': 13, 'weight': 104}, {'item_id': 14, 'weight': 120}, {'item_id': 15, 'weight': 114}], 'quadratic_pairs': [{'item_i_id': 1, 'item_j_id': 2, 'quadratic_profit': 121}, {'item_i_id': 1, 'item_j_id': 3, 'quadratic_profit': 123}, {'item_i_id': 1, 'item_j_id': 4, 'quadratic_profit': 104}, {'item_i_id': 1, 'item_j_id': 5, 'quadratic_profit': 112}, {'item_i_id': 1, 'item_j_id': 6, 'quadratic_profit': 100}, {'item_i_id': 1, 'item_j_id': 7, 'quadratic_profit': 104}, {'item_i_id': 1, 'item_j_id': 9, 'quadratic_profit': 117}, {'item_i_id': 1, 'item_j_id': 11, 'quadratic_profit': 118}, {'item_i_id': 1, 'item_j_id': 12, 'quadratic_profit': 108}, {'item_i_id': 1, 'item_j_id': 13, 'quadratic_profit': 117}, {'item_i_id': 1, 'item_j_id': 14, 'quadratic_profit': 123}, {'item_i_id': 2, 'item_j_id': 2, 'quadratic_profit': 110}, {'item_i_id': 2, 'item_j_id': 3, 'quadratic_profit': 113}, {'item_i_id': 2, 'item_j_id': 5, 'quadratic_profit': 122}, {'item_i_id': 2, 'item_j_id': 6, 'quadratic_profit': 106}, {'item_i_id': 2, 'item_j_id': 8, 'quadratic_profit': 108}, {'item_i_id': 2, 'item_j_id': 9, 'quadratic_profit': 123}, {'item_i_id': 2, 'item_j_id': 10, 'quadratic_profit': 110}, {'item_i_id': 2, 'item_j_id': 11, 'quadratic_profit': 107}, {'item_i_id': 2, 'item_j_id': 13, 'quadratic_profit': 119}, {'item_i_id': 2, 'item_j_id': 15, 'quadratic_profit': 102}, {'item_i_id': 3, 'item_j_id': 4, 'quadratic_profit': 111}, {'item_i_id': 3, 'item_j_id': 5, 'quadratic_profit': 115}, {'item_i_id': 3, 'item_j_id': 6, 'quadratic_profit': 120}, {'item_i_id': 3, 'item_j_id': 7, 'quadratic_profit': 102}, {'item_i_id': 3, 'item_j_id': 9, 'quadratic_profit': 107}, {'item_i_id': 3, 'item_j_id': 10, 'quadratic_profit': 104}, {'item_i_id': 3, 'item_j_id': 11, 'quadratic_profit': 106}, {'item_i_id': 3, 'item_j_id': 12, 'quadratic_profit': 106}, {'item_i_id': 3, 'item_j_id': 13, 'quadratic_profit': 101}, {'item_i_id': 3, 'item_j_id': 15, 'quadratic_profit': 103}, {'item_i_id': 4, 'item_j_id': 4, 'quadratic_profit': 103}, {'item_i_id': 4, 'item_j_id': 5, 'quadratic_profit': 115}, {'item_i_id': 4, 'item_j_id': 7, 'quadratic_profit': 102}, {'item_i_id': 4, 'item_j_id': 9, 'quadratic_profit': 115}, {'item_i_id': 4, 'item_j_id': 10, 'quadratic_profit': 114}, {'item_i_id': 4, 'item_j_id': 11, 'quadratic_profit': 122}, {'item_i_id': 4, 'item_j_id': 13, 'quadratic_profit': 124}, {'item_i_id': 4, 'item_j_id': 14, 'quadratic_profit': 118}, {'item_i_id': 4, 'item_j_id': 15, 'quadratic_profit': 118}, {'item_i_id': 5, 'item_j_id': 6, 'quadratic_profit': 102}, {'item_i_id': 5, 'item_j_id': 8, 'quadratic_profit': 106}, {'item_i_id': 5, 'item_j_id': 10, 'quadratic_profit': 116}, {'item_i_id': 5, 'item_j_id': 11, 'quadratic_profit': 101}, {'item_i_id': 5, 'item_j_id': 12, 'quadratic_profit': 108}, {'item_i_id': 5, 'item_j_id': 13, 'quadratic_profit': 107}, {'item_i_id': 5, 'item_j_id': 14, 'quadratic_profit': 116}, {'item_i_id': 5, 'item_j_id': 15, 'quadratic_profit': 100}, {'item_i_id': 6, 'item_j_id': 7, 'quadratic_profit': 107}, {'item_i_id': 6, 'item_j_id': 8, 'quadratic_profit': 111}, {'item_i_id': 6, 'item_j_id': 9, 'quadratic_profit': 107}, {'item_i_id': 6, 'item_j_id': 13, 'quadratic_profit': 100}, {'item_i_id': 6, 'item_j_id': 14, 'quadratic_profit': 102}, {'item_i_id': 6, 'item_j_id': 15, 'quadratic_profit': 109}, {'item_i_id': 7, 'item_j_id': 7, 'quadratic_profit': 104}, {'item_i_id': 7, 'item_j_id': 8, 'quadratic_profit': 122}, {'item_i_id': 7, 'item_j_id': 10, 'quadratic_profit': 101}, {'item_i_id': 7, 'item_j_id': 11, 'quadratic_profit': 109}, {'item_i_id': 7, 'item_j_id': 14, 'quadratic_profit': 110}, {'item_i_id': 7, 'item_j_id': 15, 'quadratic_profit': 107}, {'item_i_id': 8, 'item_j_id': 8, 'quadratic_profit': 122}, {'item_i_id': 8, 'item_j_id': 9, 'quadratic_profit': 104}, {'item_i_id': 8, 'item_j_id': 10, 'quadratic_profit': 115}, {'item_i_id': 8, 'item_j_id': 11, 'quadratic_profit': 104}, {'item_i_id': 8, 'item_j_id': 13, 'quadratic_profit': 105}, {'item_i_id': 8, 'item_j_id': 14, 'quadratic_profit': 123}, {'item_i_id': 8, 'item_j_id': 15, 'quadratic_profit': 107}, {'item_i_id': 9, 'item_j_id': 10, 'quadratic_profit': 122}, {'item_i_id': 9, 'item_j_id': 12, 'quadratic_profit': 116}, {'item_i_id': 9, 'item_j_id': 13, 'quadratic_profit': 100}, {'item_i_id': 9, 'item_j_id': 15, 'quadratic_profit': 113}, {'item_i_id': 10, 'item_j_id': 10, 'quadratic_profit': 117}, {'item_i_id': 10, 'item_j_id': 11, 'quadratic_profit': 124}, {'item_i_id': 10, 'item_j_id': 12, 'quadratic_profit': 120}, {'item_i_id': 10, 'item_j_id': 13, 'quadratic_profit': 112}, {'item_i_id': 10, 'item_j_id': 14, 'quadratic_profit': 122}, {'item_i_id': 10, 'item_j_id': 15, 'quadratic_profit': 102}, {'item_i_id': 11, 'item_j_id': 11, 'quadratic_profit': 100}, {'item_i_id': 11, 'item_j_id': 13, 'quadratic_profit': 119}, {'item_i_id': 11, 'item_j_id': 14, 'quadratic_profit': 111}, {'item_i_id': 12, 'item_j_id': 12, 'quadratic_profit': 110}, {'item_i_id': 12, 'item_j_id': 13, 'quadratic_profit': 115}, {'item_i_id': 12, 'item_j_id': 14, 'quadratic_profit': 121}, {'item_i_id': 12, 'item_j_id': 15, 'quadratic_profit': 114}, {'item_i_id': 13, 'item_j_id': 13, 'quadratic_profit': 118}, {'item_i_id': 13, 'item_j_id': 15, 'quadratic_profit': 121}, {'item_i_id': 14, 'item_j_id': 15, 'quadratic_profit': 107}, {'item_i_id': 15, 'item_j_id': 15, 'quadratic_profit': 109}]}","[4, 11, 14]",50,markdown_table,1