[ { "task_name": "MDP", "problem_type": "MDP", "instruction": "Many people on the project wanted a small selection of fabrics that covered a wide range of looks, so the job was to choose exactly m different swatches from the sample box. For every two swatches there’s a number showing how different they are, and the score for any chosen set is the sum of those numbers across all pairs inside it — the bigger that sum, the better the set for variety. No repeats; exactly m must be picked, and the detailed pairwise numbers are shown below.\n\n{\n \"total_swatches_available\": 19,\n \"swatches_to_select\": 6,\n \"swatch_identifiers\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\"\n ],\n \"data\": [\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"B\",\n \"texture_color_dissimilarity\": 6\n },\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"C\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"D\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"E\",\n \"texture_color_dissimilarity\": 4\n },\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"F\",\n \"texture_color_dissimilarity\": 8\n },\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"G\",\n \"texture_color_dissimilarity\": 7\n },\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"H\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"I\",\n \"texture_color_dissimilarity\": 8\n },\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"J\",\n \"texture_color_dissimilarity\": 3\n },\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"K\",\n \"texture_color_dissimilarity\": 0\n },\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"L\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"M\",\n \"texture_color_dissimilarity\": 5\n },\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"N\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"O\",\n \"texture_color_dissimilarity\": 8\n },\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"P\",\n \"texture_color_dissimilarity\": 0\n },\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"Q\",\n \"texture_color_dissimilarity\": 3\n },\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"R\",\n \"texture_color_dissimilarity\": 9\n },\n {\n \"swatch_id_a\": \"A\",\n \"swatch_id_b\": \"S\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"A\",\n \"texture_color_dissimilarity\": 6\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"C\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"D\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"E\",\n \"texture_color_dissimilarity\": 4\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"F\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"G\",\n \"texture_color_dissimilarity\": 4\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"H\",\n \"texture_color_dissimilarity\": 8\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"I\",\n \"texture_color_dissimilarity\": 9\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"J\",\n \"texture_color_dissimilarity\": 5\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"K\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"L\",\n \"texture_color_dissimilarity\": 7\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"M\",\n \"texture_color_dissimilarity\": 4\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"N\",\n \"texture_color_dissimilarity\": 9\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"O\",\n \"texture_color_dissimilarity\": 8\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"P\",\n \"texture_color_dissimilarity\": 8\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"Q\",\n \"texture_color_dissimilarity\": 6\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"R\",\n \"texture_color_dissimilarity\": 0\n },\n {\n \"swatch_id_a\": \"B\",\n \"swatch_id_b\": \"S\",\n \"texture_color_dissimilarity\": 5\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"A\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"B\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"D\",\n \"texture_color_dissimilarity\": 4\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"E\",\n \"texture_color_dissimilarity\": 0\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"F\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"G\",\n \"texture_color_dissimilarity\": 9\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"H\",\n \"texture_color_dissimilarity\": 5\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"I\",\n \"texture_color_dissimilarity\": 6\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"J\",\n \"texture_color_dissimilarity\": 6\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"K\",\n \"texture_color_dissimilarity\": 5\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"L\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"M\",\n \"texture_color_dissimilarity\": 7\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"N\",\n \"texture_color_dissimilarity\": 5\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"O\",\n \"texture_color_dissimilarity\": 9\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"P\",\n \"texture_color_dissimilarity\": 6\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"Q\",\n \"texture_color_dissimilarity\": 6\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"R\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"C\",\n \"swatch_id_b\": \"S\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"A\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"B\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"C\",\n \"texture_color_dissimilarity\": 4\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"E\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"F\",\n \"texture_color_dissimilarity\": 3\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"G\",\n \"texture_color_dissimilarity\": 8\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"H\",\n \"texture_color_dissimilarity\": 3\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"I\",\n \"texture_color_dissimilarity\": 3\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"J\",\n \"texture_color_dissimilarity\": 3\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"K\",\n \"texture_color_dissimilarity\": 9\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"L\",\n \"texture_color_dissimilarity\": 0\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"M\",\n \"texture_color_dissimilarity\": 3\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"N\",\n \"texture_color_dissimilarity\": 7\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"O\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"P\",\n \"texture_color_dissimilarity\": 9\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"Q\",\n \"texture_color_dissimilarity\": 5\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"R\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"D\",\n \"swatch_id_b\": \"S\",\n \"texture_color_dissimilarity\": 4\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"A\",\n \"texture_color_dissimilarity\": 4\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"B\",\n \"texture_color_dissimilarity\": 4\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"C\",\n \"texture_color_dissimilarity\": 0\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"D\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"F\",\n \"texture_color_dissimilarity\": 5\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"G\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"H\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"I\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"J\",\n \"texture_color_dissimilarity\": 8\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"K\",\n \"texture_color_dissimilarity\": 0\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"L\",\n \"texture_color_dissimilarity\": 7\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"M\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"N\",\n \"texture_color_dissimilarity\": 4\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"O\",\n \"texture_color_dissimilarity\": 6\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"P\",\n \"texture_color_dissimilarity\": 7\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"Q\",\n \"texture_color_dissimilarity\": 5\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"R\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"E\",\n \"swatch_id_b\": \"S\",\n \"texture_color_dissimilarity\": 0\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"A\",\n \"texture_color_dissimilarity\": 8\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"B\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"C\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"D\",\n \"texture_color_dissimilarity\": 3\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"E\",\n \"texture_color_dissimilarity\": 5\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"G\",\n \"texture_color_dissimilarity\": 9\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"H\",\n \"texture_color_dissimilarity\": 4\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"I\",\n \"texture_color_dissimilarity\": 7\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"J\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"K\",\n \"texture_color_dissimilarity\": 6\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"L\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"M\",\n \"texture_color_dissimilarity\": 7\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"N\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"O\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"P\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"Q\",\n \"texture_color_dissimilarity\": 0\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"R\",\n \"texture_color_dissimilarity\": 7\n },\n {\n \"swatch_id_a\": \"F\",\n \"swatch_id_b\": \"S\",\n \"texture_color_dissimilarity\": 4\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"A\",\n \"texture_color_dissimilarity\": 7\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"B\",\n \"texture_color_dissimilarity\": 4\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"C\",\n \"texture_color_dissimilarity\": 9\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"D\",\n \"texture_color_dissimilarity\": 8\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"E\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"F\",\n \"texture_color_dissimilarity\": 9\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"H\",\n \"texture_color_dissimilarity\": 0\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"I\",\n \"texture_color_dissimilarity\": 4\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"J\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"K\",\n \"texture_color_dissimilarity\": 6\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"L\",\n \"texture_color_dissimilarity\": 3\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"M\",\n \"texture_color_dissimilarity\": 0\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"N\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"O\",\n \"texture_color_dissimilarity\": 9\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"P\",\n \"texture_color_dissimilarity\": 7\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"Q\",\n \"texture_color_dissimilarity\": 7\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"R\",\n \"texture_color_dissimilarity\": 7\n },\n {\n \"swatch_id_a\": \"G\",\n \"swatch_id_b\": \"S\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"A\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"B\",\n \"texture_color_dissimilarity\": 8\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"C\",\n \"texture_color_dissimilarity\": 5\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"D\",\n \"texture_color_dissimilarity\": 3\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"E\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"F\",\n \"texture_color_dissimilarity\": 4\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"G\",\n \"texture_color_dissimilarity\": 0\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"I\",\n \"texture_color_dissimilarity\": 7\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"J\",\n \"texture_color_dissimilarity\": 6\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"K\",\n \"texture_color_dissimilarity\": 9\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"L\",\n \"texture_color_dissimilarity\": 3\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"M\",\n \"texture_color_dissimilarity\": 8\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"N\",\n \"texture_color_dissimilarity\": 3\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"O\",\n \"texture_color_dissimilarity\": 0\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"P\",\n \"texture_color_dissimilarity\": 1\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"Q\",\n \"texture_color_dissimilarity\": 4\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"R\",\n \"texture_color_dissimilarity\": 3\n },\n {\n \"swatch_id_a\": \"H\",\n \"swatch_id_b\": \"S\",\n \"texture_color_dissimilarity\": 3\n },\n {\n \"swatch_id_a\": \"I\",\n \"swatch_id_b\": \"A\",\n \"texture_color_dissimilarity\": 8\n },\n {\n \"swatch_id_a\": \"I\",\n \"swatch_id_b\": \"B\",\n \"texture_color_dissimilarity\": 9\n },\n {\n \"swatch_id_a\": \"I\",\n \"swatch_id_b\": \"C\",\n \"texture_color_dissimilarity\": 6\n },\n {\n \"swatch_id_a\": \"I\",\n \"swatch_id_b\": \"D\",\n \"texture_color_dissimilarity\": 3\n },\n {\n \"swatch_id_a\": \"I\",\n \"swatch_id_b\": \"E\",\n \"texture_color_dissimilarity\": 2\n },\n {\n \"swatch_id_a\": \"I\",\n \"swatch_id_b\": \"F\",\n \"texture_color_dissimilarity\": 7\n },\n {\n \"swatch_id_a\": \"I\",\n \"swatch_id_b\": \"G\",\n \"texture_color_dissimilarity\": 4\n },\n {\n \"swatch_id_a\": \"I\",\n \"swatch_id_b\": \"H\",\n \"texture_color_dissimilarity\": 7\n },\n {\n \"swatch_id_a\": \"I\",\n \"swatch_id_b\": \"J\",\n \"texture_color_dissimilarity\": 8\n },\n {\n \"swatch_id_a\": \"I\",\n \"swatch_id_b\": \"K\",\n \"texture_color_dissimilarity\": 8\n },\n {\n \"swatch_id_a\": \"I\",\n \"swatch_id_b\": \"L\",\n \"texture_color_dissimilarity\": 5\n },\n {\n \"swatch_id_a\": \"I\",\n \"swatch_id_b\": \"M\",\n \"texture_color_dissimilarity\": 9\n },\n {\n \"swatch_id_a\": \"I\",\n \"swatch_id_b\": \"N\",\n 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\"texture_color_dissimilarity\": 0\n },\n {\n \"swatch_id_a\": \"S\",\n \"swatch_id_b\": \"N\",\n \"texture_color_dissimilarity\": 0\n },\n {\n \"swatch_id_a\": \"S\",\n \"swatch_id_b\": \"O\",\n \"texture_color_dissimilarity\": 5\n },\n {\n \"swatch_id_a\": \"S\",\n \"swatch_id_b\": \"P\",\n \"texture_color_dissimilarity\": 0\n },\n {\n \"swatch_id_a\": \"S\",\n \"swatch_id_b\": \"Q\",\n \"texture_color_dissimilarity\": 9\n },\n {\n \"swatch_id_a\": \"S\",\n \"swatch_id_b\": \"R\",\n \"texture_color_dissimilarity\": 2\n }\n ]\n}\n\nJust a quick note — when you reply with the chosen swatches, please use this simple JSON shape so it's easy to read and parse:\n\n{\n \"solution\": [, , ...]\n}\n\nThe \"solution\" array is where you list the exactly m swatch identifiers you picked from the box — one identifier per entry. Think of each as the label that’s printed on a swatch in the sample box. This JSON is just a sketch of the shape I expect, not the actual answer itself.\n\nAlso, please use the identifiers exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as \"1\" or \"23\", single capital letters like \"A\" or \"B\", or a capital letter followed by digits like \"A1\" or \"X7\".\"", "instance": { "distance_matrix": [ [ 0, 6, 2, 2, 4, 8, 7, 2, 8, 3, 0, 2, 5, 1, 8, 0, 3, 9, 1 ], [ 6, 0, 1, 2, 4, 2, 4, 8, 9, 5, 1, 7, 4, 9, 8, 8, 6, 0, 5 ], [ 2, 1, 0, 4, 0, 2, 9, 5, 6, 6, 5, 1, 7, 5, 9, 6, 6, 2, 1 ], [ 2, 2, 4, 0, 2, 3, 8, 3, 3, 3, 9, 0, 3, 7, 2, 9, 5, 1, 4 ], [ 4, 4, 0, 2, 0, 5, 1, 1, 2, 8, 0, 7, 1, 4, 6, 7, 5, 1, 0 ], [ 8, 2, 2, 3, 5, 0, 9, 4, 7, 2, 6, 1, 7, 1, 1, 2, 0, 7, 4 ], [ 7, 4, 9, 8, 1, 9, 0, 0, 4, 1, 6, 3, 0, 2, 9, 7, 7, 7, 2 ], [ 2, 8, 5, 3, 1, 4, 0, 0, 7, 6, 9, 3, 8, 3, 0, 1, 4, 3, 3 ], [ 8, 9, 6, 3, 2, 7, 4, 7, 0, 8, 8, 5, 9, 5, 6, 7, 7, 7, 5 ], [ 3, 5, 6, 3, 8, 2, 1, 6, 8, 0, 0, 0, 1, 6, 9, 7, 0, 7, 6 ], [ 0, 1, 5, 9, 0, 6, 6, 9, 8, 0, 0, 7, 5, 4, 4, 4, 6, 9, 9 ], [ 2, 7, 1, 0, 7, 1, 3, 3, 5, 0, 7, 0, 8, 8, 1, 3, 7, 9, 0 ], [ 5, 4, 7, 3, 1, 7, 0, 8, 9, 1, 5, 8, 0, 3, 2, 8, 3, 1, 0 ], [ 1, 9, 5, 7, 4, 1, 2, 3, 5, 6, 4, 8, 3, 0, 2, 4, 1, 3, 0 ], [ 8, 8, 9, 2, 6, 1, 9, 0, 6, 9, 4, 1, 2, 2, 0, 8, 6, 8, 5 ], [ 0, 8, 6, 9, 7, 2, 7, 1, 7, 7, 4, 3, 8, 4, 8, 0, 4, 1, 0 ], [ 3, 6, 6, 5, 5, 0, 7, 4, 7, 0, 6, 7, 3, 1, 6, 4, 0, 3, 9 ], [ 9, 0, 2, 1, 1, 7, 7, 3, 7, 7, 9, 9, 1, 3, 8, 1, 3, 0, 2 ], [ 1, 5, 1, 4, 0, 4, 2, 3, 5, 6, 9, 0, 0, 0, 5, 0, 9, 2, 0 ] ], "m": 6, "objective": 105.0 }, "solution": [ 0, 5, 6, 8, 14, 17 ], "obj": 105.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 19, "m": 6, "nodes": [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S" ], "distance_pairs": [ { "from_id": "A", "to_id": "B", "distance": 6 }, { "from_id": "A", "to_id": "C", "distance": 2 }, { "from_id": "A", "to_id": "D", 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"from_id": "B", "to_id": "G", "distance": 4 }, { "from_id": "B", "to_id": "H", "distance": 8 }, { "from_id": "B", "to_id": "I", "distance": 9 }, { "from_id": "B", "to_id": "J", "distance": 5 }, { "from_id": "B", "to_id": "K", "distance": 1 }, { "from_id": "B", "to_id": "L", "distance": 7 }, { "from_id": "B", "to_id": "M", "distance": 4 }, { "from_id": "B", "to_id": "N", "distance": 9 }, { "from_id": "B", "to_id": "O", "distance": 8 }, { "from_id": "B", "to_id": "P", "distance": 8 }, { "from_id": "B", "to_id": "Q", "distance": 6 }, { "from_id": "B", "to_id": "R", "distance": 0 }, { "from_id": "B", "to_id": "S", "distance": 5 }, { "from_id": "C", "to_id": "A", "distance": 2 }, { "from_id": "C", "to_id": "B", "distance": 1 }, { "from_id": "C", "to_id": "D", "distance": 4 }, { "from_id": "C", "to_id": "E", "distance": 0 }, { "from_id": "C", "to_id": "F", "distance": 2 }, { "from_id": "C", "to_id": "G", "distance": 9 }, { "from_id": "C", "to_id": "H", "distance": 5 }, { "from_id": "C", 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"distance": 9 }, { "from_id": "D", "to_id": "L", "distance": 0 }, { "from_id": "D", "to_id": "M", "distance": 3 }, { "from_id": "D", "to_id": "N", "distance": 7 }, { "from_id": "D", "to_id": "O", "distance": 2 }, { "from_id": "D", "to_id": "P", "distance": 9 }, { "from_id": "D", "to_id": "Q", "distance": 5 }, { "from_id": "D", "to_id": "R", "distance": 1 }, { "from_id": "D", "to_id": "S", "distance": 4 }, { "from_id": "E", "to_id": "A", "distance": 4 }, { "from_id": "E", "to_id": "B", "distance": 4 }, { "from_id": "E", "to_id": "C", "distance": 0 }, { "from_id": "E", "to_id": "D", "distance": 2 }, { "from_id": "E", "to_id": "F", "distance": 5 }, { "from_id": "E", "to_id": "G", "distance": 1 }, { "from_id": "E", "to_id": "H", "distance": 1 }, { "from_id": "E", "to_id": "I", "distance": 2 }, { "from_id": "E", "to_id": "J", "distance": 8 }, { "from_id": "E", "to_id": "K", "distance": 0 }, { "from_id": "E", "to_id": "L", "distance": 7 }, { "from_id": "E", "to_id": "M", "distance": 1 }, { "from_id": "E", "to_id": "N", "distance": 4 }, { "from_id": "E", "to_id": "O", "distance": 6 }, { "from_id": "E", "to_id": "P", "distance": 7 }, { "from_id": "E", "to_id": "Q", "distance": 5 }, { "from_id": "E", "to_id": "R", "distance": 1 }, { "from_id": "E", "to_id": "S", "distance": 0 }, { "from_id": "F", "to_id": "A", "distance": 8 }, { "from_id": "F", "to_id": "B", "distance": 2 }, { "from_id": "F", "to_id": "C", "distance": 2 }, { "from_id": "F", "to_id": "D", "distance": 3 }, { "from_id": "F", "to_id": "E", "distance": 5 }, { "from_id": "F", "to_id": "G", "distance": 9 }, { "from_id": "F", "to_id": "H", "distance": 4 }, { "from_id": "F", "to_id": "I", "distance": 7 }, { "from_id": "F", "to_id": "J", "distance": 2 }, { "from_id": "F", "to_id": "K", "distance": 6 }, { "from_id": "F", "to_id": "L", "distance": 1 }, { "from_id": "F", "to_id": "M", "distance": 7 }, { "from_id": "F", "to_id": "N", "distance": 1 }, { "from_id": "F", "to_id": "O", "distance": 1 }, { "from_id": "F", 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"to_id": "K", "distance": 5 }, { "from_id": "M", "to_id": "L", "distance": 8 }, { "from_id": "M", "to_id": "N", "distance": 3 }, { "from_id": "M", "to_id": "O", "distance": 2 }, { "from_id": "M", "to_id": "P", "distance": 8 }, { "from_id": "M", "to_id": "Q", "distance": 3 }, { "from_id": "M", "to_id": "R", "distance": 1 }, { "from_id": "M", "to_id": "S", "distance": 0 }, { "from_id": "N", "to_id": "A", "distance": 1 }, { "from_id": "N", "to_id": "B", "distance": 9 }, { "from_id": "N", "to_id": "C", "distance": 5 }, { "from_id": "N", "to_id": "D", "distance": 7 }, { "from_id": "N", "to_id": "E", "distance": 4 }, { "from_id": "N", "to_id": "F", "distance": 1 }, { "from_id": "N", "to_id": "G", "distance": 2 }, { "from_id": "N", "to_id": "H", "distance": 3 }, { "from_id": "N", "to_id": "I", "distance": 5 }, { "from_id": "N", "to_id": "J", "distance": 6 }, { "from_id": "N", "to_id": "K", "distance": 4 }, { "from_id": "N", "to_id": "L", "distance": 8 }, { "from_id": "N", "to_id": "M", "distance": 3 }, { "from_id": "N", "to_id": "O", "distance": 2 }, { "from_id": "N", "to_id": "P", "distance": 4 }, { "from_id": "N", "to_id": "Q", "distance": 1 }, { "from_id": "N", "to_id": "R", "distance": 3 }, { "from_id": "N", "to_id": "S", "distance": 0 }, { "from_id": "O", "to_id": "A", "distance": 8 }, { "from_id": "O", "to_id": "B", "distance": 8 }, { "from_id": "O", "to_id": "C", "distance": 9 }, { "from_id": "O", "to_id": "D", "distance": 2 }, { "from_id": "O", "to_id": "E", "distance": 6 }, { "from_id": "O", "to_id": "F", "distance": 1 }, { "from_id": "O", "to_id": "G", "distance": 9 }, { "from_id": "O", "to_id": "H", "distance": 0 }, { "from_id": "O", "to_id": "I", "distance": 6 }, { "from_id": "O", "to_id": "J", "distance": 9 }, { "from_id": "O", "to_id": "K", "distance": 4 }, { "from_id": "O", "to_id": "L", "distance": 1 }, { "from_id": "O", "to_id": "M", "distance": 2 }, { "from_id": "O", "to_id": "N", "distance": 2 }, { "from_id": "O", "to_id": "P", "distance": 8 }, { "from_id": "O", "to_id": "Q", "distance": 6 }, { "from_id": "O", "to_id": "R", "distance": 8 }, { "from_id": "O", "to_id": "S", "distance": 5 }, { "from_id": "P", "to_id": "A", "distance": 0 }, { "from_id": "P", "to_id": "B", "distance": 8 }, { "from_id": "P", "to_id": "C", "distance": 6 }, { "from_id": "P", "to_id": "D", "distance": 9 }, { "from_id": "P", "to_id": "E", "distance": 7 }, { "from_id": "P", "to_id": "F", "distance": 2 }, { "from_id": "P", "to_id": "G", "distance": 7 }, { "from_id": "P", "to_id": "H", "distance": 1 }, { "from_id": "P", "to_id": "I", "distance": 7 }, { "from_id": "P", "to_id": "J", "distance": 7 }, { "from_id": "P", "to_id": "K", "distance": 4 }, { "from_id": "P", "to_id": "L", "distance": 3 }, { "from_id": "P", "to_id": "M", "distance": 8 }, { "from_id": "P", "to_id": "N", "distance": 4 }, { "from_id": "P", "to_id": "O", "distance": 8 }, { "from_id": "P", "to_id": "Q", "distance": 4 }, { "from_id": "P", "to_id": "R", "distance": 1 }, { "from_id": "P", "to_id": "S", "distance": 0 }, { "from_id": "Q", "to_id": "A", "distance": 3 }, { "from_id": "Q", "to_id": "B", "distance": 6 }, { "from_id": "Q", "to_id": "C", "distance": 6 }, { "from_id": "Q", "to_id": "D", "distance": 5 }, { "from_id": "Q", "to_id": "E", "distance": 5 }, { "from_id": "Q", "to_id": "F", "distance": 0 }, { "from_id": "Q", "to_id": "G", "distance": 7 }, { "from_id": "Q", "to_id": "H", "distance": 4 }, { "from_id": "Q", "to_id": "I", "distance": 7 }, { "from_id": "Q", "to_id": "J", "distance": 0 }, { "from_id": "Q", "to_id": "K", "distance": 6 }, { "from_id": "Q", "to_id": "L", "distance": 7 }, { "from_id": "Q", "to_id": "M", "distance": 3 }, { "from_id": "Q", "to_id": "N", "distance": 1 }, { "from_id": "Q", "to_id": "O", "distance": 6 }, { "from_id": "Q", "to_id": "P", "distance": 4 }, { "from_id": "Q", "to_id": "R", "distance": 3 }, { "from_id": "Q", "to_id": "S", "distance": 9 }, { "from_id": "R", "to_id": "A", "distance": 9 }, { "from_id": "R", "to_id": "B", "distance": 0 }, { "from_id": "R", "to_id": "C", "distance": 2 }, { "from_id": "R", "to_id": "D", "distance": 1 }, { "from_id": "R", "to_id": "E", "distance": 1 }, { "from_id": "R", "to_id": "F", "distance": 7 }, { "from_id": "R", "to_id": "G", "distance": 7 }, { "from_id": "R", "to_id": "H", "distance": 3 }, { "from_id": "R", "to_id": "I", "distance": 7 }, { "from_id": "R", "to_id": "J", "distance": 7 }, { "from_id": "R", "to_id": "K", "distance": 9 }, { "from_id": "R", "to_id": "L", "distance": 9 }, { "from_id": "R", "to_id": "M", "distance": 1 }, { "from_id": "R", "to_id": "N", "distance": 3 }, { "from_id": "R", "to_id": "O", "distance": 8 }, { "from_id": "R", "to_id": "P", "distance": 1 }, { "from_id": "R", "to_id": "Q", "distance": 3 }, { "from_id": "R", "to_id": "S", "distance": 2 }, { "from_id": "S", "to_id": "A", "distance": 1 }, { "from_id": "S", "to_id": "B", "distance": 5 }, { "from_id": "S", "to_id": "C", "distance": 1 }, { "from_id": "S", "to_id": "D", "distance": 4 }, { "from_id": "S", "to_id": "E", "distance": 0 }, { "from_id": "S", "to_id": "F", "distance": 4 }, { "from_id": "S", "to_id": "G", "distance": 2 }, { "from_id": "S", "to_id": "H", "distance": 3 }, { "from_id": "S", "to_id": "I", "distance": 5 }, { "from_id": "S", "to_id": "J", "distance": 6 }, { "from_id": "S", "to_id": "K", "distance": 9 }, { "from_id": "S", "to_id": "L", "distance": 0 }, { "from_id": "S", "to_id": "M", "distance": 0 }, { "from_id": "S", "to_id": "N", "distance": 0 }, { "from_id": "S", "to_id": "O", "distance": 5 }, { "from_id": "S", "to_id": "P", "distance": 0 }, { "from_id": "S", "to_id": "Q", "distance": 9 }, { "from_id": "S", "to_id": "R", "distance": 2 } ] }, "solution_variant": [ "A", "F", "G", "I", "O", "R" ], "context_index": 1, "input_format": "json", "input_index_base": "names" }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Many DJs prefer a set that jumps between sounds, so the brief here was simple: choose m distinct tracks from the candidates to get the widest possible spread. Every two tracks have a measured distance, and a selection gets scored by summing the distances for every pair among the chosen tracks — a bigger sum means a more varied playlist. You can’t pick the same track twice and the selection must be exactly m items. The concrete candidate list and pairwise distances are listed below.\n\nThere are 24 candidate tracks: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X. The brief requires choosing exactly 7 distinct tracks.\nTracks A and B — measured distance 0.\nTracks A and C — measured distance 5.\nTracks A and D — measured distance 1.\nTracks A and E — measured distance 4.\nTracks A and F — measured distance 5.\nTracks A and G — measured distance 0.\nTracks A and H — measured distance 1.\nTracks A and I — measured distance 8.\nTracks A and J — measured distance 1.\nTracks A and K — measured distance 2.\nTracks A and L — measured distance 4.\nTracks A and M — measured distance 4.\nTracks A and N — measured distance 3.\nTracks A and O — measured distance 3.\nTracks A and P — measured distance 5.\nTracks A and Q — measured distance 0.\nTracks A and R — measured distance 4.\nTracks A and S — measured distance 5.\nTracks A and T — measured distance 0.\nTracks A and U — measured distance 5.\nTracks A and V — measured distance 0.\nTracks A and W — measured distance 7.\nTracks A and X — measured distance 0.\nTracks B and A — measured distance 0.\nTracks B and C — measured distance 2.\nTracks B and D — measured distance 2.\nTracks B and E — measured distance 1.\nTracks B and F — measured distance 7.\nTracks B and G — measured distance 8.\nTracks B and H — measured distance 1.\nTracks B and I — measured distance 2.\nTracks B and J — measured distance 9.\nTracks B and K — measured distance 2.\nTracks B and L — measured distance 1.\nTracks B and M — measured distance 5.\nTracks B and N — measured distance 2.\nTracks B and O — measured distance 0.\nTracks B and P — measured distance 1.\nTracks B and Q — measured distance 4.\nTracks B and R — measured distance 8.\nTracks B and S — measured distance 4.\nTracks B and T — measured distance 1.\nTracks B and U — measured distance 4.\nTracks B and V — measured distance 4.\nTracks B and W — measured distance 3.\nTracks B and X — measured distance 8.\nTracks C and A — measured distance 5.\nTracks C and B — measured distance 2.\nTracks C and D — measured distance 4.\nTracks C and E — measured distance 2.\nTracks C and F — measured distance 0.\nTracks C and G — measured distance 2.\nTracks C and H — measured distance 9.\nTracks C and I — measured distance 7.\nTracks C and J — measured distance 0.\nTracks C and K — measured distance 7.\nTracks C and L — measured distance 3.\nTracks C and M — measured distance 9.\nTracks C and N — measured distance 1.\nTracks C and O — measured distance 1.\nTracks C and P — measured distance 0.\nTracks C and Q — measured distance 6.\nTracks C and R — measured distance 1.\nTracks C and S — measured distance 3.\nTracks C and T — measured distance 9.\nTracks C and U — measured distance 5.\nTracks C and V — measured distance 3.\nTracks C and W — measured distance 5.\nTracks C and X — measured distance 2.\nTracks D and A — measured distance 1.\nTracks D and B — measured distance 2.\nTracks D and C — measured distance 4.\nTracks D and E — measured distance 9.\nTracks D and F — measured distance 7.\nTracks D and G — measured distance 9.\nTracks D and H — measured distance 6.\nTracks D and I — measured distance 3.\nTracks D and J — measured distance 5.\nTracks D and K — measured distance 4.\nTracks D and L — measured distance 4.\nTracks D and M — measured distance 6.\nTracks D and N — measured distance 9.\nTracks D and O — measured distance 8.\nTracks D and P — measured distance 3.\nTracks D and Q — measured distance 2.\nTracks D and R — measured distance 7.\nTracks D and S — measured distance 5.\nTracks D and T — measured distance 7.\nTracks D and U — measured distance 8.\nTracks D and V — measured distance 1.\nTracks D and W — measured distance 8.\nTracks D and X — measured distance 0.\nTracks E and A — measured distance 4.\nTracks E and B — measured distance 1.\nTracks E and C — measured distance 2.\nTracks E and D — measured distance 9.\nTracks E and F — measured distance 9.\nTracks E and G — measured distance 9.\nTracks E and H — measured distance 5.\nTracks E and I — measured distance 6.\nTracks E and J — measured distance 0.\nTracks E and K — measured distance 5.\nTracks E and L — measured distance 5.\nTracks E and M — measured distance 1.\nTracks E and N — measured distance 6.\nTracks E and O — measured distance 4.\nTracks E and P — measured distance 5.\nTracks E and Q — measured distance 8.\nTracks E and R — measured distance 6.\nTracks E and S — measured distance 8.\nTracks E and T — measured distance 1.\nTracks E and U — measured distance 0.\nTracks E and V — measured distance 8.\nTracks E and W — measured distance 1.\nTracks E and X — measured distance 1.\nTracks F and A — measured distance 5.\nTracks F and B — measured distance 7.\nTracks F and C — measured distance 0.\nTracks F and D — measured distance 7.\nTracks F and E — measured distance 9.\nTracks F and G — measured distance 8.\nTracks F and H — measured distance 0.\nTracks F and I — measured distance 0.\nTracks F and J — measured distance 5.\nTracks F and K — measured distance 6.\nTracks F and L — measured distance 4.\nTracks F and M — measured distance 1.\nTracks F and N — measured distance 0.\nTracks F and O — measured distance 4.\nTracks F and P — measured distance 4.\nTracks F and Q — measured distance 7.\nTracks F and R — measured distance 4.\nTracks F and S — measured distance 8.\nTracks F and T — measured distance 9.\nTracks F and U — measured distance 0.\nTracks F and V — measured distance 1.\nTracks F and W — measured distance 7.\nTracks F and X — measured distance 0.\nTracks G and A — measured distance 0.\nTracks G and B — measured distance 8.\nTracks G and C — measured distance 2.\nTracks G and D — measured distance 9.\nTracks G and E — measured distance 9.\nTracks G and F — measured distance 8.\nTracks G and H — measured distance 0.\nTracks G and I — measured distance 6.\nTracks G and J — measured distance 5.\nTracks G and K — measured distance 6.\nTracks G and L — measured distance 2.\nTracks G and M — measured distance 6.\nTracks G and N — measured distance 1.\nTracks G and O — measured distance 3.\nTracks G and P — measured distance 3.\nTracks G and Q — measured distance 4.\nTracks G and R — measured distance 4.\nTracks G and S — measured distance 3.\nTracks G and T — measured distance 8.\nTracks G and U — measured distance 0.\nTracks G and V — measured distance 9.\nTracks G and W — measured distance 5.\nTracks G and X — measured distance 4.\nTracks H and A — measured distance 1.\nTracks H and B — measured distance 1.\nTracks H and C — measured distance 9.\nTracks H and D — measured distance 6.\nTracks H and E — measured distance 5.\nTracks H and F — measured distance 0.\nTracks H and G — measured distance 0.\nTracks H and I — measured distance 2.\nTracks H and J — measured distance 0.\nTracks H and K — measured distance 9.\nTracks H and L — measured distance 4.\nTracks H and M — measured distance 1.\nTracks H and N — measured distance 7.\nTracks H and O — measured distance 4.\nTracks H and P — measured distance 6.\nTracks H and Q — measured distance 3.\nTracks H and R — measured distance 9.\nTracks H and S — measured distance 4.\nTracks H and T — measured distance 9.\nTracks H and U — measured distance 7.\nTracks H and V — measured distance 2.\nTracks H and W — measured distance 5.\nTracks H and X — measured distance 4.\nTracks I and A — measured distance 8.\nTracks I and B — measured distance 2.\nTracks I and C — measured distance 7.\nTracks I and D — measured distance 3.\nTracks I and E — measured distance 6.\nTracks I and F — measured distance 0.\nTracks I and G — measured distance 6.\nTracks I and H — measured distance 2.\nTracks I and J — measured distance 3.\nTracks I and K — measured distance 0.\nTracks I and L — measured distance 4.\nTracks I and M — measured distance 8.\nTracks I and N — measured distance 5.\nTracks I and O — measured distance 2.\nTracks I and P — measured distance 9.\nTracks I and Q — measured distance 1.\nTracks I and R — measured distance 7.\nTracks I and S — measured distance 3.\nTracks I and T — measured distance 4.\nTracks I and U — measured distance 6.\nTracks I and V — measured distance 0.\nTracks I and W — measured distance 0.\nTracks I and X — measured distance 9.\nTracks J and A — measured distance 1.\nTracks J and B — measured distance 9.\nTracks J and C — measured distance 0.\nTracks J and D — measured distance 5.\nTracks J and E — measured distance 0.\nTracks J and F — measured distance 5.\nTracks J and G — measured distance 5.\nTracks J and H — measured distance 0.\nTracks J and I — measured distance 3.\nTracks J and K — measured distance 0.\nTracks J and L — measured distance 4.\nTracks J and M — measured distance 3.\nTracks J and N — measured distance 3.\nTracks J and O — measured distance 3.\nTracks J and P — measured distance 5.\nTracks J and Q — measured distance 0.\nTracks J and R — measured distance 5.\nTracks J and S — measured distance 4.\nTracks J and T — measured distance 5.\nTracks J and U — measured distance 0.\nTracks J and V — measured distance 0.\nTracks J and W — measured distance 1.\nTracks J and X — measured distance 8.\nTracks K and A — measured distance 2.\nTracks K and B — measured distance 2.\nTracks K and C — measured distance 7.\nTracks K and D — measured distance 4.\nTracks K and E — measured distance 5.\nTracks K and F — measured distance 6.\nTracks K and G — measured distance 6.\nTracks K and H — measured distance 9.\nTracks K and I — measured distance 0.\nTracks K and J — measured distance 0.\nTracks K and L — measured distance 4.\nTracks K and M — measured distance 9.\nTracks K and N — measured distance 6.\nTracks K and O — measured distance 2.\nTracks K and P — measured distance 1.\nTracks K and Q — measured distance 0.\nTracks K and R — measured distance 9.\nTracks K and S — measured distance 5.\nTracks K and T — measured distance 1.\nTracks K and U — measured distance 2.\nTracks K and V — measured distance 5.\nTracks K and W — measured distance 2.\nTracks K and X — measured distance 9.\nTracks L and A — measured distance 4.\nTracks L and B — measured distance 1.\nTracks L and C — measured distance 3.\nTracks L and D — measured distance 4.\nTracks L and E — measured distance 5.\nTracks L and F — measured distance 4.\nTracks L and G — measured distance 2.\nTracks L and H — measured distance 4.\nTracks L and I — measured distance 4.\nTracks L and J — measured distance 4.\nTracks L and K — measured distance 4.\nTracks L and M — measured distance 7.\nTracks L and N — measured distance 1.\nTracks L and O — measured distance 6.\nTracks L and P — measured distance 1.\nTracks L and Q — measured distance 3.\nTracks L and R — measured distance 2.\nTracks L and S — measured distance 9.\nTracks L and T — measured distance 5.\nTracks L and U — measured distance 3.\nTracks L and V — measured distance 1.\nTracks L and W — measured distance 6.\nTracks L and X — measured distance 8.\nTracks M and A — measured distance 4.\nTracks M and B — measured distance 5.\nTracks M and C — measured distance 9.\nTracks M and D — measured distance 6.\nTracks M and E — measured distance 1.\nTracks M and F — measured distance 1.\nTracks M and G — measured distance 6.\nTracks M and H — measured distance 1.\nTracks M and I — measured distance 8.\nTracks M and J — measured distance 3.\nTracks M and K — measured distance 9.\nTracks M and L — measured distance 7.\nTracks M and N — measured distance 2.\nTracks M and O — measured distance 9.\nTracks M and P — measured distance 8.\nTracks M and Q — measured distance 1.\nTracks M and R — measured distance 6.\nTracks M and S — measured distance 3.\nTracks M and T — measured distance 4.\nTracks M and U — measured distance 0.\nTracks M and V — measured distance 2.\nTracks M and W — measured distance 9.\nTracks M and X — measured distance 9.\nTracks N and A — measured distance 3.\nTracks N and B — measured distance 2.\nTracks N and C — measured distance 1.\nTracks N and D — measured distance 9.\nTracks N and E — measured distance 6.\nTracks N and F — measured distance 0.\nTracks N and G — measured distance 1.\nTracks N and H — measured distance 7.\nTracks N and I — measured distance 5.\nTracks N and J — measured distance 3.\nTracks N and K — measured distance 6.\nTracks N and L — measured distance 1.\nTracks N and M — measured distance 2.\nTracks N and O — measured distance 5.\nTracks N and P — measured distance 9.\nTracks N and Q — measured distance 2.\nTracks N and R — measured distance 0.\nTracks N and S — measured distance 1.\nTracks N and T — measured distance 9.\nTracks N and U — measured distance 9.\nTracks N and V — measured distance 0.\nTracks N and W — measured distance 9.\nTracks N and X — measured distance 4.\nTracks O and A — measured distance 3.\nTracks O and B — measured distance 0.\nTracks O and C — measured distance 1.\nTracks O and D — measured distance 8.\nTracks O and E — measured distance 4.\nTracks O and F — measured distance 4.\nTracks O and G — measured distance 3.\nTracks O and H — measured distance 4.\nTracks O and I — measured distance 2.\nTracks O and J — measured distance 3.\nTracks O and K — measured distance 2.\nTracks O and L — measured distance 6.\nTracks O and M — measured distance 9.\nTracks O and N — measured distance 5.\nTracks O and P — measured distance 8.\nTracks O and Q — measured distance 8.\nTracks O and R — measured distance 1.\nTracks O and S — measured distance 8.\nTracks O and T — measured distance 4.\nTracks O and U — measured distance 3.\nTracks O and V — measured distance 4.\nTracks O and W — measured distance 8.\nTracks O and X — measured distance 5.\nTracks P and A — measured distance 5.\nTracks P and B — measured distance 1.\nTracks P and C — measured distance 0.\nTracks P and D — measured distance 3.\nTracks P and E — measured distance 5.\nTracks P and F — measured distance 4.\nTracks P and G — measured distance 3.\nTracks P and H — measured distance 6.\nTracks P and I — measured distance 9.\nTracks P and J — measured distance 5.\nTracks P and K — measured distance 1.\nTracks P and L — measured distance 1.\nTracks P and M — measured distance 8.\nTracks P and N — measured distance 9.\nTracks P and O — measured distance 8.\nTracks P and Q — measured distance 2.\nTracks P and R — measured distance 1.\nTracks P and S — measured distance 7.\nTracks P and T — measured distance 7.\nTracks P and U — measured distance 4.\nTracks P and V — measured distance 1.\nTracks P and W — measured distance 9.\nTracks P and X — measured distance 6.\nTracks Q and A — measured distance 0.\nTracks Q and B — measured distance 4.\nTracks Q and C — measured distance 6.\nTracks Q and D — measured distance 2.\nTracks Q and E — measured distance 8.\nTracks Q and F — measured distance 7.\nTracks Q and G — measured distance 4.\nTracks Q and H — measured distance 3.\nTracks Q and I — measured distance 1.\nTracks Q and J — measured distance 0.\nTracks Q and K — measured distance 0.\nTracks Q and L — measured distance 3.\nTracks Q and M — measured distance 1.\nTracks Q and N — measured distance 2.\nTracks Q and O — measured distance 8.\nTracks Q and P — measured distance 2.\nTracks Q and R — measured distance 9.\nTracks Q and S — measured distance 9.\nTracks Q and T — measured distance 8.\nTracks Q and U — measured distance 5.\nTracks Q and V — measured distance 4.\nTracks Q and W — measured distance 9.\nTracks Q and X — measured distance 4.\nTracks R and A — measured distance 4.\nTracks R and B — measured distance 8.\nTracks R and C — measured distance 1.\nTracks R and D — measured distance 7.\nTracks R and E — measured distance 6.\nTracks R and F — measured distance 4.\nTracks R and G — measured distance 4.\nTracks R and H — measured distance 9.\nTracks R and I — measured distance 7.\nTracks R and J — measured distance 5.\nTracks R and K — measured distance 9.\nTracks R and L — measured distance 2.\nTracks R and M — measured distance 6.\nTracks R and N — measured distance 0.\nTracks R and O — measured distance 1.\nTracks R and P — measured distance 1.\nTracks R and Q — measured distance 9.\nTracks R and S — measured distance 2.\nTracks R and T — measured distance 1.\nTracks R and U — measured distance 8.\nTracks R and V — measured distance 9.\nTracks R and W — measured distance 6.\nTracks R and X — measured distance 9.\nTracks S and A — measured distance 5.\nTracks S and B — measured distance 4.\nTracks S and C — measured distance 3.\nTracks S and D — measured distance 5.\nTracks S and E — measured distance 8.\nTracks S and F — measured distance 8.\nTracks S and G — measured distance 3.\nTracks S and H — measured distance 4.\nTracks S and I — measured distance 3.\nTracks S and J — measured distance 4.\nTracks S and K — measured distance 5.\nTracks S and L — measured distance 9.\nTracks S and M — measured distance 3.\nTracks S and N — measured distance 1.\nTracks S and O — measured distance 8.\nTracks S and P — measured distance 7.\nTracks S and Q — measured distance 9.\nTracks S and R — measured distance 2.\nTracks S and T — measured distance 4.\nTracks S and U — measured distance 1.\nTracks S and V — measured distance 6.\nTracks S and W — measured distance 3.\nTracks S and X — measured distance 3.\nTracks T and A — measured distance 0.\nTracks T and B — measured distance 1.\nTracks T and C — measured distance 9.\nTracks T and D — measured distance 7.\nTracks T and E — measured distance 1.\nTracks T and F — measured distance 9.\nTracks T and G — measured distance 8.\nTracks T and H — measured distance 9.\nTracks T and I — measured distance 4.\nTracks T and J — measured distance 5.\nTracks T and K — measured distance 1.\nTracks T and L — measured distance 5.\nTracks T and M — measured distance 4.\nTracks T and N — measured distance 9.\nTracks T and O — measured distance 4.\nTracks T and P — measured distance 7.\nTracks T and Q — measured distance 8.\nTracks T and R — measured distance 1.\nTracks T and S — measured distance 4.\nTracks T and U — measured distance 9.\nTracks T and V — measured distance 5.\nTracks T and W — measured distance 7.\nTracks T and X — measured distance 2.\nTracks U and A — measured distance 5.\nTracks U and B — measured distance 4.\nTracks U and C — measured distance 5.\nTracks U and D — measured distance 8.\nTracks U and E — measured distance 0.\nTracks U and F — measured distance 0.\nTracks U and G — measured distance 0.\nTracks U and H — measured distance 7.\nTracks U and I — measured distance 6.\nTracks U and J — measured distance 0.\nTracks U and K — measured distance 2.\nTracks U and L — measured distance 3.\nTracks U and M — measured distance 0.\nTracks U and N — measured distance 9.\nTracks U and O — measured distance 3.\nTracks U and P — measured distance 4.\nTracks U and Q — measured distance 5.\nTracks U and R — measured distance 8.\nTracks U and S — measured distance 1.\nTracks U and T — measured distance 9.\nTracks U and V — measured distance 8.\nTracks U and W — measured distance 5.\nTracks U and X — measured distance 5.\nTracks V and A — measured distance 0.\nTracks V and B — measured distance 4.\nTracks V and C — measured distance 3.\nTracks V and D — measured distance 1.\nTracks V and E — measured distance 8.\nTracks V and F — measured distance 1.\nTracks V and G — measured distance 9.\nTracks V and H — measured distance 2.\nTracks V and I — measured distance 0.\nTracks V and J — measured distance 0.\nTracks V and K — measured distance 5.\nTracks V and L — measured distance 1.\nTracks V and M — measured distance 2.\nTracks V and N — measured distance 0.\nTracks V and O — measured distance 4.\nTracks V and P — measured distance 1.\nTracks V and Q — measured distance 4.\nTracks V and R — measured distance 9.\nTracks V and S — measured distance 6.\nTracks V and T — measured distance 5.\nTracks V and U — measured distance 8.\nTracks V and W — measured distance 6.\nTracks V and X — measured distance 1.\nTracks W and A — measured distance 7.\nTracks W and B — measured distance 3.\nTracks W and C — measured distance 5.\nTracks W and D — measured distance 8.\nTracks W and E — measured distance 1.\nTracks W and F — measured distance 7.\nTracks W and G — measured distance 5.\nTracks W and H — measured distance 5.\nTracks W and I — measured distance 0.\nTracks W and J — measured distance 1.\nTracks W and K — measured distance 2.\nTracks W and L — measured distance 6.\nTracks W and M — measured distance 9.\nTracks W and N — measured distance 9.\nTracks W and O — measured distance 8.\nTracks W and P — measured distance 9.\nTracks W and Q — measured distance 9.\nTracks W and R — measured distance 6.\nTracks W and S — measured distance 3.\nTracks W and T — measured distance 7.\nTracks W and U — measured distance 5.\nTracks W and V — measured distance 6.\nTracks W and X — measured distance 8.\nTracks X and A — measured distance 0.\nTracks X and B — measured distance 8.\nTracks X and C — measured distance 2.\nTracks X and D — measured distance 0.\nTracks X and E — measured distance 1.\nTracks X and F — measured distance 0.\nTracks X and G — measured distance 4.\nTracks X and H — measured distance 4.\nTracks X and I — measured distance 9.\nTracks X and J — measured distance 8.\nTracks X and K — measured distance 9.\nTracks X and L — measured distance 8.\nTracks X and M — measured distance 9.\nTracks X and N — measured distance 4.\nTracks X and O — measured distance 5.\nTracks X and P — measured distance 6.\nTracks X and Q — measured distance 4.\nTracks X and R — measured distance 9.\nTracks X and S — measured distance 3.\nTracks X and T — measured distance 2.\nTracks X and U — measured distance 5.\nTracks X and V — measured distance 1.\nTracks X and W — measured distance 8.\nUse the pairwise distances below to score any selection of 7 tracks.\n\nOh — and when you’re ready to send back the pick, please use this simple JSON layout so it’s easy to read by whatever’s checking the answers:\n\n{\n \"solution\": [, , ...]\n}\n\nPretty straightforward: the \"solution\" array is where you list the m tracks you’ve chosen (use the exact labels from the candidate list). This is just a sketch of the shape I’m expecting, not the actual final answer.\n\nPlease make sure you use the identifiers exactly as they appear in the instance input — don’t rename items or invent new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 0, 5, 1, 4, 5, 0, 1, 8, 1, 2, 4, 4, 3, 3, 5, 0, 4, 5, 0, 5, 0, 7, 0 ], [ 0, 0, 2, 2, 1, 7, 8, 1, 2, 9, 2, 1, 5, 2, 0, 1, 4, 8, 4, 1, 4, 4, 3, 8 ], [ 5, 2, 0, 4, 2, 0, 2, 9, 7, 0, 7, 3, 9, 1, 1, 0, 6, 1, 3, 9, 5, 3, 5, 2 ], [ 1, 2, 4, 0, 9, 7, 9, 6, 3, 5, 4, 4, 6, 9, 8, 3, 2, 7, 5, 7, 8, 1, 8, 0 ], [ 4, 1, 2, 9, 0, 9, 9, 5, 6, 0, 5, 5, 1, 6, 4, 5, 8, 6, 8, 1, 0, 8, 1, 1 ], [ 5, 7, 0, 7, 9, 0, 8, 0, 0, 5, 6, 4, 1, 0, 4, 4, 7, 4, 8, 9, 0, 1, 7, 0 ], [ 0, 8, 2, 9, 9, 8, 0, 0, 6, 5, 6, 2, 6, 1, 3, 3, 4, 4, 3, 8, 0, 9, 5, 4 ], [ 1, 1, 9, 6, 5, 0, 0, 0, 2, 0, 9, 4, 1, 7, 4, 6, 3, 9, 4, 9, 7, 2, 5, 4 ], [ 8, 2, 7, 3, 6, 0, 6, 2, 0, 3, 0, 4, 8, 5, 2, 9, 1, 7, 3, 4, 6, 0, 0, 9 ], [ 1, 9, 0, 5, 0, 5, 5, 0, 3, 0, 0, 4, 3, 3, 3, 5, 0, 5, 4, 5, 0, 0, 1, 8 ], [ 2, 2, 7, 4, 5, 6, 6, 9, 0, 0, 0, 4, 9, 6, 2, 1, 0, 9, 5, 1, 2, 5, 2, 9 ], [ 4, 1, 3, 4, 5, 4, 2, 4, 4, 4, 4, 0, 7, 1, 6, 1, 3, 2, 9, 5, 3, 1, 6, 8 ], [ 4, 5, 9, 6, 1, 1, 6, 1, 8, 3, 9, 7, 0, 2, 9, 8, 1, 6, 3, 4, 0, 2, 9, 9 ], [ 3, 2, 1, 9, 6, 0, 1, 7, 5, 3, 6, 1, 2, 0, 5, 9, 2, 0, 1, 9, 9, 0, 9, 4 ], [ 3, 0, 1, 8, 4, 4, 3, 4, 2, 3, 2, 6, 9, 5, 0, 8, 8, 1, 8, 4, 3, 4, 8, 5 ], [ 5, 1, 0, 3, 5, 4, 3, 6, 9, 5, 1, 1, 8, 9, 8, 0, 2, 1, 7, 7, 4, 1, 9, 6 ], [ 0, 4, 6, 2, 8, 7, 4, 3, 1, 0, 0, 3, 1, 2, 8, 2, 0, 9, 9, 8, 5, 4, 9, 4 ], [ 4, 8, 1, 7, 6, 4, 4, 9, 7, 5, 9, 2, 6, 0, 1, 1, 9, 0, 2, 1, 8, 9, 6, 9 ], [ 5, 4, 3, 5, 8, 8, 3, 4, 3, 4, 5, 9, 3, 1, 8, 7, 9, 2, 0, 4, 1, 6, 3, 3 ], [ 0, 1, 9, 7, 1, 9, 8, 9, 4, 5, 1, 5, 4, 9, 4, 7, 8, 1, 4, 0, 9, 5, 7, 2 ], [ 5, 4, 5, 8, 0, 0, 0, 7, 6, 0, 2, 3, 0, 9, 3, 4, 5, 8, 1, 9, 0, 8, 5, 5 ], [ 0, 4, 3, 1, 8, 1, 9, 2, 0, 0, 5, 1, 2, 0, 4, 1, 4, 9, 6, 5, 8, 0, 6, 1 ], [ 7, 3, 5, 8, 1, 7, 5, 5, 0, 1, 2, 6, 9, 9, 8, 9, 9, 6, 3, 7, 5, 6, 0, 8 ], [ 0, 8, 2, 0, 1, 0, 4, 4, 9, 8, 9, 8, 9, 4, 5, 6, 4, 9, 3, 2, 5, 1, 8, 0 ] ], "m": 7, "objective": 152.0 }, "solution": [ 3, 7, 13, 15, 19, 20, 22 ], "obj": 152.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 24, "m": 7, "nodes": [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S", "T", "U", "V", "W", "X" ], "distance_pairs": [ { "from_id": "A", "to_id": "B", "distance": 0 }, { "from_id": "A", "to_id": "C", "distance": 5 }, { "from_id": "A", "to_id": "D", "distance": 1 }, { "from_id": "A", "to_id": "E", "distance": 4 }, { "from_id": "A", "to_id": "F", "distance": 5 }, { "from_id": "A", "to_id": "G", "distance": 0 }, { "from_id": "A", "to_id": "H", "distance": 1 }, { "from_id": "A", "to_id": "I", "distance": 8 }, { "from_id": "A", "to_id": "J", "distance": 1 }, { "from_id": "A", "to_id": "K", "distance": 2 }, { "from_id": "A", "to_id": "L", "distance": 4 }, { "from_id": "A", "to_id": "M", "distance": 4 }, { "from_id": "A", "to_id": "N", "distance": 3 }, { "from_id": "A", "to_id": "O", "distance": 3 }, { "from_id": "A", "to_id": "P", "distance": 5 }, { "from_id": "A", "to_id": "Q", "distance": 0 }, { "from_id": "A", "to_id": "R", "distance": 4 }, { "from_id": "A", "to_id": "S", "distance": 5 }, { "from_id": "A", "to_id": "T", "distance": 0 }, { "from_id": "A", "to_id": "U", "distance": 5 }, { "from_id": "A", "to_id": "V", "distance": 0 }, { "from_id": "A", "to_id": "W", "distance": 7 }, { "from_id": "A", "to_id": "X", "distance": 0 }, { "from_id": "B", "to_id": "A", "distance": 0 }, { "from_id": "B", "to_id": "C", "distance": 2 }, { "from_id": "B", "to_id": "D", "distance": 2 }, { "from_id": "B", "to_id": "E", "distance": 1 }, { "from_id": "B", "to_id": "F", "distance": 7 }, { "from_id": "B", "to_id": "G", "distance": 8 }, { "from_id": "B", "to_id": "H", "distance": 1 }, { "from_id": "B", "to_id": "I", "distance": 2 }, { "from_id": "B", "to_id": "J", "distance": 9 }, { "from_id": "B", "to_id": "K", "distance": 2 }, { "from_id": "B", "to_id": "L", "distance": 1 }, { "from_id": "B", "to_id": "M", "distance": 5 }, { "from_id": "B", "to_id": "N", "distance": 2 }, { "from_id": "B", "to_id": "O", "distance": 0 }, { "from_id": "B", "to_id": "P", "distance": 1 }, { "from_id": "B", "to_id": "Q", "distance": 4 }, { "from_id": "B", "to_id": "R", "distance": 8 }, { "from_id": "B", "to_id": "S", "distance": 4 }, { "from_id": "B", "to_id": "T", "distance": 1 }, { "from_id": "B", "to_id": "U", "distance": 4 }, { "from_id": "B", "to_id": "V", "distance": 4 }, { "from_id": "B", "to_id": "W", "distance": 3 }, { "from_id": "B", "to_id": "X", "distance": 8 }, { "from_id": "C", "to_id": "A", "distance": 5 }, { "from_id": "C", "to_id": "B", "distance": 2 }, { "from_id": "C", "to_id": "D", "distance": 4 }, { "from_id": "C", "to_id": "E", "distance": 2 }, { "from_id": "C", "to_id": "F", "distance": 0 }, { "from_id": "C", "to_id": "G", "distance": 2 }, { "from_id": "C", "to_id": "H", "distance": 9 }, { "from_id": "C", "to_id": "I", "distance": 7 }, { "from_id": "C", "to_id": "J", "distance": 0 }, { "from_id": "C", "to_id": "K", "distance": 7 }, { "from_id": "C", "to_id": "L", "distance": 3 }, { "from_id": "C", "to_id": "M", "distance": 9 }, { "from_id": "C", "to_id": "N", "distance": 1 }, { "from_id": "C", "to_id": "O", "distance": 1 }, { "from_id": "C", "to_id": "P", "distance": 0 }, { "from_id": "C", "to_id": "Q", "distance": 6 }, { "from_id": "C", "to_id": "R", "distance": 1 }, { "from_id": "C", "to_id": "S", "distance": 3 }, { "from_id": "C", "to_id": "T", "distance": 9 }, { "from_id": "C", "to_id": "U", "distance": 5 }, { "from_id": "C", "to_id": "V", "distance": 3 }, { "from_id": "C", "to_id": "W", "distance": 5 }, { "from_id": "C", "to_id": "X", "distance": 2 }, { "from_id": "D", "to_id": "A", "distance": 1 }, { "from_id": "D", "to_id": "B", "distance": 2 }, { "from_id": "D", "to_id": "C", "distance": 4 }, { "from_id": "D", "to_id": "E", "distance": 9 }, { "from_id": "D", "to_id": "F", "distance": 7 }, { "from_id": "D", "to_id": "G", "distance": 9 }, { "from_id": "D", "to_id": "H", "distance": 6 }, { "from_id": "D", "to_id": "I", "distance": 3 }, { "from_id": "D", "to_id": "J", "distance": 5 }, { "from_id": "D", "to_id": "K", "distance": 4 }, { "from_id": "D", "to_id": "L", "distance": 4 }, { "from_id": "D", "to_id": "M", "distance": 6 }, { "from_id": "D", "to_id": "N", "distance": 9 }, { "from_id": "D", "to_id": "O", "distance": 8 }, { "from_id": "D", "to_id": "P", "distance": 3 }, { "from_id": "D", "to_id": "Q", "distance": 2 }, { "from_id": "D", "to_id": "R", "distance": 7 }, { "from_id": "D", "to_id": "S", "distance": 5 }, { "from_id": "D", "to_id": "T", "distance": 7 }, { "from_id": "D", "to_id": "U", "distance": 8 }, { "from_id": "D", "to_id": "V", "distance": 1 }, { "from_id": "D", "to_id": "W", "distance": 8 }, { "from_id": "D", "to_id": "X", "distance": 0 }, { "from_id": "E", "to_id": "A", "distance": 4 }, { "from_id": "E", "to_id": "B", "distance": 1 }, { "from_id": "E", "to_id": "C", "distance": 2 }, { "from_id": "E", "to_id": "D", "distance": 9 }, { "from_id": "E", "to_id": "F", "distance": 9 }, { "from_id": "E", "to_id": "G", "distance": 9 }, { "from_id": "E", "to_id": "H", "distance": 5 }, { "from_id": "E", "to_id": "I", "distance": 6 }, { "from_id": "E", "to_id": "J", "distance": 0 }, { "from_id": "E", "to_id": "K", "distance": 5 }, { "from_id": "E", "to_id": "L", "distance": 5 }, { "from_id": "E", "to_id": "M", "distance": 1 }, { "from_id": "E", "to_id": "N", "distance": 6 }, { "from_id": "E", "to_id": "O", "distance": 4 }, { "from_id": "E", "to_id": "P", "distance": 5 }, { "from_id": "E", "to_id": "Q", "distance": 8 }, { "from_id": "E", "to_id": "R", "distance": 6 }, { "from_id": "E", "to_id": "S", "distance": 8 }, { "from_id": "E", "to_id": "T", "distance": 1 }, { "from_id": "E", "to_id": "U", "distance": 0 }, { "from_id": "E", "to_id": "V", "distance": 8 }, { "from_id": "E", "to_id": "W", "distance": 1 }, { "from_id": "E", "to_id": "X", "distance": 1 }, { "from_id": "F", "to_id": "A", "distance": 5 }, { "from_id": "F", "to_id": "B", "distance": 7 }, { "from_id": "F", "to_id": "C", "distance": 0 }, { "from_id": "F", "to_id": "D", "distance": 7 }, { "from_id": "F", "to_id": "E", "distance": 9 }, { "from_id": "F", "to_id": "G", "distance": 8 }, { "from_id": "F", "to_id": "H", "distance": 0 }, { "from_id": "F", "to_id": "I", "distance": 0 }, { "from_id": "F", "to_id": "J", "distance": 5 }, { "from_id": "F", "to_id": "K", "distance": 6 }, { "from_id": "F", "to_id": "L", "distance": 4 }, { "from_id": "F", "to_id": "M", "distance": 1 }, { "from_id": "F", "to_id": "N", "distance": 0 }, { "from_id": "F", "to_id": "O", "distance": 4 }, { "from_id": "F", "to_id": "P", "distance": 4 }, { "from_id": "F", "to_id": "Q", "distance": 7 }, { "from_id": "F", "to_id": "R", "distance": 4 }, { "from_id": "F", "to_id": "S", "distance": 8 }, { "from_id": "F", "to_id": "T", "distance": 9 }, { "from_id": "F", "to_id": "U", "distance": 0 }, { "from_id": "F", "to_id": "V", "distance": 1 }, { "from_id": "F", "to_id": "W", "distance": 7 }, { "from_id": "F", "to_id": "X", "distance": 0 }, { "from_id": "G", "to_id": "A", "distance": 0 }, { "from_id": "G", "to_id": "B", "distance": 8 }, { "from_id": "G", "to_id": "C", "distance": 2 }, { "from_id": "G", "to_id": "D", "distance": 9 }, { "from_id": "G", "to_id": "E", "distance": 9 }, { "from_id": "G", "to_id": "F", "distance": 8 }, { "from_id": "G", "to_id": "H", "distance": 0 }, { "from_id": "G", "to_id": "I", "distance": 6 }, { "from_id": "G", "to_id": "J", "distance": 5 }, { "from_id": "G", "to_id": "K", "distance": 6 }, { "from_id": "G", "to_id": "L", "distance": 2 }, { "from_id": "G", "to_id": "M", "distance": 6 }, { "from_id": "G", "to_id": "N", "distance": 1 }, { "from_id": "G", "to_id": "O", "distance": 3 }, { "from_id": "G", "to_id": "P", "distance": 3 }, { "from_id": "G", "to_id": "Q", "distance": 4 }, { "from_id": "G", "to_id": "R", "distance": 4 }, { "from_id": "G", "to_id": "S", "distance": 3 }, { "from_id": "G", "to_id": "T", "distance": 8 }, { "from_id": "G", "to_id": "U", "distance": 0 }, { "from_id": "G", "to_id": "V", "distance": 9 }, { "from_id": "G", "to_id": "W", "distance": 5 }, { "from_id": "G", "to_id": "X", "distance": 4 }, { "from_id": "H", "to_id": "A", "distance": 1 }, { "from_id": "H", "to_id": "B", "distance": 1 }, { "from_id": "H", "to_id": "C", "distance": 9 }, { "from_id": "H", "to_id": "D", "distance": 6 }, { "from_id": "H", "to_id": "E", "distance": 5 }, { "from_id": "H", "to_id": "F", "distance": 0 }, { "from_id": "H", "to_id": "G", "distance": 0 }, { "from_id": "H", "to_id": "I", "distance": 2 }, { "from_id": "H", "to_id": "J", "distance": 0 }, { "from_id": "H", "to_id": "K", "distance": 9 }, { "from_id": "H", "to_id": "L", "distance": 4 }, { "from_id": "H", "to_id": "M", "distance": 1 }, { "from_id": "H", "to_id": "N", "distance": 7 }, { "from_id": "H", "to_id": "O", "distance": 4 }, { "from_id": "H", "to_id": "P", "distance": 6 }, { "from_id": "H", "to_id": "Q", "distance": 3 }, { "from_id": "H", "to_id": "R", "distance": 9 }, { "from_id": "H", "to_id": "S", "distance": 4 }, { "from_id": "H", "to_id": "T", "distance": 9 }, { "from_id": "H", "to_id": "U", "distance": 7 }, { "from_id": "H", "to_id": "V", "distance": 2 }, { "from_id": "H", "to_id": "W", "distance": 5 }, { "from_id": "H", "to_id": "X", "distance": 4 }, { "from_id": "I", "to_id": "A", "distance": 8 }, { "from_id": "I", "to_id": "B", "distance": 2 }, { "from_id": "I", "to_id": "C", "distance": 7 }, { "from_id": "I", "to_id": "D", "distance": 3 }, { "from_id": "I", "to_id": "E", "distance": 6 }, { "from_id": "I", "to_id": "F", "distance": 0 }, { "from_id": "I", "to_id": "G", "distance": 6 }, { "from_id": "I", "to_id": "H", "distance": 2 }, { 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m artworks that are as unlike one another as possible. For every pair there’s a number that says how different they are, and the measure of a group is the sum of those pairwise numbers across all pairs in the group. The aim is to make that sum as big as possible while choosing exactly m distinct pieces — no repeats. The detailed list of pieces and their difference scores appears below.\n\nThere are 24 artworks available (A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X) and the curator must pick exactly 10.\n\n| artwork_id_a | artwork_id_b | stylistic_distance |\n|---|---|---|\n| A | B | 181 |\n| A | C | 212 |\n| A | D | 205 |\n| A | E | 178 |\n| A | F | 143 |\n| A | G | 176 |\n| A | H | 178 |\n| A | I | 186 |\n| A | J | 152 |\n| A | K | 219 |\n| A | L | 221 |\n| A | M | 168 |\n| A | N | 215 |\n| A | O | 235 |\n| A | P | 219 |\n| A | Q | 253 |\n| A | R | 202 |\n| A | S | 186 |\n| A | T | 205 |\n| A | U | 226 |\n| A | V | 186 |\n| A | W | 210 |\n| A | X | 196 |\n| B | A | 181 |\n| B | C | 167 |\n| B | D | 166 |\n| B | E | 178 |\n| B | F | 210 |\n| B | G | 181 |\n| B | H | 222 |\n| B | I | 204 |\n| B | J | 187 |\n| B | K | 135 |\n| B | L | 182 |\n| B | M | 154 |\n| B | N | 135 |\n| B | O | 195 |\n| B | P | 171 |\n| B | Q | 180 |\n| B | R | 163 |\n| B | S | 161 |\n| B | T | 191 |\n| B | U | 204 |\n| B | V | 172 |\n| B | W | 193 |\n| B | X | 183 |\n| C | A | 212 |\n| C | B | 167 |\n| C | D | 176 |\n| C | E | 187 |\n| C | F | 187 |\n| C | G | 179 |\n| C | H | 184 |\n| C | I | 188 |\n| C | J | 173 |\n| C | K | 195 |\n| C | L | 208 |\n| C | M | 191 |\n| C | N | 145 |\n| C | O | 172 |\n| C | P | 186 |\n| C | Q | 148 |\n| C | R | 203 |\n| C | S | 151 |\n| C | T | 199 |\n| C | U | 164 |\n| C | V | 123 |\n| C | W | 186 |\n| C | X | 171 |\n| D | A | 205 |\n| D | B | 166 |\n| D | C | 176 |\n| D | E | 197 |\n| D | F | 189 |\n| D | G | 177 |\n| D | H | 196 |\n| D | I | 147 |\n| D | J | 169 |\n| D | K | 189 |\n| D | L | 167 |\n| D | M | 174 |\n| D | N | 179 |\n| D | O | 188 |\n| D | P | 169 |\n| D | Q | 212 |\n| D | R | 203 |\n| D | S | 141 |\n| D | T | 175 |\n| D | U | 162 |\n| D | V | 179 |\n| D | W | 174 |\n| D | X | 198 |\n| E | A | 178 |\n| E | B | 178 |\n| E | C | 187 |\n| E | D | 197 |\n| E | F | 195 |\n| E | G | 154 |\n| E | H | 172 |\n| E | I | 166 |\n| E | J | 168 |\n| E | K | 214 |\n| E | L | 228 |\n| E | M | 193 |\n| E | N | 205 |\n| E | O | 185 |\n| E | P | 223 |\n| E | Q | 211 |\n| E | R | 169 |\n| E | S | 149 |\n| E | T | 155 |\n| E | U | 185 |\n| E | V | 183 |\n| E | W | 185 |\n| E | X | 187 |\n| F | A | 143 |\n| F | B | 210 |\n| F | C | 187 |\n| F | D | 189 |\n| F | E | 195 |\n| F | G | 207 |\n| F | H | 144 |\n| F | I | 175 |\n| F | J | 131 |\n| F | K | 214 |\n| F | L | 189 |\n| F | M | 158 |\n| F | N | 195 |\n| F | O | 224 |\n| F | P | 225 |\n| F | Q | 204 |\n| F | R | 183 |\n| F | S | 176 |\n| F | T | 186 |\n| F | U | 173 |\n| F | V | 155 |\n| F | W | 196 |\n| F | X | 194 |\n| G | A | 176 |\n| G | B | 181 |\n| G | C | 179 |\n| G | D | 177 |\n| G | E | 154 |\n| G | F | 207 |\n| G | H | 192 |\n| G | I | 200 |\n| G | J | 214 |\n| G | K | 214 |\n| G | L | 245 |\n| G | M | 194 |\n| G | N | 219 |\n| G | O | 162 |\n| G | P | 175 |\n| G | Q | 218 |\n| G | R | 176 |\n| G | S | 185 |\n| G | T | 202 |\n| G | U | 194 |\n| G | V | 170 |\n| G | W | 218 |\n| G | X | 175 |\n| H | A | 178 |\n| H | B | 222 |\n| H | C | 184 |\n| H | D | 196 |\n| H | E | 172 |\n| H | F | 144 |\n| H | G | 192 |\n| H | I | 170 |\n| H | J | 184 |\n| H | K | 237 |\n| H | L | 229 |\n| H | M | 180 |\n| H | N | 206 |\n| H | O | 188 |\n| H | P | 209 |\n| H | Q | 183 |\n| H | R | 166 |\n| H | S | 163 |\n| H | T | 183 |\n| H | U | 182 |\n| H | V | 180 |\n| H | W | 186 |\n| H | X | 182 |\n| I | A | 186 |\n| I | B | 204 |\n| I | C | 188 |\n| I | D | 147 |\n| I | E | 166 |\n| I | F | 175 |\n| I | G | 200 |\n| I | H | 170 |\n| I | J | 125 |\n| I | K | 226 |\n| I | L | 205 |\n| I | M | 173 |\n| I | N | 205 |\n| I | O | 169 |\n| I | P | 201 |\n| I | Q | 233 |\n| I | R | 206 |\n| I | S | 146 |\n| I | T | 164 |\n| I | U | 177 |\n| I | V | 178 |\n| I | W | 191 |\n| I | X | 196 |\n| J | A | 152 |\n| J | B | 187 |\n| J | C | 173 |\n| J | D | 169 |\n| J | E | 168 |\n| J | F | 131 |\n| J | G | 214 |\n| J | H | 184 |\n| J | I | 125 |\n| J | K | 200 |\n| J | L | 163 |\n| J | M | 168 |\n| J | N | 180 |\n| J | O | 213 |\n| J | P | 205 |\n| J | Q | 232 |\n| J | R | 202 |\n| J | S | 155 |\n| J | T | 168 |\n| J | U | 188 |\n| J | V | 165 |\n| J | W | 163 |\n| J | X | 223 |\n| K | A | 219 |\n| K | B | 135 |\n| K | C | 195 |\n| K | D | 189 |\n| K | E | 214 |\n| K | F | 214 |\n| K | G | 214 |\n| K | H | 237 |\n| K | I | 226 |\n| K | J | 200 |\n| K | L | 165 |\n| K | M | 169 |\n| K | N | 137 |\n| K | O | 240 |\n| K | P | 212 |\n| K | Q | 204 |\n| K | R | 208 |\n| K | S | 179 |\n| K | T | 225 |\n| K | U | 201 |\n| K | V | 199 |\n| K | W | 228 |\n| K | X | 245 |\n| L | A | 221 |\n| L | B | 182 |\n| L | C | 208 |\n| L | D | 167 |\n| L | E | 228 |\n| L | F | 189 |\n| L | G | 245 |\n| L | H | 229 |\n| L | I | 205 |\n| L | J | 163 |\n| L | K | 165 |\n| L | M | 177 |\n| L | N | 133 |\n| L | O | 270 |\n| L | P | 183 |\n| L | Q | 215 |\n| L | R | 210 |\n| L | S | 203 |\n| L | T | 184 |\n| L | U | 199 |\n| L | V | 196 |\n| L | W | 178 |\n| L | X | 226 |\n| M | A | 168 |\n| M | B | 154 |\n| M | C | 191 |\n| M | D | 174 |\n| M | E | 193 |\n| M | F | 158 |\n| M | G | 194 |\n| M | H | 180 |\n| M | I | 173 |\n| M | J | 168 |\n| M | K | 169 |\n| M | L | 177 |\n| M | N | 152 |\n| M | O | 200 |\n| M | P | 190 |\n| M | Q | 170 |\n| M | R | 180 |\n| M | S | 166 |\n| M | T | 149 |\n| M | U | 138 |\n| M | V | 132 |\n| M | W | 186 |\n| M | X | 174 |\n| N | A | 215 |\n| N | B | 135 |\n| N | C | 145 |\n| N | D | 179 |\n| N | E | 205 |\n| N | F | 195 |\n| N | G | 219 |\n| N | H | 206 |\n| N | I | 205 |\n| N | J | 180 |\n| N | K | 137 |\n| N | L | 133 |\n| N | M | 152 |\n| N | O | 212 |\n| N | P | 177 |\n| N | Q | 147 |\n| N | R | 192 |\n| N | S | 163 |\n| N | T | 192 |\n| N | U | 160 |\n| N | V | 167 |\n| N | W | 178 |\n| N | X | 202 |\n| O | A | 235 |\n| O | B | 195 |\n| O | C | 172 |\n| O | D | 188 |\n| O | E | 185 |\n| O | F | 224 |\n| O | G | 162 |\n| O | H | 188 |\n| O | I | 169 |\n| O | J | 213 |\n| O | K | 240 |\n| O | L | 270 |\n| O | M | 200 |\n| O | N | 212 |\n| O | P | 198 |\n| O | Q | 203 |\n| O | R | 203 |\n| O | S | 177 |\n| O | T | 207 |\n| O | U | 197 |\n| O | V | 177 |\n| O | W | 226 |\n| O | X | 193 |\n| P | A | 219 |\n| P | B | 171 |\n| P | C | 186 |\n| P | D | 169 |\n| P | E | 223 |\n| P | F | 225 |\n| P | G | 175 |\n| P | H | 209 |\n| P | I | 201 |\n| P | J | 205 |\n| P | K | 212 |\n| P | L | 183 |\n| P | M | 190 |\n| P | N | 177 |\n| P | O | 198 |\n| P | Q | 209 |\n| P | R | 189 |\n| P | S | 210 |\n| P | T | 183 |\n| P | U | 218 |\n| P | V | 193 |\n| P | W | 218 |\n| P | X | 198 |\n| Q | A | 253 |\n| Q | B | 180 |\n| Q | C | 148 |\n| Q | D | 212 |\n| Q | E | 211 |\n| Q | F | 204 |\n| Q | G | 218 |\n| Q | H | 183 |\n| Q | I | 233 |\n| Q | J | 232 |\n| Q | K | 204 |\n| Q | L | 215 |\n| Q | M | 170 |\n| Q | N | 147 |\n| Q | O | 203 |\n| Q | P | 209 |\n| Q | R | 163 |\n| Q | S | 203 |\n| Q | T | 208 |\n| Q | U | 148 |\n| Q | V | 157 |\n| Q | W | 198 |\n| Q | X | 184 |\n| R | A | 202 |\n| R | B | 163 |\n| R | C | 203 |\n| R | D | 203 |\n| R | E | 169 |\n| R | F | 183 |\n| R | G | 176 |\n| R | H | 166 |\n| R | I | 206 |\n| R | J | 202 |\n| R | K | 208 |\n| R | L | 210 |\n| R | M | 180 |\n| R | N | 192 |\n| R | O | 203 |\n| R | P | 189 |\n| R | Q | 163 |\n| R | S | 195 |\n| R | T | 188 |\n| R | U | 193 |\n| R | V | 193 |\n| R | W | 178 |\n| R | X | 191 |\n| S | A | 186 |\n| S | B | 161 |\n| S | C | 151 |\n| S | D | 141 |\n| S | E | 149 |\n| S | F | 176 |\n| S | G | 185 |\n| S | H | 163 |\n| S | I | 146 |\n| S | J | 155 |\n| S | K | 179 |\n| S | L | 203 |\n| S | M | 166 |\n| S | N | 163 |\n| S | O | 177 |\n| S | P | 210 |\n| S | Q | 203 |\n| S | R | 195 |\n| S | T | 183 |\n| S | U | 154 |\n| S | V | 171 |\n| S | W | 190 |\n| S | X | 188 |\n| T | A | 205 |\n| T | B | 191 |\n| T | C | 199 |\n| T | D | 175 |\n| T | E | 155 |\n| T | F | 186 |\n| T | G | 202 |\n| T | H | 183 |\n| T | I | 164 |\n| T | J | 168 |\n| T | K | 225 |\n| T | L | 184 |\n| T | M | 149 |\n| T | N | 192 |\n| T | O | 207 |\n| T | P | 183 |\n| T | Q | 208 |\n| T | R | 188 |\n| T | S | 183 |\n| T | U | 170 |\n| T | V | 161 |\n| T | W | 178 |\n| T | X | 165 |\n| U | A | 226 |\n| U | B | 204 |\n| U | C | 164 |\n| U | D | 162 |\n| U | E | 185 |\n| U | F | 173 |\n| U | G | 194 |\n| U | H | 182 |\n| U | I | 177 |\n| U | J | 188 |\n| U | K | 201 |\n| U | L | 199 |\n| U | M | 138 |\n| U | N | 160 |\n| U | O | 197 |\n| U | P | 218 |\n| U | Q | 148 |\n| U | R | 193 |\n| U | S | 154 |\n| U | T | 170 |\n| U | V | 133 |\n| U | W | 182 |\n| U | X | 200 |\n| V | A | 186 |\n| V | B | 172 |\n| V | C | 123 |\n| V | D | 179 |\n| V | E | 183 |\n| V | F | 155 |\n| V | G | 170 |\n| V | H | 180 |\n| V | I | 178 |\n| V | J | 165 |\n| V | K | 199 |\n| V | L | 196 |\n| V | M | 132 |\n| V | N | 167 |\n| V | O | 177 |\n| V | P | 193 |\n| V | Q | 157 |\n| V | R | 193 |\n| V | S | 171 |\n| V | T | 161 |\n| V | U | 133 |\n| V | W | 179 |\n| V | X | 147 |\n| W | A | 210 |\n| W | B | 193 |\n| W | C | 186 |\n| W | D | 174 |\n| W | E | 185 |\n| W | F | 196 |\n| W | G | 218 |\n| W | H | 186 |\n| W | I | 191 |\n| W | J | 163 |\n| W | K | 228 |\n| W | L | 178 |\n| W | M | 186 |\n| W | N | 178 |\n| W | O | 226 |\n| W | P | 218 |\n| W | Q | 198 |\n| W | R | 178 |\n| W | S | 190 |\n| W | T | 178 |\n| W | U | 182 |\n| W | V | 179 |\n| W | X | 226 |\n| X | A | 196 |\n| X | B | 183 |\n| X | C | 171 |\n| X | D | 198 |\n| X | E | 187 |\n| X | F | 194 |\n| X | G | 175 |\n| X | H | 182 |\n| X | I | 196 |\n| X | J | 223 |\n| X | K | 245 |\n| X | L | 226 |\n| X | M | 174 |\n| X | N | 202 |\n| X | O | 193 |\n| X | P | 198 |\n| X | Q | 184 |\n| X | R | 191 |\n| X | S | 188 |\n| X | T | 165 |\n| X | U | 200 |\n| X | V | 147 |\n| X | W | 226 |\n\nThe aim is to maximize the sum of these pairwise distances for the chosen 10 artworks.\n\nOh, and when you give the final pick, just drop it into a tiny JSON blob like this so it's clear and easy to read:\n\n{\n \"solution\": [, , ...]\n}\n\n\"solution\" is simply the list of artwork identifiers for the pieces you want in the mini-show — one identifier per chosen piece. This JSON is just a sketch of the shape I expect, not the actual answer itself.\n\nPlease 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”.\"", "instance": { "distance_matrix": [ [ 0, 181, 212, 205, 178, 143, 176, 178, 186, 152, 219, 221, 168, 215, 235, 219, 253, 202, 186, 205, 226, 186, 210, 196 ], [ 181, 0, 167, 166, 178, 210, 181, 222, 204, 187, 135, 182, 154, 135, 195, 171, 180, 163, 161, 191, 204, 172, 193, 183 ], [ 212, 167, 0, 176, 187, 187, 179, 184, 188, 173, 195, 208, 191, 145, 172, 186, 148, 203, 151, 199, 164, 123, 186, 171 ], [ 205, 166, 176, 0, 197, 189, 177, 196, 147, 169, 189, 167, 174, 179, 188, 169, 212, 203, 141, 175, 162, 179, 174, 198 ], [ 178, 178, 187, 197, 0, 195, 154, 172, 166, 168, 214, 228, 193, 205, 185, 223, 211, 169, 149, 155, 185, 183, 185, 187 ], [ 143, 210, 187, 189, 195, 0, 207, 144, 175, 131, 214, 189, 158, 195, 224, 225, 204, 183, 176, 186, 173, 155, 196, 194 ], [ 176, 181, 179, 177, 154, 207, 0, 192, 200, 214, 214, 245, 194, 219, 162, 175, 218, 176, 185, 202, 194, 170, 218, 175 ], [ 178, 222, 184, 196, 172, 144, 192, 0, 170, 184, 237, 229, 180, 206, 188, 209, 183, 166, 163, 183, 182, 180, 186, 182 ], [ 186, 204, 188, 147, 166, 175, 200, 170, 0, 125, 226, 205, 173, 205, 169, 201, 233, 206, 146, 164, 177, 178, 191, 196 ], [ 152, 187, 173, 169, 168, 131, 214, 184, 125, 0, 200, 163, 168, 180, 213, 205, 232, 202, 155, 168, 188, 165, 163, 223 ], [ 219, 135, 195, 189, 214, 214, 214, 237, 226, 200, 0, 165, 169, 137, 240, 212, 204, 208, 179, 225, 201, 199, 228, 245 ], [ 221, 182, 208, 167, 228, 189, 245, 229, 205, 163, 165, 0, 177, 133, 270, 183, 215, 210, 203, 184, 199, 196, 178, 226 ], [ 168, 154, 191, 174, 193, 158, 194, 180, 173, 168, 169, 177, 0, 152, 200, 190, 170, 180, 166, 149, 138, 132, 186, 174 ], [ 215, 135, 145, 179, 205, 195, 219, 206, 205, 180, 137, 133, 152, 0, 212, 177, 147, 192, 163, 192, 160, 167, 178, 202 ], [ 235, 195, 172, 188, 185, 224, 162, 188, 169, 213, 240, 270, 200, 212, 0, 198, 203, 203, 177, 207, 197, 177, 226, 193 ], [ 219, 171, 186, 169, 223, 225, 175, 209, 201, 205, 212, 183, 190, 177, 198, 0, 209, 189, 210, 183, 218, 193, 218, 198 ], [ 253, 180, 148, 212, 211, 204, 218, 183, 233, 232, 204, 215, 170, 147, 203, 209, 0, 163, 203, 208, 148, 157, 198, 184 ], [ 202, 163, 203, 203, 169, 183, 176, 166, 206, 202, 208, 210, 180, 192, 203, 189, 163, 0, 195, 188, 193, 193, 178, 191 ], [ 186, 161, 151, 141, 149, 176, 185, 163, 146, 155, 179, 203, 166, 163, 177, 210, 203, 195, 0, 183, 154, 171, 190, 188 ], [ 205, 191, 199, 175, 155, 186, 202, 183, 164, 168, 225, 184, 149, 192, 207, 183, 208, 188, 183, 0, 170, 161, 178, 165 ], [ 226, 204, 164, 162, 185, 173, 194, 182, 177, 188, 201, 199, 138, 160, 197, 218, 148, 193, 154, 170, 0, 133, 182, 200 ], [ 186, 172, 123, 179, 183, 155, 170, 180, 178, 165, 199, 196, 132, 167, 177, 193, 157, 193, 171, 161, 133, 0, 179, 147 ], [ 210, 193, 186, 174, 185, 196, 218, 186, 191, 163, 228, 178, 186, 178, 226, 218, 198, 178, 190, 178, 182, 179, 0, 226 ], [ 196, 183, 171, 198, 187, 194, 175, 182, 196, 223, 245, 226, 174, 202, 193, 198, 184, 191, 188, 165, 200, 147, 226, 0 ] ], "m": 10, "objective": 9367.0 }, "solution": [ 0, 4, 5, 10, 11, 14, 15, 16, 22, 23 ], "obj": 9367.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 24, "m": 10, "nodes": [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S", "T", "U", "V", "W", "X" ], "distance_pairs": [ { "from_id": "A", "to_id": "B", "distance": 181 }, { "from_id": "A", "to_id": "C", "distance": 212 }, { "from_id": "A", "to_id": "D", "distance": 205 }, { "from_id": "A", "to_id": "E", "distance": 178 }, { "from_id": "A", "to_id": "F", "distance": 143 }, { "from_id": "A", "to_id": "G", "distance": 176 }, { "from_id": "A", "to_id": "H", "distance": 178 }, { "from_id": "A", "to_id": "I", "distance": 186 }, { "from_id": "A", "to_id": "J", "distance": 152 }, { "from_id": "A", "to_id": "K", "distance": 219 }, { "from_id": "A", "to_id": "L", "distance": 221 }, { "from_id": "A", "to_id": "M", "distance": 168 }, { "from_id": "A", "to_id": "N", "distance": 215 }, { "from_id": "A", "to_id": "O", "distance": 235 }, { "from_id": "A", "to_id": "P", "distance": 219 }, { "from_id": "A", "to_id": "Q", "distance": 253 }, { "from_id": "A", "to_id": "R", "distance": 202 }, { "from_id": "A", "to_id": "S", "distance": 186 }, { "from_id": "A", "to_id": "T", "distance": 205 }, { "from_id": "A", "to_id": "U", "distance": 226 }, { "from_id": "A", "to_id": "V", "distance": 186 }, { "from_id": "A", "to_id": "W", "distance": 210 }, { "from_id": "A", "to_id": "X", "distance": 196 }, { "from_id": "B", "to_id": "A", "distance": 181 }, { "from_id": "B", "to_id": "C", "distance": 167 }, { "from_id": "B", "to_id": "D", "distance": 166 }, { "from_id": "B", "to_id": "E", "distance": 178 }, { "from_id": "B", "to_id": "F", "distance": 210 }, { "from_id": "B", "to_id": "G", "distance": 181 }, { "from_id": "B", "to_id": "H", "distance": 222 }, { "from_id": "B", "to_id": "I", "distance": 204 }, { "from_id": "B", "to_id": "J", "distance": 187 }, { "from_id": "B", "to_id": "K", "distance": 135 }, { "from_id": "B", "to_id": "L", "distance": 182 }, { "from_id": "B", "to_id": "M", "distance": 154 }, { "from_id": "B", "to_id": "N", "distance": 135 }, { "from_id": "B", "to_id": "O", "distance": 195 }, { "from_id": "B", "to_id": "P", "distance": 171 }, { "from_id": "B", "to_id": "Q", "distance": 180 }, { "from_id": "B", "to_id": "R", "distance": 163 }, { "from_id": "B", "to_id": "S", "distance": 161 }, { "from_id": "B", "to_id": "T", "distance": 191 }, { "from_id": "B", "to_id": "U", "distance": 204 }, { "from_id": "B", "to_id": "V", "distance": 172 }, { "from_id": "B", "to_id": "W", "distance": 193 }, { "from_id": "B", "to_id": "X", "distance": 183 }, { "from_id": "C", "to_id": "A", "distance": 212 }, { "from_id": "C", "to_id": "B", "distance": 167 }, { "from_id": "C", "to_id": "D", "distance": 176 }, { "from_id": "C", "to_id": 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208 }, { "from_id": "R", "to_id": "L", "distance": 210 }, { "from_id": "R", "to_id": "M", "distance": 180 }, { "from_id": "R", "to_id": "N", "distance": 192 }, { "from_id": "R", "to_id": "O", "distance": 203 }, { "from_id": "R", "to_id": "P", "distance": 189 }, { "from_id": "R", "to_id": "Q", "distance": 163 }, { "from_id": "R", "to_id": "S", "distance": 195 }, { "from_id": "R", "to_id": "T", "distance": 188 }, { "from_id": "R", "to_id": "U", "distance": 193 }, { "from_id": "R", "to_id": "V", "distance": 193 }, { "from_id": "R", "to_id": "W", "distance": 178 }, { "from_id": "R", "to_id": "X", "distance": 191 }, { "from_id": "S", "to_id": "A", "distance": 186 }, { "from_id": "S", "to_id": "B", "distance": 161 }, { "from_id": "S", "to_id": "C", "distance": 151 }, { "from_id": "S", "to_id": "D", "distance": 141 }, { "from_id": "S", "to_id": "E", "distance": 149 }, { "from_id": "S", "to_id": "F", "distance": 176 }, { "from_id": "S", "to_id": "G", "distance": 185 }, { "from_id": "S", "to_id": "H", "distance": 163 }, { "from_id": "S", "to_id": "I", "distance": 146 }, { "from_id": "S", "to_id": "J", "distance": 155 }, { "from_id": "S", "to_id": "K", "distance": 179 }, { "from_id": "S", "to_id": "L", "distance": 203 }, { "from_id": "S", "to_id": "M", "distance": 166 }, { "from_id": "S", "to_id": "N", "distance": 163 }, { "from_id": "S", "to_id": "O", "distance": 177 }, { "from_id": "S", "to_id": "P", "distance": 210 }, { "from_id": "S", "to_id": "Q", "distance": 203 }, { "from_id": "S", "to_id": "R", "distance": 195 }, { "from_id": "S", "to_id": "T", "distance": 183 }, { "from_id": "S", "to_id": "U", "distance": 154 }, { "from_id": "S", "to_id": "V", "distance": 171 }, { "from_id": "S", "to_id": "W", "distance": 190 }, { "from_id": "S", "to_id": "X", "distance": 188 }, { "from_id": "T", "to_id": "A", "distance": 205 }, { "from_id": "T", "to_id": "B", "distance": 191 }, { "from_id": "T", "to_id": "C", "distance": 199 }, { "from_id": "T", "to_id": "D", "distance": 175 }, { "from_id": "T", "to_id": "E", "distance": 155 }, { "from_id": "T", "to_id": "F", "distance": 186 }, { "from_id": "T", "to_id": "G", "distance": 202 }, { "from_id": "T", "to_id": "H", "distance": 183 }, { "from_id": "T", "to_id": "I", "distance": 164 }, { "from_id": "T", "to_id": "J", "distance": 168 }, { "from_id": "T", "to_id": "K", "distance": 225 }, { "from_id": "T", "to_id": "L", "distance": 184 }, { "from_id": "T", "to_id": "M", "distance": 149 }, { "from_id": "T", "to_id": "N", "distance": 192 }, { "from_id": "T", "to_id": "O", "distance": 207 }, { "from_id": "T", "to_id": "P", "distance": 183 }, { "from_id": "T", "to_id": "Q", "distance": 208 }, { "from_id": "T", "to_id": "R", "distance": 188 }, { "from_id": "T", "to_id": "S", "distance": 183 }, { "from_id": "T", "to_id": "U", "distance": 170 }, { "from_id": "T", "to_id": "V", "distance": 161 }, { "from_id": "T", "to_id": "W", "distance": 178 }, { "from_id": "T", "to_id": "X", "distance": 165 }, { "from_id": "U", "to_id": "A", "distance": 226 }, { "from_id": "U", "to_id": "B", "distance": 204 }, { "from_id": "U", "to_id": "C", "distance": 164 }, { "from_id": "U", "to_id": "D", "distance": 162 }, { "from_id": "U", "to_id": "E", "distance": 185 }, { "from_id": "U", "to_id": "F", "distance": 173 }, { "from_id": "U", "to_id": "G", "distance": 194 }, { "from_id": "U", "to_id": "H", "distance": 182 }, { "from_id": "U", "to_id": "I", "distance": 177 }, { "from_id": "U", "to_id": "J", "distance": 188 }, { "from_id": "U", "to_id": "K", "distance": 201 }, { "from_id": "U", "to_id": "L", "distance": 199 }, { "from_id": "U", "to_id": "M", "distance": 138 }, { "from_id": "U", "to_id": "N", "distance": 160 }, { "from_id": "U", "to_id": "O", "distance": 197 }, { "from_id": "U", "to_id": "P", "distance": 218 }, { "from_id": "U", "to_id": "Q", "distance": 148 }, { "from_id": "U", "to_id": "R", "distance": 193 }, { "from_id": "U", "to_id": "S", "distance": 154 }, { "from_id": "U", "to_id": "T", "distance": 170 }, { "from_id": "U", "to_id": "V", "distance": 133 }, { "from_id": "U", "to_id": "W", "distance": 182 }, { "from_id": "U", "to_id": "X", "distance": 200 }, { "from_id": "V", "to_id": "A", "distance": 186 }, { "from_id": "V", "to_id": "B", "distance": 172 }, { "from_id": "V", "to_id": "C", "distance": 123 }, { "from_id": "V", "to_id": "D", "distance": 179 }, { "from_id": "V", "to_id": "E", "distance": 183 }, { "from_id": "V", "to_id": "F", "distance": 155 }, { "from_id": "V", "to_id": "G", "distance": 170 }, { "from_id": "V", "to_id": "H", "distance": 180 }, { "from_id": "V", "to_id": "I", "distance": 178 }, { "from_id": "V", "to_id": "J", "distance": 165 }, { "from_id": "V", "to_id": "K", "distance": 199 }, { "from_id": "V", "to_id": "L", "distance": 196 }, { "from_id": "V", "to_id": "M", "distance": 132 }, { "from_id": "V", "to_id": "N", "distance": 167 }, { "from_id": "V", "to_id": "O", "distance": 177 }, { "from_id": "V", "to_id": "P", "distance": 193 }, { "from_id": "V", "to_id": "Q", "distance": 157 }, { "from_id": "V", "to_id": "R", "distance": 193 }, { "from_id": "V", "to_id": "S", "distance": 171 }, { "from_id": "V", "to_id": "T", "distance": 161 }, { "from_id": "V", "to_id": "U", "distance": 133 }, { "from_id": "V", "to_id": "W", "distance": 179 }, { "from_id": "V", "to_id": "X", "distance": 147 }, { "from_id": "W", "to_id": "A", "distance": 210 }, { "from_id": "W", "to_id": "B", "distance": 193 }, { "from_id": "W", "to_id": "C", "distance": 186 }, { "from_id": "W", "to_id": "D", "distance": 174 }, { "from_id": "W", "to_id": "E", "distance": 185 }, { "from_id": "W", "to_id": "F", "distance": 196 }, { "from_id": "W", "to_id": "G", "distance": 218 }, { "from_id": "W", "to_id": "H", "distance": 186 }, { "from_id": "W", "to_id": "I", "distance": 191 }, { "from_id": "W", "to_id": "J", "distance": 163 }, { "from_id": "W", "to_id": "K", "distance": 228 }, { "from_id": "W", "to_id": "L", "distance": 178 }, { "from_id": "W", "to_id": "M", "distance": 186 }, { "from_id": "W", "to_id": "N", "distance": 178 }, { "from_id": "W", "to_id": "O", "distance": 226 }, { "from_id": "W", "to_id": "P", "distance": 218 }, { "from_id": "W", "to_id": "Q", "distance": 198 }, { "from_id": "W", "to_id": "R", "distance": 178 }, { "from_id": "W", "to_id": "S", "distance": 190 }, { "from_id": "W", "to_id": "T", "distance": 178 }, { "from_id": "W", "to_id": "U", "distance": 182 }, { "from_id": "W", "to_id": "V", "distance": 179 }, { "from_id": "W", "to_id": "X", "distance": 226 }, { "from_id": "X", "to_id": "A", "distance": 196 }, { "from_id": "X", "to_id": "B", "distance": 183 }, { "from_id": "X", "to_id": "C", "distance": 171 }, { "from_id": "X", "to_id": "D", "distance": 198 }, { "from_id": "X", "to_id": "E", "distance": 187 }, { "from_id": "X", "to_id": "F", "distance": 194 }, { "from_id": "X", "to_id": "G", "distance": 175 }, { "from_id": "X", "to_id": "H", "distance": 182 }, { "from_id": "X", "to_id": "I", "distance": 196 }, { "from_id": "X", "to_id": "J", "distance": 223 }, { "from_id": "X", "to_id": "K", "distance": 245 }, { "from_id": "X", "to_id": "L", "distance": 226 }, { "from_id": "X", "to_id": "M", "distance": 174 }, { "from_id": "X", "to_id": "N", "distance": 202 }, { "from_id": "X", "to_id": "O", "distance": 193 }, { "from_id": "X", "to_id": "P", "distance": 198 }, { "from_id": "X", "to_id": "Q", "distance": 184 }, { "from_id": "X", "to_id": "R", "distance": 191 }, { "from_id": "X", "to_id": "S", "distance": 188 }, { "from_id": "X", "to_id": "T", "distance": 165 }, { "from_id": "X", "to_id": "U", "distance": 200 }, { "from_id": "X", "to_id": "V", "distance": 147 }, { "from_id": "X", "to_id": "W", "distance": 226 } ] }, "solution_variant": [ "A", "E", "F", "K", "L", "O", "P", "Q", "W", "X" ], "context_index": 3, "input_format": "markdown_table", "input_index_base": "names" }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Many people at the tasting panel wanted one rule: make the flight feel varied from start to finish. The chefs measured how different each pair of dishes is, and the idea is to pick exactly m different plates so that if those pairwise differences are all added together, the total is as big as it can be — that total is the variety score. Every pick has to be a distinct dish and all m slots must be taken. The specific items, the number m, and their pairwise distances are given below.\n\n{\n \"total_dishes\": 21,\n \"flight_size\": 4,\n \"dish_ids\": [\n 0,\n 1,\n 2,\n 3,\n 4,\n 5,\n 6,\n 7,\n 8,\n 9,\n 10,\n 11,\n 12,\n 13,\n 14,\n 15,\n 16,\n 17,\n 18,\n 19,\n 20\n ],\n \"data\": [\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 1,\n \"flavor_distance\": 189\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 2,\n \"flavor_distance\": 158\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 3,\n \"flavor_distance\": 169\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 4,\n \"flavor_distance\": 183\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 5,\n \"flavor_distance\": 144\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 6,\n \"flavor_distance\": 179\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 7,\n \"flavor_distance\": 165\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 8,\n \"flavor_distance\": 161\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 9,\n \"flavor_distance\": 187\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 10,\n \"flavor_distance\": 212\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 11,\n \"flavor_distance\": 129\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 12,\n \"flavor_distance\": 158\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 13,\n \"flavor_distance\": 165\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 14,\n \"flavor_distance\": 171\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 15,\n \"flavor_distance\": 226\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 16,\n \"flavor_distance\": 163\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 17,\n \"flavor_distance\": 186\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 18,\n \"flavor_distance\": 203\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 19,\n \"flavor_distance\": 168\n },\n {\n \"dish_id_a\": 0,\n \"dish_id_b\": 20,\n \"flavor_distance\": 161\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 0,\n \"flavor_distance\": 189\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 2,\n \"flavor_distance\": 173\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 3,\n \"flavor_distance\": 151\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 4,\n \"flavor_distance\": 217\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 5,\n \"flavor_distance\": 152\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 6,\n \"flavor_distance\": 173\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 7,\n \"flavor_distance\": 163\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 8,\n \"flavor_distance\": 153\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 9,\n \"flavor_distance\": 151\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 10,\n \"flavor_distance\": 180\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 11,\n \"flavor_distance\": 224\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 12,\n \"flavor_distance\": 180\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 13,\n \"flavor_distance\": 139\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 14,\n \"flavor_distance\": 213\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 15,\n \"flavor_distance\": 170\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 16,\n \"flavor_distance\": 184\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 17,\n \"flavor_distance\": 162\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 18,\n \"flavor_distance\": 180\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 19,\n \"flavor_distance\": 221\n },\n {\n \"dish_id_a\": 1,\n \"dish_id_b\": 20,\n \"flavor_distance\": 169\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 0,\n \"flavor_distance\": 158\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 1,\n \"flavor_distance\": 173\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 3,\n \"flavor_distance\": 170\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 4,\n \"flavor_distance\": 170\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 5,\n \"flavor_distance\": 122\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 6,\n \"flavor_distance\": 187\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 7,\n \"flavor_distance\": 178\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 8,\n \"flavor_distance\": 152\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 9,\n \"flavor_distance\": 189\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 10,\n \"flavor_distance\": 191\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 11,\n \"flavor_distance\": 200\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 12,\n \"flavor_distance\": 109\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 13,\n \"flavor_distance\": 166\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 14,\n \"flavor_distance\": 167\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 15,\n \"flavor_distance\": 195\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 16,\n \"flavor_distance\": 138\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 17,\n \"flavor_distance\": 152\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 18,\n \"flavor_distance\": 147\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 19,\n \"flavor_distance\": 136\n },\n {\n \"dish_id_a\": 2,\n \"dish_id_b\": 20,\n \"flavor_distance\": 140\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 0,\n \"flavor_distance\": 169\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 1,\n \"flavor_distance\": 151\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 2,\n \"flavor_distance\": 170\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 4,\n \"flavor_distance\": 205\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 5,\n \"flavor_distance\": 123\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 6,\n \"flavor_distance\": 186\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 7,\n \"flavor_distance\": 143\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 8,\n \"flavor_distance\": 204\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 9,\n \"flavor_distance\": 183\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 10,\n \"flavor_distance\": 192\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 11,\n \"flavor_distance\": 196\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 12,\n \"flavor_distance\": 164\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 13,\n \"flavor_distance\": 193\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 14,\n \"flavor_distance\": 206\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 15,\n \"flavor_distance\": 202\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 16,\n \"flavor_distance\": 207\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 17,\n \"flavor_distance\": 162\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 18,\n \"flavor_distance\": 153\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 19,\n \"flavor_distance\": 199\n },\n {\n \"dish_id_a\": 3,\n \"dish_id_b\": 20,\n \"flavor_distance\": 171\n },\n {\n \"dish_id_a\": 4,\n \"dish_id_b\": 0,\n \"flavor_distance\": 183\n },\n {\n \"dish_id_a\": 4,\n \"dish_id_b\": 1,\n \"flavor_distance\": 217\n },\n {\n \"dish_id_a\": 4,\n \"dish_id_b\": 2,\n \"flavor_distance\": 170\n },\n {\n \"dish_id_a\": 4,\n \"dish_id_b\": 3,\n \"flavor_distance\": 205\n },\n {\n \"dish_id_a\": 4,\n \"dish_id_b\": 5,\n \"flavor_distance\": 169\n },\n {\n \"dish_id_a\": 4,\n \"dish_id_b\": 6,\n \"flavor_distance\": 168\n },\n {\n \"dish_id_a\": 4,\n \"dish_id_b\": 7,\n \"flavor_distance\": 168\n },\n {\n \"dish_id_a\": 4,\n 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8,\n \"flavor_distance\": 133\n },\n {\n \"dish_id_a\": 13,\n \"dish_id_b\": 9,\n \"flavor_distance\": 157\n },\n {\n \"dish_id_a\": 13,\n \"dish_id_b\": 10,\n \"flavor_distance\": 172\n },\n {\n \"dish_id_a\": 13,\n \"dish_id_b\": 11,\n \"flavor_distance\": 182\n },\n {\n \"dish_id_a\": 13,\n \"dish_id_b\": 12,\n \"flavor_distance\": 158\n },\n {\n \"dish_id_a\": 13,\n \"dish_id_b\": 14,\n \"flavor_distance\": 139\n },\n {\n \"dish_id_a\": 13,\n \"dish_id_b\": 15,\n \"flavor_distance\": 154\n },\n {\n \"dish_id_a\": 13,\n \"dish_id_b\": 16,\n \"flavor_distance\": 136\n },\n {\n \"dish_id_a\": 13,\n \"dish_id_b\": 17,\n \"flavor_distance\": 176\n },\n {\n \"dish_id_a\": 13,\n \"dish_id_b\": 18,\n \"flavor_distance\": 194\n },\n {\n \"dish_id_a\": 13,\n \"dish_id_b\": 19,\n \"flavor_distance\": 170\n },\n {\n \"dish_id_a\": 13,\n \"dish_id_b\": 20,\n \"flavor_distance\": 164\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 0,\n \"flavor_distance\": 171\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 1,\n \"flavor_distance\": 213\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 2,\n \"flavor_distance\": 167\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 3,\n \"flavor_distance\": 206\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 4,\n \"flavor_distance\": 188\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 5,\n \"flavor_distance\": 192\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 6,\n \"flavor_distance\": 169\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 7,\n \"flavor_distance\": 165\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 8,\n \"flavor_distance\": 194\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 9,\n \"flavor_distance\": 184\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 10,\n \"flavor_distance\": 184\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 11,\n \"flavor_distance\": 166\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 12,\n \"flavor_distance\": 139\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 13,\n \"flavor_distance\": 139\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 15,\n \"flavor_distance\": 194\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 16,\n \"flavor_distance\": 167\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 17,\n \"flavor_distance\": 151\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 18,\n \"flavor_distance\": 217\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 19,\n \"flavor_distance\": 159\n },\n {\n \"dish_id_a\": 14,\n \"dish_id_b\": 20,\n \"flavor_distance\": 166\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 0,\n \"flavor_distance\": 226\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 1,\n \"flavor_distance\": 170\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 2,\n \"flavor_distance\": 195\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 3,\n \"flavor_distance\": 202\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 4,\n \"flavor_distance\": 197\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 5,\n \"flavor_distance\": 205\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 6,\n \"flavor_distance\": 129\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 7,\n \"flavor_distance\": 175\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 8,\n \"flavor_distance\": 155\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 9,\n \"flavor_distance\": 188\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 10,\n \"flavor_distance\": 119\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 11,\n \"flavor_distance\": 238\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 12,\n \"flavor_distance\": 193\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 13,\n \"flavor_distance\": 154\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 14,\n \"flavor_distance\": 194\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 16,\n \"flavor_distance\": 199\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 17,\n \"flavor_distance\": 183\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 18,\n \"flavor_distance\": 202\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 19,\n \"flavor_distance\": 197\n },\n {\n \"dish_id_a\": 15,\n \"dish_id_b\": 20,\n \"flavor_distance\": 166\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 0,\n \"flavor_distance\": 163\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 1,\n \"flavor_distance\": 184\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 2,\n \"flavor_distance\": 138\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 3,\n \"flavor_distance\": 207\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 4,\n \"flavor_distance\": 143\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 5,\n \"flavor_distance\": 149\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 6,\n \"flavor_distance\": 152\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 7,\n \"flavor_distance\": 185\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 8,\n \"flavor_distance\": 115\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 9,\n \"flavor_distance\": 154\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 10,\n \"flavor_distance\": 208\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 11,\n \"flavor_distance\": 181\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 12,\n \"flavor_distance\": 153\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 13,\n \"flavor_distance\": 136\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 14,\n \"flavor_distance\": 167\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 15,\n \"flavor_distance\": 199\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 17,\n \"flavor_distance\": 152\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 18,\n \"flavor_distance\": 170\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 19,\n \"flavor_distance\": 162\n },\n {\n \"dish_id_a\": 16,\n \"dish_id_b\": 20,\n \"flavor_distance\": 142\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 0,\n \"flavor_distance\": 186\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 1,\n \"flavor_distance\": 162\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 2,\n \"flavor_distance\": 152\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 3,\n \"flavor_distance\": 162\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 4,\n \"flavor_distance\": 168\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 5,\n \"flavor_distance\": 127\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 6,\n \"flavor_distance\": 155\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 7,\n \"flavor_distance\": 118\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 8,\n \"flavor_distance\": 171\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 9,\n \"flavor_distance\": 165\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 10,\n \"flavor_distance\": 167\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 11,\n \"flavor_distance\": 177\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 12,\n \"flavor_distance\": 142\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 13,\n \"flavor_distance\": 176\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 14,\n \"flavor_distance\": 151\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 15,\n \"flavor_distance\": 183\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 16,\n \"flavor_distance\": 152\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 18,\n \"flavor_distance\": 177\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 19,\n \"flavor_distance\": 187\n },\n {\n \"dish_id_a\": 17,\n \"dish_id_b\": 20,\n \"flavor_distance\": 142\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 0,\n \"flavor_distance\": 203\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 1,\n \"flavor_distance\": 180\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 2,\n \"flavor_distance\": 147\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 3,\n \"flavor_distance\": 153\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 4,\n \"flavor_distance\": 190\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 5,\n \"flavor_distance\": 159\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 6,\n \"flavor_distance\": 187\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 7,\n \"flavor_distance\": 196\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 8,\n \"flavor_distance\": 176\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 9,\n \"flavor_distance\": 199\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 10,\n \"flavor_distance\": 191\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 11,\n \"flavor_distance\": 208\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 12,\n \"flavor_distance\": 158\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 13,\n \"flavor_distance\": 194\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 14,\n \"flavor_distance\": 217\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 15,\n \"flavor_distance\": 202\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 16,\n \"flavor_distance\": 170\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 17,\n \"flavor_distance\": 177\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 19,\n \"flavor_distance\": 180\n },\n {\n \"dish_id_a\": 18,\n \"dish_id_b\": 20,\n \"flavor_distance\": 140\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 0,\n \"flavor_distance\": 168\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 1,\n \"flavor_distance\": 221\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 2,\n \"flavor_distance\": 136\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 3,\n \"flavor_distance\": 199\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 4,\n \"flavor_distance\": 186\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 5,\n \"flavor_distance\": 156\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 6,\n \"flavor_distance\": 188\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 7,\n \"flavor_distance\": 190\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 8,\n \"flavor_distance\": 180\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 9,\n \"flavor_distance\": 223\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 10,\n \"flavor_distance\": 221\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 11,\n \"flavor_distance\": 153\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 12,\n \"flavor_distance\": 157\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 13,\n \"flavor_distance\": 170\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 14,\n \"flavor_distance\": 159\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 15,\n \"flavor_distance\": 197\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 16,\n \"flavor_distance\": 162\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 17,\n \"flavor_distance\": 187\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 18,\n \"flavor_distance\": 180\n },\n {\n \"dish_id_a\": 19,\n \"dish_id_b\": 20,\n \"flavor_distance\": 184\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 0,\n \"flavor_distance\": 161\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 1,\n \"flavor_distance\": 169\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 2,\n \"flavor_distance\": 140\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 3,\n \"flavor_distance\": 171\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 4,\n \"flavor_distance\": 97\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 5,\n \"flavor_distance\": 162\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 6,\n \"flavor_distance\": 144\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 7,\n \"flavor_distance\": 144\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 8,\n \"flavor_distance\": 114\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 9,\n \"flavor_distance\": 164\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 10,\n \"flavor_distance\": 152\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 11,\n \"flavor_distance\": 183\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 12,\n \"flavor_distance\": 145\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 13,\n \"flavor_distance\": 164\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 14,\n \"flavor_distance\": 166\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 15,\n \"flavor_distance\": 166\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 16,\n \"flavor_distance\": 142\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 17,\n \"flavor_distance\": 142\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 18,\n \"flavor_distance\": 140\n },\n {\n \"dish_id_a\": 20,\n \"dish_id_b\": 19,\n \"flavor_distance\": 184\n }\n ]\n}\n\nYou can just hand me your picked flight in a tiny JSON snippet like this — it's an easy shape for the answer so we both know what to expect.\n\n{\n \"solution\": [, , ...]\n}\n\n\"solution\" is the list of exactly m plates you want in the flight — each entry should be the dish identifier from the instance. Think of it like filling out a simple form: list the labels of the plates you picked, nothing more. This JSON is just a sketch of the shape I need, not the actual final selection.\n\nPlease make sure to use the exact identifiers from the instance input — don't rename them or add new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 189, 158, 169, 183, 144, 179, 165, 161, 187, 212, 129, 158, 165, 171, 226, 163, 186, 203, 168, 161 ], [ 189, 0, 173, 151, 217, 152, 173, 163, 153, 151, 180, 224, 180, 139, 213, 170, 184, 162, 180, 221, 169 ], [ 158, 173, 0, 170, 170, 122, 187, 178, 152, 189, 191, 200, 109, 166, 167, 195, 138, 152, 147, 136, 140 ], [ 169, 151, 170, 0, 205, 123, 186, 143, 204, 183, 192, 196, 164, 193, 206, 202, 207, 162, 153, 199, 171 ], [ 183, 217, 170, 205, 0, 169, 168, 168, 136, 202, 192, 206, 177, 203, 188, 197, 143, 168, 190, 186, 97 ], [ 144, 152, 122, 123, 169, 0, 184, 151, 170, 184, 196, 178, 136, 186, 192, 205, 149, 127, 159, 156, 162 ], [ 179, 173, 187, 186, 168, 184, 0, 164, 113, 165, 121, 179, 167, 128, 169, 129, 152, 155, 187, 188, 144 ], [ 165, 163, 178, 143, 168, 151, 164, 0, 185, 177, 178, 159, 165, 173, 165, 175, 185, 118, 196, 190, 144 ], [ 161, 153, 152, 204, 136, 170, 113, 185, 0, 167, 174, 196, 168, 133, 194, 155, 115, 171, 176, 180, 114 ], [ 187, 151, 189, 183, 202, 184, 165, 177, 167, 0, 199, 201, 175, 157, 184, 188, 154, 165, 199, 223, 164 ], [ 212, 180, 191, 192, 192, 196, 121, 178, 174, 199, 0, 233, 185, 172, 184, 119, 208, 167, 191, 221, 152 ], [ 129, 224, 200, 196, 206, 178, 179, 159, 196, 201, 233, 0, 185, 182, 166, 238, 181, 177, 208, 153, 183 ], [ 158, 180, 109, 164, 177, 136, 167, 165, 168, 175, 185, 185, 0, 158, 139, 193, 153, 142, 158, 157, 145 ], [ 165, 139, 166, 193, 203, 186, 128, 173, 133, 157, 172, 182, 158, 0, 139, 154, 136, 176, 194, 170, 164 ], [ 171, 213, 167, 206, 188, 192, 169, 165, 194, 184, 184, 166, 139, 139, 0, 194, 167, 151, 217, 159, 166 ], [ 226, 170, 195, 202, 197, 205, 129, 175, 155, 188, 119, 238, 193, 154, 194, 0, 199, 183, 202, 197, 166 ], [ 163, 184, 138, 207, 143, 149, 152, 185, 115, 154, 208, 181, 153, 136, 167, 199, 0, 152, 170, 162, 142 ], [ 186, 162, 152, 162, 168, 127, 155, 118, 171, 165, 167, 177, 142, 176, 151, 183, 152, 0, 177, 187, 142 ], [ 203, 180, 147, 153, 190, 159, 187, 196, 176, 199, 191, 208, 158, 194, 217, 202, 170, 177, 0, 180, 140 ], [ 168, 221, 136, 199, 186, 156, 188, 190, 180, 223, 221, 153, 157, 170, 159, 197, 162, 187, 180, 0, 184 ], [ 161, 169, 140, 171, 97, 162, 144, 144, 114, 164, 152, 183, 145, 164, 166, 166, 142, 142, 140, 184, 0 ] ], "m": 4, "objective": 1252.0 }, "solution": [ 1, 4, 10, 11 ], "obj": 1252.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 21, "m": 4, "nodes": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 ], "distance_pairs": [ { "from_id": 0, "to_id": 1, "distance": 189 }, { "from_id": 0, "to_id": 2, "distance": 158 }, { "from_id": 0, "to_id": 3, "distance": 169 }, { "from_id": 0, "to_id": 4, "distance": 183 }, { "from_id": 0, "to_id": 5, "distance": 144 }, { "from_id": 0, "to_id": 6, "distance": 179 }, { "from_id": 0, "to_id": 7, "distance": 165 }, { "from_id": 0, "to_id": 8, "distance": 161 }, { "from_id": 0, "to_id": 9, "distance": 187 }, { "from_id": 0, "to_id": 10, "distance": 212 }, { "from_id": 0, "to_id": 11, "distance": 129 }, { "from_id": 0, "to_id": 12, "distance": 158 }, { "from_id": 0, "to_id": 13, "distance": 165 }, { "from_id": 0, "to_id": 14, "distance": 171 }, { "from_id": 0, "to_id": 15, "distance": 226 }, { "from_id": 0, "to_id": 16, "distance": 163 }, { "from_id": 0, "to_id": 17, "distance": 186 }, { "from_id": 0, "to_id": 18, "distance": 203 }, { "from_id": 0, "to_id": 19, "distance": 168 }, { "from_id": 0, "to_id": 20, "distance": 161 }, { "from_id": 1, "to_id": 0, "distance": 189 }, { "from_id": 1, "to_id": 2, "distance": 173 }, { "from_id": 1, "to_id": 3, "distance": 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20, "to_id": 17, "distance": 142 }, { "from_id": 20, "to_id": 18, "distance": 140 }, { "from_id": 20, "to_id": 19, "distance": 184 } ] }, "solution_variant": [ 1, 4, 10, 11 ], "context_index": 4, "input_format": "json", "input_index_base": 0 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Recently the brochure editor wanted a lineup of exactly m destinations that really showcases variety. There’s a matrix of numbers that say how different each pair of cities or regions is in culture and geography, and to judge a proposed set of m distinct places the editor simply adds up every pair’s difference number inside that set. The higher that summed total, the more varied the brochure looks. The full set of pairwise dissimilarities and the chosen m will be given below.\n\nBelow are the 25 candidate destinations 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, and the editor asks for exactly 2 distinct picks.\nThe dissimilarity between 1 and 2 is 4.\nThe dissimilarity between 1 and 3 is 2.\nThe dissimilarity between 1 and 4 is 8.\nThe dissimilarity between 1 and 5 is 2.\nThe dissimilarity between 1 and 6 is 1.\nThe dissimilarity between 1 and 7 is 9.\nThe dissimilarity between 1 and 8 is 3.\nThe dissimilarity between 1 and 9 is 7.\nThe dissimilarity between 1 and 10 is 3.\nThe dissimilarity between 1 and 11 is 7.\nThe dissimilarity between 1 and 12 is 9.\nThe dissimilarity between 1 and 13 is 3.\nThe dissimilarity between 1 and 14 is 0.\nThe dissimilarity between 1 and 15 is 4.\nThe dissimilarity between 1 and 16 is 3.\nThe dissimilarity between 1 and 17 is 7.\nThe dissimilarity between 1 and 18 is 1.\nThe dissimilarity between 1 and 19 is 2.\nThe dissimilarity between 1 and 20 is 3.\nThe dissimilarity between 1 and 21 is 4.\nThe dissimilarity between 1 and 22 is 8.\nThe dissimilarity between 1 and 23 is 3.\nThe dissimilarity between 1 and 24 is 3.\nThe dissimilarity between 1 and 25 is 0.\nThe dissimilarity between 2 and 1 is 4.\nThe dissimilarity between 2 and 3 is 8.\nThe dissimilarity between 2 and 4 is 2.\nThe dissimilarity between 2 and 5 is 5.\nThe dissimilarity between 2 and 6 is 3.\nThe dissimilarity between 2 and 7 is 1.\nThe dissimilarity between 2 and 8 is 5.\nThe dissimilarity between 2 and 9 is 2.\nThe dissimilarity between 2 and 10 is 2.\nThe dissimilarity between 2 and 11 is 0.\nThe dissimilarity between 2 and 12 is 8.\nThe dissimilarity between 2 and 13 is 9.\nThe dissimilarity between 2 and 14 is 0.\nThe dissimilarity between 2 and 15 is 0.\nThe dissimilarity between 2 and 16 is 7.\nThe dissimilarity between 2 and 17 is 3.\nThe dissimilarity between 2 and 18 is 7.\nThe dissimilarity between 2 and 19 is 9.\nThe dissimilarity between 2 and 20 is 6.\nThe dissimilarity between 2 and 21 is 2.\nThe dissimilarity between 2 and 22 is 9.\nThe dissimilarity between 2 and 23 is 6.\nThe dissimilarity between 2 and 24 is 3.\nThe dissimilarity between 2 and 25 is 9.\nThe dissimilarity between 3 and 1 is 2.\nThe dissimilarity between 3 and 2 is 8.\nThe dissimilarity between 3 and 4 is 4.\nThe dissimilarity between 3 and 5 is 7.\nThe dissimilarity between 3 and 6 is 0.\nThe dissimilarity between 3 and 7 is 7.\nThe dissimilarity between 3 and 8 is 6.\nThe dissimilarity between 3 and 9 is 8.\nThe dissimilarity between 3 and 10 is 1.\nThe dissimilarity between 3 and 11 is 7.\nThe dissimilarity between 3 and 12 is 3.\nThe dissimilarity between 3 and 13 is 2.\nThe dissimilarity between 3 and 14 is 2.\nThe dissimilarity between 3 and 15 is 7.\nThe dissimilarity between 3 and 16 is 1.\nThe dissimilarity between 3 and 17 is 6.\nThe dissimilarity between 3 and 18 is 8.\nThe dissimilarity between 3 and 19 is 1.\nThe dissimilarity between 3 and 20 is 8.\nThe dissimilarity between 3 and 21 is 1.\nThe dissimilarity between 3 and 22 is 1.\nThe dissimilarity between 3 and 23 is 6.\nThe dissimilarity between 3 and 24 is 9.\nThe dissimilarity between 3 and 25 is 7.\nThe dissimilarity between 4 and 1 is 8.\nThe dissimilarity between 4 and 2 is 2.\nThe dissimilarity between 4 and 3 is 4.\nThe dissimilarity between 4 and 5 is 4.\nThe dissimilarity between 4 and 6 is 0.\nThe dissimilarity between 4 and 7 is 8.\nThe dissimilarity between 4 and 8 is 9.\nThe dissimilarity between 4 and 9 is 8.\nThe dissimilarity between 4 and 10 is 6.\nThe dissimilarity between 4 and 11 is 9.\nThe dissimilarity between 4 and 12 is 7.\nThe dissimilarity between 4 and 13 is 0.\nThe dissimilarity between 4 and 14 is 7.\nThe dissimilarity between 4 and 15 is 9.\nThe dissimilarity between 4 and 16 is 3.\nThe dissimilarity between 4 and 17 is 3.\nThe dissimilarity between 4 and 18 is 6.\nThe dissimilarity between 4 and 19 is 8.\nThe dissimilarity between 4 and 20 is 4.\nThe dissimilarity between 4 and 21 is 9.\nThe dissimilarity between 4 and 22 is 6.\nThe dissimilarity between 4 and 23 is 1.\nThe dissimilarity between 4 and 24 is 6.\nThe dissimilarity between 4 and 25 is 9.\nThe dissimilarity between 5 and 1 is 2.\nThe dissimilarity between 5 and 2 is 5.\nThe dissimilarity between 5 and 3 is 7.\nThe dissimilarity between 5 and 4 is 4.\nThe dissimilarity between 5 and 6 is 9.\nThe dissimilarity between 5 and 7 is 5.\nThe dissimilarity between 5 and 8 is 0.\nThe dissimilarity between 5 and 9 is 6.\nThe dissimilarity between 5 and 10 is 9.\nThe dissimilarity between 5 and 11 is 0.\nThe dissimilarity between 5 and 12 is 6.\nThe dissimilarity between 5 and 13 is 2.\nThe dissimilarity between 5 and 14 is 8.\nThe dissimilarity between 5 and 15 is 8.\nThe dissimilarity between 5 and 16 is 1.\nThe dissimilarity between 5 and 17 is 6.\nThe dissimilarity between 5 and 18 is 0.\nThe dissimilarity between 5 and 19 is 1.\nThe dissimilarity between 5 and 20 is 1.\nThe dissimilarity between 5 and 21 is 5.\nThe dissimilarity between 5 and 22 is 9.\nThe dissimilarity between 5 and 23 is 4.\nThe dissimilarity between 5 and 24 is 8.\nThe dissimilarity between 5 and 25 is 2.\nThe dissimilarity between 6 and 1 is 1.\nThe dissimilarity between 6 and 2 is 3.\nThe dissimilarity between 6 and 3 is 0.\nThe dissimilarity between 6 and 4 is 0.\nThe dissimilarity between 6 and 5 is 9.\nThe dissimilarity between 6 and 7 is 4.\nThe dissimilarity between 6 and 8 is 4.\nThe dissimilarity between 6 and 9 is 4.\nThe dissimilarity between 6 and 10 is 8.\nThe dissimilarity between 6 and 11 is 9.\nThe dissimilarity between 6 and 12 is 8.\nThe dissimilarity between 6 and 13 is 2.\nThe dissimilarity between 6 and 14 is 9.\nThe dissimilarity between 6 and 15 is 5.\nThe dissimilarity between 6 and 16 is 7.\nThe dissimilarity between 6 and 17 is 4.\nThe dissimilarity between 6 and 18 is 3.\nThe dissimilarity between 6 and 19 is 7.\nThe dissimilarity between 6 and 20 is 3.\nThe dissimilarity between 6 and 21 is 0.\nThe dissimilarity between 6 and 22 is 7.\nThe dissimilarity between 6 and 23 is 8.\nThe dissimilarity between 6 and 24 is 9.\nThe dissimilarity between 6 and 25 is 3.\nThe dissimilarity between 7 and 1 is 9.\nThe dissimilarity between 7 and 2 is 1.\nThe dissimilarity between 7 and 3 is 7.\nThe dissimilarity between 7 and 4 is 8.\nThe dissimilarity between 7 and 5 is 5.\nThe dissimilarity between 7 and 6 is 4.\nThe dissimilarity between 7 and 8 is 5.\nThe dissimilarity between 7 and 9 is 0.\nThe dissimilarity between 7 and 10 is 7.\nThe dissimilarity between 7 and 11 is 7.\nThe dissimilarity between 7 and 12 is 3.\nThe dissimilarity between 7 and 13 is 3.\nThe dissimilarity between 7 and 14 is 5.\nThe dissimilarity between 7 and 15 is 3.\nThe dissimilarity between 7 and 16 is 2.\nThe dissimilarity between 7 and 17 is 1.\nThe dissimilarity between 7 and 18 is 6.\nThe dissimilarity between 7 and 19 is 2.\nThe dissimilarity between 7 and 20 is 7.\nThe dissimilarity between 7 and 21 is 3.\nThe dissimilarity between 7 and 22 is 9.\nThe dissimilarity between 7 and 23 is 7.\nThe dissimilarity between 7 and 24 is 1.\nThe dissimilarity between 7 and 25 is 5.\nThe dissimilarity between 8 and 1 is 3.\nThe dissimilarity between 8 and 2 is 5.\nThe dissimilarity between 8 and 3 is 6.\nThe dissimilarity between 8 and 4 is 9.\nThe dissimilarity between 8 and 5 is 0.\nThe dissimilarity between 8 and 6 is 4.\nThe dissimilarity between 8 and 7 is 5.\nThe dissimilarity between 8 and 9 is 5.\nThe dissimilarity between 8 and 10 is 2.\nThe dissimilarity between 8 and 11 is 2.\nThe dissimilarity between 8 and 12 is 4.\nThe dissimilarity between 8 and 13 is 5.\nThe dissimilarity between 8 and 14 is 5.\nThe dissimilarity between 8 and 15 is 4.\nThe dissimilarity between 8 and 16 is 0.\nThe dissimilarity between 8 and 17 is 1.\nThe dissimilarity between 8 and 18 is 5.\nThe dissimilarity between 8 and 19 is 3.\nThe dissimilarity between 8 and 20 is 2.\nThe dissimilarity between 8 and 21 is 1.\nThe dissimilarity between 8 and 22 is 7.\nThe dissimilarity between 8 and 23 is 3.\nThe dissimilarity between 8 and 24 is 0.\nThe dissimilarity between 8 and 25 is 0.\nThe dissimilarity between 9 and 1 is 7.\nThe dissimilarity between 9 and 2 is 2.\nThe dissimilarity between 9 and 3 is 8.\nThe dissimilarity between 9 and 4 is 8.\nThe dissimilarity between 9 and 5 is 6.\nThe dissimilarity between 9 and 6 is 4.\nThe dissimilarity between 9 and 7 is 0.\nThe dissimilarity between 9 and 8 is 5.\nThe dissimilarity between 9 and 10 is 5.\nThe dissimilarity between 9 and 11 is 6.\nThe dissimilarity between 9 and 12 is 9.\nThe dissimilarity between 9 and 13 is 2.\nThe dissimilarity between 9 and 14 is 5.\nThe dissimilarity between 9 and 15 is 3.\nThe dissimilarity between 9 and 16 is 9.\nThe dissimilarity between 9 and 17 is 7.\nThe dissimilarity between 9 and 18 is 3.\nThe dissimilarity between 9 and 19 is 7.\nThe dissimilarity between 9 and 20 is 5.\nThe dissimilarity between 9 and 21 is 9.\nThe dissimilarity between 9 and 22 is 8.\nThe dissimilarity between 9 and 23 is 8.\nThe dissimilarity between 9 and 24 is 0.\nThe dissimilarity between 9 and 25 is 0.\nThe dissimilarity between 10 and 1 is 3.\nThe dissimilarity between 10 and 2 is 2.\nThe dissimilarity between 10 and 3 is 1.\nThe dissimilarity between 10 and 4 is 6.\nThe dissimilarity between 10 and 5 is 9.\nThe dissimilarity between 10 and 6 is 8.\nThe dissimilarity between 10 and 7 is 7.\nThe dissimilarity between 10 and 8 is 2.\nThe dissimilarity between 10 and 9 is 5.\nThe dissimilarity between 10 and 11 is 7.\nThe dissimilarity between 10 and 12 is 3.\nThe dissimilarity between 10 and 13 is 0.\nThe dissimilarity between 10 and 14 is 7.\nThe dissimilarity between 10 and 15 is 3.\nThe dissimilarity between 10 and 16 is 8.\nThe dissimilarity between 10 and 17 is 7.\nThe dissimilarity between 10 and 18 is 5.\nThe dissimilarity between 10 and 19 is 4.\nThe dissimilarity between 10 and 20 is 5.\nThe dissimilarity between 10 and 21 is 1.\nThe dissimilarity between 10 and 22 is 3.\nThe dissimilarity between 10 and 23 is 9.\nThe dissimilarity between 10 and 24 is 5.\nThe dissimilarity between 10 and 25 is 3.\nThe dissimilarity between 11 and 1 is 7.\nThe dissimilarity between 11 and 2 is 0.\nThe dissimilarity between 11 and 3 is 7.\nThe dissimilarity between 11 and 4 is 9.\nThe dissimilarity between 11 and 5 is 0.\nThe dissimilarity between 11 and 6 is 9.\nThe dissimilarity between 11 and 7 is 7.\nThe dissimilarity between 11 and 8 is 2.\nThe dissimilarity between 11 and 9 is 6.\nThe dissimilarity between 11 and 10 is 7.\nThe dissimilarity between 11 and 12 is 2.\nThe dissimilarity between 11 and 13 is 6.\nThe dissimilarity between 11 and 14 is 1.\nThe dissimilarity between 11 and 15 is 5.\nThe dissimilarity between 11 and 16 is 8.\nThe dissimilarity between 11 and 17 is 5.\nThe dissimilarity between 11 and 18 is 2.\nThe dissimilarity between 11 and 19 is 8.\nThe dissimilarity between 11 and 20 is 8.\nThe dissimilarity between 11 and 21 is 2.\nThe dissimilarity between 11 and 22 is 4.\nThe dissimilarity between 11 and 23 is 9.\nThe dissimilarity between 11 and 24 is 3.\nThe dissimilarity between 11 and 25 is 4.\nThe dissimilarity between 12 and 1 is 9.\nThe dissimilarity between 12 and 2 is 8.\nThe dissimilarity between 12 and 3 is 3.\nThe dissimilarity between 12 and 4 is 7.\nThe dissimilarity between 12 and 5 is 6.\nThe dissimilarity between 12 and 6 is 8.\nThe dissimilarity between 12 and 7 is 3.\nThe dissimilarity between 12 and 8 is 4.\nThe dissimilarity between 12 and 9 is 9.\nThe dissimilarity between 12 and 10 is 3.\nThe dissimilarity between 12 and 11 is 2.\nThe dissimilarity between 12 and 13 is 7.\nThe dissimilarity between 12 and 14 is 1.\nThe dissimilarity between 12 and 15 is 9.\nThe dissimilarity between 12 and 16 is 5.\nThe dissimilarity between 12 and 17 is 3.\nThe dissimilarity between 12 and 18 is 4.\nThe dissimilarity between 12 and 19 is 2.\nThe dissimilarity between 12 and 20 is 0.\nThe dissimilarity between 12 and 21 is 9.\nThe dissimilarity between 12 and 22 is 7.\nThe dissimilarity between 12 and 23 is 8.\nThe dissimilarity between 12 and 24 is 0.\nThe dissimilarity between 12 and 25 is 4.\nThe dissimilarity between 13 and 1 is 3.\nThe dissimilarity between 13 and 2 is 9.\nThe dissimilarity between 13 and 3 is 2.\nThe dissimilarity between 13 and 4 is 0.\nThe dissimilarity between 13 and 5 is 2.\nThe dissimilarity between 13 and 6 is 2.\nThe dissimilarity between 13 and 7 is 3.\nThe dissimilarity between 13 and 8 is 5.\nThe dissimilarity between 13 and 9 is 2.\nThe dissimilarity between 13 and 10 is 0.\nThe dissimilarity between 13 and 11 is 6.\nThe dissimilarity between 13 and 12 is 7.\nThe dissimilarity between 13 and 14 is 1.\nThe dissimilarity between 13 and 15 is 4.\nThe dissimilarity between 13 and 16 is 7.\nThe dissimilarity between 13 and 17 is 9.\nThe dissimilarity between 13 and 18 is 4.\nThe dissimilarity between 13 and 19 is 3.\nThe dissimilarity between 13 and 20 is 0.\nThe dissimilarity between 13 and 21 is 4.\nThe dissimilarity between 13 and 22 is 1.\nThe dissimilarity between 13 and 23 is 1.\nThe dissimilarity between 13 and 24 is 8.\nThe dissimilarity between 13 and 25 is 0.\nThe dissimilarity between 14 and 1 is 0.\nThe dissimilarity between 14 and 2 is 0.\nThe dissimilarity between 14 and 3 is 2.\nThe dissimilarity between 14 and 4 is 7.\nThe dissimilarity between 14 and 5 is 8.\nThe dissimilarity between 14 and 6 is 9.\nThe dissimilarity between 14 and 7 is 5.\nThe dissimilarity between 14 and 8 is 5.\nThe dissimilarity between 14 and 9 is 5.\nThe dissimilarity between 14 and 10 is 7.\nThe dissimilarity between 14 and 11 is 1.\nThe dissimilarity between 14 and 12 is 1.\nThe dissimilarity between 14 and 13 is 1.\nThe dissimilarity between 14 and 15 is 4.\nThe dissimilarity between 14 and 16 is 9.\nThe dissimilarity between 14 and 17 is 7.\nThe dissimilarity between 14 and 18 is 6.\nThe dissimilarity between 14 and 19 is 6.\nThe dissimilarity between 14 and 20 is 0.\nThe dissimilarity between 14 and 21 is 4.\nThe dissimilarity between 14 and 22 is 1.\nThe dissimilarity between 14 and 23 is 2.\nThe dissimilarity between 14 and 24 is 0.\nThe dissimilarity between 14 and 25 is 8.\nThe dissimilarity between 15 and 1 is 4.\nThe dissimilarity between 15 and 2 is 0.\nThe dissimilarity between 15 and 3 is 7.\nThe dissimilarity between 15 and 4 is 9.\nThe dissimilarity between 15 and 5 is 8.\nThe dissimilarity between 15 and 6 is 5.\nThe dissimilarity between 15 and 7 is 3.\nThe dissimilarity between 15 and 8 is 4.\nThe dissimilarity between 15 and 9 is 3.\nThe dissimilarity between 15 and 10 is 3.\nThe dissimilarity between 15 and 11 is 5.\nThe dissimilarity between 15 and 12 is 9.\nThe dissimilarity between 15 and 13 is 4.\nThe dissimilarity between 15 and 14 is 4.\nThe dissimilarity between 15 and 16 is 1.\nThe dissimilarity between 15 and 17 is 8.\nThe dissimilarity between 15 and 18 is 1.\nThe dissimilarity between 15 and 19 is 2.\nThe dissimilarity between 15 and 20 is 3.\nThe dissimilarity between 15 and 21 is 5.\nThe dissimilarity between 15 and 22 is 8.\nThe dissimilarity between 15 and 23 is 7.\nThe dissimilarity between 15 and 24 is 5.\nThe dissimilarity between 15 and 25 is 5.\nThe dissimilarity between 16 and 1 is 3.\nThe dissimilarity between 16 and 2 is 7.\nThe dissimilarity between 16 and 3 is 1.\nThe dissimilarity between 16 and 4 is 3.\nThe dissimilarity between 16 and 5 is 1.\nThe dissimilarity between 16 and 6 is 7.\nThe dissimilarity between 16 and 7 is 2.\nThe dissimilarity between 16 and 8 is 0.\nThe dissimilarity between 16 and 9 is 9.\nThe dissimilarity between 16 and 10 is 8.\nThe dissimilarity between 16 and 11 is 8.\nThe dissimilarity between 16 and 12 is 5.\nThe dissimilarity between 16 and 13 is 7.\nThe dissimilarity between 16 and 14 is 9.\nThe dissimilarity between 16 and 15 is 1.\nThe dissimilarity between 16 and 17 is 9.\nThe dissimilarity between 16 and 18 is 3.\nThe dissimilarity between 16 and 19 is 8.\nThe dissimilarity between 16 and 20 is 9.\nThe dissimilarity between 16 and 21 is 0.\nThe dissimilarity between 16 and 22 is 1.\nThe dissimilarity between 16 and 23 is 1.\nThe dissimilarity between 16 and 24 is 1.\nThe dissimilarity between 16 and 25 is 9.\nThe dissimilarity between 17 and 1 is 7.\nThe dissimilarity between 17 and 2 is 3.\nThe dissimilarity between 17 and 3 is 6.\nThe dissimilarity between 17 and 4 is 3.\nThe dissimilarity between 17 and 5 is 6.\nThe dissimilarity between 17 and 6 is 4.\nThe dissimilarity between 17 and 7 is 1.\nThe dissimilarity between 17 and 8 is 1.\nThe dissimilarity between 17 and 9 is 7.\nThe dissimilarity between 17 and 10 is 7.\nThe dissimilarity between 17 and 11 is 5.\nThe dissimilarity between 17 and 12 is 3.\nThe dissimilarity between 17 and 13 is 9.\nThe dissimilarity between 17 and 14 is 7.\nThe dissimilarity between 17 and 15 is 8.\nThe dissimilarity between 17 and 16 is 9.\nThe dissimilarity between 17 and 18 is 0.\nThe dissimilarity between 17 and 19 is 4.\nThe dissimilarity between 17 and 20 is 4.\nThe dissimilarity between 17 and 21 is 6.\nThe dissimilarity between 17 and 22 is 9.\nThe dissimilarity between 17 and 23 is 2.\nThe dissimilarity between 17 and 24 is 3.\nThe dissimilarity between 17 and 25 is 3.\nThe dissimilarity between 18 and 1 is 1.\nThe dissimilarity between 18 and 2 is 7.\nThe dissimilarity between 18 and 3 is 8.\nThe dissimilarity between 18 and 4 is 6.\nThe dissimilarity between 18 and 5 is 0.\nThe dissimilarity between 18 and 6 is 3.\nThe dissimilarity between 18 and 7 is 6.\nThe dissimilarity between 18 and 8 is 5.\nThe dissimilarity between 18 and 9 is 3.\nThe dissimilarity between 18 and 10 is 5.\nThe dissimilarity between 18 and 11 is 2.\nThe dissimilarity between 18 and 12 is 4.\nThe dissimilarity between 18 and 13 is 4.\nThe dissimilarity between 18 and 14 is 6.\nThe dissimilarity between 18 and 15 is 1.\nThe dissimilarity between 18 and 16 is 3.\nThe dissimilarity between 18 and 17 is 0.\nThe dissimilarity between 18 and 19 is 4.\nThe dissimilarity between 18 and 20 is 3.\nThe dissimilarity between 18 and 21 is 3.\nThe dissimilarity between 18 and 22 is 6.\nThe dissimilarity between 18 and 23 is 9.\nThe dissimilarity between 18 and 24 is 5.\nThe dissimilarity between 18 and 25 is 1.\nThe dissimilarity between 19 and 1 is 2.\nThe dissimilarity between 19 and 2 is 9.\nThe dissimilarity between 19 and 3 is 1.\nThe dissimilarity between 19 and 4 is 8.\nThe dissimilarity between 19 and 5 is 1.\nThe dissimilarity between 19 and 6 is 7.\nThe dissimilarity between 19 and 7 is 2.\nThe dissimilarity between 19 and 8 is 3.\nThe dissimilarity between 19 and 9 is 7.\nThe dissimilarity between 19 and 10 is 4.\nThe dissimilarity between 19 and 11 is 8.\nThe dissimilarity between 19 and 12 is 2.\nThe dissimilarity between 19 and 13 is 3.\nThe dissimilarity between 19 and 14 is 6.\nThe dissimilarity between 19 and 15 is 2.\nThe dissimilarity between 19 and 16 is 8.\nThe dissimilarity between 19 and 17 is 4.\nThe dissimilarity between 19 and 18 is 4.\nThe dissimilarity between 19 and 20 is 7.\nThe dissimilarity between 19 and 21 is 0.\nThe dissimilarity between 19 and 22 is 9.\nThe dissimilarity between 19 and 23 is 9.\nThe dissimilarity between 19 and 24 is 3.\nThe dissimilarity between 19 and 25 is 0.\nThe dissimilarity between 20 and 1 is 3.\nThe dissimilarity between 20 and 2 is 6.\nThe dissimilarity between 20 and 3 is 8.\nThe dissimilarity between 20 and 4 is 4.\nThe dissimilarity between 20 and 5 is 1.\nThe dissimilarity between 20 and 6 is 3.\nThe dissimilarity between 20 and 7 is 7.\nThe dissimilarity between 20 and 8 is 2.\nThe dissimilarity between 20 and 9 is 5.\nThe dissimilarity between 20 and 10 is 5.\nThe dissimilarity between 20 and 11 is 8.\nThe dissimilarity between 20 and 12 is 0.\nThe dissimilarity between 20 and 13 is 0.\nThe dissimilarity between 20 and 14 is 0.\nThe dissimilarity between 20 and 15 is 3.\nThe dissimilarity between 20 and 16 is 9.\nThe dissimilarity between 20 and 17 is 4.\nThe dissimilarity between 20 and 18 is 3.\nThe dissimilarity between 20 and 19 is 7.\nThe dissimilarity between 20 and 21 is 7.\nThe dissimilarity between 20 and 22 is 1.\nThe dissimilarity between 20 and 23 is 9.\nThe dissimilarity between 20 and 24 is 9.\nThe dissimilarity between 20 and 25 is 4.\nThe dissimilarity between 21 and 1 is 4.\nThe dissimilarity between 21 and 2 is 2.\nThe dissimilarity between 21 and 3 is 1.\nThe dissimilarity between 21 and 4 is 9.\nThe dissimilarity between 21 and 5 is 5.\nThe dissimilarity between 21 and 6 is 0.\nThe dissimilarity between 21 and 7 is 3.\nThe dissimilarity between 21 and 8 is 1.\nThe dissimilarity between 21 and 9 is 9.\nThe dissimilarity between 21 and 10 is 1.\nThe dissimilarity between 21 and 11 is 2.\nThe dissimilarity between 21 and 12 is 9.\nThe dissimilarity between 21 and 13 is 4.\nThe dissimilarity between 21 and 14 is 4.\nThe dissimilarity between 21 and 15 is 5.\nThe dissimilarity between 21 and 16 is 0.\nThe dissimilarity between 21 and 17 is 6.\nThe dissimilarity between 21 and 18 is 3.\nThe dissimilarity between 21 and 19 is 0.\nThe dissimilarity between 21 and 20 is 7.\nThe dissimilarity between 21 and 22 is 6.\nThe dissimilarity between 21 and 23 is 9.\nThe dissimilarity between 21 and 24 is 4.\nThe dissimilarity between 21 and 25 is 0.\nThe dissimilarity between 22 and 1 is 8.\nThe dissimilarity between 22 and 2 is 9.\nThe dissimilarity between 22 and 3 is 1.\nThe dissimilarity between 22 and 4 is 6.\nThe dissimilarity between 22 and 5 is 9.\nThe dissimilarity between 22 and 6 is 7.\nThe dissimilarity between 22 and 7 is 9.\nThe dissimilarity between 22 and 8 is 7.\nThe dissimilarity between 22 and 9 is 8.\nThe dissimilarity between 22 and 10 is 3.\nThe dissimilarity between 22 and 11 is 4.\nThe dissimilarity between 22 and 12 is 7.\nThe dissimilarity between 22 and 13 is 1.\nThe dissimilarity between 22 and 14 is 1.\nThe dissimilarity between 22 and 15 is 8.\nThe dissimilarity between 22 and 16 is 1.\nThe dissimilarity between 22 and 17 is 9.\nThe dissimilarity between 22 and 18 is 6.\nThe dissimilarity between 22 and 19 is 9.\nThe dissimilarity between 22 and 20 is 1.\nThe dissimilarity between 22 and 21 is 6.\nThe dissimilarity between 22 and 23 is 6.\nThe dissimilarity between 22 and 24 is 9.\nThe dissimilarity between 22 and 25 is 4.\nThe dissimilarity between 23 and 1 is 3.\nThe dissimilarity between 23 and 2 is 6.\nThe dissimilarity between 23 and 3 is 6.\nThe dissimilarity between 23 and 4 is 1.\nThe dissimilarity between 23 and 5 is 4.\nThe dissimilarity between 23 and 6 is 8.\nThe dissimilarity between 23 and 7 is 7.\nThe dissimilarity between 23 and 8 is 3.\nThe dissimilarity between 23 and 9 is 8.\nThe dissimilarity between 23 and 10 is 9.\nThe dissimilarity between 23 and 11 is 9.\nThe dissimilarity between 23 and 12 is 8.\nThe dissimilarity between 23 and 13 is 1.\nThe dissimilarity between 23 and 14 is 2.\nThe dissimilarity between 23 and 15 is 7.\nThe dissimilarity between 23 and 16 is 1.\nThe dissimilarity between 23 and 17 is 2.\nThe dissimilarity between 23 and 18 is 9.\nThe dissimilarity between 23 and 19 is 9.\nThe dissimilarity between 23 and 20 is 9.\nThe dissimilarity between 23 and 21 is 9.\nThe dissimilarity between 23 and 22 is 6.\nThe dissimilarity between 23 and 24 is 2.\nThe dissimilarity between 23 and 25 is 5.\nThe dissimilarity between 24 and 1 is 3.\nThe dissimilarity between 24 and 2 is 3.\nThe dissimilarity between 24 and 3 is 9.\nThe dissimilarity between 24 and 4 is 6.\nThe dissimilarity between 24 and 5 is 8.\nThe dissimilarity between 24 and 6 is 9.\nThe dissimilarity between 24 and 7 is 1.\nThe dissimilarity between 24 and 8 is 0.\nThe dissimilarity between 24 and 9 is 0.\nThe dissimilarity between 24 and 10 is 5.\nThe dissimilarity between 24 and 11 is 3.\nThe dissimilarity between 24 and 12 is 0.\nThe dissimilarity between 24 and 13 is 8.\nThe dissimilarity between 24 and 14 is 0.\nThe dissimilarity between 24 and 15 is 5.\nThe dissimilarity between 24 and 16 is 1.\nThe dissimilarity between 24 and 17 is 3.\nThe dissimilarity between 24 and 18 is 5.\nThe dissimilarity between 24 and 19 is 3.\nThe dissimilarity between 24 and 20 is 9.\nThe dissimilarity between 24 and 21 is 4.\nThe dissimilarity between 24 and 22 is 9.\nThe dissimilarity between 24 and 23 is 2.\nThe dissimilarity between 24 and 25 is 0.\nThe dissimilarity between 25 and 1 is 0.\nThe dissimilarity between 25 and 2 is 9.\nThe dissimilarity between 25 and 3 is 7.\nThe dissimilarity between 25 and 4 is 9.\nThe dissimilarity between 25 and 5 is 2.\nThe dissimilarity between 25 and 6 is 3.\nThe dissimilarity between 25 and 7 is 5.\nThe dissimilarity between 25 and 8 is 0.\nThe dissimilarity between 25 and 9 is 0.\nThe dissimilarity between 25 and 10 is 3.\nThe dissimilarity between 25 and 11 is 4.\nThe dissimilarity between 25 and 12 is 4.\nThe dissimilarity between 25 and 13 is 0.\nThe dissimilarity between 25 and 14 is 8.\nThe dissimilarity between 25 and 15 is 5.\nThe dissimilarity between 25 and 16 is 9.\nThe dissimilarity between 25 and 17 is 3.\nThe dissimilarity between 25 and 18 is 1.\nThe dissimilarity between 25 and 19 is 0.\nThe dissimilarity between 25 and 20 is 4.\nThe dissimilarity between 25 and 21 is 0.\nThe dissimilarity between 25 and 22 is 4.\nThe dissimilarity between 25 and 23 is 5.\nThe dissimilarity between 25 and 24 is 0.\nThese entries let the editor total pairwise dissimilarities to judge any proposed set of 2 destinations.\n\nIf it helps, just send the final picks back in a tiny JSON snippet like this:\n\n{\n \"solution\": [, , ...]\n}\n\nThis little sketch just means \"solution\" should be an array containing exactly m chosen places — each item is one identifier for a destination from the instance. Think of it as a simple form: the key is the field name and the array lists the chosen spots. This is just the shape I expect, not the real answer itself.\n\nPlease be sure to use the identifiers exactly as they show up in the instance input — no renaming and no 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”.\"", "instance": { "distance_matrix": [ [ 0, 4, 2, 8, 2, 1, 9, 3, 7, 3, 7, 9, 3, 0, 4, 3, 7, 1, 2, 3, 4, 8, 3, 3, 0 ], [ 4, 0, 8, 2, 5, 3, 1, 5, 2, 2, 0, 8, 9, 0, 0, 7, 3, 7, 9, 6, 2, 9, 6, 3, 9 ], [ 2, 8, 0, 4, 7, 0, 7, 6, 8, 1, 7, 3, 2, 2, 7, 1, 6, 8, 1, 8, 1, 1, 6, 9, 7 ], [ 8, 2, 4, 0, 4, 0, 8, 9, 8, 6, 9, 7, 0, 7, 9, 3, 3, 6, 8, 4, 9, 6, 1, 6, 9 ], [ 2, 5, 7, 4, 0, 9, 5, 0, 6, 9, 0, 6, 2, 8, 8, 1, 6, 0, 1, 1, 5, 9, 4, 8, 2 ], [ 1, 3, 0, 0, 9, 0, 4, 4, 4, 8, 9, 8, 2, 9, 5, 7, 4, 3, 7, 3, 0, 7, 8, 9, 3 ], [ 9, 1, 7, 8, 5, 4, 0, 5, 0, 7, 7, 3, 3, 5, 3, 2, 1, 6, 2, 7, 3, 9, 7, 1, 5 ], [ 3, 5, 6, 9, 0, 4, 5, 0, 5, 2, 2, 4, 5, 5, 4, 0, 1, 5, 3, 2, 1, 7, 3, 0, 0 ], [ 7, 2, 8, 8, 6, 4, 0, 5, 0, 5, 6, 9, 2, 5, 3, 9, 7, 3, 7, 5, 9, 8, 8, 0, 0 ], [ 3, 2, 1, 6, 9, 8, 7, 2, 5, 0, 7, 3, 0, 7, 3, 8, 7, 5, 4, 5, 1, 3, 9, 5, 3 ], [ 7, 0, 7, 9, 0, 9, 7, 2, 6, 7, 0, 2, 6, 1, 5, 8, 5, 2, 8, 8, 2, 4, 9, 3, 4 ], [ 9, 8, 3, 7, 6, 8, 3, 4, 9, 3, 2, 0, 7, 1, 9, 5, 3, 4, 2, 0, 9, 7, 8, 0, 4 ], [ 3, 9, 2, 0, 2, 2, 3, 5, 2, 0, 6, 7, 0, 1, 4, 7, 9, 4, 3, 0, 4, 1, 1, 8, 0 ], [ 0, 0, 2, 7, 8, 9, 5, 5, 5, 7, 1, 1, 1, 0, 4, 9, 7, 6, 6, 0, 4, 1, 2, 0, 8 ], [ 4, 0, 7, 9, 8, 5, 3, 4, 3, 3, 5, 9, 4, 4, 0, 1, 8, 1, 2, 3, 5, 8, 7, 5, 5 ], [ 3, 7, 1, 3, 1, 7, 2, 0, 9, 8, 8, 5, 7, 9, 1, 0, 9, 3, 8, 9, 0, 1, 1, 1, 9 ], [ 7, 3, 6, 3, 6, 4, 1, 1, 7, 7, 5, 3, 9, 7, 8, 9, 0, 0, 4, 4, 6, 9, 2, 3, 3 ], [ 1, 7, 8, 6, 0, 3, 6, 5, 3, 5, 2, 4, 4, 6, 1, 3, 0, 0, 4, 3, 3, 6, 9, 5, 1 ], [ 2, 9, 1, 8, 1, 7, 2, 3, 7, 4, 8, 2, 3, 6, 2, 8, 4, 4, 0, 7, 0, 9, 9, 3, 0 ], [ 3, 6, 8, 4, 1, 3, 7, 2, 5, 5, 8, 0, 0, 0, 3, 9, 4, 3, 7, 0, 7, 1, 9, 9, 4 ], [ 4, 2, 1, 9, 5, 0, 3, 1, 9, 1, 2, 9, 4, 4, 5, 0, 6, 3, 0, 7, 0, 6, 9, 4, 0 ], [ 8, 9, 1, 6, 9, 7, 9, 7, 8, 3, 4, 7, 1, 1, 8, 1, 9, 6, 9, 1, 6, 0, 6, 9, 4 ], [ 3, 6, 6, 1, 4, 8, 7, 3, 8, 9, 9, 8, 1, 2, 7, 1, 2, 9, 9, 9, 9, 6, 0, 2, 5 ], [ 3, 3, 9, 6, 8, 9, 1, 0, 0, 5, 3, 0, 8, 0, 5, 1, 3, 5, 3, 9, 4, 9, 2, 0, 0 ], [ 0, 9, 7, 9, 2, 3, 5, 0, 0, 3, 4, 4, 0, 8, 5, 9, 3, 1, 0, 4, 0, 4, 5, 0, 0 ] ], "m": 2, "objective": 9.0 }, "solution": [ 3, 7 ], "obj": 9.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 25, "m": 2, "nodes": [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 ], "distance_pairs": [ { "from_id": 1, "to_id": 2, "distance": 4 }, { "from_id": 1, "to_id": 3, "distance": 2 }, { "from_id": 1, "to_id": 4, "distance": 8 }, { "from_id": 1, "to_id": 5, "distance": 2 }, { "from_id": 1, "to_id": 6, "distance": 1 }, { "from_id": 1, "to_id": 7, "distance": 9 }, { "from_id": 1, "to_id": 8, "distance": 3 }, { "from_id": 1, "to_id": 9, "distance": 7 }, { "from_id": 1, "to_id": 10, "distance": 3 }, { "from_id": 1, "to_id": 11, "distance": 7 }, { "from_id": 1, "to_id": 12, "distance": 9 }, { "from_id": 1, "to_id": 13, "distance": 3 }, { 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There’s a distance value for each pair of species that captures ecological and appearance differences, and the border’s “variety score” comes from adding up those pairwise distances for whatever species are chosen — higher totals mean a more diverse grouping. The plan calls for exactly m distinct species, no duplicates and no empty spots. The specific species and numbers follow below.\n\n# num_species=24\n# species_to_select=19\n# species_ids=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23\nspecies_a_id,species_b_id,pairwise_distance\n0,1,66\n0,2,28\n0,3,84\n0,4,44\n0,5,30\n0,6,46\n0,7,4\n0,8,54\n0,9,29\n0,10,36\n0,11,21\n0,12,68\n0,13,75\n0,14,50\n0,15,18\n0,16,53\n0,17,44\n0,18,81\n0,19,60\n0,20,9\n0,21,72\n0,22,36\n0,23,36\n1,0,66\n1,2,86\n1,3,41\n1,4,26\n1,5,96\n1,6,33\n1,7,63\n1,8,56\n1,9,72\n1,10,99\n1,11,62\n1,12,7\n1,13,47\n1,14,30\n1,15,79\n1,16,46\n1,17,58\n1,18,95\n1,19,14\n1,20,72\n1,21,42\n1,22,39\n1,23,36\n2,0,28\n2,1,86\n2,3,92\n2,4,61\n2,5,21\n2,6,73\n2,7,33\n2,8,83\n2,9,18\n2,10,14\n2,11,49\n2,12,86\n2,13,79\n2,14,62\n2,15,36\n2,16,57\n2,17,73\n2,18,64\n2,19,84\n2,20,19\n2,21,78\n2,22,63\n2,23,50\n3,0,84\n3,1,41\n3,2,92\n3,4,41\n3,5,109\n3,6,70\n3,7,83\n3,8,94\n3,9,74\n3,10,107\n3,11,90\n3,12,33\n3,13,15\n3,14,33\n3,15,101\n3,16,35\n3,17,94\n3,18,71\n3,19,55\n3,20,86\n3,21,14\n3,22,71\n3,23,48\n4,0,44\n4,1,26\n4,2,61\n4,3,41\n4,5,73\n4,6,33\n4,7,43\n4,8,57\n4,9,46\n4,10,74\n4,11,48\n4,12,25\n4,13,38\n4,14,10\n4,15,60\n4,16,25\n4,17,54\n4,18,73\n4,19,30\n4,20,48\n4,21,34\n4,22,31\n4,23,10\n5,0,30\n5,1,96\n5,2,21\n5,3,109\n5,4,73\n5,6,75\n5,7,33\n5,8,79\n5,9,38\n5,10,13\n5,11,44\n5,12,97\n5,13,98\n5,14,77\n5,15,25\n5,16,75\n5,17,67\n5,18,86\n5,19,91\n5,20,24\n5,21,96\n5,22,65\n5,23,63\n6,0,46\n6,1,33\n6,2,73\n6,3,70\n6,4,33\n6,5,75\n6,7,42\n6,8,24\n6,9,65\n6,10,82\n6,11,33\n6,12,39\n6,13,70\n6,14,44\n6,15,53\n6,16,58\n6,17,25\n6,18,105\n6,19,20\n6,20,54\n6,21,66\n6,22,10\n6,23,37\n7,0,4\n7,1,63\n7,2,33\n7,3,83\n7,4,43\n7,5,33\n7,6,42\n7,8,50\n7,9,33\n7,10,40\n7,11,17\n7,12,66\n7,13,75\n7,14,49\n7,15,17\n7,16,54\n7,17,39\n7,18,84\n7,19,57\n7,20,13\n7,21,72\n7,22,32\n7,23,35\n8,0,54\n8,1,56\n8,2,83\n8,3,94\n8,4,57\n8,5,79\n8,6,24\n8,7,50\n8,9,80\n8,10,89\n8,11,35\n8,12,63\n8,13,95\n8,14,67\n8,15,54\n8,16,81\n8,17,11\n8,18,125\n8,19,41\n8,20,64\n8,21,90\n8,22,26\n8,23,59\n9,0,29\n9,1,72\n9,2,18\n9,3,74\n9,4,46\n9,5,38\n9,6,65\n9,7,33\n9,8,80\n9,10,33\n9,11,50\n9,12,71\n9,13,61\n9,14,45\n9,15,44\n9,16,39\n9,17,71\n9,18,52\n9,19,73\n9,20,22\n9,21,60\n9,22,56\n9,23,35\n10,0,36\n10,1,99\n10,2,14\n10,3,107\n10,4,74\n10,5,13\n10,6,82\n10,7,40\n10,8,89\n10,9,33\n10,11,54\n10,12,99\n10,13,94\n10,14,76\n10,15,37\n10,16,71\n10,17,78\n10,18,75\n10,19,96\n10,20,28\n10,21,93\n10,22,72\n10,23,64\n11,0,21\n11,1,62\n11,2,49\n11,3,90\n11,4,48\n11,5,44\n11,6,33\n11,7,17\n11,8,35\n11,9,50\n11,10,54\n11,12,66\n11,13,84\n11,14,57\n11,15,20\n11,16,65\n11,17,23\n11,18,100\n11,19,53\n11,20,30\n11,21,81\n11,22,24\n11,23,43\n12,0,68\n12,1,7\n12,2,86\n12,3,33\n12,4,25\n12,5,97\n12,6,39\n12,7,66\n12,8,63\n12,9,71\n12,10,99\n12,11,66\n12,13,40\n12,14,26\n12,15,82\n12,16,41\n12,17,64\n12,18,90\n12,19,22\n12,20,73\n12,21,35\n12,22,44\n12,23,35\n13,0,75\n13,1,47\n13,2,79\n13,3,15\n13,4,38\n13,5,98\n13,6,70\n13,7,75\n13,8,95\n13,9,61\n13,10,94\n13,11,84\n13,12,40\n13,14,28\n13,15,93\n13,16,22\n13,17,92\n13,18,56\n13,19,59\n13,20,75\n13,21,4\n13,22,69\n13,23,41\n14,0,50\n14,1,30\n14,2,62\n14,3,33\n14,4,10\n14,5,77\n14,6,44\n14,7,49\n14,8,67\n14,9,45\n14,10,76\n14,11,57\n14,12,26\n14,13,28\n14,15,67\n14,16,16\n14,17,64\n14,18,65\n14,19,37\n14,20,53\n14,21,24\n14,22,42\n14,23,14\n15,0,18\n15,1,79\n15,2,36\n15,3,101\n15,4,60\n15,5,25\n15,6,53\n15,7,17\n15,8,54\n15,9,44\n15,10,37\n15,11,20\n15,12,82\n15,13,93\n15,14,67\n15,16,71\n15,17,42\n15,18,96\n15,19,71\n15,20,21\n15,21,90\n15,22,43\n15,23,53\n16,0,53\n16,1,46\n16,2,57\n16,3,35\n16,4,25\n16,5,75\n16,6,58\n16,7,54\n16,8,81\n16,9,39\n16,10,71\n16,11,65\n16,12,41\n16,13,22\n16,14,16\n16,15,71\n16,17,77\n16,18,49\n16,19,54\n16,20,53\n16,21,21\n16,22,54\n16,23,22\n17,0,44\n17,1,58\n17,2,73\n17,3,94\n17,4,54\n17,5,67\n17,6,25\n17,7,39\n17,8,11\n17,9,71\n17,10,78\n17,11,23\n17,12,64\n17,13,92\n17,14,64\n17,15,42\n17,16,77\n17,18,118\n17,19,45\n17,20,53\n17,21,88\n17,22,22\n17,23,54\n18,0,81\n18,1,95\n18,2,64\n18,3,71\n18,4,73\n18,5,86\n18,6,105\n18,7,84\n18,8,125\n18,9,52\n18,10,75\n18,11,100\n18,12,90\n18,13,56\n18,14,65\n18,15,96\n18,16,49\n18,17,118\n18,19,103\n18,20,75\n18,21,58\n18,22,99\n18,23,67\n19,0,60\n19,1,14\n19,2,84\n19,3,55\n19,4,30\n19,5,91\n19,6,20\n19,7,57\n19,8,41\n19,9,73\n19,10,96\n19,11,53\n19,12,22\n19,13,59\n19,14,37\n19,15,71\n19,16,54\n19,17,45\n19,18,103\n19,20,68\n19,21,55\n19,22,28\n19,23,38\n20,0,9\n20,1,72\n20,2,19\n20,3,86\n20,4,48\n20,5,24\n20,6,54\n20,7,13\n20,8,64\n20,9,22\n20,10,28\n20,11,30\n20,12,73\n20,13,75\n20,14,53\n20,15,21\n20,16,53\n20,17,53\n20,18,75\n20,19,68\n20,21,73\n20,22,44\n20,23,39\n21,0,72\n21,1,42\n21,2,78\n21,3,14\n21,4,34\n21,5,96\n21,6,66\n21,7,72\n21,8,90\n21,9,60\n21,10,93\n21,11,81\n21,12,35\n21,13,4\n21,14,24\n21,15,90\n21,16,21\n21,17,88\n21,18,58\n21,19,55\n21,20,73\n21,22,66\n21,23,37\n22,0,36\n22,1,39\n22,2,63\n22,3,71\n22,4,31\n22,5,65\n22,6,10\n22,7,32\n22,8,26\n22,9,56\n22,10,72\n22,11,24\n22,12,44\n22,13,69\n22,14,42\n22,15,43\n22,16,54\n22,17,22\n22,18,99\n22,19,28\n22,20,44\n22,21,66\n22,23,32\n23,0,36\n23,1,36\n23,2,50\n23,3,48\n23,4,10\n23,5,63\n23,6,37\n23,7,35\n23,8,59\n23,9,35\n23,10,64\n23,11,43\n23,12,35\n23,13,41\n23,14,14\n23,15,53\n23,16,22\n23,17,54\n23,18,67\n23,19,38\n23,20,39\n23,21,37\n23,22,32\n\nOh, and when you send back the chosen lineup, please use this simple JSON shape so it's easy to read and parse:\n\n{\n \"solution\": [, , ...]\n}\n\nThink of \"solution\" as just the list of species you picked for the border — put their exact identifiers in that array, nothing fancy. This JSON is only a sketch of the shape I expect, not the actual selection.\n\nPlease use the identifiers exactly as they appear in the instance input — no renaming and no new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 66, 28, 84, 44, 30, 46, 4, 54, 29, 36, 21, 68, 75, 50, 18, 53, 44, 81, 60, 9, 72, 36, 36 ], [ 66, 0, 86, 41, 26, 96, 33, 63, 56, 72, 99, 62, 7, 47, 30, 79, 46, 58, 95, 14, 72, 42, 39, 36 ], [ 28, 86, 0, 92, 61, 21, 73, 33, 83, 18, 14, 49, 86, 79, 62, 36, 57, 73, 64, 84, 19, 78, 63, 50 ], [ 84, 41, 92, 0, 41, 109, 70, 83, 94, 74, 107, 90, 33, 15, 33, 101, 35, 94, 71, 55, 86, 14, 71, 48 ], [ 44, 26, 61, 41, 0, 73, 33, 43, 57, 46, 74, 48, 25, 38, 10, 60, 25, 54, 73, 30, 48, 34, 31, 10 ], [ 30, 96, 21, 109, 73, 0, 75, 33, 79, 38, 13, 44, 97, 98, 77, 25, 75, 67, 86, 91, 24, 96, 65, 63 ], [ 46, 33, 73, 70, 33, 75, 0, 42, 24, 65, 82, 33, 39, 70, 44, 53, 58, 25, 105, 20, 54, 66, 10, 37 ], [ 4, 63, 33, 83, 43, 33, 42, 0, 50, 33, 40, 17, 66, 75, 49, 17, 54, 39, 84, 57, 13, 72, 32, 35 ], [ 54, 56, 83, 94, 57, 79, 24, 50, 0, 80, 89, 35, 63, 95, 67, 54, 81, 11, 125, 41, 64, 90, 26, 59 ], [ 29, 72, 18, 74, 46, 38, 65, 33, 80, 0, 33, 50, 71, 61, 45, 44, 39, 71, 52, 73, 22, 60, 56, 35 ], [ 36, 99, 14, 107, 74, 13, 82, 40, 89, 33, 0, 54, 99, 94, 76, 37, 71, 78, 75, 96, 28, 93, 72, 64 ], [ 21, 62, 49, 90, 48, 44, 33, 17, 35, 50, 54, 0, 66, 84, 57, 20, 65, 23, 100, 53, 30, 81, 24, 43 ], [ 68, 7, 86, 33, 25, 97, 39, 66, 63, 71, 99, 66, 0, 40, 26, 82, 41, 64, 90, 22, 73, 35, 44, 35 ], [ 75, 47, 79, 15, 38, 98, 70, 75, 95, 61, 94, 84, 40, 0, 28, 93, 22, 92, 56, 59, 75, 4, 69, 41 ], [ 50, 30, 62, 33, 10, 77, 44, 49, 67, 45, 76, 57, 26, 28, 0, 67, 16, 64, 65, 37, 53, 24, 42, 14 ], [ 18, 79, 36, 101, 60, 25, 53, 17, 54, 44, 37, 20, 82, 93, 67, 0, 71, 42, 96, 71, 21, 90, 43, 53 ], [ 53, 46, 57, 35, 25, 75, 58, 54, 81, 39, 71, 65, 41, 22, 16, 71, 0, 77, 49, 54, 53, 21, 54, 22 ], [ 44, 58, 73, 94, 54, 67, 25, 39, 11, 71, 78, 23, 64, 92, 64, 42, 77, 0, 118, 45, 53, 88, 22, 54 ], [ 81, 95, 64, 71, 73, 86, 105, 84, 125, 52, 75, 100, 90, 56, 65, 96, 49, 118, 0, 103, 75, 58, 99, 67 ], [ 60, 14, 84, 55, 30, 91, 20, 57, 41, 73, 96, 53, 22, 59, 37, 71, 54, 45, 103, 0, 68, 55, 28, 38 ], [ 9, 72, 19, 86, 48, 24, 54, 13, 64, 22, 28, 30, 73, 75, 53, 21, 53, 53, 75, 68, 0, 73, 44, 39 ], [ 72, 42, 78, 14, 34, 96, 66, 72, 90, 60, 93, 81, 35, 4, 24, 90, 21, 88, 58, 55, 73, 0, 66, 37 ], [ 36, 39, 63, 71, 31, 65, 10, 32, 26, 56, 72, 24, 44, 69, 42, 43, 54, 22, 99, 28, 44, 66, 0, 32 ], [ 36, 36, 50, 48, 10, 63, 37, 35, 59, 35, 64, 43, 35, 41, 14, 53, 22, 54, 67, 38, 39, 37, 32, 0 ] ], "m": 19, "objective": 10356.0 }, "solution": [ 1, 2, 3, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22 ], "obj": 10356.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 24, "m": 19, "nodes": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 ], "distance_pairs": [ { "from_id": 0, "to_id": 1, "distance": 66 }, { "from_id": 0, "to_id": 2, "distance": 28 }, { "from_id": 0, "to_id": 3, "distance": 84 }, { "from_id": 0, "to_id": 4, "distance": 44 }, { "from_id": 0, "to_id": 5, "distance": 30 }, { "from_id": 0, "to_id": 6, "distance": 46 }, { "from_id": 0, "to_id": 7, "distance": 4 }, { "from_id": 0, "to_id": 8, "distance": 54 }, { "from_id": 0, "to_id": 9, "distance": 29 }, { "from_id": 0, "to_id": 10, "distance": 36 }, { "from_id": 0, "to_id": 11, "distance": 21 }, { "from_id": 0, "to_id": 12, "distance": 68 }, { "from_id": 0, "to_id": 13, "distance": 75 }, { "from_id": 0, "to_id": 14, "distance": 50 }, { "from_id": 0, "to_id": 15, "distance": 18 }, { "from_id": 0, "to_id": 16, "distance": 53 }, { "from_id": 0, "to_id": 17, "distance": 44 }, { "from_id": 0, "to_id": 18, "distance": 81 }, { "from_id": 0, "to_id": 19, "distance": 60 }, { "from_id": 0, "to_id": 20, "distance": 9 }, { "from_id": 0, "to_id": 21, "distance": 72 }, { "from_id": 0, "to_id": 22, "distance": 36 }, { "from_id": 0, "to_id": 23, "distance": 36 }, { "from_id": 1, "to_id": 0, "distance": 66 }, { "from_id": 1, "to_id": 2, "distance": 86 }, { "from_id": 1, "to_id": 3, "distance": 41 }, { "from_id": 1, "to_id": 4, "distance": 26 }, { "from_id": 1, "to_id": 5, "distance": 96 }, { "from_id": 1, "to_id": 6, "distance": 33 }, { "from_id": 1, "to_id": 7, "distance": 63 }, { "from_id": 1, "to_id": 8, "distance": 56 }, { "from_id": 1, "to_id": 9, "distance": 72 }, { "from_id": 1, "to_id": 10, "distance": 99 }, { "from_id": 1, "to_id": 11, "distance": 62 }, { "from_id": 1, "to_id": 12, "distance": 7 }, { "from_id": 1, "to_id": 13, "distance": 47 }, { "from_id": 1, "to_id": 14, "distance": 30 }, { "from_id": 1, "to_id": 15, "distance": 79 }, { "from_id": 1, "to_id": 16, "distance": 46 }, { "from_id": 1, "to_id": 17, "distance": 58 }, { "from_id": 1, "to_id": 18, "distance": 95 }, { "from_id": 1, "to_id": 19, "distance": 14 }, { "from_id": 1, "to_id": 20, "distance": 72 }, { "from_id": 1, "to_id": 21, "distance": 42 }, { "from_id": 1, "to_id": 22, "distance": 39 }, { "from_id": 1, "to_id": 23, "distance": 36 }, { "from_id": 2, "to_id": 0, "distance": 28 }, { "from_id": 2, "to_id": 1, "distance": 86 }, { "from_id": 2, "to_id": 3, "distance": 92 }, { "from_id": 2, "to_id": 4, "distance": 61 }, { "from_id": 2, "to_id": 5, "distance": 21 }, { "from_id": 2, "to_id": 6, "distance": 73 }, { "from_id": 2, "to_id": 7, "distance": 33 }, { "from_id": 2, "to_id": 8, "distance": 83 }, { "from_id": 2, "to_id": 9, "distance": 18 }, { "from_id": 2, "to_id": 10, "distance": 14 }, { "from_id": 2, "to_id": 11, "distance": 49 }, { "from_id": 2, "to_id": 12, "distance": 86 }, { "from_id": 2, "to_id": 13, "distance": 79 }, { "from_id": 2, "to_id": 14, "distance": 62 }, { "from_id": 2, "to_id": 15, "distance": 36 }, { "from_id": 2, "to_id": 16, "distance": 57 }, { "from_id": 2, "to_id": 17, "distance": 73 }, { "from_id": 2, "to_id": 18, "distance": 64 }, { "from_id": 2, "to_id": 19, "distance": 84 }, { "from_id": 2, "to_id": 20, "distance": 19 }, { "from_id": 2, "to_id": 21, "distance": 78 }, { "from_id": 2, "to_id": 22, "distance": 63 }, { "from_id": 2, "to_id": 23, "distance": 50 }, { "from_id": 3, "to_id": 0, "distance": 84 }, { "from_id": 3, "to_id": 1, "distance": 41 }, { "from_id": 3, "to_id": 2, "distance": 92 }, { "from_id": 3, "to_id": 4, "distance": 41 }, { "from_id": 3, "to_id": 5, "distance": 109 }, { "from_id": 3, "to_id": 6, "distance": 70 }, { "from_id": 3, "to_id": 7, "distance": 83 }, { "from_id": 3, "to_id": 8, "distance": 94 }, { "from_id": 3, "to_id": 9, "distance": 74 }, { "from_id": 3, "to_id": 10, "distance": 107 }, { "from_id": 3, "to_id": 11, "distance": 90 }, { "from_id": 3, "to_id": 12, "distance": 33 }, { "from_id": 3, "to_id": 13, "distance": 15 }, { "from_id": 3, "to_id": 14, "distance": 33 }, { 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"from_id": 23, "to_id": 18, "distance": 67 }, { "from_id": 23, "to_id": 19, "distance": 38 }, { "from_id": 23, "to_id": 20, "distance": 39 }, { "from_id": 23, "to_id": 21, "distance": 37 }, { "from_id": 23, "to_id": 22, "distance": 32 } ] }, "solution_variant": [ 1, 2, 3, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22 ], "context_index": 6, "input_format": "csv", "input_index_base": 0 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "There’s a gig where an organizer needs a fixed-size panel and someone measured how different each pair of potential speakers are on topics. The trick is to pick exactly m different speakers so that when the difference numbers for every pair inside that chosen group are totaled, the total is as large as possible — that means the panel covers the widest range of perspectives. Each spot must be filled and no one can be chosen twice. The detailed scores and the value of m appear below.\n\n{\n \"num_candidate_speakers\": 23,\n \"panel_size\": 9,\n \"speaker_ids\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\"\n ],\n \"data\": [\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"B\",\n \"topic_distance_score\": 123\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"C\",\n \"topic_distance_score\": 123\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"D\",\n \"topic_distance_score\": 165\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"E\",\n \"topic_distance_score\": 130\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"F\",\n \"topic_distance_score\": 109\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"G\",\n \"topic_distance_score\": 165\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"H\",\n \"topic_distance_score\": 148\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"I\",\n \"topic_distance_score\": 108\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"J\",\n \"topic_distance_score\": 134\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"K\",\n \"topic_distance_score\": 142\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"L\",\n \"topic_distance_score\": 132\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"M\",\n \"topic_distance_score\": 166\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"N\",\n \"topic_distance_score\": 187\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"O\",\n \"topic_distance_score\": 153\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"P\",\n \"topic_distance_score\": 189\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"Q\",\n \"topic_distance_score\": 149\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"R\",\n \"topic_distance_score\": 172\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"S\",\n \"topic_distance_score\": 140\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"T\",\n \"topic_distance_score\": 150\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"U\",\n \"topic_distance_score\": 168\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"V\",\n \"topic_distance_score\": 101\n },\n {\n \"speaker_a_id\": \"A\",\n \"speaker_b_id\": \"W\",\n \"topic_distance_score\": 122\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"A\",\n \"topic_distance_score\": 123\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"C\",\n \"topic_distance_score\": 124\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"D\",\n \"topic_distance_score\": 163\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"E\",\n \"topic_distance_score\": 116\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"F\",\n \"topic_distance_score\": 169\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"G\",\n \"topic_distance_score\": 132\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"H\",\n \"topic_distance_score\": 149\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"I\",\n \"topic_distance_score\": 144\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"J\",\n \"topic_distance_score\": 168\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"K\",\n \"topic_distance_score\": 124\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"L\",\n \"topic_distance_score\": 107\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"M\",\n \"topic_distance_score\": 122\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"N\",\n \"topic_distance_score\": 138\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"O\",\n \"topic_distance_score\": 142\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"P\",\n \"topic_distance_score\": 176\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"Q\",\n \"topic_distance_score\": 76\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"R\",\n \"topic_distance_score\": 170\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"S\",\n \"topic_distance_score\": 143\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"T\",\n \"topic_distance_score\": 117\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"U\",\n \"topic_distance_score\": 138\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"V\",\n \"topic_distance_score\": 118\n },\n {\n \"speaker_a_id\": \"B\",\n \"speaker_b_id\": \"W\",\n \"topic_distance_score\": 137\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"A\",\n \"topic_distance_score\": 123\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"B\",\n \"topic_distance_score\": 124\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"D\",\n \"topic_distance_score\": 150\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"E\",\n \"topic_distance_score\": 178\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"F\",\n \"topic_distance_score\": 170\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"G\",\n \"topic_distance_score\": 163\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"H\",\n \"topic_distance_score\": 184\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"I\",\n \"topic_distance_score\": 150\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"J\",\n \"topic_distance_score\": 170\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"K\",\n \"topic_distance_score\": 155\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"L\",\n \"topic_distance_score\": 136\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"M\",\n \"topic_distance_score\": 185\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"N\",\n \"topic_distance_score\": 173\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"O\",\n \"topic_distance_score\": 148\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"P\",\n \"topic_distance_score\": 154\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"Q\",\n \"topic_distance_score\": 108\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"R\",\n \"topic_distance_score\": 199\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"S\",\n \"topic_distance_score\": 140\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"T\",\n \"topic_distance_score\": 143\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"U\",\n \"topic_distance_score\": 176\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"V\",\n \"topic_distance_score\": 169\n },\n {\n \"speaker_a_id\": \"C\",\n \"speaker_b_id\": \"W\",\n \"topic_distance_score\": 118\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"A\",\n \"topic_distance_score\": 165\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"B\",\n \"topic_distance_score\": 163\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"C\",\n \"topic_distance_score\": 150\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"E\",\n \"topic_distance_score\": 170\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"F\",\n \"topic_distance_score\": 164\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"G\",\n \"topic_distance_score\": 150\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"H\",\n \"topic_distance_score\": 166\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"I\",\n \"topic_distance_score\": 135\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"J\",\n \"topic_distance_score\": 170\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"K\",\n \"topic_distance_score\": 153\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"L\",\n \"topic_distance_score\": 155\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"M\",\n \"topic_distance_score\": 194\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"N\",\n \"topic_distance_score\": 145\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"O\",\n \"topic_distance_score\": 138\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"P\",\n \"topic_distance_score\": 120\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"Q\",\n \"topic_distance_score\": 156\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"R\",\n \"topic_distance_score\": 155\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"S\",\n \"topic_distance_score\": 147\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"T\",\n \"topic_distance_score\": 132\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"U\",\n \"topic_distance_score\": 123\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"V\",\n \"topic_distance_score\": 171\n },\n {\n \"speaker_a_id\": \"D\",\n \"speaker_b_id\": \"W\",\n \"topic_distance_score\": 125\n },\n {\n \"speaker_a_id\": \"E\",\n \"speaker_b_id\": \"A\",\n \"topic_distance_score\": 130\n },\n {\n \"speaker_a_id\": \"E\",\n \"speaker_b_id\": \"B\",\n \"topic_distance_score\": 116\n },\n {\n \"speaker_a_id\": \"E\",\n \"speaker_b_id\": \"C\",\n \"topic_distance_score\": 178\n },\n {\n \"speaker_a_id\": \"E\",\n \"speaker_b_id\": \"D\",\n \"topic_distance_score\": 170\n },\n {\n \"speaker_a_id\": \"E\",\n \"speaker_b_id\": \"F\",\n \"topic_distance_score\": 152\n },\n {\n \"speaker_a_id\": \"E\",\n \"speaker_b_id\": \"G\",\n \"topic_distance_score\": 159\n },\n {\n \"speaker_a_id\": \"E\",\n \"speaker_b_id\": \"H\",\n \"topic_distance_score\": 115\n },\n {\n \"speaker_a_id\": \"E\",\n \"speaker_b_id\": \"I\",\n \"topic_distance_score\": 114\n },\n {\n \"speaker_a_id\": \"E\",\n \"speaker_b_id\": \"J\",\n \"topic_distance_score\": 168\n },\n {\n \"speaker_a_id\": \"E\",\n \"speaker_b_id\": \"K\",\n \"topic_distance_score\": 114\n },\n {\n \"speaker_a_id\": \"E\",\n \"speaker_b_id\": \"L\",\n \"topic_distance_score\": 112\n },\n {\n \"speaker_a_id\": \"E\",\n \"speaker_b_id\": \"M\",\n \"topic_distance_score\": 162\n },\n {\n \"speaker_a_id\": \"E\",\n \"speaker_b_id\": \"N\",\n \"topic_distance_score\": 156\n },\n {\n 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\"speaker_a_id\": \"U\",\n \"speaker_b_id\": \"S\",\n \"topic_distance_score\": 155\n },\n {\n \"speaker_a_id\": \"U\",\n \"speaker_b_id\": \"T\",\n \"topic_distance_score\": 146\n },\n {\n \"speaker_a_id\": \"U\",\n \"speaker_b_id\": \"V\",\n \"topic_distance_score\": 132\n },\n {\n \"speaker_a_id\": \"U\",\n \"speaker_b_id\": \"W\",\n \"topic_distance_score\": 165\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"A\",\n \"topic_distance_score\": 101\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"B\",\n \"topic_distance_score\": 118\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"C\",\n \"topic_distance_score\": 169\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"D\",\n \"topic_distance_score\": 171\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"E\",\n \"topic_distance_score\": 94\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"F\",\n \"topic_distance_score\": 137\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"G\",\n \"topic_distance_score\": 140\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"H\",\n \"topic_distance_score\": 115\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"I\",\n \"topic_distance_score\": 119\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"J\",\n \"topic_distance_score\": 104\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"K\",\n \"topic_distance_score\": 128\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"L\",\n \"topic_distance_score\": 109\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"M\",\n \"topic_distance_score\": 149\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"N\",\n \"topic_distance_score\": 154\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"O\",\n \"topic_distance_score\": 154\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"P\",\n \"topic_distance_score\": 183\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"Q\",\n \"topic_distance_score\": 139\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"R\",\n \"topic_distance_score\": 146\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"S\",\n \"topic_distance_score\": 131\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"T\",\n \"topic_distance_score\": 145\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"U\",\n \"topic_distance_score\": 132\n },\n {\n \"speaker_a_id\": \"V\",\n \"speaker_b_id\": \"W\",\n \"topic_distance_score\": 149\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"A\",\n \"topic_distance_score\": 122\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"B\",\n \"topic_distance_score\": 137\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"C\",\n \"topic_distance_score\": 118\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"D\",\n \"topic_distance_score\": 125\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"E\",\n \"topic_distance_score\": 176\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"F\",\n \"topic_distance_score\": 166\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"G\",\n \"topic_distance_score\": 148\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"H\",\n \"topic_distance_score\": 172\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"I\",\n \"topic_distance_score\": 136\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"J\",\n \"topic_distance_score\": 138\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"K\",\n \"topic_distance_score\": 157\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"L\",\n \"topic_distance_score\": 140\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"M\",\n \"topic_distance_score\": 148\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"N\",\n \"topic_distance_score\": 167\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"O\",\n \"topic_distance_score\": 175\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"P\",\n \"topic_distance_score\": 145\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"Q\",\n \"topic_distance_score\": 141\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"R\",\n \"topic_distance_score\": 143\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"S\",\n \"topic_distance_score\": 133\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"T\",\n \"topic_distance_score\": 143\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"U\",\n \"topic_distance_score\": 165\n },\n {\n \"speaker_a_id\": \"W\",\n \"speaker_b_id\": \"V\",\n \"topic_distance_score\": 149\n }\n ]\n}\n\nOh, and when you send back your pick, please follow this little JSON layout so it's easy to read by whatever's checking it:\n\n{\n \"solution\": [, , ...]\n}\n\nThink of \"solution\" as the list of the chosen speaker IDs — one entry per chosen seat on the panel. Each placeholder in the array stands for the exact identifier of a speaker from the instance (and you should include exactly m of them). This block is just a sketch of the shape I need, not the real answer.\n\nUse the identifiers exactly as they appear in the instance input — do not rename them or invent new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”\".", "instance": { "distance_matrix": [ [ 0, 123, 123, 165, 130, 109, 165, 148, 108, 134, 142, 132, 166, 187, 153, 189, 149, 172, 140, 150, 168, 101, 122 ], [ 123, 0, 124, 163, 116, 169, 132, 149, 144, 168, 124, 107, 122, 138, 142, 176, 76, 170, 143, 117, 138, 118, 137 ], [ 123, 124, 0, 150, 178, 170, 163, 184, 150, 170, 155, 136, 185, 173, 148, 154, 108, 199, 140, 143, 176, 169, 118 ], [ 165, 163, 150, 0, 170, 164, 150, 166, 135, 170, 153, 155, 194, 145, 138, 120, 156, 155, 147, 132, 123, 171, 125 ], [ 130, 116, 178, 170, 0, 152, 159, 115, 114, 168, 114, 112, 162, 156, 148, 183, 141, 173, 159, 117, 142, 94, 176 ], [ 109, 169, 170, 164, 152, 0, 178, 172, 113, 154, 136, 170, 183, 191, 131, 200, 188, 141, 159, 159, 155, 137, 166 ], [ 165, 132, 163, 150, 159, 178, 0, 207, 179, 172, 152, 154, 142, 144, 180, 171, 108, 151, 122, 154, 129, 140, 148 ], [ 148, 149, 184, 166, 115, 172, 207, 0, 119, 136, 153, 105, 200, 169, 168, 188, 177, 191, 194, 171, 142, 115, 172 ], [ 108, 144, 150, 135, 114, 113, 179, 119, 0, 160, 124, 146, 172, 171, 129, 145, 171, 143, 165, 129, 149, 119, 136 ], [ 134, 168, 170, 170, 168, 154, 172, 136, 160, 0, 171, 113, 192, 183, 179, 199, 174, 171, 133, 187, 151, 104, 138 ], [ 142, 124, 155, 153, 114, 136, 152, 153, 124, 171, 0, 130, 118, 105, 112, 148, 123, 129, 125, 94, 133, 128, 157 ], [ 132, 107, 136, 155, 112, 170, 154, 105, 146, 113, 130, 0, 174, 133, 167, 180, 115, 187, 128, 144, 119, 109, 140 ], [ 166, 122, 185, 194, 162, 183, 142, 200, 172, 192, 118, 174, 0, 150, 165, 173, 134, 125, 152, 144, 186, 149, 148 ], [ 187, 138, 173, 145, 156, 191, 144, 169, 171, 183, 105, 133, 150, 0, 161, 128, 111, 139, 130, 129, 119, 154, 167 ], [ 153, 142, 148, 138, 148, 131, 180, 168, 129, 179, 112, 167, 165, 161, 0, 148, 148, 154, 150, 104, 161, 154, 175 ], [ 189, 176, 154, 120, 183, 200, 171, 188, 145, 199, 148, 180, 173, 128, 148, 0, 151, 146, 149, 135, 182, 183, 145 ], [ 149, 76, 108, 156, 141, 188, 108, 177, 171, 174, 123, 115, 134, 111, 148, 151, 0, 171, 121, 113, 140, 139, 141 ], [ 172, 170, 199, 155, 173, 141, 151, 191, 143, 171, 129, 187, 125, 139, 154, 146, 171, 0, 151, 136, 151, 146, 143 ], [ 140, 143, 140, 147, 159, 159, 122, 194, 165, 133, 125, 128, 152, 130, 150, 149, 121, 151, 0, 132, 155, 131, 133 ], [ 150, 117, 143, 132, 117, 159, 154, 171, 129, 187, 94, 144, 144, 129, 104, 135, 113, 136, 132, 0, 146, 145, 143 ], [ 168, 138, 176, 123, 142, 155, 129, 142, 149, 151, 133, 119, 186, 119, 161, 182, 140, 151, 155, 146, 0, 132, 165 ], [ 101, 118, 169, 171, 94, 137, 140, 115, 119, 104, 128, 109, 149, 154, 154, 183, 139, 146, 131, 145, 132, 0, 149 ], [ 122, 137, 118, 125, 176, 166, 148, 172, 136, 138, 157, 140, 148, 167, 175, 145, 141, 143, 133, 143, 165, 149, 0 ] ], "m": 9, "objective": 6111.0 }, "solution": [ 2, 5, 6, 7, 9, 12, 13, 14, 15 ], "obj": 6111.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 23, "m": 9, "nodes": [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S", "T", "U", "V", "W" ], "distance_pairs": [ { "from_id": "A", "to_id": "B", "distance": 123 }, { "from_id": "A", "to_id": "C", "distance": 123 }, { "from_id": "A", "to_id": "D", "distance": 165 }, { "from_id": "A", "to_id": "E", "distance": 130 }, { "from_id": "A", "to_id": "F", "distance": 109 }, { "from_id": "A", "to_id": "G", "distance": 165 }, { "from_id": "A", "to_id": "H", "distance": 148 }, { "from_id": "A", "to_id": "I", "distance": 108 }, { "from_id": "A", "to_id": "J", "distance": 134 }, { "from_id": "A", "to_id": "K", "distance": 142 }, { "from_id": "A", "to_id": "L", "distance": 132 }, { "from_id": "A", "to_id": "M", "distance": 166 }, { "from_id": "A", "to_id": "N", "distance": 187 }, { "from_id": "A", "to_id": "O", "distance": 153 }, { "from_id": "A", "to_id": "P", "distance": 189 }, { "from_id": "A", "to_id": "Q", "distance": 149 }, { "from_id": "A", "to_id": "R", "distance": 172 }, { "from_id": "A", "to_id": "S", "distance": 140 }, { "from_id": "A", "to_id": "T", "distance": 150 }, { "from_id": "A", "to_id": "U", "distance": 168 }, { "from_id": "A", "to_id": "V", "distance": 101 }, { "from_id": "A", "to_id": "W", "distance": 122 }, { "from_id": "B", "to_id": "A", "distance": 123 }, { "from_id": "B", "to_id": "C", "distance": 124 }, { "from_id": "B", "to_id": "D", "distance": 163 }, { "from_id": "B", "to_id": "E", "distance": 116 }, { "from_id": "B", "to_id": "F", "distance": 169 }, { "from_id": "B", "to_id": "G", "distance": 132 }, { "from_id": "B", "to_id": "H", "distance": 149 }, { "from_id": "B", "to_id": "I", "distance": 144 }, { "from_id": "B", "to_id": "J", "distance": 168 }, { "from_id": "B", "to_id": "K", "distance": 124 }, { "from_id": "B", "to_id": "L", "distance": 107 }, { "from_id": "B", "to_id": "M", "distance": 122 }, { "from_id": "B", "to_id": "N", "distance": 138 }, { "from_id": "B", "to_id": "O", "distance": 142 }, { "from_id": "B", "to_id": "P", "distance": 176 }, { "from_id": "B", "to_id": "Q", "distance": 76 }, { "from_id": "B", "to_id": "R", "distance": 170 }, { "from_id": "B", "to_id": "S", "distance": 143 }, { "from_id": "B", "to_id": "T", "distance": 117 }, { "from_id": "B", "to_id": "U", "distance": 138 }, { "from_id": "B", "to_id": "V", "distance": 118 }, { "from_id": "B", "to_id": "W", "distance": 137 }, { "from_id": "C", "to_id": "A", "distance": 123 }, { "from_id": "C", "to_id": "B", "distance": 124 }, { "from_id": "C", "to_id": "D", "distance": 150 }, { "from_id": "C", "to_id": "E", "distance": 178 }, { "from_id": "C", "to_id": "F", "distance": 170 }, { "from_id": "C", "to_id": "G", "distance": 163 }, { "from_id": "C", "to_id": "H", "distance": 184 }, { "from_id": "C", "to_id": "I", "distance": 150 }, { "from_id": "C", "to_id": "J", "distance": 170 }, { "from_id": "C", "to_id": "K", "distance": 155 }, { "from_id": "C", "to_id": "L", "distance": 136 }, { "from_id": "C", "to_id": "M", "distance": 185 }, { "from_id": "C", "to_id": "N", "distance": 173 }, { "from_id": "C", "to_id": "O", "distance": 148 }, { "from_id": "C", "to_id": "P", "distance": 154 }, { "from_id": "C", "to_id": "Q", "distance": 108 }, { "from_id": "C", "to_id": "R", "distance": 199 }, { "from_id": "C", "to_id": "S", "distance": 140 }, { "from_id": "C", "to_id": "T", "distance": 143 }, { "from_id": "C", "to_id": "U", "distance": 176 }, { "from_id": "C", "to_id": "V", "distance": 169 }, { "from_id": "C", "to_id": "W", "distance": 118 }, { "from_id": "D", "to_id": "A", "distance": 165 }, { "from_id": "D", "to_id": "B", "distance": 163 }, { "from_id": "D", "to_id": "C", "distance": 150 }, { "from_id": "D", "to_id": "E", "distance": 170 }, { "from_id": "D", "to_id": "F", "distance": 164 }, { "from_id": "D", "to_id": "G", "distance": 150 }, { "from_id": "D", "to_id": "H", "distance": 166 }, { "from_id": "D", "to_id": "I", "distance": 135 }, { "from_id": "D", "to_id": "J", "distance": 170 }, { "from_id": "D", "to_id": "K", "distance": 153 }, { "from_id": "D", "to_id": "L", "distance": 155 }, { "from_id": "D", "to_id": "M", "distance": 194 }, { "from_id": "D", "to_id": "N", "distance": 145 }, { "from_id": "D", "to_id": "O", "distance": 138 }, { "from_id": "D", "to_id": "P", "distance": 120 }, { "from_id": "D", "to_id": "Q", "distance": 156 }, { "from_id": "D", "to_id": "R", "distance": 155 }, { "from_id": "D", "to_id": "S", "distance": 147 }, { "from_id": "D", "to_id": "T", "distance": 132 }, { "from_id": "D", "to_id": "U", "distance": 123 }, { "from_id": "D", "to_id": "V", "distance": 171 }, { "from_id": "D", "to_id": "W", "distance": 125 }, { "from_id": "E", "to_id": "A", "distance": 130 }, { "from_id": "E", "to_id": "B", "distance": 116 }, { "from_id": "E", "to_id": "C", "distance": 178 }, { "from_id": "E", "to_id": "D", "distance": 170 }, { "from_id": "E", "to_id": "F", "distance": 152 }, { "from_id": "E", "to_id": "G", "distance": 159 }, { "from_id": "E", "to_id": "H", "distance": 115 }, { "from_id": "E", "to_id": "I", "distance": 114 }, { "from_id": "E", "to_id": "J", "distance": 168 }, { "from_id": "E", "to_id": "K", "distance": 114 }, { "from_id": "E", "to_id": "L", "distance": 112 }, { "from_id": "E", "to_id": "M", "distance": 162 }, { "from_id": "E", "to_id": "N", "distance": 156 }, { "from_id": "E", "to_id": "O", "distance": 148 }, { "from_id": "E", "to_id": "P", "distance": 183 }, { "from_id": "E", "to_id": "Q", "distance": 141 }, { "from_id": "E", "to_id": "R", "distance": 173 }, { "from_id": "E", "to_id": "S", "distance": 159 }, { "from_id": "E", "to_id": "T", "distance": 117 }, { "from_id": "E", "to_id": "U", "distance": 142 }, { "from_id": "E", "to_id": "V", "distance": 94 }, { "from_id": "E", "to_id": "W", "distance": 176 }, { "from_id": "F", "to_id": "A", "distance": 109 }, { "from_id": "F", "to_id": "B", "distance": 169 }, { "from_id": "F", "to_id": "C", "distance": 170 }, { "from_id": "F", "to_id": "D", "distance": 164 }, { "from_id": "F", "to_id": "E", "distance": 152 }, { "from_id": "F", "to_id": "G", "distance": 178 }, { "from_id": "F", "to_id": "H", "distance": 172 }, { "from_id": "F", "to_id": "I", "distance": 113 }, { "from_id": "F", "to_id": "J", "distance": 154 }, { "from_id": "F", "to_id": "K", "distance": 136 }, { "from_id": "F", "to_id": "L", "distance": 170 }, { "from_id": "F", "to_id": "M", "distance": 183 }, { "from_id": "F", "to_id": "N", "distance": 191 }, { "from_id": "F", "to_id": "O", "distance": 131 }, { "from_id": "F", "to_id": "P", "distance": 200 }, { "from_id": "F", "to_id": "Q", "distance": 188 }, { "from_id": "F", "to_id": "R", "distance": 141 }, { "from_id": "F", "to_id": "S", "distance": 159 }, { "from_id": "F", "to_id": "T", "distance": 159 }, { "from_id": "F", "to_id": "U", "distance": 155 }, { "from_id": "F", "to_id": "V", "distance": 137 }, { "from_id": "F", "to_id": "W", "distance": 166 }, { "from_id": "G", "to_id": "A", "distance": 165 }, { "from_id": "G", "to_id": "B", "distance": 132 }, { "from_id": "G", "to_id": "C", "distance": 163 }, { "from_id": "G", "to_id": "D", "distance": 150 }, { "from_id": "G", "to_id": "E", "distance": 159 }, { "from_id": "G", "to_id": "F", "distance": 178 }, { "from_id": "G", "to_id": "H", "distance": 207 }, { "from_id": "G", "to_id": "I", "distance": 179 }, { "from_id": "G", "to_id": "J", "distance": 172 }, { "from_id": "G", "to_id": "K", "distance": 152 }, { "from_id": "G", "to_id": "L", "distance": 154 }, { "from_id": "G", "to_id": "M", "distance": 142 }, { "from_id": "G", "to_id": "N", "distance": 144 }, { "from_id": "G", "to_id": "O", "distance": 180 }, { "from_id": "G", "to_id": "P", "distance": 171 }, { "from_id": "G", "to_id": "Q", "distance": 108 }, { "from_id": "G", "to_id": "R", 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"from_id": "W", "to_id": "R", "distance": 143 }, { "from_id": "W", "to_id": "S", "distance": 133 }, { "from_id": "W", "to_id": "T", "distance": 143 }, { "from_id": "W", "to_id": "U", "distance": 165 }, { "from_id": "W", "to_id": "V", "distance": 149 } ] }, "solution_variant": [ "C", "F", "G", "H", "J", "M", "N", "O", "P" ], "context_index": 7, "input_format": "json", "input_index_base": "names" }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "We have to assemble a fixed set of prototypes for the next round: exactly m unique concepts and nothing doubled. The better choice is the one where the items in the set are spread far apart — you check how different every pair is, add up all those pairwise distances, and prefer the set with the largest total. The specific prototypes and their pairwise distance values are listed below.\n\n# total_prototypes_available=25\n# prototypes_to_select=2\n# prototype_ids_list=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24\nprototype_id_a,prototype_id_b,pairwise_feature_dissimilarity\n0,1,5\n0,2,4\n0,3,5\n0,4,8\n0,5,5\n0,6,2\n0,7,8\n0,8,0\n0,9,2\n0,10,7\n0,11,0\n0,12,6\n0,13,3\n0,14,0\n0,15,8\n0,16,2\n0,17,3\n0,18,7\n0,19,6\n0,20,4\n0,21,4\n0,22,1\n0,23,0\n0,24,3\n1,0,5\n1,2,2\n1,3,2\n1,4,1\n1,5,5\n1,6,4\n1,7,0\n1,8,9\n1,9,1\n1,10,8\n1,11,0\n1,12,3\n1,13,9\n1,14,7\n1,15,2\n1,16,9\n1,17,6\n1,18,5\n1,19,7\n1,20,2\n1,21,5\n1,22,4\n1,23,4\n1,24,9\n2,0,4\n2,1,2\n2,3,0\n2,4,5\n2,5,1\n2,6,0\n2,7,6\n2,8,2\n2,9,9\n2,10,6\n2,11,9\n2,12,9\n2,13,3\n2,14,3\n2,15,3\n2,16,8\n2,17,2\n2,18,3\n2,19,8\n2,20,2\n2,21,5\n2,22,6\n2,23,7\n2,24,8\n3,0,5\n3,1,2\n3,2,0\n3,4,5\n3,5,3\n3,6,2\n3,7,4\n3,8,7\n3,9,0\n3,10,2\n3,11,9\n3,12,9\n3,13,6\n3,14,0\n3,15,4\n3,16,4\n3,17,1\n3,18,2\n3,19,4\n3,20,1\n3,21,8\n3,22,3\n3,23,0\n3,24,3\n4,0,8\n4,1,1\n4,2,5\n4,3,5\n4,5,8\n4,6,0\n4,7,8\n4,8,7\n4,9,0\n4,10,6\n4,11,6\n4,12,0\n4,13,1\n4,14,3\n4,15,8\n4,16,8\n4,17,1\n4,18,2\n4,19,2\n4,20,0\n4,21,7\n4,22,9\n4,23,4\n4,24,8\n5,0,5\n5,1,5\n5,2,1\n5,3,3\n5,4,8\n5,6,1\n5,7,3\n5,8,6\n5,9,0\n5,10,0\n5,11,2\n5,12,9\n5,13,4\n5,14,4\n5,15,5\n5,16,9\n5,17,1\n5,18,0\n5,19,1\n5,20,2\n5,21,9\n5,22,6\n5,23,9\n5,24,9\n6,0,2\n6,1,4\n6,2,0\n6,3,2\n6,4,0\n6,5,1\n6,7,0\n6,8,3\n6,9,9\n6,10,9\n6,11,8\n6,12,6\n6,13,2\n6,14,5\n6,15,7\n6,16,8\n6,17,2\n6,18,1\n6,19,9\n6,20,7\n6,21,1\n6,22,4\n6,23,1\n6,24,3\n7,0,8\n7,1,0\n7,2,6\n7,3,4\n7,4,8\n7,5,3\n7,6,0\n7,8,5\n7,9,9\n7,10,3\n7,11,3\n7,12,7\n7,13,0\n7,14,9\n7,15,8\n7,16,0\n7,17,2\n7,18,6\n7,19,6\n7,20,7\n7,21,8\n7,22,9\n7,23,8\n7,24,3\n8,0,0\n8,1,9\n8,2,2\n8,3,7\n8,4,7\n8,5,6\n8,6,3\n8,7,5\n8,9,0\n8,10,3\n8,11,3\n8,12,5\n8,13,7\n8,14,9\n8,15,7\n8,16,8\n8,17,2\n8,18,4\n8,19,0\n8,20,4\n8,21,6\n8,22,8\n8,23,6\n8,24,1\n9,0,2\n9,1,1\n9,2,9\n9,3,0\n9,4,0\n9,5,0\n9,6,9\n9,7,9\n9,8,0\n9,10,8\n9,11,9\n9,12,1\n9,13,0\n9,14,7\n9,15,4\n9,16,1\n9,17,8\n9,18,5\n9,19,6\n9,20,3\n9,21,6\n9,22,0\n9,23,0\n9,24,4\n10,0,7\n10,1,8\n10,2,6\n10,3,2\n10,4,6\n10,5,0\n10,6,9\n10,7,3\n10,8,3\n10,9,8\n10,11,3\n10,12,4\n10,13,9\n10,14,9\n10,15,7\n10,16,8\n10,17,6\n10,18,3\n10,19,5\n10,20,3\n10,21,5\n10,22,9\n10,23,7\n10,24,9\n11,0,0\n11,1,0\n11,2,9\n11,3,9\n11,4,6\n11,5,2\n11,6,8\n11,7,3\n11,8,3\n11,9,9\n11,10,3\n11,12,1\n11,13,0\n11,14,1\n11,15,4\n11,16,2\n11,17,1\n11,18,5\n11,19,8\n11,20,8\n11,21,4\n11,22,4\n11,23,2\n11,24,1\n12,0,6\n12,1,3\n12,2,9\n12,3,9\n12,4,0\n12,5,9\n12,6,6\n12,7,7\n12,8,5\n12,9,1\n12,10,4\n12,11,1\n12,13,7\n12,14,0\n12,15,5\n12,16,1\n12,17,3\n12,18,4\n12,19,5\n12,20,6\n12,21,9\n12,22,6\n12,23,6\n12,24,5\n13,0,3\n13,1,9\n13,2,3\n13,3,6\n13,4,1\n13,5,4\n13,6,2\n13,7,0\n13,8,7\n13,9,0\n13,10,9\n13,11,0\n13,12,7\n13,14,0\n13,15,1\n13,16,9\n13,17,1\n13,18,4\n13,19,0\n13,20,9\n13,21,1\n13,22,2\n13,23,1\n13,24,5\n14,0,0\n14,1,7\n14,2,3\n14,3,0\n14,4,3\n14,5,4\n14,6,5\n14,7,9\n14,8,9\n14,9,7\n14,10,9\n14,11,1\n14,12,0\n14,13,0\n14,15,3\n14,16,1\n14,17,3\n14,18,8\n14,19,9\n14,20,2\n14,21,5\n14,22,8\n14,23,2\n14,24,5\n15,0,8\n15,1,2\n15,2,3\n15,3,4\n15,4,8\n15,5,5\n15,6,7\n15,7,8\n15,8,7\n15,9,4\n15,10,7\n15,11,4\n15,12,5\n15,13,1\n15,14,3\n15,16,6\n15,17,0\n15,18,5\n15,19,3\n15,20,6\n15,21,0\n15,22,2\n15,23,2\n15,24,9\n16,0,2\n16,1,9\n16,2,8\n16,3,4\n16,4,8\n16,5,9\n16,6,8\n16,7,0\n16,8,8\n16,9,1\n16,10,8\n16,11,2\n16,12,1\n16,13,9\n16,14,1\n16,15,6\n16,17,6\n16,18,7\n16,19,4\n16,20,7\n16,21,1\n16,22,5\n16,23,6\n16,24,9\n17,0,3\n17,1,6\n17,2,2\n17,3,1\n17,4,1\n17,5,1\n17,6,2\n17,7,2\n17,8,2\n17,9,8\n17,10,6\n17,11,1\n17,12,3\n17,13,1\n17,14,3\n17,15,0\n17,16,6\n17,18,7\n17,19,0\n17,20,8\n17,21,1\n17,22,4\n17,23,3\n17,24,7\n18,0,7\n18,1,5\n18,2,3\n18,3,2\n18,4,2\n18,5,0\n18,6,1\n18,7,6\n18,8,4\n18,9,5\n18,10,3\n18,11,5\n18,12,4\n18,13,4\n18,14,8\n18,15,5\n18,16,7\n18,17,7\n18,19,1\n18,20,0\n18,21,9\n18,22,4\n18,23,8\n18,24,7\n19,0,6\n19,1,7\n19,2,8\n19,3,4\n19,4,2\n19,5,1\n19,6,9\n19,7,6\n19,8,0\n19,9,6\n19,10,5\n19,11,8\n19,12,5\n19,13,0\n19,14,9\n19,15,3\n19,16,4\n19,17,0\n19,18,1\n19,20,7\n19,21,9\n19,22,4\n19,23,5\n19,24,3\n20,0,4\n20,1,2\n20,2,2\n20,3,1\n20,4,0\n20,5,2\n20,6,7\n20,7,7\n20,8,4\n20,9,3\n20,10,3\n20,11,8\n20,12,6\n20,13,9\n20,14,2\n20,15,6\n20,16,7\n20,17,8\n20,18,0\n20,19,7\n20,21,1\n20,22,1\n20,23,8\n20,24,9\n21,0,4\n21,1,5\n21,2,5\n21,3,8\n21,4,7\n21,5,9\n21,6,1\n21,7,8\n21,8,6\n21,9,6\n21,10,5\n21,11,4\n21,12,9\n21,13,1\n21,14,5\n21,15,0\n21,16,1\n21,17,1\n21,18,9\n21,19,9\n21,20,1\n21,22,9\n21,23,6\n21,24,5\n22,0,1\n22,1,4\n22,2,6\n22,3,3\n22,4,9\n22,5,6\n22,6,4\n22,7,9\n22,8,8\n22,9,0\n22,10,9\n22,11,4\n22,12,6\n22,13,2\n22,14,8\n22,15,2\n22,16,5\n22,17,4\n22,18,4\n22,19,4\n22,20,1\n22,21,9\n22,23,0\n22,24,8\n23,0,0\n23,1,4\n23,2,7\n23,3,0\n23,4,4\n23,5,9\n23,6,1\n23,7,8\n23,8,6\n23,9,0\n23,10,7\n23,11,2\n23,12,6\n23,13,1\n23,14,2\n23,15,2\n23,16,6\n23,17,3\n23,18,8\n23,19,5\n23,20,8\n23,21,6\n23,22,0\n23,24,4\n24,0,3\n24,1,9\n24,2,8\n24,3,3\n24,4,8\n24,5,9\n24,6,3\n24,7,3\n24,8,1\n24,9,4\n24,10,9\n24,11,1\n24,12,5\n24,13,5\n24,14,5\n24,15,9\n24,16,9\n24,17,7\n24,18,7\n24,19,3\n24,20,9\n24,21,5\n24,22,8\n24,23,4\n\nJust a heads-up — when you send the final selection, please use this simple JSON layout so it's easy to read and process:\n\n{\n \"solution\": [, , ...]\n}\n\nThis JSON is just a sketch of the shape I expect: \"solution\" is the list of exactly m prototypes you pick (one identifier per entry), and the array is where you put those identifiers. Think of it like filling out a short form — list the chosen items in order, that's it.\n\nPlease make sure to use the identifiers exactly as they appear in the instance input — no renaming and no new labels. \nValid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.", "instance": { "distance_matrix": [ [ 0, 5, 4, 5, 8, 5, 2, 8, 0, 2, 7, 0, 6, 3, 0, 8, 2, 3, 7, 6, 4, 4, 1, 0, 3 ], [ 5, 0, 2, 2, 1, 5, 4, 0, 9, 1, 8, 0, 3, 9, 7, 2, 9, 6, 5, 7, 2, 5, 4, 4, 9 ], [ 4, 2, 0, 0, 5, 1, 0, 6, 2, 9, 6, 9, 9, 3, 3, 3, 8, 2, 3, 8, 2, 5, 6, 7, 8 ], [ 5, 2, 0, 0, 5, 3, 2, 4, 7, 0, 2, 9, 9, 6, 0, 4, 4, 1, 2, 4, 1, 8, 3, 0, 3 ], [ 8, 1, 5, 5, 0, 8, 0, 8, 7, 0, 6, 6, 0, 1, 3, 8, 8, 1, 2, 2, 0, 7, 9, 4, 8 ], [ 5, 5, 1, 3, 8, 0, 1, 3, 6, 0, 0, 2, 9, 4, 4, 5, 9, 1, 0, 1, 2, 9, 6, 9, 9 ], [ 2, 4, 0, 2, 0, 1, 0, 0, 3, 9, 9, 8, 6, 2, 5, 7, 8, 2, 1, 9, 7, 1, 4, 1, 3 ], [ 8, 0, 6, 4, 8, 3, 0, 0, 5, 9, 3, 3, 7, 0, 9, 8, 0, 2, 6, 6, 7, 8, 9, 8, 3 ], [ 0, 9, 2, 7, 7, 6, 3, 5, 0, 0, 3, 3, 5, 7, 9, 7, 8, 2, 4, 0, 4, 6, 8, 6, 1 ], [ 2, 1, 9, 0, 0, 0, 9, 9, 0, 0, 8, 9, 1, 0, 7, 4, 1, 8, 5, 6, 3, 6, 0, 0, 4 ], [ 7, 8, 6, 2, 6, 0, 9, 3, 3, 8, 0, 3, 4, 9, 9, 7, 8, 6, 3, 5, 3, 5, 9, 7, 9 ], [ 0, 0, 9, 9, 6, 2, 8, 3, 3, 9, 3, 0, 1, 0, 1, 4, 2, 1, 5, 8, 8, 4, 4, 2, 1 ], [ 6, 3, 9, 9, 0, 9, 6, 7, 5, 1, 4, 1, 0, 7, 0, 5, 1, 3, 4, 5, 6, 9, 6, 6, 5 ], [ 3, 9, 3, 6, 1, 4, 2, 0, 7, 0, 9, 0, 7, 0, 0, 1, 9, 1, 4, 0, 9, 1, 2, 1, 5 ], [ 0, 7, 3, 0, 3, 4, 5, 9, 9, 7, 9, 1, 0, 0, 0, 3, 1, 3, 8, 9, 2, 5, 8, 2, 5 ], [ 8, 2, 3, 4, 8, 5, 7, 8, 7, 4, 7, 4, 5, 1, 3, 0, 6, 0, 5, 3, 6, 0, 2, 2, 9 ], [ 2, 9, 8, 4, 8, 9, 8, 0, 8, 1, 8, 2, 1, 9, 1, 6, 0, 6, 7, 4, 7, 1, 5, 6, 9 ], [ 3, 6, 2, 1, 1, 1, 2, 2, 2, 8, 6, 1, 3, 1, 3, 0, 6, 0, 7, 0, 8, 1, 4, 3, 7 ], [ 7, 5, 3, 2, 2, 0, 1, 6, 4, 5, 3, 5, 4, 4, 8, 5, 7, 7, 0, 1, 0, 9, 4, 8, 7 ], [ 6, 7, 8, 4, 2, 1, 9, 6, 0, 6, 5, 8, 5, 0, 9, 3, 4, 0, 1, 0, 7, 9, 4, 5, 3 ], [ 4, 2, 2, 1, 0, 2, 7, 7, 4, 3, 3, 8, 6, 9, 2, 6, 7, 8, 0, 7, 0, 1, 1, 8, 9 ], [ 4, 5, 5, 8, 7, 9, 1, 8, 6, 6, 5, 4, 9, 1, 5, 0, 1, 1, 9, 9, 1, 0, 9, 6, 5 ], [ 1, 4, 6, 3, 9, 6, 4, 9, 8, 0, 9, 4, 6, 2, 8, 2, 5, 4, 4, 4, 1, 9, 0, 0, 8 ], [ 0, 4, 7, 0, 4, 9, 1, 8, 6, 0, 7, 2, 6, 1, 2, 2, 6, 3, 8, 5, 8, 6, 0, 0, 4 ], [ 3, 9, 8, 3, 8, 9, 3, 3, 1, 4, 9, 1, 5, 5, 5, 9, 9, 7, 7, 3, 9, 5, 8, 4, 0 ] ], "m": 2, "objective": 9.0 }, "solution": [ 3, 11 ], "obj": 9.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 25, "m": 2, "nodes": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 ], "distance_pairs": [ { "from_id": 0, "to_id": 1, "distance": 5 }, { "from_id": 0, "to_id": 2, "distance": 4 }, { "from_id": 0, "to_id": 3, "distance": 5 }, { "from_id": 0, "to_id": 4, "distance": 8 }, { "from_id": 0, "to_id": 5, "distance": 5 }, { "from_id": 0, "to_id": 6, "distance": 2 }, { "from_id": 0, "to_id": 7, "distance": 8 }, { "from_id": 0, "to_id": 8, "distance": 0 }, { "from_id": 0, "to_id": 9, "distance": 2 }, { "from_id": 0, "to_id": 10, "distance": 7 }, { "from_id": 0, "to_id": 11, "distance": 0 }, { "from_id": 0, "to_id": 12, "distance": 6 }, { "from_id": 0, "to_id": 13, "distance": 3 }, { "from_id": 0, "to_id": 14, "distance": 0 }, { "from_id": 0, "to_id": 15, "distance": 8 }, { "from_id": 0, "to_id": 16, "distance": 2 }, { "from_id": 0, "to_id": 17, "distance": 3 }, { "from_id": 0, "to_id": 18, "distance": 7 }, { "from_id": 0, "to_id": 19, "distance": 6 }, { "from_id": 0, "to_id": 20, "distance": 4 }, { "from_id": 0, "to_id": 21, "distance": 4 }, { "from_id": 0, "to_id": 22, "distance": 1 }, { "from_id": 0, "to_id": 23, "distance": 0 }, { "from_id": 0, "to_id": 24, "distance": 3 }, { "from_id": 1, "to_id": 0, "distance": 5 }, { "from_id": 1, "to_id": 2, "distance": 2 }, { "from_id": 1, "to_id": 3, "distance": 2 }, { "from_id": 1, "to_id": 4, "distance": 1 }, { "from_id": 1, "to_id": 5, "distance": 5 }, { "from_id": 1, "to_id": 6, "distance": 4 }, { "from_id": 1, "to_id": 7, "distance": 0 }, { "from_id": 1, "to_id": 8, "distance": 9 }, { "from_id": 1, "to_id": 9, "distance": 1 }, { "from_id": 1, "to_id": 10, "distance": 8 }, { "from_id": 1, "to_id": 11, "distance": 0 }, { "from_id": 1, "to_id": 12, "distance": 3 }, { "from_id": 1, "to_id": 13, "distance": 9 }, { "from_id": 1, "to_id": 14, "distance": 7 }, { "from_id": 1, "to_id": 15, "distance": 2 }, { "from_id": 1, "to_id": 16, "distance": 9 }, { "from_id": 1, "to_id": 17, "distance": 6 }, { "from_id": 1, "to_id": 18, "distance": 5 }, { "from_id": 1, "to_id": 19, "distance": 7 }, { "from_id": 1, "to_id": 20, "distance": 2 }, { "from_id": 1, "to_id": 21, "distance": 5 }, { "from_id": 1, "to_id": 22, "distance": 4 }, { "from_id": 1, "to_id": 23, "distance": 4 }, { "from_id": 1, "to_id": 24, "distance": 9 }, { "from_id": 2, "to_id": 0, "distance": 4 }, { "from_id": 2, "to_id": 1, "distance": 2 }, { "from_id": 2, "to_id": 3, "distance": 0 }, { "from_id": 2, 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"to_id": 8, "distance": 4 }, { "from_id": 18, "to_id": 9, "distance": 5 }, { "from_id": 18, "to_id": 10, "distance": 3 }, { "from_id": 18, "to_id": 11, "distance": 5 }, { "from_id": 18, "to_id": 12, "distance": 4 }, { "from_id": 18, "to_id": 13, "distance": 4 }, { "from_id": 18, "to_id": 14, "distance": 8 }, { "from_id": 18, "to_id": 15, "distance": 5 }, { "from_id": 18, "to_id": 16, "distance": 7 }, { "from_id": 18, "to_id": 17, "distance": 7 }, { "from_id": 18, "to_id": 19, "distance": 1 }, { "from_id": 18, "to_id": 20, "distance": 0 }, { "from_id": 18, "to_id": 21, "distance": 9 }, { "from_id": 18, "to_id": 22, "distance": 4 }, { "from_id": 18, "to_id": 23, "distance": 8 }, { "from_id": 18, "to_id": 24, "distance": 7 }, { "from_id": 19, "to_id": 0, "distance": 6 }, { "from_id": 19, "to_id": 1, "distance": 7 }, { "from_id": 19, "to_id": 2, "distance": 8 }, { "from_id": 19, "to_id": 3, "distance": 4 }, { "from_id": 19, "to_id": 4, "distance": 2 }, { "from_id": 19, "to_id": 5, "distance": 1 }, { "from_id": 19, "to_id": 6, "distance": 9 }, { "from_id": 19, "to_id": 7, "distance": 6 }, { "from_id": 19, "to_id": 8, "distance": 0 }, { "from_id": 19, "to_id": 9, "distance": 6 }, { "from_id": 19, "to_id": 10, "distance": 5 }, { "from_id": 19, "to_id": 11, "distance": 8 }, { "from_id": 19, "to_id": 12, "distance": 5 }, { "from_id": 19, "to_id": 13, "distance": 0 }, { "from_id": 19, "to_id": 14, "distance": 9 }, { "from_id": 19, "to_id": 15, "distance": 3 }, { "from_id": 19, "to_id": 16, "distance": 4 }, { "from_id": 19, "to_id": 17, "distance": 0 }, { "from_id": 19, "to_id": 18, "distance": 1 }, { "from_id": 19, "to_id": 20, "distance": 7 }, { "from_id": 19, "to_id": 21, "distance": 9 }, { "from_id": 19, "to_id": 22, "distance": 4 }, { "from_id": 19, "to_id": 23, "distance": 5 }, { "from_id": 19, "to_id": 24, "distance": 3 }, { "from_id": 20, "to_id": 0, "distance": 4 }, { "from_id": 20, "to_id": 1, "distance": 2 }, { "from_id": 20, "to_id": 2, "distance": 2 }, { "from_id": 20, "to_id": 3, "distance": 1 }, { "from_id": 20, "to_id": 4, "distance": 0 }, { "from_id": 20, "to_id": 5, "distance": 2 }, { "from_id": 20, "to_id": 6, "distance": 7 }, { "from_id": 20, "to_id": 7, "distance": 7 }, { "from_id": 20, "to_id": 8, "distance": 4 }, { "from_id": 20, "to_id": 9, "distance": 3 }, { "from_id": 20, "to_id": 10, "distance": 3 }, { "from_id": 20, "to_id": 11, "distance": 8 }, { "from_id": 20, "to_id": 12, "distance": 6 }, { "from_id": 20, "to_id": 13, "distance": 9 }, { "from_id": 20, "to_id": 14, "distance": 2 }, { "from_id": 20, "to_id": 15, "distance": 6 }, { "from_id": 20, "to_id": 16, "distance": 7 }, { "from_id": 20, "to_id": 17, "distance": 8 }, { "from_id": 20, "to_id": 18, "distance": 0 }, { "from_id": 20, "to_id": 19, "distance": 7 }, { "from_id": 20, "to_id": 21, "distance": 1 }, { "from_id": 20, "to_id": 22, "distance": 1 }, { "from_id": 20, "to_id": 23, "distance": 8 }, { "from_id": 20, "to_id": 24, "distance": 9 }, { "from_id": 21, "to_id": 0, "distance": 4 }, { "from_id": 21, "to_id": 1, "distance": 5 }, { "from_id": 21, "to_id": 2, "distance": 5 }, { "from_id": 21, "to_id": 3, "distance": 8 }, { "from_id": 21, "to_id": 4, "distance": 7 }, { "from_id": 21, "to_id": 5, "distance": 9 }, { "from_id": 21, "to_id": 6, "distance": 1 }, { "from_id": 21, "to_id": 7, "distance": 8 }, { "from_id": 21, "to_id": 8, "distance": 6 }, { "from_id": 21, "to_id": 9, "distance": 6 }, { "from_id": 21, "to_id": 10, "distance": 5 }, { "from_id": 21, "to_id": 11, "distance": 4 }, { "from_id": 21, "to_id": 12, "distance": 9 }, { "from_id": 21, "to_id": 13, "distance": 1 }, { "from_id": 21, "to_id": 14, "distance": 5 }, { "from_id": 21, "to_id": 15, "distance": 0 }, { "from_id": 21, "to_id": 16, "distance": 1 }, { "from_id": 21, "to_id": 17, "distance": 1 }, { "from_id": 21, "to_id": 18, "distance": 9 }, { "from_id": 21, "to_id": 19, "distance": 9 }, { "from_id": 21, "to_id": 20, "distance": 1 }, { "from_id": 21, "to_id": 22, "distance": 9 }, { "from_id": 21, "to_id": 23, "distance": 6 }, { "from_id": 21, "to_id": 24, "distance": 5 }, { "from_id": 22, "to_id": 0, "distance": 1 }, { "from_id": 22, "to_id": 1, "distance": 4 }, { "from_id": 22, "to_id": 2, "distance": 6 }, { "from_id": 22, "to_id": 3, "distance": 3 }, { "from_id": 22, "to_id": 4, "distance": 9 }, { "from_id": 22, "to_id": 5, "distance": 6 }, { "from_id": 22, "to_id": 6, "distance": 4 }, { "from_id": 22, "to_id": 7, "distance": 9 }, { "from_id": 22, "to_id": 8, "distance": 8 }, { "from_id": 22, "to_id": 9, "distance": 0 }, { "from_id": 22, "to_id": 10, "distance": 9 }, { "from_id": 22, "to_id": 11, "distance": 4 }, { "from_id": 22, "to_id": 12, "distance": 6 }, { "from_id": 22, "to_id": 13, "distance": 2 }, { "from_id": 22, "to_id": 14, "distance": 8 }, { "from_id": 22, "to_id": 15, "distance": 2 }, { "from_id": 22, "to_id": 16, "distance": 5 }, { "from_id": 22, "to_id": 17, "distance": 4 }, { "from_id": 22, "to_id": 18, "distance": 4 }, { "from_id": 22, "to_id": 19, "distance": 4 }, { "from_id": 22, "to_id": 20, "distance": 1 }, { "from_id": 22, "to_id": 21, "distance": 9 }, { "from_id": 22, "to_id": 23, "distance": 0 }, { "from_id": 22, "to_id": 24, "distance": 8 }, { "from_id": 23, "to_id": 0, "distance": 0 }, { "from_id": 23, "to_id": 1, "distance": 4 }, { "from_id": 23, "to_id": 2, "distance": 7 }, { "from_id": 23, "to_id": 3, "distance": 0 }, { "from_id": 23, "to_id": 4, "distance": 4 }, { "from_id": 23, "to_id": 5, "distance": 9 }, { "from_id": 23, "to_id": 6, "distance": 1 }, { "from_id": 23, "to_id": 7, "distance": 8 }, { "from_id": 23, "to_id": 8, "distance": 6 }, { "from_id": 23, "to_id": 9, "distance": 0 }, { "from_id": 23, "to_id": 10, "distance": 7 }, { "from_id": 23, "to_id": 11, "distance": 2 }, { "from_id": 23, "to_id": 12, "distance": 6 }, { "from_id": 23, "to_id": 13, "distance": 1 }, { "from_id": 23, "to_id": 14, "distance": 2 }, { "from_id": 23, "to_id": 15, "distance": 2 }, { "from_id": 23, "to_id": 16, "distance": 6 }, { "from_id": 23, "to_id": 17, "distance": 3 }, { "from_id": 23, "to_id": 18, "distance": 8 }, { "from_id": 23, "to_id": 19, "distance": 5 }, { "from_id": 23, "to_id": 20, "distance": 8 }, { "from_id": 23, "to_id": 21, "distance": 6 }, { "from_id": 23, "to_id": 22, "distance": 0 }, { "from_id": 23, "to_id": 24, "distance": 4 }, { "from_id": 24, "to_id": 0, "distance": 3 }, { "from_id": 24, "to_id": 1, "distance": 9 }, { "from_id": 24, "to_id": 2, "distance": 8 }, { "from_id": 24, "to_id": 3, "distance": 3 }, { "from_id": 24, "to_id": 4, "distance": 8 }, { "from_id": 24, "to_id": 5, "distance": 9 }, { "from_id": 24, "to_id": 6, "distance": 3 }, { "from_id": 24, "to_id": 7, "distance": 3 }, { "from_id": 24, "to_id": 8, "distance": 1 }, { "from_id": 24, "to_id": 9, "distance": 4 }, { "from_id": 24, "to_id": 10, "distance": 9 }, { "from_id": 24, "to_id": 11, "distance": 1 }, { "from_id": 24, "to_id": 12, "distance": 5 }, { "from_id": 24, "to_id": 13, "distance": 5 }, { "from_id": 24, "to_id": 14, "distance": 5 }, { "from_id": 24, "to_id": 15, "distance": 9 }, { "from_id": 24, "to_id": 16, "distance": 9 }, { "from_id": 24, "to_id": 17, "distance": 7 }, { "from_id": 24, "to_id": 18, "distance": 7 }, { "from_id": 24, "to_id": 19, "distance": 3 }, { "from_id": 24, "to_id": 20, "distance": 9 }, { "from_id": 24, "to_id": 21, "distance": 5 }, { "from_id": 24, "to_id": 22, "distance": 8 }, { "from_id": 24, "to_id": 23, "distance": 4 } ] }, "solution_variant": [ 3, 11 ], "context_index": 8, "input_format": "csv", "input_index_base": 0 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Back in class a decision needs to be made about which student projects to spotlight: pick exactly m unique projects so the display doesn’t feel repetitive. Each possible pair of projects has a number that says how different they are, and for any chosen set those pair numbers are added up across every pair in the group; the higher that total, the more diverse the selection. The actual projects, all the pairwise difference numbers, and the value of m follow below.\n\nThere are 24 projects listed as A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, and exactly 14 must be chosen.\nBetween project A and B, the content difference is 193.\nBetween project A and C, the content difference is 166.\nBetween project A and D, the content difference is 138.\nBetween project A and E, the content difference is 133.\nBetween project A and F, the content difference is 164.\nBetween project A and G, the content difference is 161.\nBetween project A and H, the content difference is 149.\nBetween project A and I, the content difference is 137.\nBetween project A and J, the content difference is 173.\nBetween project A and K, the content difference is 122.\nBetween project A and L, the content difference is 191.\nBetween project A and M, the content difference is 86.\nBetween project A and N, the content difference is 153.\nBetween project A and O, the content difference is 144.\nBetween project A and P, the content difference is 151.\nBetween project A and Q, the content difference is 133.\nBetween project A and R, the content difference is 131.\nBetween project A and S, the content difference is 143.\nBetween project A and T, the content difference is 148.\nBetween project A and U, the content difference is 151.\nBetween project A and V, the content difference is 175.\nBetween project A and W, the content difference is 152.\nBetween project A and X, the content difference is 116.\nBetween project B and A, the content difference is 193.\nBetween project B and C, the content difference is 111.\nBetween project B and D, the content difference is 116.\nBetween project B and E, the content difference is 187.\nBetween project B and F, the content difference is 137.\nBetween project B and G, the content difference is 149.\nBetween project B and H, the content difference is 164.\nBetween project B and I, the content difference is 136.\nBetween project B and J, the content difference is 125.\nBetween project B and K, the content difference is 148.\nBetween project B and L, the content difference is 133.\nBetween project B and M, the content difference is 193.\nBetween project B and N, the content difference is 164.\nBetween project B and O, the content difference is 190.\nBetween project B and P, the content difference is 151.\nBetween project B and Q, the content difference is 126.\nBetween project B and R, the content difference is 132.\nBetween project B and S, the content difference is 180.\nBetween project B and T, the content difference is 127.\nBetween project B and U, the content difference is 133.\nBetween project B and V, the content difference is 66.\nBetween project B and W, the content difference is 157.\nBetween project B and X, the content difference is 182.\nBetween project C and A, the content difference is 166.\nBetween project C and B, the content difference is 111.\nBetween project C and D, the content difference is 145.\nBetween project C and E, the content difference is 193.\nBetween project C and F, the content difference is 114.\nBetween project C and G, the content difference is 139.\nBetween project C and H, the content difference is 165.\nBetween project C and I, the content difference is 133.\nBetween project C and J, the content difference is 127.\nBetween project C and K, the content difference is 109.\nBetween project C and L, the content difference is 182.\nBetween project C and M, the content difference is 178.\nBetween project C and N, the content difference is 122.\nBetween project C and O, the content difference is 174.\nBetween project C and P, the content difference is 144.\nBetween project C and Q, the content difference is 108.\nBetween project C and R, the content difference is 105.\nBetween project C and S, the content difference is 153.\nBetween project C and T, the content difference is 146.\nBetween project C and U, the content difference is 144.\nBetween project C and V, the content difference is 106.\nBetween project C and W, the content difference is 172.\nBetween project C and X, the content difference is 146.\nBetween project D and A, the content difference is 138.\nBetween project D and B, the content difference is 116.\nBetween project D and C, the content difference is 145.\nBetween project D and E, the content difference is 137.\nBetween project D and F, the content difference is 151.\nBetween project D and G, the content difference is 132.\nBetween project D and H, the content difference is 154.\nBetween project D and I, the content difference is 148.\nBetween project D and J, the content difference is 141.\nBetween project D and K, the content difference is 121.\nBetween project D and L, the content difference is 124.\nBetween project D and M, the content difference is 107.\nBetween project D and N, the content difference is 160.\nBetween project D and O, the content difference is 160.\nBetween project D and P, the content difference is 153.\nBetween project D and Q, the content difference is 91.\nBetween project D and R, the content difference is 137.\nBetween project D and S, the content difference is 188.\nBetween project D and T, the content difference is 125.\nBetween project D and U, the content difference is 84.\nBetween project D and V, the content difference is 86.\nBetween project D and W, the content difference is 133.\nBetween project D and X, the content difference is 161.\nBetween project E and A, the content difference is 133.\nBetween project E and B, the content difference is 187.\nBetween project E and C, the content difference is 193.\nBetween project E and D, the content difference is 137.\nBetween project E and F, the content difference is 198.\nBetween project E and G, the content difference is 145.\nBetween project E and H, the content difference is 147.\nBetween project E and I, the content difference is 155.\nBetween project E and J, the content difference is 191.\nBetween project E and K, the content difference is 138.\nBetween project E and L, the content difference is 141.\nBetween project E and M, the content difference is 118.\nBetween project E and N, the content difference is 154.\nBetween project E and O, the content difference is 155.\nBetween project E and P, the content difference is 191.\nBetween project E and Q, the content difference is 151.\nBetween project E and R, the content difference is 174.\nBetween project E and S, the content difference is 181.\nBetween project E and T, the content difference is 118.\nBetween project E and U, the content difference is 158.\nBetween project E and V, the content difference is 180.\nBetween project E and W, the content difference is 116.\nBetween project E and X, the content difference is 151.\nBetween project F and A, the content difference is 164.\nBetween project F and B, the content difference is 137.\nBetween project F and C, the content difference is 114.\nBetween project F and D, the content difference is 151.\nBetween project F and E, the content difference is 198.\nBetween project F and G, the content difference is 113.\nBetween project F and H, the content difference is 113.\nBetween project F and I, the content difference is 124.\nBetween project F and J, the content difference is 123.\nBetween project F and K, the content difference is 108.\nBetween project F and L, the content difference is 189.\nBetween project F and M, the content difference is 168.\nBetween project F and N, the content difference is 136.\nBetween project F and O, the content difference is 129.\nBetween project F and P, the content difference is 120.\nBetween project F and Q, the content difference is 142.\nBetween project F and R, the content difference is 129.\nBetween project F and S, the content difference is 130.\nBetween project F and T, the content difference is 126.\nBetween project F and U, the content difference is 129.\nBetween project F and V, the content difference is 140.\nBetween project F and W, the content difference is 155.\nBetween project F and X, the content difference is 167.\nBetween project G and A, the content difference is 161.\nBetween project G and B, the content difference is 149.\nBetween project G and C, the content difference is 139.\nBetween project G and D, the content difference is 132.\nBetween project G and E, the content difference is 145.\nBetween project G and F, the content difference is 113.\nBetween project G and H, the content difference is 123.\nBetween project G and I, the content difference is 148.\nBetween project G and J, the content difference is 126.\nBetween project G and K, the content difference is 114.\nBetween project G and L, the content difference is 129.\nBetween project G and M, the content difference is 138.\nBetween project G and N, the content difference is 140.\nBetween project G and O, the content difference is 147.\nBetween project G and P, the content difference is 114.\nBetween project G and Q, the content difference is 128.\nBetween project G and R, the content difference is 141.\nBetween project G and S, the content difference is 145.\nBetween project G and T, the content difference is 110.\nBetween project G and U, the content difference is 89.\nBetween project G and V, the content difference is 160.\nBetween project G and W, the content difference is 105.\nBetween project G and X, the content difference is 135.\nBetween project H and A, the content difference is 149.\nBetween project H and B, the content difference is 164.\nBetween project H and C, the content difference is 165.\nBetween project H and D, the content difference is 154.\nBetween project H and E, the content difference is 147.\nBetween project H and F, the content difference is 113.\nBetween project H and G, the content difference is 123.\nBetween project H and I, the content difference is 119.\nBetween project H and J, the content difference is 148.\nBetween project H and K, the content difference is 144.\nBetween project H and L, the content difference is 186.\nBetween project H and M, the content difference is 156.\nBetween project H and N, the content difference is 177.\nBetween project H and O, the content difference is 95.\nBetween project H and P, the content difference is 150.\nBetween project H and Q, the content difference is 133.\nBetween project H and R, the content difference is 139.\nBetween project H and S, the content difference is 161.\nBetween project H and T, the content difference is 102.\nBetween project H and U, the content difference is 136.\nBetween project H and V, the content difference is 167.\nBetween project H and W, the content difference is 149.\nBetween project H and X, the content difference is 181.\nBetween project I and A, the content difference is 137.\nBetween project I and B, the content difference is 136.\nBetween project I and C, the content difference is 133.\nBetween project I and D, the content difference is 148.\nBetween project I and E, the content difference is 155.\nBetween project I and F, the content difference is 124.\nBetween project I and G, the content difference is 148.\nBetween project I and H, the content difference is 119.\nBetween project I and J, the content difference is 155.\nBetween project I and K, the content difference is 133.\nBetween project I and L, the content difference is 157.\nBetween project I and M, the content difference is 165.\nBetween project I and N, the content difference is 123.\nBetween project I and O, the content difference is 139.\nBetween project I and P, the content difference is 126.\nBetween project I and Q, the content difference is 128.\nBetween project I and R, the content difference is 105.\nBetween project I and S, the content difference is 104.\nBetween project I and T, the content difference is 138.\nBetween project I and U, the content difference is 168.\nBetween project I and V, the content difference is 135.\nBetween project I and W, the content difference is 151.\nBetween project I and X, the content difference is 174.\nBetween project J and A, the content difference is 173.\nBetween project J and B, the content difference is 125.\nBetween project J and C, the content difference is 127.\nBetween project J and D, the content difference is 141.\nBetween project J and E, the content difference is 191.\nBetween project J and F, the content difference is 123.\nBetween project J and G, the content difference is 126.\nBetween project J and H, the content difference is 148.\nBetween project J and I, the content difference is 155.\nBetween project J and K, the content difference is 130.\nBetween project J and L, the content difference is 183.\nBetween project J and M, the content difference is 181.\nBetween project J and N, the content difference is 167.\nBetween project J and O, the content difference is 184.\nBetween project J and P, the content difference is 120.\nBetween project J and Q, the content difference is 129.\nBetween project J and R, the content difference is 160.\nBetween project J and S, the content difference is 155.\nBetween project J and T, the content difference is 143.\nBetween project J and U, the content difference is 123.\nBetween project J and V, the content difference is 133.\nBetween project J and W, the content difference is 161.\nBetween project J and X, the content difference is 155.\nBetween project K and A, the content difference is 122.\nBetween project K and B, the content difference is 148.\nBetween project K and C, the content difference is 109.\nBetween project K and D, the content difference is 121.\nBetween project K and E, the content difference is 138.\nBetween project K and F, the content difference is 108.\nBetween project K and G, the content difference is 114.\nBetween project K and H, the content difference is 144.\nBetween project K and I, the content difference is 133.\nBetween project K and J, the content difference is 130.\nBetween project K and L, the content difference is 164.\nBetween project K and M, the content difference is 117.\nBetween project K and N, the content difference is 82.\nBetween project K and O, the content difference is 126.\nBetween project K and P, the content difference is 132.\nBetween project K and Q, the content difference is 118.\nBetween project K and R, the content difference is 135.\nBetween project K and S, the content difference is 132.\nBetween project K and T, the content difference is 116.\nBetween project K and U, the content difference is 130.\nBetween project K and V, the content difference is 135.\nBetween project K and W, the content difference is 125.\nBetween project K and X, the content difference is 120.\nBetween project L and A, the content difference is 191.\nBetween project L and B, the content difference is 133.\nBetween project L and C, the content difference is 182.\nBetween project L and D, the content difference is 124.\nBetween project L and E, the content difference is 141.\nBetween project L and F, the content difference is 189.\nBetween project L and G, the content difference is 129.\nBetween project L and H, the content difference is 186.\nBetween project L and I, the content difference is 157.\nBetween project L and J, the content difference is 183.\nBetween project L and K, the content difference is 164.\nBetween project L and M, the content difference is 164.\nBetween project L and N, the content difference is 161.\nBetween project L and O, the content difference is 203.\nBetween project L and P, the content difference is 167.\nBetween project L and Q, the content difference is 160.\nBetween project L and R, the content difference is 169.\nBetween project L and S, the content difference is 181.\nBetween project L and T, the content difference is 130.\nBetween project L and U, the content difference is 148.\nBetween project L and V, the content difference is 150.\nBetween project L and W, the content difference is 99.\nBetween project L and X, the content difference is 176.\nBetween project M and A, the content difference is 86.\nBetween project M and B, the content difference is 193.\nBetween project M and C, the content difference is 178.\nBetween project M and D, the content difference is 107.\nBetween project M and E, the content difference is 118.\nBetween project M and F, the content difference is 168.\nBetween project M and G, the content difference is 138.\nBetween project M and H, the content difference is 156.\nBetween project M and I, the content difference is 165.\nBetween project M and J, the content difference is 181.\nBetween project M and K, the content difference is 117.\nBetween project M and L, the content difference is 164.\nBetween project M and N, the content difference is 151.\nBetween project M and O, the content difference is 133.\nBetween project M and P, the content difference is 155.\nBetween project M and Q, the content difference is 128.\nBetween project M and R, the content difference is 142.\nBetween project M and S, the content difference is 169.\nBetween project M and T, the content difference is 137.\nBetween project M and U, the content difference is 119.\nBetween project M and V, the content difference is 168.\nBetween project M and W, the content difference is 119.\nBetween project M and X, the content difference is 119.\nBetween project N and A, the content difference is 153.\nBetween project N and B, the content difference is 164.\nBetween project N and C, the content difference is 122.\nBetween project N and D, the content difference is 160.\nBetween project N and E, the content difference is 154.\nBetween project N and F, the content difference is 136.\nBetween project N and G, the content difference is 140.\nBetween project N and H, the content difference is 177.\nBetween project N and I, the content difference is 123.\nBetween project N and J, the content difference is 167.\nBetween project N and K, the content difference is 82.\nBetween project N and L, the content difference is 161.\nBetween project N and M, the content difference is 151.\nBetween project N and O, the content difference is 161.\nBetween project N and P, the content difference is 149.\nBetween project N and Q, the content difference is 158.\nBetween project N and R, the content difference is 144.\nBetween project N and S, the content difference is 103.\nBetween project N and T, the content difference is 144.\nBetween project N and U, the content difference is 178.\nBetween project N and V, the content difference is 155.\nBetween project N and W, the content difference is 124.\nBetween project N and X, the content difference is 131.\nBetween project O and A, the content difference is 144.\nBetween project O and B, the content difference is 190.\nBetween project O and C, the content difference is 174.\nBetween project O and D, the content difference is 160.\nBetween project O and E, the content difference is 155.\nBetween project O and F, the content difference is 129.\nBetween project O and G, the content difference is 147.\nBetween project O and H, the content difference is 95.\nBetween project O and I, the content difference is 139.\nBetween project O and J, the content difference is 184.\nBetween project O and K, the content difference is 126.\nBetween project O and L, the content difference is 203.\nBetween project O and M, the content difference is 133.\nBetween project O and N, the content difference is 161.\nBetween project O and P, the content difference is 148.\nBetween project O and Q, the content difference is 140.\nBetween project O and R, the content difference is 130.\nBetween project O and S, the content difference is 173.\nBetween project O and T, the content difference is 134.\nBetween project O and U, the content difference is 152.\nBetween project O and V, the content difference is 177.\nBetween project O and W, the content difference is 160.\nBetween project O and X, the content difference is 181.\nBetween project P and A, the content difference is 151.\nBetween project P and B, the content difference is 151.\nBetween project P and C, the content difference is 144.\nBetween project P and D, the content difference is 153.\nBetween project P and E, the content difference is 191.\nBetween project P and F, the content difference is 120.\nBetween project P and G, the content difference is 114.\nBetween project P and H, the content difference is 150.\nBetween project P and I, the content difference is 126.\nBetween project P and J, the content difference is 120.\nBetween project P and K, the content difference is 132.\nBetween project P and L, the content difference is 167.\nBetween project P and M, the content difference is 155.\nBetween project P and N, the content difference is 149.\nBetween project P and O, the content difference is 148.\nBetween project P and Q, the content difference is 131.\nBetween project P and R, the content difference is 109.\nBetween project P and S, the content difference is 113.\nBetween project P and T, the content difference is 162.\nBetween project P and U, the content difference is 127.\nBetween project P and V, the content difference is 159.\nBetween project P and W, the content difference is 149.\nBetween project P and X, the content difference is 142.\nBetween project Q and A, the content difference is 133.\nBetween project Q and B, the content difference is 126.\nBetween project Q and C, the content difference is 108.\nBetween project Q and D, the content difference is 91.\nBetween project Q and E, the content difference is 151.\nBetween project Q and F, the content difference is 142.\nBetween project Q and G, the content difference is 128.\nBetween project Q and H, the content difference is 133.\nBetween project Q and I, the content difference is 128.\nBetween project Q and J, the content difference is 129.\nBetween project Q and K, the content difference is 118.\nBetween project Q and L, the content difference is 160.\nBetween project Q and M, the content difference is 128.\nBetween project Q and N, the content difference is 158.\nBetween project Q and O, the content difference is 140.\nBetween project Q and P, the content difference is 131.\nBetween project Q and R, the content difference is 100.\nBetween project Q and S, the content difference is 175.\nBetween project Q and T, the content difference is 139.\nBetween project Q and U, the content difference is 102.\nBetween project Q and V, the content difference is 106.\nBetween project Q and W, the content difference is 160.\nBetween project Q and X, the content difference is 155.\nBetween project R and A, the content difference is 131.\nBetween project R and B, the content difference is 132.\nBetween project R and C, the content difference is 105.\nBetween project R and D, the content difference is 137.\nBetween project R and E, the content difference is 174.\nBetween project R and F, the content difference is 129.\nBetween project R and G, the content difference is 141.\nBetween project R and H, the content difference is 139.\nBetween project R and I, the content difference is 105.\nBetween project R and J, the content difference is 160.\nBetween project R and K, the content difference is 135.\nBetween project R and L, the content difference is 169.\nBetween project R and M, the content difference is 142.\nBetween project R and N, the content difference is 144.\nBetween project R and O, the content difference is 130.\nBetween project R and P, the content difference is 109.\nBetween project R and Q, the content difference is 100.\nBetween project R and S, the content difference is 142.\nBetween project R and T, the content difference is 146.\nBetween project R and U, the content difference is 132.\nBetween project R and V, the content difference is 124.\nBetween project R and W, the content difference is 159.\nBetween project R and X, the content difference is 141.\nBetween project S and A, the content difference is 143.\nBetween project S and B, the content difference is 180.\nBetween project S and C, the content difference is 153.\nBetween project S and D, the content difference is 188.\nBetween project S and E, the content difference is 181.\nBetween project S and F, the content difference is 130.\nBetween project S and G, the content difference is 145.\nBetween project S and H, the content difference is 161.\nBetween project S and I, the content difference is 104.\nBetween project S and J, the content difference is 155.\nBetween project S and K, the content difference is 132.\nBetween project S and L, the content difference is 181.\nBetween project S and M, the content difference is 169.\nBetween project S and N, the content difference is 103.\nBetween project S and O, the content difference is 173.\nBetween project S and P, the content difference is 113.\nBetween project S and Q, the content difference is 175.\nBetween project S and R, the content difference is 142.\nBetween project S and T, the content difference is 160.\nBetween project S and U, the content difference is 186.\nBetween project S and V, the content difference is 187.\nBetween project S and W, the content difference is 138.\nBetween project S and X, the content difference is 131.\nBetween project T and A, the content difference is 148.\nBetween project T and B, the content difference is 127.\nBetween project T and C, the content difference is 146.\nBetween project T and D, the content difference is 125.\nBetween project T and E, the content difference is 118.\nBetween project T and F, the content difference is 126.\nBetween project T and G, the content difference is 110.\nBetween project T and H, the content difference is 102.\nBetween project T and I, the content difference is 138.\nBetween project T and J, the content difference is 143.\nBetween project T and K, the content difference is 116.\nBetween project T and L, the content difference is 130.\nBetween project T and M, the content difference is 137.\nBetween project T and N, the content difference is 144.\nBetween project T and O, the content difference is 134.\nBetween project T and P, the content difference is 162.\nBetween project T and Q, the content difference is 139.\nBetween project T and R, the content difference is 146.\nBetween project T and S, the content difference is 160.\nBetween project T and U, the content difference is 126.\nBetween project T and V, the content difference is 138.\nBetween project T and W, the content difference is 85.\nBetween project T and X, the content difference is 141.\nBetween project U and A, the content difference is 151.\nBetween project U and B, the content difference is 133.\nBetween project U and C, the content difference is 144.\nBetween project U and D, the content difference is 84.\nBetween project U and E, the content difference is 158.\nBetween project U and F, the content difference is 129.\nBetween project U and G, the content difference is 89.\nBetween project U and H, the content difference is 136.\nBetween project U and I, the content difference is 168.\nBetween project U and J, the content difference is 123.\nBetween project U and K, the content difference is 130.\nBetween project U and L, the content difference is 148.\nBetween project U and M, the content difference is 119.\nBetween project U and N, the content difference is 178.\nBetween project U and O, the content difference is 152.\nBetween project U and P, the content difference is 127.\nBetween project U and Q, the content difference is 102.\nBetween project U and R, the content difference is 132.\nBetween project U and S, the content difference is 186.\nBetween project U and T, the content difference is 126.\nBetween project U and V, the content difference is 127.\nBetween project U and W, the content difference is 138.\nBetween project U and X, the content difference is 147.\nBetween project V and A, the content difference is 175.\nBetween project V and B, the content difference is 66.\nBetween project V and C, the content difference is 106.\nBetween project V and D, the content difference is 86.\nBetween project V and E, the content difference is 180.\nBetween project V and F, the content difference is 140.\nBetween project V and G, the content difference is 160.\nBetween project V and H, the content difference is 167.\nBetween project V and I, the content difference is 135.\nBetween project V and J, the content difference is 133.\nBetween project V and K, the content difference is 135.\nBetween project V and L, the content difference is 150.\nBetween project V and M, the content difference is 168.\nBetween project V and N, the content difference is 155.\nBetween project V and O, the content difference is 177.\nBetween project V and P, the content difference is 159.\nBetween project V and Q, the content difference is 106.\nBetween project V and R, the content difference is 124.\nBetween project V and S, the content difference is 187.\nBetween project V and T, the content difference is 138.\nBetween project V and U, the content difference is 127.\nBetween project V and W, the content difference is 164.\nBetween project V and X, the content difference is 183.\nBetween project W and A, the content difference is 152.\nBetween project W and B, the content difference is 157.\nBetween project W and C, the content difference is 172.\nBetween project W and D, the content difference is 133.\nBetween project W and E, the content difference is 116.\nBetween project W and F, the content difference is 155.\nBetween project W and G, the content difference is 105.\nBetween project W and H, the content difference is 149.\nBetween project W and I, the content difference is 151.\nBetween project W and J, the content difference is 161.\nBetween project W and K, the content difference is 125.\nBetween project W and L, the content difference is 99.\nBetween project W and M, the content difference is 119.\nBetween project W and N, the content difference is 124.\nBetween project W and O, the content difference is 160.\nBetween project W and P, the content difference is 149.\nBetween project W and Q, the content difference is 160.\nBetween project W and R, the content difference is 159.\nBetween project W and S, the content difference is 138.\nBetween project W and T, the content difference is 85.\nBetween project W and U, the content difference is 138.\nBetween project W and V, the content difference is 164.\nBetween project W and X, the content difference is 121.\nBetween project X and A, the content difference is 116.\nBetween project X and B, the content difference is 182.\nBetween project X and C, the content difference is 146.\nBetween project X and D, the content difference is 161.\nBetween project X and E, the content difference is 151.\nBetween project X and F, the content difference is 167.\nBetween project X and G, the content difference is 135.\nBetween project X and H, the content difference is 181.\nBetween project X and I, the content difference is 174.\nBetween project X and J, the content difference is 155.\nBetween project X and K, the content difference is 120.\nBetween project X and L, the content difference is 176.\nBetween project X and M, the content difference is 119.\nBetween project X and N, the content difference is 131.\nBetween project X and O, the content difference is 181.\nBetween project X and P, the content difference is 142.\nBetween project X and Q, the content difference is 155.\nBetween project X and R, the content difference is 141.\nBetween project X and S, the content difference is 131.\nBetween project X and T, the content difference is 141.\nBetween project X and U, the content difference is 147.\nBetween project X and V, the content difference is 183.\nBetween project X and W, the content difference is 121.\nThese lines provide the pairwise content differences to sum when evaluating any selection of 14 projects.\n\nIf you want the final pick in a tidy little package, I usually show it as a tiny JSON snippet like this so it's easy to copy/paste:\n\n{\n \"solution\": [, , ...]\n}\n\nThis just shows the shape I’ll return: \"solution\" is the list of the chosen project IDs to spotlight, and each is a placeholder for one project (I’ll replace those with the actual IDs from the instance when I give the final answer). Think of this as a sketch of the expected layout, not the actual selection.\n\nPlease make sure to use the exact identifiers from the instance input — don’t rename them or invent new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 193, 166, 138, 133, 164, 161, 149, 137, 173, 122, 191, 86, 153, 144, 151, 133, 131, 143, 148, 151, 175, 152, 116 ], [ 193, 0, 111, 116, 187, 137, 149, 164, 136, 125, 148, 133, 193, 164, 190, 151, 126, 132, 180, 127, 133, 66, 157, 182 ], [ 166, 111, 0, 145, 193, 114, 139, 165, 133, 127, 109, 182, 178, 122, 174, 144, 108, 105, 153, 146, 144, 106, 172, 146 ], [ 138, 116, 145, 0, 137, 151, 132, 154, 148, 141, 121, 124, 107, 160, 160, 153, 91, 137, 188, 125, 84, 86, 133, 161 ], [ 133, 187, 193, 137, 0, 198, 145, 147, 155, 191, 138, 141, 118, 154, 155, 191, 151, 174, 181, 118, 158, 180, 116, 151 ], [ 164, 137, 114, 151, 198, 0, 113, 113, 124, 123, 108, 189, 168, 136, 129, 120, 142, 129, 130, 126, 129, 140, 155, 167 ], [ 161, 149, 139, 132, 145, 113, 0, 123, 148, 126, 114, 129, 138, 140, 147, 114, 128, 141, 145, 110, 89, 160, 105, 135 ], [ 149, 164, 165, 154, 147, 113, 123, 0, 119, 148, 144, 186, 156, 177, 95, 150, 133, 139, 161, 102, 136, 167, 149, 181 ], [ 137, 136, 133, 148, 155, 124, 148, 119, 0, 155, 133, 157, 165, 123, 139, 126, 128, 105, 104, 138, 168, 135, 151, 174 ], [ 173, 125, 127, 141, 191, 123, 126, 148, 155, 0, 130, 183, 181, 167, 184, 120, 129, 160, 155, 143, 123, 133, 161, 155 ], [ 122, 148, 109, 121, 138, 108, 114, 144, 133, 130, 0, 164, 117, 82, 126, 132, 118, 135, 132, 116, 130, 135, 125, 120 ], [ 191, 133, 182, 124, 141, 189, 129, 186, 157, 183, 164, 0, 164, 161, 203, 167, 160, 169, 181, 130, 148, 150, 99, 176 ], [ 86, 193, 178, 107, 118, 168, 138, 156, 165, 181, 117, 164, 0, 151, 133, 155, 128, 142, 169, 137, 119, 168, 119, 119 ], [ 153, 164, 122, 160, 154, 136, 140, 177, 123, 167, 82, 161, 151, 0, 161, 149, 158, 144, 103, 144, 178, 155, 124, 131 ], [ 144, 190, 174, 160, 155, 129, 147, 95, 139, 184, 126, 203, 133, 161, 0, 148, 140, 130, 173, 134, 152, 177, 160, 181 ], [ 151, 151, 144, 153, 191, 120, 114, 150, 126, 120, 132, 167, 155, 149, 148, 0, 131, 109, 113, 162, 127, 159, 149, 142 ], [ 133, 126, 108, 91, 151, 142, 128, 133, 128, 129, 118, 160, 128, 158, 140, 131, 0, 100, 175, 139, 102, 106, 160, 155 ], [ 131, 132, 105, 137, 174, 129, 141, 139, 105, 160, 135, 169, 142, 144, 130, 109, 100, 0, 142, 146, 132, 124, 159, 141 ], [ 143, 180, 153, 188, 181, 130, 145, 161, 104, 155, 132, 181, 169, 103, 173, 113, 175, 142, 0, 160, 186, 187, 138, 131 ], [ 148, 127, 146, 125, 118, 126, 110, 102, 138, 143, 116, 130, 137, 144, 134, 162, 139, 146, 160, 0, 126, 138, 85, 141 ], [ 151, 133, 144, 84, 158, 129, 89, 136, 168, 123, 130, 148, 119, 178, 152, 127, 102, 132, 186, 126, 0, 127, 138, 147 ], [ 175, 66, 106, 86, 180, 140, 160, 167, 135, 133, 135, 150, 168, 155, 177, 159, 106, 124, 187, 138, 127, 0, 164, 183 ], [ 152, 157, 172, 133, 116, 155, 105, 149, 151, 161, 125, 99, 119, 124, 160, 149, 160, 159, 138, 85, 138, 164, 0, 121 ], [ 116, 182, 146, 161, 151, 167, 135, 181, 174, 155, 120, 176, 119, 131, 181, 142, 155, 141, 131, 141, 147, 183, 121, 0 ] ], "m": 14, "objective": 14207.0 }, "solution": [ 0, 1, 2, 4, 7, 9, 11, 12, 13, 14, 15, 18, 21, 23 ], "obj": 14207.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 24, "m": 14, "nodes": [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S", "T", "U", "V", "W", "X" ], "distance_pairs": [ { "from_id": "A", "to_id": "B", "distance": 193 }, { "from_id": "A", "to_id": "C", "distance": 166 }, { "from_id": "A", "to_id": "D", "distance": 138 }, { "from_id": "A", "to_id": "E", "distance": 133 }, { "from_id": "A", "to_id": "F", "distance": 164 }, { "from_id": "A", "to_id": "G", "distance": 161 }, { "from_id": "A", "to_id": "H", "distance": 149 }, { "from_id": "A", "to_id": "I", "distance": 137 }, { "from_id": "A", "to_id": "J", "distance": 173 }, { "from_id": "A", "to_id": "K", "distance": 122 }, { "from_id": "A", "to_id": "L", "distance": 191 }, { "from_id": "A", "to_id": "M", "distance": 86 }, { "from_id": "A", "to_id": "N", "distance": 153 }, { "from_id": "A", "to_id": "O", "distance": 144 }, { "from_id": "A", "to_id": "P", "distance": 151 }, { "from_id": "A", "to_id": "Q", "distance": 133 }, { "from_id": "A", "to_id": "R", "distance": 131 }, { "from_id": "A", "to_id": "S", "distance": 143 }, { "from_id": "A", "to_id": "T", "distance": 148 }, { "from_id": "A", "to_id": "U", "distance": 151 }, { "from_id": "A", "to_id": "V", "distance": 175 }, { "from_id": "A", "to_id": "W", "distance": 152 }, { "from_id": "A", "to_id": "X", "distance": 116 }, { "from_id": "B", "to_id": "A", "distance": 193 }, { "from_id": "B", "to_id": "C", "distance": 111 }, { "from_id": "B", "to_id": "D", "distance": 116 }, { "from_id": "B", "to_id": "E", "distance": 187 }, { "from_id": "B", "to_id": "F", "distance": 137 }, { "from_id": "B", "to_id": "G", "distance": 149 }, { "from_id": "B", "to_id": "H", "distance": 164 }, { "from_id": "B", "to_id": "I", "distance": 136 }, { "from_id": "B", "to_id": "J", "distance": 125 }, { "from_id": "B", "to_id": "K", "distance": 148 }, { "from_id": "B", "to_id": "L", "distance": 133 }, { "from_id": "B", "to_id": "M", "distance": 193 }, { "from_id": "B", "to_id": "N", "distance": 164 }, { "from_id": "B", "to_id": "O", "distance": 190 }, { "from_id": "B", "to_id": "P", "distance": 151 }, { "from_id": "B", "to_id": "Q", "distance": 126 }, { "from_id": "B", "to_id": "R", "distance": 132 }, { "from_id": "B", "to_id": "S", "distance": 180 }, { "from_id": "B", "to_id": "T", "distance": 127 }, { "from_id": "B", "to_id": "U", "distance": 133 }, { "from_id": "B", "to_id": "V", "distance": 66 }, { "from_id": "B", "to_id": "W", "distance": 157 }, { "from_id": "B", "to_id": "X", "distance": 182 }, { "from_id": "C", "to_id": "A", "distance": 166 }, { "from_id": "C", "to_id": "B", "distance": 111 }, { "from_id": "C", "to_id": "D", "distance": 145 }, { "from_id": "C", "to_id": "E", "distance": 193 }, { "from_id": "C", "to_id": "F", "distance": 114 }, { "from_id": "C", "to_id": "G", "distance": 139 }, { "from_id": "C", "to_id": "H", "distance": 165 }, { "from_id": "C", "to_id": "I", "distance": 133 }, { "from_id": "C", "to_id": "J", "distance": 127 }, { "from_id": "C", "to_id": "K", "distance": 109 }, { "from_id": "C", "to_id": "L", "distance": 182 }, { "from_id": "C", "to_id": "M", "distance": 178 }, { "from_id": "C", "to_id": "N", "distance": 122 }, { "from_id": "C", "to_id": "O", "distance": 174 }, { "from_id": "C", "to_id": "P", "distance": 144 }, { "from_id": "C", "to_id": "Q", "distance": 108 }, { "from_id": "C", "to_id": "R", "distance": 105 }, { "from_id": "C", "to_id": "S", "distance": 153 }, { "from_id": "C", "to_id": "T", "distance": 146 }, { "from_id": "C", "to_id": "U", "distance": 144 }, { "from_id": "C", "to_id": "V", "distance": 106 }, { "from_id": "C", "to_id": "W", "distance": 172 }, { "from_id": "C", "to_id": "X", "distance": 146 }, { "from_id": "D", "to_id": "A", "distance": 138 }, { "from_id": "D", "to_id": "B", "distance": 116 }, { "from_id": "D", "to_id": "C", "distance": 145 }, { "from_id": "D", "to_id": "E", "distance": 137 }, { "from_id": "D", "to_id": "F", "distance": 151 }, { "from_id": "D", "to_id": "G", "distance": 132 }, { "from_id": "D", "to_id": "H", "distance": 154 }, { "from_id": "D", "to_id": "I", "distance": 148 }, { "from_id": "D", "to_id": "J", "distance": 141 }, { "from_id": "D", "to_id": "K", "distance": 121 }, { "from_id": "D", "to_id": "L", "distance": 124 }, { "from_id": "D", "to_id": "M", "distance": 107 }, { "from_id": "D", "to_id": "N", "distance": 160 }, { "from_id": "D", "to_id": "O", "distance": 160 }, { "from_id": "D", "to_id": "P", "distance": 153 }, { "from_id": "D", "to_id": "Q", "distance": 91 }, 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148 }, { "from_id": "T", "to_id": "B", "distance": 127 }, { "from_id": "T", "to_id": "C", "distance": 146 }, { "from_id": "T", "to_id": "D", "distance": 125 }, { "from_id": "T", "to_id": "E", "distance": 118 }, { "from_id": "T", "to_id": "F", "distance": 126 }, { "from_id": "T", "to_id": "G", "distance": 110 }, { "from_id": "T", "to_id": "H", "distance": 102 }, { "from_id": "T", "to_id": "I", "distance": 138 }, { "from_id": "T", "to_id": "J", "distance": 143 }, { "from_id": "T", "to_id": "K", "distance": 116 }, { "from_id": "T", "to_id": "L", "distance": 130 }, { "from_id": "T", "to_id": "M", "distance": 137 }, { "from_id": "T", "to_id": "N", "distance": 144 }, { "from_id": "T", "to_id": "O", "distance": 134 }, { "from_id": "T", "to_id": "P", "distance": 162 }, { "from_id": "T", "to_id": "Q", "distance": 139 }, { "from_id": "T", "to_id": "R", "distance": 146 }, { "from_id": "T", "to_id": "S", "distance": 160 }, { "from_id": "T", "to_id": "U", "distance": 126 }, { "from_id": "T", 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"W", "to_id": "K", "distance": 125 }, { "from_id": "W", "to_id": "L", "distance": 99 }, { "from_id": "W", "to_id": "M", "distance": 119 }, { "from_id": "W", "to_id": "N", "distance": 124 }, { "from_id": "W", "to_id": "O", "distance": 160 }, { "from_id": "W", "to_id": "P", "distance": 149 }, { "from_id": "W", "to_id": "Q", "distance": 160 }, { "from_id": "W", "to_id": "R", "distance": 159 }, { "from_id": "W", "to_id": "S", "distance": 138 }, { "from_id": "W", "to_id": "T", "distance": 85 }, { "from_id": "W", "to_id": "U", "distance": 138 }, { "from_id": "W", "to_id": "V", "distance": 164 }, { "from_id": "W", "to_id": "X", "distance": 121 }, { "from_id": "X", "to_id": "A", "distance": 116 }, { "from_id": "X", "to_id": "B", "distance": 182 }, { "from_id": "X", "to_id": "C", "distance": 146 }, { "from_id": "X", "to_id": "D", "distance": 161 }, { "from_id": "X", "to_id": "E", "distance": 151 }, { "from_id": "X", "to_id": "F", "distance": 167 }, { "from_id": "X", "to_id": "G", "distance": 135 }, { "from_id": "X", "to_id": "H", "distance": 181 }, { "from_id": "X", "to_id": "I", "distance": 174 }, { "from_id": "X", "to_id": "J", "distance": 155 }, { "from_id": "X", "to_id": "K", "distance": 120 }, { "from_id": "X", "to_id": "L", "distance": 176 }, { "from_id": "X", "to_id": "M", "distance": 119 }, { "from_id": "X", "to_id": "N", "distance": 131 }, { "from_id": "X", "to_id": "O", "distance": 181 }, { "from_id": "X", "to_id": "P", "distance": 142 }, { "from_id": "X", "to_id": "Q", "distance": 155 }, { "from_id": "X", "to_id": "R", "distance": 141 }, { "from_id": "X", "to_id": "S", "distance": 131 }, { "from_id": "X", "to_id": "T", "distance": 141 }, { "from_id": "X", "to_id": "U", "distance": 147 }, { "from_id": "X", "to_id": "V", "distance": 183 }, { "from_id": "X", "to_id": "W", "distance": 121 } ] }, "solution_variant": [ "A", "B", "C", "E", "H", "J", "L", "M", "N", "O", "P", "S", "V", "X" ], "context_index": 9, "input_format": "nl", "input_index_base": "names" }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "We’re putting together the next cycle of books and the brief was simple: choose exactly m distinct titles that together cover the widest range of themes. There’s a chart that gives a “how different” number for each pair of books; to judge a lineup, add up those numbers for every pair inside the chosen group — a higher total means the picks are more diverse. No duplicate picks and the count has to be exactly m. The detailed titles and the pair differences are listed below.\n\n{\n \"total_titles\": 23,\n \"selection_size\": 9,\n \"title_ids\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\"\n ],\n \"data\": [\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"B\",\n \"thematic_distance\": 110\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"C\",\n \"thematic_distance\": 110\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"D\",\n \"thematic_distance\": 98\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"E\",\n \"thematic_distance\": 82\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"F\",\n \"thematic_distance\": 115\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"G\",\n \"thematic_distance\": 162\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"H\",\n \"thematic_distance\": 77\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"I\",\n \"thematic_distance\": 105\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"J\",\n \"thematic_distance\": 132\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"K\",\n \"thematic_distance\": 121\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"L\",\n \"thematic_distance\": 161\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"M\",\n \"thematic_distance\": 98\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"N\",\n \"thematic_distance\": 129\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"O\",\n \"thematic_distance\": 143\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"P\",\n \"thematic_distance\": 114\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"Q\",\n \"thematic_distance\": 125\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"R\",\n \"thematic_distance\": 102\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"S\",\n \"thematic_distance\": 93\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"T\",\n \"thematic_distance\": 129\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"U\",\n \"thematic_distance\": 105\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"V\",\n \"thematic_distance\": 126\n },\n {\n \"title_id_a\": \"A\",\n \"title_id_b\": \"W\",\n \"thematic_distance\": 143\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"A\",\n \"thematic_distance\": 110\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"C\",\n \"thematic_distance\": 116\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"D\",\n \"thematic_distance\": 128\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"E\",\n \"thematic_distance\": 140\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"F\",\n \"thematic_distance\": 68\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"G\",\n \"thematic_distance\": 126\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"H\",\n \"thematic_distance\": 136\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"I\",\n \"thematic_distance\": 143\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"J\",\n \"thematic_distance\": 135\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"K\",\n \"thematic_distance\": 124\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"L\",\n \"thematic_distance\": 106\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"M\",\n \"thematic_distance\": 132\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"N\",\n \"thematic_distance\": 150\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"O\",\n \"thematic_distance\": 148\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"P\",\n \"thematic_distance\": 90\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"Q\",\n \"thematic_distance\": 112\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"R\",\n \"thematic_distance\": 138\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"S\",\n \"thematic_distance\": 66\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"T\",\n \"thematic_distance\": 152\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"U\",\n \"thematic_distance\": 129\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"V\",\n \"thematic_distance\": 136\n },\n {\n \"title_id_a\": \"B\",\n \"title_id_b\": \"W\",\n \"thematic_distance\": 122\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"A\",\n \"thematic_distance\": 110\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"B\",\n \"thematic_distance\": 116\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"D\",\n \"thematic_distance\": 95\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"E\",\n \"thematic_distance\": 120\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"F\",\n \"thematic_distance\": 109\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"G\",\n \"thematic_distance\": 168\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"H\",\n \"thematic_distance\": 133\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"I\",\n \"thematic_distance\": 115\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"J\",\n \"thematic_distance\": 189\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"K\",\n \"thematic_distance\": 77\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"L\",\n \"thematic_distance\": 155\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"M\",\n \"thematic_distance\": 124\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"N\",\n \"thematic_distance\": 148\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"O\",\n \"thematic_distance\": 116\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"P\",\n \"thematic_distance\": 120\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"Q\",\n \"thematic_distance\": 144\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"R\",\n \"thematic_distance\": 137\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"S\",\n \"thematic_distance\": 101\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"T\",\n \"thematic_distance\": 141\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"U\",\n \"thematic_distance\": 126\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"V\",\n \"thematic_distance\": 143\n },\n {\n \"title_id_a\": \"C\",\n \"title_id_b\": \"W\",\n \"thematic_distance\": 123\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"A\",\n \"thematic_distance\": 98\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"B\",\n \"thematic_distance\": 128\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"C\",\n \"thematic_distance\": 95\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"E\",\n \"thematic_distance\": 77\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"F\",\n \"thematic_distance\": 139\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"G\",\n \"thematic_distance\": 132\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"H\",\n \"thematic_distance\": 81\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"I\",\n \"thematic_distance\": 104\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"J\",\n \"thematic_distance\": 165\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"K\",\n \"thematic_distance\": 106\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"L\",\n \"thematic_distance\": 135\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"M\",\n \"thematic_distance\": 97\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"N\",\n \"thematic_distance\": 134\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"O\",\n \"thematic_distance\": 114\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"P\",\n \"thematic_distance\": 121\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"Q\",\n \"thematic_distance\": 94\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"R\",\n \"thematic_distance\": 122\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"S\",\n \"thematic_distance\": 110\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"T\",\n \"thematic_distance\": 104\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"U\",\n \"thematic_distance\": 119\n },\n {\n \"title_id_a\": \"D\",\n \"title_id_b\": \"V\",\n 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},\n {\n \"title_id_a\": \"V\",\n \"title_id_b\": \"K\",\n \"thematic_distance\": 115\n },\n {\n \"title_id_a\": \"V\",\n \"title_id_b\": \"L\",\n \"thematic_distance\": 138\n },\n {\n \"title_id_a\": \"V\",\n \"title_id_b\": \"M\",\n \"thematic_distance\": 152\n },\n {\n \"title_id_a\": \"V\",\n \"title_id_b\": \"N\",\n \"thematic_distance\": 126\n },\n {\n \"title_id_a\": \"V\",\n \"title_id_b\": \"O\",\n \"thematic_distance\": 146\n },\n {\n \"title_id_a\": \"V\",\n \"title_id_b\": \"P\",\n \"thematic_distance\": 100\n },\n {\n \"title_id_a\": \"V\",\n \"title_id_b\": \"Q\",\n \"thematic_distance\": 121\n },\n {\n \"title_id_a\": \"V\",\n \"title_id_b\": \"R\",\n \"thematic_distance\": 89\n },\n {\n \"title_id_a\": \"V\",\n \"title_id_b\": \"S\",\n \"thematic_distance\": 143\n },\n {\n \"title_id_a\": \"V\",\n \"title_id_b\": \"T\",\n \"thematic_distance\": 163\n },\n {\n \"title_id_a\": \"V\",\n \"title_id_b\": \"U\",\n \"thematic_distance\": 127\n },\n {\n \"title_id_a\": \"V\",\n \"title_id_b\": \"W\",\n \"thematic_distance\": 115\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"A\",\n \"thematic_distance\": 143\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"B\",\n \"thematic_distance\": 122\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"C\",\n \"thematic_distance\": 123\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"D\",\n \"thematic_distance\": 110\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"E\",\n \"thematic_distance\": 162\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"F\",\n \"thematic_distance\": 130\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"G\",\n \"thematic_distance\": 168\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"H\",\n \"thematic_distance\": 161\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"I\",\n \"thematic_distance\": 169\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"J\",\n \"thematic_distance\": 185\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"K\",\n \"thematic_distance\": 132\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"L\",\n \"thematic_distance\": 124\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"M\",\n \"thematic_distance\": 152\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"N\",\n \"thematic_distance\": 171\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"O\",\n \"thematic_distance\": 159\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"P\",\n \"thematic_distance\": 121\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"Q\",\n \"thematic_distance\": 88\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"R\",\n \"thematic_distance\": 136\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"S\",\n \"thematic_distance\": 149\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"T\",\n \"thematic_distance\": 173\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"U\",\n \"thematic_distance\": 153\n },\n {\n \"title_id_a\": \"W\",\n \"title_id_b\": \"V\",\n \"thematic_distance\": 115\n }\n ]\n}\n\nWhen you hand me the final picks, just drop them into this simple JSON layout so everything is clear and machine-friendly:\n\n{\n \"solution\": [, , ...]\n}\n\nA quick, informal note on that: \"solution\" is the list where you put the identifiers of the chosen titles — one identifier per selected book. Treat the block like a little form: the array holds the exact items you're picking. This is just a sketch of the expected shape, not the actual answer.\n\nPlease use identifiers exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 110, 110, 98, 82, 115, 162, 77, 105, 132, 121, 161, 98, 129, 143, 114, 125, 102, 93, 129, 105, 126, 143 ], [ 110, 0, 116, 128, 140, 68, 126, 136, 143, 135, 124, 106, 132, 150, 148, 90, 112, 138, 66, 152, 129, 136, 122 ], [ 110, 116, 0, 95, 120, 109, 168, 133, 115, 189, 77, 155, 124, 148, 116, 120, 144, 137, 101, 141, 126, 143, 123 ], [ 98, 128, 95, 0, 77, 139, 132, 81, 104, 165, 106, 135, 97, 134, 114, 121, 94, 122, 110, 104, 119, 143, 110 ], [ 82, 140, 120, 77, 0, 132, 126, 64, 107, 143, 107, 152, 77, 123, 113, 114, 123, 109, 113, 68, 91, 145, 162 ], [ 115, 68, 109, 139, 132, 0, 131, 148, 148, 139, 99, 107, 133, 144, 126, 52, 121, 115, 93, 134, 95, 112, 130 ], [ 162, 126, 168, 132, 126, 131, 0, 142, 145, 166, 152, 90, 138, 161, 112, 122, 109, 157, 119, 96, 135, 163, 168 ], [ 77, 136, 133, 81, 64, 148, 142, 0, 109, 148, 140, 160, 95, 161, 136, 134, 132, 138, 118, 97, 134, 166, 161 ], [ 105, 143, 115, 104, 107, 148, 145, 109, 0, 205, 149, 170, 139, 161, 99, 159, 143, 155, 98, 122, 133, 163, 169 ], [ 132, 135, 189, 165, 143, 139, 166, 148, 205, 0, 154, 145, 144, 123, 194, 108, 154, 113, 142, 169, 148, 152, 185 ], [ 121, 124, 77, 106, 107, 99, 152, 140, 149, 154, 0, 145, 125, 116, 124, 86, 143, 116, 114, 132, 114, 115, 132 ], [ 161, 106, 155, 135, 152, 107, 90, 160, 170, 145, 145, 0, 137, 152, 120, 93, 88, 126, 125, 129, 133, 138, 124 ], [ 98, 132, 124, 97, 77, 133, 138, 95, 139, 144, 125, 137, 0, 134, 132, 122, 120, 104, 122, 92, 103, 152, 152 ], [ 129, 150, 148, 134, 123, 144, 161, 161, 161, 123, 116, 152, 134, 0, 153, 124, 142, 87, 120, 158, 116, 126, 171 ], [ 143, 148, 116, 114, 113, 126, 112, 136, 99, 194, 124, 120, 132, 153, 0, 125, 133, 132, 120, 84, 117, 146, 159 ], [ 114, 90, 120, 121, 114, 52, 122, 134, 159, 108, 86, 93, 122, 124, 125, 0, 104, 89, 109, 120, 88, 100, 121 ], [ 125, 112, 144, 94, 123, 121, 109, 132, 143, 154, 143, 88, 120, 142, 133, 104, 0, 110, 125, 124, 108, 121, 88 ], [ 102, 138, 137, 122, 109, 115, 157, 138, 155, 113, 116, 126, 104, 87, 132, 89, 110, 0, 132, 130, 83, 89, 136 ], [ 93, 66, 101, 110, 113, 93, 119, 118, 98, 142, 114, 125, 122, 120, 120, 109, 125, 132, 0, 133, 119, 143, 149 ], [ 129, 152, 141, 104, 68, 134, 96, 97, 122, 169, 132, 129, 92, 158, 84, 120, 124, 130, 133, 0, 100, 163, 173 ], [ 105, 129, 126, 119, 91, 95, 135, 134, 133, 148, 114, 133, 103, 116, 117, 88, 108, 83, 119, 100, 0, 127, 153 ], [ 126, 136, 143, 143, 145, 112, 163, 166, 163, 152, 115, 138, 152, 126, 146, 100, 121, 89, 143, 163, 127, 0, 115 ], [ 143, 122, 123, 110, 162, 130, 168, 161, 169, 185, 132, 124, 152, 171, 159, 121, 88, 136, 149, 173, 153, 115, 0 ] ], "m": 9, "objective": 5413.0 }, "solution": [ 2, 6, 7, 8, 9, 11, 13, 21, 22 ], "obj": 5413.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 23, "m": 9, "nodes": [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S", "T", "U", "V", "W" ], "distance_pairs": [ { "from_id": "A", "to_id": "B", "distance": 110 }, { "from_id": "A", "to_id": "C", "distance": 110 }, { "from_id": "A", "to_id": "D", "distance": 98 }, { "from_id": "A", "to_id": "E", "distance": 82 }, { "from_id": "A", "to_id": "F", "distance": 115 }, { "from_id": "A", "to_id": "G", "distance": 162 }, { "from_id": "A", "to_id": "H", "distance": 77 }, { "from_id": "A", "to_id": "I", "distance": 105 }, { "from_id": "A", "to_id": "J", "distance": 132 }, { "from_id": "A", "to_id": "K", "distance": 121 }, { "from_id": "A", "to_id": "L", "distance": 161 }, { "from_id": "A", "to_id": "M", "distance": 98 }, { "from_id": "A", "to_id": "N", "distance": 129 }, { "from_id": "A", "to_id": "O", "distance": 143 }, { "from_id": "A", "to_id": "P", "distance": 114 }, { "from_id": "A", "to_id": "Q", "distance": 125 }, { "from_id": "A", "to_id": "R", "distance": 102 }, { "from_id": "A", "to_id": "S", "distance": 93 }, { "from_id": "A", "to_id": "T", "distance": 129 }, { "from_id": "A", "to_id": "U", "distance": 105 }, { "from_id": "A", "to_id": "V", "distance": 126 }, { "from_id": "A", "to_id": "W", "distance": 143 }, { "from_id": "B", "to_id": "A", "distance": 110 }, { "from_id": "B", "to_id": "C", "distance": 116 }, { "from_id": "B", "to_id": "D", "distance": 128 }, { "from_id": "B", "to_id": "E", "distance": 140 }, { "from_id": "B", "to_id": "F", "distance": 68 }, { "from_id": "B", "to_id": "G", "distance": 126 }, { "from_id": "B", "to_id": "H", "distance": 136 }, { "from_id": "B", "to_id": "I", "distance": 143 }, { "from_id": "B", "to_id": "J", "distance": 135 }, { "from_id": "B", "to_id": "K", "distance": 124 }, { "from_id": "B", "to_id": "L", "distance": 106 }, { "from_id": "B", "to_id": "M", "distance": 132 }, { "from_id": "B", "to_id": "N", "distance": 150 }, { "from_id": "B", "to_id": "O", "distance": 148 }, { "from_id": "B", "to_id": "P", "distance": 90 }, { "from_id": "B", "to_id": "Q", "distance": 112 }, { "from_id": "B", "to_id": "R", "distance": 138 }, { "from_id": "B", "to_id": "S", "distance": 66 }, { "from_id": "B", "to_id": "T", "distance": 152 }, { "from_id": "B", "to_id": "U", "distance": 129 }, { "from_id": "B", "to_id": "V", "distance": 136 }, { "from_id": "B", "to_id": "W", "distance": 122 }, { "from_id": "C", "to_id": "A", "distance": 110 }, { "from_id": "C", "to_id": "B", "distance": 116 }, { "from_id": "C", "to_id": "D", "distance": 95 }, { "from_id": "C", "to_id": "E", "distance": 120 }, { "from_id": "C", "to_id": "F", "distance": 109 }, { "from_id": "C", "to_id": "G", "distance": 168 }, { "from_id": "C", "to_id": "H", "distance": 133 }, { "from_id": "C", "to_id": "I", "distance": 115 }, { "from_id": "C", "to_id": "J", "distance": 189 }, { "from_id": "C", "to_id": "K", "distance": 77 }, { "from_id": "C", "to_id": "L", "distance": 155 }, { "from_id": "C", "to_id": "M", "distance": 124 }, { "from_id": "C", "to_id": "N", "distance": 148 }, { "from_id": "C", "to_id": "O", "distance": 116 }, { "from_id": "C", "to_id": "P", "distance": 120 }, { "from_id": "C", "to_id": "Q", "distance": 144 }, { "from_id": "C", "to_id": "R", "distance": 137 }, { "from_id": "C", "to_id": "S", "distance": 101 }, { "from_id": "C", "to_id": "T", "distance": 141 }, { "from_id": "C", "to_id": "U", "distance": 126 }, { "from_id": "C", "to_id": "V", "distance": 143 }, { "from_id": "C", "to_id": "W", "distance": 123 }, { "from_id": "D", "to_id": "A", "distance": 98 }, { "from_id": "D", "to_id": "B", "distance": 128 }, { "from_id": "D", "to_id": "C", "distance": 95 }, { "from_id": "D", "to_id": "E", "distance": 77 }, { "from_id": "D", "to_id": "F", "distance": 139 }, { "from_id": "D", "to_id": "G", "distance": 132 }, { "from_id": "D", "to_id": "H", "distance": 81 }, { "from_id": "D", "to_id": "I", "distance": 104 }, { "from_id": "D", "to_id": "J", "distance": 165 }, { "from_id": "D", "to_id": "K", "distance": 106 }, { "from_id": "D", "to_id": "L", "distance": 135 }, { "from_id": "D", "to_id": "M", "distance": 97 }, { "from_id": "D", "to_id": "N", "distance": 134 }, { "from_id": "D", "to_id": "O", "distance": 114 }, { "from_id": "D", "to_id": "P", "distance": 121 }, { "from_id": "D", "to_id": "Q", "distance": 94 }, { "from_id": "D", "to_id": "R", "distance": 122 }, { "from_id": "D", "to_id": "S", "distance": 110 }, { "from_id": "D", "to_id": "T", "distance": 104 }, { "from_id": "D", "to_id": "U", "distance": 119 }, { "from_id": "D", "to_id": "V", "distance": 143 }, { "from_id": "D", "to_id": "W", "distance": 110 }, { "from_id": "E", "to_id": "A", "distance": 82 }, { "from_id": "E", "to_id": "B", "distance": 140 }, { "from_id": "E", "to_id": "C", "distance": 120 }, { "from_id": "E", "to_id": "D", "distance": 77 }, { "from_id": "E", "to_id": "F", "distance": 132 }, { "from_id": "E", "to_id": "G", "distance": 126 }, { "from_id": "E", "to_id": "H", "distance": 64 }, { "from_id": "E", "to_id": "I", "distance": 107 }, { "from_id": "E", "to_id": "J", "distance": 143 }, { "from_id": "E", "to_id": "K", "distance": 107 }, { "from_id": "E", "to_id": "L", "distance": 152 }, { "from_id": "E", "to_id": "M", "distance": 77 }, { "from_id": "E", "to_id": "N", "distance": 123 }, { "from_id": "E", "to_id": "O", "distance": 113 }, { "from_id": "E", "to_id": "P", "distance": 114 }, { "from_id": "E", "to_id": "Q", "distance": 123 }, { "from_id": "E", "to_id": "R", "distance": 109 }, { "from_id": "E", "to_id": "S", "distance": 113 }, { "from_id": "E", "to_id": "T", "distance": 68 }, { "from_id": "E", "to_id": "U", "distance": 91 }, { "from_id": "E", "to_id": "V", "distance": 145 }, { "from_id": "E", "to_id": "W", "distance": 162 }, { "from_id": "F", "to_id": "A", "distance": 115 }, { "from_id": "F", "to_id": "B", "distance": 68 }, { "from_id": "F", "to_id": "C", "distance": 109 }, { "from_id": "F", "to_id": "D", "distance": 139 }, { "from_id": "F", "to_id": "E", "distance": 132 }, { "from_id": "F", "to_id": "G", "distance": 131 }, { "from_id": "F", "to_id": "H", "distance": 148 }, { "from_id": "F", "to_id": "I", "distance": 148 }, { "from_id": "F", "to_id": "J", "distance": 139 }, { "from_id": "F", "to_id": "K", "distance": 99 }, { "from_id": "F", "to_id": "L", "distance": 107 }, { "from_id": "F", "to_id": "M", "distance": 133 }, { "from_id": "F", "to_id": "N", "distance": 144 }, { "from_id": "F", "to_id": "O", "distance": 126 }, { "from_id": "F", "to_id": "P", "distance": 52 }, { "from_id": "F", "to_id": "Q", "distance": 121 }, { "from_id": "F", "to_id": "R", "distance": 115 }, { "from_id": "F", "to_id": "S", "distance": 93 }, { "from_id": "F", "to_id": "T", "distance": 134 }, { "from_id": "F", "to_id": "U", "distance": 95 }, { "from_id": "F", "to_id": "V", "distance": 112 }, { "from_id": "F", "to_id": "W", "distance": 130 }, { "from_id": "G", "to_id": "A", "distance": 162 }, { "from_id": "G", "to_id": "B", "distance": 126 }, { "from_id": "G", "to_id": "C", "distance": 168 }, { "from_id": "G", "to_id": "D", "distance": 132 }, { "from_id": "G", "to_id": "E", "distance": 126 }, { "from_id": "G", "to_id": "F", "distance": 131 }, { "from_id": "G", "to_id": "H", "distance": 142 }, { "from_id": "G", "to_id": "I", "distance": 145 }, { "from_id": "G", "to_id": "J", "distance": 166 }, { "from_id": "G", "to_id": "K", "distance": 152 }, { "from_id": "G", "to_id": "L", "distance": 90 }, { "from_id": "G", "to_id": "M", "distance": 138 }, { "from_id": "G", "to_id": "N", "distance": 161 }, { "from_id": "G", "to_id": "O", "distance": 112 }, { "from_id": "G", "to_id": "P", "distance": 122 }, { "from_id": "G", "to_id": "Q", "distance": 109 }, { "from_id": "G", "to_id": "R", "distance": 157 }, { "from_id": "G", "to_id": "S", "distance": 119 }, { "from_id": "G", "to_id": "T", "distance": 96 }, 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"to_id": "U", "distance": 153 }, { "from_id": "W", "to_id": "V", "distance": 115 } ] }, "solution_variant": [ "C", "G", "H", "I", "J", "L", "N", "V", "W" ], "context_index": 10, "input_format": "json", "input_index_base": "names" }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Recently the hiring lead asked for a final-interview list of exactly m candidates and gave a matrix of how much each pair differs in background and skills — the aim was a lineup that’s as spread out as possible. The simple check for a lineup’s variety is to add every pair’s difference inside that lineup; the larger the sum, the better the spread. The selection must include precisely m different people. The specific candidates and their pairwise dissimilarities are provided below.\n\n# total_candidates=20\n# finalist_count=16\n# candidate_ids=A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T\ncandidate_id_a,candidate_id_b,dissimilarity_score\nA,B,81\nA,C,110\nA,D,106\nA,E,113\nA,F,123\nA,G,88\nA,H,57\nA,I,69\nA,J,55\nA,K,120\nA,L,92\nA,M,80\nA,N,122\nA,O,83\nA,P,100\nA,Q,120\nA,R,109\nA,S,108\nA,T,70\nB,A,81\nB,C,126\nB,D,109\nB,E,105\nB,F,87\nB,G,91\nB,H,67\nB,I,107\nB,J,68\nB,K,111\nB,L,53\nB,M,109\nB,N,92\nB,O,107\nB,P,139\nB,Q,106\nB,R,112\nB,S,102\nB,T,102\nC,A,110\nC,B,126\nC,D,112\nC,E,93\nC,F,94\nC,G,76\nC,H,125\nC,I,94\nC,J,104\nC,K,46\nC,L,109\nC,M,91\nC,N,95\nC,O,78\nC,P,101\nC,Q,70\nC,R,52\nC,S,58\nC,T,83\nD,A,106\nD,B,109\nD,C,112\nD,E,87\nD,F,92\nD,G,134\nD,H,133\nD,I,116\nD,J,106\nD,K,107\nD,L,100\nD,M,59\nD,N,114\nD,O,71\nD,P,57\nD,Q,102\nD,R,106\nD,S,114\nD,T,82\nE,A,113\nE,B,105\nE,C,93\nE,D,87\nE,F,85\nE,G,106\nE,H,127\nE,I,94\nE,J,83\nE,K,97\nE,L,78\nE,M,81\nE,N,96\nE,O,53\nE,P,109\nE,Q,103\nE,R,48\nE,S,70\nE,T,111\nF,A,123\nF,B,87\nF,C,94\nF,D,92\nF,E,85\nF,G,93\nF,H,121\nF,I,107\nF,J,112\nF,K,60\nF,L,54\nF,M,114\nF,N,67\nF,O,103\nF,P,128\nF,Q,64\nF,R,83\nF,S,62\nF,T,89\nG,A,88\nG,B,91\nG,C,76\nG,D,134\nG,E,106\nG,F,93\nG,H,97\nG,I,66\nG,J,94\nG,K,70\nG,L,87\nG,M,109\nG,N,67\nG,O,98\nG,P,139\nG,Q,71\nG,R,79\nG,S,64\nG,T,89\nH,A,57\nH,B,67\nH,C,125\nH,D,133\nH,E,127\nH,F,121\nH,G,97\nH,I,101\nH,J,58\nH,K,128\nH,L,80\nH,M,122\nH,N,135\nH,O,118\nH,P,138\nH,Q,136\nH,R,120\nH,S,109\nH,T,95\nI,A,69\nI,B,107\nI,C,94\nI,D,116\nI,E,94\nI,F,107\nI,G,66\nI,H,101\nI,J,90\nI,K,100\nI,L,90\nI,M,91\nI,N,99\nI,O,78\nI,P,113\nI,Q,101\nI,R,85\nI,S,78\nI,T,73\nJ,A,55\nJ,B,68\nJ,C,104\nJ,D,106\nJ,E,83\nJ,F,112\nJ,G,94\nJ,H,58\nJ,I,90\nJ,K,116\nJ,L,76\nJ,M,82\nJ,N,118\nJ,O,73\nJ,P,111\nJ,Q,122\nJ,R,85\nJ,S,94\nJ,T,98\nK,A,120\nK,B,111\nK,C,46\nK,D,107\nK,E,97\nK,F,60\nK,G,70\nK,H,128\nK,I,100\nK,J,116\nK,L,91\nK,M,105\nK,N,66\nK,O,95\nK,P,117\nK,Q,37\nK,R,66\nK,S,52\nK,T,80\nL,A,92\nL,B,53\nL,C,109\nL,D,100\nL,E,78\nL,F,54\nL,G,87\nL,H,80\nL,I,90\nL,J,76\nL,K,91\nL,M,110\nL,N,88\nL,O,97\nL,P,131\nL,Q,97\nL,R,85\nL,S,67\nL,T,89\nM,A,80\nM,B,109\nM,C,91\nM,D,59\nM,E,81\nM,F,114\nM,G,109\nM,H,122\nM,I,91\nM,J,82\nM,K,105\nM,L,110\nM,N,106\nM,O,33\nM,P,53\nM,Q,98\nM,R,87\nM,S,108\nM,T,84\nN,A,122\nN,B,92\nN,C,95\nN,D,114\nN,E,96\nN,F,67\nN,G,67\nN,H,135\nN,I,99\nN,J,118\nN,K,66\nN,L,88\nN,M,106\nN,O,99\nN,P,141\nN,Q,42\nN,R,87\nN,S,79\nN,T,108\nO,A,83\nO,B,107\nO,C,78\nO,D,71\nO,E,53\nO,F,103\nO,G,98\nO,H,118\nO,I,78\nO,J,73\nO,K,95\nO,L,97\nO,M,33\nO,N,99\nO,P,70\nO,Q,96\nO,R,59\nO,S,85\nO,T,88\nP,A,100\nP,B,139\nP,C,101\nP,D,57\nP,E,109\nP,F,128\nP,G,139\nP,H,138\nP,I,113\nP,J,111\nP,K,117\nP,L,131\nP,M,53\nP,N,141\nP,O,70\nP,Q,119\nP,R,111\nP,S,125\nP,T,77\nQ,A,120\nQ,B,106\nQ,C,70\nQ,D,102\nQ,E,103\nQ,F,64\nQ,G,71\nQ,H,136\nQ,I,101\nQ,J,122\nQ,K,37\nQ,L,97\nQ,M,98\nQ,N,42\nQ,O,96\nQ,P,119\nQ,R,84\nQ,S,75\nQ,T,86\nR,A,109\nR,B,112\nR,C,52\nR,D,106\nR,E,48\nR,F,83\nR,G,79\nR,H,120\nR,I,85\nR,J,85\nR,K,66\nR,L,85\nR,M,87\nR,N,87\nR,O,59\nR,P,111\nR,Q,84\nR,S,39\nR,T,99\nS,A,108\nS,B,102\nS,C,58\nS,D,114\nS,E,70\nS,F,62\nS,G,64\nS,H,109\nS,I,78\nS,J,94\nS,K,52\nS,L,67\nS,M,108\nS,N,79\nS,O,85\nS,P,125\nS,Q,75\nS,R,39\nS,T,87\nT,A,70\nT,B,102\nT,C,83\nT,D,82\nT,E,111\nT,F,89\nT,G,89\nT,H,95\nT,I,73\nT,J,98\nT,K,80\nT,L,89\nT,M,84\nT,N,108\nT,O,88\nT,P,77\nT,Q,86\nT,R,99\nT,S,87\n\nIf you want to hand me the final shortlist, just use the tiny JSON layout below — super simple and easy for me to read.\n\n{\n \"solution\": [, , ...]\n}\n\nThis shows the shape I need: \"solution\" is an array containing the exact candidate IDs you choose for the final interview lineup (exactly m of them). Think of it like filling in a short form — list the people you picked in that array. This JSON block is only a sketch of the expected shape, not the actual answer.\n\nPlease use the identifiers exactly as they appear in the instance input — do not rename them or invent new labels. For example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 81, 110, 106, 113, 123, 88, 57, 69, 55, 120, 92, 80, 122, 83, 100, 120, 109, 108, 70 ], [ 81, 0, 126, 109, 105, 87, 91, 67, 107, 68, 111, 53, 109, 92, 107, 139, 106, 112, 102, 102 ], [ 110, 126, 0, 112, 93, 94, 76, 125, 94, 104, 46, 109, 91, 95, 78, 101, 70, 52, 58, 83 ], [ 106, 109, 112, 0, 87, 92, 134, 133, 116, 106, 107, 100, 59, 114, 71, 57, 102, 106, 114, 82 ], [ 113, 105, 93, 87, 0, 85, 106, 127, 94, 83, 97, 78, 81, 96, 53, 109, 103, 48, 70, 111 ], [ 123, 87, 94, 92, 85, 0, 93, 121, 107, 112, 60, 54, 114, 67, 103, 128, 64, 83, 62, 89 ], [ 88, 91, 76, 134, 106, 93, 0, 97, 66, 94, 70, 87, 109, 67, 98, 139, 71, 79, 64, 89 ], [ 57, 67, 125, 133, 127, 121, 97, 0, 101, 58, 128, 80, 122, 135, 118, 138, 136, 120, 109, 95 ], [ 69, 107, 94, 116, 94, 107, 66, 101, 0, 90, 100, 90, 91, 99, 78, 113, 101, 85, 78, 73 ], [ 55, 68, 104, 106, 83, 112, 94, 58, 90, 0, 116, 76, 82, 118, 73, 111, 122, 85, 94, 98 ], [ 120, 111, 46, 107, 97, 60, 70, 128, 100, 116, 0, 91, 105, 66, 95, 117, 37, 66, 52, 80 ], [ 92, 53, 109, 100, 78, 54, 87, 80, 90, 76, 91, 0, 110, 88, 97, 131, 97, 85, 67, 89 ], [ 80, 109, 91, 59, 81, 114, 109, 122, 91, 82, 105, 110, 0, 106, 33, 53, 98, 87, 108, 84 ], [ 122, 92, 95, 114, 96, 67, 67, 135, 99, 118, 66, 88, 106, 0, 99, 141, 42, 87, 79, 108 ], [ 83, 107, 78, 71, 53, 103, 98, 118, 78, 73, 95, 97, 33, 99, 0, 70, 96, 59, 85, 88 ], [ 100, 139, 101, 57, 109, 128, 139, 138, 113, 111, 117, 131, 53, 141, 70, 0, 119, 111, 125, 77 ], [ 120, 106, 70, 102, 103, 64, 71, 136, 101, 122, 37, 97, 98, 42, 96, 119, 0, 84, 75, 86 ], [ 109, 112, 52, 106, 48, 83, 79, 120, 85, 85, 66, 85, 87, 87, 59, 111, 84, 0, 39, 99 ], [ 108, 102, 58, 114, 70, 62, 64, 109, 78, 94, 52, 67, 108, 79, 85, 125, 75, 39, 0, 87 ], [ 70, 102, 83, 82, 111, 89, 89, 95, 73, 98, 80, 89, 84, 108, 88, 77, 86, 99, 87, 0 ] ], "m": 16, "objective": 11610.0 }, "solution": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16 ], "obj": 11610.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 20, "m": 16, "nodes": [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S", "T" ], "distance_pairs": [ { "from_id": "A", "to_id": "B", "distance": 81 }, { "from_id": "A", "to_id": "C", "distance": 110 }, { "from_id": "A", "to_id": "D", "distance": 106 }, { "from_id": "A", "to_id": "E", "distance": 113 }, { "from_id": "A", "to_id": "F", "distance": 123 }, { "from_id": "A", "to_id": "G", "distance": 88 }, { "from_id": "A", "to_id": "H", "distance": 57 }, { "from_id": "A", "to_id": "I", "distance": 69 }, { "from_id": "A", "to_id": "J", "distance": 55 }, { "from_id": "A", "to_id": "K", "distance": 120 }, { "from_id": "A", "to_id": "L", "distance": 92 }, { "from_id": "A", "to_id": "M", "distance": 80 }, { "from_id": "A", "to_id": "N", "distance": 122 }, { "from_id": "A", "to_id": "O", "distance": 83 }, { "from_id": "A", "to_id": "P", "distance": 100 }, { "from_id": "A", "to_id": "Q", "distance": 120 }, { "from_id": "A", "to_id": "R", "distance": 109 }, { "from_id": "A", "to_id": "S", "distance": 108 }, { "from_id": "A", "to_id": "T", "distance": 70 }, { "from_id": "B", "to_id": "A", "distance": 81 }, { "from_id": "B", "to_id": "C", "distance": 126 }, { "from_id": "B", "to_id": "D", "distance": 109 }, { "from_id": "B", "to_id": "E", "distance": 105 }, { "from_id": "B", "to_id": "F", "distance": 87 }, { "from_id": "B", "to_id": "G", "distance": 91 }, { "from_id": "B", "to_id": "H", "distance": 67 }, { "from_id": "B", "to_id": "I", "distance": 107 }, { "from_id": "B", "to_id": "J", "distance": 68 }, { "from_id": "B", "to_id": "K", "distance": 111 }, { "from_id": "B", "to_id": "L", "distance": 53 }, { "from_id": "B", "to_id": "M", "distance": 109 }, { "from_id": "B", "to_id": "N", "distance": 92 }, { "from_id": "B", "to_id": "O", "distance": 107 }, { "from_id": "B", "to_id": "P", "distance": 139 }, { "from_id": "B", "to_id": "Q", "distance": 106 }, { "from_id": "B", "to_id": "R", "distance": 112 }, { "from_id": "B", "to_id": "S", "distance": 102 }, { "from_id": "B", "to_id": "T", "distance": 102 }, { "from_id": "C", "to_id": "A", "distance": 110 }, { "from_id": "C", "to_id": "B", "distance": 126 }, { "from_id": "C", "to_id": "D", "distance": 112 }, { "from_id": "C", "to_id": "E", "distance": 93 }, { "from_id": "C", "to_id": "F", "distance": 94 }, { "from_id": "C", "to_id": "G", "distance": 76 }, { "from_id": "C", "to_id": "H", "distance": 125 }, { "from_id": "C", "to_id": "I", "distance": 94 }, { "from_id": "C", "to_id": "J", "distance": 104 }, { "from_id": "C", "to_id": "K", "distance": 46 }, { "from_id": "C", "to_id": "L", "distance": 109 }, { "from_id": "C", "to_id": "M", "distance": 91 }, { "from_id": "C", "to_id": "N", "distance": 95 }, { "from_id": "C", "to_id": "O", "distance": 78 }, { "from_id": "C", "to_id": "P", "distance": 101 }, { "from_id": "C", "to_id": "Q", "distance": 70 }, { "from_id": "C", "to_id": "R", "distance": 52 }, { "from_id": "C", "to_id": "S", "distance": 58 }, { "from_id": "C", "to_id": "T", "distance": 83 }, { "from_id": "D", "to_id": "A", "distance": 106 }, { "from_id": "D", "to_id": "B", "distance": 109 }, { "from_id": "D", "to_id": "C", "distance": 112 }, { "from_id": "D", "to_id": "E", "distance": 87 }, { "from_id": "D", "to_id": "F", "distance": 92 }, { "from_id": "D", "to_id": "G", "distance": 134 }, { "from_id": "D", "to_id": "H", "distance": 133 }, { "from_id": "D", "to_id": "I", "distance": 116 }, { "from_id": "D", "to_id": "J", "distance": 106 }, { "from_id": "D", "to_id": "K", "distance": 107 }, { "from_id": "D", "to_id": "L", "distance": 100 }, { "from_id": "D", "to_id": "M", "distance": 59 }, { "from_id": "D", "to_id": "N", "distance": 114 }, { "from_id": "D", "to_id": "O", "distance": 71 }, { "from_id": "D", "to_id": "P", "distance": 57 }, { "from_id": "D", "to_id": "Q", "distance": 102 }, { "from_id": "D", "to_id": "R", "distance": 106 }, { "from_id": "D", "to_id": "S", "distance": 114 }, { "from_id": "D", "to_id": "T", "distance": 82 }, { "from_id": "E", "to_id": "A", "distance": 113 }, { "from_id": "E", "to_id": "B", "distance": 105 }, { "from_id": "E", "to_id": "C", "distance": 93 }, { "from_id": "E", "to_id": "D", "distance": 87 }, { "from_id": "E", "to_id": "F", "distance": 85 }, { "from_id": "E", "to_id": "G", "distance": 106 }, { "from_id": "E", "to_id": "H", "distance": 127 }, { "from_id": "E", "to_id": "I", "distance": 94 }, { "from_id": "E", "to_id": "J", "distance": 83 }, { "from_id": "E", "to_id": "K", "distance": 97 }, { "from_id": "E", "to_id": "L", "distance": 78 }, { "from_id": "E", "to_id": "M", "distance": 81 }, { "from_id": "E", "to_id": "N", "distance": 96 }, { "from_id": "E", "to_id": "O", "distance": 53 }, { "from_id": "E", "to_id": "P", "distance": 109 }, { "from_id": "E", "to_id": "Q", "distance": 103 }, { "from_id": "E", "to_id": "R", "distance": 48 }, { "from_id": "E", "to_id": "S", "distance": 70 }, { "from_id": "E", "to_id": "T", "distance": 111 }, { "from_id": "F", "to_id": "A", "distance": 123 }, { "from_id": "F", "to_id": "B", "distance": 87 }, { "from_id": "F", "to_id": "C", "distance": 94 }, { "from_id": "F", "to_id": "D", "distance": 92 }, { "from_id": "F", "to_id": "E", "distance": 85 }, { "from_id": "F", "to_id": "G", "distance": 93 }, { "from_id": "F", "to_id": "H", "distance": 121 }, { "from_id": "F", "to_id": "I", "distance": 107 }, { "from_id": "F", "to_id": "J", "distance": 112 }, { "from_id": "F", "to_id": "K", "distance": 60 }, { "from_id": "F", "to_id": "L", "distance": 54 }, { "from_id": "F", "to_id": "M", "distance": 114 }, { "from_id": "F", "to_id": "N", "distance": 67 }, { "from_id": "F", "to_id": "O", "distance": 103 }, { "from_id": "F", "to_id": "P", "distance": 128 }, { "from_id": "F", "to_id": "Q", "distance": 64 }, { "from_id": "F", "to_id": "R", "distance": 83 }, { "from_id": "F", "to_id": "S", "distance": 62 }, { "from_id": "F", "to_id": "T", "distance": 89 }, { "from_id": "G", "to_id": "A", "distance": 88 }, { "from_id": "G", "to_id": "B", "distance": 91 }, { "from_id": "G", "to_id": "C", "distance": 76 }, { "from_id": "G", "to_id": "D", "distance": 134 }, { "from_id": "G", "to_id": "E", "distance": 106 }, { "from_id": "G", "to_id": "F", "distance": 93 }, { "from_id": "G", "to_id": "H", "distance": 97 }, { "from_id": "G", "to_id": "I", "distance": 66 }, { "from_id": "G", "to_id": "J", "distance": 94 }, { "from_id": "G", "to_id": "K", "distance": 70 }, { "from_id": "G", "to_id": "L", "distance": 87 }, { "from_id": "G", "to_id": "M", "distance": 109 }, { "from_id": "G", "to_id": "N", "distance": 67 }, { "from_id": "G", "to_id": "O", "distance": 98 }, { "from_id": "G", "to_id": "P", "distance": 139 }, { "from_id": "G", "to_id": "Q", "distance": 71 }, { "from_id": "G", "to_id": "R", "distance": 79 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"distance": 109 }, { "from_id": "H", "to_id": "T", "distance": 95 }, { "from_id": "I", "to_id": "A", "distance": 69 }, { "from_id": "I", "to_id": "B", "distance": 107 }, { "from_id": "I", "to_id": "C", "distance": 94 }, { "from_id": "I", "to_id": "D", "distance": 116 }, { "from_id": "I", "to_id": "E", "distance": 94 }, { "from_id": "I", "to_id": "F", "distance": 107 }, { "from_id": "I", "to_id": "G", "distance": 66 }, { "from_id": "I", "to_id": "H", "distance": 101 }, { "from_id": "I", "to_id": "J", "distance": 90 }, { "from_id": "I", "to_id": "K", "distance": 100 }, { "from_id": "I", "to_id": "L", "distance": 90 }, { "from_id": "I", "to_id": "M", "distance": 91 }, { "from_id": "I", "to_id": "N", "distance": 99 }, { "from_id": "I", "to_id": "O", "distance": 78 }, { "from_id": "I", "to_id": "P", "distance": 113 }, { "from_id": "I", "to_id": "Q", "distance": 101 }, { "from_id": "I", "to_id": "R", "distance": 85 }, { "from_id": "I", "to_id": "S", "distance": 78 }, { "from_id": "I", 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"from_id": "T", "to_id": "H", "distance": 95 }, { "from_id": "T", "to_id": "I", "distance": 73 }, { "from_id": "T", "to_id": "J", "distance": 98 }, { "from_id": "T", "to_id": "K", "distance": 80 }, { "from_id": "T", "to_id": "L", "distance": 89 }, { "from_id": "T", "to_id": "M", "distance": 84 }, { "from_id": "T", "to_id": "N", "distance": 108 }, { "from_id": "T", "to_id": "O", "distance": 88 }, { "from_id": "T", "to_id": "P", "distance": 77 }, { "from_id": "T", "to_id": "Q", "distance": 86 }, { "from_id": "T", "to_id": "R", "distance": 99 }, { "from_id": "T", "to_id": "S", "distance": 87 } ] }, "solution_variant": [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "P", "Q" ], "context_index": 11, "input_format": "csv", "input_index_base": "names" }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "We need to put together a tight block of exactly m short films for a festival night. We’ve got a matrix that tells us how far apart any two shorts are in tone and topic, and the goal is a lineup where the films are as spread out from one another as possible. The way to judge a lineup is simple: add the distance between every pair of films in that selection — bigger sums mean a more diverse program. Each pick must be a different film, and the specific distances and the value of m are listed below.\n\n{\n \"total_shorts_available\": 24,\n \"showcase_lineup_size\": 7,\n \"short_film_ids\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"data\": [\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"B\",\n \"style_subject_dissimilarity\": 0\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"C\",\n \"style_subject_dissimilarity\": 8\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"D\",\n \"style_subject_dissimilarity\": 8\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"E\",\n \"style_subject_dissimilarity\": 2\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"F\",\n \"style_subject_dissimilarity\": 4\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"G\",\n \"style_subject_dissimilarity\": 1\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"H\",\n \"style_subject_dissimilarity\": 1\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"I\",\n \"style_subject_dissimilarity\": 6\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"J\",\n \"style_subject_dissimilarity\": 7\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"K\",\n \"style_subject_dissimilarity\": 8\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"L\",\n \"style_subject_dissimilarity\": 4\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"M\",\n \"style_subject_dissimilarity\": 0\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"N\",\n \"style_subject_dissimilarity\": 6\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"O\",\n \"style_subject_dissimilarity\": 7\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"P\",\n \"style_subject_dissimilarity\": 7\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"Q\",\n \"style_subject_dissimilarity\": 3\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"R\",\n \"style_subject_dissimilarity\": 1\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"S\",\n \"style_subject_dissimilarity\": 6\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"T\",\n \"style_subject_dissimilarity\": 7\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"U\",\n \"style_subject_dissimilarity\": 0\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"V\",\n \"style_subject_dissimilarity\": 8\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"W\",\n \"style_subject_dissimilarity\": 6\n },\n {\n \"short_id_a\": \"A\",\n \"short_id_b\": \"X\",\n \"style_subject_dissimilarity\": 4\n },\n {\n \"short_id_a\": \"B\",\n \"short_id_b\": \"A\",\n \"style_subject_dissimilarity\": 0\n },\n {\n \"short_id_a\": \"B\",\n \"short_id_b\": \"C\",\n \"style_subject_dissimilarity\": 2\n },\n {\n \"short_id_a\": \"B\",\n \"short_id_b\": \"D\",\n \"style_subject_dissimilarity\": 2\n },\n {\n \"short_id_a\": \"B\",\n \"short_id_b\": \"E\",\n 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\"short_id_b\": \"O\",\n \"style_subject_dissimilarity\": 8\n },\n {\n \"short_id_a\": \"X\",\n \"short_id_b\": \"P\",\n \"style_subject_dissimilarity\": 8\n },\n {\n \"short_id_a\": \"X\",\n \"short_id_b\": \"Q\",\n \"style_subject_dissimilarity\": 2\n },\n {\n \"short_id_a\": \"X\",\n \"short_id_b\": \"R\",\n \"style_subject_dissimilarity\": 5\n },\n {\n \"short_id_a\": \"X\",\n \"short_id_b\": \"S\",\n \"style_subject_dissimilarity\": 5\n },\n {\n \"short_id_a\": \"X\",\n \"short_id_b\": \"T\",\n \"style_subject_dissimilarity\": 5\n },\n {\n \"short_id_a\": \"X\",\n \"short_id_b\": \"U\",\n \"style_subject_dissimilarity\": 0\n },\n {\n \"short_id_a\": \"X\",\n \"short_id_b\": \"V\",\n \"style_subject_dissimilarity\": 2\n },\n {\n \"short_id_a\": \"X\",\n \"short_id_b\": \"W\",\n \"style_subject_dissimilarity\": 4\n }\n ]\n}\n\nAlso, when you reply, please stick to this simple JSON shape so it's easy to parse:\n\n{\n \"solution\": [, , ...]\n}\n\nHere \"solution\" is just a list of the film identifiers you pick for the festival night. Each is a placeholder for one exact film ID from the instance (replace it with the actual ID when you give your answer). Think of the JSON above as a quick form — it's the shape I need, not the final picks.\n\nPlease use the identifiers exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 0, 8, 8, 2, 4, 1, 1, 6, 7, 8, 4, 0, 6, 7, 7, 3, 1, 6, 7, 0, 8, 6, 4 ], [ 0, 0, 2, 2, 7, 1, 8, 5, 3, 7, 8, 4, 8, 5, 0, 0, 5, 0, 0, 1, 9, 5, 5, 0 ], [ 8, 2, 0, 2, 0, 2, 0, 4, 1, 9, 6, 5, 6, 0, 7, 4, 8, 0, 8, 2, 3, 3, 2, 6 ], [ 8, 2, 2, 0, 6, 7, 2, 2, 9, 9, 8, 8, 0, 4, 8, 8, 7, 8, 2, 6, 7, 2, 7, 3 ], [ 2, 7, 0, 6, 0, 8, 3, 9, 5, 3, 8, 0, 2, 2, 9, 0, 9, 5, 3, 1, 2, 4, 3, 0 ], [ 4, 1, 2, 7, 8, 0, 6, 3, 4, 0, 0, 1, 6, 0, 5, 3, 6, 9, 4, 9, 3, 2, 6, 8 ], [ 1, 8, 0, 2, 3, 6, 0, 2, 3, 3, 4, 8, 1, 0, 1, 7, 4, 2, 9, 6, 3, 3, 8, 4 ], [ 1, 5, 4, 2, 9, 3, 2, 0, 3, 7, 9, 6, 2, 6, 9, 6, 9, 2, 8, 3, 4, 7, 5, 3 ], [ 6, 3, 1, 9, 5, 4, 3, 3, 0, 4, 7, 2, 8, 5, 9, 4, 4, 7, 9, 6, 4, 8, 1, 4 ], [ 7, 7, 9, 9, 3, 0, 3, 7, 4, 0, 9, 9, 4, 5, 9, 6, 4, 3, 4, 3, 6, 6, 6, 2 ], [ 8, 8, 6, 8, 8, 0, 4, 9, 7, 9, 0, 3, 3, 5, 7, 3, 0, 3, 7, 1, 0, 5, 2, 9 ], [ 4, 4, 5, 8, 0, 1, 8, 6, 2, 9, 3, 0, 6, 0, 5, 2, 4, 1, 2, 9, 0, 2, 9, 3 ], [ 0, 8, 6, 0, 2, 6, 1, 2, 8, 4, 3, 6, 0, 4, 7, 5, 6, 7, 4, 2, 8, 2, 6, 0 ], [ 6, 5, 0, 4, 2, 0, 0, 6, 5, 5, 5, 0, 4, 0, 0, 9, 4, 0, 4, 5, 9, 1, 6, 2 ], [ 7, 0, 7, 8, 9, 5, 1, 9, 9, 9, 7, 5, 7, 0, 0, 8, 3, 1, 5, 7, 9, 3, 6, 8 ], [ 7, 0, 4, 8, 0, 3, 7, 6, 4, 6, 3, 2, 5, 9, 8, 0, 2, 2, 4, 8, 0, 1, 0, 8 ], [ 3, 5, 8, 7, 9, 6, 4, 9, 4, 4, 0, 4, 6, 4, 3, 2, 0, 5, 8, 1, 1, 0, 7, 2 ], [ 1, 0, 0, 8, 5, 9, 2, 2, 7, 3, 3, 1, 7, 0, 1, 2, 5, 0, 0, 5, 7, 2, 3, 5 ], [ 6, 0, 8, 2, 3, 4, 9, 8, 9, 4, 7, 2, 4, 4, 5, 4, 8, 0, 0, 4, 1, 1, 5, 5 ], [ 7, 1, 2, 6, 1, 9, 6, 3, 6, 3, 1, 9, 2, 5, 7, 8, 1, 5, 4, 0, 3, 7, 9, 5 ], [ 0, 9, 3, 7, 2, 3, 3, 4, 4, 6, 0, 0, 8, 9, 9, 0, 1, 7, 1, 3, 0, 6, 3, 0 ], [ 8, 5, 3, 2, 4, 2, 3, 7, 8, 6, 5, 2, 2, 1, 3, 1, 0, 2, 1, 7, 6, 0, 3, 2 ], [ 6, 5, 2, 7, 3, 6, 8, 5, 1, 6, 2, 9, 6, 6, 6, 0, 7, 3, 5, 9, 3, 3, 0, 4 ], [ 4, 0, 6, 3, 0, 8, 4, 3, 4, 2, 9, 3, 0, 2, 8, 8, 2, 5, 5, 5, 0, 2, 4, 0 ] ], "m": 7, "objective": 151.0 }, "solution": [ 0, 3, 8, 9, 10, 14, 15 ], "obj": 151.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 24, "m": 7, "nodes": [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S", "T", "U", "V", "W", "X" ], "distance_pairs": [ { "from_id": "A", "to_id": "B", "distance": 0 }, { "from_id": "A", "to_id": "C", "distance": 8 }, { "from_id": "A", "to_id": "D", "distance": 8 }, { "from_id": "A", "to_id": "E", "distance": 2 }, { "from_id": "A", "to_id": "F", "distance": 4 }, { "from_id": "A", "to_id": "G", "distance": 1 }, { "from_id": "A", "to_id": "H", "distance": 1 }, { "from_id": "A", "to_id": "I", "distance": 6 }, { "from_id": "A", "to_id": "J", "distance": 7 }, { "from_id": "A", "to_id": "K", "distance": 8 }, { "from_id": "A", "to_id": "L", "distance": 4 }, { "from_id": "A", "to_id": "M", "distance": 0 }, { "from_id": "A", "to_id": "N", "distance": 6 }, { "from_id": "A", "to_id": "O", "distance": 7 }, { "from_id": "A", "to_id": "P", "distance": 7 }, { "from_id": "A", "to_id": "Q", "distance": 3 }, { "from_id": "A", "to_id": "R", "distance": 1 }, { "from_id": "A", "to_id": "S", "distance": 6 }, { "from_id": "A", "to_id": "T", "distance": 7 }, { "from_id": "A", "to_id": "U", "distance": 0 }, { "from_id": "A", "to_id": "V", "distance": 8 }, { "from_id": "A", "to_id": "W", "distance": 6 }, { "from_id": "A", "to_id": "X", "distance": 4 }, { "from_id": "B", "to_id": "A", "distance": 0 }, { "from_id": "B", "to_id": "C", "distance": 2 }, { "from_id": "B", "to_id": "D", "distance": 2 }, { "from_id": "B", "to_id": "E", "distance": 7 }, { "from_id": "B", "to_id": "F", "distance": 1 }, { "from_id": "B", "to_id": "G", "distance": 8 }, { "from_id": "B", "to_id": "H", "distance": 5 }, { "from_id": "B", "to_id": "I", "distance": 3 }, { "from_id": "B", "to_id": "J", "distance": 7 }, { "from_id": "B", "to_id": "K", "distance": 8 }, { "from_id": "B", "to_id": "L", "distance": 4 }, { "from_id": "B", "to_id": "M", "distance": 8 }, { "from_id": "B", "to_id": "N", "distance": 5 }, { "from_id": "B", "to_id": "O", "distance": 0 }, { "from_id": "B", "to_id": "P", "distance": 0 }, { "from_id": "B", "to_id": "Q", "distance": 5 }, { "from_id": "B", "to_id": "R", "distance": 0 }, { "from_id": "B", "to_id": "S", "distance": 0 }, { "from_id": "B", "to_id": "T", "distance": 1 }, { "from_id": "B", "to_id": "U", "distance": 9 }, { "from_id": "B", "to_id": "V", "distance": 5 }, { "from_id": "B", "to_id": "W", "distance": 5 }, { "from_id": "B", "to_id": "X", "distance": 0 }, { "from_id": "C", "to_id": "A", "distance": 8 }, { "from_id": "C", "to_id": "B", "distance": 2 }, { "from_id": "C", "to_id": "D", "distance": 2 }, { "from_id": "C", "to_id": "E", "distance": 0 }, { "from_id": "C", "to_id": "F", "distance": 2 }, { "from_id": "C", "to_id": "G", "distance": 0 }, { "from_id": "C", "to_id": "H", "distance": 4 }, { "from_id": "C", "to_id": "I", "distance": 1 }, { "from_id": "C", "to_id": "J", "distance": 9 }, { "from_id": "C", "to_id": "K", "distance": 6 }, { "from_id": "C", "to_id": "L", "distance": 5 }, { "from_id": "C", "to_id": "M", "distance": 6 }, { "from_id": "C", "to_id": "N", "distance": 0 }, { "from_id": "C", "to_id": "O", "distance": 7 }, { "from_id": "C", "to_id": "P", "distance": 4 }, { "from_id": "C", "to_id": "Q", "distance": 8 }, { "from_id": "C", "to_id": "R", "distance": 0 }, { "from_id": "C", "to_id": "S", "distance": 8 }, { "from_id": "C", "to_id": "T", "distance": 2 }, { "from_id": "C", "to_id": "U", "distance": 3 }, { "from_id": "C", "to_id": "V", "distance": 3 }, { "from_id": "C", "to_id": "W", "distance": 2 }, { "from_id": "C", "to_id": "X", "distance": 6 }, { "from_id": "D", "to_id": "A", "distance": 8 }, { "from_id": "D", "to_id": "B", "distance": 2 }, { "from_id": "D", "to_id": "C", "distance": 2 }, { "from_id": "D", "to_id": "E", "distance": 6 }, { "from_id": "D", "to_id": "F", "distance": 7 }, { "from_id": "D", "to_id": "G", "distance": 2 }, { "from_id": "D", "to_id": "H", "distance": 2 }, { "from_id": "D", "to_id": "I", "distance": 9 }, { "from_id": "D", "to_id": "J", "distance": 9 }, { "from_id": "D", "to_id": "K", "distance": 8 }, { "from_id": "D", "to_id": "L", "distance": 8 }, { "from_id": "D", "to_id": "M", "distance": 0 }, { "from_id": "D", "to_id": "N", "distance": 4 }, { "from_id": "D", "to_id": "O", "distance": 8 }, { "from_id": "D", "to_id": "P", "distance": 8 }, { "from_id": "D", "to_id": "Q", "distance": 7 }, { "from_id": "D", "to_id": "R", "distance": 8 }, { "from_id": "D", "to_id": "S", "distance": 2 }, { "from_id": "D", "to_id": "T", "distance": 6 }, { "from_id": "D", "to_id": "U", "distance": 7 }, { "from_id": "D", "to_id": "V", "distance": 2 }, { "from_id": "D", "to_id": "W", 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"from_id": "E", "to_id": "U", "distance": 2 }, { "from_id": "E", "to_id": "V", "distance": 4 }, { "from_id": "E", "to_id": "W", "distance": 3 }, { "from_id": "E", "to_id": "X", "distance": 0 }, { "from_id": "F", "to_id": "A", "distance": 4 }, { "from_id": "F", "to_id": "B", "distance": 1 }, { "from_id": "F", "to_id": "C", "distance": 2 }, { "from_id": "F", "to_id": "D", "distance": 7 }, { "from_id": "F", "to_id": "E", "distance": 8 }, { "from_id": "F", "to_id": "G", "distance": 6 }, { "from_id": "F", "to_id": "H", "distance": 3 }, { "from_id": "F", "to_id": "I", "distance": 4 }, { "from_id": "F", "to_id": "J", "distance": 0 }, { "from_id": "F", "to_id": "K", "distance": 0 }, { "from_id": "F", "to_id": "L", "distance": 1 }, { "from_id": "F", "to_id": "M", "distance": 6 }, { "from_id": "F", "to_id": "N", "distance": 0 }, { "from_id": "F", "to_id": "O", "distance": 5 }, { "from_id": "F", "to_id": "P", "distance": 3 }, { "from_id": "F", "to_id": "Q", "distance": 6 }, { "from_id": "F", "to_id": "R", "distance": 9 }, { "from_id": "F", "to_id": "S", "distance": 4 }, { "from_id": "F", "to_id": "T", "distance": 9 }, { "from_id": "F", "to_id": "U", "distance": 3 }, { "from_id": "F", "to_id": "V", "distance": 2 }, { "from_id": "F", "to_id": "W", "distance": 6 }, { "from_id": "F", "to_id": "X", "distance": 8 }, { "from_id": "G", "to_id": "A", "distance": 1 }, { "from_id": "G", "to_id": "B", "distance": 8 }, { "from_id": "G", "to_id": "C", "distance": 0 }, { "from_id": "G", "to_id": "D", "distance": 2 }, { "from_id": "G", "to_id": "E", "distance": 3 }, { "from_id": "G", "to_id": "F", "distance": 6 }, { "from_id": "G", "to_id": "H", "distance": 2 }, { "from_id": "G", "to_id": "I", "distance": 3 }, { "from_id": "G", "to_id": "J", "distance": 3 }, { "from_id": "G", "to_id": "K", "distance": 4 }, { "from_id": "G", "to_id": "L", "distance": 8 }, { "from_id": "G", "to_id": "M", "distance": 1 }, { "from_id": "G", "to_id": "N", "distance": 0 }, { "from_id": "G", "to_id": "O", 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each pair’s followers and content styles are. The task is to pick exactly m influencers, no repeats, so that if one sums up the “difference” number for every pair inside that group the total comes out as large as possible — that total distinctness is what makes one selection better than another. The exact numbers and choices to consider are shown below.\n\n# total_influencer_candidates=19\n# selection_size=6\n# influencer_ids=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19\ninfluencer_a_id,influencer_b_id,audience_content_distance\n1,2,4\n1,3,0\n1,4,5\n1,5,1\n1,6,0\n1,7,1\n1,8,4\n1,9,4\n1,10,6\n1,11,3\n1,12,3\n1,13,3\n1,14,0\n1,15,5\n1,16,4\n1,17,5\n1,18,8\n1,19,0\n2,1,4\n2,3,4\n2,4,5\n2,5,4\n2,6,7\n2,7,9\n2,8,8\n2,9,4\n2,10,7\n2,11,7\n2,12,8\n2,13,5\n2,14,8\n2,15,2\n2,16,8\n2,17,3\n2,18,1\n2,19,5\n3,1,0\n3,2,4\n3,4,7\n3,5,1\n3,6,1\n3,7,3\n3,8,5\n3,9,5\n3,10,7\n3,11,2\n3,12,0\n3,13,4\n3,14,4\n3,15,8\n3,16,8\n3,17,4\n3,18,2\n3,19,8\n4,1,5\n4,2,5\n4,3,7\n4,5,0\n4,6,3\n4,7,6\n4,8,7\n4,9,1\n4,10,9\n4,11,0\n4,12,4\n4,13,9\n4,14,7\n4,15,8\n4,16,4\n4,17,8\n4,18,7\n4,19,0\n5,1,1\n5,2,4\n5,3,1\n5,4,0\n5,6,6\n5,7,7\n5,8,8\n5,9,1\n5,10,5\n5,11,7\n5,12,4\n5,13,9\n5,14,3\n5,15,0\n5,16,9\n5,17,4\n5,18,7\n5,19,4\n6,1,0\n6,2,7\n6,3,1\n6,4,3\n6,5,6\n6,7,6\n6,8,1\n6,9,7\n6,10,0\n6,11,0\n6,12,8\n6,13,8\n6,14,8\n6,15,3\n6,16,1\n6,17,0\n6,18,0\n6,19,8\n7,1,1\n7,2,9\n7,3,3\n7,4,6\n7,5,7\n7,6,6\n7,8,9\n7,9,7\n7,10,1\n7,11,2\n7,12,3\n7,13,4\n7,14,0\n7,15,7\n7,16,8\n7,17,4\n7,18,7\n7,19,0\n8,1,4\n8,2,8\n8,3,5\n8,4,7\n8,5,8\n8,6,1\n8,7,9\n8,9,4\n8,10,9\n8,11,3\n8,12,0\n8,13,2\n8,14,1\n8,15,3\n8,16,0\n8,17,9\n8,18,1\n8,19,5\n9,1,4\n9,2,4\n9,3,5\n9,4,1\n9,5,1\n9,6,7\n9,7,7\n9,8,4\n9,10,0\n9,11,2\n9,12,9\n9,13,1\n9,14,1\n9,15,5\n9,16,6\n9,17,3\n9,18,2\n9,19,9\n10,1,6\n10,2,7\n10,3,7\n10,4,9\n10,5,5\n10,6,0\n10,7,1\n10,8,9\n10,9,0\n10,11,2\n10,12,6\n10,13,3\n10,14,9\n10,15,1\n10,16,0\n10,17,6\n10,18,0\n10,19,8\n11,1,3\n11,2,7\n11,3,2\n11,4,0\n11,5,7\n11,6,0\n11,7,2\n11,8,3\n11,9,2\n11,10,2\n11,12,5\n11,13,0\n11,14,2\n11,15,1\n11,16,0\n11,17,1\n11,18,8\n11,19,4\n12,1,3\n12,2,8\n12,3,0\n12,4,4\n12,5,4\n12,6,8\n12,7,3\n12,8,0\n12,9,9\n12,10,6\n12,11,5\n12,13,6\n12,14,8\n12,15,5\n12,16,1\n12,17,8\n12,18,8\n12,19,5\n13,1,3\n13,2,5\n13,3,4\n13,4,9\n13,5,9\n13,6,8\n13,7,4\n13,8,2\n13,9,1\n13,10,3\n13,11,0\n13,12,6\n13,14,0\n13,15,8\n13,16,4\n13,17,1\n13,18,2\n13,19,7\n14,1,0\n14,2,8\n14,3,4\n14,4,7\n14,5,3\n14,6,8\n14,7,0\n14,8,1\n14,9,1\n14,10,9\n14,11,2\n14,12,8\n14,13,0\n14,15,3\n14,16,9\n14,17,9\n14,18,0\n14,19,4\n15,1,5\n15,2,2\n15,3,8\n15,4,8\n15,5,0\n15,6,3\n15,7,7\n15,8,3\n15,9,5\n15,10,1\n15,11,1\n15,12,5\n15,13,8\n15,14,3\n15,16,2\n15,17,0\n15,18,5\n15,19,3\n16,1,4\n16,2,8\n16,3,8\n16,4,4\n16,5,9\n16,6,1\n16,7,8\n16,8,0\n16,9,6\n16,10,0\n16,11,0\n16,12,1\n16,13,4\n16,14,9\n16,15,2\n16,17,2\n16,18,3\n16,19,9\n17,1,5\n17,2,3\n17,3,4\n17,4,8\n17,5,4\n17,6,0\n17,7,4\n17,8,9\n17,9,3\n17,10,6\n17,11,1\n17,12,8\n17,13,1\n17,14,9\n17,15,0\n17,16,2\n17,18,7\n17,19,3\n18,1,8\n18,2,1\n18,3,2\n18,4,7\n18,5,7\n18,6,0\n18,7,7\n18,8,1\n18,9,2\n18,10,0\n18,11,8\n18,12,8\n18,13,2\n18,14,0\n18,15,5\n18,16,3\n18,17,7\n18,19,7\n19,1,0\n19,2,5\n19,3,8\n19,4,0\n19,5,4\n19,6,8\n19,7,0\n19,8,5\n19,9,9\n19,10,8\n19,11,4\n19,12,5\n19,13,7\n19,14,4\n19,15,3\n19,16,9\n19,17,3\n19,18,7\n\nI'll put the chosen creators into a simple JSON sketch so it's easy to paste into whatever you're using:\n\n{\n \"solution\": [, , ...]\n}\n\nThink of \"solution\" as just the list of influencers you pick — each placeholder like stands for one identifier from the input (one per selected creator). This is just the expected shape of the reply, not the actual selection.\n\nPlease make sure to use the identifiers exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 4, 0, 5, 1, 0, 1, 4, 4, 6, 3, 3, 3, 0, 5, 4, 5, 8, 0 ], [ 4, 0, 4, 5, 4, 7, 9, 8, 4, 7, 7, 8, 5, 8, 2, 8, 3, 1, 5 ], [ 0, 4, 0, 7, 1, 1, 3, 5, 5, 7, 2, 0, 4, 4, 8, 8, 4, 2, 8 ], [ 5, 5, 7, 0, 0, 3, 6, 7, 1, 9, 0, 4, 9, 7, 8, 4, 8, 7, 0 ], [ 1, 4, 1, 0, 0, 6, 7, 8, 1, 5, 7, 4, 9, 3, 0, 9, 4, 7, 4 ], [ 0, 7, 1, 3, 6, 0, 6, 1, 7, 0, 0, 8, 8, 8, 3, 1, 0, 0, 8 ], [ 1, 9, 3, 6, 7, 6, 0, 9, 7, 1, 2, 3, 4, 0, 7, 8, 4, 7, 0 ], [ 4, 8, 5, 7, 8, 1, 9, 0, 4, 9, 3, 0, 2, 1, 3, 0, 9, 1, 5 ], [ 4, 4, 5, 1, 1, 7, 7, 4, 0, 0, 2, 9, 1, 1, 5, 6, 3, 2, 9 ], [ 6, 7, 7, 9, 5, 0, 1, 9, 0, 0, 2, 6, 3, 9, 1, 0, 6, 0, 8 ], [ 3, 7, 2, 0, 7, 0, 2, 3, 2, 2, 0, 5, 0, 2, 1, 0, 1, 8, 4 ], [ 3, 8, 0, 4, 4, 8, 3, 0, 9, 6, 5, 0, 6, 8, 5, 1, 8, 8, 5 ], [ 3, 5, 4, 9, 9, 8, 4, 2, 1, 3, 0, 6, 0, 0, 8, 4, 1, 2, 7 ], [ 0, 8, 4, 7, 3, 8, 0, 1, 1, 9, 2, 8, 0, 0, 3, 9, 9, 0, 4 ], [ 5, 2, 8, 8, 0, 3, 7, 3, 5, 1, 1, 5, 8, 3, 0, 2, 0, 5, 3 ], [ 4, 8, 8, 4, 9, 1, 8, 0, 6, 0, 0, 1, 4, 9, 2, 0, 2, 3, 9 ], [ 5, 3, 4, 8, 4, 0, 4, 9, 3, 6, 1, 8, 1, 9, 0, 2, 0, 7, 3 ], [ 8, 1, 2, 7, 7, 0, 7, 1, 2, 0, 8, 8, 2, 0, 5, 3, 7, 0, 7 ], [ 0, 5, 8, 0, 4, 8, 0, 5, 9, 8, 4, 5, 7, 4, 3, 9, 3, 7, 0 ] ], "m": 6, "objective": 105.0 }, "solution": [ 1, 3, 9, 11, 13, 16 ], "obj": 105.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 19, "m": 6, "nodes": [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 ], "distance_pairs": [ { "from_id": 1, "to_id": 2, "distance": 4 }, { "from_id": 1, "to_id": 3, "distance": 0 }, { "from_id": 1, "to_id": 4, "distance": 5 }, { "from_id": 1, "to_id": 5, "distance": 1 }, { "from_id": 1, "to_id": 6, "distance": 0 }, { "from_id": 1, "to_id": 7, "distance": 1 }, { "from_id": 1, "to_id": 8, "distance": 4 }, { "from_id": 1, "to_id": 9, "distance": 4 }, { "from_id": 1, "to_id": 10, "distance": 6 }, { "from_id": 1, "to_id": 11, "distance": 3 }, { "from_id": 1, "to_id": 12, "distance": 3 }, { "from_id": 1, "to_id": 13, "distance": 3 }, { "from_id": 1, "to_id": 14, "distance": 0 }, { "from_id": 1, "to_id": 15, "distance": 5 }, { "from_id": 1, "to_id": 16, "distance": 4 }, { "from_id": 1, "to_id": 17, "distance": 5 }, { "from_id": 1, "to_id": 18, "distance": 8 }, { "from_id": 1, "to_id": 19, "distance": 0 }, { "from_id": 2, "to_id": 1, "distance": 4 }, { "from_id": 2, "to_id": 3, "distance": 4 }, { "from_id": 2, "to_id": 4, "distance": 5 }, { "from_id": 2, "to_id": 5, "distance": 4 }, { "from_id": 2, "to_id": 6, "distance": 7 }, { "from_id": 2, "to_id": 7, "distance": 9 }, { "from_id": 2, "to_id": 8, "distance": 8 }, { "from_id": 2, "to_id": 9, "distance": 4 }, { "from_id": 2, "to_id": 10, "distance": 7 }, { "from_id": 2, "to_id": 11, "distance": 7 }, { "from_id": 2, "to_id": 12, "distance": 8 }, { "from_id": 2, "to_id": 13, "distance": 5 }, { "from_id": 2, "to_id": 14, "distance": 8 }, { "from_id": 2, "to_id": 15, "distance": 2 }, { "from_id": 2, "to_id": 16, "distance": 8 }, { "from_id": 2, "to_id": 17, "distance": 3 }, { "from_id": 2, "to_id": 18, "distance": 1 }, { "from_id": 2, "to_id": 19, "distance": 5 }, { "from_id": 3, "to_id": 1, "distance": 0 }, { "from_id": 3, "to_id": 2, "distance": 4 }, { "from_id": 3, "to_id": 4, "distance": 7 }, { "from_id": 3, "to_id": 5, "distance": 1 }, { "from_id": 3, "to_id": 6, "distance": 1 }, { "from_id": 3, "to_id": 7, "distance": 3 }, { "from_id": 3, "to_id": 8, "distance": 5 }, { "from_id": 3, "to_id": 9, "distance": 5 }, { "from_id": 3, "to_id": 10, "distance": 7 }, { "from_id": 3, "to_id": 11, "distance": 2 }, { "from_id": 3, "to_id": 12, "distance": 0 }, { "from_id": 3, "to_id": 13, "distance": 4 }, { "from_id": 3, "to_id": 14, "distance": 4 }, { "from_id": 3, "to_id": 15, "distance": 8 }, { "from_id": 3, "to_id": 16, "distance": 8 }, { "from_id": 3, "to_id": 17, "distance": 4 }, { "from_id": 3, "to_id": 18, "distance": 2 }, { "from_id": 3, "to_id": 19, "distance": 8 }, { "from_id": 4, "to_id": 1, "distance": 5 }, { "from_id": 4, "to_id": 2, "distance": 5 }, { "from_id": 4, "to_id": 3, "distance": 7 }, { "from_id": 4, "to_id": 5, "distance": 0 }, { "from_id": 4, "to_id": 6, "distance": 3 }, { "from_id": 4, "to_id": 7, "distance": 6 }, { "from_id": 4, "to_id": 8, "distance": 7 }, { "from_id": 4, "to_id": 9, "distance": 1 }, { "from_id": 4, "to_id": 10, "distance": 9 }, { "from_id": 4, "to_id": 11, "distance": 0 }, { "from_id": 4, "to_id": 12, "distance": 4 }, { "from_id": 4, "to_id": 13, "distance": 9 }, { "from_id": 4, "to_id": 14, "distance": 7 }, { "from_id": 4, "to_id": 15, "distance": 8 }, { "from_id": 4, "to_id": 16, "distance": 4 }, { "from_id": 4, "to_id": 17, "distance": 8 }, { "from_id": 4, "to_id": 18, "distance": 7 }, { "from_id": 4, "to_id": 19, "distance": 0 }, { "from_id": 5, "to_id": 1, "distance": 1 }, { "from_id": 5, "to_id": 2, "distance": 4 }, { "from_id": 5, "to_id": 3, "distance": 1 }, { "from_id": 5, "to_id": 4, "distance": 0 }, { "from_id": 5, "to_id": 6, "distance": 6 }, { "from_id": 5, "to_id": 7, "distance": 7 }, { "from_id": 5, "to_id": 8, "distance": 8 }, { "from_id": 5, "to_id": 9, "distance": 1 }, { "from_id": 5, "to_id": 10, "distance": 5 }, { "from_id": 5, "to_id": 11, "distance": 7 }, { "from_id": 5, "to_id": 12, "distance": 4 }, { "from_id": 5, "to_id": 13, "distance": 9 }, { "from_id": 5, "to_id": 14, "distance": 3 }, { "from_id": 5, "to_id": 15, "distance": 0 }, { "from_id": 5, "to_id": 16, "distance": 9 }, { "from_id": 5, "to_id": 17, "distance": 4 }, { "from_id": 5, "to_id": 18, "distance": 7 }, { "from_id": 5, "to_id": 19, "distance": 4 }, { "from_id": 6, "to_id": 1, "distance": 0 }, { "from_id": 6, "to_id": 2, "distance": 7 }, { "from_id": 6, "to_id": 3, "distance": 1 }, { "from_id": 6, "to_id": 4, "distance": 3 }, { "from_id": 6, "to_id": 5, "distance": 6 }, { "from_id": 6, "to_id": 7, "distance": 6 }, { "from_id": 6, "to_id": 8, "distance": 1 }, { "from_id": 6, "to_id": 9, "distance": 7 }, { "from_id": 6, "to_id": 10, "distance": 0 }, { "from_id": 6, "to_id": 11, "distance": 0 }, { "from_id": 6, "to_id": 12, "distance": 8 }, { "from_id": 6, "to_id": 13, "distance": 8 }, { "from_id": 6, "to_id": 14, "distance": 8 }, { "from_id": 6, "to_id": 15, "distance": 3 }, { "from_id": 6, "to_id": 16, "distance": 1 }, { "from_id": 6, "to_id": 17, "distance": 0 }, { "from_id": 6, "to_id": 18, "distance": 0 }, { "from_id": 6, "to_id": 19, "distance": 8 }, { "from_id": 7, "to_id": 1, "distance": 1 }, { "from_id": 7, "to_id": 2, "distance": 9 }, { "from_id": 7, "to_id": 3, "distance": 3 }, { "from_id": 7, "to_id": 4, "distance": 6 }, { "from_id": 7, "to_id": 5, "distance": 7 }, { "from_id": 7, "to_id": 6, "distance": 6 }, { "from_id": 7, "to_id": 8, "distance": 9 }, { "from_id": 7, "to_id": 9, "distance": 7 }, { "from_id": 7, "to_id": 10, "distance": 1 }, { "from_id": 7, "to_id": 11, "distance": 2 }, { "from_id": 7, "to_id": 12, "distance": 3 }, { "from_id": 7, "to_id": 13, "distance": 4 }, { "from_id": 7, "to_id": 14, "distance": 0 }, { "from_id": 7, "to_id": 15, "distance": 7 }, { "from_id": 7, "to_id": 16, "distance": 8 }, { "from_id": 7, "to_id": 17, "distance": 4 }, { "from_id": 7, "to_id": 18, "distance": 7 }, { "from_id": 7, "to_id": 19, "distance": 0 }, { "from_id": 8, "to_id": 1, "distance": 4 }, { "from_id": 8, "to_id": 2, "distance": 8 }, { "from_id": 8, "to_id": 3, "distance": 5 }, { "from_id": 8, "to_id": 4, "distance": 7 }, { "from_id": 8, "to_id": 5, "distance": 8 }, { "from_id": 8, "to_id": 6, "distance": 1 }, { "from_id": 8, "to_id": 7, "distance": 9 }, { "from_id": 8, "to_id": 9, "distance": 4 }, { "from_id": 8, "to_id": 10, "distance": 9 }, { "from_id": 8, "to_id": 11, "distance": 3 }, { "from_id": 8, "to_id": 12, "distance": 0 }, { "from_id": 8, "to_id": 13, "distance": 2 }, { "from_id": 8, "to_id": 14, "distance": 1 }, { "from_id": 8, "to_id": 15, "distance": 3 }, { "from_id": 8, "to_id": 16, "distance": 0 }, { "from_id": 8, "to_id": 17, "distance": 9 }, { "from_id": 8, "to_id": 18, "distance": 1 }, { "from_id": 8, "to_id": 19, "distance": 5 }, { "from_id": 9, "to_id": 1, "distance": 4 }, { "from_id": 9, "to_id": 2, "distance": 4 }, { "from_id": 9, "to_id": 3, "distance": 5 }, { "from_id": 9, "to_id": 4, "distance": 1 }, { "from_id": 9, "to_id": 5, "distance": 1 }, { "from_id": 9, "to_id": 6, "distance": 7 }, { "from_id": 9, "to_id": 7, "distance": 7 }, { "from_id": 9, "to_id": 8, "distance": 4 }, { "from_id": 9, "to_id": 10, "distance": 0 }, { "from_id": 9, "to_id": 11, "distance": 2 }, { "from_id": 9, "to_id": 12, "distance": 9 }, { "from_id": 9, "to_id": 13, "distance": 1 }, { "from_id": 9, "to_id": 14, "distance": 1 }, { "from_id": 9, "to_id": 15, "distance": 5 }, { "from_id": 9, "to_id": 16, "distance": 6 }, { "from_id": 9, "to_id": 17, "distance": 3 }, { "from_id": 9, "to_id": 18, "distance": 2 }, { "from_id": 9, "to_id": 19, "distance": 9 }, { "from_id": 10, "to_id": 1, "distance": 6 }, { "from_id": 10, "to_id": 2, "distance": 7 }, { "from_id": 10, "to_id": 3, "distance": 7 }, { "from_id": 10, "to_id": 4, "distance": 9 }, { "from_id": 10, "to_id": 5, "distance": 5 }, { "from_id": 10, "to_id": 6, "distance": 0 }, { "from_id": 10, "to_id": 7, "distance": 1 }, { "from_id": 10, "to_id": 8, "distance": 9 }, { "from_id": 10, "to_id": 9, "distance": 0 }, { "from_id": 10, "to_id": 11, "distance": 2 }, { "from_id": 10, "to_id": 12, "distance": 6 }, { "from_id": 10, "to_id": 13, "distance": 3 }, { "from_id": 10, "to_id": 14, "distance": 9 }, { "from_id": 10, "to_id": 15, "distance": 1 }, { "from_id": 10, "to_id": 16, "distance": 0 }, { "from_id": 10, "to_id": 17, "distance": 6 }, { "from_id": 10, "to_id": 18, "distance": 0 }, { "from_id": 10, "to_id": 19, "distance": 8 }, { "from_id": 11, "to_id": 1, "distance": 3 }, { "from_id": 11, "to_id": 2, "distance": 7 }, { "from_id": 11, "to_id": 3, "distance": 2 }, { "from_id": 11, "to_id": 4, "distance": 0 }, { "from_id": 11, "to_id": 5, "distance": 7 }, { "from_id": 11, "to_id": 6, "distance": 0 }, { "from_id": 11, "to_id": 7, "distance": 2 }, { "from_id": 11, "to_id": 8, "distance": 3 }, { "from_id": 11, "to_id": 9, "distance": 2 }, { "from_id": 11, "to_id": 10, "distance": 2 }, { "from_id": 11, "to_id": 12, "distance": 5 }, { "from_id": 11, "to_id": 13, "distance": 0 }, { "from_id": 11, "to_id": 14, "distance": 2 }, { "from_id": 11, "to_id": 15, "distance": 1 }, { "from_id": 11, "to_id": 16, "distance": 0 }, { "from_id": 11, "to_id": 17, "distance": 1 }, { "from_id": 11, "to_id": 18, "distance": 8 }, { "from_id": 11, "to_id": 19, "distance": 4 }, { "from_id": 12, "to_id": 1, "distance": 3 }, { "from_id": 12, "to_id": 2, "distance": 8 }, { "from_id": 12, "to_id": 3, "distance": 0 }, { "from_id": 12, "to_id": 4, "distance": 4 }, { "from_id": 12, "to_id": 5, "distance": 4 }, { "from_id": 12, "to_id": 6, "distance": 8 }, { "from_id": 12, "to_id": 7, "distance": 3 }, { "from_id": 12, "to_id": 8, "distance": 0 }, { "from_id": 12, "to_id": 9, "distance": 9 }, { "from_id": 12, "to_id": 10, "distance": 6 }, { "from_id": 12, "to_id": 11, "distance": 5 }, { "from_id": 12, "to_id": 13, "distance": 6 }, { "from_id": 12, "to_id": 14, "distance": 8 }, { "from_id": 12, "to_id": 15, "distance": 5 }, { "from_id": 12, "to_id": 16, "distance": 1 }, { "from_id": 12, "to_id": 17, "distance": 8 }, { "from_id": 12, "to_id": 18, "distance": 8 }, { "from_id": 12, "to_id": 19, "distance": 5 }, { "from_id": 13, "to_id": 1, "distance": 3 }, { "from_id": 13, "to_id": 2, "distance": 5 }, { "from_id": 13, "to_id": 3, "distance": 4 }, { "from_id": 13, "to_id": 4, "distance": 9 }, { "from_id": 13, "to_id": 5, "distance": 9 }, { "from_id": 13, "to_id": 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5, "distance": 4 }, { "from_id": 19, "to_id": 6, "distance": 8 }, { "from_id": 19, "to_id": 7, "distance": 0 }, { "from_id": 19, "to_id": 8, "distance": 5 }, { "from_id": 19, "to_id": 9, "distance": 9 }, { "from_id": 19, "to_id": 10, "distance": 8 }, { "from_id": 19, "to_id": 11, "distance": 4 }, { "from_id": 19, "to_id": 12, "distance": 5 }, { "from_id": 19, "to_id": 13, "distance": 7 }, { "from_id": 19, "to_id": 14, "distance": 4 }, { "from_id": 19, "to_id": 15, "distance": 3 }, { "from_id": 19, "to_id": 16, "distance": 9 }, { "from_id": 19, "to_id": 17, "distance": 3 }, { "from_id": 19, "to_id": 18, "distance": 7 } ] }, "solution_variant": [ 2, 4, 10, 12, 14, 17 ], "context_index": 13, "input_format": "csv", "input_index_base": 1 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Recently a team needed a shortlist of sampling locations from a larger grid, with the idea that the shortlist should capture the widest variety of environmental conditions. Every pair of sites has a given distance value showing how different they are, and the shortlist’s quality is the sum of those values across every pair inside it — larger sums signal greater variety. The shortlist has to be exactly m unique sites, nothing left out and no duplicates. The exact pairwise distances and the value of m appear below.\n\n{\n \"total_survey_locations\": 24,\n \"sites_to_select\": 7,\n \"location_ids\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\",\n \"T\",\n \"U\",\n \"V\",\n \"W\",\n \"X\"\n ],\n \"data\": [\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"B\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"C\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"D\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"E\",\n \"env_distance_between_sites\": 3\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"F\",\n \"env_distance_between_sites\": 5\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"G\",\n \"env_distance_between_sites\": 1\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"H\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"I\",\n \"env_distance_between_sites\": 5\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"J\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"K\",\n \"env_distance_between_sites\": 7\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"L\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"M\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"N\",\n \"env_distance_between_sites\": 5\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"O\",\n \"env_distance_between_sites\": 7\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"P\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"Q\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"R\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"S\",\n \"env_distance_between_sites\": 7\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"T\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"U\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"V\",\n \"env_distance_between_sites\": 6\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"W\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"A\",\n \"site_id_b\": \"X\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"B\",\n \"site_id_b\": \"A\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"B\",\n \"site_id_b\": \"C\",\n \"env_distance_between_sites\": 7\n },\n {\n \"site_id_a\": \"B\",\n \"site_id_b\": \"D\",\n \"env_distance_between_sites\": 0\n },\n {\n \"site_id_a\": \"B\",\n \"site_id_b\": \"E\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"B\",\n \"site_id_b\": \"F\",\n \"env_distance_between_sites\": 1\n },\n {\n \"site_id_a\": \"B\",\n \"site_id_b\": \"G\",\n \"env_distance_between_sites\": 6\n },\n {\n \"site_id_a\": \"B\",\n \"site_id_b\": \"H\",\n \"env_distance_between_sites\": 1\n },\n {\n \"site_id_a\": \"B\",\n \"site_id_b\": \"I\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"B\",\n \"site_id_b\": \"J\",\n \"env_distance_between_sites\": 6\n },\n {\n \"site_id_a\": \"B\",\n \"site_id_b\": \"K\",\n \"env_distance_between_sites\": 6\n },\n {\n \"site_id_a\": \"B\",\n \"site_id_b\": \"L\",\n \"env_distance_between_sites\": 7\n },\n {\n \"site_id_a\": \"B\",\n \"site_id_b\": \"M\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"B\",\n \"site_id_b\": \"N\",\n \"env_distance_between_sites\": 0\n },\n {\n \"site_id_a\": \"B\",\n \"site_id_b\": \"O\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"B\",\n \"site_id_b\": \"P\",\n \"env_distance_between_sites\": 5\n },\n {\n \"site_id_a\": \"B\",\n \"site_id_b\": \"Q\",\n 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\"site_id_a\": \"T\",\n \"site_id_b\": \"H\",\n \"env_distance_between_sites\": 0\n },\n {\n \"site_id_a\": \"T\",\n \"site_id_b\": \"I\",\n \"env_distance_between_sites\": 2\n },\n {\n \"site_id_a\": \"T\",\n \"site_id_b\": \"J\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"T\",\n \"site_id_b\": \"K\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"T\",\n \"site_id_b\": \"L\",\n \"env_distance_between_sites\": 0\n },\n {\n \"site_id_a\": \"T\",\n \"site_id_b\": \"M\",\n \"env_distance_between_sites\": 7\n },\n {\n \"site_id_a\": \"T\",\n \"site_id_b\": \"N\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"T\",\n \"site_id_b\": \"O\",\n \"env_distance_between_sites\": 7\n },\n {\n \"site_id_a\": \"T\",\n \"site_id_b\": \"P\",\n \"env_distance_between_sites\": 0\n },\n {\n \"site_id_a\": \"T\",\n \"site_id_b\": \"Q\",\n \"env_distance_between_sites\": 1\n },\n {\n \"site_id_a\": \"T\",\n \"site_id_b\": \"R\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"T\",\n \"site_id_b\": \"S\",\n \"env_distance_between_sites\": 7\n },\n {\n \"site_id_a\": \"T\",\n \"site_id_b\": \"U\",\n \"env_distance_between_sites\": 0\n },\n {\n \"site_id_a\": \"T\",\n \"site_id_b\": \"V\",\n \"env_distance_between_sites\": 2\n },\n {\n \"site_id_a\": \"T\",\n \"site_id_b\": \"W\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"T\",\n \"site_id_b\": \"X\",\n \"env_distance_between_sites\": 2\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"A\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"B\",\n \"env_distance_between_sites\": 7\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"C\",\n \"env_distance_between_sites\": 3\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"D\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"E\",\n \"env_distance_between_sites\": 7\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"F\",\n \"env_distance_between_sites\": 7\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"G\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"H\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"I\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"J\",\n \"env_distance_between_sites\": 6\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"K\",\n \"env_distance_between_sites\": 1\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"L\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"M\",\n \"env_distance_between_sites\": 5\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"N\",\n \"env_distance_between_sites\": 0\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"O\",\n \"env_distance_between_sites\": 3\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"P\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"Q\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"R\",\n \"env_distance_between_sites\": 1\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"S\",\n \"env_distance_between_sites\": 2\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"T\",\n \"env_distance_between_sites\": 0\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"V\",\n \"env_distance_between_sites\": 3\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"W\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"U\",\n \"site_id_b\": \"X\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"A\",\n \"env_distance_between_sites\": 6\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"B\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"C\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"D\",\n \"env_distance_between_sites\": 7\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"E\",\n \"env_distance_between_sites\": 2\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"F\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"G\",\n \"env_distance_between_sites\": 3\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"H\",\n \"env_distance_between_sites\": 2\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"I\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"J\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"K\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"L\",\n \"env_distance_between_sites\": 6\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"M\",\n \"env_distance_between_sites\": 2\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"N\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"O\",\n \"env_distance_between_sites\": 0\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"P\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"Q\",\n \"env_distance_between_sites\": 5\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"R\",\n \"env_distance_between_sites\": 7\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"S\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"T\",\n \"env_distance_between_sites\": 2\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"U\",\n \"env_distance_between_sites\": 3\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"W\",\n \"env_distance_between_sites\": 5\n },\n {\n \"site_id_a\": \"V\",\n \"site_id_b\": \"X\",\n \"env_distance_between_sites\": 0\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"A\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"B\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"C\",\n \"env_distance_between_sites\": 7\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"D\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"E\",\n \"env_distance_between_sites\": 7\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"F\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"G\",\n \"env_distance_between_sites\": 5\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"H\",\n \"env_distance_between_sites\": 5\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"I\",\n \"env_distance_between_sites\": 7\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"J\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"K\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"L\",\n \"env_distance_between_sites\": 1\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"M\",\n \"env_distance_between_sites\": 0\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"N\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"O\",\n \"env_distance_between_sites\": 7\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"P\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"Q\",\n \"env_distance_between_sites\": 1\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"R\",\n \"env_distance_between_sites\": 1\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"S\",\n \"env_distance_between_sites\": 2\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"T\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"U\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"V\",\n \"env_distance_between_sites\": 5\n },\n {\n \"site_id_a\": \"W\",\n \"site_id_b\": \"X\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"A\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"B\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"C\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"D\",\n \"env_distance_between_sites\": 4\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"E\",\n \"env_distance_between_sites\": 3\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"F\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"G\",\n \"env_distance_between_sites\": 5\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"H\",\n \"env_distance_between_sites\": 6\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"I\",\n \"env_distance_between_sites\": 8\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"J\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"K\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"L\",\n \"env_distance_between_sites\": 0\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"M\",\n \"env_distance_between_sites\": 0\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"N\",\n \"env_distance_between_sites\": 6\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"O\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"P\",\n \"env_distance_between_sites\": 2\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"Q\",\n \"env_distance_between_sites\": 6\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"R\",\n \"env_distance_between_sites\": 0\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"S\",\n \"env_distance_between_sites\": 6\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"T\",\n \"env_distance_between_sites\": 2\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"U\",\n \"env_distance_between_sites\": 9\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"V\",\n \"env_distance_between_sites\": 0\n },\n {\n \"site_id_a\": \"X\",\n \"site_id_b\": \"W\",\n \"env_distance_between_sites\": 4\n }\n ]\n}\n\nWhen you send the shortlist back, just use this simple JSON shape so it's easy to read programmatically:\n\n{\n \"solution\": [, , ...]\n}\n\nThink of \"solution\" as the list of the m sites you picked — just drop each site identifier in that array, exactly as it appears in the instance. This snippet is only a sketch of the expected shape, not the actual answer.\n\nPlease make sure you use the identifiers exactly as they show up in the input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 4, 8, 4, 3, 5, 1, 9, 5, 9, 7, 9, 9, 5, 7, 4, 8, 4, 7, 9, 8, 6, 8, 4 ], [ 4, 0, 7, 0, 9, 1, 6, 1, 4, 6, 6, 7, 4, 0, 9, 5, 2, 4, 4, 3, 7, 9, 9, 8 ], [ 8, 7, 0, 8, 5, 1, 5, 2, 4, 1, 2, 3, 4, 2, 3, 6, 0, 0, 7, 2, 3, 4, 7, 8 ], [ 4, 0, 8, 0, 3, 7, 3, 1, 4, 1, 6, 4, 1, 4, 6, 1, 4, 9, 3, 1, 8, 7, 4, 4 ], [ 3, 9, 5, 3, 0, 6, 3, 0, 0, 9, 4, 0, 7, 6, 0, 3, 4, 0, 6, 2, 7, 2, 7, 3 ], [ 5, 1, 1, 7, 6, 0, 9, 3, 3, 4, 8, 7, 7, 3, 9, 2, 9, 9, 2, 5, 7, 4, 8, 9 ], [ 1, 6, 5, 3, 3, 9, 0, 1, 6, 7, 2, 2, 7, 8, 0, 8, 8, 1, 6, 3, 9, 3, 5, 5 ], [ 9, 1, 2, 1, 0, 3, 1, 0, 7, 0, 3, 9, 6, 6, 0, 4, 6, 2, 3, 0, 8, 2, 5, 6 ], [ 5, 4, 4, 4, 0, 3, 6, 7, 0, 7, 5, 2, 2, 4, 2, 8, 9, 2, 6, 2, 4, 8, 7, 8 ], [ 9, 6, 1, 1, 9, 4, 7, 0, 7, 0, 7, 6, 1, 9, 6, 2, 6, 5, 7, 4, 6, 8, 9, 9 ], [ 7, 6, 2, 6, 4, 8, 2, 3, 5, 7, 0, 3, 5, 4, 2, 3, 0, 7, 4, 8, 1, 4, 8, 9 ], [ 9, 7, 3, 4, 0, 7, 2, 9, 2, 6, 3, 0, 4, 0, 7, 1, 1, 1, 9, 0, 9, 6, 1, 0 ], [ 9, 4, 4, 1, 7, 7, 7, 6, 2, 1, 5, 4, 0, 5, 3, 2, 1, 4, 2, 7, 5, 2, 0, 0 ], [ 5, 0, 2, 4, 6, 3, 8, 6, 4, 9, 4, 0, 5, 0, 2, 0, 9, 3, 4, 9, 0, 9, 9, 6 ], [ 7, 9, 3, 6, 0, 9, 0, 0, 2, 6, 2, 7, 3, 2, 0, 3, 2, 0, 6, 7, 3, 0, 7, 9 ], [ 4, 5, 6, 1, 3, 2, 8, 4, 8, 2, 3, 1, 2, 0, 3, 0, 2, 3, 5, 0, 4, 4, 4, 2 ], [ 8, 2, 0, 4, 4, 9, 8, 6, 9, 6, 0, 1, 1, 9, 2, 2, 0, 2, 4, 1, 8, 5, 1, 6 ], [ 4, 4, 0, 9, 0, 9, 1, 2, 2, 5, 7, 1, 4, 3, 0, 3, 2, 0, 5, 9, 1, 7, 1, 0 ], [ 7, 4, 7, 3, 6, 2, 6, 3, 6, 7, 4, 9, 2, 4, 6, 5, 4, 5, 0, 7, 2, 8, 2, 6 ], [ 9, 3, 2, 1, 2, 5, 3, 0, 2, 4, 8, 0, 7, 9, 7, 0, 1, 9, 7, 0, 0, 2, 9, 2 ], [ 8, 7, 3, 8, 7, 7, 9, 8, 4, 6, 1, 9, 5, 0, 3, 4, 8, 1, 2, 0, 0, 3, 9, 9 ], [ 6, 9, 4, 7, 2, 4, 3, 2, 8, 8, 4, 6, 2, 9, 0, 4, 5, 7, 8, 2, 3, 0, 5, 0 ], [ 8, 9, 7, 4, 7, 8, 5, 5, 7, 9, 8, 1, 0, 9, 7, 4, 1, 1, 2, 9, 9, 5, 0, 4 ], [ 4, 8, 8, 4, 3, 9, 5, 6, 8, 9, 9, 0, 0, 6, 9, 2, 6, 0, 6, 2, 9, 0, 4, 0 ] ], "m": 7, "objective": 150.0 }, "solution": [ 0, 1, 9, 14, 20, 22, 23 ], "obj": 150.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 24, "m": 7, "nodes": [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S", "T", "U", "V", "W", "X" ], "distance_pairs": [ { "from_id": "A", "to_id": "B", "distance": 4 }, { "from_id": "A", "to_id": "C", "distance": 8 }, { "from_id": "A", "to_id": "D", "distance": 4 }, { "from_id": "A", "to_id": "E", "distance": 3 }, { "from_id": "A", "to_id": "F", "distance": 5 }, { "from_id": "A", "to_id": "G", "distance": 1 }, { "from_id": "A", "to_id": "H", "distance": 9 }, { "from_id": "A", "to_id": "I", "distance": 5 }, { "from_id": "A", "to_id": "J", "distance": 9 }, { "from_id": 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The way to judge a playlist is to look at every pair inside it, add up those “different” scores, and prefer the playlist with the biggest total — that means the tracks are spread out in style. Each song can only be used once and the playlist must contain precisely m tracks. The exact songs and their pairwise difference numbers appear below.\n\n# total_tracks_available=24\n# playlist_size=2\n# track_ids=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23\ntrack_id_1,track_id_2,mood_genre_dissimilarity\n0,1,5\n0,2,6\n0,3,8\n0,4,9\n0,5,7\n0,6,7\n0,7,6\n0,8,2\n0,9,3\n0,10,0\n0,11,0\n0,12,7\n0,13,1\n0,14,9\n0,15,6\n0,16,6\n0,17,8\n0,18,0\n0,19,8\n0,20,6\n0,21,8\n0,22,0\n0,23,6\n1,0,5\n1,2,1\n1,3,7\n1,4,1\n1,5,6\n1,6,7\n1,7,1\n1,8,9\n1,9,3\n1,10,1\n1,11,5\n1,12,7\n1,13,1\n1,14,8\n1,15,4\n1,16,4\n1,17,3\n1,18,5\n1,19,2\n1,20,9\n1,21,4\n1,22,6\n1,23,4\n2,0,6\n2,1,1\n2,3,6\n2,4,9\n2,5,3\n2,6,1\n2,7,0\n2,8,8\n2,9,0\n2,10,2\n2,11,5\n2,12,4\n2,13,0\n2,14,7\n2,15,9\n2,16,2\n2,17,4\n2,18,8\n2,19,6\n2,20,6\n2,21,0\n2,22,7\n2,23,3\n3,0,8\n3,1,7\n3,2,6\n3,4,9\n3,5,4\n3,6,8\n3,7,3\n3,8,1\n3,9,7\n3,10,7\n3,11,9\n3,12,4\n3,13,6\n3,14,6\n3,15,9\n3,16,3\n3,17,4\n3,18,2\n3,19,0\n3,20,9\n3,21,8\n3,22,9\n3,23,7\n4,0,9\n4,1,1\n4,2,9\n4,3,9\n4,5,4\n4,6,7\n4,7,7\n4,8,8\n4,9,7\n4,10,7\n4,11,3\n4,12,6\n4,13,1\n4,14,2\n4,15,6\n4,16,8\n4,17,2\n4,18,4\n4,19,3\n4,20,2\n4,21,9\n4,22,5\n4,23,4\n5,0,7\n5,1,6\n5,2,3\n5,3,4\n5,4,4\n5,6,5\n5,7,0\n5,8,7\n5,9,9\n5,10,2\n5,11,0\n5,12,4\n5,13,8\n5,14,1\n5,15,1\n5,16,1\n5,17,3\n5,18,4\n5,19,2\n5,20,8\n5,21,2\n5,22,1\n5,23,2\n6,0,7\n6,1,7\n6,2,1\n6,3,8\n6,4,7\n6,5,5\n6,7,3\n6,8,8\n6,9,8\n6,10,9\n6,11,4\n6,12,6\n6,13,6\n6,14,1\n6,15,1\n6,16,0\n6,17,4\n6,18,1\n6,19,2\n6,20,2\n6,21,3\n6,22,8\n6,23,0\n7,0,6\n7,1,1\n7,2,0\n7,3,3\n7,4,7\n7,5,0\n7,6,3\n7,8,1\n7,9,6\n7,10,1\n7,11,3\n7,12,7\n7,13,1\n7,14,2\n7,15,8\n7,16,2\n7,17,6\n7,18,2\n7,19,8\n7,20,2\n7,21,3\n7,22,3\n7,23,4\n8,0,2\n8,1,9\n8,2,8\n8,3,1\n8,4,8\n8,5,7\n8,6,8\n8,7,1\n8,9,4\n8,10,3\n8,11,7\n8,12,4\n8,13,4\n8,14,4\n8,15,7\n8,16,5\n8,17,4\n8,18,8\n8,19,8\n8,20,0\n8,21,7\n8,22,5\n8,23,0\n9,0,3\n9,1,3\n9,2,0\n9,3,7\n9,4,7\n9,5,9\n9,6,8\n9,7,6\n9,8,4\n9,10,5\n9,11,6\n9,12,7\n9,13,8\n9,14,8\n9,15,2\n9,16,2\n9,17,8\n9,18,9\n9,19,7\n9,20,1\n9,21,9\n9,22,9\n9,23,4\n10,0,0\n10,1,1\n10,2,2\n10,3,7\n10,4,7\n10,5,2\n10,6,9\n10,7,1\n10,8,3\n10,9,5\n10,11,7\n10,12,0\n10,13,3\n10,14,5\n10,15,3\n10,16,6\n10,17,2\n10,18,2\n10,19,0\n10,20,9\n10,21,9\n10,22,7\n10,23,8\n11,0,0\n11,1,5\n11,2,5\n11,3,9\n11,4,3\n11,5,0\n11,6,4\n11,7,3\n11,8,7\n11,9,6\n11,10,7\n11,12,1\n11,13,2\n11,14,8\n11,15,6\n11,16,5\n11,17,9\n11,18,5\n11,19,0\n11,20,0\n11,21,4\n11,22,3\n11,23,7\n12,0,7\n12,1,7\n12,2,4\n12,3,4\n12,4,6\n12,5,4\n12,6,6\n12,7,7\n12,8,4\n12,9,7\n12,10,0\n12,11,1\n12,13,9\n12,14,8\n12,15,9\n12,16,3\n12,17,4\n12,18,1\n12,19,7\n12,20,8\n12,21,7\n12,22,7\n12,23,9\n13,0,1\n13,1,1\n13,2,0\n13,3,6\n13,4,1\n13,5,8\n13,6,6\n13,7,1\n13,8,4\n13,9,8\n13,10,3\n13,11,2\n13,12,9\n13,14,7\n13,15,3\n13,16,5\n13,17,4\n13,18,9\n13,19,8\n13,20,3\n13,21,9\n13,22,3\n13,23,7\n14,0,9\n14,1,8\n14,2,7\n14,3,6\n14,4,2\n14,5,1\n14,6,1\n14,7,2\n14,8,4\n14,9,8\n14,10,5\n14,11,8\n14,12,8\n14,13,7\n14,15,8\n14,16,3\n14,17,7\n14,18,0\n14,19,7\n14,20,5\n14,21,8\n14,22,4\n14,23,8\n15,0,6\n15,1,4\n15,2,9\n15,3,9\n15,4,6\n15,5,1\n15,6,1\n15,7,8\n15,8,7\n15,9,2\n15,10,3\n15,11,6\n15,12,9\n15,13,3\n15,14,8\n15,16,9\n15,17,3\n15,18,1\n15,19,6\n15,20,8\n15,21,6\n15,22,9\n15,23,7\n16,0,6\n16,1,4\n16,2,2\n16,3,3\n16,4,8\n16,5,1\n16,6,0\n16,7,2\n16,8,5\n16,9,2\n16,10,6\n16,11,5\n16,12,3\n16,13,5\n16,14,3\n16,15,9\n16,17,1\n16,18,1\n16,19,1\n16,20,4\n16,21,8\n16,22,0\n16,23,0\n17,0,8\n17,1,3\n17,2,4\n17,3,4\n17,4,2\n17,5,3\n17,6,4\n17,7,6\n17,8,4\n17,9,8\n17,10,2\n17,11,9\n17,12,4\n17,13,4\n17,14,7\n17,15,3\n17,16,1\n17,18,5\n17,19,7\n17,20,3\n17,21,6\n17,22,9\n17,23,7\n18,0,0\n18,1,5\n18,2,8\n18,3,2\n18,4,4\n18,5,4\n18,6,1\n18,7,2\n18,8,8\n18,9,9\n18,10,2\n18,11,5\n18,12,1\n18,13,9\n18,14,0\n18,15,1\n18,16,1\n18,17,5\n18,19,4\n18,20,2\n18,21,2\n18,22,1\n18,23,7\n19,0,8\n19,1,2\n19,2,6\n19,3,0\n19,4,3\n19,5,2\n19,6,2\n19,7,8\n19,8,8\n19,9,7\n19,10,0\n19,11,0\n19,12,7\n19,13,8\n19,14,7\n19,15,6\n19,16,1\n19,17,7\n19,18,4\n19,20,4\n19,21,2\n19,22,2\n19,23,8\n20,0,6\n20,1,9\n20,2,6\n20,3,9\n20,4,2\n20,5,8\n20,6,2\n20,7,2\n20,8,0\n20,9,1\n20,10,9\n20,11,0\n20,12,8\n20,13,3\n20,14,5\n20,15,8\n20,16,4\n20,17,3\n20,18,2\n20,19,4\n20,21,5\n20,22,1\n20,23,8\n21,0,8\n21,1,4\n21,2,0\n21,3,8\n21,4,9\n21,5,2\n21,6,3\n21,7,3\n21,8,7\n21,9,9\n21,10,9\n21,11,4\n21,12,7\n21,13,9\n21,14,8\n21,15,6\n21,16,8\n21,17,6\n21,18,2\n21,19,2\n21,20,5\n21,22,6\n21,23,6\n22,0,0\n22,1,6\n22,2,7\n22,3,9\n22,4,5\n22,5,1\n22,6,8\n22,7,3\n22,8,5\n22,9,9\n22,10,7\n22,11,3\n22,12,7\n22,13,3\n22,14,4\n22,15,9\n22,16,0\n22,17,9\n22,18,1\n22,19,2\n22,20,1\n22,21,6\n22,23,4\n23,0,6\n23,1,4\n23,2,3\n23,3,7\n23,4,4\n23,5,2\n23,6,0\n23,7,4\n23,8,0\n23,9,4\n23,10,8\n23,11,7\n23,12,9\n23,13,7\n23,14,8\n23,15,7\n23,16,0\n23,17,7\n23,18,7\n23,19,8\n23,20,8\n23,21,6\n23,22,4\n\nAlso, when you’re ready to send back your chosen playlist, just drop it into a tiny JSON snippet like this so it’s easy to read and parse:\n\n{\n \"solution\": [, , ...]\n}\n\nHere \"solution\" is just the list of the m songs you picked — put each song’s identifier exactly as it appears in the instance (that is, the labels from the song list). Think of this JSON as a simple form: the key is the field name and the array is where you list the selected tracks. This is only a sketch of the expected shape, not the actual answer.\n\nPlease use the identifiers exactly as given in the input — no renaming and no new labels. \nValid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.", "instance": { "distance_matrix": [ [ 0, 5, 6, 8, 9, 7, 7, 6, 2, 3, 0, 0, 7, 1, 9, 6, 6, 8, 0, 8, 6, 8, 0, 6 ], [ 5, 0, 1, 7, 1, 6, 7, 1, 9, 3, 1, 5, 7, 1, 8, 4, 4, 3, 5, 2, 9, 4, 6, 4 ], [ 6, 1, 0, 6, 9, 3, 1, 0, 8, 0, 2, 5, 4, 0, 7, 9, 2, 4, 8, 6, 6, 0, 7, 3 ], [ 8, 7, 6, 0, 9, 4, 8, 3, 1, 7, 7, 9, 4, 6, 6, 9, 3, 4, 2, 0, 9, 8, 9, 7 ], [ 9, 1, 9, 9, 0, 4, 7, 7, 8, 7, 7, 3, 6, 1, 2, 6, 8, 2, 4, 3, 2, 9, 5, 4 ], [ 7, 6, 3, 4, 4, 0, 5, 0, 7, 9, 2, 0, 4, 8, 1, 1, 1, 3, 4, 2, 8, 2, 1, 2 ], [ 7, 7, 1, 8, 7, 5, 0, 3, 8, 8, 9, 4, 6, 6, 1, 1, 0, 4, 1, 2, 2, 3, 8, 0 ], [ 6, 1, 0, 3, 7, 0, 3, 0, 1, 6, 1, 3, 7, 1, 2, 8, 2, 6, 2, 8, 2, 3, 3, 4 ], [ 2, 9, 8, 1, 8, 7, 8, 1, 0, 4, 3, 7, 4, 4, 4, 7, 5, 4, 8, 8, 0, 7, 5, 0 ], [ 3, 3, 0, 7, 7, 9, 8, 6, 4, 0, 5, 6, 7, 8, 8, 2, 2, 8, 9, 7, 1, 9, 9, 4 ], [ 0, 1, 2, 7, 7, 2, 9, 1, 3, 5, 0, 7, 0, 3, 5, 3, 6, 2, 2, 0, 9, 9, 7, 8 ], [ 0, 5, 5, 9, 3, 0, 4, 3, 7, 6, 7, 0, 1, 2, 8, 6, 5, 9, 5, 0, 0, 4, 3, 7 ], [ 7, 7, 4, 4, 6, 4, 6, 7, 4, 7, 0, 1, 0, 9, 8, 9, 3, 4, 1, 7, 8, 7, 7, 9 ], [ 1, 1, 0, 6, 1, 8, 6, 1, 4, 8, 3, 2, 9, 0, 7, 3, 5, 4, 9, 8, 3, 9, 3, 7 ], [ 9, 8, 7, 6, 2, 1, 1, 2, 4, 8, 5, 8, 8, 7, 0, 8, 3, 7, 0, 7, 5, 8, 4, 8 ], [ 6, 4, 9, 9, 6, 1, 1, 8, 7, 2, 3, 6, 9, 3, 8, 0, 9, 3, 1, 6, 8, 6, 9, 7 ], [ 6, 4, 2, 3, 8, 1, 0, 2, 5, 2, 6, 5, 3, 5, 3, 9, 0, 1, 1, 1, 4, 8, 0, 0 ], [ 8, 3, 4, 4, 2, 3, 4, 6, 4, 8, 2, 9, 4, 4, 7, 3, 1, 0, 5, 7, 3, 6, 9, 7 ], [ 0, 5, 8, 2, 4, 4, 1, 2, 8, 9, 2, 5, 1, 9, 0, 1, 1, 5, 0, 4, 2, 2, 1, 7 ], [ 8, 2, 6, 0, 3, 2, 2, 8, 8, 7, 0, 0, 7, 8, 7, 6, 1, 7, 4, 0, 4, 2, 2, 8 ], [ 6, 9, 6, 9, 2, 8, 2, 2, 0, 1, 9, 0, 8, 3, 5, 8, 4, 3, 2, 4, 0, 5, 1, 8 ], [ 8, 4, 0, 8, 9, 2, 3, 3, 7, 9, 9, 4, 7, 9, 8, 6, 8, 6, 2, 2, 5, 0, 6, 6 ], [ 0, 6, 7, 9, 5, 1, 8, 3, 5, 9, 7, 3, 7, 3, 4, 9, 0, 9, 1, 2, 1, 6, 0, 4 ], [ 6, 4, 3, 7, 4, 2, 0, 4, 0, 4, 8, 7, 9, 7, 8, 7, 0, 7, 7, 8, 8, 6, 4, 0 ] ], "m": 2, "objective": 9.0 }, "solution": [ 12, 23 ], "obj": 9.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 24, "m": 2, "nodes": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 ], "distance_pairs": [ { "from_id": 0, "to_id": 1, "distance": 5 }, { "from_id": 0, "to_id": 2, "distance": 6 }, { "from_id": 0, "to_id": 3, "distance": 8 }, { "from_id": 0, "to_id": 4, "distance": 9 }, { "from_id": 0, "to_id": 5, "distance": 7 }, { "from_id": 0, "to_id": 6, "distance": 7 }, { "from_id": 0, "to_id": 7, "distance": 6 }, { "from_id": 0, "to_id": 8, "distance": 2 }, { "from_id": 0, "to_id": 9, "distance": 3 }, { "from_id": 0, "to_id": 10, "distance": 0 }, { "from_id": 0, "to_id": 11, "distance": 0 }, { "from_id": 0, "to_id": 12, "distance": 7 }, { "from_id": 0, "to_id": 13, "distance": 1 }, { "from_id": 0, "to_id": 14, "distance": 9 }, { "from_id": 0, "to_id": 15, "distance": 6 }, { "from_id": 0, "to_id": 16, "distance": 6 }, { "from_id": 0, "to_id": 17, "distance": 8 }, { "from_id": 0, "to_id": 18, "distance": 0 }, { "from_id": 0, "to_id": 19, "distance": 8 }, { "from_id": 0, "to_id": 20, "distance": 6 }, { "from_id": 0, "to_id": 21, "distance": 8 }, { "from_id": 0, "to_id": 22, "distance": 0 }, { "from_id": 0, "to_id": 23, "distance": 6 }, { "from_id": 1, "to_id": 0, "distance": 5 }, { "from_id": 1, "to_id": 2, "distance": 1 }, { "from_id": 1, "to_id": 3, "distance": 7 }, { "from_id": 1, "to_id": 4, "distance": 1 }, { "from_id": 1, "to_id": 5, "distance": 6 }, { "from_id": 1, "to_id": 6, "distance": 7 }, { "from_id": 1, "to_id": 7, "distance": 1 }, { "from_id": 1, "to_id": 8, "distance": 9 }, { "from_id": 1, "to_id": 9, "distance": 3 }, { "from_id": 1, "to_id": 10, "distance": 1 }, { "from_id": 1, "to_id": 11, "distance": 5 }, { "from_id": 1, "to_id": 12, "distance": 7 }, { "from_id": 1, "to_id": 13, "distance": 1 }, { "from_id": 1, "to_id": 14, "distance": 8 }, { "from_id": 1, "to_id": 15, "distance": 4 }, { "from_id": 1, "to_id": 16, "distance": 4 }, { "from_id": 1, "to_id": 17, "distance": 3 }, { "from_id": 1, "to_id": 18, "distance": 5 }, { "from_id": 1, "to_id": 19, "distance": 2 }, { "from_id": 1, "to_id": 20, "distance": 9 }, { "from_id": 1, "to_id": 21, "distance": 4 }, { "from_id": 1, "to_id": 22, "distance": 6 }, { "from_id": 1, "to_id": 23, "distance": 4 }, { "from_id": 2, "to_id": 0, "distance": 6 }, { "from_id": 2, "to_id": 1, "distance": 1 }, { "from_id": 2, "to_id": 3, "distance": 6 }, { "from_id": 2, "to_id": 4, "distance": 9 }, { "from_id": 2, "to_id": 5, "distance": 3 }, { "from_id": 2, "to_id": 6, "distance": 1 }, { "from_id": 2, "to_id": 7, "distance": 0 }, { "from_id": 2, "to_id": 8, "distance": 8 }, { "from_id": 2, "to_id": 9, "distance": 0 }, { "from_id": 2, "to_id": 10, "distance": 2 }, { "from_id": 2, "to_id": 11, "distance": 5 }, { "from_id": 2, "to_id": 12, "distance": 4 }, { "from_id": 2, "to_id": 13, "distance": 0 }, { "from_id": 2, "to_id": 14, "distance": 7 }, { "from_id": 2, "to_id": 15, "distance": 9 }, { "from_id": 2, "to_id": 16, "distance": 2 }, { "from_id": 2, "to_id": 17, "distance": 4 }, { "from_id": 2, "to_id": 18, "distance": 8 }, { "from_id": 2, "to_id": 19, "distance": 6 }, { "from_id": 2, "to_id": 20, "distance": 6 }, { "from_id": 2, "to_id": 21, "distance": 0 }, { "from_id": 2, "to_id": 22, "distance": 7 }, { "from_id": 2, "to_id": 23, "distance": 3 }, { "from_id": 3, "to_id": 0, "distance": 8 }, { "from_id": 3, "to_id": 1, "distance": 7 }, { "from_id": 3, "to_id": 2, "distance": 6 }, { "from_id": 3, "to_id": 4, "distance": 9 }, { "from_id": 3, "to_id": 5, "distance": 4 }, { "from_id": 3, "to_id": 6, "distance": 8 }, { "from_id": 3, "to_id": 7, "distance": 3 }, { "from_id": 3, "to_id": 8, "distance": 1 }, { "from_id": 3, "to_id": 9, "distance": 7 }, 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"solution_variant": [ 12, 23 ], "context_index": 15, "input_format": "csv", "input_index_base": 0 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "We were tasked by a buyer to assemble a tight capsule of a set size, and the brief was simple: make the picks as varied as possible. Each pair of garments has a given style dissimilarity, and the quality of the capsule is judged by adding up those dissimilarities for every pair among the chosen pieces — higher total means a more spread-out collection. The selection must be exactly m distinct items, nothing doubled up, and the item-by-item distances to use are listed below.\n\n{\n \"total_garments_available\": 19,\n \"capsule_size\": 15,\n \"garment_ids\": [\n \"A\",\n \"B\",\n \"C\",\n \"D\",\n \"E\",\n \"F\",\n \"G\",\n \"H\",\n \"I\",\n \"J\",\n \"K\",\n \"L\",\n \"M\",\n \"N\",\n \"O\",\n \"P\",\n \"Q\",\n \"R\",\n \"S\"\n ],\n \"data\": [\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"B\",\n \"style_dissimilarity\": 160\n },\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"C\",\n \"style_dissimilarity\": 203\n },\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"D\",\n \"style_dissimilarity\": 165\n },\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"E\",\n \"style_dissimilarity\": 205\n },\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"F\",\n \"style_dissimilarity\": 142\n },\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"G\",\n \"style_dissimilarity\": 184\n },\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"H\",\n \"style_dissimilarity\": 188\n },\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"I\",\n \"style_dissimilarity\": 147\n },\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"J\",\n \"style_dissimilarity\": 179\n },\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"K\",\n \"style_dissimilarity\": 154\n },\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"L\",\n \"style_dissimilarity\": 195\n },\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"M\",\n \"style_dissimilarity\": 202\n },\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"N\",\n \"style_dissimilarity\": 154\n },\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"O\",\n \"style_dissimilarity\": 175\n },\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"P\",\n \"style_dissimilarity\": 162\n },\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"Q\",\n \"style_dissimilarity\": 170\n },\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"R\",\n \"style_dissimilarity\": 144\n },\n {\n \"garment_id_a\": \"A\",\n \"garment_id_b\": \"S\",\n \"style_dissimilarity\": 148\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"A\",\n \"style_dissimilarity\": 160\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"C\",\n \"style_dissimilarity\": 176\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"D\",\n \"style_dissimilarity\": 153\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"E\",\n \"style_dissimilarity\": 173\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"F\",\n \"style_dissimilarity\": 169\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"G\",\n \"style_dissimilarity\": 155\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"H\",\n \"style_dissimilarity\": 177\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"I\",\n \"style_dissimilarity\": 159\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"J\",\n \"style_dissimilarity\": 143\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"K\",\n \"style_dissimilarity\": 183\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"L\",\n \"style_dissimilarity\": 150\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"M\",\n \"style_dissimilarity\": 178\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"N\",\n \"style_dissimilarity\": 196\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"O\",\n \"style_dissimilarity\": 187\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"P\",\n \"style_dissimilarity\": 202\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"Q\",\n \"style_dissimilarity\": 180\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"R\",\n \"style_dissimilarity\": 132\n },\n {\n \"garment_id_a\": \"B\",\n \"garment_id_b\": \"S\",\n \"style_dissimilarity\": 174\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"A\",\n \"style_dissimilarity\": 203\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"B\",\n \"style_dissimilarity\": 176\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"D\",\n \"style_dissimilarity\": 147\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"E\",\n \"style_dissimilarity\": 169\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"F\",\n \"style_dissimilarity\": 208\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"G\",\n \"style_dissimilarity\": 152\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"H\",\n \"style_dissimilarity\": 189\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"I\",\n \"style_dissimilarity\": 190\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"J\",\n \"style_dissimilarity\": 190\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"K\",\n \"style_dissimilarity\": 160\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"L\",\n \"style_dissimilarity\": 164\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"M\",\n \"style_dissimilarity\": 149\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"N\",\n \"style_dissimilarity\": 223\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"O\",\n \"style_dissimilarity\": 214\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"P\",\n \"style_dissimilarity\": 191\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"Q\",\n \"style_dissimilarity\": 173\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"R\",\n \"style_dissimilarity\": 196\n },\n {\n \"garment_id_a\": \"C\",\n \"garment_id_b\": \"S\",\n \"style_dissimilarity\": 199\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"A\",\n \"style_dissimilarity\": 165\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"B\",\n \"style_dissimilarity\": 153\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"C\",\n \"style_dissimilarity\": 147\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"E\",\n \"style_dissimilarity\": 180\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"F\",\n \"style_dissimilarity\": 169\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"G\",\n \"style_dissimilarity\": 124\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"H\",\n \"style_dissimilarity\": 179\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"I\",\n \"style_dissimilarity\": 203\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"J\",\n \"style_dissimilarity\": 140\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"K\",\n \"style_dissimilarity\": 177\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"L\",\n \"style_dissimilarity\": 206\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"M\",\n \"style_dissimilarity\": 145\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"N\",\n \"style_dissimilarity\": 190\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"O\",\n \"style_dissimilarity\": 153\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"P\",\n \"style_dissimilarity\": 178\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"Q\",\n \"style_dissimilarity\": 179\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"R\",\n \"style_dissimilarity\": 150\n },\n {\n \"garment_id_a\": \"D\",\n \"garment_id_b\": \"S\",\n \"style_dissimilarity\": 165\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"A\",\n \"style_dissimilarity\": 205\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"B\",\n \"style_dissimilarity\": 173\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"C\",\n \"style_dissimilarity\": 169\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"D\",\n \"style_dissimilarity\": 180\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"F\",\n \"style_dissimilarity\": 166\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"G\",\n \"style_dissimilarity\": 182\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"H\",\n \"style_dissimilarity\": 210\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"I\",\n \"style_dissimilarity\": 154\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"J\",\n \"style_dissimilarity\": 182\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"K\",\n \"style_dissimilarity\": 155\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"L\",\n \"style_dissimilarity\": 186\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"M\",\n \"style_dissimilarity\": 152\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"N\",\n \"style_dissimilarity\": 196\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"O\",\n \"style_dissimilarity\": 199\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"P\",\n \"style_dissimilarity\": 226\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"Q\",\n \"style_dissimilarity\": 179\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"R\",\n \"style_dissimilarity\": 157\n },\n {\n \"garment_id_a\": \"E\",\n \"garment_id_b\": \"S\",\n \"style_dissimilarity\": 178\n },\n {\n \"garment_id_a\": \"F\",\n \"garment_id_b\": \"A\",\n \"style_dissimilarity\": 142\n },\n {\n \"garment_id_a\": \"F\",\n \"garment_id_b\": \"B\",\n \"style_dissimilarity\": 169\n },\n {\n \"garment_id_a\": \"F\",\n \"garment_id_b\": \"C\",\n \"style_dissimilarity\": 208\n },\n {\n \"garment_id_a\": \"F\",\n \"garment_id_b\": \"D\",\n \"style_dissimilarity\": 169\n },\n {\n \"garment_id_a\": \"F\",\n \"garment_id_b\": \"E\",\n \"style_dissimilarity\": 166\n },\n {\n \"garment_id_a\": \"F\",\n \"garment_id_b\": \"G\",\n \"style_dissimilarity\": 180\n },\n {\n \"garment_id_a\": \"F\",\n \"garment_id_b\": \"H\",\n \"style_dissimilarity\": 159\n },\n {\n \"garment_id_a\": \"F\",\n \"garment_id_b\": \"I\",\n \"style_dissimilarity\": 140\n },\n {\n \"garment_id_a\": \"F\",\n \"garment_id_b\": \"J\",\n \"style_dissimilarity\": 134\n },\n {\n \"garment_id_a\": \"F\",\n \"garment_id_b\": \"K\",\n \"style_dissimilarity\": 136\n },\n {\n \"garment_id_a\": \"F\",\n \"garment_id_b\": \"L\",\n \"style_dissimilarity\": 198\n },\n {\n \"garment_id_a\": \"F\",\n \"garment_id_b\": \"M\",\n 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\"E\",\n \"style_dissimilarity\": 179\n },\n {\n \"garment_id_a\": \"Q\",\n \"garment_id_b\": \"F\",\n \"style_dissimilarity\": 155\n },\n {\n \"garment_id_a\": \"Q\",\n \"garment_id_b\": \"G\",\n \"style_dissimilarity\": 178\n },\n {\n \"garment_id_a\": \"Q\",\n \"garment_id_b\": \"H\",\n \"style_dissimilarity\": 194\n },\n {\n \"garment_id_a\": \"Q\",\n \"garment_id_b\": \"I\",\n \"style_dissimilarity\": 166\n },\n {\n \"garment_id_a\": \"Q\",\n \"garment_id_b\": \"J\",\n \"style_dissimilarity\": 161\n },\n {\n \"garment_id_a\": \"Q\",\n \"garment_id_b\": \"K\",\n \"style_dissimilarity\": 178\n },\n {\n \"garment_id_a\": \"Q\",\n \"garment_id_b\": \"L\",\n \"style_dissimilarity\": 181\n },\n {\n \"garment_id_a\": \"Q\",\n \"garment_id_b\": \"M\",\n \"style_dissimilarity\": 175\n },\n {\n \"garment_id_a\": \"Q\",\n \"garment_id_b\": \"N\",\n \"style_dissimilarity\": 189\n },\n {\n \"garment_id_a\": \"Q\",\n \"garment_id_b\": \"O\",\n \"style_dissimilarity\": 213\n },\n {\n \"garment_id_a\": \"Q\",\n \"garment_id_b\": \"P\",\n \"style_dissimilarity\": 164\n },\n {\n \"garment_id_a\": \"Q\",\n \"garment_id_b\": \"R\",\n \"style_dissimilarity\": 159\n },\n {\n \"garment_id_a\": \"Q\",\n \"garment_id_b\": \"S\",\n \"style_dissimilarity\": 153\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"A\",\n \"style_dissimilarity\": 144\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"B\",\n \"style_dissimilarity\": 132\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"C\",\n \"style_dissimilarity\": 196\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"D\",\n \"style_dissimilarity\": 150\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"E\",\n \"style_dissimilarity\": 157\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"F\",\n \"style_dissimilarity\": 124\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"G\",\n \"style_dissimilarity\": 172\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"H\",\n \"style_dissimilarity\": 193\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"I\",\n \"style_dissimilarity\": 147\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"J\",\n \"style_dissimilarity\": 115\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"K\",\n \"style_dissimilarity\": 167\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"L\",\n \"style_dissimilarity\": 171\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"M\",\n \"style_dissimilarity\": 188\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"N\",\n \"style_dissimilarity\": 132\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"O\",\n \"style_dissimilarity\": 171\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"P\",\n \"style_dissimilarity\": 209\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"Q\",\n \"style_dissimilarity\": 159\n },\n {\n \"garment_id_a\": \"R\",\n \"garment_id_b\": \"S\",\n \"style_dissimilarity\": 162\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"A\",\n \"style_dissimilarity\": 148\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"B\",\n \"style_dissimilarity\": 174\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"C\",\n \"style_dissimilarity\": 199\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"D\",\n \"style_dissimilarity\": 165\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"E\",\n \"style_dissimilarity\": 178\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"F\",\n \"style_dissimilarity\": 70\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"G\",\n \"style_dissimilarity\": 176\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"H\",\n \"style_dissimilarity\": 164\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"I\",\n \"style_dissimilarity\": 156\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"J\",\n \"style_dissimilarity\": 158\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"K\",\n \"style_dissimilarity\": 137\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"L\",\n \"style_dissimilarity\": 193\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"M\",\n \"style_dissimilarity\": 168\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"N\",\n \"style_dissimilarity\": 190\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"O\",\n \"style_dissimilarity\": 176\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"P\",\n \"style_dissimilarity\": 190\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"Q\",\n \"style_dissimilarity\": 153\n },\n {\n \"garment_id_a\": \"S\",\n \"garment_id_b\": \"R\",\n \"style_dissimilarity\": 162\n }\n ]\n}\n\nI'll keep the final reply in a tiny, predictable JSON sketch so you (or a script) can grab the picks easily.\n\n{\n \"solution\": [, , ...]\n}\n\nThis just shows the shape I’ll use: \"solution\" is the array that will hold the m chosen items. Think of it like a little shopping list — each entry should be one item identifier from the instance. The block above is only a sketch of the expected shape, not the actual answer.\n\nPlease use the exact identifiers from the instance input — do not rename them or add new labels. For example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 160, 203, 165, 205, 142, 184, 188, 147, 179, 154, 195, 202, 154, 175, 162, 170, 144, 148 ], [ 160, 0, 176, 153, 173, 169, 155, 177, 159, 143, 183, 150, 178, 196, 187, 202, 180, 132, 174 ], [ 203, 176, 0, 147, 169, 208, 152, 189, 190, 190, 160, 164, 149, 223, 214, 191, 173, 196, 199 ], [ 165, 153, 147, 0, 180, 169, 124, 179, 203, 140, 177, 206, 145, 190, 153, 178, 179, 150, 165 ], [ 205, 173, 169, 180, 0, 166, 182, 210, 154, 182, 155, 186, 152, 196, 199, 226, 179, 157, 178 ], [ 142, 169, 208, 169, 166, 0, 180, 159, 140, 134, 136, 198, 179, 162, 180, 189, 155, 124, 70 ], [ 184, 155, 152, 124, 182, 180, 0, 194, 205, 128, 189, 191, 144, 213, 138, 161, 178, 172, 176 ], [ 188, 177, 189, 179, 210, 159, 194, 0, 196, 174, 136, 216, 177, 175, 189, 170, 194, 193, 164 ], [ 147, 159, 190, 203, 154, 140, 205, 196, 0, 188, 149, 160, 196, 160, 201, 211, 166, 147, 156 ], [ 179, 143, 190, 140, 182, 134, 128, 174, 188, 0, 172, 197, 145, 172, 162, 173, 161, 115, 158 ], [ 154, 183, 160, 177, 155, 136, 189, 136, 149, 172, 0, 193, 163, 162, 182, 178, 178, 167, 137 ], [ 195, 150, 164, 206, 186, 198, 191, 216, 160, 197, 193, 0, 180, 197, 227, 214, 181, 171, 193 ], [ 202, 178, 149, 145, 152, 179, 144, 177, 196, 145, 163, 180, 0, 204, 162, 162, 175, 188, 168 ], [ 154, 196, 223, 190, 196, 162, 213, 175, 160, 172, 162, 197, 204, 0, 197, 189, 189, 132, 190 ], [ 175, 187, 214, 153, 199, 180, 138, 189, 201, 162, 182, 227, 162, 197, 0, 207, 213, 171, 176 ], [ 162, 202, 191, 178, 226, 189, 161, 170, 211, 173, 178, 214, 162, 189, 207, 0, 164, 209, 190 ], [ 170, 180, 173, 179, 179, 155, 178, 194, 166, 161, 178, 181, 175, 189, 213, 164, 0, 159, 153 ], [ 144, 132, 196, 150, 157, 124, 172, 193, 147, 115, 167, 171, 188, 132, 171, 209, 159, 0, 162 ], [ 148, 174, 199, 165, 178, 70, 176, 164, 156, 158, 137, 193, 168, 190, 176, 190, 153, 162, 0 ] ], "m": 15, "objective": 18904.0 }, "solution": [ 0, 1, 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 15, 16, 18 ], "obj": 18904.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 19, "m": 15, "nodes": [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S" ], "distance_pairs": [ { "from_id": "A", "to_id": "B", "distance": 160 }, { "from_id": "A", "to_id": "C", "distance": 203 }, { "from_id": "A", "to_id": "D", "distance": 165 }, { "from_id": "A", "to_id": "E", "distance": 205 }, { "from_id": "A", "to_id": "F", "distance": 142 }, { "from_id": "A", "to_id": "G", "distance": 184 }, { "from_id": "A", "to_id": "H", "distance": 188 }, { "from_id": "A", "to_id": "I", "distance": 147 }, { "from_id": "A", "to_id": "J", "distance": 179 }, { "from_id": "A", "to_id": "K", "distance": 154 }, { "from_id": "A", "to_id": "L", "distance": 195 }, { "from_id": "A", "to_id": "M", "distance": 202 }, { "from_id": "A", "to_id": "N", "distance": 154 }, { "from_id": "A", "to_id": "O", "distance": 175 }, { "from_id": "A", "to_id": "P", "distance": 162 }, { "from_id": "A", "to_id": "Q", "distance": 170 }, { "from_id": "A", "to_id": "R", "distance": 144 }, { "from_id": "A", "to_id": "S", "distance": 148 }, { "from_id": "B", "to_id": "A", "distance": 160 }, { "from_id": "B", "to_id": "C", "distance": 176 }, { "from_id": "B", "to_id": "D", "distance": 153 }, { "from_id": "B", "to_id": "E", "distance": 173 }, { "from_id": "B", "to_id": "F", "distance": 169 }, { "from_id": "B", "to_id": "G", "distance": 155 }, { "from_id": "B", "to_id": "H", "distance": 177 }, { "from_id": "B", "to_id": "I", "distance": 159 }, { "from_id": "B", "to_id": "J", "distance": 143 }, { "from_id": "B", "to_id": "K", "distance": 183 }, { "from_id": "B", "to_id": "L", "distance": 150 }, { "from_id": "B", "to_id": "M", "distance": 178 }, { "from_id": "B", "to_id": "N", "distance": 196 }, { "from_id": "B", "to_id": "O", "distance": 187 }, { "from_id": "B", "to_id": "P", "distance": 202 }, { "from_id": "B", "to_id": "Q", "distance": 180 }, { "from_id": "B", "to_id": "R", "distance": 132 }, { "from_id": "B", "to_id": "S", "distance": 174 }, { "from_id": "C", "to_id": "A", "distance": 203 }, { "from_id": "C", "to_id": "B", "distance": 176 }, { "from_id": "C", "to_id": "D", "distance": 147 }, { "from_id": "C", "to_id": "E", "distance": 169 }, { "from_id": "C", "to_id": "F", "distance": 208 }, { "from_id": "C", "to_id": "G", "distance": 152 }, { "from_id": "C", "to_id": "H", "distance": 189 }, { "from_id": "C", "to_id": "I", "distance": 190 }, { "from_id": "C", "to_id": "J", "distance": 190 }, { "from_id": "C", "to_id": "K", "distance": 160 }, { "from_id": "C", "to_id": "L", "distance": 164 }, { "from_id": "C", "to_id": "M", "distance": 149 }, { "from_id": "C", "to_id": "N", "distance": 223 }, { "from_id": "C", "to_id": "O", "distance": 214 }, { "from_id": "C", "to_id": "P", "distance": 191 }, { "from_id": "C", "to_id": "Q", "distance": 173 }, { "from_id": "C", "to_id": "R", "distance": 196 }, { "from_id": "C", "to_id": "S", "distance": 199 }, { "from_id": "D", "to_id": "A", "distance": 165 }, { "from_id": "D", "to_id": "B", "distance": 153 }, { "from_id": "D", "to_id": "C", "distance": 147 }, { "from_id": "D", "to_id": "E", "distance": 180 }, { "from_id": "D", "to_id": "F", "distance": 169 }, { "from_id": "D", "to_id": "G", "distance": 124 }, { "from_id": "D", "to_id": "H", "distance": 179 }, { "from_id": "D", "to_id": "I", "distance": 203 }, { "from_id": "D", "to_id": "J", "distance": 140 }, { "from_id": "D", "to_id": "K", "distance": 177 }, { "from_id": "D", "to_id": "L", "distance": 206 }, { "from_id": "D", "to_id": "M", "distance": 145 }, { "from_id": "D", "to_id": "N", "distance": 190 }, { "from_id": "D", "to_id": "O", "distance": 153 }, { "from_id": "D", "to_id": "P", "distance": 178 }, { "from_id": "D", "to_id": "Q", "distance": 179 }, { "from_id": "D", "to_id": "R", "distance": 150 }, { "from_id": "D", "to_id": "S", "distance": 165 }, { "from_id": "E", "to_id": "A", "distance": 205 }, { "from_id": "E", "to_id": "B", "distance": 173 }, { "from_id": "E", "to_id": "C", "distance": 169 }, { "from_id": "E", "to_id": "D", "distance": 180 }, { "from_id": "E", "to_id": "F", "distance": 166 }, { "from_id": "E", "to_id": "G", "distance": 182 }, { "from_id": "E", "to_id": "H", "distance": 210 }, { "from_id": "E", "to_id": "I", "distance": 154 }, { "from_id": "E", "to_id": "J", "distance": 182 }, { "from_id": "E", "to_id": "K", "distance": 155 }, { "from_id": "E", "to_id": "L", "distance": 186 }, { "from_id": "E", "to_id": "M", "distance": 152 }, { "from_id": "E", "to_id": "N", "distance": 196 }, { "from_id": "E", "to_id": "O", "distance": 199 }, { "from_id": "E", "to_id": "P", "distance": 226 }, { "from_id": "E", "to_id": "Q", "distance": 179 }, { "from_id": "E", "to_id": "R", "distance": 157 }, { "from_id": "E", "to_id": "S", "distance": 178 }, { "from_id": "F", "to_id": "A", "distance": 142 }, { "from_id": "F", "to_id": "B", "distance": 169 }, { "from_id": "F", "to_id": "C", "distance": 208 }, { "from_id": "F", "to_id": "D", "distance": 169 }, { "from_id": "F", "to_id": "E", "distance": 166 }, { "from_id": "F", "to_id": "G", "distance": 180 }, { "from_id": "F", "to_id": "H", "distance": 159 }, { "from_id": "F", "to_id": "I", "distance": 140 }, { "from_id": "F", "to_id": "J", "distance": 134 }, { "from_id": "F", "to_id": "K", "distance": 136 }, { "from_id": "F", "to_id": "L", "distance": 198 }, { "from_id": "F", "to_id": "M", "distance": 179 }, { "from_id": "F", "to_id": "N", "distance": 162 }, { "from_id": "F", "to_id": "O", "distance": 180 }, { "from_id": "F", "to_id": "P", "distance": 189 }, { "from_id": "F", "to_id": "Q", "distance": 155 }, { "from_id": "F", "to_id": "R", "distance": 124 }, { "from_id": "F", "to_id": "S", "distance": 70 }, { "from_id": "G", "to_id": "A", "distance": 184 }, { "from_id": "G", "to_id": "B", "distance": 155 }, { "from_id": "G", "to_id": "C", "distance": 152 }, { "from_id": "G", "to_id": "D", "distance": 124 }, { "from_id": "G", "to_id": "E", "distance": 182 }, { "from_id": "G", "to_id": "F", "distance": 180 }, { "from_id": "G", "to_id": "H", "distance": 194 }, { "from_id": "G", "to_id": "I", "distance": 205 }, { "from_id": "G", "to_id": "J", "distance": 128 }, { "from_id": "G", "to_id": "K", "distance": 189 }, { "from_id": "G", "to_id": "L", "distance": 191 }, { "from_id": "G", "to_id": "M", "distance": 144 }, { "from_id": "G", "to_id": "N", "distance": 213 }, { "from_id": "G", "to_id": "O", "distance": 138 }, { "from_id": "G", "to_id": "P", "distance": 161 }, { "from_id": "G", "to_id": "Q", "distance": 178 }, { "from_id": "G", "to_id": "R", "distance": 172 }, { "from_id": "G", "to_id": "S", "distance": 176 }, { "from_id": "H", "to_id": "A", "distance": 188 }, { "from_id": "H", "to_id": "B", "distance": 177 }, { "from_id": "H", "to_id": "C", "distance": 189 }, { "from_id": "H", "to_id": "D", "distance": 179 }, { "from_id": "H", "to_id": "E", "distance": 210 }, { "from_id": "H", "to_id": "F", "distance": 159 }, { "from_id": "H", "to_id": "G", "distance": 194 }, { "from_id": "H", "to_id": "I", "distance": 196 }, { "from_id": "H", "to_id": "J", "distance": 174 }, { "from_id": "H", "to_id": "K", "distance": 136 }, { "from_id": "H", "to_id": "L", "distance": 216 }, { "from_id": "H", "to_id": "M", "distance": 177 }, { "from_id": "H", "to_id": "N", "distance": 175 }, { "from_id": "H", "to_id": "O", "distance": 189 }, { "from_id": "H", "to_id": "P", "distance": 170 }, { "from_id": "H", "to_id": "Q", "distance": 194 }, { "from_id": "H", "to_id": "R", "distance": 193 }, { "from_id": "H", "to_id": "S", "distance": 164 }, { "from_id": "I", "to_id": "A", "distance": 147 }, { "from_id": "I", "to_id": "B", "distance": 159 }, { "from_id": "I", "to_id": "C", "distance": 190 }, { "from_id": "I", "to_id": "D", "distance": 203 }, { "from_id": "I", "to_id": "E", "distance": 154 }, { "from_id": "I", "to_id": "F", "distance": 140 }, { "from_id": "I", "to_id": "G", "distance": 205 }, { "from_id": "I", "to_id": "H", "distance": 196 }, { "from_id": "I", "to_id": "J", "distance": 188 }, { "from_id": "I", "to_id": "K", "distance": 149 }, { "from_id": "I", "to_id": "L", "distance": 160 }, { "from_id": "I", "to_id": "M", "distance": 196 }, { "from_id": "I", "to_id": "N", "distance": 160 }, { "from_id": "I", "to_id": "O", "distance": 201 }, { "from_id": "I", "to_id": "P", "distance": 211 }, { "from_id": "I", "to_id": "Q", "distance": 166 }, { "from_id": "I", "to_id": "R", "distance": 147 }, { "from_id": "I", "to_id": "S", "distance": 156 }, { "from_id": "J", "to_id": "A", "distance": 179 }, { "from_id": "J", "to_id": "B", "distance": 143 }, { "from_id": "J", "to_id": "C", "distance": 190 }, { "from_id": "J", "to_id": "D", "distance": 140 }, { "from_id": "J", "to_id": "E", "distance": 182 }, { "from_id": "J", "to_id": "F", "distance": 134 }, { "from_id": "J", "to_id": "G", "distance": 128 }, { "from_id": "J", "to_id": "H", "distance": 174 }, { "from_id": "J", "to_id": "I", "distance": 188 }, { "from_id": "J", "to_id": "K", "distance": 172 }, { "from_id": "J", "to_id": "L", "distance": 197 }, { "from_id": "J", "to_id": "M", "distance": 145 }, { "from_id": "J", "to_id": "N", "distance": 172 }, { "from_id": "J", "to_id": "O", "distance": 162 }, { "from_id": "J", "to_id": "P", "distance": 173 }, { "from_id": "J", "to_id": "Q", "distance": 161 }, { "from_id": "J", "to_id": "R", "distance": 115 }, { "from_id": "J", "to_id": "S", "distance": 158 }, { "from_id": "K", "to_id": "A", "distance": 154 }, { "from_id": "K", "to_id": "B", "distance": 183 }, { "from_id": "K", "to_id": "C", "distance": 160 }, { "from_id": "K", "to_id": "D", "distance": 177 }, { "from_id": "K", "to_id": "E", "distance": 155 }, { "from_id": "K", "to_id": "F", "distance": 136 }, { "from_id": "K", "to_id": "G", "distance": 189 }, { "from_id": "K", "to_id": "H", "distance": 136 }, { "from_id": "K", "to_id": "I", "distance": 149 }, { "from_id": "K", "to_id": "J", "distance": 172 }, { "from_id": "K", "to_id": "L", "distance": 193 }, { "from_id": "K", "to_id": "M", "distance": 163 }, { "from_id": "K", "to_id": "N", "distance": 162 }, { "from_id": "K", "to_id": "O", "distance": 182 }, { "from_id": "K", "to_id": "P", "distance": 178 }, { "from_id": "K", "to_id": "Q", "distance": 178 }, { "from_id": "K", "to_id": "R", "distance": 167 }, { "from_id": "K", "to_id": "S", "distance": 137 }, { "from_id": "L", "to_id": "A", "distance": 195 }, { "from_id": "L", "to_id": "B", "distance": 150 }, { "from_id": "L", "to_id": "C", "distance": 164 }, { "from_id": "L", "to_id": "D", "distance": 206 }, { "from_id": "L", "to_id": "E", "distance": 186 }, { "from_id": "L", "to_id": "F", "distance": 198 }, { "from_id": "L", "to_id": "G", "distance": 191 }, { "from_id": "L", "to_id": "H", "distance": 216 }, { "from_id": "L", "to_id": "I", "distance": 160 }, { "from_id": "L", "to_id": "J", "distance": 197 }, { "from_id": "L", "to_id": "K", "distance": 193 }, { "from_id": "L", "to_id": "M", "distance": 180 }, { "from_id": "L", "to_id": "N", "distance": 197 }, { "from_id": "L", "to_id": "O", "distance": 227 }, { "from_id": "L", "to_id": "P", "distance": 214 }, { "from_id": "L", "to_id": "Q", "distance": 181 }, { "from_id": "L", "to_id": "R", "distance": 171 }, { "from_id": "L", "to_id": "S", "distance": 193 }, { "from_id": "M", "to_id": "A", "distance": 202 }, { "from_id": "M", "to_id": "B", "distance": 178 }, { "from_id": "M", "to_id": "C", "distance": 149 }, { "from_id": "M", "to_id": "D", "distance": 145 }, { "from_id": "M", "to_id": "E", "distance": 152 }, { "from_id": "M", "to_id": "F", "distance": 179 }, { "from_id": "M", "to_id": "G", "distance": 144 }, { "from_id": "M", "to_id": "H", "distance": 177 }, { "from_id": "M", "to_id": "I", "distance": 196 }, { "from_id": "M", "to_id": "J", "distance": 145 }, { "from_id": "M", "to_id": "K", "distance": 163 }, { "from_id": "M", "to_id": "L", "distance": 180 }, { "from_id": "M", "to_id": "N", "distance": 204 }, { "from_id": "M", "to_id": "O", "distance": 162 }, { "from_id": "M", "to_id": "P", "distance": 162 }, { "from_id": "M", "to_id": "Q", "distance": 175 }, { "from_id": "M", "to_id": "R", "distance": 188 }, { "from_id": "M", "to_id": "S", "distance": 168 }, { "from_id": "N", "to_id": "A", "distance": 154 }, { "from_id": "N", "to_id": "B", "distance": 196 }, { "from_id": "N", "to_id": "C", "distance": 223 }, { "from_id": "N", "to_id": "D", "distance": 190 }, { "from_id": "N", "to_id": "E", "distance": 196 }, { "from_id": "N", "to_id": "F", "distance": 162 }, { "from_id": "N", "to_id": "G", "distance": 213 }, { "from_id": "N", "to_id": "H", "distance": 175 }, { "from_id": "N", "to_id": "I", "distance": 160 }, { "from_id": "N", "to_id": "J", "distance": 172 }, { "from_id": "N", "to_id": "K", "distance": 162 }, { "from_id": "N", "to_id": "L", "distance": 197 }, { "from_id": "N", "to_id": "M", "distance": 204 }, { "from_id": "N", "to_id": "O", "distance": 197 }, { "from_id": "N", "to_id": "P", "distance": 189 }, { "from_id": "N", "to_id": "Q", "distance": 189 }, { "from_id": "N", "to_id": "R", "distance": 132 }, { "from_id": "N", "to_id": "S", "distance": 190 }, { "from_id": "O", "to_id": "A", "distance": 175 }, { "from_id": "O", "to_id": "B", "distance": 187 }, { "from_id": "O", "to_id": "C", "distance": 214 }, { "from_id": "O", "to_id": "D", "distance": 153 }, { "from_id": "O", "to_id": "E", "distance": 199 }, { "from_id": "O", "to_id": "F", "distance": 180 }, { "from_id": "O", "to_id": "G", "distance": 138 }, { "from_id": "O", "to_id": "H", "distance": 189 }, { "from_id": "O", "to_id": "I", "distance": 201 }, { "from_id": "O", "to_id": "J", "distance": 162 }, { "from_id": "O", "to_id": "K", "distance": 182 }, { "from_id": "O", "to_id": "L", "distance": 227 }, { "from_id": "O", "to_id": "M", "distance": 162 }, { "from_id": "O", "to_id": "N", "distance": 197 }, { "from_id": "O", "to_id": "P", "distance": 207 }, { "from_id": "O", "to_id": "Q", "distance": 213 }, { "from_id": "O", "to_id": "R", "distance": 171 }, { "from_id": "O", "to_id": "S", "distance": 176 }, { "from_id": "P", "to_id": "A", "distance": 162 }, { "from_id": "P", "to_id": "B", "distance": 202 }, { "from_id": "P", "to_id": "C", "distance": 191 }, { "from_id": "P", "to_id": "D", "distance": 178 }, { "from_id": "P", "to_id": "E", "distance": 226 }, { "from_id": "P", "to_id": "F", "distance": 189 }, { "from_id": "P", "to_id": "G", "distance": 161 }, { "from_id": "P", "to_id": "H", "distance": 170 }, { "from_id": "P", "to_id": "I", "distance": 211 }, { "from_id": "P", "to_id": "J", "distance": 173 }, { "from_id": "P", "to_id": "K", "distance": 178 }, { "from_id": "P", "to_id": "L", "distance": 214 }, { "from_id": "P", "to_id": "M", "distance": 162 }, { "from_id": "P", "to_id": "N", "distance": 189 }, { "from_id": "P", "to_id": "O", "distance": 207 }, { "from_id": "P", "to_id": "Q", "distance": 164 }, { "from_id": "P", "to_id": "R", "distance": 209 }, { "from_id": "P", "to_id": "S", "distance": 190 }, { "from_id": "Q", "to_id": "A", "distance": 170 }, { "from_id": "Q", "to_id": "B", "distance": 180 }, { "from_id": "Q", "to_id": "C", "distance": 173 }, { "from_id": "Q", "to_id": "D", "distance": 179 }, { "from_id": "Q", "to_id": "E", "distance": 179 }, { "from_id": "Q", "to_id": "F", "distance": 155 }, { "from_id": "Q", "to_id": "G", "distance": 178 }, { "from_id": "Q", "to_id": "H", "distance": 194 }, { "from_id": "Q", "to_id": "I", "distance": 166 }, { "from_id": "Q", "to_id": "J", "distance": 161 }, { "from_id": "Q", "to_id": "K", "distance": 178 }, { "from_id": "Q", "to_id": "L", "distance": 181 }, { "from_id": "Q", "to_id": "M", "distance": 175 }, { "from_id": "Q", "to_id": "N", "distance": 189 }, { "from_id": "Q", "to_id": "O", "distance": 213 }, { "from_id": "Q", "to_id": "P", "distance": 164 }, { "from_id": "Q", "to_id": "R", "distance": 159 }, { "from_id": "Q", "to_id": "S", "distance": 153 }, { "from_id": "R", "to_id": "A", "distance": 144 }, { "from_id": "R", "to_id": "B", "distance": 132 }, { "from_id": "R", "to_id": "C", "distance": 196 }, { "from_id": "R", "to_id": "D", "distance": 150 }, { "from_id": "R", "to_id": "E", "distance": 157 }, { "from_id": "R", "to_id": "F", "distance": 124 }, { "from_id": "R", "to_id": "G", "distance": 172 }, { "from_id": "R", "to_id": "H", "distance": 193 }, { "from_id": "R", "to_id": "I", "distance": 147 }, { "from_id": "R", "to_id": "J", "distance": 115 }, { "from_id": "R", "to_id": "K", "distance": 167 }, { "from_id": "R", "to_id": "L", "distance": 171 }, { "from_id": "R", "to_id": "M", "distance": 188 }, { "from_id": "R", "to_id": "N", "distance": 132 }, { "from_id": "R", "to_id": "O", "distance": 171 }, { "from_id": "R", "to_id": "P", "distance": 209 }, { "from_id": "R", "to_id": "Q", "distance": 159 }, { "from_id": "R", "to_id": "S", "distance": 162 }, { "from_id": "S", "to_id": "A", "distance": 148 }, { "from_id": "S", "to_id": "B", "distance": 174 }, { "from_id": "S", "to_id": "C", "distance": 199 }, { "from_id": "S", "to_id": "D", "distance": 165 }, { "from_id": "S", "to_id": "E", "distance": 178 }, { "from_id": "S", "to_id": "F", "distance": 70 }, { "from_id": "S", "to_id": "G", "distance": 176 }, { "from_id": "S", "to_id": "H", "distance": 164 }, { "from_id": "S", "to_id": "I", "distance": 156 }, { "from_id": "S", "to_id": "J", "distance": 158 }, { "from_id": "S", "to_id": "K", "distance": 137 }, { "from_id": "S", "to_id": "L", "distance": 193 }, { "from_id": "S", "to_id": "M", "distance": 168 }, { "from_id": "S", "to_id": "N", "distance": 190 }, { "from_id": "S", "to_id": "O", "distance": 176 }, { "from_id": "S", "to_id": "P", "distance": 190 }, { "from_id": "S", "to_id": "Q", "distance": 153 }, { "from_id": "S", "to_id": "R", "distance": 162 } ] }, "solution_variant": [ "A", "B", "C", "D", "E", "G", "H", "I", "L", "M", "N", "O", "P", "Q", "S" ], "context_index": 16, "input_format": "json", "input_index_base": "names" }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Recently a museum educator wanted a compact display of exactly m objects that would showcase cultural variety, not redundancy. There’s a table that gives a distance-like score for each pair of objects that says how different they are, and the way to judge any candidate selection is to add together those pair scores for every pair in the group — the bigger the total, the more diverse the display will feel. The pick must include exactly m unique artifacts. The item list and the pairwise scores follow below.\n\nThe catalog held 19 artifacts labeled 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, and the educator requested exactly 2 artifacts to comprise the compact display.\n\n| artifact_a_id | artifact_b_id | cultural_distance |\n|---|---|---|\n| 1 | 2 | 4 |\n| 1 | 3 | 2 |\n| 1 | 4 | 2 |\n| 1 | 5 | 6 |\n| 1 | 6 | 8 |\n| 1 | 7 | 1 |\n| 1 | 8 | 8 |\n| 1 | 9 | 9 |\n| 1 | 10 | 7 |\n| 1 | 11 | 4 |\n| 1 | 12 | 0 |\n| 1 | 13 | 9 |\n| 1 | 14 | 5 |\n| 1 | 15 | 0 |\n| 1 | 16 | 6 |\n| 1 | 17 | 7 |\n| 1 | 18 | 1 |\n| 1 | 19 | 0 |\n| 2 | 1 | 4 |\n| 2 | 3 | 6 |\n| 2 | 4 | 3 |\n| 2 | 5 | 0 |\n| 2 | 6 | 9 |\n| 2 | 7 | 9 |\n| 2 | 8 | 4 |\n| 2 | 9 | 3 |\n| 2 | 10 | 1 |\n| 2 | 11 | 2 |\n| 2 | 12 | 9 |\n| 2 | 13 | 9 |\n| 2 | 14 | 1 |\n| 2 | 15 | 6 |\n| 2 | 16 | 0 |\n| 2 | 17 | 6 |\n| 2 | 18 | 2 |\n| 2 | 19 | 1 |\n| 3 | 1 | 2 |\n| 3 | 2 | 6 |\n| 3 | 4 | 6 |\n| 3 | 5 | 8 |\n| 3 | 6 | 8 |\n| 3 | 7 | 1 |\n| 3 | 8 | 5 |\n| 3 | 9 | 3 |\n| 3 | 10 | 7 |\n| 3 | 11 | 1 |\n| 3 | 12 | 2 |\n| 3 | 13 | 6 |\n| 3 | 14 | 1 |\n| 3 | 15 | 9 |\n| 3 | 16 | 7 |\n| 3 | 17 | 0 |\n| 3 | 18 | 6 |\n| 3 | 19 | 8 |\n| 4 | 1 | 2 |\n| 4 | 2 | 3 |\n| 4 | 3 | 6 |\n| 4 | 5 | 5 |\n| 4 | 6 | 4 |\n| 4 | 7 | 6 |\n| 4 | 8 | 8 |\n| 4 | 9 | 7 |\n| 4 | 10 | 9 |\n| 4 | 11 | 8 |\n| 4 | 12 | 8 |\n| 4 | 13 | 2 |\n| 4 | 14 | 9 |\n| 4 | 15 | 3 |\n| 4 | 16 | 0 |\n| 4 | 17 | 8 |\n| 4 | 18 | 6 |\n| 4 | 19 | 8 |\n| 5 | 1 | 6 |\n| 5 | 2 | 0 |\n| 5 | 3 | 8 |\n| 5 | 4 | 5 |\n| 5 | 6 | 0 |\n| 5 | 7 | 9 |\n| 5 | 8 | 4 |\n| 5 | 9 | 6 |\n| 5 | 10 | 2 |\n| 5 | 11 | 7 |\n| 5 | 12 | 9 |\n| 5 | 13 | 8 |\n| 5 | 14 | 8 |\n| 5 | 15 | 2 |\n| 5 | 16 | 4 |\n| 5 | 17 | 0 |\n| 5 | 18 | 7 |\n| 5 | 19 | 7 |\n| 6 | 1 | 8 |\n| 6 | 2 | 9 |\n| 6 | 3 | 8 |\n| 6 | 4 | 4 |\n| 6 | 5 | 0 |\n| 6 | 7 | 7 |\n| 6 | 8 | 7 |\n| 6 | 9 | 0 |\n| 6 | 10 | 8 |\n| 6 | 11 | 8 |\n| 6 | 12 | 5 |\n| 6 | 13 | 0 |\n| 6 | 14 | 7 |\n| 6 | 15 | 9 |\n| 6 | 16 | 7 |\n| 6 | 17 | 9 |\n| 6 | 18 | 6 |\n| 6 | 19 | 1 |\n| 7 | 1 | 1 |\n| 7 | 2 | 9 |\n| 7 | 3 | 1 |\n| 7 | 4 | 6 |\n| 7 | 5 | 9 |\n| 7 | 6 | 7 |\n| 7 | 8 | 7 |\n| 7 | 9 | 0 |\n| 7 | 10 | 0 |\n| 7 | 11 | 1 |\n| 7 | 12 | 6 |\n| 7 | 13 | 8 |\n| 7 | 14 | 6 |\n| 7 | 15 | 2 |\n| 7 | 16 | 2 |\n| 7 | 17 | 8 |\n| 7 | 18 | 3 |\n| 7 | 19 | 6 |\n| 8 | 1 | 8 |\n| 8 | 2 | 4 |\n| 8 | 3 | 5 |\n| 8 | 4 | 8 |\n| 8 | 5 | 4 |\n| 8 | 6 | 7 |\n| 8 | 7 | 7 |\n| 8 | 9 | 2 |\n| 8 | 10 | 3 |\n| 8 | 11 | 6 |\n| 8 | 12 | 6 |\n| 8 | 13 | 0 |\n| 8 | 14 | 0 |\n| 8 | 15 | 9 |\n| 8 | 16 | 9 |\n| 8 | 17 | 8 |\n| 8 | 18 | 2 |\n| 8 | 19 | 1 |\n| 9 | 1 | 9 |\n| 9 | 2 | 3 |\n| 9 | 3 | 3 |\n| 9 | 4 | 7 |\n| 9 | 5 | 6 |\n| 9 | 6 | 0 |\n| 9 | 7 | 0 |\n| 9 | 8 | 2 |\n| 9 | 10 | 0 |\n| 9 | 11 | 0 |\n| 9 | 12 | 6 |\n| 9 | 13 | 8 |\n| 9 | 14 | 9 |\n| 9 | 15 | 6 |\n| 9 | 16 | 6 |\n| 9 | 17 | 3 |\n| 9 | 18 | 0 |\n| 9 | 19 | 9 |\n| 10 | 1 | 7 |\n| 10 | 2 | 1 |\n| 10 | 3 | 7 |\n| 10 | 4 | 9 |\n| 10 | 5 | 2 |\n| 10 | 6 | 8 |\n| 10 | 7 | 0 |\n| 10 | 8 | 3 |\n| 10 | 9 | 0 |\n| 10 | 11 | 3 |\n| 10 | 12 | 9 |\n| 10 | 13 | 4 |\n| 10 | 14 | 9 |\n| 10 | 15 | 1 |\n| 10 | 16 | 8 |\n| 10 | 17 | 9 |\n| 10 | 18 | 6 |\n| 10 | 19 | 0 |\n| 11 | 1 | 4 |\n| 11 | 2 | 2 |\n| 11 | 3 | 1 |\n| 11 | 4 | 8 |\n| 11 | 5 | 7 |\n| 11 | 6 | 8 |\n| 11 | 7 | 1 |\n| 11 | 8 | 6 |\n| 11 | 9 | 0 |\n| 11 | 10 | 3 |\n| 11 | 12 | 7 |\n| 11 | 13 | 1 |\n| 11 | 14 | 6 |\n| 11 | 15 | 2 |\n| 11 | 16 | 7 |\n| 11 | 17 | 0 |\n| 11 | 18 | 4 |\n| 11 | 19 | 3 |\n| 12 | 1 | 0 |\n| 12 | 2 | 9 |\n| 12 | 3 | 2 |\n| 12 | 4 | 8 |\n| 12 | 5 | 9 |\n| 12 | 6 | 5 |\n| 12 | 7 | 6 |\n| 12 | 8 | 6 |\n| 12 | 9 | 6 |\n| 12 | 10 | 9 |\n| 12 | 11 | 7 |\n| 12 | 13 | 9 |\n| 12 | 14 | 0 |\n| 12 | 15 | 3 |\n| 12 | 16 | 0 |\n| 12 | 17 | 6 |\n| 12 | 18 | 9 |\n| 12 | 19 | 0 |\n| 13 | 1 | 9 |\n| 13 | 2 | 9 |\n| 13 | 3 | 6 |\n| 13 | 4 | 2 |\n| 13 | 5 | 8 |\n| 13 | 6 | 0 |\n| 13 | 7 | 8 |\n| 13 | 8 | 0 |\n| 13 | 9 | 8 |\n| 13 | 10 | 4 |\n| 13 | 11 | 1 |\n| 13 | 12 | 9 |\n| 13 | 14 | 4 |\n| 13 | 15 | 2 |\n| 13 | 16 | 2 |\n| 13 | 17 | 1 |\n| 13 | 18 | 2 |\n| 13 | 19 | 3 |\n| 14 | 1 | 5 |\n| 14 | 2 | 1 |\n| 14 | 3 | 1 |\n| 14 | 4 | 9 |\n| 14 | 5 | 8 |\n| 14 | 6 | 7 |\n| 14 | 7 | 6 |\n| 14 | 8 | 0 |\n| 14 | 9 | 9 |\n| 14 | 10 | 9 |\n| 14 | 11 | 6 |\n| 14 | 12 | 0 |\n| 14 | 13 | 4 |\n| 14 | 15 | 9 |\n| 14 | 16 | 8 |\n| 14 | 17 | 9 |\n| 14 | 18 | 1 |\n| 14 | 19 | 1 |\n| 15 | 1 | 0 |\n| 15 | 2 | 6 |\n| 15 | 3 | 9 |\n| 15 | 4 | 3 |\n| 15 | 5 | 2 |\n| 15 | 6 | 9 |\n| 15 | 7 | 2 |\n| 15 | 8 | 9 |\n| 15 | 9 | 6 |\n| 15 | 10 | 1 |\n| 15 | 11 | 2 |\n| 15 | 12 | 3 |\n| 15 | 13 | 2 |\n| 15 | 14 | 9 |\n| 15 | 16 | 2 |\n| 15 | 17 | 0 |\n| 15 | 18 | 6 |\n| 15 | 19 | 6 |\n| 16 | 1 | 6 |\n| 16 | 2 | 0 |\n| 16 | 3 | 7 |\n| 16 | 4 | 0 |\n| 16 | 5 | 4 |\n| 16 | 6 | 7 |\n| 16 | 7 | 2 |\n| 16 | 8 | 9 |\n| 16 | 9 | 6 |\n| 16 | 10 | 8 |\n| 16 | 11 | 7 |\n| 16 | 12 | 0 |\n| 16 | 13 | 2 |\n| 16 | 14 | 8 |\n| 16 | 15 | 2 |\n| 16 | 17 | 2 |\n| 16 | 18 | 9 |\n| 16 | 19 | 8 |\n| 17 | 1 | 7 |\n| 17 | 2 | 6 |\n| 17 | 3 | 0 |\n| 17 | 4 | 8 |\n| 17 | 5 | 0 |\n| 17 | 6 | 9 |\n| 17 | 7 | 8 |\n| 17 | 8 | 8 |\n| 17 | 9 | 3 |\n| 17 | 10 | 9 |\n| 17 | 11 | 0 |\n| 17 | 12 | 6 |\n| 17 | 13 | 1 |\n| 17 | 14 | 9 |\n| 17 | 15 | 0 |\n| 17 | 16 | 2 |\n| 17 | 18 | 4 |\n| 17 | 19 | 5 |\n| 18 | 1 | 1 |\n| 18 | 2 | 2 |\n| 18 | 3 | 6 |\n| 18 | 4 | 6 |\n| 18 | 5 | 7 |\n| 18 | 6 | 6 |\n| 18 | 7 | 3 |\n| 18 | 8 | 2 |\n| 18 | 9 | 0 |\n| 18 | 10 | 6 |\n| 18 | 11 | 4 |\n| 18 | 12 | 9 |\n| 18 | 13 | 2 |\n| 18 | 14 | 1 |\n| 18 | 15 | 6 |\n| 18 | 16 | 9 |\n| 18 | 17 | 4 |\n| 18 | 19 | 7 |\n| 19 | 1 | 0 |\n| 19 | 2 | 1 |\n| 19 | 3 | 8 |\n| 19 | 4 | 8 |\n| 19 | 5 | 7 |\n| 19 | 6 | 1 |\n| 19 | 7 | 6 |\n| 19 | 8 | 1 |\n| 19 | 9 | 9 |\n| 19 | 10 | 0 |\n| 19 | 11 | 3 |\n| 19 | 12 | 0 |\n| 19 | 13 | 3 |\n| 19 | 14 | 1 |\n| 19 | 15 | 6 |\n| 19 | 16 | 8 |\n| 19 | 17 | 5 |\n| 19 | 18 | 7 |\n\nThese pairwise cultural distances will be summed to judge any candidate selection of exactly 2 from the 19 artifacts.\n\nAlso, when you send back your pick, please use a simple JSON layout like this so it's easy to read and parse:\n\n{\n \"solution\": [, , ...]\n}\n\nThis little block just means \"solution\" should be the list of exactly m artifacts you pick — each entry is the identifier for one object from the instance. It's just a sketch of the expected shape, not the actual answer.\n\nPlease make sure all identifiers in that list are used exactly as they appear in the instance input — no renaming and no made-up labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 4, 2, 2, 6, 8, 1, 8, 9, 7, 4, 0, 9, 5, 0, 6, 7, 1, 0 ], [ 4, 0, 6, 3, 0, 9, 9, 4, 3, 1, 2, 9, 9, 1, 6, 0, 6, 2, 1 ], [ 2, 6, 0, 6, 8, 8, 1, 5, 3, 7, 1, 2, 6, 1, 9, 7, 0, 6, 8 ], [ 2, 3, 6, 0, 5, 4, 6, 8, 7, 9, 8, 8, 2, 9, 3, 0, 8, 6, 8 ], [ 6, 0, 8, 5, 0, 0, 9, 4, 6, 2, 7, 9, 8, 8, 2, 4, 0, 7, 7 ], [ 8, 9, 8, 4, 0, 0, 7, 7, 0, 8, 8, 5, 0, 7, 9, 7, 9, 6, 1 ], [ 1, 9, 1, 6, 9, 7, 0, 7, 0, 0, 1, 6, 8, 6, 2, 2, 8, 3, 6 ], [ 8, 4, 5, 8, 4, 7, 7, 0, 2, 3, 6, 6, 0, 0, 9, 9, 8, 2, 1 ], [ 9, 3, 3, 7, 6, 0, 0, 2, 0, 0, 0, 6, 8, 9, 6, 6, 3, 0, 9 ], [ 7, 1, 7, 9, 2, 8, 0, 3, 0, 0, 3, 9, 4, 9, 1, 8, 9, 6, 0 ], [ 4, 2, 1, 8, 7, 8, 1, 6, 0, 3, 0, 7, 1, 6, 2, 7, 0, 4, 3 ], [ 0, 9, 2, 8, 9, 5, 6, 6, 6, 9, 7, 0, 9, 0, 3, 0, 6, 9, 0 ], [ 9, 9, 6, 2, 8, 0, 8, 0, 8, 4, 1, 9, 0, 4, 2, 2, 1, 2, 3 ], [ 5, 1, 1, 9, 8, 7, 6, 0, 9, 9, 6, 0, 4, 0, 9, 8, 9, 1, 1 ], [ 0, 6, 9, 3, 2, 9, 2, 9, 6, 1, 2, 3, 2, 9, 0, 2, 0, 6, 6 ], [ 6, 0, 7, 0, 4, 7, 2, 9, 6, 8, 7, 0, 2, 8, 2, 0, 2, 9, 8 ], [ 7, 6, 0, 8, 0, 9, 8, 8, 3, 9, 0, 6, 1, 9, 0, 2, 0, 4, 5 ], [ 1, 2, 6, 6, 7, 6, 3, 2, 0, 6, 4, 9, 2, 1, 6, 9, 4, 0, 7 ], [ 0, 1, 8, 8, 7, 1, 6, 1, 9, 0, 3, 0, 3, 1, 6, 8, 5, 7, 0 ] ], "m": 2, "objective": 9.0 }, "solution": [ 1, 11 ], "obj": 9.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 19, "m": 2, "nodes": [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 ], "distance_pairs": [ { "from_id": 1, "to_id": 2, "distance": 4 }, { "from_id": 1, "to_id": 3, "distance": 2 }, { "from_id": 1, "to_id": 4, "distance": 2 }, { "from_id": 1, "to_id": 5, "distance": 6 }, { "from_id": 1, "to_id": 6, "distance": 8 }, { "from_id": 1, "to_id": 7, "distance": 1 }, { "from_id": 1, "to_id": 8, "distance": 8 }, { "from_id": 1, "to_id": 9, "distance": 9 }, { "from_id": 1, "to_id": 10, "distance": 7 }, { "from_id": 1, "to_id": 11, "distance": 4 }, { "from_id": 1, "to_id": 12, "distance": 0 }, { "from_id": 1, "to_id": 13, "distance": 9 }, { "from_id": 1, "to_id": 14, "distance": 5 }, { "from_id": 1, "to_id": 15, "distance": 0 }, { "from_id": 1, "to_id": 16, "distance": 6 }, { "from_id": 1, "to_id": 17, "distance": 7 }, { "from_id": 1, "to_id": 18, "distance": 1 }, { "from_id": 1, "to_id": 19, "distance": 0 }, { "from_id": 2, "to_id": 1, "distance": 4 }, { "from_id": 2, "to_id": 3, "distance": 6 }, { "from_id": 2, "to_id": 4, "distance": 3 }, { "from_id": 2, "to_id": 5, "distance": 0 }, { "from_id": 2, "to_id": 6, "distance": 9 }, { "from_id": 2, "to_id": 7, "distance": 9 }, { "from_id": 2, "to_id": 8, "distance": 4 }, { "from_id": 2, "to_id": 9, "distance": 3 }, { 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}, { "from_id": 3, "to_id": 14, "distance": 1 }, { "from_id": 3, "to_id": 15, "distance": 9 }, { "from_id": 3, "to_id": 16, "distance": 7 }, { "from_id": 3, "to_id": 17, "distance": 0 }, { "from_id": 3, "to_id": 18, "distance": 6 }, { "from_id": 3, "to_id": 19, "distance": 8 }, { "from_id": 4, "to_id": 1, "distance": 2 }, { "from_id": 4, "to_id": 2, "distance": 3 }, { "from_id": 4, "to_id": 3, "distance": 6 }, { "from_id": 4, "to_id": 5, "distance": 5 }, { "from_id": 4, "to_id": 6, "distance": 4 }, { "from_id": 4, "to_id": 7, "distance": 6 }, { "from_id": 4, "to_id": 8, "distance": 8 }, { "from_id": 4, "to_id": 9, "distance": 7 }, { "from_id": 4, "to_id": 10, "distance": 9 }, { "from_id": 4, "to_id": 11, "distance": 8 }, { "from_id": 4, "to_id": 12, "distance": 8 }, { "from_id": 4, "to_id": 13, "distance": 2 }, { "from_id": 4, "to_id": 14, "distance": 9 }, { "from_id": 4, "to_id": 15, "distance": 3 }, { "from_id": 4, "to_id": 16, "distance": 0 }, { "from_id": 4, "to_id": 17, 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"from_id": 16, "to_id": 18, "distance": 9 }, { "from_id": 16, "to_id": 19, "distance": 8 }, { "from_id": 17, "to_id": 1, "distance": 7 }, { "from_id": 17, "to_id": 2, "distance": 6 }, { "from_id": 17, "to_id": 3, "distance": 0 }, { "from_id": 17, "to_id": 4, "distance": 8 }, { "from_id": 17, "to_id": 5, "distance": 0 }, { "from_id": 17, "to_id": 6, "distance": 9 }, { "from_id": 17, "to_id": 7, "distance": 8 }, { "from_id": 17, "to_id": 8, "distance": 8 }, { "from_id": 17, "to_id": 9, "distance": 3 }, { "from_id": 17, "to_id": 10, "distance": 9 }, { "from_id": 17, "to_id": 11, "distance": 0 }, { "from_id": 17, "to_id": 12, "distance": 6 }, { "from_id": 17, "to_id": 13, "distance": 1 }, { "from_id": 17, "to_id": 14, "distance": 9 }, { "from_id": 17, "to_id": 15, "distance": 0 }, { "from_id": 17, "to_id": 16, "distance": 2 }, { "from_id": 17, "to_id": 18, "distance": 4 }, { "from_id": 17, "to_id": 19, "distance": 5 }, { "from_id": 18, "to_id": 1, "distance": 1 }, { "from_id": 18, "to_id": 2, "distance": 2 }, { "from_id": 18, "to_id": 3, "distance": 6 }, { "from_id": 18, "to_id": 4, "distance": 6 }, { "from_id": 18, "to_id": 5, "distance": 7 }, { "from_id": 18, "to_id": 6, "distance": 6 }, { "from_id": 18, "to_id": 7, "distance": 3 }, { "from_id": 18, "to_id": 8, "distance": 2 }, { "from_id": 18, "to_id": 9, "distance": 0 }, { "from_id": 18, "to_id": 10, "distance": 6 }, { "from_id": 18, "to_id": 11, "distance": 4 }, { "from_id": 18, "to_id": 12, "distance": 9 }, { "from_id": 18, "to_id": 13, "distance": 2 }, { "from_id": 18, "to_id": 14, "distance": 1 }, { "from_id": 18, "to_id": 15, "distance": 6 }, { "from_id": 18, "to_id": 16, "distance": 9 }, { "from_id": 18, "to_id": 17, "distance": 4 }, { "from_id": 18, "to_id": 19, "distance": 7 }, { "from_id": 19, "to_id": 1, "distance": 0 }, { "from_id": 19, "to_id": 2, "distance": 1 }, { "from_id": 19, "to_id": 3, "distance": 8 }, { "from_id": 19, "to_id": 4, "distance": 8 }, { "from_id": 19, "to_id": 5, "distance": 7 }, { "from_id": 19, "to_id": 6, "distance": 1 }, { "from_id": 19, "to_id": 7, "distance": 6 }, { "from_id": 19, "to_id": 8, "distance": 1 }, { "from_id": 19, "to_id": 9, "distance": 9 }, { "from_id": 19, "to_id": 10, "distance": 0 }, { "from_id": 19, "to_id": 11, "distance": 3 }, { "from_id": 19, "to_id": 12, "distance": 0 }, { "from_id": 19, "to_id": 13, "distance": 3 }, { "from_id": 19, "to_id": 14, "distance": 1 }, { "from_id": 19, "to_id": 15, "distance": 6 }, { "from_id": 19, "to_id": 16, "distance": 8 }, { "from_id": 19, "to_id": 17, "distance": 5 }, { "from_id": 19, "to_id": 18, "distance": 7 } ] }, "solution_variant": [ 2, 12 ], "context_index": 17, "input_format": "markdown_table", "input_index_base": 1 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "There’s a trainer who wants a tightly-packed session of exactly m exercises that doesn’t feel repetitive. Every pair of candidate exercises has a distance value that captures how different their movements and muscle targets are, and the session’s total variety comes from summing those pairwise distances for the chosen exercises. Exactly m distinct exercises must be picked — nothing duplicated or left blank — and the full list with the actual distances is shown below.\n\nThe trainer is considering 20 candidate exercises 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 and wants a tightly-packed session of exactly 6 exercises.\nBetween 1 and 2: movement-muscle dissimilarity 4.\nBetween 1 and 3: movement-muscle dissimilarity 9.\nBetween 1 and 4: movement-muscle dissimilarity 6.\nBetween 1 and 5: movement-muscle dissimilarity 8.\nBetween 1 and 6: movement-muscle dissimilarity 4.\nBetween 1 and 7: movement-muscle dissimilarity 8.\nBetween 1 and 8: movement-muscle dissimilarity 4.\nBetween 1 and 9: movement-muscle dissimilarity 6.\nBetween 1 and 10: movement-muscle dissimilarity 4.\nBetween 1 and 11: movement-muscle dissimilarity 4.\nBetween 1 and 12: movement-muscle dissimilarity 7.\nBetween 1 and 13: movement-muscle dissimilarity 9.\nBetween 1 and 14: movement-muscle dissimilarity 0.\nBetween 1 and 15: movement-muscle dissimilarity 2.\nBetween 1 and 16: movement-muscle dissimilarity 0.\nBetween 1 and 17: movement-muscle dissimilarity 8.\nBetween 1 and 18: movement-muscle dissimilarity 7.\nBetween 1 and 19: movement-muscle dissimilarity 6.\nBetween 1 and 20: movement-muscle dissimilarity 9.\nBetween 2 and 1: movement-muscle dissimilarity 4.\nBetween 2 and 3: movement-muscle dissimilarity 5.\nBetween 2 and 4: movement-muscle dissimilarity 2.\nBetween 2 and 5: movement-muscle dissimilarity 5.\nBetween 2 and 6: movement-muscle dissimilarity 2.\nBetween 2 and 7: movement-muscle dissimilarity 5.\nBetween 2 and 8: movement-muscle dissimilarity 6.\nBetween 2 and 9: movement-muscle dissimilarity 7.\nBetween 2 and 10: movement-muscle dissimilarity 9.\nBetween 2 and 11: movement-muscle dissimilarity 7.\nBetween 2 and 12: movement-muscle dissimilarity 7.\nBetween 2 and 13: movement-muscle dissimilarity 3.\nBetween 2 and 14: movement-muscle dissimilarity 0.\nBetween 2 and 15: movement-muscle dissimilarity 9.\nBetween 2 and 16: movement-muscle dissimilarity 2.\nBetween 2 and 17: movement-muscle dissimilarity 8.\nBetween 2 and 18: movement-muscle dissimilarity 3.\nBetween 2 and 19: movement-muscle dissimilarity 4.\nBetween 2 and 20: movement-muscle dissimilarity 8.\nBetween 3 and 1: movement-muscle dissimilarity 9.\nBetween 3 and 2: movement-muscle dissimilarity 5.\nBetween 3 and 4: movement-muscle dissimilarity 1.\nBetween 3 and 5: movement-muscle dissimilarity 0.\nBetween 3 and 6: movement-muscle dissimilarity 1.\nBetween 3 and 7: movement-muscle dissimilarity 9.\nBetween 3 and 8: movement-muscle dissimilarity 1.\nBetween 3 and 9: movement-muscle dissimilarity 8.\nBetween 3 and 10: movement-muscle dissimilarity 9.\nBetween 3 and 11: movement-muscle dissimilarity 4.\nBetween 3 and 12: movement-muscle dissimilarity 2.\nBetween 3 and 13: movement-muscle dissimilarity 0.\nBetween 3 and 14: movement-muscle dissimilarity 7.\nBetween 3 and 15: movement-muscle dissimilarity 2.\nBetween 3 and 16: movement-muscle dissimilarity 6.\nBetween 3 and 17: movement-muscle dissimilarity 0.\nBetween 3 and 18: movement-muscle dissimilarity 1.\nBetween 3 and 19: movement-muscle dissimilarity 7.\nBetween 3 and 20: movement-muscle dissimilarity 3.\nBetween 4 and 1: movement-muscle dissimilarity 6.\nBetween 4 and 2: movement-muscle dissimilarity 2.\nBetween 4 and 3: movement-muscle dissimilarity 1.\nBetween 4 and 5: movement-muscle dissimilarity 3.\nBetween 4 and 6: movement-muscle dissimilarity 8.\nBetween 4 and 7: movement-muscle dissimilarity 2.\nBetween 4 and 8: movement-muscle dissimilarity 6.\nBetween 4 and 9: movement-muscle dissimilarity 0.\nBetween 4 and 10: movement-muscle dissimilarity 3.\nBetween 4 and 11: movement-muscle dissimilarity 5.\nBetween 4 and 12: movement-muscle dissimilarity 7.\nBetween 4 and 13: movement-muscle dissimilarity 2.\nBetween 4 and 14: movement-muscle dissimilarity 5.\nBetween 4 and 15: movement-muscle dissimilarity 1.\nBetween 4 and 16: movement-muscle dissimilarity 3.\nBetween 4 and 17: movement-muscle dissimilarity 1.\nBetween 4 and 18: movement-muscle dissimilarity 9.\nBetween 4 and 19: movement-muscle dissimilarity 4.\nBetween 4 and 20: movement-muscle dissimilarity 6.\nBetween 5 and 1: movement-muscle dissimilarity 8.\nBetween 5 and 2: movement-muscle dissimilarity 5.\nBetween 5 and 3: movement-muscle dissimilarity 0.\nBetween 5 and 4: movement-muscle dissimilarity 3.\nBetween 5 and 6: movement-muscle dissimilarity 9.\nBetween 5 and 7: movement-muscle dissimilarity 2.\nBetween 5 and 8: movement-muscle dissimilarity 7.\nBetween 5 and 9: movement-muscle dissimilarity 2.\nBetween 5 and 10: movement-muscle dissimilarity 1.\nBetween 5 and 11: movement-muscle dissimilarity 2.\nBetween 5 and 12: movement-muscle dissimilarity 7.\nBetween 5 and 13: movement-muscle dissimilarity 7.\nBetween 5 and 14: movement-muscle dissimilarity 1.\nBetween 5 and 15: movement-muscle dissimilarity 6.\nBetween 5 and 16: movement-muscle dissimilarity 4.\nBetween 5 and 17: movement-muscle dissimilarity 9.\nBetween 5 and 18: movement-muscle dissimilarity 8.\nBetween 5 and 19: movement-muscle dissimilarity 8.\nBetween 5 and 20: movement-muscle dissimilarity 1.\nBetween 6 and 1: movement-muscle dissimilarity 4.\nBetween 6 and 2: movement-muscle dissimilarity 2.\nBetween 6 and 3: movement-muscle dissimilarity 1.\nBetween 6 and 4: movement-muscle dissimilarity 8.\nBetween 6 and 5: movement-muscle dissimilarity 9.\nBetween 6 and 7: movement-muscle dissimilarity 7.\nBetween 6 and 8: movement-muscle dissimilarity 5.\nBetween 6 and 9: movement-muscle dissimilarity 1.\nBetween 6 and 10: movement-muscle dissimilarity 4.\nBetween 6 and 11: movement-muscle dissimilarity 8.\nBetween 6 and 12: movement-muscle dissimilarity 0.\nBetween 6 and 13: movement-muscle dissimilarity 5.\nBetween 6 and 14: movement-muscle dissimilarity 2.\nBetween 6 and 15: movement-muscle dissimilarity 2.\nBetween 6 and 16: movement-muscle dissimilarity 2.\nBetween 6 and 17: movement-muscle dissimilarity 6.\nBetween 6 and 18: movement-muscle dissimilarity 3.\nBetween 6 and 19: movement-muscle dissimilarity 8.\nBetween 6 and 20: movement-muscle dissimilarity 4.\nBetween 7 and 1: movement-muscle dissimilarity 8.\nBetween 7 and 2: movement-muscle dissimilarity 5.\nBetween 7 and 3: movement-muscle dissimilarity 9.\nBetween 7 and 4: movement-muscle dissimilarity 2.\nBetween 7 and 5: movement-muscle dissimilarity 2.\nBetween 7 and 6: movement-muscle dissimilarity 7.\nBetween 7 and 8: movement-muscle dissimilarity 6.\nBetween 7 and 9: movement-muscle dissimilarity 4.\nBetween 7 and 10: movement-muscle dissimilarity 4.\nBetween 7 and 11: movement-muscle dissimilarity 6.\nBetween 7 and 12: movement-muscle dissimilarity 5.\nBetween 7 and 13: movement-muscle dissimilarity 2.\nBetween 7 and 14: movement-muscle dissimilarity 0.\nBetween 7 and 15: movement-muscle dissimilarity 5.\nBetween 7 and 16: movement-muscle dissimilarity 6.\nBetween 7 and 17: movement-muscle dissimilarity 0.\nBetween 7 and 18: movement-muscle dissimilarity 9.\nBetween 7 and 19: movement-muscle dissimilarity 7.\nBetween 7 and 20: movement-muscle dissimilarity 8.\nBetween 8 and 1: movement-muscle dissimilarity 4.\nBetween 8 and 2: movement-muscle dissimilarity 6.\nBetween 8 and 3: movement-muscle dissimilarity 1.\nBetween 8 and 4: movement-muscle dissimilarity 6.\nBetween 8 and 5: movement-muscle dissimilarity 7.\nBetween 8 and 6: movement-muscle dissimilarity 5.\nBetween 8 and 7: movement-muscle dissimilarity 6.\nBetween 8 and 9: movement-muscle dissimilarity 6.\nBetween 8 and 10: movement-muscle dissimilarity 0.\nBetween 8 and 11: movement-muscle dissimilarity 4.\nBetween 8 and 12: movement-muscle dissimilarity 6.\nBetween 8 and 13: movement-muscle dissimilarity 4.\nBetween 8 and 14: movement-muscle dissimilarity 8.\nBetween 8 and 15: movement-muscle dissimilarity 9.\nBetween 8 and 16: movement-muscle dissimilarity 8.\nBetween 8 and 17: movement-muscle dissimilarity 1.\nBetween 8 and 18: movement-muscle dissimilarity 2.\nBetween 8 and 19: movement-muscle dissimilarity 2.\nBetween 8 and 20: movement-muscle dissimilarity 9.\nBetween 9 and 1: movement-muscle dissimilarity 6.\nBetween 9 and 2: movement-muscle dissimilarity 7.\nBetween 9 and 3: movement-muscle dissimilarity 8.\nBetween 9 and 4: movement-muscle dissimilarity 0.\nBetween 9 and 5: movement-muscle dissimilarity 2.\nBetween 9 and 6: movement-muscle dissimilarity 1.\nBetween 9 and 7: movement-muscle dissimilarity 4.\nBetween 9 and 8: movement-muscle dissimilarity 6.\nBetween 9 and 10: movement-muscle dissimilarity 9.\nBetween 9 and 11: movement-muscle dissimilarity 5.\nBetween 9 and 12: movement-muscle dissimilarity 3.\nBetween 9 and 13: movement-muscle dissimilarity 3.\nBetween 9 and 14: movement-muscle dissimilarity 9.\nBetween 9 and 15: movement-muscle dissimilarity 4.\nBetween 9 and 16: movement-muscle dissimilarity 8.\nBetween 9 and 17: movement-muscle dissimilarity 9.\nBetween 9 and 18: movement-muscle dissimilarity 0.\nBetween 9 and 19: movement-muscle dissimilarity 7.\nBetween 9 and 20: movement-muscle dissimilarity 0.\nBetween 10 and 1: movement-muscle dissimilarity 4.\nBetween 10 and 2: movement-muscle dissimilarity 9.\nBetween 10 and 3: movement-muscle dissimilarity 9.\nBetween 10 and 4: movement-muscle dissimilarity 3.\nBetween 10 and 5: movement-muscle dissimilarity 1.\nBetween 10 and 6: movement-muscle dissimilarity 4.\nBetween 10 and 7: movement-muscle dissimilarity 4.\nBetween 10 and 8: movement-muscle dissimilarity 0.\nBetween 10 and 9: movement-muscle dissimilarity 9.\nBetween 10 and 11: movement-muscle dissimilarity 4.\nBetween 10 and 12: movement-muscle dissimilarity 1.\nBetween 10 and 13: movement-muscle dissimilarity 0.\nBetween 10 and 14: movement-muscle dissimilarity 0.\nBetween 10 and 15: movement-muscle dissimilarity 7.\nBetween 10 and 16: movement-muscle dissimilarity 7.\nBetween 10 and 17: movement-muscle dissimilarity 5.\nBetween 10 and 18: movement-muscle dissimilarity 9.\nBetween 10 and 19: movement-muscle dissimilarity 0.\nBetween 10 and 20: movement-muscle dissimilarity 6.\nBetween 11 and 1: movement-muscle dissimilarity 4.\nBetween 11 and 2: movement-muscle dissimilarity 7.\nBetween 11 and 3: movement-muscle dissimilarity 4.\nBetween 11 and 4: movement-muscle dissimilarity 5.\nBetween 11 and 5: movement-muscle dissimilarity 2.\nBetween 11 and 6: movement-muscle dissimilarity 8.\nBetween 11 and 7: movement-muscle dissimilarity 6.\nBetween 11 and 8: movement-muscle dissimilarity 4.\nBetween 11 and 9: movement-muscle dissimilarity 5.\nBetween 11 and 10: movement-muscle dissimilarity 4.\nBetween 11 and 12: movement-muscle dissimilarity 7.\nBetween 11 and 13: movement-muscle dissimilarity 3.\nBetween 11 and 14: movement-muscle dissimilarity 7.\nBetween 11 and 15: movement-muscle dissimilarity 4.\nBetween 11 and 16: movement-muscle dissimilarity 5.\nBetween 11 and 17: movement-muscle dissimilarity 3.\nBetween 11 and 18: movement-muscle dissimilarity 9.\nBetween 11 and 19: movement-muscle dissimilarity 9.\nBetween 11 and 20: movement-muscle dissimilarity 3.\nBetween 12 and 1: movement-muscle dissimilarity 7.\nBetween 12 and 2: movement-muscle dissimilarity 7.\nBetween 12 and 3: movement-muscle dissimilarity 2.\nBetween 12 and 4: movement-muscle dissimilarity 7.\nBetween 12 and 5: movement-muscle dissimilarity 7.\nBetween 12 and 6: movement-muscle dissimilarity 0.\nBetween 12 and 7: movement-muscle dissimilarity 5.\nBetween 12 and 8: movement-muscle dissimilarity 6.\nBetween 12 and 9: movement-muscle dissimilarity 3.\nBetween 12 and 10: movement-muscle dissimilarity 1.\nBetween 12 and 11: movement-muscle dissimilarity 7.\nBetween 12 and 13: movement-muscle dissimilarity 3.\nBetween 12 and 14: movement-muscle dissimilarity 3.\nBetween 12 and 15: movement-muscle dissimilarity 8.\nBetween 12 and 16: movement-muscle dissimilarity 3.\nBetween 12 and 17: movement-muscle dissimilarity 1.\nBetween 12 and 18: movement-muscle dissimilarity 8.\nBetween 12 and 19: movement-muscle dissimilarity 3.\nBetween 12 and 20: movement-muscle dissimilarity 8.\nBetween 13 and 1: movement-muscle dissimilarity 9.\nBetween 13 and 2: movement-muscle dissimilarity 3.\nBetween 13 and 3: movement-muscle dissimilarity 0.\nBetween 13 and 4: movement-muscle dissimilarity 2.\nBetween 13 and 5: movement-muscle dissimilarity 7.\nBetween 13 and 6: movement-muscle dissimilarity 5.\nBetween 13 and 7: movement-muscle dissimilarity 2.\nBetween 13 and 8: movement-muscle dissimilarity 4.\nBetween 13 and 9: movement-muscle dissimilarity 3.\nBetween 13 and 10: movement-muscle dissimilarity 0.\nBetween 13 and 11: movement-muscle dissimilarity 3.\nBetween 13 and 12: movement-muscle dissimilarity 3.\nBetween 13 and 14: movement-muscle dissimilarity 9.\nBetween 13 and 15: movement-muscle dissimilarity 4.\nBetween 13 and 16: movement-muscle dissimilarity 0.\nBetween 13 and 17: movement-muscle dissimilarity 1.\nBetween 13 and 18: movement-muscle dissimilarity 1.\nBetween 13 and 19: movement-muscle dissimilarity 6.\nBetween 13 and 20: movement-muscle dissimilarity 3.\nBetween 14 and 1: movement-muscle dissimilarity 0.\nBetween 14 and 2: movement-muscle dissimilarity 0.\nBetween 14 and 3: movement-muscle dissimilarity 7.\nBetween 14 and 4: movement-muscle dissimilarity 5.\nBetween 14 and 5: movement-muscle dissimilarity 1.\nBetween 14 and 6: movement-muscle dissimilarity 2.\nBetween 14 and 7: movement-muscle dissimilarity 0.\nBetween 14 and 8: movement-muscle dissimilarity 8.\nBetween 14 and 9: movement-muscle dissimilarity 9.\nBetween 14 and 10: movement-muscle dissimilarity 0.\nBetween 14 and 11: movement-muscle dissimilarity 7.\nBetween 14 and 12: movement-muscle dissimilarity 3.\nBetween 14 and 13: movement-muscle dissimilarity 9.\nBetween 14 and 15: movement-muscle dissimilarity 4.\nBetween 14 and 16: movement-muscle dissimilarity 8.\nBetween 14 and 17: movement-muscle dissimilarity 4.\nBetween 14 and 18: movement-muscle dissimilarity 7.\nBetween 14 and 19: movement-muscle dissimilarity 0.\nBetween 14 and 20: movement-muscle dissimilarity 6.\nBetween 15 and 1: movement-muscle dissimilarity 2.\nBetween 15 and 2: movement-muscle dissimilarity 9.\nBetween 15 and 3: movement-muscle dissimilarity 2.\nBetween 15 and 4: movement-muscle dissimilarity 1.\nBetween 15 and 5: movement-muscle dissimilarity 6.\nBetween 15 and 6: movement-muscle dissimilarity 2.\nBetween 15 and 7: movement-muscle dissimilarity 5.\nBetween 15 and 8: movement-muscle dissimilarity 9.\nBetween 15 and 9: movement-muscle dissimilarity 4.\nBetween 15 and 10: movement-muscle dissimilarity 7.\nBetween 15 and 11: movement-muscle dissimilarity 4.\nBetween 15 and 12: movement-muscle dissimilarity 8.\nBetween 15 and 13: movement-muscle dissimilarity 4.\nBetween 15 and 14: movement-muscle dissimilarity 4.\nBetween 15 and 16: movement-muscle dissimilarity 7.\nBetween 15 and 17: movement-muscle dissimilarity 2.\nBetween 15 and 18: movement-muscle dissimilarity 7.\nBetween 15 and 19: movement-muscle dissimilarity 8.\nBetween 15 and 20: movement-muscle dissimilarity 4.\nBetween 16 and 1: movement-muscle dissimilarity 0.\nBetween 16 and 2: movement-muscle dissimilarity 2.\nBetween 16 and 3: movement-muscle dissimilarity 6.\nBetween 16 and 4: movement-muscle dissimilarity 3.\nBetween 16 and 5: movement-muscle dissimilarity 4.\nBetween 16 and 6: movement-muscle dissimilarity 2.\nBetween 16 and 7: movement-muscle dissimilarity 6.\nBetween 16 and 8: movement-muscle dissimilarity 8.\nBetween 16 and 9: movement-muscle dissimilarity 8.\nBetween 16 and 10: movement-muscle dissimilarity 7.\nBetween 16 and 11: movement-muscle dissimilarity 5.\nBetween 16 and 12: movement-muscle dissimilarity 3.\nBetween 16 and 13: movement-muscle dissimilarity 0.\nBetween 16 and 14: movement-muscle dissimilarity 8.\nBetween 16 and 15: movement-muscle dissimilarity 7.\nBetween 16 and 17: movement-muscle dissimilarity 2.\nBetween 16 and 18: movement-muscle dissimilarity 0.\nBetween 16 and 19: movement-muscle dissimilarity 7.\nBetween 16 and 20: movement-muscle dissimilarity 1.\nBetween 17 and 1: movement-muscle dissimilarity 8.\nBetween 17 and 2: movement-muscle dissimilarity 8.\nBetween 17 and 3: movement-muscle dissimilarity 0.\nBetween 17 and 4: movement-muscle dissimilarity 1.\nBetween 17 and 5: movement-muscle dissimilarity 9.\nBetween 17 and 6: movement-muscle dissimilarity 6.\nBetween 17 and 7: movement-muscle dissimilarity 0.\nBetween 17 and 8: movement-muscle dissimilarity 1.\nBetween 17 and 9: movement-muscle dissimilarity 9.\nBetween 17 and 10: movement-muscle dissimilarity 5.\nBetween 17 and 11: movement-muscle dissimilarity 3.\nBetween 17 and 12: movement-muscle dissimilarity 1.\nBetween 17 and 13: movement-muscle dissimilarity 1.\nBetween 17 and 14: movement-muscle dissimilarity 4.\nBetween 17 and 15: movement-muscle dissimilarity 2.\nBetween 17 and 16: movement-muscle dissimilarity 2.\nBetween 17 and 18: movement-muscle dissimilarity 9.\nBetween 17 and 19: movement-muscle dissimilarity 8.\nBetween 17 and 20: movement-muscle dissimilarity 6.\nBetween 18 and 1: movement-muscle dissimilarity 7.\nBetween 18 and 2: movement-muscle dissimilarity 3.\nBetween 18 and 3: movement-muscle dissimilarity 1.\nBetween 18 and 4: movement-muscle dissimilarity 9.\nBetween 18 and 5: movement-muscle dissimilarity 8.\nBetween 18 and 6: movement-muscle dissimilarity 3.\nBetween 18 and 7: movement-muscle dissimilarity 9.\nBetween 18 and 8: movement-muscle dissimilarity 2.\nBetween 18 and 9: movement-muscle dissimilarity 0.\nBetween 18 and 10: movement-muscle dissimilarity 9.\nBetween 18 and 11: movement-muscle dissimilarity 9.\nBetween 18 and 12: movement-muscle dissimilarity 8.\nBetween 18 and 13: movement-muscle dissimilarity 1.\nBetween 18 and 14: movement-muscle dissimilarity 7.\nBetween 18 and 15: movement-muscle dissimilarity 7.\nBetween 18 and 16: movement-muscle dissimilarity 0.\nBetween 18 and 17: movement-muscle dissimilarity 9.\nBetween 18 and 19: movement-muscle dissimilarity 4.\nBetween 18 and 20: movement-muscle dissimilarity 3.\nBetween 19 and 1: movement-muscle dissimilarity 6.\nBetween 19 and 2: movement-muscle dissimilarity 4.\nBetween 19 and 3: movement-muscle dissimilarity 7.\nBetween 19 and 4: movement-muscle dissimilarity 4.\nBetween 19 and 5: movement-muscle dissimilarity 8.\nBetween 19 and 6: movement-muscle dissimilarity 8.\nBetween 19 and 7: movement-muscle dissimilarity 7.\nBetween 19 and 8: movement-muscle dissimilarity 2.\nBetween 19 and 9: movement-muscle dissimilarity 7.\nBetween 19 and 10: movement-muscle dissimilarity 0.\nBetween 19 and 11: movement-muscle dissimilarity 9.\nBetween 19 and 12: movement-muscle dissimilarity 3.\nBetween 19 and 13: movement-muscle dissimilarity 6.\nBetween 19 and 14: movement-muscle dissimilarity 0.\nBetween 19 and 15: movement-muscle dissimilarity 8.\nBetween 19 and 16: movement-muscle dissimilarity 7.\nBetween 19 and 17: movement-muscle dissimilarity 8.\nBetween 19 and 18: movement-muscle dissimilarity 4.\nBetween 19 and 20: movement-muscle dissimilarity 5.\nBetween 20 and 1: movement-muscle dissimilarity 9.\nBetween 20 and 2: movement-muscle dissimilarity 8.\nBetween 20 and 3: movement-muscle dissimilarity 3.\nBetween 20 and 4: movement-muscle dissimilarity 6.\nBetween 20 and 5: movement-muscle dissimilarity 1.\nBetween 20 and 6: movement-muscle dissimilarity 4.\nBetween 20 and 7: movement-muscle dissimilarity 8.\nBetween 20 and 8: movement-muscle dissimilarity 9.\nBetween 20 and 9: movement-muscle dissimilarity 0.\nBetween 20 and 10: movement-muscle dissimilarity 6.\nBetween 20 and 11: movement-muscle dissimilarity 3.\nBetween 20 and 12: movement-muscle dissimilarity 8.\nBetween 20 and 13: movement-muscle dissimilarity 3.\nBetween 20 and 14: movement-muscle dissimilarity 6.\nBetween 20 and 15: movement-muscle dissimilarity 4.\nBetween 20 and 16: movement-muscle dissimilarity 1.\nBetween 20 and 17: movement-muscle dissimilarity 6.\nBetween 20 and 18: movement-muscle dissimilarity 3.\nBetween 20 and 19: movement-muscle dissimilarity 5.\nThe trainer must pick exactly 6 distinct exercises to form the session.\n\nAlso, when you give the final picks, please put them in this simple JSON shape so it's easy to read and machine-friendly:\n\n{\n \"solution\": [, , ...]\n}\n\nHere \"solution\" is just the list of exactly m exercise identifiers (the ones you pick for the session). The angle-bracket placeholders show the shape — you'll replace each placeholder with one of the actual exercise IDs from the instance. This is just a sketch of the expected shape, not the actual filled-in answer.\n\nPlease make sure to use the identifiers exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 4, 9, 6, 8, 4, 8, 4, 6, 4, 4, 7, 9, 0, 2, 0, 8, 7, 6, 9 ], [ 4, 0, 5, 2, 5, 2, 5, 6, 7, 9, 7, 7, 3, 0, 9, 2, 8, 3, 4, 8 ], [ 9, 5, 0, 1, 0, 1, 9, 1, 8, 9, 4, 2, 0, 7, 2, 6, 0, 1, 7, 3 ], [ 6, 2, 1, 0, 3, 8, 2, 6, 0, 3, 5, 7, 2, 5, 1, 3, 1, 9, 4, 6 ], [ 8, 5, 0, 3, 0, 9, 2, 7, 2, 1, 2, 7, 7, 1, 6, 4, 9, 8, 8, 1 ], [ 4, 2, 1, 8, 9, 0, 7, 5, 1, 4, 8, 0, 5, 2, 2, 2, 6, 3, 8, 4 ], [ 8, 5, 9, 2, 2, 7, 0, 6, 4, 4, 6, 5, 2, 0, 5, 6, 0, 9, 7, 8 ], [ 4, 6, 1, 6, 7, 5, 6, 0, 6, 0, 4, 6, 4, 8, 9, 8, 1, 2, 2, 9 ], [ 6, 7, 8, 0, 2, 1, 4, 6, 0, 9, 5, 3, 3, 9, 4, 8, 9, 0, 7, 0 ], [ 4, 9, 9, 3, 1, 4, 4, 0, 9, 0, 4, 1, 0, 0, 7, 7, 5, 9, 0, 6 ], [ 4, 7, 4, 5, 2, 8, 6, 4, 5, 4, 0, 7, 3, 7, 4, 5, 3, 9, 9, 3 ], [ 7, 7, 2, 7, 7, 0, 5, 6, 3, 1, 7, 0, 3, 3, 8, 3, 1, 8, 3, 8 ], [ 9, 3, 0, 2, 7, 5, 2, 4, 3, 0, 3, 3, 0, 9, 4, 0, 1, 1, 6, 3 ], [ 0, 0, 7, 5, 1, 2, 0, 8, 9, 0, 7, 3, 9, 0, 4, 8, 4, 7, 0, 6 ], [ 2, 9, 2, 1, 6, 2, 5, 9, 4, 7, 4, 8, 4, 4, 0, 7, 2, 7, 8, 4 ], [ 0, 2, 6, 3, 4, 2, 6, 8, 8, 7, 5, 3, 0, 8, 7, 0, 2, 0, 7, 1 ], [ 8, 8, 0, 1, 9, 6, 0, 1, 9, 5, 3, 1, 1, 4, 2, 2, 0, 9, 8, 6 ], [ 7, 3, 1, 9, 8, 3, 9, 2, 0, 9, 9, 8, 1, 7, 7, 0, 9, 0, 4, 3 ], [ 6, 4, 7, 4, 8, 8, 7, 2, 7, 0, 9, 3, 6, 0, 8, 7, 8, 4, 0, 5 ], [ 9, 8, 3, 6, 1, 4, 8, 9, 0, 6, 3, 8, 3, 6, 4, 1, 6, 3, 5, 0 ] ], "m": 6, "objective": 105.0 }, "solution": [ 0, 4, 5, 16, 17, 18 ], "obj": 105.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 20, "m": 6, "nodes": [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 ], "distance_pairs": [ { "from_id": 1, "to_id": 2, "distance": 4 }, { 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"to_id": 1, "distance": 8 }, { "from_id": 17, "to_id": 2, "distance": 8 }, { "from_id": 17, "to_id": 3, "distance": 0 }, { "from_id": 17, "to_id": 4, "distance": 1 }, { "from_id": 17, "to_id": 5, "distance": 9 }, { "from_id": 17, "to_id": 6, "distance": 6 }, { "from_id": 17, "to_id": 7, "distance": 0 }, { "from_id": 17, "to_id": 8, "distance": 1 }, { "from_id": 17, "to_id": 9, "distance": 9 }, { "from_id": 17, "to_id": 10, "distance": 5 }, { "from_id": 17, "to_id": 11, "distance": 3 }, { "from_id": 17, "to_id": 12, "distance": 1 }, { "from_id": 17, "to_id": 13, "distance": 1 }, { "from_id": 17, "to_id": 14, "distance": 4 }, { "from_id": 17, "to_id": 15, "distance": 2 }, { "from_id": 17, "to_id": 16, "distance": 2 }, { "from_id": 17, "to_id": 18, "distance": 9 }, { "from_id": 17, "to_id": 19, "distance": 8 }, { "from_id": 17, "to_id": 20, "distance": 6 }, { "from_id": 18, "to_id": 1, "distance": 7 }, { "from_id": 18, "to_id": 2, "distance": 3 }, { "from_id": 18, "to_id": 3, "distance": 1 }, { "from_id": 18, "to_id": 4, "distance": 9 }, { "from_id": 18, "to_id": 5, "distance": 8 }, { "from_id": 18, "to_id": 6, "distance": 3 }, { "from_id": 18, "to_id": 7, "distance": 9 }, { "from_id": 18, "to_id": 8, "distance": 2 }, { "from_id": 18, "to_id": 9, "distance": 0 }, { "from_id": 18, "to_id": 10, "distance": 9 }, { "from_id": 18, "to_id": 11, "distance": 9 }, { "from_id": 18, "to_id": 12, "distance": 8 }, { "from_id": 18, "to_id": 13, "distance": 1 }, { "from_id": 18, "to_id": 14, "distance": 7 }, { "from_id": 18, "to_id": 15, "distance": 7 }, { "from_id": 18, "to_id": 16, "distance": 0 }, { "from_id": 18, "to_id": 17, "distance": 9 }, { "from_id": 18, "to_id": 19, "distance": 4 }, { "from_id": 18, "to_id": 20, "distance": 3 }, { "from_id": 19, "to_id": 1, "distance": 6 }, { "from_id": 19, "to_id": 2, "distance": 4 }, { "from_id": 19, "to_id": 3, "distance": 7 }, { "from_id": 19, "to_id": 4, "distance": 4 }, { "from_id": 19, "to_id": 5, "distance": 8 }, { "from_id": 19, "to_id": 6, "distance": 8 }, { "from_id": 19, "to_id": 7, "distance": 7 }, { "from_id": 19, "to_id": 8, "distance": 2 }, { "from_id": 19, "to_id": 9, "distance": 7 }, { "from_id": 19, "to_id": 10, "distance": 0 }, { "from_id": 19, "to_id": 11, "distance": 9 }, { "from_id": 19, "to_id": 12, "distance": 3 }, { "from_id": 19, "to_id": 13, "distance": 6 }, { "from_id": 19, "to_id": 14, "distance": 0 }, { "from_id": 19, "to_id": 15, "distance": 8 }, { "from_id": 19, "to_id": 16, "distance": 7 }, { "from_id": 19, "to_id": 17, "distance": 8 }, { "from_id": 19, "to_id": 18, "distance": 4 }, { "from_id": 19, "to_id": 20, "distance": 5 }, { "from_id": 20, "to_id": 1, "distance": 9 }, { "from_id": 20, "to_id": 2, "distance": 8 }, { "from_id": 20, "to_id": 3, "distance": 3 }, { "from_id": 20, "to_id": 4, "distance": 6 }, { "from_id": 20, "to_id": 5, "distance": 1 }, { "from_id": 20, "to_id": 6, "distance": 4 }, { "from_id": 20, "to_id": 7, "distance": 8 }, { "from_id": 20, "to_id": 8, "distance": 9 }, { "from_id": 20, "to_id": 9, "distance": 0 }, { "from_id": 20, "to_id": 10, "distance": 6 }, { "from_id": 20, "to_id": 11, "distance": 3 }, { "from_id": 20, "to_id": 12, "distance": 8 }, { "from_id": 20, "to_id": 13, "distance": 3 }, { "from_id": 20, "to_id": 14, "distance": 6 }, { "from_id": 20, "to_id": 15, "distance": 4 }, { "from_id": 20, "to_id": 16, "distance": 1 }, { "from_id": 20, "to_id": 17, "distance": 6 }, { "from_id": 20, "to_id": 18, "distance": 3 }, { "from_id": 20, "to_id": 19, "distance": 5 } ] }, "solution_variant": [ 1, 5, 6, 17, 18, 19 ], "context_index": 18, "input_format": "nl", "input_index_base": 1 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Many people respond better to variety, so the realtor wants a brochure with exactly m uniquely chosen listings that feel distinct from each other. There’s a given number for every pair of properties saying how unlike they are, and the aim is to pick those m properties so the sum of all pairwise numbers inside that set comes out as large as possible. No duplicates allowed and the count is fixed at m. The precise distances and the count are provided below.\n\nThere are 25 total listings identified as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, and the realtor will select exactly 7 for the brochure.\nPair 0 and 1 have dissimilarity 1.\nPair 0 and 2 have dissimilarity 1.\nPair 0 and 3 have dissimilarity 0.\nPair 0 and 4 have dissimilarity 6.\nPair 0 and 5 have dissimilarity 7.\nPair 0 and 6 have dissimilarity 3.\nPair 0 and 7 have dissimilarity 6.\nPair 0 and 8 have dissimilarity 5.\nPair 0 and 9 have dissimilarity 0.\nPair 0 and 10 have dissimilarity 5.\nPair 0 and 11 have dissimilarity 7.\nPair 0 and 12 have dissimilarity 8.\nPair 0 and 13 have dissimilarity 4.\nPair 0 and 14 have dissimilarity 3.\nPair 0 and 15 have dissimilarity 4.\nPair 0 and 16 have dissimilarity 8.\nPair 0 and 17 have dissimilarity 5.\nPair 0 and 18 have dissimilarity 4.\nPair 0 and 19 have dissimilarity 8.\nPair 0 and 20 have dissimilarity 6.\nPair 0 and 21 have dissimilarity 1.\nPair 0 and 22 have dissimilarity 8.\nPair 0 and 23 have dissimilarity 3.\nPair 0 and 24 have dissimilarity 0.\nPair 1 and 0 have dissimilarity 1.\nPair 1 and 2 have dissimilarity 6.\nPair 1 and 3 have dissimilarity 7.\nPair 1 and 4 have dissimilarity 1.\nPair 1 and 5 have dissimilarity 9.\nPair 1 and 6 have dissimilarity 9.\nPair 1 and 7 have dissimilarity 4.\nPair 1 and 8 have dissimilarity 5.\nPair 1 and 9 have dissimilarity 8.\nPair 1 and 10 have dissimilarity 8.\nPair 1 and 11 have dissimilarity 5.\nPair 1 and 12 have dissimilarity 7.\nPair 1 and 13 have dissimilarity 5.\nPair 1 and 14 have dissimilarity 8.\nPair 1 and 15 have dissimilarity 3.\nPair 1 and 16 have dissimilarity 2.\nPair 1 and 17 have dissimilarity 2.\nPair 1 and 18 have dissimilarity 5.\nPair 1 and 19 have dissimilarity 8.\nPair 1 and 20 have dissimilarity 1.\nPair 1 and 21 have dissimilarity 0.\nPair 1 and 22 have dissimilarity 6.\nPair 1 and 23 have dissimilarity 0.\nPair 1 and 24 have dissimilarity 5.\nPair 2 and 0 have dissimilarity 1.\nPair 2 and 1 have dissimilarity 6.\nPair 2 and 3 have dissimilarity 1.\nPair 2 and 4 have dissimilarity 4.\nPair 2 and 5 have dissimilarity 2.\nPair 2 and 6 have dissimilarity 3.\nPair 2 and 7 have dissimilarity 9.\nPair 2 and 8 have dissimilarity 2.\nPair 2 and 9 have dissimilarity 4.\nPair 2 and 10 have dissimilarity 4.\nPair 2 and 11 have dissimilarity 8.\nPair 2 and 12 have dissimilarity 1.\nPair 2 and 13 have dissimilarity 4.\nPair 2 and 14 have dissimilarity 7.\nPair 2 and 15 have dissimilarity 6.\nPair 2 and 16 have dissimilarity 6.\nPair 2 and 17 have dissimilarity 6.\nPair 2 and 18 have dissimilarity 1.\nPair 2 and 19 have dissimilarity 2.\nPair 2 and 20 have dissimilarity 4.\nPair 2 and 21 have dissimilarity 6.\nPair 2 and 22 have dissimilarity 8.\nPair 2 and 23 have dissimilarity 4.\nPair 2 and 24 have dissimilarity 9.\nPair 3 and 0 have dissimilarity 0.\nPair 3 and 1 have dissimilarity 7.\nPair 3 and 2 have dissimilarity 1.\nPair 3 and 4 have dissimilarity 9.\nPair 3 and 5 have dissimilarity 5.\nPair 3 and 6 have dissimilarity 6.\nPair 3 and 7 have dissimilarity 6.\nPair 3 and 8 have dissimilarity 5.\nPair 3 and 9 have dissimilarity 8.\nPair 3 and 10 have dissimilarity 8.\nPair 3 and 11 have dissimilarity 0.\nPair 3 and 12 have dissimilarity 8.\nPair 3 and 13 have dissimilarity 9.\nPair 3 and 14 have dissimilarity 1.\nPair 3 and 15 have dissimilarity 3.\nPair 3 and 16 have dissimilarity 6.\nPair 3 and 17 have dissimilarity 9.\nPair 3 and 18 have dissimilarity 3.\nPair 3 and 19 have dissimilarity 0.\nPair 3 and 20 have dissimilarity 0.\nPair 3 and 21 have dissimilarity 7.\nPair 3 and 22 have dissimilarity 5.\nPair 3 and 23 have dissimilarity 0.\nPair 3 and 24 have dissimilarity 6.\nPair 4 and 0 have dissimilarity 6.\nPair 4 and 1 have dissimilarity 1.\nPair 4 and 2 have dissimilarity 4.\nPair 4 and 3 have dissimilarity 9.\nPair 4 and 5 have dissimilarity 5.\nPair 4 and 6 have dissimilarity 6.\nPair 4 and 7 have dissimilarity 3.\nPair 4 and 8 have dissimilarity 2.\nPair 4 and 9 have dissimilarity 4.\nPair 4 and 10 have dissimilarity 6.\nPair 4 and 11 have dissimilarity 8.\nPair 4 and 12 have dissimilarity 8.\nPair 4 and 13 have dissimilarity 8.\nPair 4 and 14 have dissimilarity 0.\nPair 4 and 15 have dissimilarity 1.\nPair 4 and 16 have dissimilarity 1.\nPair 4 and 17 have dissimilarity 7.\nPair 4 and 18 have dissimilarity 4.\nPair 4 and 19 have dissimilarity 1.\nPair 4 and 20 have dissimilarity 6.\nPair 4 and 21 have dissimilarity 9.\nPair 4 and 22 have dissimilarity 2.\nPair 4 and 23 have dissimilarity 5.\nPair 4 and 24 have dissimilarity 2.\nPair 5 and 0 have dissimilarity 7.\nPair 5 and 1 have dissimilarity 9.\nPair 5 and 2 have dissimilarity 2.\nPair 5 and 3 have dissimilarity 5.\nPair 5 and 4 have dissimilarity 5.\nPair 5 and 6 have dissimilarity 8.\nPair 5 and 7 have dissimilarity 9.\nPair 5 and 8 have dissimilarity 5.\nPair 5 and 9 have dissimilarity 5.\nPair 5 and 10 have dissimilarity 0.\nPair 5 and 11 have dissimilarity 2.\nPair 5 and 12 have dissimilarity 6.\nPair 5 and 13 have dissimilarity 9.\nPair 5 and 14 have dissimilarity 8.\nPair 5 and 15 have dissimilarity 1.\nPair 5 and 16 have dissimilarity 7.\nPair 5 and 17 have dissimilarity 0.\nPair 5 and 18 have dissimilarity 6.\nPair 5 and 19 have dissimilarity 8.\nPair 5 and 20 have dissimilarity 7.\nPair 5 and 21 have dissimilarity 4.\nPair 5 and 22 have dissimilarity 2.\nPair 5 and 23 have dissimilarity 2.\nPair 5 and 24 have dissimilarity 0.\nPair 6 and 0 have dissimilarity 3.\nPair 6 and 1 have dissimilarity 9.\nPair 6 and 2 have dissimilarity 3.\nPair 6 and 3 have dissimilarity 6.\nPair 6 and 4 have dissimilarity 6.\nPair 6 and 5 have dissimilarity 8.\nPair 6 and 7 have dissimilarity 3.\nPair 6 and 8 have dissimilarity 1.\nPair 6 and 9 have dissimilarity 3.\nPair 6 and 10 have dissimilarity 9.\nPair 6 and 11 have dissimilarity 0.\nPair 6 and 12 have dissimilarity 2.\nPair 6 and 13 have dissimilarity 4.\nPair 6 and 14 have dissimilarity 5.\nPair 6 and 15 have dissimilarity 8.\nPair 6 and 16 have dissimilarity 1.\nPair 6 and 17 have dissimilarity 1.\nPair 6 and 18 have dissimilarity 3.\nPair 6 and 19 have dissimilarity 4.\nPair 6 and 20 have dissimilarity 4.\nPair 6 and 21 have dissimilarity 0.\nPair 6 and 22 have dissimilarity 9.\nPair 6 and 23 have dissimilarity 5.\nPair 6 and 24 have dissimilarity 3.\nPair 7 and 0 have dissimilarity 6.\nPair 7 and 1 have dissimilarity 4.\nPair 7 and 2 have dissimilarity 9.\nPair 7 and 3 have dissimilarity 6.\nPair 7 and 4 have dissimilarity 3.\nPair 7 and 5 have dissimilarity 9.\nPair 7 and 6 have dissimilarity 3.\nPair 7 and 8 have dissimilarity 0.\nPair 7 and 9 have dissimilarity 5.\nPair 7 and 10 have dissimilarity 9.\nPair 7 and 11 have dissimilarity 3.\nPair 7 and 12 have dissimilarity 2.\nPair 7 and 13 have dissimilarity 6.\nPair 7 and 14 have dissimilarity 8.\nPair 7 and 15 have dissimilarity 1.\nPair 7 and 16 have dissimilarity 8.\nPair 7 and 17 have dissimilarity 7.\nPair 7 and 18 have dissimilarity 8.\nPair 7 and 19 have dissimilarity 5.\nPair 7 and 20 have dissimilarity 4.\nPair 7 and 21 have dissimilarity 4.\nPair 7 and 22 have dissimilarity 6.\nPair 7 and 23 have dissimilarity 6.\nPair 7 and 24 have dissimilarity 7.\nPair 8 and 0 have dissimilarity 5.\nPair 8 and 1 have dissimilarity 5.\nPair 8 and 2 have dissimilarity 2.\nPair 8 and 3 have dissimilarity 5.\nPair 8 and 4 have dissimilarity 2.\nPair 8 and 5 have dissimilarity 5.\nPair 8 and 6 have dissimilarity 1.\nPair 8 and 7 have dissimilarity 0.\nPair 8 and 9 have dissimilarity 5.\nPair 8 and 10 have dissimilarity 9.\nPair 8 and 11 have dissimilarity 4.\nPair 8 and 12 have dissimilarity 2.\nPair 8 and 13 have dissimilarity 8.\nPair 8 and 14 have dissimilarity 8.\nPair 8 and 15 have dissimilarity 7.\nPair 8 and 16 have dissimilarity 8.\nPair 8 and 17 have dissimilarity 5.\nPair 8 and 18 have dissimilarity 5.\nPair 8 and 19 have dissimilarity 7.\nPair 8 and 20 have dissimilarity 5.\nPair 8 and 21 have dissimilarity 4.\nPair 8 and 22 have dissimilarity 9.\nPair 8 and 23 have dissimilarity 5.\nPair 8 and 24 have dissimilarity 4.\nPair 9 and 0 have dissimilarity 0.\nPair 9 and 1 have dissimilarity 8.\nPair 9 and 2 have dissimilarity 4.\nPair 9 and 3 have dissimilarity 8.\nPair 9 and 4 have dissimilarity 4.\nPair 9 and 5 have dissimilarity 5.\nPair 9 and 6 have dissimilarity 3.\nPair 9 and 7 have dissimilarity 5.\nPair 9 and 8 have dissimilarity 5.\nPair 9 and 10 have dissimilarity 9.\nPair 9 and 11 have dissimilarity 5.\nPair 9 and 12 have dissimilarity 4.\nPair 9 and 13 have dissimilarity 9.\nPair 9 and 14 have dissimilarity 9.\nPair 9 and 15 have dissimilarity 5.\nPair 9 and 16 have dissimilarity 1.\nPair 9 and 17 have dissimilarity 7.\nPair 9 and 18 have dissimilarity 2.\nPair 9 and 19 have dissimilarity 5.\nPair 9 and 20 have dissimilarity 5.\nPair 9 and 21 have dissimilarity 7.\nPair 9 and 22 have dissimilarity 9.\nPair 9 and 23 have dissimilarity 7.\nPair 9 and 24 have dissimilarity 9.\nPair 10 and 0 have dissimilarity 5.\nPair 10 and 1 have dissimilarity 8.\nPair 10 and 2 have dissimilarity 4.\nPair 10 and 3 have dissimilarity 8.\nPair 10 and 4 have dissimilarity 6.\nPair 10 and 5 have dissimilarity 0.\nPair 10 and 6 have dissimilarity 9.\nPair 10 and 7 have dissimilarity 9.\nPair 10 and 8 have dissimilarity 9.\nPair 10 and 9 have dissimilarity 9.\nPair 10 and 11 have dissimilarity 3.\nPair 10 and 12 have dissimilarity 9.\nPair 10 and 13 have dissimilarity 1.\nPair 10 and 14 have dissimilarity 2.\nPair 10 and 15 have dissimilarity 7.\nPair 10 and 16 have dissimilarity 2.\nPair 10 and 17 have dissimilarity 2.\nPair 10 and 18 have dissimilarity 5.\nPair 10 and 19 have dissimilarity 9.\nPair 10 and 20 have dissimilarity 2.\nPair 10 and 21 have dissimilarity 9.\nPair 10 and 22 have dissimilarity 1.\nPair 10 and 23 have dissimilarity 9.\nPair 10 and 24 have dissimilarity 6.\nPair 11 and 0 have dissimilarity 7.\nPair 11 and 1 have dissimilarity 5.\nPair 11 and 2 have dissimilarity 8.\nPair 11 and 3 have dissimilarity 0.\nPair 11 and 4 have dissimilarity 8.\nPair 11 and 5 have dissimilarity 2.\nPair 11 and 6 have dissimilarity 0.\nPair 11 and 7 have dissimilarity 3.\nPair 11 and 8 have dissimilarity 4.\nPair 11 and 9 have dissimilarity 5.\nPair 11 and 10 have dissimilarity 3.\nPair 11 and 12 have dissimilarity 3.\nPair 11 and 13 have dissimilarity 8.\nPair 11 and 14 have dissimilarity 2.\nPair 11 and 15 have dissimilarity 9.\nPair 11 and 16 have dissimilarity 9.\nPair 11 and 17 have dissimilarity 6.\nPair 11 and 18 have dissimilarity 4.\nPair 11 and 19 have dissimilarity 8.\nPair 11 and 20 have dissimilarity 2.\nPair 11 and 21 have dissimilarity 7.\nPair 11 and 22 have dissimilarity 5.\nPair 11 and 23 have dissimilarity 6.\nPair 11 and 24 have dissimilarity 4.\nPair 12 and 0 have dissimilarity 8.\nPair 12 and 1 have dissimilarity 7.\nPair 12 and 2 have dissimilarity 1.\nPair 12 and 3 have dissimilarity 8.\nPair 12 and 4 have dissimilarity 8.\nPair 12 and 5 have dissimilarity 6.\nPair 12 and 6 have dissimilarity 2.\nPair 12 and 7 have dissimilarity 2.\nPair 12 and 8 have dissimilarity 2.\nPair 12 and 9 have dissimilarity 4.\nPair 12 and 10 have dissimilarity 9.\nPair 12 and 11 have dissimilarity 3.\nPair 12 and 13 have dissimilarity 4.\nPair 12 and 14 have dissimilarity 3.\nPair 12 and 15 have dissimilarity 3.\nPair 12 and 16 have dissimilarity 1.\nPair 12 and 17 have dissimilarity 3.\nPair 12 and 18 have dissimilarity 0.\nPair 12 and 19 have dissimilarity 6.\nPair 12 and 20 have dissimilarity 5.\nPair 12 and 21 have dissimilarity 2.\nPair 12 and 22 have dissimilarity 0.\nPair 12 and 23 have dissimilarity 8.\nPair 12 and 24 have dissimilarity 5.\nPair 13 and 0 have dissimilarity 4.\nPair 13 and 1 have dissimilarity 5.\nPair 13 and 2 have dissimilarity 4.\nPair 13 and 3 have dissimilarity 9.\nPair 13 and 4 have dissimilarity 8.\nPair 13 and 5 have dissimilarity 9.\nPair 13 and 6 have dissimilarity 4.\nPair 13 and 7 have dissimilarity 6.\nPair 13 and 8 have dissimilarity 8.\nPair 13 and 9 have dissimilarity 9.\nPair 13 and 10 have dissimilarity 1.\nPair 13 and 11 have dissimilarity 8.\nPair 13 and 12 have dissimilarity 4.\nPair 13 and 14 have dissimilarity 2.\nPair 13 and 15 have dissimilarity 2.\nPair 13 and 16 have dissimilarity 6.\nPair 13 and 17 have dissimilarity 5.\nPair 13 and 18 have dissimilarity 8.\nPair 13 and 19 have dissimilarity 3.\nPair 13 and 20 have dissimilarity 5.\nPair 13 and 21 have dissimilarity 9.\nPair 13 and 22 have dissimilarity 9.\nPair 13 and 23 have dissimilarity 9.\nPair 13 and 24 have dissimilarity 4.\nPair 14 and 0 have dissimilarity 3.\nPair 14 and 1 have dissimilarity 8.\nPair 14 and 2 have dissimilarity 7.\nPair 14 and 3 have dissimilarity 1.\nPair 14 and 4 have dissimilarity 0.\nPair 14 and 5 have dissimilarity 8.\nPair 14 and 6 have dissimilarity 5.\nPair 14 and 7 have dissimilarity 8.\nPair 14 and 8 have dissimilarity 8.\nPair 14 and 9 have dissimilarity 9.\nPair 14 and 10 have dissimilarity 2.\nPair 14 and 11 have dissimilarity 2.\nPair 14 and 12 have dissimilarity 3.\nPair 14 and 13 have dissimilarity 2.\nPair 14 and 15 have dissimilarity 8.\nPair 14 and 16 have dissimilarity 5.\nPair 14 and 17 have dissimilarity 4.\nPair 14 and 18 have dissimilarity 8.\nPair 14 and 19 have dissimilarity 5.\nPair 14 and 20 have dissimilarity 5.\nPair 14 and 21 have dissimilarity 5.\nPair 14 and 22 have dissimilarity 3.\nPair 14 and 23 have dissimilarity 2.\nPair 14 and 24 have dissimilarity 7.\nPair 15 and 0 have dissimilarity 4.\nPair 15 and 1 have dissimilarity 3.\nPair 15 and 2 have dissimilarity 6.\nPair 15 and 3 have dissimilarity 3.\nPair 15 and 4 have dissimilarity 1.\nPair 15 and 5 have dissimilarity 1.\nPair 15 and 6 have dissimilarity 8.\nPair 15 and 7 have dissimilarity 1.\nPair 15 and 8 have dissimilarity 7.\nPair 15 and 9 have dissimilarity 5.\nPair 15 and 10 have dissimilarity 7.\nPair 15 and 11 have dissimilarity 9.\nPair 15 and 12 have dissimilarity 3.\nPair 15 and 13 have dissimilarity 2.\nPair 15 and 14 have dissimilarity 8.\nPair 15 and 16 have dissimilarity 0.\nPair 15 and 17 have dissimilarity 4.\nPair 15 and 18 have dissimilarity 4.\nPair 15 and 19 have dissimilarity 8.\nPair 15 and 20 have dissimilarity 0.\nPair 15 and 21 have dissimilarity 5.\nPair 15 and 22 have dissimilarity 0.\nPair 15 and 23 have dissimilarity 0.\nPair 15 and 24 have dissimilarity 1.\nPair 16 and 0 have dissimilarity 8.\nPair 16 and 1 have dissimilarity 2.\nPair 16 and 2 have dissimilarity 6.\nPair 16 and 3 have dissimilarity 6.\nPair 16 and 4 have dissimilarity 1.\nPair 16 and 5 have dissimilarity 7.\nPair 16 and 6 have dissimilarity 1.\nPair 16 and 7 have dissimilarity 8.\nPair 16 and 8 have dissimilarity 8.\nPair 16 and 9 have dissimilarity 1.\nPair 16 and 10 have dissimilarity 2.\nPair 16 and 11 have dissimilarity 9.\nPair 16 and 12 have dissimilarity 1.\nPair 16 and 13 have dissimilarity 6.\nPair 16 and 14 have dissimilarity 5.\nPair 16 and 15 have dissimilarity 0.\nPair 16 and 17 have dissimilarity 0.\nPair 16 and 18 have dissimilarity 2.\nPair 16 and 19 have dissimilarity 1.\nPair 16 and 20 have dissimilarity 4.\nPair 16 and 21 have dissimilarity 4.\nPair 16 and 22 have dissimilarity 9.\nPair 16 and 23 have dissimilarity 3.\nPair 16 and 24 have dissimilarity 9.\nPair 17 and 0 have dissimilarity 5.\nPair 17 and 1 have dissimilarity 2.\nPair 17 and 2 have dissimilarity 6.\nPair 17 and 3 have dissimilarity 9.\nPair 17 and 4 have dissimilarity 7.\nPair 17 and 5 have dissimilarity 0.\nPair 17 and 6 have dissimilarity 1.\nPair 17 and 7 have dissimilarity 7.\nPair 17 and 8 have dissimilarity 5.\nPair 17 and 9 have dissimilarity 7.\nPair 17 and 10 have dissimilarity 2.\nPair 17 and 11 have dissimilarity 6.\nPair 17 and 12 have dissimilarity 3.\nPair 17 and 13 have dissimilarity 5.\nPair 17 and 14 have dissimilarity 4.\nPair 17 and 15 have dissimilarity 4.\nPair 17 and 16 have dissimilarity 0.\nPair 17 and 18 have dissimilarity 1.\nPair 17 and 19 have dissimilarity 0.\nPair 17 and 20 have dissimilarity 2.\nPair 17 and 21 have dissimilarity 4.\nPair 17 and 22 have dissimilarity 7.\nPair 17 and 23 have dissimilarity 6.\nPair 17 and 24 have dissimilarity 3.\nPair 18 and 0 have dissimilarity 4.\nPair 18 and 1 have dissimilarity 5.\nPair 18 and 2 have dissimilarity 1.\nPair 18 and 3 have dissimilarity 3.\nPair 18 and 4 have dissimilarity 4.\nPair 18 and 5 have dissimilarity 6.\nPair 18 and 6 have dissimilarity 3.\nPair 18 and 7 have dissimilarity 8.\nPair 18 and 8 have dissimilarity 5.\nPair 18 and 9 have dissimilarity 2.\nPair 18 and 10 have dissimilarity 5.\nPair 18 and 11 have dissimilarity 4.\nPair 18 and 12 have dissimilarity 0.\nPair 18 and 13 have dissimilarity 8.\nPair 18 and 14 have dissimilarity 8.\nPair 18 and 15 have dissimilarity 4.\nPair 18 and 16 have dissimilarity 2.\nPair 18 and 17 have dissimilarity 1.\nPair 18 and 19 have dissimilarity 1.\nPair 18 and 20 have dissimilarity 2.\nPair 18 and 21 have dissimilarity 2.\nPair 18 and 22 have dissimilarity 8.\nPair 18 and 23 have dissimilarity 4.\nPair 18 and 24 have dissimilarity 6.\nPair 19 and 0 have dissimilarity 8.\nPair 19 and 1 have dissimilarity 8.\nPair 19 and 2 have dissimilarity 2.\nPair 19 and 3 have dissimilarity 0.\nPair 19 and 4 have dissimilarity 1.\nPair 19 and 5 have dissimilarity 8.\nPair 19 and 6 have dissimilarity 4.\nPair 19 and 7 have dissimilarity 5.\nPair 19 and 8 have dissimilarity 7.\nPair 19 and 9 have dissimilarity 5.\nPair 19 and 10 have dissimilarity 9.\nPair 19 and 11 have dissimilarity 8.\nPair 19 and 12 have dissimilarity 6.\nPair 19 and 13 have dissimilarity 3.\nPair 19 and 14 have dissimilarity 5.\nPair 19 and 15 have dissimilarity 8.\nPair 19 and 16 have dissimilarity 1.\nPair 19 and 17 have dissimilarity 0.\nPair 19 and 18 have dissimilarity 1.\nPair 19 and 20 have dissimilarity 6.\nPair 19 and 21 have dissimilarity 4.\nPair 19 and 22 have dissimilarity 3.\nPair 19 and 23 have dissimilarity 5.\nPair 19 and 24 have dissimilarity 9.\nPair 20 and 0 have dissimilarity 6.\nPair 20 and 1 have dissimilarity 1.\nPair 20 and 2 have dissimilarity 4.\nPair 20 and 3 have dissimilarity 0.\nPair 20 and 4 have dissimilarity 6.\nPair 20 and 5 have dissimilarity 7.\nPair 20 and 6 have dissimilarity 4.\nPair 20 and 7 have dissimilarity 4.\nPair 20 and 8 have dissimilarity 5.\nPair 20 and 9 have dissimilarity 5.\nPair 20 and 10 have dissimilarity 2.\nPair 20 and 11 have dissimilarity 2.\nPair 20 and 12 have dissimilarity 5.\nPair 20 and 13 have dissimilarity 5.\nPair 20 and 14 have dissimilarity 5.\nPair 20 and 15 have dissimilarity 0.\nPair 20 and 16 have dissimilarity 4.\nPair 20 and 17 have dissimilarity 2.\nPair 20 and 18 have dissimilarity 2.\nPair 20 and 19 have dissimilarity 6.\nPair 20 and 21 have dissimilarity 4.\nPair 20 and 22 have dissimilarity 0.\nPair 20 and 23 have dissimilarity 7.\nPair 20 and 24 have dissimilarity 3.\nPair 21 and 0 have dissimilarity 1.\nPair 21 and 1 have dissimilarity 0.\nPair 21 and 2 have dissimilarity 6.\nPair 21 and 3 have dissimilarity 7.\nPair 21 and 4 have dissimilarity 9.\nPair 21 and 5 have dissimilarity 4.\nPair 21 and 6 have dissimilarity 0.\nPair 21 and 7 have dissimilarity 4.\nPair 21 and 8 have dissimilarity 4.\nPair 21 and 9 have dissimilarity 7.\nPair 21 and 10 have dissimilarity 9.\nPair 21 and 11 have dissimilarity 7.\nPair 21 and 12 have dissimilarity 2.\nPair 21 and 13 have dissimilarity 9.\nPair 21 and 14 have dissimilarity 5.\nPair 21 and 15 have dissimilarity 5.\nPair 21 and 16 have dissimilarity 4.\nPair 21 and 17 have dissimilarity 4.\nPair 21 and 18 have dissimilarity 2.\nPair 21 and 19 have dissimilarity 4.\nPair 21 and 20 have dissimilarity 4.\nPair 21 and 22 have dissimilarity 4.\nPair 21 and 23 have dissimilarity 4.\nPair 21 and 24 have dissimilarity 0.\nPair 22 and 0 have dissimilarity 8.\nPair 22 and 1 have dissimilarity 6.\nPair 22 and 2 have dissimilarity 8.\nPair 22 and 3 have dissimilarity 5.\nPair 22 and 4 have dissimilarity 2.\nPair 22 and 5 have dissimilarity 2.\nPair 22 and 6 have dissimilarity 9.\nPair 22 and 7 have dissimilarity 6.\nPair 22 and 8 have dissimilarity 9.\nPair 22 and 9 have dissimilarity 9.\nPair 22 and 10 have dissimilarity 1.\nPair 22 and 11 have dissimilarity 5.\nPair 22 and 12 have dissimilarity 0.\nPair 22 and 13 have dissimilarity 9.\nPair 22 and 14 have dissimilarity 3.\nPair 22 and 15 have dissimilarity 0.\nPair 22 and 16 have dissimilarity 9.\nPair 22 and 17 have dissimilarity 7.\nPair 22 and 18 have dissimilarity 8.\nPair 22 and 19 have dissimilarity 3.\nPair 22 and 20 have dissimilarity 0.\nPair 22 and 21 have dissimilarity 4.\nPair 22 and 23 have dissimilarity 1.\nPair 22 and 24 have dissimilarity 5.\nPair 23 and 0 have dissimilarity 3.\nPair 23 and 1 have dissimilarity 0.\nPair 23 and 2 have dissimilarity 4.\nPair 23 and 3 have dissimilarity 0.\nPair 23 and 4 have dissimilarity 5.\nPair 23 and 5 have dissimilarity 2.\nPair 23 and 6 have dissimilarity 5.\nPair 23 and 7 have dissimilarity 6.\nPair 23 and 8 have dissimilarity 5.\nPair 23 and 9 have dissimilarity 7.\nPair 23 and 10 have dissimilarity 9.\nPair 23 and 11 have dissimilarity 6.\nPair 23 and 12 have dissimilarity 8.\nPair 23 and 13 have dissimilarity 9.\nPair 23 and 14 have dissimilarity 2.\nPair 23 and 15 have dissimilarity 0.\nPair 23 and 16 have dissimilarity 3.\nPair 23 and 17 have dissimilarity 6.\nPair 23 and 18 have dissimilarity 4.\nPair 23 and 19 have dissimilarity 5.\nPair 23 and 20 have dissimilarity 7.\nPair 23 and 21 have dissimilarity 4.\nPair 23 and 22 have dissimilarity 1.\nPair 23 and 24 have dissimilarity 8.\nPair 24 and 0 have dissimilarity 0.\nPair 24 and 1 have dissimilarity 5.\nPair 24 and 2 have dissimilarity 9.\nPair 24 and 3 have dissimilarity 6.\nPair 24 and 4 have dissimilarity 2.\nPair 24 and 5 have dissimilarity 0.\nPair 24 and 6 have dissimilarity 3.\nPair 24 and 7 have dissimilarity 7.\nPair 24 and 8 have dissimilarity 4.\nPair 24 and 9 have dissimilarity 9.\nPair 24 and 10 have dissimilarity 6.\nPair 24 and 11 have dissimilarity 4.\nPair 24 and 12 have dissimilarity 5.\nPair 24 and 13 have dissimilarity 4.\nPair 24 and 14 have dissimilarity 7.\nPair 24 and 15 have dissimilarity 1.\nPair 24 and 16 have dissimilarity 9.\nPair 24 and 17 have dissimilarity 3.\nPair 24 and 18 have dissimilarity 6.\nPair 24 and 19 have dissimilarity 9.\nPair 24 and 20 have dissimilarity 3.\nPair 24 and 21 have dissimilarity 0.\nPair 24 and 22 have dissimilarity 5.\nPair 24 and 23 have dissimilarity 8.\nThese pairwise scores let the realtor pick 7 listings that maximize the brochure's overall distinctness.\n\nAlso, when you send back the chosen listings, please use this simple JSON layout so I can read it easily:\n\n{\n \"solution\": [, , ...]\n}\n\nThink of \"solution\" as the list of the m property identifiers you picked for the brochure — each entry is one listing. This block is just a sketch of the shape I expect, not the actual answer itself, so you'll replace those placeholders with the real identifiers from the instance.\n\nPlease be sure to use the identifiers exactly as they appear in the input — no renaming and no new labels. \n\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 1, 1, 0, 6, 7, 3, 6, 5, 0, 5, 7, 8, 4, 3, 4, 8, 5, 4, 8, 6, 1, 8, 3, 0 ], [ 1, 0, 6, 7, 1, 9, 9, 4, 5, 8, 8, 5, 7, 5, 8, 3, 2, 2, 5, 8, 1, 0, 6, 0, 5 ], [ 1, 6, 0, 1, 4, 2, 3, 9, 2, 4, 4, 8, 1, 4, 7, 6, 6, 6, 1, 2, 4, 6, 8, 4, 9 ], [ 0, 7, 1, 0, 9, 5, 6, 6, 5, 8, 8, 0, 8, 9, 1, 3, 6, 9, 3, 0, 0, 7, 5, 0, 6 ], [ 6, 1, 4, 9, 0, 5, 6, 3, 2, 4, 6, 8, 8, 8, 0, 1, 1, 7, 4, 1, 6, 9, 2, 5, 2 ], [ 7, 9, 2, 5, 5, 0, 8, 9, 5, 5, 0, 2, 6, 9, 8, 1, 7, 0, 6, 8, 7, 4, 2, 2, 0 ], [ 3, 9, 3, 6, 6, 8, 0, 3, 1, 3, 9, 0, 2, 4, 5, 8, 1, 1, 3, 4, 4, 0, 9, 5, 3 ], [ 6, 4, 9, 6, 3, 9, 3, 0, 0, 5, 9, 3, 2, 6, 8, 1, 8, 7, 8, 5, 4, 4, 6, 6, 7 ], [ 5, 5, 2, 5, 2, 5, 1, 0, 0, 5, 9, 4, 2, 8, 8, 7, 8, 5, 5, 7, 5, 4, 9, 5, 4 ], [ 0, 8, 4, 8, 4, 5, 3, 5, 5, 0, 9, 5, 4, 9, 9, 5, 1, 7, 2, 5, 5, 7, 9, 7, 9 ], [ 5, 8, 4, 8, 6, 0, 9, 9, 9, 9, 0, 3, 9, 1, 2, 7, 2, 2, 5, 9, 2, 9, 1, 9, 6 ], [ 7, 5, 8, 0, 8, 2, 0, 3, 4, 5, 3, 0, 3, 8, 2, 9, 9, 6, 4, 8, 2, 7, 5, 6, 4 ], [ 8, 7, 1, 8, 8, 6, 2, 2, 2, 4, 9, 3, 0, 4, 3, 3, 1, 3, 0, 6, 5, 2, 0, 8, 5 ], [ 4, 5, 4, 9, 8, 9, 4, 6, 8, 9, 1, 8, 4, 0, 2, 2, 6, 5, 8, 3, 5, 9, 9, 9, 4 ], [ 3, 8, 7, 1, 0, 8, 5, 8, 8, 9, 2, 2, 3, 2, 0, 8, 5, 4, 8, 5, 5, 5, 3, 2, 7 ], [ 4, 3, 6, 3, 1, 1, 8, 1, 7, 5, 7, 9, 3, 2, 8, 0, 0, 4, 4, 8, 0, 5, 0, 0, 1 ], [ 8, 2, 6, 6, 1, 7, 1, 8, 8, 1, 2, 9, 1, 6, 5, 0, 0, 0, 2, 1, 4, 4, 9, 3, 9 ], [ 5, 2, 6, 9, 7, 0, 1, 7, 5, 7, 2, 6, 3, 5, 4, 4, 0, 0, 1, 0, 2, 4, 7, 6, 3 ], [ 4, 5, 1, 3, 4, 6, 3, 8, 5, 2, 5, 4, 0, 8, 8, 4, 2, 1, 0, 1, 2, 2, 8, 4, 6 ], [ 8, 8, 2, 0, 1, 8, 4, 5, 7, 5, 9, 8, 6, 3, 5, 8, 1, 0, 1, 0, 6, 4, 3, 5, 9 ], [ 6, 1, 4, 0, 6, 7, 4, 4, 5, 5, 2, 2, 5, 5, 5, 0, 4, 2, 2, 6, 0, 4, 0, 7, 3 ], [ 1, 0, 6, 7, 9, 4, 0, 4, 4, 7, 9, 7, 2, 9, 5, 5, 4, 4, 2, 4, 4, 0, 4, 4, 0 ], [ 8, 6, 8, 5, 2, 2, 9, 6, 9, 9, 1, 5, 0, 9, 3, 0, 9, 7, 8, 3, 0, 4, 0, 1, 5 ], [ 3, 0, 4, 0, 5, 2, 5, 6, 5, 7, 9, 6, 8, 9, 2, 0, 3, 6, 4, 5, 7, 4, 1, 0, 8 ], [ 0, 5, 9, 6, 2, 0, 3, 7, 4, 9, 6, 4, 5, 4, 7, 1, 9, 3, 6, 9, 3, 0, 5, 8, 0 ] ], "m": 7, "objective": 147.0 }, "solution": [ 3, 4, 9, 13, 17, 21, 22 ], "obj": 147.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 25, "m": 7, "nodes": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 ], "distance_pairs": [ { "from_id": 0, "to_id": 1, "distance": 1 }, { "from_id": 0, "to_id": 2, "distance": 1 }, { "from_id": 0, "to_id": 3, "distance": 0 }, { "from_id": 0, "to_id": 4, "distance": 6 }, { "from_id": 0, "to_id": 5, "distance": 7 }, { "from_id": 0, "to_id": 6, "distance": 3 }, { "from_id": 0, "to_id": 7, "distance": 6 }, { "from_id": 0, "to_id": 8, "distance": 5 }, { "from_id": 0, "to_id": 9, "distance": 0 }, { "from_id": 0, "to_id": 10, "distance": 5 }, { "from_id": 0, "to_id": 11, "distance": 7 }, { "from_id": 0, "to_id": 12, "distance": 8 }, { "from_id": 0, "to_id": 13, "distance": 4 }, { "from_id": 0, "to_id": 14, "distance": 3 }, { "from_id": 0, "to_id": 15, "distance": 4 }, { "from_id": 0, "to_id": 16, "distance": 8 }, { "from_id": 0, "to_id": 17, "distance": 5 }, { "from_id": 0, "to_id": 18, "distance": 4 }, { "from_id": 0, "to_id": 19, "distance": 8 }, { "from_id": 0, "to_id": 20, "distance": 6 }, { "from_id": 0, "to_id": 21, "distance": 1 }, { "from_id": 0, "to_id": 22, "distance": 8 }, { "from_id": 0, "to_id": 23, "distance": 3 }, { "from_id": 0, "to_id": 24, "distance": 0 }, { "from_id": 1, "to_id": 0, "distance": 1 }, { "from_id": 1, "to_id": 2, "distance": 6 }, { "from_id": 1, "to_id": 3, "distance": 7 }, { "from_id": 1, "to_id": 4, "distance": 1 }, { "from_id": 1, "to_id": 5, "distance": 9 }, { "from_id": 1, "to_id": 6, "distance": 9 }, { "from_id": 1, "to_id": 7, "distance": 4 }, { "from_id": 1, "to_id": 8, "distance": 5 }, { "from_id": 1, "to_id": 9, "distance": 8 }, { "from_id": 1, "to_id": 10, "distance": 8 }, { "from_id": 1, "to_id": 11, "distance": 5 }, { "from_id": 1, "to_id": 12, "distance": 7 }, { "from_id": 1, "to_id": 13, "distance": 5 }, { "from_id": 1, "to_id": 14, "distance": 8 }, { "from_id": 1, "to_id": 15, "distance": 3 }, { "from_id": 1, "to_id": 16, "distance": 2 }, { "from_id": 1, "to_id": 17, "distance": 2 }, { "from_id": 1, "to_id": 18, "distance": 5 }, { "from_id": 1, "to_id": 19, "distance": 8 }, { "from_id": 1, "to_id": 20, "distance": 1 }, { "from_id": 1, "to_id": 21, "distance": 0 }, { "from_id": 1, "to_id": 22, "distance": 6 }, { "from_id": 1, "to_id": 23, "distance": 0 }, { "from_id": 1, "to_id": 24, "distance": 5 }, { "from_id": 2, "to_id": 0, "distance": 1 }, { "from_id": 2, "to_id": 1, "distance": 6 }, { "from_id": 2, "to_id": 3, "distance": 1 }, { "from_id": 2, "to_id": 4, "distance": 4 }, { "from_id": 2, "to_id": 5, "distance": 2 }, { "from_id": 2, "to_id": 6, "distance": 3 }, { "from_id": 2, "to_id": 7, "distance": 9 }, { "from_id": 2, "to_id": 8, "distance": 2 }, { "from_id": 2, "to_id": 9, "distance": 4 }, { "from_id": 2, "to_id": 10, "distance": 4 }, { "from_id": 2, "to_id": 11, "distance": 8 }, { "from_id": 2, "to_id": 12, "distance": 1 }, { "from_id": 2, "to_id": 13, "distance": 4 }, { "from_id": 2, "to_id": 14, "distance": 7 }, { "from_id": 2, "to_id": 15, "distance": 6 }, { "from_id": 2, "to_id": 16, "distance": 6 }, { "from_id": 2, "to_id": 17, "distance": 6 }, { "from_id": 2, "to_id": 18, "distance": 1 }, { "from_id": 2, "to_id": 19, "distance": 2 }, { "from_id": 2, "to_id": 20, "distance": 4 }, { "from_id": 2, "to_id": 21, "distance": 6 }, { "from_id": 2, "to_id": 22, "distance": 8 }, { "from_id": 2, "to_id": 23, "distance": 4 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The way to tell which selection wins is simple — for each pair of picked plates note how different they are, add all those pair differences together, and the selection with the biggest total is the most diverse. The detailed list of options and their measured differences is shown below.\n\nThere are 21 distinct appetizer options (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21), and exactly 2 must be chosen.\n\n| appetizer_id_first | appetizer_id_second | flavor_presentation_difference |\n|---|---|---|\n| 1 | 2 | 0 |\n| 1 | 3 | 4 |\n| 1 | 4 | 7 |\n| 1 | 5 | 1 |\n| 1 | 6 | 4 |\n| 1 | 7 | 4 |\n| 1 | 8 | 0 |\n| 1 | 9 | 0 |\n| 1 | 10 | 5 |\n| 1 | 11 | 8 |\n| 1 | 12 | 3 |\n| 1 | 13 | 5 |\n| 1 | 14 | 9 |\n| 1 | 15 | 7 |\n| 1 | 16 | 9 |\n| 1 | 17 | 6 |\n| 1 | 18 | 1 |\n| 1 | 19 | 7 |\n| 1 | 20 | 9 |\n| 1 | 21 | 9 |\n| 2 | 1 | 0 |\n| 2 | 3 | 8 |\n| 2 | 4 | 4 |\n| 2 | 5 | 0 |\n| 2 | 6 | 3 |\n| 2 | 7 | 3 |\n| 2 | 8 | 1 |\n| 2 | 9 | 6 |\n| 2 | 10 | 3 |\n| 2 | 11 | 2 |\n| 2 | 12 | 6 |\n| 2 | 13 | 1 |\n| 2 | 14 | 3 |\n| 2 | 15 | 1 |\n| 2 | 16 | 1 |\n| 2 | 17 | 4 |\n| 2 | 18 | 0 |\n| 2 | 19 | 8 |\n| 2 | 20 | 9 |\n| 2 | 21 | 7 |\n| 3 | 1 | 4 |\n| 3 | 2 | 8 |\n| 3 | 4 | 2 |\n| 3 | 5 | 6 |\n| 3 | 6 | 5 |\n| 3 | 7 | 3 |\n| 3 | 8 | 5 |\n| 3 | 9 | 9 |\n| 3 | 10 | 4 |\n| 3 | 11 | 5 |\n| 3 | 12 | 1 |\n| 3 | 13 | 7 |\n| 3 | 14 | 0 |\n| 3 | 15 | 4 |\n| 3 | 16 | 3 |\n| 3 | 17 | 3 |\n| 3 | 18 | 1 |\n| 3 | 19 | 9 |\n| 3 | 20 | 3 |\n| 3 | 21 | 7 |\n| 4 | 1 | 7 |\n| 4 | 2 | 4 |\n| 4 | 3 | 2 |\n| 4 | 5 | 6 |\n| 4 | 6 | 1 |\n| 4 | 7 | 0 |\n| 4 | 8 | 6 |\n| 4 | 9 | 7 |\n| 4 | 10 | 7 |\n| 4 | 11 | 3 |\n| 4 | 12 | 0 |\n| 4 | 13 | 7 |\n| 4 | 14 | 2 |\n| 4 | 15 | 7 |\n| 4 | 16 | 2 |\n| 4 | 17 | 5 |\n| 4 | 18 | 8 |\n| 4 | 19 | 7 |\n| 4 | 20 | 7 |\n| 4 | 21 | 9 |\n| 5 | 1 | 1 |\n| 5 | 2 | 0 |\n| 5 | 3 | 6 |\n| 5 | 4 | 6 |\n| 5 | 6 | 1 |\n| 5 | 7 | 1 |\n| 5 | 8 | 1 |\n| 5 | 9 | 1 |\n| 5 | 10 | 4 |\n| 5 | 11 | 5 |\n| 5 | 12 | 3 |\n| 5 | 13 | 8 |\n| 5 | 14 | 6 |\n| 5 | 15 | 7 |\n| 5 | 16 | 8 |\n| 5 | 17 | 3 |\n| 5 | 18 | 3 |\n| 5 | 19 | 8 |\n| 5 | 20 | 7 |\n| 5 | 21 | 1 |\n| 6 | 1 | 4 |\n| 6 | 2 | 3 |\n| 6 | 3 | 5 |\n| 6 | 4 | 1 |\n| 6 | 5 | 1 |\n| 6 | 7 | 6 |\n| 6 | 8 | 9 |\n| 6 | 9 | 9 |\n| 6 | 10 | 5 |\n| 6 | 11 | 3 |\n| 6 | 12 | 5 |\n| 6 | 13 | 1 |\n| 6 | 14 | 3 |\n| 6 | 15 | 7 |\n| 6 | 16 | 4 |\n| 6 | 17 | 9 |\n| 6 | 18 | 3 |\n| 6 | 19 | 8 |\n| 6 | 20 | 2 |\n| 6 | 21 | 9 |\n| 7 | 1 | 4 |\n| 7 | 2 | 3 |\n| 7 | 3 | 3 |\n| 7 | 4 | 0 |\n| 7 | 5 | 1 |\n| 7 | 6 | 6 |\n| 7 | 8 | 8 |\n| 7 | 9 | 9 |\n| 7 | 10 | 9 |\n| 7 | 11 | 2 |\n| 7 | 12 | 8 |\n| 7 | 13 | 1 |\n| 7 | 14 | 4 |\n| 7 | 15 | 0 |\n| 7 | 16 | 7 |\n| 7 | 17 | 7 |\n| 7 | 18 | 2 |\n| 7 | 19 | 0 |\n| 7 | 20 | 1 |\n| 7 | 21 | 9 |\n| 8 | 1 | 0 |\n| 8 | 2 | 1 |\n| 8 | 3 | 5 |\n| 8 | 4 | 6 |\n| 8 | 5 | 1 |\n| 8 | 6 | 9 |\n| 8 | 7 | 8 |\n| 8 | 9 | 2 |\n| 8 | 10 | 3 |\n| 8 | 11 | 7 |\n| 8 | 12 | 2 |\n| 8 | 13 | 5 |\n| 8 | 14 | 9 |\n| 8 | 15 | 0 |\n| 8 | 16 | 7 |\n| 8 | 17 | 2 |\n| 8 | 18 | 7 |\n| 8 | 19 | 1 |\n| 8 | 20 | 6 |\n| 8 | 21 | 1 |\n| 9 | 1 | 0 |\n| 9 | 2 | 6 |\n| 9 | 3 | 9 |\n| 9 | 4 | 7 |\n| 9 | 5 | 1 |\n| 9 | 6 | 9 |\n| 9 | 7 | 9 |\n| 9 | 8 | 2 |\n| 9 | 10 | 4 |\n| 9 | 11 | 3 |\n| 9 | 12 | 1 |\n| 9 | 13 | 2 |\n| 9 | 14 | 2 |\n| 9 | 15 | 5 |\n| 9 | 16 | 3 |\n| 9 | 17 | 5 |\n| 9 | 18 | 0 |\n| 9 | 19 | 3 |\n| 9 | 20 | 8 |\n| 9 | 21 | 7 |\n| 10 | 1 | 5 |\n| 10 | 2 | 3 |\n| 10 | 3 | 4 |\n| 10 | 4 | 7 |\n| 10 | 5 | 4 |\n| 10 | 6 | 5 |\n| 10 | 7 | 9 |\n| 10 | 8 | 3 |\n| 10 | 9 | 4 |\n| 10 | 11 | 8 |\n| 10 | 12 | 3 |\n| 10 | 13 | 5 |\n| 10 | 14 | 2 |\n| 10 | 15 | 5 |\n| 10 | 16 | 0 |\n| 10 | 17 | 1 |\n| 10 | 18 | 7 |\n| 10 | 19 | 1 |\n| 10 | 20 | 9 |\n| 10 | 21 | 0 |\n| 11 | 1 | 8 |\n| 11 | 2 | 2 |\n| 11 | 3 | 5 |\n| 11 | 4 | 3 |\n| 11 | 5 | 5 |\n| 11 | 6 | 3 |\n| 11 | 7 | 2 |\n| 11 | 8 | 7 |\n| 11 | 9 | 3 |\n| 11 | 10 | 8 |\n| 11 | 12 | 2 |\n| 11 | 13 | 0 |\n| 11 | 14 | 8 |\n| 11 | 15 | 5 |\n| 11 | 16 | 1 |\n| 11 | 17 | 6 |\n| 11 | 18 | 3 |\n| 11 | 19 | 1 |\n| 11 | 20 | 8 |\n| 11 | 21 | 1 |\n| 12 | 1 | 3 |\n| 12 | 2 | 6 |\n| 12 | 3 | 1 |\n| 12 | 4 | 0 |\n| 12 | 5 | 3 |\n| 12 | 6 | 5 |\n| 12 | 7 | 8 |\n| 12 | 8 | 2 |\n| 12 | 9 | 1 |\n| 12 | 10 | 3 |\n| 12 | 11 | 2 |\n| 12 | 13 | 3 |\n| 12 | 14 | 1 |\n| 12 | 15 | 8 |\n| 12 | 16 | 7 |\n| 12 | 17 | 0 |\n| 12 | 18 | 3 |\n| 12 | 19 | 1 |\n| 12 | 20 | 8 |\n| 12 | 21 | 0 |\n| 13 | 1 | 5 |\n| 13 | 2 | 1 |\n| 13 | 3 | 7 |\n| 13 | 4 | 7 |\n| 13 | 5 | 8 |\n| 13 | 6 | 1 |\n| 13 | 7 | 1 |\n| 13 | 8 | 5 |\n| 13 | 9 | 2 |\n| 13 | 10 | 5 |\n| 13 | 11 | 0 |\n| 13 | 12 | 3 |\n| 13 | 14 | 4 |\n| 13 | 15 | 8 |\n| 13 | 16 | 4 |\n| 13 | 17 | 1 |\n| 13 | 18 | 5 |\n| 13 | 19 | 9 |\n| 13 | 20 | 1 |\n| 13 | 21 | 0 |\n| 14 | 1 | 9 |\n| 14 | 2 | 3 |\n| 14 | 3 | 0 |\n| 14 | 4 | 2 |\n| 14 | 5 | 6 |\n| 14 | 6 | 3 |\n| 14 | 7 | 4 |\n| 14 | 8 | 9 |\n| 14 | 9 | 2 |\n| 14 | 10 | 2 |\n| 14 | 11 | 8 |\n| 14 | 12 | 1 |\n| 14 | 13 | 4 |\n| 14 | 15 | 1 |\n| 14 | 16 | 1 |\n| 14 | 17 | 4 |\n| 14 | 18 | 1 |\n| 14 | 19 | 0 |\n| 14 | 20 | 1 |\n| 14 | 21 | 3 |\n| 15 | 1 | 7 |\n| 15 | 2 | 1 |\n| 15 | 3 | 4 |\n| 15 | 4 | 7 |\n| 15 | 5 | 7 |\n| 15 | 6 | 7 |\n| 15 | 7 | 0 |\n| 15 | 8 | 0 |\n| 15 | 9 | 5 |\n| 15 | 10 | 5 |\n| 15 | 11 | 5 |\n| 15 | 12 | 8 |\n| 15 | 13 | 8 |\n| 15 | 14 | 1 |\n| 15 | 16 | 5 |\n| 15 | 17 | 4 |\n| 15 | 18 | 0 |\n| 15 | 19 | 9 |\n| 15 | 20 | 8 |\n| 15 | 21 | 3 |\n| 16 | 1 | 9 |\n| 16 | 2 | 1 |\n| 16 | 3 | 3 |\n| 16 | 4 | 2 |\n| 16 | 5 | 8 |\n| 16 | 6 | 4 |\n| 16 | 7 | 7 |\n| 16 | 8 | 7 |\n| 16 | 9 | 3 |\n| 16 | 10 | 0 |\n| 16 | 11 | 1 |\n| 16 | 12 | 7 |\n| 16 | 13 | 4 |\n| 16 | 14 | 1 |\n| 16 | 15 | 5 |\n| 16 | 17 | 1 |\n| 16 | 18 | 5 |\n| 16 | 19 | 7 |\n| 16 | 20 | 7 |\n| 16 | 21 | 3 |\n| 17 | 1 | 6 |\n| 17 | 2 | 4 |\n| 17 | 3 | 3 |\n| 17 | 4 | 5 |\n| 17 | 5 | 3 |\n| 17 | 6 | 9 |\n| 17 | 7 | 7 |\n| 17 | 8 | 2 |\n| 17 | 9 | 5 |\n| 17 | 10 | 1 |\n| 17 | 11 | 6 |\n| 17 | 12 | 0 |\n| 17 | 13 | 1 |\n| 17 | 14 | 4 |\n| 17 | 15 | 4 |\n| 17 | 16 | 1 |\n| 17 | 18 | 6 |\n| 17 | 19 | 7 |\n| 17 | 20 | 8 |\n| 17 | 21 | 9 |\n| 18 | 1 | 1 |\n| 18 | 2 | 0 |\n| 18 | 3 | 1 |\n| 18 | 4 | 8 |\n| 18 | 5 | 3 |\n| 18 | 6 | 3 |\n| 18 | 7 | 2 |\n| 18 | 8 | 7 |\n| 18 | 9 | 0 |\n| 18 | 10 | 7 |\n| 18 | 11 | 3 |\n| 18 | 12 | 3 |\n| 18 | 13 | 5 |\n| 18 | 14 | 1 |\n| 18 | 15 | 0 |\n| 18 | 16 | 5 |\n| 18 | 17 | 6 |\n| 18 | 19 | 1 |\n| 18 | 20 | 2 |\n| 18 | 21 | 3 |\n| 19 | 1 | 7 |\n| 19 | 2 | 8 |\n| 19 | 3 | 9 |\n| 19 | 4 | 7 |\n| 19 | 5 | 8 |\n| 19 | 6 | 8 |\n| 19 | 7 | 0 |\n| 19 | 8 | 1 |\n| 19 | 9 | 3 |\n| 19 | 10 | 1 |\n| 19 | 11 | 1 |\n| 19 | 12 | 1 |\n| 19 | 13 | 9 |\n| 19 | 14 | 0 |\n| 19 | 15 | 9 |\n| 19 | 16 | 7 |\n| 19 | 17 | 7 |\n| 19 | 18 | 1 |\n| 19 | 20 | 0 |\n| 19 | 21 | 5 |\n| 20 | 1 | 9 |\n| 20 | 2 | 9 |\n| 20 | 3 | 3 |\n| 20 | 4 | 7 |\n| 20 | 5 | 7 |\n| 20 | 6 | 2 |\n| 20 | 7 | 1 |\n| 20 | 8 | 6 |\n| 20 | 9 | 8 |\n| 20 | 10 | 9 |\n| 20 | 11 | 8 |\n| 20 | 12 | 8 |\n| 20 | 13 | 1 |\n| 20 | 14 | 1 |\n| 20 | 15 | 8 |\n| 20 | 16 | 7 |\n| 20 | 17 | 8 |\n| 20 | 18 | 2 |\n| 20 | 19 | 0 |\n| 20 | 21 | 3 |\n| 21 | 1 | 9 |\n| 21 | 2 | 7 |\n| 21 | 3 | 7 |\n| 21 | 4 | 9 |\n| 21 | 5 | 1 |\n| 21 | 6 | 9 |\n| 21 | 7 | 9 |\n| 21 | 8 | 1 |\n| 21 | 9 | 7 |\n| 21 | 10 | 0 |\n| 21 | 11 | 1 |\n| 21 | 12 | 0 |\n| 21 | 13 | 0 |\n| 21 | 14 | 3 |\n| 21 | 15 | 3 |\n| 21 | 16 | 3 |\n| 21 | 17 | 9 |\n| 21 | 18 | 3 |\n| 21 | 19 | 5 |\n| 21 | 20 | 3 |\n\nUse these pairwise differences to total each candidate menu and select the 2 that yields the largest sum.\n\nYou can just hand the final pick back as a tiny JSON snippet like this — nothing fancy, just a list of the plates you chose:\n\n{\n \"solution\": [, , ...]\n}\n\nHere \"solution\" holds the exact items you want on the menu — each placeholder is one picked plate. Consider the JSON above a little form template: replace each with the actual identifier from the instance when you reply. This is just the shape I expect, not your actual answer.\n\nPlease be sure to use the identifiers exactly as they appear in the instance input — do not rename them or invent new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 0, 4, 7, 1, 4, 4, 0, 0, 5, 8, 3, 5, 9, 7, 9, 6, 1, 7, 9, 9 ], [ 0, 0, 8, 4, 0, 3, 3, 1, 6, 3, 2, 6, 1, 3, 1, 1, 4, 0, 8, 9, 7 ], [ 4, 8, 0, 2, 6, 5, 3, 5, 9, 4, 5, 1, 7, 0, 4, 3, 3, 1, 9, 3, 7 ], [ 7, 4, 2, 0, 6, 1, 0, 6, 7, 7, 3, 0, 7, 2, 7, 2, 5, 8, 7, 7, 9 ], [ 1, 0, 6, 6, 0, 1, 1, 1, 1, 4, 5, 3, 8, 6, 7, 8, 3, 3, 8, 7, 1 ], [ 4, 3, 5, 1, 1, 0, 6, 9, 9, 5, 3, 5, 1, 3, 7, 4, 9, 3, 8, 2, 9 ], [ 4, 3, 3, 0, 1, 6, 0, 8, 9, 9, 2, 8, 1, 4, 0, 7, 7, 2, 0, 1, 9 ], [ 0, 1, 5, 6, 1, 9, 8, 0, 2, 3, 7, 2, 5, 9, 0, 7, 2, 7, 1, 6, 1 ], [ 0, 6, 9, 7, 1, 9, 9, 2, 0, 4, 3, 1, 2, 2, 5, 3, 5, 0, 3, 8, 7 ], [ 5, 3, 4, 7, 4, 5, 9, 3, 4, 0, 8, 3, 5, 2, 5, 0, 1, 7, 1, 9, 0 ], [ 8, 2, 5, 3, 5, 3, 2, 7, 3, 8, 0, 2, 0, 8, 5, 1, 6, 3, 1, 8, 1 ], [ 3, 6, 1, 0, 3, 5, 8, 2, 1, 3, 2, 0, 3, 1, 8, 7, 0, 3, 1, 8, 0 ], [ 5, 1, 7, 7, 8, 1, 1, 5, 2, 5, 0, 3, 0, 4, 8, 4, 1, 5, 9, 1, 0 ], [ 9, 3, 0, 2, 6, 3, 4, 9, 2, 2, 8, 1, 4, 0, 1, 1, 4, 1, 0, 1, 3 ], [ 7, 1, 4, 7, 7, 7, 0, 0, 5, 5, 5, 8, 8, 1, 0, 5, 4, 0, 9, 8, 3 ], [ 9, 1, 3, 2, 8, 4, 7, 7, 3, 0, 1, 7, 4, 1, 5, 0, 1, 5, 7, 7, 3 ], [ 6, 4, 3, 5, 3, 9, 7, 2, 5, 1, 6, 0, 1, 4, 4, 1, 0, 6, 7, 8, 9 ], [ 1, 0, 1, 8, 3, 3, 2, 7, 0, 7, 3, 3, 5, 1, 0, 5, 6, 0, 1, 2, 3 ], [ 7, 8, 9, 7, 8, 8, 0, 1, 3, 1, 1, 1, 9, 0, 9, 7, 7, 1, 0, 0, 5 ], [ 9, 9, 3, 7, 7, 2, 1, 6, 8, 9, 8, 8, 1, 1, 8, 7, 8, 2, 0, 0, 3 ], [ 9, 7, 7, 9, 1, 9, 9, 1, 7, 0, 1, 0, 0, 3, 3, 3, 9, 3, 5, 3, 0 ] ], "m": 2, "objective": 9.0 }, "solution": [ 16, 20 ], "obj": 9.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 21, "m": 2, "nodes": [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 ], "distance_pairs": [ { "from_id": 1, "to_id": 2, "distance": 0 }, { "from_id": 1, "to_id": 3, "distance": 4 }, { "from_id": 1, "to_id": 4, "distance": 7 }, { "from_id": 1, "to_id": 5, "distance": 1 }, { "from_id": 1, "to_id": 6, "distance": 4 }, { "from_id": 1, "to_id": 7, "distance": 4 }, { "from_id": 1, "to_id": 8, "distance": 0 }, { "from_id": 1, "to_id": 9, "distance": 0 }, { "from_id": 1, "to_id": 10, "distance": 5 }, { "from_id": 1, "to_id": 11, "distance": 8 }, { "from_id": 1, "to_id": 12, "distance": 3 }, { "from_id": 1, "to_id": 13, "distance": 5 }, { "from_id": 1, "to_id": 14, "distance": 9 }, { "from_id": 1, "to_id": 15, "distance": 7 }, { "from_id": 1, "to_id": 16, "distance": 9 }, { "from_id": 1, "to_id": 17, "distance": 6 }, { "from_id": 1, "to_id": 18, "distance": 1 }, { "from_id": 1, "to_id": 19, "distance": 7 }, { "from_id": 1, "to_id": 20, "distance": 9 }, { "from_id": 1, "to_id": 21, "distance": 9 }, { "from_id": 2, "to_id": 1, "distance": 0 }, { "from_id": 2, "to_id": 3, "distance": 8 }, { "from_id": 2, "to_id": 4, "distance": 4 }, { "from_id": 2, "to_id": 5, "distance": 0 }, { "from_id": 2, "to_id": 6, "distance": 3 }, { "from_id": 2, "to_id": 7, "distance": 3 }, { "from_id": 2, "to_id": 8, "distance": 1 }, { "from_id": 2, "to_id": 9, "distance": 6 }, { "from_id": 2, "to_id": 10, "distance": 3 }, { "from_id": 2, "to_id": 11, "distance": 2 }, { "from_id": 2, "to_id": 12, "distance": 6 }, { "from_id": 2, "to_id": 13, "distance": 1 }, { "from_id": 2, "to_id": 14, "distance": 3 }, { "from_id": 2, "to_id": 15, "distance": 1 }, { "from_id": 2, "to_id": 16, "distance": 1 }, { "from_id": 2, "to_id": 17, "distance": 4 }, { "from_id": 2, "to_id": 18, "distance": 0 }, { "from_id": 2, "to_id": 19, "distance": 8 }, { "from_id": 2, "to_id": 20, "distance": 9 }, { "from_id": 2, "to_id": 21, "distance": 7 }, { "from_id": 3, "to_id": 1, "distance": 4 }, { "from_id": 3, "to_id": 2, "distance": 8 }, { "from_id": 3, "to_id": 4, "distance": 2 }, { "from_id": 3, "to_id": 5, "distance": 6 }, { "from_id": 3, "to_id": 6, "distance": 5 }, { "from_id": 3, "to_id": 7, "distance": 3 }, { "from_id": 3, "to_id": 8, "distance": 5 }, { "from_id": 3, "to_id": 9, "distance": 9 }, { "from_id": 3, "to_id": 10, "distance": 4 }, { "from_id": 3, "to_id": 11, "distance": 5 }, { "from_id": 3, "to_id": 12, "distance": 1 }, { "from_id": 3, "to_id": 13, "distance": 7 }, { "from_id": 3, "to_id": 14, "distance": 0 }, { "from_id": 3, "to_id": 15, "distance": 4 }, { "from_id": 3, "to_id": 16, "distance": 3 }, { "from_id": 3, "to_id": 17, "distance": 3 }, { "from_id": 3, "to_id": 18, "distance": 1 }, { "from_id": 3, "to_id": 19, "distance": 9 }, { "from_id": 3, "to_id": 20, "distance": 3 }, { "from_id": 3, "to_id": 21, "distance": 7 }, { "from_id": 4, "to_id": 1, "distance": 7 }, { "from_id": 4, "to_id": 2, "distance": 4 }, { "from_id": 4, "to_id": 3, "distance": 2 }, { "from_id": 4, "to_id": 5, "distance": 6 }, { "from_id": 4, "to_id": 6, "distance": 1 }, { "from_id": 4, "to_id": 7, "distance": 0 }, { "from_id": 4, "to_id": 8, "distance": 6 }, { "from_id": 4, "to_id": 9, "distance": 7 }, { "from_id": 4, "to_id": 10, "distance": 7 }, { "from_id": 4, "to_id": 11, "distance": 3 }, { "from_id": 4, "to_id": 12, "distance": 0 }, { "from_id": 4, "to_id": 13, "distance": 7 }, { "from_id": 4, "to_id": 14, "distance": 2 }, { "from_id": 4, "to_id": 15, "distance": 7 }, { "from_id": 4, "to_id": 16, "distance": 2 }, { "from_id": 4, "to_id": 17, "distance": 5 }, { "from_id": 4, "to_id": 18, "distance": 8 }, { "from_id": 4, "to_id": 19, "distance": 7 }, { "from_id": 4, "to_id": 20, "distance": 7 }, { "from_id": 4, "to_id": 21, "distance": 9 }, { "from_id": 5, "to_id": 1, "distance": 1 }, { "from_id": 5, "to_id": 2, "distance": 0 }, { "from_id": 5, "to_id": 3, "distance": 6 }, { "from_id": 5, "to_id": 4, "distance": 6 }, { "from_id": 5, "to_id": 6, "distance": 1 }, { "from_id": 5, "to_id": 7, "distance": 1 }, { "from_id": 5, "to_id": 8, "distance": 1 }, { "from_id": 5, "to_id": 9, "distance": 1 }, { "from_id": 5, "to_id": 10, "distance": 4 }, { "from_id": 5, "to_id": 11, "distance": 5 }, { "from_id": 5, "to_id": 12, "distance": 3 }, { "from_id": 5, "to_id": 13, "distance": 8 }, { "from_id": 5, "to_id": 14, "distance": 6 }, { "from_id": 5, "to_id": 15, "distance": 7 }, { "from_id": 5, "to_id": 16, "distance": 8 }, { "from_id": 5, "to_id": 17, "distance": 3 }, { "from_id": 5, "to_id": 18, "distance": 3 }, { "from_id": 5, "to_id": 19, "distance": 8 }, { "from_id": 5, "to_id": 20, "distance": 7 }, { "from_id": 5, "to_id": 21, "distance": 1 }, { "from_id": 6, "to_id": 1, "distance": 4 }, { "from_id": 6, "to_id": 2, "distance": 3 }, { "from_id": 6, "to_id": 3, "distance": 5 }, { "from_id": 6, "to_id": 4, "distance": 1 }, { "from_id": 6, "to_id": 5, "distance": 1 }, { "from_id": 6, "to_id": 7, "distance": 6 }, { "from_id": 6, "to_id": 8, "distance": 9 }, { "from_id": 6, "to_id": 9, "distance": 9 }, { 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"to_id": 10, "distance": 0 }, { "from_id": 21, "to_id": 11, "distance": 1 }, { "from_id": 21, "to_id": 12, "distance": 0 }, { "from_id": 21, "to_id": 13, "distance": 0 }, { "from_id": 21, "to_id": 14, "distance": 3 }, { "from_id": 21, "to_id": 15, "distance": 3 }, { "from_id": 21, "to_id": 16, "distance": 3 }, { "from_id": 21, "to_id": 17, "distance": 9 }, { "from_id": 21, "to_id": 18, "distance": 3 }, { "from_id": 21, "to_id": 19, "distance": 5 }, { "from_id": 21, "to_id": 20, "distance": 3 } ] }, "solution_variant": [ 17, 21 ], "context_index": 20, "input_format": "markdown_table", "input_index_base": 1 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "I’m the person putting together a playlist for a party, and the brief is simple: pick exactly the number of tracks they asked for — no more, no fewer, and no repeats — and make sure the whole thing feels wide-ranging. To do that, every pair of songs has a little “how different” score, so the trick is to pick songs whose pairwise differences add up to as big a number as possible; the bigger that total, the more variety across the set. The concrete track list and those difference numbers are shown below.\n\n# num_tracks=22\n# selection_size=2\n# track_ids=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21\ntrack_a_id,track_b_id,dissimilarity_score\n0,1,5\n0,2,5\n0,3,6\n0,4,2\n0,5,2\n0,6,0\n0,7,2\n0,8,7\n0,9,5\n0,10,2\n0,11,2\n0,12,0\n0,13,9\n0,14,6\n0,15,5\n0,16,6\n0,17,4\n0,18,4\n0,19,1\n0,20,3\n0,21,7\n1,0,5\n1,2,4\n1,3,5\n1,4,1\n1,5,9\n1,6,3\n1,7,4\n1,8,2\n1,9,8\n1,10,2\n1,11,4\n1,12,9\n1,13,3\n1,14,4\n1,15,9\n1,16,6\n1,17,6\n1,18,1\n1,19,0\n1,20,8\n1,21,7\n2,0,5\n2,1,4\n2,3,1\n2,4,3\n2,5,7\n2,6,1\n2,7,1\n2,8,1\n2,9,7\n2,10,7\n2,11,0\n2,12,6\n2,13,7\n2,14,6\n2,15,7\n2,16,2\n2,17,6\n2,18,4\n2,19,5\n2,20,7\n2,21,4\n3,0,6\n3,1,5\n3,2,1\n3,4,5\n3,5,0\n3,6,9\n3,7,0\n3,8,3\n3,9,5\n3,10,1\n3,11,7\n3,12,5\n3,13,1\n3,14,3\n3,15,7\n3,16,1\n3,17,5\n3,18,7\n3,19,2\n3,20,3\n3,21,1\n4,0,2\n4,1,1\n4,2,3\n4,3,5\n4,5,5\n4,6,4\n4,7,0\n4,8,5\n4,9,7\n4,10,7\n4,11,0\n4,12,4\n4,13,7\n4,14,5\n4,15,6\n4,16,4\n4,17,9\n4,18,9\n4,19,7\n4,20,5\n4,21,9\n5,0,2\n5,1,9\n5,2,7\n5,3,0\n5,4,5\n5,6,0\n5,7,4\n5,8,7\n5,9,8\n5,10,4\n5,11,0\n5,12,5\n5,13,2\n5,14,9\n5,15,5\n5,16,2\n5,17,9\n5,18,1\n5,19,8\n5,20,0\n5,21,3\n6,0,0\n6,1,3\n6,2,1\n6,3,9\n6,4,4\n6,5,0\n6,7,1\n6,8,2\n6,9,4\n6,10,1\n6,11,5\n6,12,5\n6,13,5\n6,14,9\n6,15,1\n6,16,9\n6,17,5\n6,18,5\n6,19,1\n6,20,5\n6,21,6\n7,0,2\n7,1,4\n7,2,1\n7,3,0\n7,4,0\n7,5,4\n7,6,1\n7,8,7\n7,9,3\n7,10,7\n7,11,9\n7,12,8\n7,13,4\n7,14,8\n7,15,5\n7,16,9\n7,17,2\n7,18,6\n7,19,9\n7,20,2\n7,21,1\n8,0,7\n8,1,2\n8,2,1\n8,3,3\n8,4,5\n8,5,7\n8,6,2\n8,7,7\n8,9,6\n8,10,6\n8,11,7\n8,12,9\n8,13,6\n8,14,0\n8,15,0\n8,16,6\n8,17,4\n8,18,3\n8,19,2\n8,20,8\n8,21,8\n9,0,5\n9,1,8\n9,2,7\n9,3,5\n9,4,7\n9,5,8\n9,6,4\n9,7,3\n9,8,6\n9,10,5\n9,11,0\n9,12,8\n9,13,1\n9,14,4\n9,15,1\n9,16,3\n9,17,6\n9,18,6\n9,19,4\n9,20,7\n9,21,9\n10,0,2\n10,1,2\n10,2,7\n10,3,1\n10,4,7\n10,5,4\n10,6,1\n10,7,7\n10,8,6\n10,9,5\n10,11,5\n10,12,6\n10,13,3\n10,14,6\n10,15,9\n10,16,7\n10,17,2\n10,18,0\n10,19,3\n10,20,7\n10,21,1\n11,0,2\n11,1,4\n11,2,0\n11,3,7\n11,4,0\n11,5,0\n11,6,5\n11,7,9\n11,8,7\n11,9,0\n11,10,5\n11,12,1\n11,13,7\n11,14,9\n11,15,2\n11,16,6\n11,17,9\n11,18,8\n11,19,8\n11,20,5\n11,21,1\n12,0,0\n12,1,9\n12,2,6\n12,3,5\n12,4,4\n12,5,5\n12,6,5\n12,7,8\n12,8,9\n12,9,8\n12,10,6\n12,11,1\n12,13,7\n12,14,5\n12,15,6\n12,16,4\n12,17,9\n12,18,9\n12,19,9\n12,20,8\n12,21,6\n13,0,9\n13,1,3\n13,2,7\n13,3,1\n13,4,7\n13,5,2\n13,6,5\n13,7,4\n13,8,6\n13,9,1\n13,10,3\n13,11,7\n13,12,7\n13,14,5\n13,15,1\n13,16,7\n13,17,0\n13,18,7\n13,19,8\n13,20,8\n13,21,6\n14,0,6\n14,1,4\n14,2,6\n14,3,3\n14,4,5\n14,5,9\n14,6,9\n14,7,8\n14,8,0\n14,9,4\n14,10,6\n14,11,9\n14,12,5\n14,13,5\n14,15,6\n14,16,3\n14,17,7\n14,18,1\n14,19,7\n14,20,7\n14,21,8\n15,0,5\n15,1,9\n15,2,7\n15,3,7\n15,4,6\n15,5,5\n15,6,1\n15,7,5\n15,8,0\n15,9,1\n15,10,9\n15,11,2\n15,12,6\n15,13,1\n15,14,6\n15,16,3\n15,17,1\n15,18,8\n15,19,7\n15,20,2\n15,21,3\n16,0,6\n16,1,6\n16,2,2\n16,3,1\n16,4,4\n16,5,2\n16,6,9\n16,7,9\n16,8,6\n16,9,3\n16,10,7\n16,11,6\n16,12,4\n16,13,7\n16,14,3\n16,15,3\n16,17,6\n16,18,9\n16,19,2\n16,20,0\n16,21,0\n17,0,4\n17,1,6\n17,2,6\n17,3,5\n17,4,9\n17,5,9\n17,6,5\n17,7,2\n17,8,4\n17,9,6\n17,10,2\n17,11,9\n17,12,9\n17,13,0\n17,14,7\n17,15,1\n17,16,6\n17,18,1\n17,19,8\n17,20,6\n17,21,0\n18,0,4\n18,1,1\n18,2,4\n18,3,7\n18,4,9\n18,5,1\n18,6,5\n18,7,6\n18,8,3\n18,9,6\n18,10,0\n18,11,8\n18,12,9\n18,13,7\n18,14,1\n18,15,8\n18,16,9\n18,17,1\n18,19,2\n18,20,9\n18,21,6\n19,0,1\n19,1,0\n19,2,5\n19,3,2\n19,4,7\n19,5,8\n19,6,1\n19,7,9\n19,8,2\n19,9,4\n19,10,3\n19,11,8\n19,12,9\n19,13,8\n19,14,7\n19,15,7\n19,16,2\n19,17,8\n19,18,2\n19,20,3\n19,21,9\n20,0,3\n20,1,8\n20,2,7\n20,3,3\n20,4,5\n20,5,0\n20,6,5\n20,7,2\n20,8,8\n20,9,7\n20,10,7\n20,11,5\n20,12,8\n20,13,8\n20,14,7\n20,15,2\n20,16,0\n20,17,6\n20,18,9\n20,19,3\n20,21,2\n21,0,7\n21,1,7\n21,2,4\n21,3,1\n21,4,9\n21,5,3\n21,6,6\n21,7,1\n21,8,8\n21,9,9\n21,10,1\n21,11,1\n21,12,6\n21,13,6\n21,14,8\n21,15,3\n21,16,0\n21,17,0\n21,18,6\n21,19,9\n21,20,2\n\nAlso, when you send back the final pick, please put it in this little JSON shape so it's easy to parse:\n\n{\n \"solution\": [, , ...]\n}\n\nThink of \"solution\" as the playlist you picked — a simple array of the track identifiers (exactly the ones from the input). Each is just a placeholder for whatever ID the instance uses for a song. This block is just a sketch of the format I expect you to follow, not the actual answer itself.\n\nPlease make sure to use the exact identifiers from the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 5, 5, 6, 2, 2, 0, 2, 7, 5, 2, 2, 0, 9, 6, 5, 6, 4, 4, 1, 3, 7 ], [ 5, 0, 4, 5, 1, 9, 3, 4, 2, 8, 2, 4, 9, 3, 4, 9, 6, 6, 1, 0, 8, 7 ], [ 5, 4, 0, 1, 3, 7, 1, 1, 1, 7, 7, 0, 6, 7, 6, 7, 2, 6, 4, 5, 7, 4 ], [ 6, 5, 1, 0, 5, 0, 9, 0, 3, 5, 1, 7, 5, 1, 3, 7, 1, 5, 7, 2, 3, 1 ], [ 2, 1, 3, 5, 0, 5, 4, 0, 5, 7, 7, 0, 4, 7, 5, 6, 4, 9, 9, 7, 5, 9 ], [ 2, 9, 7, 0, 5, 0, 0, 4, 7, 8, 4, 0, 5, 2, 9, 5, 2, 9, 1, 8, 0, 3 ], [ 0, 3, 1, 9, 4, 0, 0, 1, 2, 4, 1, 5, 5, 5, 9, 1, 9, 5, 5, 1, 5, 6 ], [ 2, 4, 1, 0, 0, 4, 1, 0, 7, 3, 7, 9, 8, 4, 8, 5, 9, 2, 6, 9, 2, 1 ], [ 7, 2, 1, 3, 5, 7, 2, 7, 0, 6, 6, 7, 9, 6, 0, 0, 6, 4, 3, 2, 8, 8 ], [ 5, 8, 7, 5, 7, 8, 4, 3, 6, 0, 5, 0, 8, 1, 4, 1, 3, 6, 6, 4, 7, 9 ], [ 2, 2, 7, 1, 7, 4, 1, 7, 6, 5, 0, 5, 6, 3, 6, 9, 7, 2, 0, 3, 7, 1 ], [ 2, 4, 0, 7, 0, 0, 5, 9, 7, 0, 5, 0, 1, 7, 9, 2, 6, 9, 8, 8, 5, 1 ], [ 0, 9, 6, 5, 4, 5, 5, 8, 9, 8, 6, 1, 0, 7, 5, 6, 4, 9, 9, 9, 8, 6 ], [ 9, 3, 7, 1, 7, 2, 5, 4, 6, 1, 3, 7, 7, 0, 5, 1, 7, 0, 7, 8, 8, 6 ], [ 6, 4, 6, 3, 5, 9, 9, 8, 0, 4, 6, 9, 5, 5, 0, 6, 3, 7, 1, 7, 7, 8 ], [ 5, 9, 7, 7, 6, 5, 1, 5, 0, 1, 9, 2, 6, 1, 6, 0, 3, 1, 8, 7, 2, 3 ], [ 6, 6, 2, 1, 4, 2, 9, 9, 6, 3, 7, 6, 4, 7, 3, 3, 0, 6, 9, 2, 0, 0 ], [ 4, 6, 6, 5, 9, 9, 5, 2, 4, 6, 2, 9, 9, 0, 7, 1, 6, 0, 1, 8, 6, 0 ], [ 4, 1, 4, 7, 9, 1, 5, 6, 3, 6, 0, 8, 9, 7, 1, 8, 9, 1, 0, 2, 9, 6 ], [ 1, 0, 5, 2, 7, 8, 1, 9, 2, 4, 3, 8, 9, 8, 7, 7, 2, 8, 2, 0, 3, 9 ], [ 3, 8, 7, 3, 5, 0, 5, 2, 8, 7, 7, 5, 8, 8, 7, 2, 0, 6, 9, 3, 0, 2 ], [ 7, 7, 4, 1, 9, 3, 6, 1, 8, 9, 1, 1, 6, 6, 8, 3, 0, 0, 6, 9, 2, 0 ] ], "m": 2, "objective": 9.0 }, "solution": [ 4, 21 ], "obj": 9.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 22, "m": 2, "nodes": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 ], "distance_pairs": [ { "from_id": 0, "to_id": 1, "distance": 5 }, { "from_id": 0, "to_id": 2, "distance": 5 }, { "from_id": 0, "to_id": 3, "distance": 6 }, { "from_id": 0, "to_id": 4, "distance": 2 }, { "from_id": 0, "to_id": 5, "distance": 2 }, { "from_id": 0, "to_id": 6, "distance": 0 }, { "from_id": 0, "to_id": 7, "distance": 2 }, { "from_id": 0, "to_id": 8, "distance": 7 }, { "from_id": 0, "to_id": 9, "distance": 5 }, { "from_id": 0, "to_id": 10, "distance": 2 }, { "from_id": 0, "to_id": 11, "distance": 2 }, { "from_id": 0, "to_id": 12, "distance": 0 }, { "from_id": 0, "to_id": 13, "distance": 9 }, { "from_id": 0, "to_id": 14, "distance": 6 }, { "from_id": 0, "to_id": 15, "distance": 5 }, { "from_id": 0, "to_id": 16, "distance": 6 }, { "from_id": 0, "to_id": 17, "distance": 4 }, { "from_id": 0, "to_id": 18, "distance": 4 }, { "from_id": 0, "to_id": 19, "distance": 1 }, { "from_id": 0, "to_id": 20, "distance": 3 }, { "from_id": 0, "to_id": 21, "distance": 7 }, { "from_id": 1, "to_id": 0, "distance": 5 }, { "from_id": 1, "to_id": 2, "distance": 4 }, { "from_id": 1, "to_id": 3, "distance": 5 }, { "from_id": 1, "to_id": 4, "distance": 1 }, { "from_id": 1, "to_id": 5, "distance": 9 }, { "from_id": 1, "to_id": 6, "distance": 3 }, { "from_id": 1, "to_id": 7, "distance": 4 }, { "from_id": 1, "to_id": 8, "distance": 2 }, { "from_id": 1, "to_id": 9, "distance": 8 }, { "from_id": 1, "to_id": 10, "distance": 2 }, { "from_id": 1, "to_id": 11, "distance": 4 }, { "from_id": 1, "to_id": 12, "distance": 9 }, { "from_id": 1, "to_id": 13, "distance": 3 }, { "from_id": 1, "to_id": 14, "distance": 4 }, { "from_id": 1, "to_id": 15, "distance": 9 }, { "from_id": 1, "to_id": 16, "distance": 6 }, { "from_id": 1, "to_id": 17, "distance": 6 }, { "from_id": 1, "to_id": 18, "distance": 1 }, { "from_id": 1, "to_id": 19, "distance": 0 }, { "from_id": 1, "to_id": 20, "distance": 8 }, { "from_id": 1, "to_id": 21, "distance": 7 }, { "from_id": 2, "to_id": 0, "distance": 5 }, { "from_id": 2, "to_id": 1, "distance": 4 }, { "from_id": 2, "to_id": 3, "distance": 1 }, { "from_id": 2, "to_id": 4, "distance": 3 }, { "from_id": 2, "to_id": 5, "distance": 7 }, { "from_id": 2, "to_id": 6, "distance": 1 }, { "from_id": 2, "to_id": 7, "distance": 1 }, { "from_id": 2, "to_id": 8, "distance": 1 }, { "from_id": 2, "to_id": 9, "distance": 7 }, { "from_id": 2, "to_id": 10, "distance": 7 }, { "from_id": 2, "to_id": 11, "distance": 0 }, { "from_id": 2, "to_id": 12, "distance": 6 }, { "from_id": 2, "to_id": 13, "distance": 7 }, { "from_id": 2, "to_id": 14, "distance": 6 }, { "from_id": 2, "to_id": 15, "distance": 7 }, { "from_id": 2, "to_id": 16, "distance": 2 }, { "from_id": 2, "to_id": 17, "distance": 6 }, { "from_id": 2, "to_id": 18, "distance": 4 }, { "from_id": 2, "to_id": 19, "distance": 5 }, { "from_id": 2, "to_id": 20, "distance": 7 }, { "from_id": 2, "to_id": 21, "distance": 4 }, { "from_id": 3, "to_id": 0, "distance": 6 }, { "from_id": 3, "to_id": 1, "distance": 5 }, { "from_id": 3, "to_id": 2, "distance": 1 }, { "from_id": 3, "to_id": 4, "distance": 5 }, { "from_id": 3, "to_id": 5, "distance": 0 }, { "from_id": 3, "to_id": 6, "distance": 9 }, { "from_id": 3, "to_id": 7, "distance": 0 }, { "from_id": 3, "to_id": 8, "distance": 3 }, { "from_id": 3, "to_id": 9, "distance": 5 }, { "from_id": 3, "to_id": 10, "distance": 1 }, { "from_id": 3, "to_id": 11, "distance": 7 }, { "from_id": 3, "to_id": 12, "distance": 5 }, { "from_id": 3, "to_id": 13, "distance": 1 }, { "from_id": 3, "to_id": 14, "distance": 3 }, { "from_id": 3, "to_id": 15, "distance": 7 }, { "from_id": 3, "to_id": 16, "distance": 1 }, { "from_id": 3, "to_id": 17, "distance": 5 }, { "from_id": 3, "to_id": 18, "distance": 7 }, { "from_id": 3, "to_id": 19, "distance": 2 }, { "from_id": 3, "to_id": 20, "distance": 3 }, { "from_id": 3, "to_id": 21, "distance": 1 }, { "from_id": 4, "to_id": 0, "distance": 2 }, { "from_id": 4, "to_id": 1, "distance": 1 }, { "from_id": 4, "to_id": 2, "distance": 3 }, { "from_id": 4, "to_id": 3, "distance": 5 }, { "from_id": 4, "to_id": 5, "distance": 5 }, { "from_id": 4, "to_id": 6, "distance": 4 }, { "from_id": 4, "to_id": 7, "distance": 0 }, { "from_id": 4, "to_id": 8, "distance": 5 }, { "from_id": 4, "to_id": 9, "distance": 7 }, { "from_id": 4, "to_id": 10, "distance": 7 }, { "from_id": 4, "to_id": 11, "distance": 0 }, { "from_id": 4, "to_id": 12, "distance": 4 }, { "from_id": 4, "to_id": 13, "distance": 7 }, { "from_id": 4, "to_id": 14, "distance": 5 }, { "from_id": 4, "to_id": 15, "distance": 6 }, { "from_id": 4, "to_id": 16, "distance": 4 }, { "from_id": 4, "to_id": 17, "distance": 9 }, { "from_id": 4, "to_id": 18, "distance": 9 }, { "from_id": 4, "to_id": 19, "distance": 7 }, { "from_id": 4, "to_id": 20, "distance": 5 }, { "from_id": 4, "to_id": 21, 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To see which lineup is best, the planner looks at how different every pair of chosen activities is and totals those differences across all pairs; the higher that total, the better the mix. No activity can be repeated and the selection size is fixed. The concrete details of the activities and their pairwise differences appear below.\n\nBelow are the 21 activities (A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U) and the recorded pairwise dissimilarities to use when selecting exactly 2.\n\n| activity_id_a | activity_id_b | dissimilarity_score |\n|---|---|---|\n| A | B | 7 |\n| A | C | 1 |\n| A | D | 9 |\n| A | E | 0 |\n| A | F | 4 |\n| A | G | 9 |\n| A | H | 9 |\n| A | I | 7 |\n| A | J | 8 |\n| A | K | 1 |\n| A | L | 8 |\n| A | M | 8 |\n| A | N | 3 |\n| A | O | 9 |\n| A | P | 0 |\n| A | Q | 4 |\n| A | R | 5 |\n| A | S | 7 |\n| A | T | 6 |\n| A | U | 7 |\n| B | A | 7 |\n| B | C | 7 |\n| B | D | 9 |\n| B | E | 5 |\n| B | F | 8 |\n| B | G | 9 |\n| B | H | 3 |\n| B | I | 7 |\n| B | J | 8 |\n| B | K | 5 |\n| B | L | 8 |\n| B | M | 0 |\n| B | N | 7 |\n| B | O | 7 |\n| B | P | 8 |\n| B | Q | 5 |\n| B | R | 8 |\n| B | S | 1 |\n| B | T | 8 |\n| B | U | 7 |\n| C | A | 1 |\n| C | B | 7 |\n| C | D | 0 |\n| C | E | 4 |\n| C | F | 2 |\n| C | G | 4 |\n| C | H | 8 |\n| C | I | 9 |\n| C | J | 5 |\n| C | K | 7 |\n| C | L | 1 |\n| C | M | 8 |\n| C | N | 8 |\n| C | O | 3 |\n| C | P | 3 |\n| C | Q | 5 |\n| C | R | 1 |\n| C | S | 5 |\n| C | T | 1 |\n| C | U | 3 |\n| D | A | 9 |\n| D | B | 9 |\n| D | C | 0 |\n| D | E | 0 |\n| D | F | 7 |\n| D | G | 5 |\n| D | H | 8 |\n| D | I | 5 |\n| D | J | 7 |\n| D | K | 0 |\n| D | L | 2 |\n| D | M | 7 |\n| D | N | 6 |\n| D | O | 6 |\n| D | P | 1 |\n| D | Q | 6 |\n| D | R | 2 |\n| D | S | 0 |\n| D | T | 3 |\n| D | U | 6 |\n| E | A | 0 |\n| E | B | 5 |\n| E | C | 4 |\n| E | D | 0 |\n| E | F | 9 |\n| E | G | 5 |\n| E | H | 3 |\n| E | I | 8 |\n| E | J | 1 |\n| E | K | 8 |\n| E | L | 8 |\n| E | M | 3 |\n| E | N | 4 |\n| E | O | 3 |\n| E | P | 3 |\n| E | Q | 4 |\n| E | R | 8 |\n| E | S | 2 |\n| E | T | 9 |\n| E | U | 7 |\n| F | A | 4 |\n| F | B | 8 |\n| F | C | 2 |\n| F | D | 7 |\n| F | E | 9 |\n| F | G | 8 |\n| F | H | 0 |\n| F | I | 8 |\n| F | J | 6 |\n| F | K | 8 |\n| F | L | 7 |\n| F | M | 9 |\n| F | N | 7 |\n| F | O | 5 |\n| F | P | 3 |\n| F | Q | 7 |\n| F | R | 8 |\n| F | S | 2 |\n| F | T | 0 |\n| F | U | 0 |\n| G | A | 9 |\n| G | B | 9 |\n| G | C | 4 |\n| G | D | 5 |\n| G | E | 5 |\n| G | F | 8 |\n| G | H | 5 |\n| G | I | 1 |\n| G | J | 4 |\n| G | K | 9 |\n| G | L | 2 |\n| G | M | 7 |\n| G | N | 4 |\n| G | O | 6 |\n| G | P | 6 |\n| G | Q | 3 |\n| G | R | 8 |\n| G | S | 6 |\n| G | T | 6 |\n| G | U | 2 |\n| H | A | 9 |\n| H | B | 3 |\n| H | C | 8 |\n| H | D | 8 |\n| H | E | 3 |\n| H | F | 0 |\n| H | G | 5 |\n| H | I | 7 |\n| H | J | 2 |\n| H | K | 5 |\n| H | L | 6 |\n| H | M | 2 |\n| H | N | 1 |\n| H | O | 3 |\n| H | P | 1 |\n| H | Q | 0 |\n| H | R | 7 |\n| H | S | 9 |\n| H | T | 2 |\n| H | U | 8 |\n| I | A | 7 |\n| I | B | 7 |\n| I | C | 9 |\n| I | D | 5 |\n| I | E | 8 |\n| I | F | 8 |\n| I | G | 1 |\n| I | H | 7 |\n| I | J | 2 |\n| I | K | 8 |\n| I | L | 8 |\n| I | M | 9 |\n| I | N | 6 |\n| I | O | 0 |\n| I | P | 9 |\n| I | Q | 4 |\n| I | R | 6 |\n| I | S | 4 |\n| I | T | 7 |\n| I | U | 6 |\n| J | A | 8 |\n| J | B | 8 |\n| J | C | 5 |\n| J | D | 7 |\n| J | E | 1 |\n| J | F | 6 |\n| J | G | 4 |\n| J | H | 2 |\n| J | I | 2 |\n| J | K | 4 |\n| J | L | 1 |\n| J | M | 4 |\n| J | N | 7 |\n| J | O | 9 |\n| J | P | 3 |\n| J | Q | 1 |\n| J | R | 4 |\n| J | S | 8 |\n| J | T | 1 |\n| J | U | 4 |\n| K | A | 1 |\n| K | B | 5 |\n| K | C | 7 |\n| K | D | 0 |\n| K | E | 8 |\n| K | F | 8 |\n| K | G | 9 |\n| K | H | 5 |\n| K | I | 8 |\n| K | J | 4 |\n| K | L | 7 |\n| K | M | 9 |\n| K | N | 7 |\n| K | O | 4 |\n| K | P | 2 |\n| K | Q | 7 |\n| K | R | 0 |\n| K | S | 8 |\n| K | T | 1 |\n| K | U | 5 |\n| L | A | 8 |\n| L | B | 8 |\n| L | C | 1 |\n| L | D | 2 |\n| L | E | 8 |\n| L | F | 7 |\n| L | G | 2 |\n| L | H | 6 |\n| L | I | 8 |\n| L | J | 1 |\n| L | K | 7 |\n| L | M | 0 |\n| L | N | 2 |\n| L | O | 3 |\n| L | P | 5 |\n| L | Q | 4 |\n| L | R | 6 |\n| L | S | 3 |\n| L | T | 1 |\n| L | U | 6 |\n| M | A | 8 |\n| M | B | 0 |\n| M | C | 8 |\n| M | D | 7 |\n| M | E | 3 |\n| M | F | 9 |\n| M | G | 7 |\n| M | H | 2 |\n| M | I | 9 |\n| M | J | 4 |\n| M | K | 9 |\n| M | L | 0 |\n| M | N | 0 |\n| M | O | 0 |\n| M | P | 1 |\n| M | Q | 0 |\n| M | R | 2 |\n| M | S | 0 |\n| M | T | 2 |\n| M | U | 8 |\n| N | A | 3 |\n| N | B | 7 |\n| N | C | 8 |\n| N | D | 6 |\n| N | E | 4 |\n| N | F | 7 |\n| N | G | 4 |\n| N | H | 1 |\n| N | I | 6 |\n| N | J | 7 |\n| N | K | 7 |\n| N | L | 2 |\n| N | M | 0 |\n| N | O | 9 |\n| N | P | 6 |\n| N | Q | 8 |\n| N | R | 9 |\n| N | S | 7 |\n| N | T | 4 |\n| N | U | 9 |\n| O | A | 9 |\n| O | B | 7 |\n| O | C | 3 |\n| O | D | 6 |\n| O | E | 3 |\n| O | F | 5 |\n| O | G | 6 |\n| O | H | 3 |\n| O | I | 0 |\n| O | J | 9 |\n| O | K | 4 |\n| O | L | 3 |\n| O | M | 0 |\n| O | N | 9 |\n| O | P | 4 |\n| O | Q | 2 |\n| O | R | 3 |\n| O | S | 9 |\n| O | T | 6 |\n| O | U | 7 |\n| P | A | 0 |\n| P | B | 8 |\n| P | C | 3 |\n| P | D | 1 |\n| P | E | 3 |\n| P | F | 3 |\n| P | G | 6 |\n| P | H | 1 |\n| P | I | 9 |\n| P | J | 3 |\n| P | K | 2 |\n| P | L | 5 |\n| P | M | 1 |\n| P | N | 6 |\n| P | O | 4 |\n| P | Q | 6 |\n| P | R | 0 |\n| P | S | 9 |\n| P | T | 7 |\n| P | U | 7 |\n| Q | A | 4 |\n| Q | B | 5 |\n| Q | C | 5 |\n| Q | D | 6 |\n| Q | E | 4 |\n| Q | F | 7 |\n| Q | G | 3 |\n| Q | H | 0 |\n| Q | I | 4 |\n| Q | J | 1 |\n| Q | K | 7 |\n| Q | L | 4 |\n| Q | M | 0 |\n| Q | N | 8 |\n| Q | O | 2 |\n| Q | P | 6 |\n| Q | R | 8 |\n| Q | S | 7 |\n| Q | T | 1 |\n| Q | U | 3 |\n| R | A | 5 |\n| R | B | 8 |\n| R | C | 1 |\n| R | D | 2 |\n| R | E | 8 |\n| R | F | 8 |\n| R | G | 8 |\n| R | H | 7 |\n| R | I | 6 |\n| R | J | 4 |\n| R | K | 0 |\n| R | L | 6 |\n| R | M | 2 |\n| R | N | 9 |\n| R | O | 3 |\n| R | P | 0 |\n| R | Q | 8 |\n| R | S | 7 |\n| R | T | 4 |\n| R | U | 6 |\n| S | A | 7 |\n| S | B | 1 |\n| S | C | 5 |\n| S | D | 0 |\n| S | E | 2 |\n| S | F | 2 |\n| S | G | 6 |\n| S | H | 9 |\n| S | I | 4 |\n| S | J | 8 |\n| S | K | 8 |\n| S | L | 3 |\n| S | M | 0 |\n| S | N | 7 |\n| S | O | 9 |\n| S | P | 9 |\n| S | Q | 7 |\n| S | R | 7 |\n| S | T | 5 |\n| S | U | 6 |\n| T | A | 6 |\n| T | B | 8 |\n| T | C | 1 |\n| T | D | 3 |\n| T | E | 9 |\n| T | F | 0 |\n| T | G | 6 |\n| T | H | 2 |\n| T | I | 7 |\n| T | J | 1 |\n| T | K | 1 |\n| T | L | 1 |\n| T | M | 2 |\n| T | N | 4 |\n| T | O | 6 |\n| T | P | 7 |\n| T | Q | 1 |\n| T | R | 4 |\n| T | S | 5 |\n| T | U | 3 |\n| U | A | 7 |\n| U | B | 7 |\n| U | C | 3 |\n| U | D | 6 |\n| U | E | 7 |\n| U | F | 0 |\n| U | G | 2 |\n| U | H | 8 |\n| U | I | 6 |\n| U | J | 4 |\n| U | K | 5 |\n| U | L | 6 |\n| U | M | 8 |\n| U | N | 9 |\n| U | O | 7 |\n| U | P | 7 |\n| U | Q | 3 |\n| U | R | 6 |\n| U | S | 6 |\n| U | T | 3 |\n\nThe planner will sum the listed dissimilarities across all chosen pairs to evaluate any 2-activity lineup.\n\nAlso, when you send the actual pick back, a simple JSON object like this is perfect — nothing fancy, just the list of chosen activities in the field called \"solution\":\n\n{\n \"solution\": [, , ...]\n}\n\nThink of \"solution\" as the lineup of activities the teacher will run — exactly m items — and each placeholder in the array is an activity identifier taken straight from the instance. This JSON is just a sketch of the shape I expect, not the final selection.\n\nPlease make sure to use the identifiers exactly as they appear in the instance input — do not rename them or invent new labels. \n\nFor example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 7, 1, 9, 0, 4, 9, 9, 7, 8, 1, 8, 8, 3, 9, 0, 4, 5, 7, 6, 7 ], [ 7, 0, 7, 9, 5, 8, 9, 3, 7, 8, 5, 8, 0, 7, 7, 8, 5, 8, 1, 8, 7 ], [ 1, 7, 0, 0, 4, 2, 4, 8, 9, 5, 7, 1, 8, 8, 3, 3, 5, 1, 5, 1, 3 ], [ 9, 9, 0, 0, 0, 7, 5, 8, 5, 7, 0, 2, 7, 6, 6, 1, 6, 2, 0, 3, 6 ], [ 0, 5, 4, 0, 0, 9, 5, 3, 8, 1, 8, 8, 3, 4, 3, 3, 4, 8, 2, 9, 7 ], [ 4, 8, 2, 7, 9, 0, 8, 0, 8, 6, 8, 7, 9, 7, 5, 3, 7, 8, 2, 0, 0 ], [ 9, 9, 4, 5, 5, 8, 0, 5, 1, 4, 9, 2, 7, 4, 6, 6, 3, 8, 6, 6, 2 ], [ 9, 3, 8, 8, 3, 0, 5, 0, 7, 2, 5, 6, 2, 1, 3, 1, 0, 7, 9, 2, 8 ], [ 7, 7, 9, 5, 8, 8, 1, 7, 0, 2, 8, 8, 9, 6, 0, 9, 4, 6, 4, 7, 6 ], [ 8, 8, 5, 7, 1, 6, 4, 2, 2, 0, 4, 1, 4, 7, 9, 3, 1, 4, 8, 1, 4 ], [ 1, 5, 7, 0, 8, 8, 9, 5, 8, 4, 0, 7, 9, 7, 4, 2, 7, 0, 8, 1, 5 ], [ 8, 8, 1, 2, 8, 7, 2, 6, 8, 1, 7, 0, 0, 2, 3, 5, 4, 6, 3, 1, 6 ], [ 8, 0, 8, 7, 3, 9, 7, 2, 9, 4, 9, 0, 0, 0, 0, 1, 0, 2, 0, 2, 8 ], [ 3, 7, 8, 6, 4, 7, 4, 1, 6, 7, 7, 2, 0, 0, 9, 6, 8, 9, 7, 4, 9 ], [ 9, 7, 3, 6, 3, 5, 6, 3, 0, 9, 4, 3, 0, 9, 0, 4, 2, 3, 9, 6, 7 ], [ 0, 8, 3, 1, 3, 3, 6, 1, 9, 3, 2, 5, 1, 6, 4, 0, 6, 0, 9, 7, 7 ], [ 4, 5, 5, 6, 4, 7, 3, 0, 4, 1, 7, 4, 0, 8, 2, 6, 0, 8, 7, 1, 3 ], [ 5, 8, 1, 2, 8, 8, 8, 7, 6, 4, 0, 6, 2, 9, 3, 0, 8, 0, 7, 4, 6 ], [ 7, 1, 5, 0, 2, 2, 6, 9, 4, 8, 8, 3, 0, 7, 9, 9, 7, 7, 0, 5, 6 ], [ 6, 8, 1, 3, 9, 0, 6, 2, 7, 1, 1, 1, 2, 4, 6, 7, 1, 4, 5, 0, 3 ], [ 7, 7, 3, 6, 7, 0, 2, 8, 6, 4, 5, 6, 8, 9, 7, 7, 3, 6, 6, 3, 0 ] ], "m": 2, "objective": 9.0 }, "solution": [ 15, 18 ], "obj": 9.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 21, "m": 2, "nodes": [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S", "T", "U" ], "distance_pairs": [ { "from_id": "A", "to_id": "B", "distance": 7 }, { "from_id": "A", "to_id": "C", "distance": 1 }, { "from_id": "A", "to_id": "D", "distance": 9 }, { "from_id": "A", "to_id": "E", "distance": 0 }, { "from_id": "A", "to_id": "F", "distance": 4 }, { "from_id": "A", "to_id": "G", "distance": 9 }, { "from_id": "A", "to_id": "H", "distance": 9 }, { "from_id": "A", "to_id": "I", "distance": 7 }, { "from_id": "A", "to_id": "J", "distance": 8 }, { "from_id": "A", "to_id": "K", "distance": 1 }, { "from_id": "A", "to_id": "L", "distance": 8 }, { "from_id": "A", "to_id": "M", "distance": 8 }, { "from_id": "A", "to_id": "N", "distance": 3 }, { "from_id": "A", "to_id": "O", "distance": 9 }, { "from_id": "A", "to_id": "P", "distance": 0 }, { "from_id": "A", "to_id": "Q", "distance": 4 }, { "from_id": "A", "to_id": "R", "distance": 5 }, { "from_id": "A", "to_id": "S", "distance": 7 }, { "from_id": "A", "to_id": "T", "distance": 6 }, { "from_id": "A", "to_id": "U", "distance": 7 }, { "from_id": "B", "to_id": "A", "distance": 7 }, { "from_id": "B", "to_id": "C", "distance": 7 }, { "from_id": "B", "to_id": "D", "distance": 9 }, { "from_id": "B", "to_id": "E", "distance": 5 }, { "from_id": "B", "to_id": "F", "distance": 8 }, { "from_id": "B", "to_id": "G", "distance": 9 }, { "from_id": "B", "to_id": "H", "distance": 3 }, { "from_id": "B", "to_id": "I", "distance": 7 }, { "from_id": "B", "to_id": "J", "distance": 8 }, { "from_id": "B", "to_id": "K", "distance": 5 }, { "from_id": "B", "to_id": "L", "distance": 8 }, { "from_id": "B", "to_id": "M", "distance": 0 }, { "from_id": "B", "to_id": "N", "distance": 7 }, { "from_id": "B", "to_id": "O", "distance": 7 }, { "from_id": "B", "to_id": "P", "distance": 8 }, { "from_id": "B", "to_id": "Q", "distance": 5 }, { "from_id": "B", "to_id": "R", "distance": 8 }, { "from_id": "B", "to_id": "S", "distance": 1 }, { "from_id": "B", "to_id": "T", "distance": 8 }, { "from_id": "B", "to_id": "U", "distance": 7 }, { "from_id": "C", "to_id": "A", "distance": 1 }, { "from_id": "C", "to_id": "B", "distance": 7 }, { "from_id": "C", "to_id": "D", "distance": 0 }, { "from_id": "C", "to_id": "E", "distance": 4 }, { "from_id": "C", "to_id": "F", "distance": 2 }, { "from_id": "C", "to_id": "G", "distance": 4 }, { "from_id": "C", "to_id": "H", "distance": 8 }, { "from_id": "C", "to_id": "I", "distance": 9 }, { "from_id": "C", "to_id": "J", "distance": 5 }, { "from_id": "C", "to_id": "K", "distance": 7 }, { "from_id": "C", "to_id": "L", "distance": 1 }, { "from_id": "C", "to_id": "M", "distance": 8 }, { "from_id": "C", "to_id": "N", "distance": 8 }, { "from_id": "C", "to_id": "O", "distance": 3 }, { "from_id": "C", "to_id": "P", "distance": 3 }, { "from_id": "C", "to_id": "Q", "distance": 5 }, { "from_id": "C", "to_id": "R", "distance": 1 }, { "from_id": "C", "to_id": "S", "distance": 5 }, { "from_id": "C", "to_id": "T", "distance": 1 }, { "from_id": "C", "to_id": "U", "distance": 3 }, { "from_id": "D", "to_id": "A", "distance": 9 }, { "from_id": "D", "to_id": "B", "distance": 9 }, { "from_id": "D", "to_id": "C", "distance": 0 }, { "from_id": "D", "to_id": "E", "distance": 0 }, { "from_id": "D", "to_id": "F", "distance": 7 }, { "from_id": "D", "to_id": "G", "distance": 5 }, { "from_id": "D", "to_id": "H", "distance": 8 }, { "from_id": "D", "to_id": "I", "distance": 5 }, { "from_id": "D", "to_id": "J", "distance": 7 }, { "from_id": "D", "to_id": "K", "distance": 0 }, { "from_id": "D", "to_id": "L", "distance": 2 }, { "from_id": "D", "to_id": "M", "distance": 7 }, { "from_id": "D", "to_id": "N", "distance": 6 }, { "from_id": "D", "to_id": "O", "distance": 6 }, { "from_id": "D", "to_id": "P", "distance": 1 }, { "from_id": "D", "to_id": "Q", "distance": 6 }, { "from_id": "D", "to_id": "R", "distance": 2 }, { "from_id": "D", "to_id": "S", "distance": 0 }, { "from_id": "D", "to_id": "T", "distance": 3 }, { "from_id": "D", "to_id": "U", "distance": 6 }, { "from_id": "E", "to_id": "A", "distance": 0 }, { "from_id": "E", "to_id": "B", "distance": 5 }, { "from_id": "E", "to_id": "C", "distance": 4 }, { "from_id": "E", "to_id": "D", "distance": 0 }, { "from_id": "E", "to_id": "F", "distance": 9 }, { "from_id": "E", "to_id": "G", "distance": 5 }, { "from_id": "E", "to_id": "H", "distance": 3 }, { "from_id": "E", "to_id": "I", "distance": 8 }, { "from_id": "E", "to_id": "J", "distance": 1 }, { "from_id": "E", "to_id": "K", "distance": 8 }, { "from_id": "E", "to_id": "L", "distance": 8 }, { "from_id": "E", "to_id": "M", "distance": 3 }, { "from_id": "E", "to_id": "N", "distance": 4 }, { "from_id": "E", "to_id": "O", "distance": 3 }, { "from_id": "E", "to_id": "P", "distance": 3 }, { "from_id": "E", "to_id": "Q", "distance": 4 }, { "from_id": "E", "to_id": "R", "distance": 8 }, { "from_id": "E", "to_id": "S", "distance": 2 }, { "from_id": "E", "to_id": "T", "distance": 9 }, { "from_id": "E", 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The idea was to make the portfolio feel varied, so the best choice is the bunch where each photo pairs with very different others — you can judge that by summing the differences for every photo pair and seeing which total is biggest. The specific photos and the required number are listed below.\n\n# num_photos=24\n# portfolio_size=7\n# photo_ids=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24\nphoto_id_a,photo_id_b,stylistic_dissimilarity\n1,2,196\n1,3,231\n1,4,225\n1,5,186\n1,6,193\n1,7,191\n1,8,212\n1,9,193\n1,10,203\n1,11,216\n1,12,180\n1,13,209\n1,14,136\n1,15,177\n1,16,140\n1,17,192\n1,18,210\n1,19,197\n1,20,209\n1,21,195\n1,22,188\n1,23,177\n1,24,215\n2,1,196\n2,3,171\n2,4,186\n2,5,156\n2,6,152\n2,7,177\n2,8,150\n2,9,178\n2,10,139\n2,11,204\n2,12,188\n2,13,168\n2,14,178\n2,15,226\n2,16,168\n2,17,188\n2,18,187\n2,19,163\n2,20,183\n2,21,111\n2,22,144\n2,23,141\n2,24,194\n3,1,231\n3,2,171\n3,4,133\n3,5,196\n3,6,141\n3,7,190\n3,8,153\n3,9,197\n3,10,169\n3,11,169\n3,12,156\n3,13,111\n3,14,175\n3,15,170\n3,16,150\n3,17,165\n3,18,168\n3,19,176\n3,20,156\n3,21,166\n3,22,171\n3,23,171\n3,24,179\n4,1,225\n4,2,186\n4,3,133\n4,5,213\n4,6,160\n4,7,200\n4,8,105\n4,9,181\n4,10,164\n4,11,208\n4,12,193\n4,13,147\n4,14,194\n4,15,166\n4,16,177\n4,17,164\n4,18,137\n4,19,175\n4,20,143\n4,21,184\n4,22,188\n4,23,209\n4,24,161\n5,1,186\n5,2,156\n5,3,196\n5,4,213\n5,6,186\n5,7,176\n5,8,161\n5,9,216\n5,10,200\n5,11,177\n5,12,167\n5,13,194\n5,14,162\n5,15,210\n5,16,200\n5,17,200\n5,18,196\n5,19,188\n5,20,164\n5,21,154\n5,22,159\n5,23,179\n5,24,175\n6,1,193\n6,2,152\n6,3,141\n6,4,160\n6,5,186\n6,7,158\n6,8,155\n6,9,171\n6,10,134\n6,11,157\n6,12,152\n6,13,123\n6,14,153\n6,15,163\n6,16,157\n6,17,157\n6,18,171\n6,19,147\n6,20,166\n6,21,182\n6,22,168\n6,23,128\n6,24,125\n7,1,191\n7,2,177\n7,3,190\n7,4,200\n7,5,176\n7,6,158\n7,8,153\n7,9,145\n7,10,166\n7,11,178\n7,12,169\n7,13,186\n7,14,164\n7,15,176\n7,16,147\n7,17,107\n7,18,173\n7,19,160\n7,20,154\n7,21,178\n7,22,133\n7,23,169\n7,24,159\n8,1,212\n8,2,150\n8,3,153\n8,4,105\n8,5,161\n8,6,155\n8,7,153\n8,9,147\n8,10,139\n8,11,178\n8,12,176\n8,13,179\n8,14,169\n8,15,172\n8,16,168\n8,17,144\n8,18,133\n8,19,129\n8,20,142\n8,21,134\n8,22,145\n8,23,170\n8,24,131\n9,1,193\n9,2,178\n9,3,197\n9,4,181\n9,5,216\n9,6,171\n9,7,145\n9,8,147\n9,10,134\n9,11,211\n9,12,174\n9,13,198\n9,14,138\n9,15,182\n9,16,148\n9,17,152\n9,18,165\n9,19,178\n9,20,163\n9,21,183\n9,22,153\n9,23,155\n9,24,181\n10,1,203\n10,2,139\n10,3,169\n10,4,164\n10,5,200\n10,6,134\n10,7,166\n10,8,139\n10,9,134\n10,11,185\n10,12,187\n10,13,169\n10,14,179\n10,15,201\n10,16,142\n10,17,187\n10,18,175\n10,19,141\n10,20,188\n10,21,139\n10,22,184\n10,23,141\n10,24,179\n11,1,216\n11,2,204\n11,3,169\n11,4,208\n11,5,177\n11,6,157\n11,7,178\n11,8,178\n11,9,211\n11,10,185\n11,12,144\n11,13,168\n11,14,150\n11,15,183\n11,16,183\n11,17,173\n11,18,209\n11,19,189\n11,20,204\n11,21,197\n11,22,160\n11,23,145\n11,24,185\n12,1,180\n12,2,188\n12,3,156\n12,4,193\n12,5,167\n12,6,152\n12,7,169\n12,8,176\n12,9,174\n12,10,187\n12,11,144\n12,13,139\n12,14,103\n12,15,187\n12,16,153\n12,17,159\n12,18,201\n12,19,180\n12,20,149\n12,21,199\n12,22,133\n12,23,113\n12,24,174\n13,1,209\n13,2,168\n13,3,111\n13,4,147\n13,5,194\n13,6,123\n13,7,186\n13,8,179\n13,9,198\n13,10,169\n13,11,168\n13,12,139\n13,14,162\n13,15,172\n13,16,164\n13,17,172\n13,18,180\n13,19,173\n13,20,143\n13,21,194\n13,22,175\n13,23,167\n13,24,187\n14,1,136\n14,2,178\n14,3,175\n14,4,194\n14,5,162\n14,6,153\n14,7,164\n14,8,169\n14,9,138\n14,10,179\n14,11,150\n14,12,103\n14,13,162\n14,15,159\n14,16,131\n14,17,164\n14,18,165\n14,19,177\n14,20,162\n14,21,179\n14,22,140\n14,23,123\n14,24,180\n15,1,177\n15,2,226\n15,3,170\n15,4,166\n15,5,210\n15,6,163\n15,7,176\n15,8,172\n15,9,182\n15,10,201\n15,11,183\n15,12,187\n15,13,172\n15,14,159\n15,16,163\n15,17,142\n15,18,130\n15,19,168\n15,20,144\n15,21,202\n15,22,187\n15,23,198\n15,24,167\n16,1,140\n16,2,168\n16,3,150\n16,4,177\n16,5,200\n16,6,157\n16,7,147\n16,8,168\n16,9,148\n16,10,142\n16,11,183\n16,12,153\n16,13,164\n16,14,131\n16,15,163\n16,17,165\n16,18,179\n16,19,149\n16,20,185\n16,21,164\n16,22,154\n16,23,136\n16,24,199\n17,1,192\n17,2,188\n17,3,165\n17,4,164\n17,5,200\n17,6,157\n17,7,107\n17,8,144\n17,9,152\n17,10,187\n17,11,173\n17,12,159\n17,13,172\n17,14,164\n17,15,142\n17,16,165\n17,18,162\n17,19,182\n17,20,137\n17,21,191\n17,22,117\n17,23,180\n17,24,143\n18,1,210\n18,2,187\n18,3,168\n18,4,137\n18,5,196\n18,6,171\n18,7,173\n18,8,133\n18,9,165\n18,10,175\n18,11,209\n18,12,201\n18,13,180\n18,14,165\n18,15,130\n18,16,179\n18,17,162\n18,19,168\n18,20,124\n18,21,152\n18,22,193\n18,23,208\n18,24,165\n19,1,197\n19,2,163\n19,3,176\n19,4,175\n19,5,188\n19,6,147\n19,7,160\n19,8,129\n19,9,178\n19,10,141\n19,11,189\n19,12,180\n19,13,173\n19,14,177\n19,15,168\n19,16,149\n19,17,182\n19,18,168\n19,20,181\n19,21,148\n19,22,186\n19,23,147\n19,24,164\n20,1,209\n20,2,183\n20,3,156\n20,4,143\n20,5,164\n20,6,166\n20,7,154\n20,8,142\n20,9,163\n20,10,188\n20,11,204\n20,12,149\n20,13,143\n20,14,162\n20,15,144\n20,16,185\n20,17,137\n20,18,124\n20,19,181\n20,21,179\n20,22,148\n20,23,192\n20,24,161\n21,1,195\n21,2,111\n21,3,166\n21,4,184\n21,5,154\n21,6,182\n21,7,178\n21,8,134\n21,9,183\n21,10,139\n21,11,197\n21,12,199\n21,13,194\n21,14,179\n21,15,202\n21,16,164\n21,17,191\n21,18,152\n21,19,148\n21,20,179\n21,22,178\n21,23,167\n21,24,205\n22,1,188\n22,2,144\n22,3,171\n22,4,188\n22,5,159\n22,6,168\n22,7,133\n22,8,145\n22,9,153\n22,10,184\n22,11,160\n22,12,133\n22,13,175\n22,14,140\n22,15,187\n22,16,154\n22,17,117\n22,18,193\n22,19,186\n22,20,148\n22,21,178\n22,23,139\n22,24,183\n23,1,177\n23,2,141\n23,3,171\n23,4,209\n23,5,179\n23,6,128\n23,7,169\n23,8,170\n23,9,155\n23,10,141\n23,11,145\n23,12,113\n23,13,167\n23,14,123\n23,15,198\n23,16,136\n23,17,180\n23,18,208\n23,19,147\n23,20,192\n23,21,167\n23,22,139\n23,24,175\n24,1,215\n24,2,194\n24,3,179\n24,4,161\n24,5,175\n24,6,125\n24,7,159\n24,8,131\n24,9,181\n24,10,179\n24,11,185\n24,12,174\n24,13,187\n24,14,180\n24,15,167\n24,16,199\n24,17,143\n24,18,165\n24,19,164\n24,20,161\n24,21,205\n24,22,183\n24,23,175\n\nAlso, when you send back the final pick, please use this simple JSON layout so it's easy to read:\n\n{\n \"solution\": [, , ...]\n}\n\nThis just means \"solution\" holds the list of chosen photo IDs (one ID per selected image). It's a little form: replace each placeholder with the actual identifiers from the instance input. The JSON above is only a sketch of the shape I want, not the actual selection.\n\nMake sure all identifiers are used exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 196, 231, 225, 186, 193, 191, 212, 193, 203, 216, 180, 209, 136, 177, 140, 192, 210, 197, 209, 195, 188, 177, 215 ], [ 196, 0, 171, 186, 156, 152, 177, 150, 178, 139, 204, 188, 168, 178, 226, 168, 188, 187, 163, 183, 111, 144, 141, 194 ], [ 231, 171, 0, 133, 196, 141, 190, 153, 197, 169, 169, 156, 111, 175, 170, 150, 165, 168, 176, 156, 166, 171, 171, 179 ], [ 225, 186, 133, 0, 213, 160, 200, 105, 181, 164, 208, 193, 147, 194, 166, 177, 164, 137, 175, 143, 184, 188, 209, 161 ], [ 186, 156, 196, 213, 0, 186, 176, 161, 216, 200, 177, 167, 194, 162, 210, 200, 200, 196, 188, 164, 154, 159, 179, 175 ], [ 193, 152, 141, 160, 186, 0, 158, 155, 171, 134, 157, 152, 123, 153, 163, 157, 157, 171, 147, 166, 182, 168, 128, 125 ], [ 191, 177, 190, 200, 176, 158, 0, 153, 145, 166, 178, 169, 186, 164, 176, 147, 107, 173, 160, 154, 178, 133, 169, 159 ], [ 212, 150, 153, 105, 161, 155, 153, 0, 147, 139, 178, 176, 179, 169, 172, 168, 144, 133, 129, 142, 134, 145, 170, 131 ], [ 193, 178, 197, 181, 216, 171, 145, 147, 0, 134, 211, 174, 198, 138, 182, 148, 152, 165, 178, 163, 183, 153, 155, 181 ], [ 203, 139, 169, 164, 200, 134, 166, 139, 134, 0, 185, 187, 169, 179, 201, 142, 187, 175, 141, 188, 139, 184, 141, 179 ], [ 216, 204, 169, 208, 177, 157, 178, 178, 211, 185, 0, 144, 168, 150, 183, 183, 173, 209, 189, 204, 197, 160, 145, 185 ], [ 180, 188, 156, 193, 167, 152, 169, 176, 174, 187, 144, 0, 139, 103, 187, 153, 159, 201, 180, 149, 199, 133, 113, 174 ], [ 209, 168, 111, 147, 194, 123, 186, 179, 198, 169, 168, 139, 0, 162, 172, 164, 172, 180, 173, 143, 194, 175, 167, 187 ], [ 136, 178, 175, 194, 162, 153, 164, 169, 138, 179, 150, 103, 162, 0, 159, 131, 164, 165, 177, 162, 179, 140, 123, 180 ], [ 177, 226, 170, 166, 210, 163, 176, 172, 182, 201, 183, 187, 172, 159, 0, 163, 142, 130, 168, 144, 202, 187, 198, 167 ], [ 140, 168, 150, 177, 200, 157, 147, 168, 148, 142, 183, 153, 164, 131, 163, 0, 165, 179, 149, 185, 164, 154, 136, 199 ], [ 192, 188, 165, 164, 200, 157, 107, 144, 152, 187, 173, 159, 172, 164, 142, 165, 0, 162, 182, 137, 191, 117, 180, 143 ], [ 210, 187, 168, 137, 196, 171, 173, 133, 165, 175, 209, 201, 180, 165, 130, 179, 162, 0, 168, 124, 152, 193, 208, 165 ], [ 197, 163, 176, 175, 188, 147, 160, 129, 178, 141, 189, 180, 173, 177, 168, 149, 182, 168, 0, 181, 148, 186, 147, 164 ], [ 209, 183, 156, 143, 164, 166, 154, 142, 163, 188, 204, 149, 143, 162, 144, 185, 137, 124, 181, 0, 179, 148, 192, 161 ], [ 195, 111, 166, 184, 154, 182, 178, 134, 183, 139, 197, 199, 194, 179, 202, 164, 191, 152, 148, 179, 0, 178, 167, 205 ], [ 188, 144, 171, 188, 159, 168, 133, 145, 153, 184, 160, 133, 175, 140, 187, 154, 117, 193, 186, 148, 178, 0, 139, 183 ], [ 177, 141, 171, 209, 179, 128, 169, 170, 155, 141, 145, 113, 167, 123, 198, 136, 180, 208, 147, 192, 167, 139, 0, 175 ], [ 215, 194, 179, 161, 175, 125, 159, 131, 181, 179, 185, 174, 187, 180, 167, 199, 143, 165, 164, 161, 205, 183, 175, 0 ] ], "m": 7, "objective": 4090.0 }, "solution": [ 0, 1, 3, 4, 8, 10, 14 ], "obj": 4090.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 24, "m": 7, "nodes": [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 ], "distance_pairs": [ { "from_id": 1, "to_id": 2, "distance": 196 }, { "from_id": 1, "to_id": 3, "distance": 231 }, { "from_id": 1, "to_id": 4, "distance": 225 }, { "from_id": 1, "to_id": 5, "distance": 186 }, { "from_id": 1, "to_id": 6, "distance": 193 }, { "from_id": 1, "to_id": 7, "distance": 191 }, { "from_id": 1, "to_id": 8, "distance": 212 }, { "from_id": 1, "to_id": 9, "distance": 193 }, { "from_id": 1, "to_id": 10, "distance": 203 }, { "from_id": 1, "to_id": 11, "distance": 216 }, { "from_id": 1, "to_id": 12, "distance": 180 }, { "from_id": 1, "to_id": 13, "distance": 209 }, { "from_id": 1, "to_id": 14, "distance": 136 }, { "from_id": 1, "to_id": 15, "distance": 177 }, { "from_id": 1, "to_id": 16, "distance": 140 }, { "from_id": 1, "to_id": 17, "distance": 192 }, { "from_id": 1, "to_id": 18, "distance": 210 }, { "from_id": 1, "to_id": 19, "distance": 197 }, { "from_id": 1, "to_id": 20, "distance": 209 }, { "from_id": 1, "to_id": 21, "distance": 195 }, { "from_id": 1, "to_id": 22, "distance": 188 }, { "from_id": 1, "to_id": 23, "distance": 177 }, { "from_id": 1, "to_id": 24, "distance": 215 }, { "from_id": 2, "to_id": 1, "distance": 196 }, { "from_id": 2, "to_id": 3, "distance": 171 }, { "from_id": 2, "to_id": 4, "distance": 186 }, { "from_id": 2, "to_id": 5, "distance": 156 }, { "from_id": 2, "to_id": 6, "distance": 152 }, { "from_id": 2, "to_id": 7, "distance": 177 }, { "from_id": 2, "to_id": 8, "distance": 150 }, { "from_id": 2, "to_id": 9, "distance": 178 }, { "from_id": 2, "to_id": 10, "distance": 139 }, { "from_id": 2, "to_id": 11, "distance": 204 }, { "from_id": 2, "to_id": 12, "distance": 188 }, { "from_id": 2, "to_id": 13, "distance": 168 }, { "from_id": 2, "to_id": 14, "distance": 178 }, { "from_id": 2, "to_id": 15, "distance": 226 }, { "from_id": 2, "to_id": 16, "distance": 168 }, { "from_id": 2, "to_id": 17, "distance": 188 }, { "from_id": 2, "to_id": 18, "distance": 187 }, { "from_id": 2, "to_id": 19, "distance": 163 }, { "from_id": 2, "to_id": 20, "distance": 183 }, { "from_id": 2, "to_id": 21, "distance": 111 }, { "from_id": 2, "to_id": 22, "distance": 144 }, { "from_id": 2, "to_id": 23, "distance": 141 }, { "from_id": 2, "to_id": 24, "distance": 194 }, { "from_id": 3, "to_id": 1, "distance": 231 }, { "from_id": 3, "to_id": 2, "distance": 171 }, { "from_id": 3, "to_id": 4, "distance": 133 }, { "from_id": 3, "to_id": 5, "distance": 196 }, { "from_id": 3, "to_id": 6, "distance": 141 }, { "from_id": 3, "to_id": 7, "distance": 190 }, { "from_id": 3, "to_id": 8, "distance": 153 }, { "from_id": 3, "to_id": 9, "distance": 197 }, { "from_id": 3, "to_id": 10, "distance": 169 }, { "from_id": 3, "to_id": 11, "distance": 169 }, { "from_id": 3, "to_id": 12, "distance": 156 }, { "from_id": 3, "to_id": 13, "distance": 111 }, { "from_id": 3, "to_id": 14, "distance": 175 }, { "from_id": 3, "to_id": 15, "distance": 170 }, { "from_id": 3, "to_id": 16, "distance": 150 }, { "from_id": 3, "to_id": 17, "distance": 165 }, { "from_id": 3, "to_id": 18, "distance": 168 }, { "from_id": 3, "to_id": 19, "distance": 176 }, { "from_id": 3, "to_id": 20, "distance": 156 }, { "from_id": 3, "to_id": 21, "distance": 166 }, { "from_id": 3, "to_id": 22, "distance": 171 }, { "from_id": 3, "to_id": 23, "distance": 171 }, { "from_id": 3, "to_id": 24, "distance": 179 }, { "from_id": 4, "to_id": 1, "distance": 225 }, { "from_id": 4, "to_id": 2, "distance": 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"distance": 184 }, { "from_id": 21, "to_id": 5, "distance": 154 }, { "from_id": 21, "to_id": 6, "distance": 182 }, { "from_id": 21, "to_id": 7, "distance": 178 }, { "from_id": 21, "to_id": 8, "distance": 134 }, { "from_id": 21, "to_id": 9, "distance": 183 }, { "from_id": 21, "to_id": 10, "distance": 139 }, { "from_id": 21, "to_id": 11, "distance": 197 }, { "from_id": 21, "to_id": 12, "distance": 199 }, { "from_id": 21, "to_id": 13, "distance": 194 }, { "from_id": 21, "to_id": 14, "distance": 179 }, { "from_id": 21, "to_id": 15, "distance": 202 }, { "from_id": 21, "to_id": 16, "distance": 164 }, { "from_id": 21, "to_id": 17, "distance": 191 }, { "from_id": 21, "to_id": 18, "distance": 152 }, { "from_id": 21, "to_id": 19, "distance": 148 }, { "from_id": 21, "to_id": 20, "distance": 179 }, { "from_id": 21, "to_id": 22, "distance": 178 }, { "from_id": 21, "to_id": 23, "distance": 167 }, { "from_id": 21, "to_id": 24, "distance": 205 }, { "from_id": 22, "to_id": 1, "distance": 188 }, { "from_id": 22, "to_id": 2, "distance": 144 }, { "from_id": 22, "to_id": 3, "distance": 171 }, { "from_id": 22, "to_id": 4, "distance": 188 }, { "from_id": 22, "to_id": 5, "distance": 159 }, { "from_id": 22, "to_id": 6, "distance": 168 }, { "from_id": 22, "to_id": 7, "distance": 133 }, { "from_id": 22, "to_id": 8, "distance": 145 }, { "from_id": 22, "to_id": 9, "distance": 153 }, { "from_id": 22, "to_id": 10, "distance": 184 }, { "from_id": 22, "to_id": 11, "distance": 160 }, { "from_id": 22, "to_id": 12, "distance": 133 }, { "from_id": 22, "to_id": 13, "distance": 175 }, { "from_id": 22, "to_id": 14, "distance": 140 }, { "from_id": 22, "to_id": 15, "distance": 187 }, { "from_id": 22, "to_id": 16, "distance": 154 }, { "from_id": 22, "to_id": 17, "distance": 117 }, { "from_id": 22, "to_id": 18, "distance": 193 }, { "from_id": 22, "to_id": 19, "distance": 186 }, { "from_id": 22, "to_id": 20, "distance": 148 }, { "from_id": 22, "to_id": 21, "distance": 178 }, { "from_id": 22, "to_id": 23, "distance": 139 }, { "from_id": 22, "to_id": 24, "distance": 183 }, { "from_id": 23, "to_id": 1, "distance": 177 }, { "from_id": 23, "to_id": 2, "distance": 141 }, { "from_id": 23, "to_id": 3, "distance": 171 }, { "from_id": 23, "to_id": 4, "distance": 209 }, { "from_id": 23, "to_id": 5, "distance": 179 }, { "from_id": 23, "to_id": 6, "distance": 128 }, { "from_id": 23, "to_id": 7, "distance": 169 }, { "from_id": 23, "to_id": 8, "distance": 170 }, { "from_id": 23, "to_id": 9, "distance": 155 }, { "from_id": 23, "to_id": 10, "distance": 141 }, { "from_id": 23, "to_id": 11, "distance": 145 }, { "from_id": 23, "to_id": 12, "distance": 113 }, { "from_id": 23, "to_id": 13, "distance": 167 }, { "from_id": 23, "to_id": 14, "distance": 123 }, { "from_id": 23, "to_id": 15, "distance": 198 }, { "from_id": 23, "to_id": 16, "distance": 136 }, { "from_id": 23, "to_id": 17, "distance": 180 }, { "from_id": 23, "to_id": 18, "distance": 208 }, { "from_id": 23, "to_id": 19, "distance": 147 }, { "from_id": 23, "to_id": 20, "distance": 192 }, { "from_id": 23, "to_id": 21, "distance": 167 }, { "from_id": 23, "to_id": 22, "distance": 139 }, { "from_id": 23, "to_id": 24, "distance": 175 }, { "from_id": 24, "to_id": 1, "distance": 215 }, { "from_id": 24, "to_id": 2, "distance": 194 }, { "from_id": 24, "to_id": 3, "distance": 179 }, { "from_id": 24, "to_id": 4, "distance": 161 }, { "from_id": 24, "to_id": 5, "distance": 175 }, { "from_id": 24, "to_id": 6, "distance": 125 }, { "from_id": 24, "to_id": 7, "distance": 159 }, { "from_id": 24, "to_id": 8, "distance": 131 }, { "from_id": 24, "to_id": 9, "distance": 181 }, { "from_id": 24, "to_id": 10, "distance": 179 }, { "from_id": 24, "to_id": 11, "distance": 185 }, { "from_id": 24, "to_id": 12, "distance": 174 }, { "from_id": 24, "to_id": 13, "distance": 187 }, { "from_id": 24, "to_id": 14, "distance": 180 }, { "from_id": 24, "to_id": 15, "distance": 167 }, { "from_id": 24, "to_id": 16, "distance": 199 }, { "from_id": 24, "to_id": 17, "distance": 143 }, { "from_id": 24, "to_id": 18, "distance": 165 }, { "from_id": 24, "to_id": 19, "distance": 164 }, { "from_id": 24, "to_id": 20, "distance": 161 }, { "from_id": 24, "to_id": 21, "distance": 205 }, { "from_id": 24, "to_id": 22, "distance": 183 }, { "from_id": 24, "to_id": 23, "distance": 175 } ] }, "solution_variant": [ 1, 2, 4, 5, 9, 11, 15 ], "context_index": 23, "input_format": "csv", "input_index_base": 1 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Someone handed an interior designer a strict brief: choose the exact number of furniture pieces required and make the final grouping feel diverse and visually surprising. The practical rule was clear—pick exactly that many distinct items, nothing duplicated. To decide which grouping is better, the designer compares every chosen piece with every other chosen piece, notes how different each pair is, and totals those pairwise differences; the higher the total, the more varied the room feels. The detailed inventory and pairwise differences will appear below.\n\n# total_furniture_items=21\n# pieces_to_choose=17\n# furniture_item_ids=A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U\nfurniture_id_first,furniture_id_second,form_texture_dissimilarity\nA,B,54\nA,C,71\nA,D,85\nA,E,84\nA,F,82\nA,G,74\nA,H,70\nA,I,39\nA,J,89\nA,K,43\nA,L,63\nA,M,73\nA,N,72\nA,O,77\nA,P,98\nA,Q,91\nA,R,70\nA,S,80\nA,T,39\nA,U,101\nB,A,54\nB,C,89\nB,D,72\nB,E,59\nB,F,88\nB,G,25\nB,H,17\nB,I,55\nB,J,80\nB,K,47\nB,L,34\nB,M,59\nB,N,39\nB,O,57\nB,P,55\nB,Q,98\nB,R,48\nB,S,45\nB,T,30\nB,U,73\nC,A,71\nC,B,89\nC,D,56\nC,E,62\nC,F,58\nC,G,90\nC,H,95\nC,I,49\nC,J,49\nC,K,83\nC,L,96\nC,M,108\nC,N,73\nC,O,109\nC,P,97\nC,Q,25\nC,R,63\nC,S,97\nC,T,94\nC,U,77\nD,A,85\nD,B,72\nD,C,56\nD,E,47\nD,F,35\nD,G,61\nD,H,71\nD,I,48\nD,J,13\nD,K,96\nD,L,94\nD,M,76\nD,N,60\nD,O,74\nD,P,50\nD,Q,45\nD,R,60\nD,S,54\nD,T,94\nD,U,26\nE,A,84\nE,B,59\nE,C,62\nE,D,47\nE,F,78\nE,G,45\nE,H,53\nE,I,64\nE,J,45\nE,K,72\nE,L,64\nE,M,97\nE,N,24\nE,O,94\nE,P,49\nE,Q,60\nE,R,21\nE,S,68\nE,T,78\nE,U,41\nF,A,82\nF,B,88\nF,C,58\nF,D,35\nF,E,78\nF,G,85\nF,H,92\nF,I,44\nF,J,39\nF,K,108\nF,L,113\nF,M,73\nF,N,87\nF,O,74\nF,P,79\nF,Q,51\nF,R,86\nF,S,68\nF,T,103\nF,U,60\nG,A,74\nG,B,25\nG,C,90\nG,D,61\nG,E,45\nG,F,85\nG,H,12\nG,I,63\nG,J,68\nG,K,65\nG,L,46\nG,M,66\nG,N,29\nG,O,61\nG,P,32\nG,Q,93\nG,R,42\nG,S,37\nG,T,55\nG,U,53\nH,A,70\nH,B,17\nH,C,95\nH,D,71\nH,E,53\nH,F,92\nH,G,12\nH,I,65\nH,J,78\nH,K,57\nH,L,37\nH,M,66\nH,N,33\nH,O,62\nH,P,43\nH,Q,100\nH,R,45\nH,S,43\nH,T,45\nH,U,65\nI,A,39\nI,B,55\nI,C,49\nI,D,48\nI,E,64\nI,F,44\nI,G,63\nI,H,65\nI,J,53\nI,K,66\nI,L,76\nI,M,60\nI,N,62\nI,O,62\nI,P,74\nI,Q,61\nI,R,60\nI,S,58\nI,T,61\nI,U,69\nJ,A,89\nJ,B,80\nJ,C,49\nJ,D,13\nJ,E,45\nJ,F,39\nJ,G,68\nJ,H,78\nJ,I,53\nJ,K,98\nJ,L,98\nJ,M,89\nJ,N,62\nJ,O,87\nJ,P,59\nJ,Q,34\nJ,R,59\nJ,S,67\nJ,T,99\nJ,U,31\nK,A,43\nK,B,47\nK,C,83\nK,D,96\nK,E,72\nK,F,108\nK,G,65\nK,H,57\nK,I,66\nK,J,98\nK,L,29\nK,M,96\nK,N,54\nK,O,97\nK,P,95\nK,Q,101\nK,R,51\nK,S,89\nK,T,27\nK,U,103\nL,A,63\nL,B,34\nL,C,96\nL,D,94\nL,E,64\nL,F,113\nL,G,46\nL,H,37\nL,I,76\nL,J,98\nL,K,29\nL,M,92\nL,N,41\nL,O,91\nL,P,78\nL,Q,109\nL,R,45\nL,S,78\nL,T,31\nL,U,93\nM,A,73\nM,B,59\nM,C,108\nM,D,76\nM,E,97\nM,F,73\nM,G,66\nM,H,66\nM,I,60\nM,J,89\nM,K,96\nM,L,92\nM,N,88\nM,O,8\nM,P,71\nM,Q,112\nM,R,96\nM,S,36\nM,T,74\nM,U,84\nN,A,72\nN,B,39\nN,C,73\nN,D,60\nN,E,24\nN,F,87\nN,G,29\nN,H,33\nN,I,62\nN,J,62\nN,K,54\nN,L,41\nN,M,88\nN,O,85\nN,P,49\nN,Q,78\nN,R,14\nN,S,62\nN,T,57\nN,U,56\nO,A,77\nO,B,57\nO,C,109\nO,D,74\nO,E,94\nO,F,74\nO,G,61\nO,H,62\nO,I,62\nO,J,87\nO,K,97\nO,L,91\nO,M,8\nO,N,85\nO,P,64\nO,Q,112\nO,R,93\nO,S,29\nO,T,75\nO,U,79\nP,A,98\nP,B,55\nP,C,97\nP,D,50\nP,E,49\nP,F,79\nP,G,32\nP,H,43\nP,I,74\nP,J,59\nP,K,95\nP,L,78\nP,M,71\nP,N,49\nP,O,64\nP,Q,91\nP,R,60\nP,S,34\nP,T,86\nP,U,32\nQ,A,91\nQ,B,98\nQ,C,25\nQ,D,45\nQ,E,60\nQ,F,51\nQ,G,93\nQ,H,100\nQ,I,61\nQ,J,34\nQ,K,101\nQ,L,109\nQ,M,112\nQ,N,78\nQ,O,112\nQ,P,91\nQ,R,69\nQ,S,96\nQ,T,110\nQ,U,64\nR,A,70\nR,B,48\nR,C,63\nR,D,60\nR,E,21\nR,F,86\nR,G,42\nR,H,45\nR,I,60\nR,J,59\nR,K,51\nR,L,45\nR,M,96\nR,N,14\nR,O,93\nR,P,60\nR,Q,69\nR,S,71\nR,T,61\nR,U,60\nS,A,80\nS,B,45\nS,C,97\nS,D,54\nS,E,68\nS,F,68\nS,G,37\nS,H,43\nS,I,58\nS,J,67\nS,K,89\nS,L,78\nS,M,36\nS,N,62\nS,O,29\nS,P,34\nS,Q,96\nS,R,71\nS,T,72\nS,U,52\nT,A,39\nT,B,30\nT,C,94\nT,D,94\nT,E,78\nT,F,103\nT,G,55\nT,H,45\nT,I,61\nT,J,99\nT,K,27\nT,L,31\nT,M,74\nT,N,57\nT,O,75\nT,P,86\nT,Q,110\nT,R,61\nT,S,72\nT,U,100\nU,A,101\nU,B,73\nU,C,77\nU,D,26\nU,E,41\nU,F,60\nU,G,53\nU,H,65\nU,I,69\nU,J,31\nU,K,103\nU,L,93\nU,M,84\nU,N,56\nU,O,79\nU,P,32\nU,Q,64\nU,R,60\nU,S,52\nU,T,100\n\nOh, and to keep things machine-friendly (and so I can quickly check your pick), please return the chosen pieces in this simple JSON layout:\n\n{\n \"solution\": [, , ...]\n}\n\nThink of this as a tiny form: \"solution\" is the list of furniture items you’re selecting for the room — drop the exact item identifiers from the instance into that array, and that’s all the structure I need. This JSON is just a sketch of the shape I expect, not the actual answer itself.\n\nPlease 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”.\"", "instance": { "distance_matrix": [ [ 0, 54, 71, 85, 84, 82, 74, 70, 39, 89, 43, 63, 73, 72, 77, 98, 91, 70, 80, 39, 101 ], [ 54, 0, 89, 72, 59, 88, 25, 17, 55, 80, 47, 34, 59, 39, 57, 55, 98, 48, 45, 30, 73 ], [ 71, 89, 0, 56, 62, 58, 90, 95, 49, 49, 83, 96, 108, 73, 109, 97, 25, 63, 97, 94, 77 ], [ 85, 72, 56, 0, 47, 35, 61, 71, 48, 13, 96, 94, 76, 60, 74, 50, 45, 60, 54, 94, 26 ], [ 84, 59, 62, 47, 0, 78, 45, 53, 64, 45, 72, 64, 97, 24, 94, 49, 60, 21, 68, 78, 41 ], [ 82, 88, 58, 35, 78, 0, 85, 92, 44, 39, 108, 113, 73, 87, 74, 79, 51, 86, 68, 103, 60 ], [ 74, 25, 90, 61, 45, 85, 0, 12, 63, 68, 65, 46, 66, 29, 61, 32, 93, 42, 37, 55, 53 ], [ 70, 17, 95, 71, 53, 92, 12, 0, 65, 78, 57, 37, 66, 33, 62, 43, 100, 45, 43, 45, 65 ], [ 39, 55, 49, 48, 64, 44, 63, 65, 0, 53, 66, 76, 60, 62, 62, 74, 61, 60, 58, 61, 69 ], [ 89, 80, 49, 13, 45, 39, 68, 78, 53, 0, 98, 98, 89, 62, 87, 59, 34, 59, 67, 99, 31 ], [ 43, 47, 83, 96, 72, 108, 65, 57, 66, 98, 0, 29, 96, 54, 97, 95, 101, 51, 89, 27, 103 ], [ 63, 34, 96, 94, 64, 113, 46, 37, 76, 98, 29, 0, 92, 41, 91, 78, 109, 45, 78, 31, 93 ], [ 73, 59, 108, 76, 97, 73, 66, 66, 60, 89, 96, 92, 0, 88, 8, 71, 112, 96, 36, 74, 84 ], [ 72, 39, 73, 60, 24, 87, 29, 33, 62, 62, 54, 41, 88, 0, 85, 49, 78, 14, 62, 57, 56 ], [ 77, 57, 109, 74, 94, 74, 61, 62, 62, 87, 97, 91, 8, 85, 0, 64, 112, 93, 29, 75, 79 ], [ 98, 55, 97, 50, 49, 79, 32, 43, 74, 59, 95, 78, 71, 49, 64, 0, 91, 60, 34, 86, 32 ], [ 91, 98, 25, 45, 60, 51, 93, 100, 61, 34, 101, 109, 112, 78, 112, 91, 0, 69, 96, 110, 64 ], [ 70, 48, 63, 60, 21, 86, 42, 45, 60, 59, 51, 45, 96, 14, 93, 60, 69, 0, 71, 61, 60 ], [ 80, 45, 97, 54, 68, 68, 37, 43, 58, 67, 89, 78, 36, 62, 29, 34, 96, 71, 0, 72, 52 ], [ 39, 30, 94, 94, 78, 103, 55, 45, 61, 99, 27, 31, 74, 57, 75, 86, 110, 61, 72, 0, 100 ], [ 101, 73, 77, 26, 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}, { "from_id": "R", "to_id": "L", "distance": 45 }, { "from_id": "R", "to_id": "M", "distance": 96 }, { "from_id": "R", "to_id": "N", "distance": 14 }, { "from_id": "R", "to_id": "O", "distance": 93 }, { "from_id": "R", "to_id": "P", "distance": 60 }, { "from_id": "R", "to_id": "Q", "distance": 69 }, { "from_id": "R", "to_id": "S", "distance": 71 }, { "from_id": "R", "to_id": "T", "distance": 61 }, { "from_id": "R", "to_id": "U", "distance": 60 }, { "from_id": "S", "to_id": "A", "distance": 80 }, { "from_id": "S", "to_id": "B", "distance": 45 }, { "from_id": "S", "to_id": "C", "distance": 97 }, { "from_id": "S", "to_id": "D", "distance": 54 }, { "from_id": "S", "to_id": "E", "distance": 68 }, { "from_id": "S", "to_id": "F", "distance": 68 }, { "from_id": "S", "to_id": "G", "distance": 37 }, { "from_id": "S", "to_id": "H", "distance": 43 }, { "from_id": "S", "to_id": "I", "distance": 58 }, { "from_id": "S", "to_id": "J", "distance": 67 }, { "from_id": "S", "to_id": "K", "distance": 89 }, { "from_id": "S", "to_id": "L", "distance": 78 }, { "from_id": "S", "to_id": "M", "distance": 36 }, { "from_id": "S", "to_id": "N", "distance": 62 }, { "from_id": "S", "to_id": "O", "distance": 29 }, { "from_id": "S", "to_id": "P", "distance": 34 }, { "from_id": "S", "to_id": "Q", "distance": 96 }, { "from_id": "S", "to_id": "R", "distance": 71 }, { "from_id": "S", "to_id": "T", "distance": 72 }, { "from_id": "S", "to_id": "U", "distance": 52 }, { "from_id": "T", "to_id": "A", "distance": 39 }, { "from_id": "T", "to_id": "B", "distance": 30 }, { "from_id": "T", "to_id": "C", "distance": 94 }, { "from_id": "T", "to_id": "D", "distance": 94 }, { "from_id": "T", "to_id": "E", "distance": 78 }, { "from_id": "T", "to_id": "F", "distance": 103 }, { "from_id": "T", "to_id": "G", "distance": 55 }, { "from_id": "T", "to_id": "H", "distance": 45 }, { "from_id": "T", "to_id": "I", "distance": 61 }, { "from_id": "T", "to_id": "J", "distance": 99 }, { "from_id": "T", "to_id": "K", "distance": 27 }, { "from_id": "T", "to_id": "L", "distance": 31 }, { "from_id": "T", "to_id": "M", "distance": 74 }, { "from_id": "T", "to_id": "N", "distance": 57 }, { "from_id": "T", "to_id": "O", "distance": 75 }, { "from_id": "T", "to_id": "P", "distance": 86 }, { "from_id": "T", "to_id": "Q", "distance": 110 }, { "from_id": "T", "to_id": "R", "distance": 61 }, { "from_id": "T", "to_id": "S", "distance": 72 }, { "from_id": "T", "to_id": "U", "distance": 100 }, { "from_id": "U", "to_id": "A", "distance": 101 }, { "from_id": "U", "to_id": "B", "distance": 73 }, { "from_id": "U", "to_id": "C", "distance": 77 }, { "from_id": "U", "to_id": "D", "distance": 26 }, { "from_id": "U", "to_id": "E", "distance": 41 }, { "from_id": "U", "to_id": "F", "distance": 60 }, { "from_id": "U", "to_id": "G", "distance": 53 }, { "from_id": "U", "to_id": "H", "distance": 65 }, { "from_id": "U", "to_id": "I", "distance": 69 }, { "from_id": "U", "to_id": "J", "distance": 31 }, { "from_id": "U", "to_id": "K", "distance": 103 }, { "from_id": "U", "to_id": "L", "distance": 93 }, { "from_id": "U", "to_id": "M", "distance": 84 }, { "from_id": "U", "to_id": "N", "distance": 56 }, { "from_id": "U", "to_id": "O", "distance": 79 }, { "from_id": "U", "to_id": "P", "distance": 32 }, { "from_id": "U", "to_id": "Q", "distance": 64 }, { "from_id": "U", "to_id": "R", "distance": 60 }, { "from_id": "U", "to_id": "S", "distance": 52 }, { "from_id": "U", "to_id": "T", "distance": 100 } ] }, "solution_variant": [ "A", "C", "D", "E", "F", "H", "J", "K", "L", "M", "O", "P", "Q", "R", "S", "T", "U" ], "context_index": 24, "input_format": "csv", "input_index_base": "names" }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Many people on the floor like to experiment with contrast, so the assignment became: choose exactly N dishes so the lineup feels as diverse as possible. There’s a handy reference showing how different each dish is compared to every other one; to rate a menu you sum those pairwise differences for the chosen plates — larger sums equal more diversity. Each dish is a single choice (no repeats) and the number of dishes is fixed. The concrete list and numbers are shown below.\n\nThe reference lists 22 dishes — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 — and you must pick exactly 9.\n\n| dish_id_a | dish_id_b | flavor_distance |\n|---|---|---|\n| 1 | 2 | 174 |\n| 1 | 3 | 162 |\n| 1 | 4 | 172 |\n| 1 | 5 | 178 |\n| 1 | 6 | 189 |\n| 1 | 7 | 150 |\n| 1 | 8 | 199 |\n| 1 | 9 | 213 |\n| 1 | 10 | 203 |\n| 1 | 11 | 196 |\n| 1 | 12 | 161 |\n| 1 | 13 | 179 |\n| 1 | 14 | 193 |\n| 1 | 15 | 182 |\n| 1 | 16 | 145 |\n| 1 | 17 | 165 |\n| 1 | 18 | 202 |\n| 1 | 19 | 231 |\n| 1 | 20 | 146 |\n| 1 | 21 | 217 |\n| 1 | 22 | 203 |\n| 2 | 1 | 174 |\n| 2 | 3 | 169 |\n| 2 | 4 | 116 |\n| 2 | 5 | 151 |\n| 2 | 6 | 171 |\n| 2 | 7 | 153 |\n| 2 | 8 | 146 |\n| 2 | 9 | 205 |\n| 2 | 10 | 180 |\n| 2 | 11 | 147 |\n| 2 | 12 | 156 |\n| 2 | 13 | 167 |\n| 2 | 14 | 165 |\n| 2 | 15 | 164 |\n| 2 | 16 | 141 |\n| 2 | 17 | 121 |\n| 2 | 18 | 116 |\n| 2 | 19 | 196 |\n| 2 | 20 | 107 |\n| 2 | 21 | 208 |\n| 2 | 22 | 168 |\n| 3 | 1 | 162 |\n| 3 | 2 | 169 |\n| 3 | 4 | 178 |\n| 3 | 5 | 153 |\n| 3 | 6 | 177 |\n| 3 | 7 | 191 |\n| 3 | 8 | 165 |\n| 3 | 9 | 196 |\n| 3 | 10 | 159 |\n| 3 | 11 | 177 |\n| 3 | 12 | 157 |\n| 3 | 13 | 178 |\n| 3 | 14 | 185 |\n| 3 | 15 | 165 |\n| 3 | 16 | 128 |\n| 3 | 17 | 145 |\n| 3 | 18 | 185 |\n| 3 | 19 | 165 |\n| 3 | 20 | 117 |\n| 3 | 21 | 198 |\n| 3 | 22 | 200 |\n| 4 | 1 | 172 |\n| 4 | 2 | 116 |\n| 4 | 3 | 178 |\n| 4 | 5 | 180 |\n| 4 | 6 | 190 |\n| 4 | 7 | 194 |\n| 4 | 8 | 182 |\n| 4 | 9 | 253 |\n| 4 | 10 | 178 |\n| 4 | 11 | 115 |\n| 4 | 12 | 164 |\n| 4 | 13 | 183 |\n| 4 | 14 | 148 |\n| 4 | 15 | 168 |\n| 4 | 16 | 154 |\n| 4 | 17 | 162 |\n| 4 | 18 | 155 |\n| 4 | 19 | 198 |\n| 4 | 20 | 145 |\n| 4 | 21 | 210 |\n| 4 | 22 | 186 |\n| 5 | 1 | 178 |\n| 5 | 2 | 151 |\n| 5 | 3 | 153 |\n| 5 | 4 | 180 |\n| 5 | 6 | 207 |\n| 5 | 7 | 181 |\n| 5 | 8 | 162 |\n| 5 | 9 | 170 |\n| 5 | 10 | 211 |\n| 5 | 11 | 184 |\n| 5 | 12 | 175 |\n| 5 | 13 | 195 |\n| 5 | 14 | 152 |\n| 5 | 15 | 186 |\n| 5 | 16 | 169 |\n| 5 | 17 | 161 |\n| 5 | 18 | 170 |\n| 5 | 19 | 189 |\n| 5 | 20 | 155 |\n| 5 | 21 | 205 |\n| 5 | 22 | 195 |\n| 6 | 1 | 189 |\n| 6 | 2 | 171 |\n| 6 | 3 | 177 |\n| 6 | 4 | 190 |\n| 6 | 5 | 207 |\n| 6 | 7 | 144 |\n| 6 | 8 | 232 |\n| 6 | 9 | 233 |\n| 6 | 10 | 181 |\n| 6 | 11 | 201 |\n| 6 | 12 | 117 |\n| 6 | 13 | 176 |\n| 6 | 14 | 226 |\n| 6 | 15 | 182 |\n| 6 | 16 | 150 |\n| 6 | 17 | 170 |\n| 6 | 18 | 208 |\n| 6 | 19 | 201 |\n| 6 | 20 | 151 |\n| 6 | 21 | 212 |\n| 6 | 22 | 215 |\n| 7 | 1 | 150 |\n| 7 | 2 | 153 |\n| 7 | 3 | 191 |\n| 7 | 4 | 194 |\n| 7 | 5 | 181 |\n| 7 | 6 | 144 |\n| 7 | 8 | 188 |\n| 7 | 9 | 200 |\n| 7 | 10 | 179 |\n| 7 | 11 | 208 |\n| 7 | 12 | 149 |\n| 7 | 13 | 159 |\n| 7 | 14 | 199 |\n| 7 | 15 | 204 |\n| 7 | 16 | 149 |\n| 7 | 17 | 146 |\n| 7 | 18 | 202 |\n| 7 | 19 | 229 |\n| 7 | 20 | 163 |\n| 7 | 21 | 181 |\n| 7 | 22 | 204 |\n| 8 | 1 | 199 |\n| 8 | 2 | 146 |\n| 8 | 3 | 165 |\n| 8 | 4 | 182 |\n| 8 | 5 | 162 |\n| 8 | 6 | 232 |\n| 8 | 7 | 188 |\n| 8 | 9 | 186 |\n| 8 | 10 | 192 |\n| 8 | 11 | 200 |\n| 8 | 12 | 223 |\n| 8 | 13 | 203 |\n| 8 | 14 | 183 |\n| 8 | 15 | 212 |\n| 8 | 16 | 165 |\n| 8 | 17 | 147 |\n| 8 | 18 | 147 |\n| 8 | 19 | 205 |\n| 8 | 20 | 184 |\n| 8 | 21 | 197 |\n| 8 | 22 | 201 |\n| 9 | 1 | 213 |\n| 9 | 2 | 205 |\n| 9 | 3 | 196 |\n| 9 | 4 | 253 |\n| 9 | 5 | 170 |\n| 9 | 6 | 233 |\n| 9 | 7 | 200 |\n| 9 | 8 | 186 |\n| 9 | 10 | 197 |\n| 9 | 11 | 202 |\n| 9 | 12 | 223 |\n| 9 | 13 | 186 |\n| 9 | 14 | 221 |\n| 9 | 15 | 198 |\n| 9 | 16 | 179 |\n| 9 | 17 | 204 |\n| 9 | 18 | 179 |\n| 9 | 19 | 168 |\n| 9 | 20 | 200 |\n| 9 | 21 | 190 |\n| 9 | 22 | 212 |\n| 10 | 1 | 203 |\n| 10 | 2 | 180 |\n| 10 | 3 | 159 |\n| 10 | 4 | 178 |\n| 10 | 5 | 211 |\n| 10 | 6 | 181 |\n| 10 | 7 | 179 |\n| 10 | 8 | 192 |\n| 10 | 9 | 197 |\n| 10 | 11 | 155 |\n| 10 | 12 | 162 |\n| 10 | 13 | 165 |\n| 10 | 14 | 177 |\n| 10 | 15 | 194 |\n| 10 | 16 | 151 |\n| 10 | 17 | 167 |\n| 10 | 18 | 199 |\n| 10 | 19 | 136 |\n| 10 | 20 | 160 |\n| 10 | 21 | 154 |\n| 10 | 22 | 225 |\n| 11 | 1 | 196 |\n| 11 | 2 | 147 |\n| 11 | 3 | 177 |\n| 11 | 4 | 115 |\n| 11 | 5 | 184 |\n| 11 | 6 | 201 |\n| 11 | 7 | 208 |\n| 11 | 8 | 200 |\n| 11 | 9 | 202 |\n| 11 | 10 | 155 |\n| 11 | 12 | 160 |\n| 11 | 13 | 181 |\n| 11 | 14 | 141 |\n| 11 | 15 | 154 |\n| 11 | 16 | 137 |\n| 11 | 17 | 174 |\n| 11 | 18 | 144 |\n| 11 | 19 | 138 |\n| 11 | 20 | 149 |\n| 11 | 21 | 176 |\n| 11 | 22 | 189 |\n| 12 | 1 | 161 |\n| 12 | 2 | 156 |\n| 12 | 3 | 157 |\n| 12 | 4 | 164 |\n| 12 | 5 | 175 |\n| 12 | 6 | 117 |\n| 12 | 7 | 149 |\n| 12 | 8 | 223 |\n| 12 | 9 | 223 |\n| 12 | 10 | 162 |\n| 12 | 11 | 160 |\n| 12 | 13 | 165 |\n| 12 | 14 | 174 |\n| 12 | 15 | 182 |\n| 12 | 16 | 124 |\n| 12 | 17 | 147 |\n| 12 | 18 | 189 |\n| 12 | 19 | 190 |\n| 12 | 20 | 114 |\n| 12 | 21 | 202 |\n| 12 | 22 | 221 |\n| 13 | 1 | 179 |\n| 13 | 2 | 167 |\n| 13 | 3 | 178 |\n| 13 | 4 | 183 |\n| 13 | 5 | 195 |\n| 13 | 6 | 176 |\n| 13 | 7 | 159 |\n| 13 | 8 | 203 |\n| 13 | 9 | 186 |\n| 13 | 10 | 165 |\n| 13 | 11 | 181 |\n| 13 | 12 | 165 |\n| 13 | 14 | 199 |\n| 13 | 15 | 142 |\n| 13 | 16 | 125 |\n| 13 | 17 | 183 |\n| 13 | 18 | 191 |\n| 13 | 19 | 183 |\n| 13 | 20 | 143 |\n| 13 | 21 | 183 |\n| 13 | 22 | 144 |\n| 14 | 1 | 193 |\n| 14 | 2 | 165 |\n| 14 | 3 | 185 |\n| 14 | 4 | 148 |\n| 14 | 5 | 152 |\n| 14 | 6 | 226 |\n| 14 | 7 | 199 |\n| 14 | 8 | 183 |\n| 14 | 9 | 221 |\n| 14 | 10 | 177 |\n| 14 | 11 | 141 |\n| 14 | 12 | 174 |\n| 14 | 13 | 199 |\n| 14 | 15 | 191 |\n| 14 | 16 | 160 |\n| 14 | 17 | 186 |\n| 14 | 18 | 192 |\n| 14 | 19 | 192 |\n| 14 | 20 | 162 |\n| 14 | 21 | 198 |\n| 14 | 22 | 218 |\n| 15 | 1 | 182 |\n| 15 | 2 | 164 |\n| 15 | 3 | 165 |\n| 15 | 4 | 168 |\n| 15 | 5 | 186 |\n| 15 | 6 | 182 |\n| 15 | 7 | 204 |\n| 15 | 8 | 212 |\n| 15 | 9 | 198 |\n| 15 | 10 | 194 |\n| 15 | 11 | 154 |\n| 15 | 12 | 182 |\n| 15 | 13 | 142 |\n| 15 | 14 | 191 |\n| 15 | 16 | 128 |\n| 15 | 17 | 192 |\n| 15 | 18 | 166 |\n| 15 | 19 | 174 |\n| 15 | 20 | 130 |\n| 15 | 21 | 219 |\n| 15 | 22 | 170 |\n| 16 | 1 | 145 |\n| 16 | 2 | 141 |\n| 16 | 3 | 128 |\n| 16 | 4 | 154 |\n| 16 | 5 | 169 |\n| 16 | 6 | 150 |\n| 16 | 7 | 149 |\n| 16 | 8 | 165 |\n| 16 | 9 | 179 |\n| 16 | 10 | 151 |\n| 16 | 11 | 137 |\n| 16 | 12 | 124 |\n| 16 | 13 | 125 |\n| 16 | 14 | 160 |\n| 16 | 15 | 128 |\n| 16 | 17 | 117 |\n| 16 | 18 | 165 |\n| 16 | 19 | 144 |\n| 16 | 20 | 96 |\n| 16 | 21 | 141 |\n| 16 | 22 | 144 |\n| 17 | 1 | 165 |\n| 17 | 2 | 121 |\n| 17 | 3 | 145 |\n| 17 | 4 | 162 |\n| 17 | 5 | 161 |\n| 17 | 6 | 170 |\n| 17 | 7 | 146 |\n| 17 | 8 | 147 |\n| 17 | 9 | 204 |\n| 17 | 10 | 167 |\n| 17 | 11 | 174 |\n| 17 | 12 | 147 |\n| 17 | 13 | 183 |\n| 17 | 14 | 186 |\n| 17 | 15 | 192 |\n| 17 | 16 | 117 |\n| 17 | 18 | 149 |\n| 17 | 19 | 170 |\n| 17 | 20 | 120 |\n| 17 | 21 | 189 |\n| 17 | 22 | 172 |\n| 18 | 1 | 202 |\n| 18 | 2 | 116 |\n| 18 | 3 | 185 |\n| 18 | 4 | 155 |\n| 18 | 5 | 170 |\n| 18 | 6 | 208 |\n| 18 | 7 | 202 |\n| 18 | 8 | 147 |\n| 18 | 9 | 179 |\n| 18 | 10 | 199 |\n| 18 | 11 | 144 |\n| 18 | 12 | 189 |\n| 18 | 13 | 191 |\n| 18 | 14 | 192 |\n| 18 | 15 | 166 |\n| 18 | 16 | 165 |\n| 18 | 17 | 149 |\n| 18 | 19 | 179 |\n| 18 | 20 | 155 |\n| 18 | 21 | 239 |\n| 18 | 22 | 201 |\n| 19 | 1 | 231 |\n| 19 | 2 | 196 |\n| 19 | 3 | 165 |\n| 19 | 4 | 198 |\n| 19 | 5 | 189 |\n| 19 | 6 | 201 |\n| 19 | 7 | 229 |\n| 19 | 8 | 205 |\n| 19 | 9 | 168 |\n| 19 | 10 | 136 |\n| 19 | 11 | 138 |\n| 19 | 12 | 190 |\n| 19 | 13 | 183 |\n| 19 | 14 | 192 |\n| 19 | 15 | 174 |\n| 19 | 16 | 144 |\n| 19 | 17 | 170 |\n| 19 | 18 | 179 |\n| 19 | 20 | 165 |\n| 19 | 21 | 155 |\n| 19 | 22 | 187 |\n| 20 | 1 | 146 |\n| 20 | 2 | 107 |\n| 20 | 3 | 117 |\n| 20 | 4 | 145 |\n| 20 | 5 | 155 |\n| 20 | 6 | 151 |\n| 20 | 7 | 163 |\n| 20 | 8 | 184 |\n| 20 | 9 | 200 |\n| 20 | 10 | 160 |\n| 20 | 11 | 149 |\n| 20 | 12 | 114 |\n| 20 | 13 | 143 |\n| 20 | 14 | 162 |\n| 20 | 15 | 130 |\n| 20 | 16 | 96 |\n| 20 | 17 | 120 |\n| 20 | 18 | 155 |\n| 20 | 19 | 165 |\n| 20 | 21 | 199 |\n| 20 | 22 | 170 |\n| 21 | 1 | 217 |\n| 21 | 2 | 208 |\n| 21 | 3 | 198 |\n| 21 | 4 | 210 |\n| 21 | 5 | 205 |\n| 21 | 6 | 212 |\n| 21 | 7 | 181 |\n| 21 | 8 | 197 |\n| 21 | 9 | 190 |\n| 21 | 10 | 154 |\n| 21 | 11 | 176 |\n| 21 | 12 | 202 |\n| 21 | 13 | 183 |\n| 21 | 14 | 198 |\n| 21 | 15 | 219 |\n| 21 | 16 | 141 |\n| 21 | 17 | 189 |\n| 21 | 18 | 239 |\n| 21 | 19 | 155 |\n| 21 | 20 | 199 |\n| 21 | 22 | 193 |\n| 22 | 1 | 203 |\n| 22 | 2 | 168 |\n| 22 | 3 | 200 |\n| 22 | 4 | 186 |\n| 22 | 5 | 195 |\n| 22 | 6 | 215 |\n| 22 | 7 | 204 |\n| 22 | 8 | 201 |\n| 22 | 9 | 212 |\n| 22 | 10 | 225 |\n| 22 | 11 | 189 |\n| 22 | 12 | 221 |\n| 22 | 13 | 144 |\n| 22 | 14 | 218 |\n| 22 | 15 | 170 |\n| 22 | 16 | 144 |\n| 22 | 17 | 172 |\n| 22 | 18 | 201 |\n| 22 | 19 | 187 |\n| 22 | 20 | 170 |\n| 22 | 21 | 193 |\n\nSum the reported pairwise distances to score any 9-dish lineup from the 22 dishes.\n\nOh, and when you send the chosen lineup back, please follow this simple JSON layout:\n\n{\n \"solution\": [, , ...]\n}\n\nThink of it like a little form: \"solution\" is the list of dishes you picked for the menu (just the dish identifiers, nothing else). The JSON above is just a sketch of the shape I expect — not the actual selection itself.\n\nPlease use the identifiers exactly as they appear in the instance input — don’t rename them or invent new labels. For example:\n- plain numbers such as “1” or “23”\n- single capital letters like “A” or “B”\n- a capital letter followed by digits like “A1” or “X7”", "instance": { "distance_matrix": [ [ 0, 174, 162, 172, 178, 189, 150, 199, 213, 203, 196, 161, 179, 193, 182, 145, 165, 202, 231, 146, 217, 203 ], [ 174, 0, 169, 116, 151, 171, 153, 146, 205, 180, 147, 156, 167, 165, 164, 141, 121, 116, 196, 107, 208, 168 ], [ 162, 169, 0, 178, 153, 177, 191, 165, 196, 159, 177, 157, 178, 185, 165, 128, 145, 185, 165, 117, 198, 200 ], [ 172, 116, 178, 0, 180, 190, 194, 182, 253, 178, 115, 164, 183, 148, 168, 154, 162, 155, 198, 145, 210, 186 ], [ 178, 151, 153, 180, 0, 207, 181, 162, 170, 211, 184, 175, 195, 152, 186, 169, 161, 170, 189, 155, 205, 195 ], [ 189, 171, 177, 190, 207, 0, 144, 232, 233, 181, 201, 117, 176, 226, 182, 150, 170, 208, 201, 151, 212, 215 ], [ 150, 153, 191, 194, 181, 144, 0, 188, 200, 179, 208, 149, 159, 199, 204, 149, 146, 202, 229, 163, 181, 204 ], [ 199, 146, 165, 182, 162, 232, 188, 0, 186, 192, 200, 223, 203, 183, 212, 165, 147, 147, 205, 184, 197, 201 ], [ 213, 205, 196, 253, 170, 233, 200, 186, 0, 197, 202, 223, 186, 221, 198, 179, 204, 179, 168, 200, 190, 212 ], [ 203, 180, 159, 178, 211, 181, 179, 192, 197, 0, 155, 162, 165, 177, 194, 151, 167, 199, 136, 160, 154, 225 ], [ 196, 147, 177, 115, 184, 201, 208, 200, 202, 155, 0, 160, 181, 141, 154, 137, 174, 144, 138, 149, 176, 189 ], [ 161, 156, 157, 164, 175, 117, 149, 223, 223, 162, 160, 0, 165, 174, 182, 124, 147, 189, 190, 114, 202, 221 ], [ 179, 167, 178, 183, 195, 176, 159, 203, 186, 165, 181, 165, 0, 199, 142, 125, 183, 191, 183, 143, 183, 144 ], [ 193, 165, 185, 148, 152, 226, 199, 183, 221, 177, 141, 174, 199, 0, 191, 160, 186, 192, 192, 162, 198, 218 ], [ 182, 164, 165, 168, 186, 182, 204, 212, 198, 194, 154, 182, 142, 191, 0, 128, 192, 166, 174, 130, 219, 170 ], [ 145, 141, 128, 154, 169, 150, 149, 165, 179, 151, 137, 124, 125, 160, 128, 0, 117, 165, 144, 96, 141, 144 ], [ 165, 121, 145, 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"from_id": 18, "to_id": 20, "distance": 155 }, { "from_id": 18, "to_id": 21, "distance": 239 }, { "from_id": 18, "to_id": 22, "distance": 201 }, { "from_id": 19, "to_id": 1, "distance": 231 }, { "from_id": 19, "to_id": 2, "distance": 196 }, { "from_id": 19, "to_id": 3, "distance": 165 }, { "from_id": 19, "to_id": 4, "distance": 198 }, { "from_id": 19, "to_id": 5, "distance": 189 }, { "from_id": 19, "to_id": 6, "distance": 201 }, { "from_id": 19, "to_id": 7, "distance": 229 }, { "from_id": 19, "to_id": 8, "distance": 205 }, { "from_id": 19, "to_id": 9, "distance": 168 }, { "from_id": 19, "to_id": 10, "distance": 136 }, { "from_id": 19, "to_id": 11, "distance": 138 }, { "from_id": 19, "to_id": 12, "distance": 190 }, { "from_id": 19, "to_id": 13, "distance": 183 }, { "from_id": 19, "to_id": 14, "distance": 192 }, { "from_id": 19, "to_id": 15, "distance": 174 }, { "from_id": 19, "to_id": 16, "distance": 144 }, { "from_id": 19, "to_id": 17, "distance": 170 }, { "from_id": 19, "to_id": 18, "distance": 179 }, { "from_id": 19, "to_id": 20, "distance": 165 }, { "from_id": 19, "to_id": 21, "distance": 155 }, { "from_id": 19, "to_id": 22, "distance": 187 }, { "from_id": 20, "to_id": 1, "distance": 146 }, { "from_id": 20, "to_id": 2, "distance": 107 }, { "from_id": 20, "to_id": 3, "distance": 117 }, { "from_id": 20, "to_id": 4, "distance": 145 }, { "from_id": 20, "to_id": 5, "distance": 155 }, { "from_id": 20, "to_id": 6, "distance": 151 }, { "from_id": 20, "to_id": 7, "distance": 163 }, { "from_id": 20, "to_id": 8, "distance": 184 }, { "from_id": 20, "to_id": 9, "distance": 200 }, { "from_id": 20, "to_id": 10, "distance": 160 }, { "from_id": 20, "to_id": 11, "distance": 149 }, { "from_id": 20, "to_id": 12, "distance": 114 }, { "from_id": 20, "to_id": 13, "distance": 143 }, { "from_id": 20, "to_id": 14, "distance": 162 }, { "from_id": 20, "to_id": 15, "distance": 130 }, { "from_id": 20, "to_id": 16, "distance": 96 }, { "from_id": 20, "to_id": 17, "distance": 120 }, { "from_id": 20, "to_id": 18, "distance": 155 }, { "from_id": 20, "to_id": 19, "distance": 165 }, { "from_id": 20, "to_id": 21, "distance": 199 }, { "from_id": 20, "to_id": 22, "distance": 170 }, { "from_id": 21, "to_id": 1, "distance": 217 }, { "from_id": 21, "to_id": 2, "distance": 208 }, { "from_id": 21, "to_id": 3, "distance": 198 }, { "from_id": 21, "to_id": 4, "distance": 210 }, { "from_id": 21, "to_id": 5, "distance": 205 }, { "from_id": 21, "to_id": 6, "distance": 212 }, { "from_id": 21, "to_id": 7, "distance": 181 }, { "from_id": 21, "to_id": 8, "distance": 197 }, { "from_id": 21, "to_id": 9, "distance": 190 }, { "from_id": 21, "to_id": 10, "distance": 154 }, { "from_id": 21, "to_id": 11, "distance": 176 }, { "from_id": 21, "to_id": 12, "distance": 202 }, { "from_id": 21, "to_id": 13, "distance": 183 }, { "from_id": 21, "to_id": 14, "distance": 198 }, { "from_id": 21, "to_id": 15, "distance": 219 }, { "from_id": 21, "to_id": 16, "distance": 141 }, { "from_id": 21, "to_id": 17, "distance": 189 }, { "from_id": 21, "to_id": 18, "distance": 239 }, { "from_id": 21, "to_id": 19, "distance": 155 }, { "from_id": 21, "to_id": 20, "distance": 199 }, { "from_id": 21, "to_id": 22, "distance": 193 }, { "from_id": 22, "to_id": 1, "distance": 203 }, { "from_id": 22, "to_id": 2, "distance": 168 }, { "from_id": 22, "to_id": 3, "distance": 200 }, { "from_id": 22, "to_id": 4, "distance": 186 }, { "from_id": 22, "to_id": 5, "distance": 195 }, { "from_id": 22, "to_id": 6, "distance": 215 }, { "from_id": 22, "to_id": 7, "distance": 204 }, { "from_id": 22, "to_id": 8, "distance": 201 }, { "from_id": 22, "to_id": 9, "distance": 212 }, { "from_id": 22, "to_id": 10, "distance": 225 }, { "from_id": 22, "to_id": 11, "distance": 189 }, { "from_id": 22, "to_id": 12, "distance": 221 }, { "from_id": 22, "to_id": 13, "distance": 144 }, { "from_id": 22, "to_id": 14, "distance": 218 }, { "from_id": 22, "to_id": 15, "distance": 170 }, { "from_id": 22, "to_id": 16, "distance": 144 }, { "from_id": 22, "to_id": 17, "distance": 172 }, { "from_id": 22, "to_id": 18, "distance": 201 }, { "from_id": 22, "to_id": 19, "distance": 187 }, { "from_id": 22, "to_id": 20, "distance": 170 }, { "from_id": 22, "to_id": 21, "distance": 193 } ] }, "solution_variant": [ 1, 6, 8, 9, 10, 14, 18, 21, 22 ], "context_index": 25, "input_format": "markdown_table", "input_index_base": 1 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Someone in visual merchandising handed over a fixed number of display spots and asked for models to fill them — exactly that many, and each must be distinct. The aim is to make the lineup as spread-out as possible, which you can judge by checking every pair of chosen pieces, scoring how different each pair is, and summing those scores — the higher that total, the better the assortment. In short: pick the required count, no repeats, and try to maximize the total of all pairwise differences. The detailed list and pairwise values are shown below.\n\nThere are 22 distinct product models available (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22), and exactly 2 must be selected to fill the display.\nBetween product 1 and product 2, the model-pair dissimilarity is 3.\nBetween product 1 and product 3, the model-pair dissimilarity is 4.\nBetween product 1 and product 4, the model-pair dissimilarity is 7.\nBetween product 1 and product 5, the model-pair dissimilarity is 0.\nBetween product 1 and product 6, the model-pair dissimilarity is 6.\nBetween product 1 and product 7, the model-pair dissimilarity is 2.\nBetween product 1 and product 8, the model-pair dissimilarity is 6.\nBetween product 1 and product 9, the model-pair dissimilarity is 8.\nBetween product 1 and product 10, the model-pair dissimilarity is 4.\nBetween product 1 and product 11, the model-pair dissimilarity is 4.\nBetween product 1 and product 12, the model-pair dissimilarity is 0.\nBetween product 1 and product 13, the model-pair dissimilarity is 0.\nBetween product 1 and product 14, the model-pair dissimilarity is 0.\nBetween product 1 and product 15, the model-pair dissimilarity is 7.\nBetween product 1 and product 16, the model-pair dissimilarity is 2.\nBetween product 1 and product 17, the model-pair dissimilarity is 5.\nBetween product 1 and product 18, the model-pair dissimilarity is 7.\nBetween product 1 and product 19, the model-pair dissimilarity is 4.\nBetween product 1 and product 20, the model-pair dissimilarity is 1.\nBetween product 1 and product 21, the model-pair dissimilarity is 7.\nBetween product 1 and product 22, the model-pair dissimilarity is 6.\nBetween product 2 and product 1, the model-pair dissimilarity is 3.\nBetween product 2 and product 3, the model-pair dissimilarity is 8.\nBetween product 2 and product 4, the model-pair dissimilarity is 6.\nBetween product 2 and product 5, the model-pair dissimilarity is 2.\nBetween product 2 and product 6, the model-pair dissimilarity is 2.\nBetween product 2 and product 7, the model-pair dissimilarity is 0.\nBetween product 2 and product 8, the model-pair dissimilarity is 4.\nBetween product 2 and product 9, the model-pair dissimilarity is 0.\nBetween product 2 and product 10, the model-pair dissimilarity is 5.\nBetween product 2 and product 11, the model-pair dissimilarity is 6.\nBetween product 2 and product 12, the model-pair dissimilarity is 0.\nBetween product 2 and product 13, the model-pair dissimilarity is 3.\nBetween product 2 and product 14, the model-pair dissimilarity is 2.\nBetween product 2 and product 15, the model-pair dissimilarity is 3.\nBetween product 2 and product 16, the model-pair dissimilarity is 4.\nBetween product 2 and product 17, the model-pair dissimilarity is 5.\nBetween product 2 and product 18, the model-pair dissimilarity is 1.\nBetween product 2 and product 19, the model-pair dissimilarity is 3.\nBetween product 2 and product 20, the model-pair dissimilarity is 1.\nBetween product 2 and product 21, the model-pair dissimilarity is 8.\nBetween product 2 and product 22, the model-pair dissimilarity is 3.\nBetween product 3 and product 1, the model-pair dissimilarity is 4.\nBetween product 3 and product 2, the model-pair dissimilarity is 8.\nBetween product 3 and product 4, the model-pair dissimilarity is 6.\nBetween product 3 and product 5, the model-pair dissimilarity is 2.\nBetween product 3 and product 6, the model-pair dissimilarity is 1.\nBetween product 3 and product 7, the model-pair dissimilarity is 2.\nBetween product 3 and product 8, the model-pair dissimilarity is 1.\nBetween product 3 and product 9, the model-pair dissimilarity is 5.\nBetween product 3 and product 10, the model-pair dissimilarity is 2.\nBetween product 3 and product 11, the model-pair dissimilarity is 8.\nBetween product 3 and product 12, the model-pair dissimilarity is 2.\nBetween product 3 and product 13, the model-pair dissimilarity is 1.\nBetween product 3 and product 14, the model-pair dissimilarity is 9.\nBetween product 3 and product 15, the model-pair dissimilarity is 1.\nBetween product 3 and product 16, the model-pair dissimilarity is 9.\nBetween product 3 and product 17, the model-pair dissimilarity is 9.\nBetween product 3 and product 18, the model-pair dissimilarity is 3.\nBetween product 3 and product 19, the model-pair dissimilarity is 1.\nBetween product 3 and product 20, the model-pair dissimilarity is 8.\nBetween product 3 and product 21, the model-pair dissimilarity is 2.\nBetween product 3 and product 22, the model-pair dissimilarity is 3.\nBetween product 4 and product 1, the model-pair dissimilarity is 7.\nBetween product 4 and product 2, the model-pair dissimilarity is 6.\nBetween product 4 and product 3, the model-pair dissimilarity is 6.\nBetween product 4 and product 5, the model-pair dissimilarity is 5.\nBetween product 4 and product 6, the model-pair dissimilarity is 4.\nBetween product 4 and product 7, the model-pair dissimilarity is 3.\nBetween product 4 and product 8, the model-pair dissimilarity is 5.\nBetween product 4 and product 9, the model-pair dissimilarity is 4.\nBetween product 4 and product 10, the model-pair dissimilarity is 8.\nBetween product 4 and product 11, the model-pair dissimilarity is 5.\nBetween product 4 and product 12, the model-pair dissimilarity is 2.\nBetween product 4 and product 13, the model-pair dissimilarity is 2.\nBetween product 4 and product 14, the model-pair dissimilarity is 5.\nBetween product 4 and product 15, the model-pair dissimilarity is 0.\nBetween product 4 and product 16, the model-pair dissimilarity is 9.\nBetween product 4 and product 17, the model-pair dissimilarity is 7.\nBetween product 4 and product 18, the model-pair dissimilarity is 9.\nBetween product 4 and product 19, the model-pair dissimilarity is 7.\nBetween product 4 and product 20, the model-pair dissimilarity is 8.\nBetween product 4 and product 21, the model-pair dissimilarity is 8.\nBetween product 4 and product 22, the model-pair dissimilarity is 4.\nBetween product 5 and product 1, the model-pair dissimilarity is 0.\nBetween product 5 and product 2, the model-pair dissimilarity is 2.\nBetween product 5 and product 3, the model-pair dissimilarity is 2.\nBetween product 5 and product 4, the model-pair dissimilarity is 5.\nBetween product 5 and product 6, the model-pair dissimilarity is 6.\nBetween product 5 and product 7, the model-pair dissimilarity is 2.\nBetween product 5 and product 8, the model-pair dissimilarity is 7.\nBetween product 5 and product 9, the model-pair dissimilarity is 7.\nBetween product 5 and product 10, the model-pair dissimilarity is 9.\nBetween product 5 and product 11, the model-pair dissimilarity is 5.\nBetween product 5 and product 12, the model-pair dissimilarity is 0.\nBetween product 5 and product 13, the model-pair dissimilarity is 3.\nBetween product 5 and product 14, the model-pair dissimilarity is 8.\nBetween product 5 and product 15, the model-pair dissimilarity is 7.\nBetween product 5 and product 16, the model-pair dissimilarity is 7.\nBetween product 5 and product 17, the model-pair dissimilarity is 5.\nBetween product 5 and product 18, the model-pair dissimilarity is 5.\nBetween product 5 and product 19, the model-pair dissimilarity is 0.\nBetween product 5 and product 20, the model-pair dissimilarity is 1.\nBetween product 5 and product 21, the model-pair dissimilarity is 4.\nBetween product 5 and product 22, the model-pair dissimilarity is 2.\nBetween product 6 and product 1, the model-pair dissimilarity is 6.\nBetween product 6 and product 2, the model-pair dissimilarity is 2.\nBetween product 6 and product 3, the model-pair dissimilarity is 1.\nBetween product 6 and product 4, the model-pair dissimilarity is 4.\nBetween product 6 and product 5, the model-pair dissimilarity is 6.\nBetween product 6 and product 7, the model-pair dissimilarity is 4.\nBetween product 6 and product 8, the model-pair dissimilarity is 5.\nBetween product 6 and product 9, the model-pair dissimilarity is 6.\nBetween product 6 and product 10, the model-pair dissimilarity is 5.\nBetween product 6 and product 11, the model-pair dissimilarity is 2.\nBetween product 6 and product 12, the model-pair dissimilarity is 5.\nBetween product 6 and product 13, the model-pair dissimilarity is 1.\nBetween product 6 and product 14, the model-pair dissimilarity is 2.\nBetween product 6 and product 15, the model-pair dissimilarity is 3.\nBetween product 6 and product 16, the model-pair dissimilarity is 7.\nBetween product 6 and product 17, the model-pair dissimilarity is 0.\nBetween product 6 and product 18, the model-pair dissimilarity is 3.\nBetween product 6 and product 19, the model-pair dissimilarity is 7.\nBetween product 6 and product 20, the model-pair dissimilarity is 0.\nBetween product 6 and product 21, the model-pair dissimilarity is 3.\nBetween product 6 and product 22, the model-pair dissimilarity is 2.\nBetween product 7 and product 1, the model-pair dissimilarity is 2.\nBetween product 7 and product 2, the model-pair dissimilarity is 0.\nBetween product 7 and product 3, the model-pair dissimilarity is 2.\nBetween product 7 and product 4, the model-pair dissimilarity is 3.\nBetween product 7 and product 5, the model-pair dissimilarity is 2.\nBetween product 7 and product 6, the model-pair dissimilarity is 4.\nBetween product 7 and product 8, the model-pair dissimilarity is 0.\nBetween product 7 and product 9, the model-pair dissimilarity is 8.\nBetween product 7 and product 10, the model-pair dissimilarity is 3.\nBetween product 7 and product 11, the model-pair dissimilarity is 3.\nBetween product 7 and product 12, the model-pair dissimilarity is 8.\nBetween product 7 and product 13, the model-pair dissimilarity is 8.\nBetween product 7 and product 14, the model-pair dissimilarity is 0.\nBetween product 7 and product 15, the model-pair dissimilarity is 0.\nBetween product 7 and product 16, the model-pair dissimilarity is 9.\nBetween product 7 and product 17, the model-pair dissimilarity is 7.\nBetween product 7 and product 18, the model-pair dissimilarity is 3.\nBetween product 7 and product 19, the model-pair dissimilarity is 1.\nBetween product 7 and product 20, the model-pair dissimilarity is 8.\nBetween product 7 and product 21, the model-pair dissimilarity is 2.\nBetween product 7 and product 22, the model-pair dissimilarity is 9.\nBetween product 8 and product 1, the model-pair dissimilarity is 6.\nBetween product 8 and product 2, the model-pair dissimilarity is 4.\nBetween product 8 and product 3, the model-pair dissimilarity is 1.\nBetween product 8 and product 4, the model-pair dissimilarity is 5.\nBetween product 8 and product 5, the model-pair dissimilarity is 7.\nBetween product 8 and product 6, the model-pair dissimilarity is 5.\nBetween product 8 and product 7, the model-pair dissimilarity is 0.\nBetween product 8 and product 9, the model-pair dissimilarity is 9.\nBetween product 8 and product 10, the model-pair dissimilarity is 7.\nBetween product 8 and product 11, the model-pair dissimilarity is 8.\nBetween product 8 and product 12, the model-pair dissimilarity is 3.\nBetween product 8 and product 13, the model-pair dissimilarity is 3.\nBetween product 8 and product 14, the model-pair dissimilarity is 3.\nBetween product 8 and product 15, the model-pair dissimilarity is 2.\nBetween product 8 and product 16, the model-pair dissimilarity is 7.\nBetween product 8 and product 17, the model-pair dissimilarity is 2.\nBetween product 8 and product 18, the model-pair dissimilarity is 9.\nBetween product 8 and product 19, the model-pair dissimilarity is 7.\nBetween product 8 and product 20, the model-pair dissimilarity is 3.\nBetween product 8 and product 21, the model-pair dissimilarity is 0.\nBetween product 8 and product 22, the model-pair dissimilarity is 4.\nBetween product 9 and product 1, the model-pair dissimilarity is 8.\nBetween product 9 and product 2, the model-pair dissimilarity is 0.\nBetween product 9 and product 3, the model-pair dissimilarity is 5.\nBetween product 9 and product 4, the model-pair dissimilarity is 4.\nBetween product 9 and product 5, the model-pair dissimilarity is 7.\nBetween product 9 and product 6, the model-pair dissimilarity is 6.\nBetween product 9 and product 7, the model-pair dissimilarity is 8.\nBetween product 9 and product 8, the model-pair dissimilarity is 9.\nBetween product 9 and product 10, the model-pair dissimilarity is 1.\nBetween product 9 and product 11, the model-pair dissimilarity is 7.\nBetween product 9 and product 12, the model-pair dissimilarity is 8.\nBetween product 9 and product 13, the model-pair dissimilarity is 1.\nBetween product 9 and product 14, the model-pair dissimilarity is 9.\nBetween product 9 and product 15, the model-pair dissimilarity is 1.\nBetween product 9 and product 16, the model-pair dissimilarity is 2.\nBetween product 9 and product 17, the model-pair dissimilarity is 4.\nBetween product 9 and product 18, the model-pair dissimilarity is 3.\nBetween product 9 and product 19, the model-pair dissimilarity is 9.\nBetween product 9 and product 20, the model-pair dissimilarity is 7.\nBetween product 9 and product 21, the model-pair dissimilarity is 8.\nBetween product 9 and product 22, the model-pair dissimilarity is 7.\nBetween product 10 and product 1, the model-pair dissimilarity is 4.\nBetween product 10 and product 2, the model-pair dissimilarity is 5.\nBetween product 10 and product 3, the model-pair dissimilarity is 2.\nBetween product 10 and product 4, the model-pair dissimilarity is 8.\nBetween product 10 and product 5, the model-pair dissimilarity is 9.\nBetween product 10 and product 6, the model-pair dissimilarity is 5.\nBetween product 10 and product 7, the model-pair dissimilarity is 3.\nBetween product 10 and product 8, the model-pair dissimilarity is 7.\nBetween product 10 and product 9, the model-pair dissimilarity is 1.\nBetween product 10 and product 11, the model-pair dissimilarity is 3.\nBetween product 10 and product 12, the model-pair dissimilarity is 4.\nBetween product 10 and product 13, the model-pair dissimilarity is 9.\nBetween product 10 and product 14, the model-pair dissimilarity is 8.\nBetween product 10 and product 15, the model-pair dissimilarity is 4.\nBetween product 10 and product 16, the model-pair dissimilarity is 5.\nBetween product 10 and product 17, the model-pair dissimilarity is 4.\nBetween product 10 and product 18, the model-pair dissimilarity is 5.\nBetween product 10 and product 19, the model-pair dissimilarity is 6.\nBetween product 10 and product 20, the model-pair dissimilarity is 2.\nBetween product 10 and product 21, the model-pair dissimilarity is 7.\nBetween product 10 and product 22, the model-pair dissimilarity is 1.\nBetween product 11 and product 1, the model-pair dissimilarity is 4.\nBetween product 11 and product 2, the model-pair dissimilarity is 6.\nBetween product 11 and product 3, the model-pair dissimilarity is 8.\nBetween product 11 and product 4, the model-pair dissimilarity is 5.\nBetween product 11 and product 5, the model-pair dissimilarity is 5.\nBetween product 11 and product 6, the model-pair dissimilarity is 2.\nBetween product 11 and product 7, the model-pair dissimilarity is 3.\nBetween product 11 and product 8, the model-pair dissimilarity is 8.\nBetween product 11 and product 9, the model-pair dissimilarity is 7.\nBetween product 11 and product 10, the model-pair dissimilarity is 3.\nBetween product 11 and product 12, the model-pair dissimilarity is 4.\nBetween product 11 and product 13, the model-pair dissimilarity is 0.\nBetween product 11 and product 14, the model-pair dissimilarity is 2.\nBetween product 11 and product 15, the model-pair dissimilarity is 5.\nBetween product 11 and product 16, the model-pair dissimilarity is 2.\nBetween product 11 and product 17, the model-pair dissimilarity is 8.\nBetween product 11 and product 18, the model-pair dissimilarity is 9.\nBetween product 11 and product 19, the model-pair dissimilarity is 6.\nBetween product 11 and product 20, the model-pair dissimilarity is 3.\nBetween product 11 and product 21, the model-pair dissimilarity is 4.\nBetween product 11 and product 22, the model-pair dissimilarity is 9.\nBetween product 12 and product 1, the model-pair dissimilarity is 0.\nBetween product 12 and product 2, the model-pair dissimilarity is 0.\nBetween product 12 and product 3, the model-pair dissimilarity is 2.\nBetween product 12 and product 4, the model-pair dissimilarity is 2.\nBetween product 12 and product 5, the model-pair dissimilarity is 0.\nBetween product 12 and product 6, the model-pair dissimilarity is 5.\nBetween product 12 and product 7, the model-pair dissimilarity is 8.\nBetween product 12 and product 8, the model-pair dissimilarity is 3.\nBetween product 12 and product 9, the model-pair dissimilarity is 8.\nBetween product 12 and product 10, the model-pair dissimilarity is 4.\nBetween product 12 and product 11, the model-pair dissimilarity is 4.\nBetween product 12 and product 13, the model-pair dissimilarity is 9.\nBetween product 12 and product 14, the model-pair dissimilarity is 9.\nBetween product 12 and product 15, the model-pair dissimilarity is 9.\nBetween product 12 and product 16, the model-pair dissimilarity is 4.\nBetween product 12 and product 17, the model-pair dissimilarity is 2.\nBetween product 12 and product 18, the model-pair dissimilarity is 7.\nBetween product 12 and product 19, the model-pair dissimilarity is 8.\nBetween product 12 and product 20, the model-pair dissimilarity is 2.\nBetween product 12 and product 21, the model-pair dissimilarity is 4.\nBetween product 12 and product 22, the model-pair dissimilarity is 4.\nBetween product 13 and product 1, the model-pair dissimilarity is 0.\nBetween product 13 and product 2, the model-pair dissimilarity is 3.\nBetween product 13 and product 3, the model-pair dissimilarity is 1.\nBetween product 13 and product 4, the model-pair dissimilarity is 2.\nBetween product 13 and product 5, the model-pair dissimilarity is 3.\nBetween product 13 and product 6, the model-pair dissimilarity is 1.\nBetween product 13 and product 7, the model-pair dissimilarity is 8.\nBetween product 13 and product 8, the model-pair dissimilarity is 3.\nBetween product 13 and product 9, the model-pair dissimilarity is 1.\nBetween product 13 and product 10, the model-pair dissimilarity is 9.\nBetween product 13 and product 11, the model-pair dissimilarity is 0.\nBetween product 13 and product 12, the model-pair dissimilarity is 9.\nBetween product 13 and product 14, the model-pair dissimilarity is 7.\nBetween product 13 and product 15, the model-pair dissimilarity is 9.\nBetween product 13 and product 16, the model-pair dissimilarity is 2.\nBetween product 13 and product 17, the model-pair dissimilarity is 9.\nBetween product 13 and product 18, the model-pair dissimilarity is 9.\nBetween product 13 and product 19, the model-pair dissimilarity is 7.\nBetween product 13 and product 20, the model-pair dissimilarity is 7.\nBetween product 13 and product 21, the model-pair dissimilarity is 8.\nBetween product 13 and product 22, the model-pair dissimilarity is 1.\nBetween product 14 and product 1, the model-pair dissimilarity is 0.\nBetween product 14 and product 2, the model-pair dissimilarity is 2.\nBetween product 14 and product 3, the model-pair dissimilarity is 9.\nBetween product 14 and product 4, the model-pair dissimilarity is 5.\nBetween product 14 and product 5, the model-pair dissimilarity is 8.\nBetween product 14 and product 6, the model-pair dissimilarity is 2.\nBetween product 14 and product 7, the model-pair dissimilarity is 0.\nBetween product 14 and product 8, the model-pair dissimilarity is 3.\nBetween product 14 and product 9, the model-pair dissimilarity is 9.\nBetween product 14 and product 10, the model-pair dissimilarity is 8.\nBetween product 14 and product 11, the model-pair dissimilarity is 2.\nBetween product 14 and product 12, the model-pair dissimilarity is 9.\nBetween product 14 and product 13, the model-pair dissimilarity is 7.\nBetween product 14 and product 15, the model-pair dissimilarity is 1.\nBetween product 14 and product 16, the model-pair dissimilarity is 2.\nBetween product 14 and product 17, the model-pair dissimilarity is 2.\nBetween product 14 and product 18, the model-pair dissimilarity is 5.\nBetween product 14 and product 19, the model-pair dissimilarity is 9.\nBetween product 14 and product 20, the model-pair dissimilarity is 6.\nBetween product 14 and product 21, the model-pair dissimilarity is 1.\nBetween product 14 and product 22, the model-pair dissimilarity is 1.\nBetween product 15 and product 1, the model-pair dissimilarity is 7.\nBetween product 15 and product 2, the model-pair dissimilarity is 3.\nBetween product 15 and product 3, the model-pair dissimilarity is 1.\nBetween product 15 and product 4, the model-pair dissimilarity is 0.\nBetween product 15 and product 5, the model-pair dissimilarity is 7.\nBetween product 15 and product 6, the model-pair dissimilarity is 3.\nBetween product 15 and product 7, the model-pair dissimilarity is 0.\nBetween product 15 and product 8, the model-pair dissimilarity is 2.\nBetween product 15 and product 9, the model-pair dissimilarity is 1.\nBetween product 15 and product 10, the model-pair dissimilarity is 4.\nBetween product 15 and product 11, the model-pair dissimilarity is 5.\nBetween product 15 and product 12, the model-pair dissimilarity is 9.\nBetween product 15 and product 13, the model-pair dissimilarity is 9.\nBetween product 15 and product 14, the model-pair dissimilarity is 1.\nBetween product 15 and product 16, the model-pair dissimilarity is 1.\nBetween product 15 and product 17, the model-pair dissimilarity is 5.\nBetween product 15 and product 18, the model-pair dissimilarity is 0.\nBetween product 15 and product 19, the model-pair dissimilarity is 2.\nBetween product 15 and product 20, the model-pair dissimilarity is 4.\nBetween product 15 and product 21, the model-pair dissimilarity is 4.\nBetween product 15 and product 22, the model-pair dissimilarity is 0.\nBetween product 16 and product 1, the model-pair dissimilarity is 2.\nBetween product 16 and product 2, the model-pair dissimilarity is 4.\nBetween product 16 and product 3, the model-pair dissimilarity is 9.\nBetween product 16 and product 4, the model-pair dissimilarity is 9.\nBetween product 16 and product 5, the model-pair dissimilarity is 7.\nBetween product 16 and product 6, the model-pair dissimilarity is 7.\nBetween product 16 and product 7, the model-pair dissimilarity is 9.\nBetween product 16 and product 8, the model-pair dissimilarity is 7.\nBetween product 16 and product 9, the model-pair dissimilarity is 2.\nBetween product 16 and product 10, the model-pair dissimilarity is 5.\nBetween product 16 and product 11, the model-pair dissimilarity is 2.\nBetween product 16 and product 12, the model-pair dissimilarity is 4.\nBetween product 16 and product 13, the model-pair dissimilarity is 2.\nBetween product 16 and product 14, the model-pair dissimilarity is 2.\nBetween product 16 and product 15, the model-pair dissimilarity is 1.\nBetween product 16 and product 17, the model-pair dissimilarity is 7.\nBetween product 16 and product 18, the model-pair dissimilarity is 8.\nBetween product 16 and product 19, the model-pair dissimilarity is 5.\nBetween product 16 and product 20, the model-pair dissimilarity is 8.\nBetween product 16 and product 21, the model-pair dissimilarity is 4.\nBetween product 16 and product 22, the model-pair dissimilarity is 0.\nBetween product 17 and product 1, the model-pair dissimilarity is 5.\nBetween product 17 and product 2, the model-pair dissimilarity is 5.\nBetween product 17 and product 3, the model-pair dissimilarity is 9.\nBetween product 17 and product 4, the model-pair dissimilarity is 7.\nBetween product 17 and product 5, the model-pair dissimilarity is 5.\nBetween product 17 and product 6, the model-pair dissimilarity is 0.\nBetween product 17 and product 7, the model-pair dissimilarity is 7.\nBetween product 17 and product 8, the model-pair dissimilarity is 2.\nBetween product 17 and product 9, the model-pair dissimilarity is 4.\nBetween product 17 and product 10, the model-pair dissimilarity is 4.\nBetween product 17 and product 11, the model-pair dissimilarity is 8.\nBetween product 17 and product 12, the model-pair dissimilarity is 2.\nBetween product 17 and product 13, the model-pair dissimilarity is 9.\nBetween product 17 and product 14, the model-pair dissimilarity is 2.\nBetween product 17 and product 15, the model-pair dissimilarity is 5.\nBetween product 17 and product 16, the model-pair dissimilarity is 7.\nBetween product 17 and product 18, the model-pair dissimilarity is 9.\nBetween product 17 and product 19, the model-pair dissimilarity is 1.\nBetween product 17 and product 20, the model-pair dissimilarity is 2.\nBetween product 17 and product 21, the model-pair dissimilarity is 2.\nBetween product 17 and product 22, the model-pair dissimilarity is 5.\nBetween product 18 and product 1, the model-pair dissimilarity is 7.\nBetween product 18 and product 2, the model-pair dissimilarity is 1.\nBetween product 18 and product 3, the model-pair dissimilarity is 3.\nBetween product 18 and product 4, the model-pair dissimilarity is 9.\nBetween product 18 and product 5, the model-pair dissimilarity is 5.\nBetween product 18 and product 6, the model-pair dissimilarity is 3.\nBetween product 18 and product 7, the model-pair dissimilarity is 3.\nBetween product 18 and product 8, the model-pair dissimilarity is 9.\nBetween product 18 and product 9, the model-pair dissimilarity is 3.\nBetween product 18 and product 10, the model-pair dissimilarity is 5.\nBetween product 18 and product 11, the model-pair dissimilarity is 9.\nBetween product 18 and product 12, the model-pair dissimilarity is 7.\nBetween product 18 and product 13, the model-pair dissimilarity is 9.\nBetween product 18 and product 14, the model-pair dissimilarity is 5.\nBetween product 18 and product 15, the model-pair dissimilarity is 0.\nBetween product 18 and product 16, the model-pair dissimilarity is 8.\nBetween product 18 and product 17, the model-pair dissimilarity is 9.\nBetween product 18 and product 19, the model-pair dissimilarity is 0.\nBetween product 18 and product 20, the model-pair dissimilarity is 4.\nBetween product 18 and product 21, the model-pair dissimilarity is 1.\nBetween product 18 and product 22, the model-pair dissimilarity is 2.\nBetween product 19 and product 1, the model-pair dissimilarity is 4.\nBetween product 19 and product 2, the model-pair dissimilarity is 3.\nBetween product 19 and product 3, the model-pair dissimilarity is 1.\nBetween product 19 and product 4, the model-pair dissimilarity is 7.\nBetween product 19 and product 5, the model-pair dissimilarity is 0.\nBetween product 19 and product 6, the model-pair dissimilarity is 7.\nBetween product 19 and product 7, the model-pair dissimilarity is 1.\nBetween product 19 and product 8, the model-pair dissimilarity is 7.\nBetween product 19 and product 9, the model-pair dissimilarity is 9.\nBetween product 19 and product 10, the model-pair dissimilarity is 6.\nBetween product 19 and product 11, the model-pair dissimilarity is 6.\nBetween product 19 and product 12, the model-pair dissimilarity is 8.\nBetween product 19 and product 13, the model-pair dissimilarity is 7.\nBetween product 19 and product 14, the model-pair dissimilarity is 9.\nBetween product 19 and product 15, the model-pair dissimilarity is 2.\nBetween product 19 and product 16, the model-pair dissimilarity is 5.\nBetween product 19 and product 17, the model-pair dissimilarity is 1.\nBetween product 19 and product 18, the model-pair dissimilarity is 0.\nBetween product 19 and product 20, the model-pair dissimilarity is 5.\nBetween product 19 and product 21, the model-pair dissimilarity is 7.\nBetween product 19 and product 22, the model-pair dissimilarity is 3.\nBetween product 20 and product 1, the model-pair dissimilarity is 1.\nBetween product 20 and product 2, the model-pair dissimilarity is 1.\nBetween product 20 and product 3, the model-pair dissimilarity is 8.\nBetween product 20 and product 4, the model-pair dissimilarity is 8.\nBetween product 20 and product 5, the model-pair dissimilarity is 1.\nBetween product 20 and product 6, the model-pair dissimilarity is 0.\nBetween product 20 and product 7, the model-pair dissimilarity is 8.\nBetween product 20 and product 8, the model-pair dissimilarity is 3.\nBetween product 20 and product 9, the model-pair dissimilarity is 7.\nBetween product 20 and product 10, the model-pair dissimilarity is 2.\nBetween product 20 and product 11, the model-pair dissimilarity is 3.\nBetween product 20 and product 12, the model-pair dissimilarity is 2.\nBetween product 20 and product 13, the model-pair dissimilarity is 7.\nBetween product 20 and product 14, the model-pair dissimilarity is 6.\nBetween product 20 and product 15, the model-pair dissimilarity is 4.\nBetween product 20 and product 16, the model-pair dissimilarity is 8.\nBetween product 20 and product 17, the model-pair dissimilarity is 2.\nBetween product 20 and product 18, the model-pair dissimilarity is 4.\nBetween product 20 and product 19, the model-pair dissimilarity is 5.\nBetween product 20 and product 21, the model-pair dissimilarity is 6.\nBetween product 20 and product 22, the model-pair dissimilarity is 7.\nBetween product 21 and product 1, the model-pair dissimilarity is 7.\nBetween product 21 and product 2, the model-pair dissimilarity is 8.\nBetween product 21 and product 3, the model-pair dissimilarity is 2.\nBetween product 21 and product 4, the model-pair dissimilarity is 8.\nBetween product 21 and product 5, the model-pair dissimilarity is 4.\nBetween product 21 and product 6, the model-pair dissimilarity is 3.\nBetween product 21 and product 7, the model-pair dissimilarity is 2.\nBetween product 21 and product 8, the model-pair dissimilarity is 0.\nBetween product 21 and product 9, the model-pair dissimilarity is 8.\nBetween product 21 and product 10, the model-pair dissimilarity is 7.\nBetween product 21 and product 11, the model-pair dissimilarity is 4.\nBetween product 21 and product 12, the model-pair dissimilarity is 4.\nBetween product 21 and product 13, the model-pair dissimilarity is 8.\nBetween product 21 and product 14, the model-pair dissimilarity is 1.\nBetween product 21 and product 15, the model-pair dissimilarity is 4.\nBetween product 21 and product 16, the model-pair dissimilarity is 4.\nBetween product 21 and product 17, the model-pair dissimilarity is 2.\nBetween product 21 and product 18, the model-pair dissimilarity is 1.\nBetween product 21 and product 19, the model-pair dissimilarity is 7.\nBetween product 21 and product 20, the model-pair dissimilarity is 6.\nBetween product 21 and product 22, the model-pair dissimilarity is 6.\nBetween product 22 and product 1, the model-pair dissimilarity is 6.\nBetween product 22 and product 2, the model-pair dissimilarity is 3.\nBetween product 22 and product 3, the model-pair dissimilarity is 3.\nBetween product 22 and product 4, the model-pair dissimilarity is 4.\nBetween product 22 and product 5, the model-pair dissimilarity is 2.\nBetween product 22 and product 6, the model-pair dissimilarity is 2.\nBetween product 22 and product 7, the model-pair dissimilarity is 9.\nBetween product 22 and product 8, the model-pair dissimilarity is 4.\nBetween product 22 and product 9, the model-pair dissimilarity is 7.\nBetween product 22 and product 10, the model-pair dissimilarity is 1.\nBetween product 22 and product 11, the model-pair dissimilarity is 9.\nBetween product 22 and product 12, the model-pair dissimilarity is 4.\nBetween product 22 and product 13, the model-pair dissimilarity is 1.\nBetween product 22 and product 14, the model-pair dissimilarity is 1.\nBetween product 22 and product 15, the model-pair dissimilarity is 0.\nBetween product 22 and product 16, the model-pair dissimilarity is 0.\nBetween product 22 and product 17, the model-pair dissimilarity is 5.\nBetween product 22 and product 18, the model-pair dissimilarity is 2.\nBetween product 22 and product 19, the model-pair dissimilarity is 3.\nBetween product 22 and product 20, the model-pair dissimilarity is 7.\nBetween product 22 and product 21, the model-pair dissimilarity is 6.\nUse the listed pairwise dissimilarities to evaluate diversity and aim to maximize the total summed score among the 2 chosen.\n\nAlso, when you want to give me the chosen lineup, please stick to this little JSON shape so it's easy to read and parse:\n\n{\n \"solution\": [, , ...]\n}\n\nPretty simple: \"solution\" is just an array listing the items (the models that will occupy the display spots). Think of it like a short form you’d fill in: the chosen item identifiers go in that list, in any order. This block is only a sketch of the expected shape — not the actual selection.\n\nPlease make sure to use each identifier exactly as it appears in the instance input — no renaming and no adding new labels. \nfor example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 3, 4, 7, 0, 6, 2, 6, 8, 4, 4, 0, 0, 0, 7, 2, 5, 7, 4, 1, 7, 6 ], [ 3, 0, 8, 6, 2, 2, 0, 4, 0, 5, 6, 0, 3, 2, 3, 4, 5, 1, 3, 1, 8, 3 ], [ 4, 8, 0, 6, 2, 1, 2, 1, 5, 2, 8, 2, 1, 9, 1, 9, 9, 3, 1, 8, 2, 3 ], [ 7, 6, 6, 0, 5, 4, 3, 5, 4, 8, 5, 2, 2, 5, 0, 9, 7, 9, 7, 8, 8, 4 ], [ 0, 2, 2, 5, 0, 6, 2, 7, 7, 9, 5, 0, 3, 8, 7, 7, 5, 5, 0, 1, 4, 2 ], [ 6, 2, 1, 4, 6, 0, 4, 5, 6, 5, 2, 5, 1, 2, 3, 7, 0, 3, 7, 0, 3, 2 ], [ 2, 0, 2, 3, 2, 4, 0, 0, 8, 3, 3, 8, 8, 0, 0, 9, 7, 3, 1, 8, 2, 9 ], [ 6, 4, 1, 5, 7, 5, 0, 0, 9, 7, 8, 3, 3, 3, 2, 7, 2, 9, 7, 3, 0, 4 ], [ 8, 0, 5, 4, 7, 6, 8, 9, 0, 1, 7, 8, 1, 9, 1, 2, 4, 3, 9, 7, 8, 7 ], [ 4, 5, 2, 8, 9, 5, 3, 7, 1, 0, 3, 4, 9, 8, 4, 5, 4, 5, 6, 2, 7, 1 ], [ 4, 6, 8, 5, 5, 2, 3, 8, 7, 3, 0, 4, 0, 2, 5, 2, 8, 9, 6, 3, 4, 9 ], [ 0, 0, 2, 2, 0, 5, 8, 3, 8, 4, 4, 0, 9, 9, 9, 4, 2, 7, 8, 2, 4, 4 ], [ 0, 3, 1, 2, 3, 1, 8, 3, 1, 9, 0, 9, 0, 7, 9, 2, 9, 9, 7, 7, 8, 1 ], [ 0, 2, 9, 5, 8, 2, 0, 3, 9, 8, 2, 9, 7, 0, 1, 2, 2, 5, 9, 6, 1, 1 ], [ 7, 3, 1, 0, 7, 3, 0, 2, 1, 4, 5, 9, 9, 1, 0, 1, 5, 0, 2, 4, 4, 0 ], [ 2, 4, 9, 9, 7, 7, 9, 7, 2, 5, 2, 4, 2, 2, 1, 0, 7, 8, 5, 8, 4, 0 ], [ 5, 5, 9, 7, 5, 0, 7, 2, 4, 4, 8, 2, 9, 2, 5, 7, 0, 9, 1, 2, 2, 5 ], [ 7, 1, 3, 9, 5, 3, 3, 9, 3, 5, 9, 7, 9, 5, 0, 8, 9, 0, 0, 4, 1, 2 ], [ 4, 3, 1, 7, 0, 7, 1, 7, 9, 6, 6, 8, 7, 9, 2, 5, 1, 0, 0, 5, 7, 3 ], [ 1, 1, 8, 8, 1, 0, 8, 3, 7, 2, 3, 2, 7, 6, 4, 8, 2, 4, 5, 0, 6, 7 ], [ 7, 8, 2, 8, 4, 3, 2, 0, 8, 7, 4, 4, 8, 1, 4, 4, 2, 1, 7, 6, 0, 6 ], [ 6, 3, 3, 4, 2, 2, 9, 4, 7, 1, 9, 4, 1, 1, 0, 0, 5, 2, 3, 7, 6, 0 ] ], "m": 2, "objective": 9.0 }, "solution": [ 2, 13 ], "obj": 9.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 22, "m": 2, "nodes": [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 ], "distance_pairs": [ { "from_id": 1, "to_id": 2, "distance": 3 }, { "from_id": 1, "to_id": 3, "distance": 4 }, { "from_id": 1, "to_id": 4, "distance": 7 }, { "from_id": 1, "to_id": 5, "distance": 0 }, { "from_id": 1, "to_id": 6, "distance": 6 }, { "from_id": 1, "to_id": 7, "distance": 2 }, { "from_id": 1, "to_id": 8, "distance": 6 }, { "from_id": 1, "to_id": 9, "distance": 8 }, { "from_id": 1, "to_id": 10, "distance": 4 }, { "from_id": 1, "to_id": 11, "distance": 4 }, { "from_id": 1, "to_id": 12, "distance": 0 }, { "from_id": 1, "to_id": 13, "distance": 0 }, { "from_id": 1, "to_id": 14, "distance": 0 }, { "from_id": 1, "to_id": 15, "distance": 7 }, { "from_id": 1, "to_id": 16, "distance": 2 }, { "from_id": 1, "to_id": 17, "distance": 5 }, { "from_id": 1, "to_id": 18, "distance": 7 }, { "from_id": 1, "to_id": 19, "distance": 4 }, { "from_id": 1, "to_id": 20, "distance": 1 }, { "from_id": 1, "to_id": 21, "distance": 7 }, { "from_id": 1, "to_id": 22, "distance": 6 }, { "from_id": 2, "to_id": 1, "distance": 3 }, { "from_id": 2, "to_id": 3, "distance": 8 }, { "from_id": 2, "to_id": 4, "distance": 6 }, { "from_id": 2, "to_id": 5, "distance": 2 }, { "from_id": 2, "to_id": 6, "distance": 2 }, { "from_id": 2, "to_id": 7, "distance": 0 }, { "from_id": 2, "to_id": 8, "distance": 4 }, { "from_id": 2, "to_id": 9, "distance": 0 }, { "from_id": 2, "to_id": 10, "distance": 5 }, { "from_id": 2, "to_id": 11, "distance": 6 }, { "from_id": 2, "to_id": 12, "distance": 0 }, { "from_id": 2, "to_id": 13, "distance": 3 }, { "from_id": 2, "to_id": 14, "distance": 2 }, { "from_id": 2, "to_id": 15, "distance": 3 }, { "from_id": 2, "to_id": 16, "distance": 4 }, { "from_id": 2, "to_id": 17, "distance": 5 }, { "from_id": 2, "to_id": 18, "distance": 1 }, { "from_id": 2, "to_id": 19, "distance": 3 }, { "from_id": 2, "to_id": 20, "distance": 1 }, { "from_id": 2, "to_id": 21, "distance": 8 }, { "from_id": 2, "to_id": 22, "distance": 3 }, { "from_id": 3, "to_id": 1, "distance": 4 }, { "from_id": 3, "to_id": 2, "distance": 8 }, { "from_id": 3, "to_id": 4, "distance": 6 }, { "from_id": 3, "to_id": 5, "distance": 2 }, { "from_id": 3, "to_id": 6, "distance": 1 }, { "from_id": 3, "to_id": 7, "distance": 2 }, { "from_id": 3, "to_id": 8, "distance": 1 }, { "from_id": 3, "to_id": 9, "distance": 5 }, { "from_id": 3, "to_id": 10, "distance": 2 }, { "from_id": 3, "to_id": 11, "distance": 8 }, { "from_id": 3, "to_id": 12, "distance": 2 }, { "from_id": 3, "to_id": 13, "distance": 1 }, { "from_id": 3, "to_id": 14, "distance": 9 }, { "from_id": 3, "to_id": 15, "distance": 1 }, { "from_id": 3, "to_id": 16, "distance": 9 }, { "from_id": 3, "to_id": 17, "distance": 9 }, { "from_id": 3, "to_id": 18, "distance": 3 }, { "from_id": 3, "to_id": 19, "distance": 1 }, { "from_id": 3, "to_id": 20, "distance": 8 }, { "from_id": 3, "to_id": 21, "distance": 2 }, { "from_id": 3, "to_id": 22, "distance": 3 }, { "from_id": 4, "to_id": 1, "distance": 7 }, { "from_id": 4, "to_id": 2, "distance": 6 }, { "from_id": 4, "to_id": 3, "distance": 6 }, { "from_id": 4, "to_id": 5, "distance": 5 }, { "from_id": 4, "to_id": 6, "distance": 4 }, { "from_id": 4, "to_id": 7, "distance": 3 }, { "from_id": 4, "to_id": 8, "distance": 5 }, { "from_id": 4, "to_id": 9, "distance": 4 }, { "from_id": 4, "to_id": 10, "distance": 8 }, { "from_id": 4, "to_id": 11, "distance": 5 }, { "from_id": 4, "to_id": 12, "distance": 2 }, { "from_id": 4, "to_id": 13, "distance": 2 }, { "from_id": 4, "to_id": 14, "distance": 5 }, { "from_id": 4, "to_id": 15, "distance": 0 }, { "from_id": 4, "to_id": 16, "distance": 9 }, { "from_id": 4, "to_id": 17, "distance": 7 }, { "from_id": 4, "to_id": 18, "distance": 9 }, { "from_id": 4, "to_id": 19, "distance": 7 }, { "from_id": 4, "to_id": 20, "distance": 8 }, { "from_id": 4, "to_id": 21, "distance": 8 }, { "from_id": 4, "to_id": 22, "distance": 4 }, { "from_id": 5, "to_id": 1, "distance": 0 }, { "from_id": 5, "to_id": 2, "distance": 2 }, { "from_id": 5, "to_id": 3, "distance": 2 }, { "from_id": 5, "to_id": 4, "distance": 5 }, { "from_id": 5, "to_id": 6, "distance": 6 }, { "from_id": 5, "to_id": 7, "distance": 2 }, { "from_id": 5, "to_id": 8, "distance": 7 }, { "from_id": 5, "to_id": 9, "distance": 7 }, { "from_id": 5, "to_id": 10, "distance": 9 }, { "from_id": 5, "to_id": 11, "distance": 5 }, { "from_id": 5, "to_id": 12, "distance": 0 }, { "from_id": 5, "to_id": 13, "distance": 3 }, { "from_id": 5, "to_id": 14, "distance": 8 }, { "from_id": 5, "to_id": 15, "distance": 7 }, { "from_id": 5, "to_id": 16, "distance": 7 }, { "from_id": 5, "to_id": 17, "distance": 5 }, { "from_id": 5, "to_id": 18, "distance": 5 }, { "from_id": 5, "to_id": 19, "distance": 0 }, { "from_id": 5, "to_id": 20, "distance": 1 }, { "from_id": 5, "to_id": 21, "distance": 4 }, { "from_id": 5, "to_id": 22, "distance": 2 }, { "from_id": 6, "to_id": 1, "distance": 6 }, { "from_id": 6, "to_id": 2, "distance": 2 }, { "from_id": 6, "to_id": 3, "distance": 1 }, { "from_id": 6, "to_id": 4, "distance": 4 }, { "from_id": 6, "to_id": 5, "distance": 6 }, { "from_id": 6, "to_id": 7, "distance": 4 }, { "from_id": 6, "to_id": 8, "distance": 5 }, { "from_id": 6, "to_id": 9, "distance": 6 }, { "from_id": 6, "to_id": 10, "distance": 5 }, { "from_id": 6, "to_id": 11, "distance": 2 }, { "from_id": 6, "to_id": 12, "distance": 5 }, { "from_id": 6, "to_id": 13, "distance": 1 }, { "from_id": 6, "to_id": 14, "distance": 2 }, { "from_id": 6, "to_id": 15, "distance": 3 }, { "from_id": 6, "to_id": 16, "distance": 7 }, { "from_id": 6, "to_id": 17, "distance": 0 }, { "from_id": 6, "to_id": 18, "distance": 3 }, { "from_id": 6, "to_id": 19, "distance": 7 }, { "from_id": 6, "to_id": 20, "distance": 0 }, { "from_id": 6, "to_id": 21, "distance": 3 }, { "from_id": 6, "to_id": 22, "distance": 2 }, { "from_id": 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"from_id": 22, "to_id": 8, "distance": 4 }, { "from_id": 22, "to_id": 9, "distance": 7 }, { "from_id": 22, "to_id": 10, "distance": 1 }, { "from_id": 22, "to_id": 11, "distance": 9 }, { "from_id": 22, "to_id": 12, "distance": 4 }, { "from_id": 22, "to_id": 13, "distance": 1 }, { "from_id": 22, "to_id": 14, "distance": 1 }, { "from_id": 22, "to_id": 15, "distance": 0 }, { "from_id": 22, "to_id": 16, "distance": 0 }, { "from_id": 22, "to_id": 17, "distance": 5 }, { "from_id": 22, "to_id": 18, "distance": 2 }, { "from_id": 22, "to_id": 19, "distance": 3 }, { "from_id": 22, "to_id": 20, "distance": 7 }, { "from_id": 22, "to_id": 21, "distance": 6 } ] }, "solution_variant": [ 3, 14 ], "context_index": 26, "input_format": "nl", "input_index_base": 1 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Back at the shop, the task was clear: choose exactly the predetermined number of bottles for the seasonal shelf, but make sure they cover different flavors and regions. The way the buyer compares options is by taking every two bottles in a candidate set, adding up how dissimilar they are, and preferring the set with the larger overall sum. The selection must consist of unique bottles, and the exact count has to be met. The exact distances between bottles and the number to pick are shown below.\n\n# total_bottles_available=19\n# bottles_to_stock=8\n# bottle_ids=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19\nbottle_id_1,bottle_id_2,dissimilarity_score\n1,2,197\n1,3,177\n1,4,178\n1,5,147\n1,6,169\n1,7,189\n1,8,167\n1,9,174\n1,10,179\n1,11,188\n1,12,169\n1,13,203\n1,14,171\n1,15,175\n1,16,162\n1,17,179\n1,18,150\n1,19,174\n2,1,197\n2,3,154\n2,4,146\n2,5,166\n2,6,168\n2,7,214\n2,8,228\n2,9,193\n2,10,205\n2,11,185\n2,12,223\n2,13,169\n2,14,214\n2,15,155\n2,16,185\n2,17,183\n2,18,194\n2,19,185\n3,1,177\n3,2,154\n3,4,193\n3,5,200\n3,6,214\n3,7,214\n3,8,245\n3,9,194\n3,10,219\n3,11,162\n3,12,175\n3,13,176\n3,14,174\n3,15,202\n3,16,194\n3,17,170\n3,18,209\n3,19,218\n4,1,178\n4,2,146\n4,3,193\n4,5,181\n4,6,189\n4,7,186\n4,8,182\n4,9,191\n4,10,194\n4,11,219\n4,12,187\n4,13,181\n4,14,215\n4,15,176\n4,16,195\n4,17,213\n4,18,190\n4,19,211\n5,1,147\n5,2,166\n5,3,200\n5,4,181\n5,6,125\n5,7,226\n5,8,205\n5,9,173\n5,10,205\n5,11,169\n5,12,201\n5,13,206\n5,14,176\n5,15,164\n5,16,177\n5,17,178\n5,18,184\n5,19,191\n6,1,169\n6,2,168\n6,3,214\n6,4,189\n6,5,125\n6,7,200\n6,8,163\n6,9,168\n6,10,180\n6,11,213\n6,12,205\n6,13,202\n6,14,224\n6,15,168\n6,16,188\n6,17,165\n6,18,153\n6,19,163\n7,1,189\n7,2,214\n7,3,214\n7,4,186\n7,5,226\n7,6,200\n7,8,165\n7,9,169\n7,10,137\n7,11,240\n7,12,212\n7,13,208\n7,14,221\n7,15,225\n7,16,201\n7,17,199\n7,18,202\n7,19,228\n8,1,167\n8,2,228\n8,3,245\n8,4,182\n8,5,205\n8,6,163\n8,7,165\n8,9,177\n8,10,133\n8,11,270\n8,12,183\n8,13,210\n8,14,213\n8,15,184\n8,16,199\n8,17,196\n8,18,136\n8,19,178\n9,1,174\n9,2,193\n9,3,194\n9,4,191\n9,5,173\n9,6,168\n9,7,169\n9,8,177\n9,10,152\n9,11,200\n9,12,190\n9,13,180\n9,14,176\n9,15,149\n9,16,138\n9,17,132\n9,18,188\n9,19,186\n10,1,179\n10,2,205\n10,3,219\n10,4,194\n10,5,205\n10,6,180\n10,7,137\n10,8,133\n10,9,152\n10,11,212\n10,12,177\n10,13,192\n10,14,225\n10,15,192\n10,16,160\n10,17,167\n10,18,167\n10,19,178\n11,1,188\n11,2,185\n11,3,162\n11,4,219\n11,5,169\n11,6,213\n11,7,240\n11,8,270\n11,9,200\n11,10,212\n11,12,198\n11,13,203\n11,14,205\n11,15,207\n11,16,197\n11,17,177\n11,18,235\n11,19,226\n12,1,169\n12,2,223\n12,3,175\n12,4,187\n12,5,201\n12,6,205\n12,7,212\n12,8,183\n12,9,190\n12,10,177\n12,11,198\n12,13,189\n12,14,187\n12,15,183\n12,16,218\n12,17,193\n12,18,169\n12,19,218\n13,1,203\n13,2,169\n13,3,176\n13,4,181\n13,5,206\n13,6,202\n13,7,208\n13,8,210\n13,9,180\n13,10,192\n13,11,203\n13,12,189\n13,14,171\n13,15,188\n13,16,193\n13,17,193\n13,18,216\n13,19,178\n14,1,171\n14,2,214\n14,3,174\n14,4,215\n14,5,176\n14,6,224\n14,7,221\n14,8,213\n14,9,176\n14,10,225\n14,11,205\n14,12,187\n14,13,171\n14,15,178\n14,16,192\n14,17,180\n14,18,197\n14,19,212\n15,1,175\n15,2,155\n15,3,202\n15,4,176\n15,5,164\n15,6,168\n15,7,225\n15,8,184\n15,9,149\n15,10,192\n15,11,207\n15,12,183\n15,13,188\n15,14,178\n15,16,170\n15,17,161\n15,18,134\n15,19,178\n16,1,162\n16,2,185\n16,3,194\n16,4,195\n16,5,177\n16,6,188\n16,7,201\n16,8,199\n16,9,138\n16,10,160\n16,11,197\n16,12,218\n16,13,193\n16,14,192\n16,15,170\n16,17,133\n16,18,190\n16,19,182\n17,1,179\n17,2,183\n17,3,170\n17,4,213\n17,5,178\n17,6,165\n17,7,199\n17,8,196\n17,9,132\n17,10,167\n17,11,177\n17,12,193\n17,13,193\n17,14,180\n17,15,161\n17,16,133\n17,18,154\n17,19,179\n18,1,150\n18,2,194\n18,3,209\n18,4,190\n18,5,184\n18,6,153\n18,7,202\n18,8,136\n18,9,188\n18,10,167\n18,11,235\n18,12,169\n18,13,216\n18,14,197\n18,15,134\n18,16,190\n18,17,154\n18,19,170\n19,1,174\n19,2,185\n19,3,218\n19,4,211\n19,5,191\n19,6,163\n19,7,228\n19,8,178\n19,9,186\n19,10,178\n19,11,226\n19,12,218\n19,13,178\n19,14,212\n19,15,178\n19,16,182\n19,17,179\n19,18,170\n\nIf you want to hand the result back to me in a neat, machine-friendly way, just follow this simple JSON shape — nothing fancy, just a list of the chosen bottles:\n\n{\n \"solution\": [\"\", \"\", \"\"]\n}\n\nPretty straightforward: \"solution\" is the array of the bottles you picked for the shelf, and each placeholder like stands in for a bottle identifier from the instance. Think of this JSON as the little form to fill out — it's an example of the shape I expect, not the real answer.\n\nImportant: use the identifiers exactly as they appear in the instance input — no renaming and no new labels.\n\nFor example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 197, 177, 178, 147, 169, 189, 167, 174, 179, 188, 169, 203, 171, 175, 162, 179, 150, 174 ], [ 197, 0, 154, 146, 166, 168, 214, 228, 193, 205, 185, 223, 169, 214, 155, 185, 183, 194, 185 ], [ 177, 154, 0, 193, 200, 214, 214, 245, 194, 219, 162, 175, 176, 174, 202, 194, 170, 209, 218 ], [ 178, 146, 193, 0, 181, 189, 186, 182, 191, 194, 219, 187, 181, 215, 176, 195, 213, 190, 211 ], [ 147, 166, 200, 181, 0, 125, 226, 205, 173, 205, 169, 201, 206, 176, 164, 177, 178, 184, 191 ], [ 169, 168, 214, 189, 125, 0, 200, 163, 168, 180, 213, 205, 202, 224, 168, 188, 165, 153, 163 ], [ 189, 214, 214, 186, 226, 200, 0, 165, 169, 137, 240, 212, 208, 221, 225, 201, 199, 202, 228 ], [ 167, 228, 245, 182, 205, 163, 165, 0, 177, 133, 270, 183, 210, 213, 184, 199, 196, 136, 178 ], [ 174, 193, 194, 191, 173, 168, 169, 177, 0, 152, 200, 190, 180, 176, 149, 138, 132, 188, 186 ], [ 179, 205, 219, 194, 205, 180, 137, 133, 152, 0, 212, 177, 192, 225, 192, 160, 167, 167, 178 ], [ 188, 185, 162, 219, 169, 213, 240, 270, 200, 212, 0, 198, 203, 205, 207, 197, 177, 235, 226 ], [ 169, 223, 175, 187, 201, 205, 212, 183, 190, 177, 198, 0, 189, 187, 183, 218, 193, 169, 218 ], [ 203, 169, 176, 181, 206, 202, 208, 210, 180, 192, 203, 189, 0, 171, 188, 193, 193, 216, 178 ], [ 171, 214, 174, 215, 176, 224, 221, 213, 176, 225, 205, 187, 171, 0, 178, 192, 180, 197, 212 ], [ 175, 155, 202, 176, 164, 168, 225, 184, 149, 192, 207, 183, 188, 178, 0, 170, 161, 134, 178 ], [ 162, 185, 194, 195, 177, 188, 201, 199, 138, 160, 197, 218, 193, 192, 170, 0, 133, 190, 182 ], [ 179, 183, 170, 213, 178, 165, 199, 196, 132, 167, 177, 193, 193, 180, 161, 133, 0, 154, 179 ], [ 150, 194, 209, 190, 184, 153, 202, 136, 188, 167, 235, 169, 216, 197, 134, 190, 154, 0, 170 ], [ 174, 185, 218, 211, 191, 163, 228, 178, 186, 178, 226, 218, 178, 212, 178, 182, 179, 170, 0 ] ], "m": 8, "objective": 5779.0 }, "solution": [ 1, 6, 7, 10, 11, 13, 15, 18 ], "obj": 5779.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 19, "m": 8, "nodes": [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 ], "distance_pairs": [ { "from_id": 1, "to_id": 2, "distance": 197 }, { "from_id": 1, "to_id": 3, "distance": 177 }, { "from_id": 1, "to_id": 4, "distance": 178 }, { "from_id": 1, "to_id": 5, "distance": 147 }, { "from_id": 1, "to_id": 6, "distance": 169 }, { "from_id": 1, "to_id": 7, "distance": 189 }, { "from_id": 1, "to_id": 8, "distance": 167 }, { "from_id": 1, "to_id": 9, "distance": 174 }, { "from_id": 1, "to_id": 10, "distance": 179 }, { "from_id": 1, "to_id": 11, "distance": 188 }, { "from_id": 1, "to_id": 12, "distance": 169 }, { "from_id": 1, "to_id": 13, "distance": 203 }, { "from_id": 1, "to_id": 14, "distance": 171 }, { "from_id": 1, "to_id": 15, "distance": 175 }, { "from_id": 1, "to_id": 16, "distance": 162 }, { "from_id": 1, "to_id": 17, "distance": 179 }, { "from_id": 1, "to_id": 18, "distance": 150 }, { "from_id": 1, "to_id": 19, "distance": 174 }, { "from_id": 2, "to_id": 1, "distance": 197 }, { "from_id": 2, "to_id": 3, "distance": 154 }, { "from_id": 2, "to_id": 4, "distance": 146 }, { "from_id": 2, "to_id": 5, "distance": 166 }, { "from_id": 2, "to_id": 6, "distance": 168 }, { "from_id": 2, "to_id": 7, "distance": 214 }, { "from_id": 2, "to_id": 8, "distance": 228 }, { "from_id": 2, "to_id": 9, "distance": 193 }, { "from_id": 2, "to_id": 10, "distance": 205 }, { "from_id": 2, "to_id": 11, "distance": 185 }, { "from_id": 2, "to_id": 12, "distance": 223 }, { "from_id": 2, "to_id": 13, "distance": 169 }, { "from_id": 2, "to_id": 14, "distance": 214 }, { "from_id": 2, "to_id": 15, "distance": 155 }, { "from_id": 2, "to_id": 16, "distance": 185 }, { "from_id": 2, "to_id": 17, "distance": 183 }, { "from_id": 2, "to_id": 18, "distance": 194 }, { "from_id": 2, "to_id": 19, "distance": 185 }, { "from_id": 3, "to_id": 1, "distance": 177 }, { "from_id": 3, "to_id": 2, "distance": 154 }, { "from_id": 3, "to_id": 4, "distance": 193 }, { "from_id": 3, "to_id": 5, "distance": 200 }, { "from_id": 3, "to_id": 6, "distance": 214 }, { "from_id": 3, "to_id": 7, "distance": 214 }, { "from_id": 3, "to_id": 8, "distance": 245 }, { "from_id": 3, "to_id": 9, "distance": 194 }, { "from_id": 3, "to_id": 10, "distance": 219 }, { "from_id": 3, "to_id": 11, "distance": 162 }, { "from_id": 3, "to_id": 12, "distance": 175 }, { "from_id": 3, "to_id": 13, "distance": 176 }, { "from_id": 3, "to_id": 14, "distance": 174 }, { "from_id": 3, "to_id": 15, "distance": 202 }, { "from_id": 3, "to_id": 16, "distance": 194 }, { "from_id": 3, "to_id": 17, "distance": 170 }, { "from_id": 3, "to_id": 18, "distance": 209 }, { "from_id": 3, "to_id": 19, "distance": 218 }, { "from_id": 4, "to_id": 1, "distance": 178 }, { "from_id": 4, "to_id": 2, "distance": 146 }, { "from_id": 4, "to_id": 3, "distance": 193 }, { "from_id": 4, "to_id": 5, "distance": 181 }, { "from_id": 4, "to_id": 6, "distance": 189 }, { "from_id": 4, "to_id": 7, "distance": 186 }, { "from_id": 4, "to_id": 8, "distance": 182 }, { "from_id": 4, "to_id": 9, "distance": 191 }, { "from_id": 4, "to_id": 10, "distance": 194 }, { "from_id": 4, "to_id": 11, "distance": 219 }, { "from_id": 4, "to_id": 12, "distance": 187 }, { "from_id": 4, "to_id": 13, "distance": 181 }, { "from_id": 4, "to_id": 14, "distance": 215 }, { "from_id": 4, "to_id": 15, "distance": 176 }, { "from_id": 4, "to_id": 16, "distance": 195 }, { "from_id": 4, "to_id": 17, "distance": 213 }, { "from_id": 4, "to_id": 18, "distance": 190 }, { "from_id": 4, "to_id": 19, "distance": 211 }, { "from_id": 5, "to_id": 1, "distance": 147 }, { "from_id": 5, "to_id": 2, "distance": 166 }, { "from_id": 5, "to_id": 3, "distance": 200 }, { "from_id": 5, "to_id": 4, "distance": 181 }, { "from_id": 5, "to_id": 6, "distance": 125 }, { "from_id": 5, "to_id": 7, "distance": 226 }, { "from_id": 5, "to_id": 8, "distance": 205 }, { "from_id": 5, "to_id": 9, "distance": 173 }, { "from_id": 5, "to_id": 10, "distance": 205 }, { "from_id": 5, "to_id": 11, "distance": 169 }, { "from_id": 5, "to_id": 12, "distance": 201 }, { "from_id": 5, "to_id": 13, "distance": 206 }, { "from_id": 5, "to_id": 14, "distance": 176 }, { "from_id": 5, "to_id": 15, "distance": 164 }, { "from_id": 5, "to_id": 16, "distance": 177 }, { "from_id": 5, "to_id": 17, "distance": 178 }, { "from_id": 5, "to_id": 18, "distance": 184 }, { "from_id": 5, "to_id": 19, "distance": 191 }, { "from_id": 6, "to_id": 1, "distance": 169 }, { "from_id": 6, "to_id": 2, "distance": 168 }, { "from_id": 6, "to_id": 3, "distance": 214 }, { "from_id": 6, "to_id": 4, "distance": 189 }, { "from_id": 6, "to_id": 5, "distance": 125 }, { "from_id": 6, "to_id": 7, "distance": 200 }, { "from_id": 6, "to_id": 8, "distance": 163 }, { "from_id": 6, "to_id": 9, "distance": 168 }, { "from_id": 6, "to_id": 10, "distance": 180 }, { "from_id": 6, "to_id": 11, "distance": 213 }, { "from_id": 6, "to_id": 12, "distance": 205 }, { "from_id": 6, "to_id": 13, "distance": 202 }, { "from_id": 6, "to_id": 14, "distance": 224 }, { "from_id": 6, "to_id": 15, "distance": 168 }, { "from_id": 6, "to_id": 16, "distance": 188 }, { "from_id": 6, "to_id": 17, "distance": 165 }, { "from_id": 6, "to_id": 18, "distance": 153 }, { "from_id": 6, "to_id": 19, "distance": 163 }, { "from_id": 7, "to_id": 1, "distance": 189 }, { "from_id": 7, "to_id": 2, "distance": 214 }, { "from_id": 7, "to_id": 3, "distance": 214 }, { "from_id": 7, "to_id": 4, "distance": 186 }, { "from_id": 7, "to_id": 5, "distance": 226 }, { "from_id": 7, "to_id": 6, "distance": 200 }, { "from_id": 7, "to_id": 8, "distance": 165 }, { "from_id": 7, "to_id": 9, "distance": 169 }, { "from_id": 7, "to_id": 10, "distance": 137 }, { "from_id": 7, "to_id": 11, "distance": 240 }, { "from_id": 7, "to_id": 12, "distance": 212 }, { "from_id": 7, "to_id": 13, "distance": 208 }, { "from_id": 7, "to_id": 14, "distance": 221 }, { "from_id": 7, "to_id": 15, "distance": 225 }, { "from_id": 7, "to_id": 16, "distance": 201 }, { "from_id": 7, "to_id": 17, "distance": 199 }, { "from_id": 7, "to_id": 18, "distance": 202 }, { "from_id": 7, "to_id": 19, "distance": 228 }, { "from_id": 8, "to_id": 1, "distance": 167 }, { "from_id": 8, "to_id": 2, "distance": 228 }, { "from_id": 8, "to_id": 3, "distance": 245 }, { "from_id": 8, "to_id": 4, "distance": 182 }, { "from_id": 8, "to_id": 5, "distance": 205 }, { "from_id": 8, "to_id": 6, "distance": 163 }, { "from_id": 8, "to_id": 7, "distance": 165 }, { "from_id": 8, "to_id": 9, "distance": 177 }, { "from_id": 8, "to_id": 10, "distance": 133 }, { "from_id": 8, "to_id": 11, "distance": 270 }, { "from_id": 8, "to_id": 12, "distance": 183 }, { "from_id": 8, "to_id": 13, "distance": 210 }, { "from_id": 8, "to_id": 14, "distance": 213 }, { "from_id": 8, "to_id": 15, "distance": 184 }, { "from_id": 8, "to_id": 16, "distance": 199 }, { "from_id": 8, "to_id": 17, "distance": 196 }, { "from_id": 8, "to_id": 18, "distance": 136 }, { "from_id": 8, "to_id": 19, "distance": 178 }, { "from_id": 9, "to_id": 1, "distance": 174 }, { "from_id": 9, "to_id": 2, "distance": 193 }, { "from_id": 9, "to_id": 3, "distance": 194 }, { "from_id": 9, "to_id": 4, "distance": 191 }, { "from_id": 9, "to_id": 5, "distance": 173 }, { "from_id": 9, "to_id": 6, "distance": 168 }, { "from_id": 9, "to_id": 7, "distance": 169 }, { "from_id": 9, "to_id": 8, "distance": 177 }, { "from_id": 9, "to_id": 10, "distance": 152 }, { "from_id": 9, "to_id": 11, "distance": 200 }, { "from_id": 9, "to_id": 12, "distance": 190 }, { "from_id": 9, "to_id": 13, "distance": 180 }, { "from_id": 9, "to_id": 14, 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"distance": 170 } ] }, "solution_variant": [ 2, 7, 8, 11, 12, 14, 16, 19 ], "context_index": 27, "input_format": "csv", "input_index_base": 1 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Someone in the flat is playing curator, choosing a small selection of the shared things to keep for the next place. The ideal mini-collection has variety — pieces that vary in use and style so the group doesn’t feel repetitive. To see which mini-collection wins, tally how different each pair of chosen items is and add those numbers up; the choice with the biggest sum is the most diverse. 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\"O\",\n \"item_id_b\": \"M\",\n \"pair_dissimilarity_score\": 2\n },\n {\n \"item_id_a\": \"O\",\n \"item_id_b\": \"N\",\n \"pair_dissimilarity_score\": 4\n },\n {\n \"item_id_a\": \"O\",\n \"item_id_b\": \"P\",\n \"pair_dissimilarity_score\": 7\n },\n {\n \"item_id_a\": \"O\",\n \"item_id_b\": \"Q\",\n \"pair_dissimilarity_score\": 5\n },\n {\n \"item_id_a\": \"O\",\n \"item_id_b\": \"R\",\n \"pair_dissimilarity_score\": 9\n },\n {\n \"item_id_a\": \"O\",\n \"item_id_b\": \"S\",\n \"pair_dissimilarity_score\": 9\n },\n {\n \"item_id_a\": \"O\",\n \"item_id_b\": \"T\",\n \"pair_dissimilarity_score\": 0\n },\n {\n \"item_id_a\": \"O\",\n \"item_id_b\": \"U\",\n \"pair_dissimilarity_score\": 9\n },\n {\n \"item_id_a\": \"P\",\n \"item_id_b\": \"A\",\n \"pair_dissimilarity_score\": 8\n },\n {\n \"item_id_a\": \"P\",\n \"item_id_b\": \"B\",\n \"pair_dissimilarity_score\": 3\n },\n {\n \"item_id_a\": \"P\",\n \"item_id_b\": \"C\",\n \"pair_dissimilarity_score\": 2\n },\n {\n \"item_id_a\": 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\"P\",\n \"item_id_b\": \"O\",\n \"pair_dissimilarity_score\": 7\n },\n {\n \"item_id_a\": \"P\",\n \"item_id_b\": \"Q\",\n \"pair_dissimilarity_score\": 0\n },\n {\n \"item_id_a\": \"P\",\n \"item_id_b\": \"R\",\n \"pair_dissimilarity_score\": 1\n },\n {\n \"item_id_a\": \"P\",\n \"item_id_b\": \"S\",\n \"pair_dissimilarity_score\": 0\n },\n {\n \"item_id_a\": \"P\",\n \"item_id_b\": \"T\",\n \"pair_dissimilarity_score\": 8\n },\n {\n \"item_id_a\": \"P\",\n \"item_id_b\": \"U\",\n \"pair_dissimilarity_score\": 7\n },\n {\n \"item_id_a\": \"Q\",\n \"item_id_b\": \"A\",\n \"pair_dissimilarity_score\": 4\n },\n {\n \"item_id_a\": \"Q\",\n \"item_id_b\": \"B\",\n \"pair_dissimilarity_score\": 4\n },\n {\n \"item_id_a\": \"Q\",\n \"item_id_b\": \"C\",\n \"pair_dissimilarity_score\": 4\n },\n {\n \"item_id_a\": \"Q\",\n \"item_id_b\": \"D\",\n \"pair_dissimilarity_score\": 5\n },\n {\n \"item_id_a\": \"Q\",\n \"item_id_b\": \"E\",\n \"pair_dissimilarity_score\": 9\n },\n {\n \"item_id_a\": \"Q\",\n \"item_id_b\": \"F\",\n \"pair_dissimilarity_score\": 6\n },\n {\n \"item_id_a\": \"Q\",\n \"item_id_b\": \"G\",\n \"pair_dissimilarity_score\": 0\n },\n {\n \"item_id_a\": \"Q\",\n \"item_id_b\": \"H\",\n \"pair_dissimilarity_score\": 0\n },\n {\n \"item_id_a\": \"Q\",\n \"item_id_b\": \"I\",\n \"pair_dissimilarity_score\": 9\n },\n {\n \"item_id_a\": \"Q\",\n \"item_id_b\": \"J\",\n \"pair_dissimilarity_score\": 0\n },\n {\n \"item_id_a\": \"Q\",\n \"item_id_b\": \"K\",\n \"pair_dissimilarity_score\": 3\n },\n {\n \"item_id_a\": \"Q\",\n \"item_id_b\": \"L\",\n \"pair_dissimilarity_score\": 3\n },\n {\n \"item_id_a\": \"Q\",\n \"item_id_b\": \"M\",\n \"pair_dissimilarity_score\": 6\n },\n {\n \"item_id_a\": \"Q\",\n \"item_id_b\": \"N\",\n \"pair_dissimilarity_score\": 3\n },\n {\n \"item_id_a\": \"Q\",\n \"item_id_b\": \"O\",\n \"pair_dissimilarity_score\": 5\n },\n {\n \"item_id_a\": \"Q\",\n \"item_id_b\": \"P\",\n \"pair_dissimilarity_score\": 0\n },\n {\n \"item_id_a\": 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\"R\",\n \"item_id_b\": \"H\",\n \"pair_dissimilarity_score\": 2\n },\n {\n \"item_id_a\": \"R\",\n \"item_id_b\": \"I\",\n \"pair_dissimilarity_score\": 2\n },\n {\n \"item_id_a\": \"R\",\n \"item_id_b\": \"J\",\n \"pair_dissimilarity_score\": 5\n },\n {\n \"item_id_a\": \"R\",\n \"item_id_b\": \"K\",\n \"pair_dissimilarity_score\": 2\n },\n {\n \"item_id_a\": \"R\",\n \"item_id_b\": \"L\",\n \"pair_dissimilarity_score\": 7\n },\n {\n \"item_id_a\": \"R\",\n \"item_id_b\": \"M\",\n \"pair_dissimilarity_score\": 6\n },\n {\n \"item_id_a\": \"R\",\n \"item_id_b\": \"N\",\n \"pair_dissimilarity_score\": 2\n },\n {\n \"item_id_a\": \"R\",\n \"item_id_b\": \"O\",\n \"pair_dissimilarity_score\": 9\n },\n {\n \"item_id_a\": \"R\",\n \"item_id_b\": \"P\",\n \"pair_dissimilarity_score\": 1\n },\n {\n \"item_id_a\": \"R\",\n \"item_id_b\": \"Q\",\n \"pair_dissimilarity_score\": 2\n },\n {\n \"item_id_a\": \"R\",\n \"item_id_b\": \"S\",\n \"pair_dissimilarity_score\": 1\n },\n {\n \"item_id_a\": 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\"S\",\n \"item_id_b\": \"J\",\n \"pair_dissimilarity_score\": 4\n },\n {\n \"item_id_a\": \"S\",\n \"item_id_b\": \"K\",\n \"pair_dissimilarity_score\": 3\n },\n {\n \"item_id_a\": \"S\",\n \"item_id_b\": \"L\",\n \"pair_dissimilarity_score\": 1\n },\n {\n \"item_id_a\": \"S\",\n \"item_id_b\": \"M\",\n \"pair_dissimilarity_score\": 5\n },\n {\n \"item_id_a\": \"S\",\n \"item_id_b\": \"N\",\n \"pair_dissimilarity_score\": 9\n },\n {\n \"item_id_a\": \"S\",\n \"item_id_b\": \"O\",\n \"pair_dissimilarity_score\": 9\n },\n {\n \"item_id_a\": \"S\",\n \"item_id_b\": \"P\",\n \"pair_dissimilarity_score\": 0\n },\n {\n \"item_id_a\": \"S\",\n \"item_id_b\": \"Q\",\n \"pair_dissimilarity_score\": 1\n },\n {\n \"item_id_a\": \"S\",\n \"item_id_b\": \"R\",\n \"pair_dissimilarity_score\": 1\n },\n {\n \"item_id_a\": \"S\",\n \"item_id_b\": \"T\",\n \"pair_dissimilarity_score\": 4\n },\n {\n \"item_id_a\": \"S\",\n \"item_id_b\": \"U\",\n \"pair_dissimilarity_score\": 3\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"A\",\n \"pair_dissimilarity_score\": 8\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"B\",\n \"pair_dissimilarity_score\": 6\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"C\",\n \"pair_dissimilarity_score\": 2\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"D\",\n \"pair_dissimilarity_score\": 3\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"E\",\n \"pair_dissimilarity_score\": 7\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"F\",\n \"pair_dissimilarity_score\": 7\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"G\",\n \"pair_dissimilarity_score\": 7\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"H\",\n \"pair_dissimilarity_score\": 4\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"I\",\n \"pair_dissimilarity_score\": 1\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"J\",\n \"pair_dissimilarity_score\": 9\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"K\",\n \"pair_dissimilarity_score\": 2\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"L\",\n \"pair_dissimilarity_score\": 2\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"M\",\n \"pair_dissimilarity_score\": 8\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"N\",\n \"pair_dissimilarity_score\": 6\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"O\",\n \"pair_dissimilarity_score\": 0\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"P\",\n \"pair_dissimilarity_score\": 8\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"Q\",\n \"pair_dissimilarity_score\": 0\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"R\",\n \"pair_dissimilarity_score\": 4\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"S\",\n \"pair_dissimilarity_score\": 4\n },\n {\n \"item_id_a\": \"T\",\n \"item_id_b\": \"U\",\n \"pair_dissimilarity_score\": 3\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"A\",\n \"pair_dissimilarity_score\": 0\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"B\",\n \"pair_dissimilarity_score\": 2\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"C\",\n \"pair_dissimilarity_score\": 7\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"D\",\n \"pair_dissimilarity_score\": 1\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"E\",\n \"pair_dissimilarity_score\": 6\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"F\",\n \"pair_dissimilarity_score\": 0\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"G\",\n \"pair_dissimilarity_score\": 8\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"H\",\n \"pair_dissimilarity_score\": 0\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"I\",\n \"pair_dissimilarity_score\": 1\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"J\",\n \"pair_dissimilarity_score\": 8\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"K\",\n \"pair_dissimilarity_score\": 2\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"L\",\n \"pair_dissimilarity_score\": 5\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"M\",\n \"pair_dissimilarity_score\": 9\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"N\",\n \"pair_dissimilarity_score\": 6\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"O\",\n \"pair_dissimilarity_score\": 9\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"P\",\n \"pair_dissimilarity_score\": 7\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"Q\",\n \"pair_dissimilarity_score\": 7\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"R\",\n \"pair_dissimilarity_score\": 3\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"S\",\n \"pair_dissimilarity_score\": 3\n },\n {\n \"item_id_a\": \"U\",\n \"item_id_b\": \"T\",\n \"pair_dissimilarity_score\": 3\n }\n ]\n}\n\nWhen you tell me which pieces make the cut, just drop them into a tiny JSON object like this:\n\n{\n \"solution\": [, , ...]\n}\n\nThe \"solution\" array is where you list the exact items you want to keep — think of it as the mini-collection in one neat line. This is just a sketch of the shape I expect, not the actual answer itself.\n\nPlease make sure you use the item identifiers exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 8, 9, 2, 0, 8, 1, 7, 4, 3, 8, 7, 8, 2, 8, 8, 4, 4, 6, 8, 0 ], [ 8, 0, 8, 2, 4, 1, 5, 2, 3, 6, 1, 3, 1, 9, 7, 3, 4, 5, 8, 6, 2 ], [ 9, 8, 0, 6, 1, 9, 4, 5, 5, 4, 8, 7, 6, 5, 2, 2, 4, 3, 5, 2, 7 ], [ 2, 2, 6, 0, 9, 7, 8, 3, 3, 1, 8, 8, 2, 8, 0, 8, 5, 8, 6, 3, 1 ], [ 0, 4, 1, 9, 0, 1, 5, 1, 2, 3, 0, 2, 8, 9, 8, 8, 9, 1, 8, 7, 6 ], [ 8, 1, 9, 7, 1, 0, 0, 6, 9, 1, 2, 3, 8, 9, 4, 6, 6, 6, 8, 7, 0 ], [ 1, 5, 4, 8, 5, 0, 0, 5, 5, 5, 4, 5, 9, 0, 8, 4, 0, 1, 8, 7, 8 ], [ 7, 2, 5, 3, 1, 6, 5, 0, 8, 0, 9, 8, 4, 3, 7, 3, 0, 2, 1, 4, 0 ], [ 4, 3, 5, 3, 2, 9, 5, 8, 0, 8, 2, 0, 7, 3, 6, 4, 9, 2, 9, 1, 1 ], [ 3, 6, 4, 1, 3, 1, 5, 0, 8, 0, 6, 2, 7, 2, 6, 3, 0, 5, 4, 9, 8 ], [ 8, 1, 8, 8, 0, 2, 4, 9, 2, 6, 0, 5, 7, 0, 8, 0, 3, 2, 3, 2, 2 ], [ 7, 3, 7, 8, 2, 3, 5, 8, 0, 2, 5, 0, 9, 7, 3, 3, 3, 7, 1, 2, 5 ], [ 8, 1, 6, 2, 8, 8, 9, 4, 7, 7, 7, 9, 0, 9, 2, 1, 6, 6, 5, 8, 9 ], [ 2, 9, 5, 8, 9, 9, 0, 3, 3, 2, 0, 7, 9, 0, 4, 8, 3, 2, 9, 6, 6 ], [ 8, 7, 2, 0, 8, 4, 8, 7, 6, 6, 8, 3, 2, 4, 0, 7, 5, 9, 9, 0, 9 ], [ 8, 3, 2, 8, 8, 6, 4, 3, 4, 3, 0, 3, 1, 8, 7, 0, 0, 1, 0, 8, 7 ], [ 4, 4, 4, 5, 9, 6, 0, 0, 9, 0, 3, 3, 6, 3, 5, 0, 0, 2, 1, 0, 7 ], [ 4, 5, 3, 8, 1, 6, 1, 2, 2, 5, 2, 7, 6, 2, 9, 1, 2, 0, 1, 4, 3 ], [ 6, 8, 5, 6, 8, 8, 8, 1, 9, 4, 3, 1, 5, 9, 9, 0, 1, 1, 0, 4, 3 ], [ 8, 6, 2, 3, 7, 7, 7, 4, 1, 9, 2, 2, 8, 6, 0, 8, 0, 4, 4, 0, 3 ], [ 0, 2, 7, 1, 6, 0, 8, 0, 1, 8, 2, 5, 9, 6, 9, 7, 7, 3, 3, 3, 0 ] ], "m": 2, "objective": 9.0 }, "solution": [ 14, 17 ], "obj": 9.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 21, "m": 2, "nodes": [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S", "T", "U" ], "distance_pairs": [ { "from_id": "A", "to_id": "B", "distance": 8 }, { "from_id": "A", "to_id": "C", "distance": 9 }, { "from_id": "A", "to_id": "D", "distance": 2 }, { "from_id": "A", "to_id": "E", "distance": 0 }, { "from_id": "A", "to_id": "F", "distance": 8 }, { "from_id": "A", "to_id": "G", "distance": 1 }, { "from_id": "A", "to_id": "H", "distance": 7 }, { "from_id": "A", "to_id": "I", "distance": 4 }, { "from_id": "A", "to_id": "J", "distance": 3 }, { "from_id": "A", "to_id": "K", "distance": 8 }, { "from_id": "A", "to_id": "L", "distance": 7 }, { "from_id": "A", "to_id": "M", "distance": 8 }, { "from_id": "A", "to_id": "N", "distance": 2 }, { "from_id": "A", "to_id": "O", "distance": 8 }, { "from_id": "A", "to_id": "P", "distance": 8 }, { "from_id": "A", "to_id": "Q", "distance": 4 }, { "from_id": "A", "to_id": "R", "distance": 4 }, { "from_id": "A", "to_id": "S", "distance": 6 }, { "from_id": "A", "to_id": "T", "distance": 8 }, { "from_id": "A", "to_id": "U", "distance": 0 }, { "from_id": "B", "to_id": "A", "distance": 8 }, { "from_id": "B", "to_id": "C", "distance": 8 }, { "from_id": "B", "to_id": "D", "distance": 2 }, { "from_id": "B", "to_id": "E", "distance": 4 }, { "from_id": "B", "to_id": "F", "distance": 1 }, { "from_id": "B", "to_id": "G", "distance": 5 }, { "from_id": "B", "to_id": "H", "distance": 2 }, { "from_id": "B", "to_id": "I", "distance": 3 }, { "from_id": "B", "to_id": "J", "distance": 6 }, { "from_id": "B", "to_id": "K", "distance": 1 }, { "from_id": "B", "to_id": "L", "distance": 3 }, { "from_id": "B", "to_id": "M", "distance": 1 }, { "from_id": "B", "to_id": "N", "distance": 9 }, { "from_id": "B", "to_id": "O", "distance": 7 }, { "from_id": "B", "to_id": "P", "distance": 3 }, { "from_id": "B", "to_id": "Q", "distance": 4 }, { "from_id": "B", "to_id": "R", "distance": 5 }, { "from_id": "B", "to_id": "S", "distance": 8 }, { "from_id": "B", "to_id": "T", "distance": 6 }, { "from_id": "B", "to_id": "U", "distance": 2 }, { "from_id": "C", "to_id": "A", "distance": 9 }, { "from_id": "C", "to_id": "B", "distance": 8 }, { "from_id": "C", "to_id": "D", "distance": 6 }, { "from_id": "C", "to_id": "E", "distance": 1 }, { "from_id": "C", "to_id": "F", "distance": 9 }, { "from_id": "C", "to_id": "G", "distance": 4 }, { "from_id": "C", "to_id": "H", "distance": 5 }, { "from_id": "C", "to_id": "I", "distance": 5 }, { "from_id": "C", "to_id": "J", "distance": 4 }, { "from_id": "C", "to_id": "K", "distance": 8 }, { "from_id": "C", "to_id": "L", "distance": 7 }, { "from_id": "C", "to_id": "M", "distance": 6 }, { "from_id": "C", "to_id": "N", "distance": 5 }, { "from_id": "C", "to_id": "O", "distance": 2 }, { "from_id": "C", "to_id": "P", "distance": 2 }, { "from_id": "C", "to_id": "Q", "distance": 4 }, { "from_id": "C", "to_id": "R", "distance": 3 }, { "from_id": "C", "to_id": "S", "distance": 5 }, { "from_id": "C", "to_id": "T", "distance": 2 }, { "from_id": "C", "to_id": "U", "distance": 7 }, { "from_id": "D", "to_id": "A", "distance": 2 }, { "from_id": "D", "to_id": "B", "distance": 2 }, { "from_id": "D", "to_id": "C", "distance": 6 }, { "from_id": "D", "to_id": "E", "distance": 9 }, { "from_id": "D", "to_id": "F", "distance": 7 }, { "from_id": "D", "to_id": "G", "distance": 8 }, { "from_id": "D", "to_id": "H", "distance": 3 }, { "from_id": "D", "to_id": "I", "distance": 3 }, { "from_id": "D", "to_id": "J", "distance": 1 }, { "from_id": "D", "to_id": "K", "distance": 8 }, { "from_id": "D", "to_id": "L", "distance": 8 }, { "from_id": "D", "to_id": "M", "distance": 2 }, { "from_id": "D", "to_id": "N", "distance": 8 }, { "from_id": "D", "to_id": "O", "distance": 0 }, { "from_id": "D", "to_id": "P", "distance": 8 }, { "from_id": "D", "to_id": "Q", "distance": 5 }, { "from_id": "D", "to_id": "R", "distance": 8 }, { "from_id": "D", "to_id": "S", "distance": 6 }, { "from_id": "D", "to_id": "T", "distance": 3 }, { "from_id": "D", "to_id": "U", "distance": 1 }, { "from_id": "E", "to_id": "A", "distance": 0 }, { "from_id": "E", "to_id": "B", "distance": 4 }, { "from_id": "E", "to_id": "C", "distance": 1 }, { "from_id": "E", "to_id": "D", "distance": 9 }, { "from_id": "E", "to_id": "F", "distance": 1 }, { "from_id": "E", "to_id": "G", "distance": 5 }, { "from_id": "E", "to_id": "H", "distance": 1 }, { "from_id": "E", "to_id": "I", "distance": 2 }, { "from_id": "E", "to_id": "J", "distance": 3 }, { "from_id": "E", "to_id": "K", "distance": 0 }, { "from_id": "E", "to_id": "L", "distance": 2 }, { "from_id": "E", "to_id": "M", "distance": 8 }, { "from_id": "E", "to_id": "N", "distance": 9 }, { "from_id": "E", "to_id": "O", "distance": 8 }, { "from_id": "E", "to_id": "P", "distance": 8 }, { "from_id": "E", "to_id": "Q", "distance": 9 }, { "from_id": "E", "to_id": "R", "distance": 1 }, { "from_id": "E", "to_id": "S", "distance": 8 }, { "from_id": "E", "to_id": "T", "distance": 7 }, { "from_id": "E", "to_id": "U", "distance": 6 }, { "from_id": "F", "to_id": "A", "distance": 8 }, { "from_id": "F", "to_id": "B", "distance": 1 }, { "from_id": "F", "to_id": "C", "distance": 9 }, { "from_id": "F", "to_id": "D", "distance": 7 }, { "from_id": "F", "to_id": "E", "distance": 1 }, { "from_id": "F", "to_id": "G", "distance": 0 }, { "from_id": "F", "to_id": "H", "distance": 6 }, { "from_id": "F", "to_id": "I", "distance": 9 }, { "from_id": "F", "to_id": "J", "distance": 1 }, { "from_id": "F", "to_id": "K", "distance": 2 }, { "from_id": "F", "to_id": "L", "distance": 3 }, { "from_id": "F", "to_id": "M", "distance": 8 }, { "from_id": "F", "to_id": "N", "distance": 9 }, { "from_id": "F", "to_id": "O", "distance": 4 }, { "from_id": "F", "to_id": "P", "distance": 6 }, { "from_id": "F", "to_id": "Q", "distance": 6 }, { "from_id": "F", "to_id": "R", "distance": 6 }, { "from_id": "F", "to_id": "S", "distance": 8 }, { "from_id": "F", "to_id": "T", "distance": 7 }, { "from_id": "F", "to_id": "U", "distance": 0 }, { "from_id": "G", "to_id": "A", "distance": 1 }, { "from_id": "G", "to_id": "B", "distance": 5 }, { "from_id": "G", "to_id": "C", "distance": 4 }, { "from_id": "G", "to_id": "D", "distance": 8 }, { "from_id": "G", "to_id": "E", "distance": 5 }, { "from_id": "G", "to_id": "F", "distance": 0 }, { "from_id": "G", "to_id": "H", "distance": 5 }, { "from_id": "G", "to_id": "I", "distance": 5 }, { "from_id": "G", "to_id": "J", "distance": 5 }, { "from_id": "G", "to_id": "K", "distance": 4 }, { "from_id": "G", "to_id": "L", "distance": 5 }, { "from_id": "G", "to_id": "M", "distance": 9 }, { "from_id": "G", "to_id": "N", "distance": 0 }, { "from_id": "G", "to_id": "O", "distance": 8 }, { "from_id": "G", "to_id": "P", "distance": 4 }, { "from_id": "G", "to_id": "Q", "distance": 0 }, { "from_id": "G", "to_id": "R", "distance": 1 }, { "from_id": "G", "to_id": "S", "distance": 8 }, { "from_id": "G", "to_id": "T", "distance": 7 }, { "from_id": "G", "to_id": "U", "distance": 8 }, { "from_id": "H", "to_id": "A", "distance": 7 }, { "from_id": "H", "to_id": "B", "distance": 2 }, { "from_id": "H", "to_id": "C", "distance": 5 }, { "from_id": "H", "to_id": "D", "distance": 3 }, { "from_id": "H", "to_id": "E", "distance": 1 }, { "from_id": "H", "to_id": "F", "distance": 6 }, { "from_id": "H", "to_id": "G", "distance": 5 }, { "from_id": "H", "to_id": "I", "distance": 8 }, { "from_id": "H", "to_id": "J", "distance": 0 }, { "from_id": "H", "to_id": "K", "distance": 9 }, { "from_id": "H", "to_id": "L", "distance": 8 }, { "from_id": "H", "to_id": "M", "distance": 4 }, { "from_id": "H", "to_id": "N", "distance": 3 }, { "from_id": "H", "to_id": "O", "distance": 7 }, { "from_id": "H", "to_id": "P", "distance": 3 }, { "from_id": "H", "to_id": "Q", "distance": 0 }, { "from_id": "H", "to_id": "R", "distance": 2 }, { "from_id": "H", "to_id": "S", "distance": 1 }, { "from_id": "H", "to_id": "T", "distance": 4 }, { "from_id": "H", "to_id": "U", "distance": 0 }, { "from_id": "I", "to_id": "A", "distance": 4 }, { "from_id": "I", "to_id": "B", "distance": 3 }, { "from_id": "I", "to_id": "C", "distance": 5 }, { "from_id": "I", "to_id": "D", "distance": 3 }, { "from_id": "I", "to_id": "E", "distance": 2 }, { "from_id": "I", "to_id": "F", "distance": 9 }, { "from_id": "I", "to_id": "G", "distance": 5 }, { "from_id": "I", "to_id": "H", "distance": 8 }, { "from_id": "I", "to_id": "J", "distance": 8 }, { "from_id": "I", "to_id": "K", "distance": 2 }, { "from_id": "I", "to_id": "L", "distance": 0 }, { "from_id": "I", "to_id": "M", "distance": 7 }, { "from_id": "I", "to_id": "N", "distance": 3 }, { "from_id": "I", "to_id": "O", "distance": 6 }, { "from_id": "I", "to_id": "P", "distance": 4 }, { "from_id": "I", "to_id": "Q", "distance": 9 }, { "from_id": "I", "to_id": "R", "distance": 2 }, { "from_id": "I", "to_id": "S", "distance": 9 }, { "from_id": "I", "to_id": "T", "distance": 1 }, { "from_id": "I", "to_id": "U", "distance": 1 }, { "from_id": "J", "to_id": "A", "distance": 3 }, { "from_id": "J", "to_id": "B", "distance": 6 }, { "from_id": "J", "to_id": "C", "distance": 4 }, { "from_id": "J", "to_id": "D", "distance": 1 }, { "from_id": "J", "to_id": "E", "distance": 3 }, { "from_id": "J", "to_id": "F", "distance": 1 }, { "from_id": "J", "to_id": "G", "distance": 5 }, { "from_id": "J", "to_id": "H", "distance": 0 }, { "from_id": "J", "to_id": "I", "distance": 8 }, { "from_id": "J", "to_id": "K", "distance": 6 }, { "from_id": "J", "to_id": "L", "distance": 2 }, { "from_id": "J", "to_id": "M", "distance": 7 }, { "from_id": "J", "to_id": "N", "distance": 2 }, { "from_id": "J", "to_id": "O", "distance": 6 }, { "from_id": "J", "to_id": "P", "distance": 3 }, { "from_id": "J", "to_id": "Q", "distance": 0 }, { "from_id": "J", "to_id": "R", "distance": 5 }, { "from_id": "J", "to_id": "S", "distance": 4 }, { "from_id": "J", "to_id": "T", "distance": 9 }, { "from_id": "J", "to_id": "U", "distance": 8 }, { "from_id": "K", "to_id": "A", "distance": 8 }, { "from_id": "K", "to_id": "B", "distance": 1 }, { "from_id": "K", "to_id": "C", "distance": 8 }, { "from_id": "K", "to_id": "D", "distance": 8 }, { "from_id": "K", "to_id": "E", "distance": 0 }, { "from_id": "K", "to_id": "F", "distance": 2 }, { "from_id": "K", "to_id": "G", "distance": 4 }, { "from_id": "K", "to_id": "H", "distance": 9 }, { "from_id": "K", "to_id": "I", "distance": 2 }, { "from_id": "K", "to_id": "J", "distance": 6 }, { "from_id": "K", "to_id": "L", "distance": 5 }, { "from_id": "K", "to_id": "M", "distance": 7 }, { "from_id": "K", "to_id": "N", "distance": 0 }, { "from_id": "K", "to_id": "O", "distance": 8 }, { "from_id": "K", "to_id": "P", "distance": 0 }, { "from_id": "K", "to_id": "Q", "distance": 3 }, { "from_id": "K", "to_id": "R", "distance": 2 }, { "from_id": "K", "to_id": "S", "distance": 3 }, { "from_id": "K", "to_id": "T", "distance": 2 }, { "from_id": "K", "to_id": "U", "distance": 2 }, { "from_id": "L", "to_id": "A", "distance": 7 }, { "from_id": "L", "to_id": "B", "distance": 3 }, { "from_id": "L", "to_id": "C", "distance": 7 }, { "from_id": "L", "to_id": "D", "distance": 8 }, { "from_id": "L", "to_id": "E", "distance": 2 }, { "from_id": "L", "to_id": "F", "distance": 3 }, { "from_id": "L", "to_id": "G", "distance": 5 }, { "from_id": "L", "to_id": "H", "distance": 8 }, { "from_id": "L", "to_id": "I", "distance": 0 }, { "from_id": "L", "to_id": "J", "distance": 2 }, { "from_id": "L", "to_id": "K", "distance": 5 }, { "from_id": "L", "to_id": "M", "distance": 9 }, { "from_id": "L", "to_id": "N", "distance": 7 }, { "from_id": "L", "to_id": "O", "distance": 3 }, { "from_id": "L", "to_id": "P", "distance": 3 }, { "from_id": "L", "to_id": "Q", "distance": 3 }, { "from_id": "L", "to_id": "R", "distance": 7 }, { "from_id": "L", "to_id": "S", "distance": 1 }, { "from_id": "L", "to_id": "T", "distance": 2 }, { "from_id": "L", "to_id": "U", "distance": 5 }, { "from_id": "M", "to_id": "A", "distance": 8 }, { "from_id": "M", "to_id": "B", "distance": 1 }, { "from_id": "M", "to_id": "C", "distance": 6 }, { "from_id": "M", "to_id": "D", "distance": 2 }, { "from_id": "M", "to_id": "E", "distance": 8 }, { "from_id": "M", "to_id": "F", "distance": 8 }, { "from_id": "M", "to_id": "G", "distance": 9 }, { "from_id": "M", "to_id": "H", "distance": 4 }, { "from_id": "M", "to_id": "I", "distance": 7 }, { "from_id": "M", "to_id": "J", "distance": 7 }, { "from_id": "M", "to_id": "K", "distance": 7 }, { "from_id": "M", "to_id": "L", "distance": 9 }, { "from_id": "M", "to_id": "N", "distance": 9 }, { "from_id": "M", "to_id": "O", "distance": 2 }, { "from_id": "M", "to_id": "P", "distance": 1 }, { "from_id": "M", "to_id": "Q", "distance": 6 }, { "from_id": "M", "to_id": "R", "distance": 6 }, { "from_id": "M", "to_id": "S", "distance": 5 }, { "from_id": "M", "to_id": "T", "distance": 8 }, { "from_id": "M", "to_id": "U", "distance": 9 }, { "from_id": "N", "to_id": "A", "distance": 2 }, { "from_id": "N", "to_id": "B", "distance": 9 }, { "from_id": "N", "to_id": "C", "distance": 5 }, { "from_id": "N", "to_id": "D", "distance": 8 }, { "from_id": "N", "to_id": "E", "distance": 9 }, { "from_id": "N", "to_id": "F", "distance": 9 }, { "from_id": "N", "to_id": "G", "distance": 0 }, { "from_id": "N", "to_id": "H", "distance": 3 }, { "from_id": "N", "to_id": "I", "distance": 3 }, { "from_id": "N", "to_id": "J", "distance": 2 }, { "from_id": "N", "to_id": "K", "distance": 0 }, { "from_id": "N", "to_id": "L", "distance": 7 }, { "from_id": "N", "to_id": "M", "distance": 9 }, { "from_id": "N", "to_id": "O", "distance": 4 }, { "from_id": "N", "to_id": "P", "distance": 8 }, { "from_id": "N", "to_id": "Q", "distance": 3 }, { "from_id": "N", "to_id": "R", "distance": 2 }, { "from_id": "N", "to_id": "S", "distance": 9 }, { "from_id": "N", "to_id": "T", "distance": 6 }, { "from_id": "N", "to_id": "U", "distance": 6 }, { "from_id": "O", "to_id": "A", "distance": 8 }, { "from_id": "O", "to_id": "B", "distance": 7 }, { "from_id": "O", "to_id": "C", "distance": 2 }, { "from_id": "O", "to_id": "D", "distance": 0 }, { "from_id": "O", "to_id": "E", "distance": 8 }, { "from_id": "O", "to_id": "F", "distance": 4 }, { "from_id": "O", "to_id": "G", "distance": 8 }, { "from_id": "O", "to_id": "H", "distance": 7 }, { "from_id": "O", "to_id": "I", "distance": 6 }, { "from_id": "O", "to_id": "J", "distance": 6 }, { "from_id": "O", "to_id": "K", "distance": 8 }, { "from_id": "O", "to_id": "L", "distance": 3 }, { "from_id": "O", "to_id": "M", "distance": 2 }, { "from_id": "O", "to_id": "N", "distance": 4 }, { "from_id": "O", "to_id": "P", "distance": 7 }, { "from_id": "O", "to_id": "Q", "distance": 5 }, { "from_id": "O", "to_id": "R", "distance": 9 }, { "from_id": "O", "to_id": "S", "distance": 9 }, { "from_id": "O", "to_id": "T", "distance": 0 }, { "from_id": "O", "to_id": "U", "distance": 9 }, { "from_id": "P", "to_id": "A", "distance": 8 }, { "from_id": "P", "to_id": "B", "distance": 3 }, { "from_id": "P", "to_id": "C", "distance": 2 }, { "from_id": "P", "to_id": "D", "distance": 8 }, { "from_id": "P", "to_id": "E", "distance": 8 }, { "from_id": "P", "to_id": "F", "distance": 6 }, { "from_id": "P", "to_id": "G", "distance": 4 }, { "from_id": "P", "to_id": "H", "distance": 3 }, { "from_id": "P", "to_id": "I", "distance": 4 }, { "from_id": "P", "to_id": "J", "distance": 3 }, { "from_id": "P", "to_id": "K", "distance": 0 }, { "from_id": "P", "to_id": "L", "distance": 3 }, { "from_id": "P", "to_id": "M", "distance": 1 }, { "from_id": "P", "to_id": "N", "distance": 8 }, { "from_id": "P", "to_id": "O", "distance": 7 }, { "from_id": "P", "to_id": "Q", "distance": 0 }, { "from_id": "P", "to_id": "R", "distance": 1 }, { "from_id": "P", "to_id": "S", "distance": 0 }, { "from_id": "P", "to_id": "T", "distance": 8 }, { "from_id": "P", "to_id": "U", "distance": 7 }, { "from_id": "Q", "to_id": "A", "distance": 4 }, { "from_id": "Q", "to_id": "B", "distance": 4 }, { "from_id": "Q", "to_id": "C", "distance": 4 }, { "from_id": "Q", "to_id": "D", "distance": 5 }, { "from_id": "Q", "to_id": "E", "distance": 9 }, { "from_id": "Q", "to_id": "F", "distance": 6 }, { "from_id": "Q", "to_id": "G", "distance": 0 }, { "from_id": "Q", "to_id": "H", "distance": 0 }, { "from_id": "Q", "to_id": "I", "distance": 9 }, { "from_id": "Q", "to_id": "J", "distance": 0 }, { "from_id": "Q", "to_id": "K", "distance": 3 }, { "from_id": "Q", "to_id": "L", "distance": 3 }, { "from_id": "Q", "to_id": "M", "distance": 6 }, { "from_id": "Q", "to_id": "N", "distance": 3 }, { "from_id": "Q", "to_id": "O", "distance": 5 }, { "from_id": "Q", "to_id": "P", "distance": 0 }, { "from_id": "Q", "to_id": "R", "distance": 2 }, { "from_id": "Q", "to_id": "S", "distance": 1 }, { "from_id": "Q", "to_id": "T", "distance": 0 }, { "from_id": "Q", "to_id": "U", "distance": 7 }, { "from_id": "R", "to_id": "A", "distance": 4 }, { "from_id": "R", "to_id": "B", "distance": 5 }, { "from_id": "R", "to_id": "C", "distance": 3 }, { "from_id": "R", "to_id": "D", "distance": 8 }, { "from_id": "R", "to_id": "E", "distance": 1 }, { "from_id": "R", "to_id": "F", "distance": 6 }, { "from_id": "R", "to_id": "G", "distance": 1 }, { "from_id": "R", "to_id": "H", "distance": 2 }, { "from_id": "R", "to_id": "I", "distance": 2 }, { "from_id": "R", "to_id": "J", "distance": 5 }, { "from_id": "R", "to_id": "K", "distance": 2 }, { "from_id": "R", "to_id": "L", "distance": 7 }, { "from_id": "R", "to_id": "M", "distance": 6 }, { "from_id": "R", "to_id": "N", "distance": 2 }, { "from_id": "R", "to_id": "O", "distance": 9 }, { "from_id": "R", "to_id": "P", "distance": 1 }, { "from_id": "R", "to_id": "Q", "distance": 2 }, { "from_id": "R", "to_id": "S", "distance": 1 }, { "from_id": "R", "to_id": "T", "distance": 4 }, { "from_id": "R", "to_id": "U", "distance": 3 }, { "from_id": "S", "to_id": "A", "distance": 6 }, { "from_id": "S", "to_id": "B", "distance": 8 }, { "from_id": "S", "to_id": "C", "distance": 5 }, { "from_id": "S", "to_id": "D", "distance": 6 }, { "from_id": "S", "to_id": "E", "distance": 8 }, { "from_id": "S", "to_id": "F", "distance": 8 }, { "from_id": "S", "to_id": "G", "distance": 8 }, { "from_id": "S", "to_id": "H", "distance": 1 }, { "from_id": "S", "to_id": "I", "distance": 9 }, { "from_id": "S", "to_id": "J", "distance": 4 }, { "from_id": "S", "to_id": "K", "distance": 3 }, { "from_id": "S", "to_id": "L", "distance": 1 }, { "from_id": "S", "to_id": "M", "distance": 5 }, { "from_id": "S", "to_id": "N", "distance": 9 }, { "from_id": "S", "to_id": "O", "distance": 9 }, { "from_id": "S", "to_id": "P", "distance": 0 }, { "from_id": "S", "to_id": "Q", "distance": 1 }, { "from_id": "S", "to_id": "R", "distance": 1 }, { "from_id": "S", "to_id": "T", "distance": 4 }, { "from_id": "S", "to_id": "U", "distance": 3 }, { "from_id": "T", "to_id": "A", "distance": 8 }, { "from_id": "T", "to_id": "B", "distance": 6 }, { "from_id": "T", "to_id": "C", "distance": 2 }, { "from_id": "T", "to_id": "D", "distance": 3 }, { "from_id": "T", "to_id": "E", "distance": 7 }, { "from_id": "T", "to_id": "F", "distance": 7 }, { "from_id": "T", "to_id": "G", "distance": 7 }, { "from_id": "T", "to_id": "H", "distance": 4 }, { "from_id": "T", "to_id": "I", "distance": 1 }, { "from_id": "T", "to_id": "J", "distance": 9 }, { "from_id": "T", "to_id": "K", "distance": 2 }, { "from_id": "T", "to_id": "L", "distance": 2 }, { "from_id": "T", "to_id": "M", "distance": 8 }, { "from_id": "T", "to_id": "N", "distance": 6 }, { "from_id": "T", "to_id": "O", "distance": 0 }, { "from_id": "T", "to_id": "P", "distance": 8 }, { "from_id": "T", "to_id": "Q", "distance": 0 }, { "from_id": "T", "to_id": "R", "distance": 4 }, { "from_id": "T", "to_id": "S", "distance": 4 }, { "from_id": "T", "to_id": "U", "distance": 3 }, { "from_id": "U", "to_id": "A", "distance": 0 }, { "from_id": "U", "to_id": "B", "distance": 2 }, { "from_id": "U", "to_id": "C", "distance": 7 }, { "from_id": "U", "to_id": "D", "distance": 1 }, { "from_id": "U", "to_id": "E", "distance": 6 }, { "from_id": "U", "to_id": "F", "distance": 0 }, { "from_id": "U", "to_id": "G", "distance": 8 }, { "from_id": "U", "to_id": "H", "distance": 0 }, { "from_id": "U", "to_id": "I", "distance": 1 }, { "from_id": "U", "to_id": "J", "distance": 8 }, { "from_id": "U", "to_id": "K", "distance": 2 }, { "from_id": "U", "to_id": "L", "distance": 5 }, { "from_id": "U", "to_id": "M", "distance": 9 }, { "from_id": "U", "to_id": "N", "distance": 6 }, { "from_id": "U", "to_id": "O", "distance": 9 }, { "from_id": "U", "to_id": "P", "distance": 7 }, { "from_id": "U", "to_id": "Q", "distance": 7 }, { "from_id": "U", "to_id": "R", "distance": 3 }, { "from_id": "U", "to_id": "S", "distance": 3 }, { "from_id": "U", "to_id": "T", "distance": 3 } ] }, "solution_variant": [ "O", "R" ], "context_index": 28, "input_format": "json", "input_index_base": "names" }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Someone in the lab is trying to put together a compact, varied panel of molecules: the assignment is to pick exactly the stated number of compounds and make sure the chosen ones are as unlike each other as possible. You judge any choice by summing the pairwise dissimilarities between all chosen molecules — the higher the sum, the better the spread. No repeats and no changing the required count. The exact compounds and distance values are shown below.\n\nYou must pick exactly 6 distinct compounds from the 19 available (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19).\nBetween 1 and 2, the pairwise dissimilarity is 191.\nBetween 1 and 3, the pairwise dissimilarity is 189.\nBetween 1 and 4, the pairwise dissimilarity is 199.\nBetween 1 and 5, the pairwise dissimilarity is 220.\nBetween 1 and 6, the pairwise dissimilarity is 226.\nBetween 1 and 7, the pairwise dissimilarity is 231.\nBetween 1 and 8, the pairwise dissimilarity is 155.\nBetween 1 and 9, the pairwise dissimilarity is 188.\nBetween 1 and 10, the pairwise dissimilarity is 186.\nBetween 1 and 11, the pairwise dissimilarity is 187.\nBetween 1 and 12, the pairwise dissimilarity is 193.\nBetween 1 and 13, the pairwise dissimilarity is 212.\nBetween 1 and 14, the pairwise dissimilarity is 256.\nBetween 1 and 15, the pairwise dissimilarity is 216.\nBetween 1 and 16, the pairwise dissimilarity is 192.\nBetween 1 and 17, the pairwise dissimilarity is 203.\nBetween 1 and 18, the pairwise dissimilarity is 189.\nBetween 1 and 19, the pairwise dissimilarity is 205.\nBetween 2 and 1, the pairwise dissimilarity is 191.\nBetween 2 and 3, the pairwise dissimilarity is 220.\nBetween 2 and 4, the pairwise dissimilarity is 203.\nBetween 2 and 5, the pairwise dissimilarity is 179.\nBetween 2 and 6, the pairwise dissimilarity is 217.\nBetween 2 and 7, the pairwise dissimilarity is 152.\nBetween 2 and 8, the pairwise dissimilarity is 141.\nBetween 2 and 9, the pairwise dissimilarity is 161.\nBetween 2 and 10, the pairwise dissimilarity is 154.\nBetween 2 and 11, the pairwise dissimilarity is 173.\nBetween 2 and 12, the pairwise dissimilarity is 178.\nBetween 2 and 13, the pairwise dissimilarity is 178.\nBetween 2 and 14, the pairwise dissimilarity is 225.\nBetween 2 and 15, the pairwise dissimilarity is 192.\nBetween 2 and 16, the pairwise dissimilarity is 197.\nBetween 2 and 17, the pairwise dissimilarity is 161.\nBetween 2 and 18, the pairwise dissimilarity is 188.\nBetween 2 and 19, the pairwise dissimilarity is 169.\nBetween 3 and 1, the pairwise dissimilarity is 189.\nBetween 3 and 2, the pairwise dissimilarity is 220.\nBetween 3 and 4, the pairwise dissimilarity is 200.\nBetween 3 and 5, the pairwise dissimilarity is 227.\nBetween 3 and 6, the pairwise dissimilarity is 238.\nBetween 3 and 7, the pairwise dissimilarity is 205.\nBetween 3 and 8, the pairwise dissimilarity is 161.\nBetween 3 and 9, the pairwise dissimilarity is 172.\nBetween 3 and 10, the pairwise dissimilarity is 218.\nBetween 3 and 11, the pairwise dissimilarity is 219.\nBetween 3 and 12, the pairwise dissimilarity is 178.\nBetween 3 and 13, the pairwise dissimilarity is 201.\nBetween 3 and 14, the pairwise dissimilarity is 205.\nBetween 3 and 15, the pairwise dissimilarity is 204.\nBetween 3 and 16, the pairwise dissimilarity is 196.\nBetween 3 and 17, the pairwise dissimilarity is 177.\nBetween 3 and 18, the pairwise dissimilarity is 174.\nBetween 3 and 19, the pairwise dissimilarity is 211.\nBetween 4 and 1, the pairwise dissimilarity is 199.\nBetween 4 and 2, the pairwise dissimilarity is 203.\nBetween 4 and 3, the pairwise dissimilarity is 200.\nBetween 4 and 5, the pairwise dissimilarity is 165.\nBetween 4 and 6, the pairwise dissimilarity is 205.\nBetween 4 and 7, the pairwise dissimilarity is 181.\nBetween 4 and 8, the pairwise dissimilarity is 168.\nBetween 4 and 9, the pairwise dissimilarity is 205.\nBetween 4 and 10, the pairwise dissimilarity is 228.\nBetween 4 and 11, the pairwise dissimilarity is 204.\nBetween 4 and 12, the pairwise dissimilarity is 200.\nBetween 4 and 13, the pairwise dissimilarity is 189.\nBetween 4 and 14, the pairwise dissimilarity is 206.\nBetween 4 and 15, the pairwise dissimilarity is 183.\nBetween 4 and 16, the pairwise dissimilarity is 150.\nBetween 4 and 17, the pairwise dissimilarity is 183.\nBetween 4 and 18, the pairwise dissimilarity is 174.\nBetween 4 and 19, the pairwise dissimilarity is 116.\nBetween 5 and 1, the pairwise dissimilarity is 220.\nBetween 5 and 2, the pairwise dissimilarity is 179.\nBetween 5 and 3, the pairwise dissimilarity is 227.\nBetween 5 and 4, the pairwise dissimilarity is 165.\nBetween 5 and 6, the pairwise dissimilarity is 152.\nBetween 5 and 7, the pairwise dissimilarity is 117.\nBetween 5 and 8, the pairwise dissimilarity is 163.\nBetween 5 and 9, the pairwise dissimilarity is 166.\nBetween 5 and 10, the pairwise dissimilarity is 162.\nBetween 5 and 11, the pairwise dissimilarity is 160.\nBetween 5 and 12, the pairwise dissimilarity is 165.\nBetween 5 and 13, the pairwise dissimilarity is 190.\nBetween 5 and 14, the pairwise dissimilarity is 184.\nBetween 5 and 15, the pairwise dissimilarity is 142.\nBetween 5 and 16, the pairwise dissimilarity is 205.\nBetween 5 and 17, the pairwise dissimilarity is 170.\nBetween 5 and 18, the pairwise dissimilarity is 183.\nBetween 5 and 19, the pairwise dissimilarity is 158.\nBetween 6 and 1, the pairwise dissimilarity is 226.\nBetween 6 and 2, the pairwise dissimilarity is 217.\nBetween 6 and 3, the pairwise dissimilarity is 238.\nBetween 6 and 4, the pairwise dissimilarity is 205.\nBetween 6 and 5, the pairwise dissimilarity is 152.\nBetween 6 and 7, the pairwise dissimilarity is 173.\nBetween 6 and 8, the pairwise dissimilarity is 226.\nBetween 6 and 9, the pairwise dissimilarity is 185.\nBetween 6 and 10, the pairwise dissimilarity is 192.\nBetween 6 and 11, the pairwise dissimilarity is 194.\nBetween 6 and 12, the pairwise dissimilarity is 176.\nBetween 6 and 13, the pairwise dissimilarity is 208.\nBetween 6 and 14, the pairwise dissimilarity is 211.\nBetween 6 and 15, the pairwise dissimilarity is 171.\nBetween 6 and 16, the pairwise dissimilarity is 218.\nBetween 6 and 17, the pairwise dissimilarity is 200.\nBetween 6 and 18, the pairwise dissimilarity is 215.\nBetween 6 and 19, the pairwise dissimilarity is 209.\nBetween 7 and 1, the pairwise dissimilarity is 231.\nBetween 7 and 2, the pairwise dissimilarity is 152.\nBetween 7 and 3, the pairwise dissimilarity is 205.\nBetween 7 and 4, the pairwise dissimilarity is 181.\nBetween 7 and 5, the pairwise dissimilarity is 117.\nBetween 7 and 6, the pairwise dissimilarity is 173.\nBetween 7 and 8, the pairwise dissimilarity is 146.\nBetween 7 and 9, the pairwise dissimilarity is 148.\nBetween 7 and 10, the pairwise dissimilarity is 147.\nBetween 7 and 11, the pairwise dissimilarity is 155.\nBetween 7 and 12, the pairwise dissimilarity is 133.\nBetween 7 and 13, the pairwise dissimilarity is 167.\nBetween 7 and 14, the pairwise dissimilarity is 175.\nBetween 7 and 15, the pairwise dissimilarity is 152.\nBetween 7 and 16, the pairwise dissimilarity is 205.\nBetween 7 and 17, the pairwise dissimilarity is 134.\nBetween 7 and 18, the pairwise dissimilarity is 136.\nBetween 7 and 19, the pairwise dissimilarity is 137.\nBetween 8 and 1, the pairwise dissimilarity is 155.\nBetween 8 and 2, the pairwise dissimilarity is 141.\nBetween 8 and 3, the pairwise dissimilarity is 161.\nBetween 8 and 4, the pairwise dissimilarity is 168.\nBetween 8 and 5, the pairwise dissimilarity is 163.\nBetween 8 and 6, the pairwise dissimilarity is 226.\nBetween 8 and 7, the pairwise dissimilarity is 146.\nBetween 8 and 9, the pairwise dissimilarity is 163.\nBetween 8 and 10, the pairwise dissimilarity is 159.\nBetween 8 and 11, the pairwise dissimilarity is 125.\nBetween 8 and 12, the pairwise dissimilarity is 168.\nBetween 8 and 13, the pairwise dissimilarity is 175.\nBetween 8 and 14, the pairwise dissimilarity is 189.\nBetween 8 and 15, the pairwise dissimilarity is 178.\nBetween 8 and 16, the pairwise dissimilarity is 158.\nBetween 8 and 17, the pairwise dissimilarity is 144.\nBetween 8 and 18, the pairwise dissimilarity is 133.\nBetween 8 and 19, the pairwise dissimilarity is 157.\nBetween 9 and 1, the pairwise dissimilarity is 188.\nBetween 9 and 2, the pairwise dissimilarity is 161.\nBetween 9 and 3, the pairwise dissimilarity is 172.\nBetween 9 and 4, the pairwise dissimilarity is 205.\nBetween 9 and 5, the pairwise dissimilarity is 166.\nBetween 9 and 6, the pairwise dissimilarity is 185.\nBetween 9 and 7, the pairwise dissimilarity is 148.\nBetween 9 and 8, the pairwise dissimilarity is 163.\nBetween 9 and 10, the pairwise dissimilarity is 169.\nBetween 9 and 11, the pairwise dissimilarity is 160.\nBetween 9 and 12, the pairwise dissimilarity is 106.\nBetween 9 and 13, the pairwise dissimilarity is 177.\nBetween 9 and 14, the pairwise dissimilarity is 182.\nBetween 9 and 15, the pairwise dissimilarity is 170.\nBetween 9 and 16, the pairwise dissimilarity is 194.\nBetween 9 and 17, the pairwise dissimilarity is 129.\nBetween 9 and 18, the pairwise dissimilarity is 148.\nBetween 9 and 19, the pairwise dissimilarity is 156.\nBetween 10 and 1, the pairwise dissimilarity is 186.\nBetween 10 and 2, the pairwise dissimilarity is 154.\nBetween 10 and 3, the pairwise dissimilarity is 218.\nBetween 10 and 4, the pairwise dissimilarity is 228.\nBetween 10 and 5, the pairwise dissimilarity is 162.\nBetween 10 and 6, the pairwise dissimilarity is 192.\nBetween 10 and 7, the pairwise dissimilarity is 147.\nBetween 10 and 8, the pairwise dissimilarity is 159.\nBetween 10 and 9, the pairwise dissimilarity is 169.\nBetween 10 and 11, the pairwise dissimilarity is 149.\nBetween 10 and 12, the pairwise dissimilarity is 154.\nBetween 10 and 13, the pairwise dissimilarity is 160.\nBetween 10 and 14, the pairwise dissimilarity is 193.\nBetween 10 and 15, the pairwise dissimilarity is 180.\nBetween 10 and 16, the pairwise dissimilarity is 201.\nBetween 10 and 17, the pairwise dissimilarity is 160.\nBetween 10 and 18, the pairwise dissimilarity is 146.\nBetween 10 and 19, the pairwise dissimilarity is 190.\nBetween 11 and 1, the pairwise dissimilarity is 187.\nBetween 11 and 2, the pairwise dissimilarity is 173.\nBetween 11 and 3, the pairwise dissimilarity is 219.\nBetween 11 and 4, the pairwise dissimilarity is 204.\nBetween 11 and 5, the pairwise dissimilarity is 160.\nBetween 11 and 6, the pairwise dissimilarity is 194.\nBetween 11 and 7, the pairwise dissimilarity is 155.\nBetween 11 and 8, the pairwise dissimilarity is 125.\nBetween 11 and 9, the pairwise dissimilarity is 160.\nBetween 11 and 10, the pairwise dissimilarity is 149.\nBetween 11 and 12, the pairwise dissimilarity is 161.\nBetween 11 and 13, the pairwise dissimilarity is 187.\nBetween 11 and 14, the pairwise dissimilarity is 198.\nBetween 11 and 15, the pairwise dissimilarity is 179.\nBetween 11 and 16, the pairwise dissimilarity is 180.\nBetween 11 and 17, the pairwise dissimilarity is 159.\nBetween 11 and 18, the pairwise dissimilarity is 146.\nBetween 11 and 19, the pairwise dissimilarity is 195.\nBetween 12 and 1, the pairwise dissimilarity is 193.\nBetween 12 and 2, the pairwise dissimilarity is 178.\nBetween 12 and 3, the pairwise dissimilarity is 178.\nBetween 12 and 4, the pairwise dissimilarity is 200.\nBetween 12 and 5, the pairwise dissimilarity is 165.\nBetween 12 and 6, the pairwise dissimilarity is 176.\nBetween 12 and 7, the pairwise dissimilarity is 133.\nBetween 12 and 8, the pairwise dissimilarity is 168.\nBetween 12 and 9, the pairwise dissimilarity is 106.\nBetween 12 and 10, the pairwise dissimilarity is 154.\nBetween 12 and 11, the pairwise dissimilarity is 161.\nBetween 12 and 13, the pairwise dissimilarity is 183.\nBetween 12 and 14, the pairwise dissimilarity is 164.\nBetween 12 and 15, the pairwise dissimilarity is 201.\nBetween 12 and 16, the pairwise dissimilarity is 185.\nBetween 12 and 17, the pairwise dissimilarity is 151.\nBetween 12 and 18, the pairwise dissimilarity is 112.\nBetween 12 and 19, the pairwise dissimilarity is 160.\nBetween 13 and 1, the pairwise dissimilarity is 212.\nBetween 13 and 2, the pairwise dissimilarity is 178.\nBetween 13 and 3, the pairwise dissimilarity is 201.\nBetween 13 and 4, the pairwise dissimilarity is 189.\nBetween 13 and 5, the pairwise dissimilarity is 190.\nBetween 13 and 6, the pairwise dissimilarity is 208.\nBetween 13 and 7, the pairwise dissimilarity is 167.\nBetween 13 and 8, the pairwise dissimilarity is 175.\nBetween 13 and 9, the pairwise dissimilarity is 177.\nBetween 13 and 10, the pairwise dissimilarity is 160.\nBetween 13 and 11, the pairwise dissimilarity is 187.\nBetween 13 and 12, the pairwise dissimilarity is 183.\nBetween 13 and 14, the pairwise dissimilarity is 208.\nBetween 13 and 15, the pairwise dissimilarity is 210.\nBetween 13 and 16, the pairwise dissimilarity is 174.\nBetween 13 and 17, the pairwise dissimilarity is 131.\nBetween 13 and 18, the pairwise dissimilarity is 142.\nBetween 13 and 19, the pairwise dissimilarity is 181.\nBetween 14 and 1, the pairwise dissimilarity is 256.\nBetween 14 and 2, the pairwise dissimilarity is 225.\nBetween 14 and 3, the pairwise dissimilarity is 205.\nBetween 14 and 4, the pairwise dissimilarity is 206.\nBetween 14 and 5, the pairwise dissimilarity is 184.\nBetween 14 and 6, the pairwise dissimilarity is 211.\nBetween 14 and 7, the pairwise dissimilarity is 175.\nBetween 14 and 8, the pairwise dissimilarity is 189.\nBetween 14 and 9, the pairwise dissimilarity is 182.\nBetween 14 and 10, the pairwise dissimilarity is 193.\nBetween 14 and 11, the pairwise dissimilarity is 198.\nBetween 14 and 12, the pairwise dissimilarity is 164.\nBetween 14 and 13, the pairwise dissimilarity is 208.\nBetween 14 and 15, the pairwise dissimilarity is 211.\nBetween 14 and 16, the pairwise dissimilarity is 161.\nBetween 14 and 17, the pairwise dissimilarity is 146.\nBetween 14 and 18, the pairwise dissimilarity is 164.\nBetween 14 and 19, the pairwise dissimilarity is 191.\nBetween 15 and 1, the pairwise dissimilarity is 216.\nBetween 15 and 2, the pairwise dissimilarity is 192.\nBetween 15 and 3, the pairwise dissimilarity is 204.\nBetween 15 and 4, the pairwise dissimilarity is 183.\nBetween 15 and 5, the pairwise dissimilarity is 142.\nBetween 15 and 6, the pairwise dissimilarity is 171.\nBetween 15 and 7, the pairwise dissimilarity is 152.\nBetween 15 and 8, the pairwise dissimilarity is 178.\nBetween 15 and 9, the pairwise dissimilarity is 170.\nBetween 15 and 10, the pairwise dissimilarity is 180.\nBetween 15 and 11, the pairwise dissimilarity is 179.\nBetween 15 and 12, the pairwise dissimilarity is 201.\nBetween 15 and 13, the pairwise dissimilarity is 210.\nBetween 15 and 14, the pairwise dissimilarity is 211.\nBetween 15 and 16, the pairwise dissimilarity is 233.\nBetween 15 and 17, the pairwise dissimilarity is 180.\nBetween 15 and 18, the pairwise dissimilarity is 196.\nBetween 15 and 19, the pairwise dissimilarity is 173.\nBetween 16 and 1, the pairwise dissimilarity is 192.\nBetween 16 and 2, the pairwise dissimilarity is 197.\nBetween 16 and 3, the pairwise dissimilarity is 196.\nBetween 16 and 4, the pairwise dissimilarity is 150.\nBetween 16 and 5, the pairwise dissimilarity is 205.\nBetween 16 and 6, the pairwise dissimilarity is 218.\nBetween 16 and 7, the pairwise dissimilarity is 205.\nBetween 16 and 8, the pairwise dissimilarity is 158.\nBetween 16 and 9, the pairwise dissimilarity is 194.\nBetween 16 and 10, the pairwise dissimilarity is 201.\nBetween 16 and 11, the pairwise dissimilarity is 180.\nBetween 16 and 12, the pairwise dissimilarity is 185.\nBetween 16 and 13, the pairwise dissimilarity is 174.\nBetween 16 and 14, the pairwise dissimilarity is 161.\nBetween 16 and 15, the pairwise dissimilarity is 233.\nBetween 16 and 17, the pairwise dissimilarity is 161.\nBetween 16 and 18, the pairwise dissimilarity is 155.\nBetween 16 and 19, the pairwise dissimilarity is 169.\nBetween 17 and 1, the pairwise dissimilarity is 203.\nBetween 17 and 2, the pairwise dissimilarity is 161.\nBetween 17 and 3, the pairwise dissimilarity is 177.\nBetween 17 and 4, the pairwise dissimilarity is 183.\nBetween 17 and 5, the pairwise dissimilarity is 170.\nBetween 17 and 6, the pairwise dissimilarity is 200.\nBetween 17 and 7, the pairwise dissimilarity is 134.\nBetween 17 and 8, the pairwise dissimilarity is 144.\nBetween 17 and 9, the pairwise dissimilarity is 129.\nBetween 17 and 10, the pairwise dissimilarity is 160.\nBetween 17 and 11, the pairwise dissimilarity is 159.\nBetween 17 and 12, the pairwise dissimilarity is 151.\nBetween 17 and 13, the pairwise dissimilarity is 131.\nBetween 17 and 14, the pairwise dissimilarity is 146.\nBetween 17 and 15, the pairwise dissimilarity is 180.\nBetween 17 and 16, the pairwise dissimilarity is 161.\nBetween 17 and 18, the pairwise dissimilarity is 137.\nBetween 17 and 19, the pairwise dissimilarity is 157.\nBetween 18 and 1, the pairwise dissimilarity is 189.\nBetween 18 and 2, the pairwise dissimilarity is 188.\nBetween 18 and 3, the pairwise dissimilarity is 174.\nBetween 18 and 4, the pairwise dissimilarity is 174.\nBetween 18 and 5, the pairwise dissimilarity is 183.\nBetween 18 and 6, the pairwise dissimilarity is 215.\nBetween 18 and 7, the pairwise dissimilarity is 136.\nBetween 18 and 8, the pairwise dissimilarity is 133.\nBetween 18 and 9, the pairwise dissimilarity is 148.\nBetween 18 and 10, the pairwise dissimilarity is 146.\nBetween 18 and 11, the pairwise dissimilarity is 146.\nBetween 18 and 12, the pairwise dissimilarity is 112.\nBetween 18 and 13, the pairwise dissimilarity is 142.\nBetween 18 and 14, the pairwise dissimilarity is 164.\nBetween 18 and 15, the pairwise dissimilarity is 196.\nBetween 18 and 16, the pairwise dissimilarity is 155.\nBetween 18 and 17, the pairwise dissimilarity is 137.\nBetween 18 and 19, the pairwise dissimilarity is 132.\nBetween 19 and 1, the pairwise dissimilarity is 205.\nBetween 19 and 2, the pairwise dissimilarity is 169.\nBetween 19 and 3, the pairwise dissimilarity is 211.\nBetween 19 and 4, the pairwise dissimilarity is 116.\nBetween 19 and 5, the pairwise dissimilarity is 158.\nBetween 19 and 6, the pairwise dissimilarity is 209.\nBetween 19 and 7, the pairwise dissimilarity is 137.\nBetween 19 and 8, the pairwise dissimilarity is 157.\nBetween 19 and 9, the pairwise dissimilarity is 156.\nBetween 19 and 10, the pairwise dissimilarity is 190.\nBetween 19 and 11, the pairwise dissimilarity is 195.\nBetween 19 and 12, the pairwise dissimilarity is 160.\nBetween 19 and 13, the pairwise dissimilarity is 181.\nBetween 19 and 14, the pairwise dissimilarity is 191.\nBetween 19 and 15, the pairwise dissimilarity is 173.\nBetween 19 and 16, the pairwise dissimilarity is 169.\nBetween 19 and 17, the pairwise dissimilarity is 157.\nBetween 19 and 18, the pairwise dissimilarity is 132.\nSelect exactly 6 compounds that maximize the summed pairwise dissimilarities.\n\nAlso, just so we're on the same page, please give the chosen set in this simple JSON shape when you reply — nice and tidy so I can read it automatically.\n\n{\n \"solution\": [, , ...]\n}\n\nThis little block is just a sketch of the shape I expect: \"solution\" is the list of the m compounds you picked (use the exact identifiers from the instance). It's informal — think of it like filling out a short form, not a complicated file.\n\nQuick reminder: all identifiers must be used exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 191, 189, 199, 220, 226, 231, 155, 188, 186, 187, 193, 212, 256, 216, 192, 203, 189, 205 ], [ 191, 0, 220, 203, 179, 217, 152, 141, 161, 154, 173, 178, 178, 225, 192, 197, 161, 188, 169 ], [ 189, 220, 0, 200, 227, 238, 205, 161, 172, 218, 219, 178, 201, 205, 204, 196, 177, 174, 211 ], [ 199, 203, 200, 0, 165, 205, 181, 168, 205, 228, 204, 200, 189, 206, 183, 150, 183, 174, 116 ], [ 220, 179, 227, 165, 0, 152, 117, 163, 166, 162, 160, 165, 190, 184, 142, 205, 170, 183, 158 ], [ 226, 217, 238, 205, 152, 0, 173, 226, 185, 192, 194, 176, 208, 211, 171, 218, 200, 215, 209 ], [ 231, 152, 205, 181, 117, 173, 0, 146, 148, 147, 155, 133, 167, 175, 152, 205, 134, 136, 137 ], [ 155, 141, 161, 168, 163, 226, 146, 0, 163, 159, 125, 168, 175, 189, 178, 158, 144, 133, 157 ], [ 188, 161, 172, 205, 166, 185, 148, 163, 0, 169, 160, 106, 177, 182, 170, 194, 129, 148, 156 ], [ 186, 154, 218, 228, 162, 192, 147, 159, 169, 0, 149, 154, 160, 193, 180, 201, 160, 146, 190 ], [ 187, 173, 219, 204, 160, 194, 155, 125, 160, 149, 0, 161, 187, 198, 179, 180, 159, 146, 195 ], [ 193, 178, 178, 200, 165, 176, 133, 168, 106, 154, 161, 0, 183, 164, 201, 185, 151, 112, 160 ], [ 212, 178, 201, 189, 190, 208, 167, 175, 177, 160, 187, 183, 0, 208, 210, 174, 131, 142, 181 ], [ 256, 225, 205, 206, 184, 211, 175, 189, 182, 193, 198, 164, 208, 0, 211, 161, 146, 164, 191 ], [ 216, 192, 204, 183, 142, 171, 152, 178, 170, 180, 179, 201, 210, 211, 0, 233, 180, 196, 173 ], [ 192, 197, 196, 150, 205, 218, 205, 158, 194, 201, 180, 185, 174, 161, 233, 0, 161, 155, 169 ], [ 203, 161, 177, 183, 170, 200, 134, 144, 129, 160, 159, 151, 131, 146, 180, 161, 0, 137, 157 ], [ 189, 188, 174, 174, 183, 215, 136, 133, 148, 146, 146, 112, 142, 164, 196, 155, 137, 0, 132 ], [ 205, 169, 211, 116, 158, 209, 137, 157, 156, 190, 195, 160, 181, 191, 173, 169, 157, 132, 0 ] ], "m": 6, "objective": 3191.0 }, "solution": [ 0, 1, 2, 3, 5, 13 ], "obj": 3191.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 19, "m": 6, "nodes": [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 ], "distance_pairs": [ { "from_id": 1, "to_id": 2, "distance": 191 }, { "from_id": 1, "to_id": 3, "distance": 189 }, { "from_id": 1, "to_id": 4, "distance": 199 }, { "from_id": 1, "to_id": 5, "distance": 220 }, { "from_id": 1, "to_id": 6, "distance": 226 }, { "from_id": 1, "to_id": 7, "distance": 231 }, { "from_id": 1, "to_id": 8, "distance": 155 }, { "from_id": 1, "to_id": 9, "distance": 188 }, { "from_id": 1, "to_id": 10, "distance": 186 }, { "from_id": 1, "to_id": 11, "distance": 187 }, { "from_id": 1, "to_id": 12, "distance": 193 }, { "from_id": 1, "to_id": 13, "distance": 212 }, { "from_id": 1, "to_id": 14, "distance": 256 }, { "from_id": 1, "to_id": 15, "distance": 216 }, { "from_id": 1, "to_id": 16, "distance": 192 }, { "from_id": 1, "to_id": 17, "distance": 203 }, { "from_id": 1, "to_id": 18, "distance": 189 }, { "from_id": 1, "to_id": 19, "distance": 205 }, { "from_id": 2, "to_id": 1, "distance": 191 }, { "from_id": 2, "to_id": 3, "distance": 220 }, { "from_id": 2, "to_id": 4, "distance": 203 }, { "from_id": 2, "to_id": 5, "distance": 179 }, { "from_id": 2, "to_id": 6, "distance": 217 }, { "from_id": 2, "to_id": 7, "distance": 152 }, { "from_id": 2, "to_id": 8, "distance": 141 }, { "from_id": 2, "to_id": 9, "distance": 161 }, { "from_id": 2, "to_id": 10, "distance": 154 }, { "from_id": 2, "to_id": 11, "distance": 173 }, { "from_id": 2, "to_id": 12, "distance": 178 }, { "from_id": 2, "to_id": 13, "distance": 178 }, { "from_id": 2, "to_id": 14, "distance": 225 }, { "from_id": 2, "to_id": 15, "distance": 192 }, { "from_id": 2, "to_id": 16, "distance": 197 }, { "from_id": 2, "to_id": 17, "distance": 161 }, { "from_id": 2, "to_id": 18, "distance": 188 }, { "from_id": 2, "to_id": 19, "distance": 169 }, { "from_id": 3, "to_id": 1, "distance": 189 }, { "from_id": 3, "to_id": 2, "distance": 220 }, { "from_id": 3, "to_id": 4, "distance": 200 }, { "from_id": 3, "to_id": 5, "distance": 227 }, { "from_id": 3, "to_id": 6, "distance": 238 }, { "from_id": 3, "to_id": 7, "distance": 205 }, { "from_id": 3, "to_id": 8, "distance": 161 }, { "from_id": 3, "to_id": 9, "distance": 172 }, { "from_id": 3, "to_id": 10, "distance": 218 }, { "from_id": 3, "to_id": 11, "distance": 219 }, { "from_id": 3, "to_id": 12, "distance": 178 }, { "from_id": 3, "to_id": 13, "distance": 201 }, { "from_id": 3, "to_id": 14, "distance": 205 }, { "from_id": 3, "to_id": 15, "distance": 204 }, { "from_id": 3, "to_id": 16, "distance": 196 }, { "from_id": 3, "to_id": 17, "distance": 177 }, { "from_id": 3, "to_id": 18, "distance": 174 }, { "from_id": 3, "to_id": 19, "distance": 211 }, { "from_id": 4, "to_id": 1, "distance": 199 }, { "from_id": 4, "to_id": 2, "distance": 203 }, { "from_id": 4, "to_id": 3, "distance": 200 }, { "from_id": 4, "to_id": 5, "distance": 165 }, { "from_id": 4, "to_id": 6, "distance": 205 }, { "from_id": 4, "to_id": 7, "distance": 181 }, { "from_id": 4, "to_id": 8, "distance": 168 }, { "from_id": 4, "to_id": 9, "distance": 205 }, { "from_id": 4, "to_id": 10, "distance": 228 }, { "from_id": 4, "to_id": 11, "distance": 204 }, { "from_id": 4, "to_id": 12, "distance": 200 }, { "from_id": 4, "to_id": 13, "distance": 189 }, { "from_id": 4, "to_id": 14, "distance": 206 }, { "from_id": 4, "to_id": 15, "distance": 183 }, { "from_id": 4, "to_id": 16, "distance": 150 }, { "from_id": 4, "to_id": 17, "distance": 183 }, { "from_id": 4, "to_id": 18, "distance": 174 }, { "from_id": 4, "to_id": 19, "distance": 116 }, { "from_id": 5, "to_id": 1, "distance": 220 }, { "from_id": 5, "to_id": 2, "distance": 179 }, { "from_id": 5, "to_id": 3, "distance": 227 }, { "from_id": 5, "to_id": 4, "distance": 165 }, { "from_id": 5, "to_id": 6, "distance": 152 }, { "from_id": 5, "to_id": 7, "distance": 117 }, { "from_id": 5, "to_id": 8, "distance": 163 }, { "from_id": 5, "to_id": 9, "distance": 166 }, { "from_id": 5, "to_id": 10, "distance": 162 }, { "from_id": 5, "to_id": 11, "distance": 160 }, { "from_id": 5, "to_id": 12, "distance": 165 }, { "from_id": 5, "to_id": 13, "distance": 190 }, { "from_id": 5, "to_id": 14, "distance": 184 }, { "from_id": 5, "to_id": 15, "distance": 142 }, { "from_id": 5, "to_id": 16, "distance": 205 }, { "from_id": 5, "to_id": 17, "distance": 170 }, { "from_id": 5, "to_id": 18, "distance": 183 }, { "from_id": 5, "to_id": 19, "distance": 158 }, { "from_id": 6, "to_id": 1, "distance": 226 }, { "from_id": 6, "to_id": 2, "distance": 217 }, { "from_id": 6, "to_id": 3, "distance": 238 }, { "from_id": 6, "to_id": 4, "distance": 205 }, { "from_id": 6, "to_id": 5, "distance": 152 }, { "from_id": 6, "to_id": 7, "distance": 173 }, { "from_id": 6, "to_id": 8, "distance": 226 }, { "from_id": 6, "to_id": 9, "distance": 185 }, { "from_id": 6, "to_id": 10, "distance": 192 }, { "from_id": 6, "to_id": 11, "distance": 194 }, { "from_id": 6, "to_id": 12, "distance": 176 }, { "from_id": 6, "to_id": 13, "distance": 208 }, { "from_id": 6, "to_id": 14, "distance": 211 }, { "from_id": 6, "to_id": 15, "distance": 171 }, { "from_id": 6, "to_id": 16, "distance": 218 }, { "from_id": 6, "to_id": 17, "distance": 200 }, { "from_id": 6, "to_id": 18, "distance": 215 }, { "from_id": 6, "to_id": 19, "distance": 209 }, { "from_id": 7, "to_id": 1, "distance": 231 }, { "from_id": 7, "to_id": 2, "distance": 152 }, { "from_id": 7, "to_id": 3, "distance": 205 }, { "from_id": 7, "to_id": 4, "distance": 181 }, { "from_id": 7, "to_id": 5, "distance": 117 }, { "from_id": 7, "to_id": 6, "distance": 173 }, { "from_id": 7, "to_id": 8, "distance": 146 }, { "from_id": 7, "to_id": 9, "distance": 148 }, { "from_id": 7, "to_id": 10, "distance": 147 }, { "from_id": 7, "to_id": 11, "distance": 155 }, { "from_id": 7, "to_id": 12, "distance": 133 }, { "from_id": 7, "to_id": 13, "distance": 167 }, { "from_id": 7, "to_id": 14, "distance": 175 }, { "from_id": 7, "to_id": 15, "distance": 152 }, { "from_id": 7, "to_id": 16, "distance": 205 }, { "from_id": 7, "to_id": 17, "distance": 134 }, { "from_id": 7, "to_id": 18, "distance": 136 }, { "from_id": 7, "to_id": 19, "distance": 137 }, { "from_id": 8, "to_id": 1, "distance": 155 }, { "from_id": 8, "to_id": 2, "distance": 141 }, { "from_id": 8, "to_id": 3, "distance": 161 }, { "from_id": 8, "to_id": 4, "distance": 168 }, { "from_id": 8, "to_id": 5, "distance": 163 }, { "from_id": 8, "to_id": 6, "distance": 226 }, { "from_id": 8, "to_id": 7, "distance": 146 }, { "from_id": 8, "to_id": 9, "distance": 163 }, { "from_id": 8, "to_id": 10, "distance": 159 }, { "from_id": 8, "to_id": 11, "distance": 125 }, { "from_id": 8, "to_id": 12, "distance": 168 }, { "from_id": 8, "to_id": 13, "distance": 175 }, { "from_id": 8, "to_id": 14, "distance": 189 }, { "from_id": 8, "to_id": 15, "distance": 178 }, { "from_id": 8, "to_id": 16, "distance": 158 }, { "from_id": 8, "to_id": 17, "distance": 144 }, { "from_id": 8, "to_id": 18, "distance": 133 }, { "from_id": 8, "to_id": 19, "distance": 157 }, { "from_id": 9, "to_id": 1, "distance": 188 }, { "from_id": 9, "to_id": 2, "distance": 161 }, { "from_id": 9, "to_id": 3, "distance": 172 }, { "from_id": 9, "to_id": 4, "distance": 205 }, { "from_id": 9, "to_id": 5, "distance": 166 }, { "from_id": 9, "to_id": 6, "distance": 185 }, { "from_id": 9, "to_id": 7, "distance": 148 }, { "from_id": 9, "to_id": 8, "distance": 163 }, { "from_id": 9, "to_id": 10, "distance": 169 }, { "from_id": 9, "to_id": 11, "distance": 160 }, { "from_id": 9, "to_id": 12, "distance": 106 }, { "from_id": 9, "to_id": 13, "distance": 177 }, { "from_id": 9, "to_id": 14, "distance": 182 }, { "from_id": 9, "to_id": 15, "distance": 170 }, { "from_id": 9, "to_id": 16, "distance": 194 }, { "from_id": 9, "to_id": 17, "distance": 129 }, { "from_id": 9, "to_id": 18, "distance": 148 }, { "from_id": 9, "to_id": 19, "distance": 156 }, { "from_id": 10, "to_id": 1, "distance": 186 }, { "from_id": 10, "to_id": 2, "distance": 154 }, { "from_id": 10, "to_id": 3, "distance": 218 }, { "from_id": 10, "to_id": 4, "distance": 228 }, { "from_id": 10, "to_id": 5, "distance": 162 }, { "from_id": 10, "to_id": 6, "distance": 192 }, { "from_id": 10, "to_id": 7, "distance": 147 }, { "from_id": 10, "to_id": 8, "distance": 159 }, { "from_id": 10, "to_id": 9, "distance": 169 }, { "from_id": 10, "to_id": 11, "distance": 149 }, { "from_id": 10, "to_id": 12, "distance": 154 }, { "from_id": 10, "to_id": 13, "distance": 160 }, { "from_id": 10, "to_id": 14, "distance": 193 }, { "from_id": 10, "to_id": 15, "distance": 180 }, { "from_id": 10, "to_id": 16, "distance": 201 }, { "from_id": 10, "to_id": 17, "distance": 160 }, { "from_id": 10, "to_id": 18, "distance": 146 }, { "from_id": 10, "to_id": 19, "distance": 190 }, { "from_id": 11, "to_id": 1, "distance": 187 }, { "from_id": 11, "to_id": 2, "distance": 173 }, { "from_id": 11, "to_id": 3, "distance": 219 }, { "from_id": 11, "to_id": 4, "distance": 204 }, { "from_id": 11, "to_id": 5, "distance": 160 }, { "from_id": 11, "to_id": 6, "distance": 194 }, { "from_id": 11, "to_id": 7, "distance": 155 }, { "from_id": 11, "to_id": 8, "distance": 125 }, { "from_id": 11, "to_id": 9, "distance": 160 }, { "from_id": 11, "to_id": 10, "distance": 149 }, { "from_id": 11, "to_id": 12, "distance": 161 }, { "from_id": 11, "to_id": 13, "distance": 187 }, { "from_id": 11, "to_id": 14, "distance": 198 }, { "from_id": 11, "to_id": 15, "distance": 179 }, { "from_id": 11, "to_id": 16, "distance": 180 }, { "from_id": 11, "to_id": 17, "distance": 159 }, { "from_id": 11, "to_id": 18, "distance": 146 }, { "from_id": 11, "to_id": 19, "distance": 195 }, { "from_id": 12, "to_id": 1, "distance": 193 }, { "from_id": 12, "to_id": 2, "distance": 178 }, { "from_id": 12, "to_id": 3, "distance": 178 }, { "from_id": 12, "to_id": 4, "distance": 200 }, { "from_id": 12, "to_id": 5, "distance": 165 }, { "from_id": 12, "to_id": 6, "distance": 176 }, { "from_id": 12, "to_id": 7, "distance": 133 }, { "from_id": 12, "to_id": 8, "distance": 168 }, { "from_id": 12, "to_id": 9, "distance": 106 }, { "from_id": 12, "to_id": 10, "distance": 154 }, { "from_id": 12, "to_id": 11, "distance": 161 }, { "from_id": 12, "to_id": 13, "distance": 183 }, { "from_id": 12, "to_id": 14, "distance": 164 }, { "from_id": 12, "to_id": 15, "distance": 201 }, { "from_id": 12, "to_id": 16, "distance": 185 }, { "from_id": 12, "to_id": 17, "distance": 151 }, { "from_id": 12, "to_id": 18, "distance": 112 }, { "from_id": 12, "to_id": 19, "distance": 160 }, { "from_id": 13, "to_id": 1, "distance": 212 }, { "from_id": 13, "to_id": 2, "distance": 178 }, { "from_id": 13, "to_id": 3, "distance": 201 }, { "from_id": 13, "to_id": 4, "distance": 189 }, { "from_id": 13, "to_id": 5, "distance": 190 }, { "from_id": 13, "to_id": 6, "distance": 208 }, { "from_id": 13, "to_id": 7, "distance": 167 }, { "from_id": 13, "to_id": 8, "distance": 175 }, { "from_id": 13, "to_id": 9, "distance": 177 }, { "from_id": 13, "to_id": 10, "distance": 160 }, { "from_id": 13, "to_id": 11, "distance": 187 }, { "from_id": 13, "to_id": 12, "distance": 183 }, { "from_id": 13, "to_id": 14, "distance": 208 }, { "from_id": 13, "to_id": 15, "distance": 210 }, { "from_id": 13, "to_id": 16, "distance": 174 }, { "from_id": 13, "to_id": 17, "distance": 131 }, { "from_id": 13, "to_id": 18, "distance": 142 }, { "from_id": 13, "to_id": 19, "distance": 181 }, { "from_id": 14, "to_id": 1, "distance": 256 }, { "from_id": 14, "to_id": 2, "distance": 225 }, { "from_id": 14, "to_id": 3, "distance": 205 }, { "from_id": 14, "to_id": 4, "distance": 206 }, { "from_id": 14, "to_id": 5, "distance": 184 }, { "from_id": 14, "to_id": 6, "distance": 211 }, { "from_id": 14, "to_id": 7, "distance": 175 }, { "from_id": 14, "to_id": 8, "distance": 189 }, { "from_id": 14, "to_id": 9, "distance": 182 }, { "from_id": 14, "to_id": 10, "distance": 193 }, { "from_id": 14, "to_id": 11, "distance": 198 }, { "from_id": 14, "to_id": 12, "distance": 164 }, { "from_id": 14, "to_id": 13, "distance": 208 }, { "from_id": 14, "to_id": 15, "distance": 211 }, { "from_id": 14, "to_id": 16, "distance": 161 }, { "from_id": 14, "to_id": 17, "distance": 146 }, { "from_id": 14, "to_id": 18, "distance": 164 }, { "from_id": 14, "to_id": 19, "distance": 191 }, { "from_id": 15, "to_id": 1, "distance": 216 }, { "from_id": 15, "to_id": 2, "distance": 192 }, { "from_id": 15, "to_id": 3, "distance": 204 }, { "from_id": 15, "to_id": 4, "distance": 183 }, { "from_id": 15, "to_id": 5, "distance": 142 }, { "from_id": 15, "to_id": 6, "distance": 171 }, { "from_id": 15, "to_id": 7, "distance": 152 }, { "from_id": 15, "to_id": 8, "distance": 178 }, { "from_id": 15, "to_id": 9, "distance": 170 }, { "from_id": 15, "to_id": 10, "distance": 180 }, { "from_id": 15, "to_id": 11, "distance": 179 }, { "from_id": 15, "to_id": 12, "distance": 201 }, { "from_id": 15, "to_id": 13, "distance": 210 }, { "from_id": 15, "to_id": 14, "distance": 211 }, { "from_id": 15, "to_id": 16, "distance": 233 }, { "from_id": 15, "to_id": 17, "distance": 180 }, { "from_id": 15, "to_id": 18, "distance": 196 }, { "from_id": 15, "to_id": 19, "distance": 173 }, { "from_id": 16, "to_id": 1, "distance": 192 }, { "from_id": 16, "to_id": 2, "distance": 197 }, { "from_id": 16, "to_id": 3, "distance": 196 }, { "from_id": 16, "to_id": 4, "distance": 150 }, { "from_id": 16, "to_id": 5, "distance": 205 }, { "from_id": 16, "to_id": 6, "distance": 218 }, { "from_id": 16, "to_id": 7, "distance": 205 }, { "from_id": 16, "to_id": 8, "distance": 158 }, { "from_id": 16, "to_id": 9, "distance": 194 }, { "from_id": 16, "to_id": 10, "distance": 201 }, { "from_id": 16, "to_id": 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"from_id": 17, "to_id": 14, "distance": 146 }, { "from_id": 17, "to_id": 15, "distance": 180 }, { "from_id": 17, "to_id": 16, "distance": 161 }, { "from_id": 17, "to_id": 18, "distance": 137 }, { "from_id": 17, "to_id": 19, "distance": 157 }, { "from_id": 18, "to_id": 1, "distance": 189 }, { "from_id": 18, "to_id": 2, "distance": 188 }, { "from_id": 18, "to_id": 3, "distance": 174 }, { "from_id": 18, "to_id": 4, "distance": 174 }, { "from_id": 18, "to_id": 5, "distance": 183 }, { "from_id": 18, "to_id": 6, "distance": 215 }, { "from_id": 18, "to_id": 7, "distance": 136 }, { "from_id": 18, "to_id": 8, "distance": 133 }, { "from_id": 18, "to_id": 9, "distance": 148 }, { "from_id": 18, "to_id": 10, "distance": 146 }, { "from_id": 18, "to_id": 11, "distance": 146 }, { "from_id": 18, "to_id": 12, "distance": 112 }, { "from_id": 18, "to_id": 13, "distance": 142 }, { "from_id": 18, "to_id": 14, "distance": 164 }, { "from_id": 18, "to_id": 15, "distance": 196 }, { "from_id": 18, "to_id": 16, "distance": 155 }, { "from_id": 18, "to_id": 17, "distance": 137 }, { "from_id": 18, "to_id": 19, "distance": 132 }, { "from_id": 19, "to_id": 1, "distance": 205 }, { "from_id": 19, "to_id": 2, "distance": 169 }, { "from_id": 19, "to_id": 3, "distance": 211 }, { "from_id": 19, "to_id": 4, "distance": 116 }, { "from_id": 19, "to_id": 5, "distance": 158 }, { "from_id": 19, "to_id": 6, "distance": 209 }, { "from_id": 19, "to_id": 7, "distance": 137 }, { "from_id": 19, "to_id": 8, "distance": 157 }, { "from_id": 19, "to_id": 9, "distance": 156 }, { "from_id": 19, "to_id": 10, "distance": 190 }, { "from_id": 19, "to_id": 11, "distance": 195 }, { "from_id": 19, "to_id": 12, "distance": 160 }, { "from_id": 19, "to_id": 13, "distance": 181 }, { "from_id": 19, "to_id": 14, "distance": 191 }, { "from_id": 19, "to_id": 15, "distance": 173 }, { "from_id": 19, "to_id": 16, "distance": 169 }, { "from_id": 19, "to_id": 17, "distance": 157 }, { "from_id": 19, "to_id": 18, "distance": 132 } ] }, "solution_variant": [ 1, 2, 3, 4, 6, 14 ], "context_index": 29, "input_format": "nl", "input_index_base": 1 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "I’m putting together a short list of hikes for a weekend guide and have to pick exactly the required number of trails — no repeats, no extras — so the group feels excitingly varied. To judge any lineup, I look at every pair of chosen trails, note how different each pair is in terrain and vibe, and add those pairwise differences up; the lineup with the biggest total difference is the one I want. The concrete trail options and the pairwise difference details are shown below.\n\n# total_trail_options=25\n# trails_to_recommend_count=8\n# trail_identifiers=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24\ntrail_id_first,trail_id_second,terrain_vibe_difference_score\n0,1,5\n0,2,5\n0,3,5\n0,4,2\n0,5,8\n0,6,5\n0,7,1\n0,8,4\n0,9,4\n0,10,4\n0,11,1\n0,12,6\n0,13,1\n0,14,8\n0,15,0\n0,16,5\n0,17,2\n0,18,4\n0,19,5\n0,20,0\n0,21,6\n0,22,2\n0,23,3\n0,24,4\n1,0,5\n1,2,2\n1,3,1\n1,4,6\n1,5,5\n1,6,9\n1,7,5\n1,8,4\n1,9,5\n1,10,3\n1,11,0\n1,12,7\n1,13,9\n1,14,3\n1,15,7\n1,16,5\n1,17,6\n1,18,3\n1,19,2\n1,20,2\n1,21,6\n1,22,9\n1,23,5\n1,24,2\n2,0,5\n2,1,2\n2,3,9\n2,4,7\n2,5,6\n2,6,2\n2,7,3\n2,8,7\n2,9,2\n2,10,0\n2,11,9\n2,12,8\n2,13,7\n2,14,7\n2,15,1\n2,16,9\n2,17,4\n2,18,7\n2,19,9\n2,20,4\n2,21,3\n2,22,9\n2,23,1\n2,24,1\n3,0,5\n3,1,1\n3,2,9\n3,4,5\n3,5,2\n3,6,9\n3,7,0\n3,8,7\n3,9,1\n3,10,0\n3,11,9\n3,12,2\n3,13,3\n3,14,6\n3,15,2\n3,16,7\n3,17,4\n3,18,6\n3,19,5\n3,20,7\n3,21,1\n3,22,1\n3,23,3\n3,24,1\n4,0,2\n4,1,6\n4,2,7\n4,3,5\n4,5,2\n4,6,3\n4,7,5\n4,8,6\n4,9,0\n4,10,9\n4,11,5\n4,12,0\n4,13,1\n4,14,1\n4,15,6\n4,16,8\n4,17,0\n4,18,9\n4,19,2\n4,20,6\n4,21,1\n4,22,5\n4,23,8\n4,24,7\n5,0,8\n5,1,5\n5,2,6\n5,3,2\n5,4,2\n5,6,2\n5,7,6\n5,8,5\n5,9,8\n5,10,3\n5,11,2\n5,12,1\n5,13,9\n5,14,2\n5,15,6\n5,16,9\n5,17,7\n5,18,4\n5,19,4\n5,20,1\n5,21,7\n5,22,0\n5,23,4\n5,24,4\n6,0,5\n6,1,9\n6,2,2\n6,3,9\n6,4,3\n6,5,2\n6,7,7\n6,8,2\n6,9,7\n6,10,7\n6,11,1\n6,12,2\n6,13,7\n6,14,4\n6,15,6\n6,16,2\n6,17,8\n6,18,0\n6,19,1\n6,20,6\n6,21,3\n6,22,3\n6,23,0\n6,24,4\n7,0,1\n7,1,5\n7,2,3\n7,3,0\n7,4,5\n7,5,6\n7,6,7\n7,8,4\n7,9,2\n7,10,0\n7,11,8\n7,12,8\n7,13,7\n7,14,2\n7,15,8\n7,16,6\n7,17,8\n7,18,1\n7,19,2\n7,20,9\n7,21,9\n7,22,1\n7,23,6\n7,24,5\n8,0,4\n8,1,4\n8,2,7\n8,3,7\n8,4,6\n8,5,5\n8,6,2\n8,7,4\n8,9,1\n8,10,4\n8,11,0\n8,12,9\n8,13,3\n8,14,5\n8,15,0\n8,16,8\n8,17,4\n8,18,0\n8,19,1\n8,20,5\n8,21,3\n8,22,6\n8,23,3\n8,24,1\n9,0,4\n9,1,5\n9,2,2\n9,3,1\n9,4,0\n9,5,8\n9,6,7\n9,7,2\n9,8,1\n9,10,4\n9,11,6\n9,12,8\n9,13,7\n9,14,0\n9,15,0\n9,16,0\n9,17,2\n9,18,5\n9,19,3\n9,20,8\n9,21,5\n9,22,9\n9,23,3\n9,24,1\n10,0,4\n10,1,3\n10,2,0\n10,3,0\n10,4,9\n10,5,3\n10,6,7\n10,7,0\n10,8,4\n10,9,4\n10,11,8\n10,12,9\n10,13,2\n10,14,8\n10,15,9\n10,16,7\n10,17,5\n10,18,8\n10,19,9\n10,20,9\n10,21,1\n10,22,1\n10,23,0\n10,24,1\n11,0,1\n11,1,0\n11,2,9\n11,3,9\n11,4,5\n11,5,2\n11,6,1\n11,7,8\n11,8,0\n11,9,6\n11,10,8\n11,12,8\n11,13,0\n11,14,1\n11,15,5\n11,16,4\n11,17,0\n11,18,7\n11,19,6\n11,20,8\n11,21,9\n11,22,1\n11,23,8\n11,24,9\n12,0,6\n12,1,7\n12,2,8\n12,3,2\n12,4,0\n12,5,1\n12,6,2\n12,7,8\n12,8,9\n12,9,8\n12,10,9\n12,11,8\n12,13,1\n12,14,2\n12,15,2\n12,16,6\n12,17,4\n12,18,2\n12,19,2\n12,20,6\n12,21,1\n12,22,8\n12,23,3\n12,24,7\n13,0,1\n13,1,9\n13,2,7\n13,3,3\n13,4,1\n13,5,9\n13,6,7\n13,7,7\n13,8,3\n13,9,7\n13,10,2\n13,11,0\n13,12,1\n13,14,5\n13,15,5\n13,16,5\n13,17,3\n13,18,5\n13,19,9\n13,20,3\n13,21,9\n13,22,0\n13,23,3\n13,24,8\n14,0,8\n14,1,3\n14,2,7\n14,3,6\n14,4,1\n14,5,2\n14,6,4\n14,7,2\n14,8,5\n14,9,0\n14,10,8\n14,11,1\n14,12,2\n14,13,5\n14,15,9\n14,16,7\n14,17,2\n14,18,6\n14,19,3\n14,20,6\n14,21,0\n14,22,9\n14,23,7\n14,24,0\n15,0,0\n15,1,7\n15,2,1\n15,3,2\n15,4,6\n15,5,6\n15,6,6\n15,7,8\n15,8,0\n15,9,0\n15,10,9\n15,11,5\n15,12,2\n15,13,5\n15,14,9\n15,16,4\n15,17,5\n15,18,6\n15,19,9\n15,20,6\n15,21,1\n15,22,8\n15,23,0\n15,24,4\n16,0,5\n16,1,5\n16,2,9\n16,3,7\n16,4,8\n16,5,9\n16,6,2\n16,7,6\n16,8,8\n16,9,0\n16,10,7\n16,11,4\n16,12,6\n16,13,5\n16,14,7\n16,15,4\n16,17,0\n16,18,2\n16,19,1\n16,20,0\n16,21,6\n16,22,5\n16,23,9\n16,24,4\n17,0,2\n17,1,6\n17,2,4\n17,3,4\n17,4,0\n17,5,7\n17,6,8\n17,7,8\n17,8,4\n17,9,2\n17,10,5\n17,11,0\n17,12,4\n17,13,3\n17,14,2\n17,15,5\n17,16,0\n17,18,3\n17,19,8\n17,20,4\n17,21,2\n17,22,5\n17,23,5\n17,24,0\n18,0,4\n18,1,3\n18,2,7\n18,3,6\n18,4,9\n18,5,4\n18,6,0\n18,7,1\n18,8,0\n18,9,5\n18,10,8\n18,11,7\n18,12,2\n18,13,5\n18,14,6\n18,15,6\n18,16,2\n18,17,3\n18,19,7\n18,20,2\n18,21,0\n18,22,8\n18,23,0\n18,24,2\n19,0,5\n19,1,2\n19,2,9\n19,3,5\n19,4,2\n19,5,4\n19,6,1\n19,7,2\n19,8,1\n19,9,3\n19,10,9\n19,11,6\n19,12,2\n19,13,9\n19,14,3\n19,15,9\n19,16,1\n19,17,8\n19,18,7\n19,20,6\n19,21,2\n19,22,3\n19,23,4\n19,24,7\n20,0,0\n20,1,2\n20,2,4\n20,3,7\n20,4,6\n20,5,1\n20,6,6\n20,7,9\n20,8,5\n20,9,8\n20,10,9\n20,11,8\n20,12,6\n20,13,3\n20,14,6\n20,15,6\n20,16,0\n20,17,4\n20,18,2\n20,19,6\n20,21,8\n20,22,4\n20,23,4\n20,24,2\n21,0,6\n21,1,6\n21,2,3\n21,3,1\n21,4,1\n21,5,7\n21,6,3\n21,7,9\n21,8,3\n21,9,5\n21,10,1\n21,11,9\n21,12,1\n21,13,9\n21,14,0\n21,15,1\n21,16,6\n21,17,2\n21,18,0\n21,19,2\n21,20,8\n21,22,8\n21,23,4\n21,24,0\n22,0,2\n22,1,9\n22,2,9\n22,3,1\n22,4,5\n22,5,0\n22,6,3\n22,7,1\n22,8,6\n22,9,9\n22,10,1\n22,11,1\n22,12,8\n22,13,0\n22,14,9\n22,15,8\n22,16,5\n22,17,5\n22,18,8\n22,19,3\n22,20,4\n22,21,8\n22,23,0\n22,24,7\n23,0,3\n23,1,5\n23,2,1\n23,3,3\n23,4,8\n23,5,4\n23,6,0\n23,7,6\n23,8,3\n23,9,3\n23,10,0\n23,11,8\n23,12,3\n23,13,3\n23,14,7\n23,15,0\n23,16,9\n23,17,5\n23,18,0\n23,19,4\n23,20,4\n23,21,4\n23,22,0\n23,24,0\n24,0,4\n24,1,2\n24,2,1\n24,3,1\n24,4,7\n24,5,4\n24,6,4\n24,7,5\n24,8,1\n24,9,1\n24,10,1\n24,11,9\n24,12,7\n24,13,8\n24,14,0\n24,15,4\n24,16,4\n24,17,0\n24,18,2\n24,19,7\n24,20,2\n24,21,0\n24,22,7\n24,23,0\n\nOh, and one small request — when you send the final pick, please use this little JSON shape so it's easy to read and import:\n\n{\n \"solution\": [\"\", \"\", ...]\n}\n\nQuick, friendly rundown: \"solution\" is the list of the trails I picked for the weekend; put each trail's identifier in the array (exactly m of them), no duplicates, and use the same identifiers you see in the instance. This JSON is just a sketch of the shape I expect you to reply in — you'll fill in the real trail IDs.\n\nAll identifiers must be used exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”. \"", "instance": { "distance_matrix": [ [ 0, 5, 5, 5, 2, 8, 5, 1, 4, 4, 4, 1, 6, 1, 8, 0, 5, 2, 4, 5, 0, 6, 2, 3, 4 ], [ 5, 0, 2, 1, 6, 5, 9, 5, 4, 5, 3, 0, 7, 9, 3, 7, 5, 6, 3, 2, 2, 6, 9, 5, 2 ], [ 5, 2, 0, 9, 7, 6, 2, 3, 7, 2, 0, 9, 8, 7, 7, 1, 9, 4, 7, 9, 4, 3, 9, 1, 1 ], [ 5, 1, 9, 0, 5, 2, 9, 0, 7, 1, 0, 9, 2, 3, 6, 2, 7, 4, 6, 5, 7, 1, 1, 3, 1 ], [ 2, 6, 7, 5, 0, 2, 3, 5, 6, 0, 9, 5, 0, 1, 1, 6, 8, 0, 9, 2, 6, 1, 5, 8, 7 ], [ 8, 5, 6, 2, 2, 0, 2, 6, 5, 8, 3, 2, 1, 9, 2, 6, 9, 7, 4, 4, 1, 7, 0, 4, 4 ], [ 5, 9, 2, 9, 3, 2, 0, 7, 2, 7, 7, 1, 2, 7, 4, 6, 2, 8, 0, 1, 6, 3, 3, 0, 4 ], [ 1, 5, 3, 0, 5, 6, 7, 0, 4, 2, 0, 8, 8, 7, 2, 8, 6, 8, 1, 2, 9, 9, 1, 6, 5 ], [ 4, 4, 7, 7, 6, 5, 2, 4, 0, 1, 4, 0, 9, 3, 5, 0, 8, 4, 0, 1, 5, 3, 6, 3, 1 ], [ 4, 5, 2, 1, 0, 8, 7, 2, 1, 0, 4, 6, 8, 7, 0, 0, 0, 2, 5, 3, 8, 5, 9, 3, 1 ], [ 4, 3, 0, 0, 9, 3, 7, 0, 4, 4, 0, 8, 9, 2, 8, 9, 7, 5, 8, 9, 9, 1, 1, 0, 1 ], [ 1, 0, 9, 9, 5, 2, 1, 8, 0, 6, 8, 0, 8, 0, 1, 5, 4, 0, 7, 6, 8, 9, 1, 8, 9 ], [ 6, 7, 8, 2, 0, 1, 2, 8, 9, 8, 9, 8, 0, 1, 2, 2, 6, 4, 2, 2, 6, 1, 8, 3, 7 ], [ 1, 9, 7, 3, 1, 9, 7, 7, 3, 7, 2, 0, 1, 0, 5, 5, 5, 3, 5, 9, 3, 9, 0, 3, 8 ], [ 8, 3, 7, 6, 1, 2, 4, 2, 5, 0, 8, 1, 2, 5, 0, 9, 7, 2, 6, 3, 6, 0, 9, 7, 0 ], [ 0, 7, 1, 2, 6, 6, 6, 8, 0, 0, 9, 5, 2, 5, 9, 0, 4, 5, 6, 9, 6, 1, 8, 0, 4 ], [ 5, 5, 9, 7, 8, 9, 2, 6, 8, 0, 7, 4, 6, 5, 7, 4, 0, 0, 2, 1, 0, 6, 5, 9, 4 ], [ 2, 6, 4, 4, 0, 7, 8, 8, 4, 2, 5, 0, 4, 3, 2, 5, 0, 0, 3, 8, 4, 2, 5, 5, 0 ], [ 4, 3, 7, 6, 9, 4, 0, 1, 0, 5, 8, 7, 2, 5, 6, 6, 2, 3, 0, 7, 2, 0, 8, 0, 2 ], [ 5, 2, 9, 5, 2, 4, 1, 2, 1, 3, 9, 6, 2, 9, 3, 9, 1, 8, 7, 0, 6, 2, 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\"pairwise_dissimilarity_score\": 78\n },\n {\n \"creative_id_a\": 18,\n \"creative_id_b\": 11,\n \"pairwise_dissimilarity_score\": 142\n },\n {\n \"creative_id_a\": 18,\n \"creative_id_b\": 12,\n \"pairwise_dissimilarity_score\": 141\n },\n {\n \"creative_id_a\": 18,\n \"creative_id_b\": 13,\n \"pairwise_dissimilarity_score\": 130\n },\n {\n \"creative_id_a\": 18,\n \"creative_id_b\": 14,\n \"pairwise_dissimilarity_score\": 105\n },\n {\n \"creative_id_a\": 18,\n \"creative_id_b\": 15,\n \"pairwise_dissimilarity_score\": 119\n },\n {\n \"creative_id_a\": 18,\n \"creative_id_b\": 16,\n \"pairwise_dissimilarity_score\": 109\n },\n {\n \"creative_id_a\": 18,\n \"creative_id_b\": 17,\n \"pairwise_dissimilarity_score\": 144\n },\n {\n \"creative_id_a\": 18,\n \"creative_id_b\": 19,\n \"pairwise_dissimilarity_score\": 105\n },\n {\n \"creative_id_a\": 18,\n \"creative_id_b\": 20,\n \"pairwise_dissimilarity_score\": 115\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 0,\n \"pairwise_dissimilarity_score\": 105\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 1,\n \"pairwise_dissimilarity_score\": 80\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 2,\n \"pairwise_dissimilarity_score\": 75\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 3,\n \"pairwise_dissimilarity_score\": 63\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 4,\n \"pairwise_dissimilarity_score\": 68\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 5,\n \"pairwise_dissimilarity_score\": 77\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 6,\n \"pairwise_dissimilarity_score\": 81\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 7,\n \"pairwise_dissimilarity_score\": 71\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 8,\n \"pairwise_dissimilarity_score\": 67\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 9,\n \"pairwise_dissimilarity_score\": 107\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 10,\n \"pairwise_dissimilarity_score\": 57\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 11,\n \"pairwise_dissimilarity_score\": 97\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 12,\n \"pairwise_dissimilarity_score\": 84\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 13,\n \"pairwise_dissimilarity_score\": 87\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 14,\n \"pairwise_dissimilarity_score\": 40\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 15,\n \"pairwise_dissimilarity_score\": 121\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 16,\n \"pairwise_dissimilarity_score\": 75\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 17,\n \"pairwise_dissimilarity_score\": 92\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 18,\n \"pairwise_dissimilarity_score\": 105\n },\n {\n \"creative_id_a\": 19,\n \"creative_id_b\": 20,\n \"pairwise_dissimilarity_score\": 85\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 0,\n \"pairwise_dissimilarity_score\": 81\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 1,\n \"pairwise_dissimilarity_score\": 77\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 2,\n \"pairwise_dissimilarity_score\": 72\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 3,\n \"pairwise_dissimilarity_score\": 95\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 4,\n \"pairwise_dissimilarity_score\": 97\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 5,\n \"pairwise_dissimilarity_score\": 92\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 6,\n \"pairwise_dissimilarity_score\": 64\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 7,\n \"pairwise_dissimilarity_score\": 84\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 8,\n \"pairwise_dissimilarity_score\": 74\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 9,\n \"pairwise_dissimilarity_score\": 118\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 10,\n \"pairwise_dissimilarity_score\": 75\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 11,\n \"pairwise_dissimilarity_score\": 115\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 12,\n \"pairwise_dissimilarity_score\": 102\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 13,\n \"pairwise_dissimilarity_score\": 75\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 14,\n \"pairwise_dissimilarity_score\": 75\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 15,\n \"pairwise_dissimilarity_score\": 96\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 16,\n \"pairwise_dissimilarity_score\": 113\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 17,\n \"pairwise_dissimilarity_score\": 100\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 18,\n \"pairwise_dissimilarity_score\": 115\n },\n {\n \"creative_id_a\": 20,\n \"creative_id_b\": 19,\n \"pairwise_dissimilarity_score\": 85\n }\n ]\n}\n\nI'll just show the shape of the output we want — a tiny JSON with the chosen creatives listed inside. Nothing fancy, just put the selected creative identifiers in the array.\n\n{\n \"solution\": [, , ...]\n}\n\n\"solution\" is the list where you drop the exact identifiers of the creatives you picked (one identifier per selected creative). This JSON is just a sketch of the expected shape — not the final answer itself.\n\nAll identifiers must be used exactly as they appear in the instance input — no renaming and no new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 85, 91, 90, 143, 93, 115, 121, 69, 126, 94, 148, 51, 76, 96, 102, 143, 74, 148, 105, 81 ], [ 85, 0, 53, 80, 124, 66, 94, 101, 84, 88, 72, 101, 79, 64, 83, 83, 119, 68, 95, 80, 77 ], [ 91, 53, 0, 90, 104, 100, 63, 101, 70, 125, 46, 126, 92, 92, 75, 93, 109, 101, 60, 75, 72 ], [ 90, 80, 90, 0, 104, 83, 89, 102, 81, 132, 63, 137, 75, 96, 56, 146, 109, 99, 133, 63, 95 ], [ 143, 124, 104, 104, 0, 135, 79, 56, 84, 149, 73, 129, 131, 118, 55, 149, 46, 142, 110, 68, 97 ], [ 93, 66, 100, 83, 135, 0, 122, 114, 105, 70, 107, 79, 82, 73, 97, 108, 136, 58, 145, 77, 92 ], [ 115, 94, 63, 89, 79, 122, 0, 101, 88, 158, 53, 145, 125, 120, 70, 138, 107, 139, 89, 81, 64 ], [ 121, 101, 101, 102, 56, 114, 101, 0, 66, 108, 77, 103, 106, 73, 53, 112, 39, 103, 115, 71, 84 ], [ 69, 84, 70, 81, 84, 105, 88, 66, 0, 120, 52, 130, 59, 68, 47, 98, 78, 84, 109, 67, 74 ], [ 126, 88, 125, 132, 149, 70, 158, 108, 120, 0, 134, 51, 107, 65, 122, 83, 133, 58, 151, 107, 118 ], [ 94, 72, 46, 63, 73, 107, 53, 77, 52, 134, 0, 134, 85, 92, 37, 119, 78, 107, 78, 57, 75 ], [ 148, 101, 126, 137, 129, 79, 145, 103, 130, 51, 134, 0, 133, 91, 120, 104, 125, 95, 142, 97, 115 ], [ 51, 79, 92, 75, 131, 82, 125, 106, 59, 107, 85, 133, 0, 67, 80, 105, 120, 52, 141, 84, 102 ], [ 76, 64, 92, 96, 118, 73, 120, 73, 68, 65, 92, 91, 67, 0, 81, 69, 104, 40, 130, 87, 75 ], [ 96, 83, 75, 56, 55, 97, 70, 53, 47, 122, 37, 120, 80, 81, 0, 124, 58, 98, 105, 40, 75 ], [ 102, 83, 93, 146, 149, 108, 138, 112, 98, 83, 119, 104, 105, 69, 124, 0, 137, 77, 119, 121, 96 ], [ 143, 119, 109, 109, 46, 136, 107, 39, 78, 133, 78, 125, 120, 104, 58, 137, 0, 127, 109, 75, 113 ], [ 74, 68, 101, 99, 142, 58, 139, 103, 84, 58, 107, 95, 52, 40, 98, 77, 127, 0, 144, 92, 100 ], [ 148, 95, 60, 133, 110, 145, 89, 115, 109, 151, 78, 142, 141, 130, 105, 119, 109, 144, 0, 105, 115 ], [ 105, 80, 75, 63, 68, 77, 81, 71, 67, 107, 57, 97, 84, 87, 40, 121, 75, 92, 105, 0, 85 ], [ 81, 77, 72, 95, 97, 92, 64, 84, 74, 118, 75, 115, 102, 75, 75, 96, 113, 100, 115, 85, 0 ] ], "m": 4, "objective": 827.0 }, "solution": [ 0, 4, 9, 18 ], "obj": 827.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 21, "m": 4, "nodes": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 ], "distance_pairs": [ { "from_id": 0, "to_id": 1, "distance": 85 }, { "from_id": 0, "to_id": 2, "distance": 91 }, { "from_id": 0, "to_id": 3, "distance": 90 }, { "from_id": 0, "to_id": 4, "distance": 143 }, { "from_id": 0, "to_id": 5, "distance": 93 }, { "from_id": 0, "to_id": 6, "distance": 115 }, { "from_id": 0, "to_id": 7, "distance": 121 }, { "from_id": 0, "to_id": 8, "distance": 69 }, { "from_id": 0, "to_id": 9, "distance": 126 }, { "from_id": 0, "to_id": 10, "distance": 94 }, { "from_id": 0, "to_id": 11, "distance": 148 }, { "from_id": 0, "to_id": 12, "distance": 51 }, { "from_id": 0, "to_id": 13, "distance": 76 }, { "from_id": 0, "to_id": 14, "distance": 96 }, { "from_id": 0, "to_id": 15, "distance": 102 }, { "from_id": 0, "to_id": 16, "distance": 143 }, { "from_id": 0, "to_id": 17, "distance": 74 }, { "from_id": 0, "to_id": 18, "distance": 148 }, { "from_id": 0, "to_id": 19, "distance": 105 }, { "from_id": 0, "to_id": 20, "distance": 81 }, { "from_id": 1, "to_id": 0, "distance": 85 }, { "from_id": 1, "to_id": 2, "distance": 53 }, { "from_id": 1, "to_id": 3, "distance": 80 }, { "from_id": 1, "to_id": 4, "distance": 124 }, { "from_id": 1, "to_id": 5, "distance": 66 }, { "from_id": 1, "to_id": 6, "distance": 94 }, { "from_id": 1, "to_id": 7, "distance": 101 }, { "from_id": 1, "to_id": 8, "distance": 84 }, { "from_id": 1, "to_id": 9, "distance": 88 }, { "from_id": 1, "to_id": 10, "distance": 72 }, { "from_id": 1, "to_id": 11, "distance": 101 }, { "from_id": 1, "to_id": 12, "distance": 79 }, { "from_id": 1, "to_id": 13, "distance": 64 }, { "from_id": 1, "to_id": 14, "distance": 83 }, { "from_id": 1, "to_id": 15, "distance": 83 }, { "from_id": 1, "to_id": 16, "distance": 119 }, { "from_id": 1, "to_id": 17, "distance": 68 }, { "from_id": 1, "to_id": 18, "distance": 95 }, { "from_id": 1, "to_id": 19, "distance": 80 }, { "from_id": 1, "to_id": 20, "distance": 77 }, { "from_id": 2, "to_id": 0, "distance": 91 }, { "from_id": 2, "to_id": 1, "distance": 53 }, { "from_id": 2, "to_id": 3, "distance": 90 }, { "from_id": 2, "to_id": 4, "distance": 104 }, { "from_id": 2, "to_id": 5, "distance": 100 }, { "from_id": 2, "to_id": 6, "distance": 63 }, { "from_id": 2, "to_id": 7, "distance": 101 }, { "from_id": 2, "to_id": 8, "distance": 70 }, { "from_id": 2, "to_id": 9, "distance": 125 }, { "from_id": 2, "to_id": 10, "distance": 46 }, { "from_id": 2, "to_id": 11, "distance": 126 }, { "from_id": 2, "to_id": 12, "distance": 92 }, { "from_id": 2, "to_id": 13, "distance": 92 }, { "from_id": 2, "to_id": 14, "distance": 75 }, { "from_id": 2, "to_id": 15, "distance": 93 }, { "from_id": 2, "to_id": 16, "distance": 109 }, { "from_id": 2, "to_id": 17, "distance": 101 }, { "from_id": 2, "to_id": 18, "distance": 60 }, { "from_id": 2, "to_id": 19, "distance": 75 }, { "from_id": 2, "to_id": 20, "distance": 72 }, { "from_id": 3, "to_id": 0, "distance": 90 }, { "from_id": 3, "to_id": 1, "distance": 80 }, { "from_id": 3, "to_id": 2, "distance": 90 }, { "from_id": 3, "to_id": 4, "distance": 104 }, { "from_id": 3, "to_id": 5, "distance": 83 }, { "from_id": 3, "to_id": 6, "distance": 89 }, { "from_id": 3, "to_id": 7, "distance": 102 }, { "from_id": 3, "to_id": 8, "distance": 81 }, { "from_id": 3, "to_id": 9, "distance": 132 }, { "from_id": 3, "to_id": 10, "distance": 63 }, { "from_id": 3, "to_id": 11, "distance": 137 }, { "from_id": 3, "to_id": 12, "distance": 75 }, { "from_id": 3, "to_id": 13, "distance": 96 }, { "from_id": 3, "to_id": 14, "distance": 56 }, { "from_id": 3, "to_id": 15, "distance": 146 }, { "from_id": 3, "to_id": 16, "distance": 109 }, { "from_id": 3, "to_id": 17, "distance": 99 }, { "from_id": 3, "to_id": 18, "distance": 133 }, { "from_id": 3, "to_id": 19, "distance": 63 }, { "from_id": 3, "to_id": 20, "distance": 95 }, { "from_id": 4, "to_id": 0, "distance": 143 }, { "from_id": 4, "to_id": 1, "distance": 124 }, { "from_id": 4, "to_id": 2, "distance": 104 }, { "from_id": 4, "to_id": 3, "distance": 104 }, { "from_id": 4, "to_id": 5, "distance": 135 }, { "from_id": 4, "to_id": 6, "distance": 79 }, { "from_id": 4, "to_id": 7, "distance": 56 }, { "from_id": 4, "to_id": 8, "distance": 84 }, { "from_id": 4, "to_id": 9, "distance": 149 }, { "from_id": 4, "to_id": 10, "distance": 73 }, { "from_id": 4, "to_id": 11, "distance": 129 }, { "from_id": 4, "to_id": 12, "distance": 131 }, { "from_id": 4, "to_id": 13, "distance": 118 }, { "from_id": 4, "to_id": 14, "distance": 55 }, { "from_id": 4, "to_id": 15, "distance": 149 }, { "from_id": 4, "to_id": 16, "distance": 46 }, { "from_id": 4, "to_id": 17, "distance": 142 }, { "from_id": 4, "to_id": 18, "distance": 110 }, { "from_id": 4, "to_id": 19, "distance": 68 }, { "from_id": 4, "to_id": 20, "distance": 97 }, { "from_id": 5, "to_id": 0, "distance": 93 }, { "from_id": 5, "to_id": 1, "distance": 66 }, { "from_id": 5, "to_id": 2, "distance": 100 }, { "from_id": 5, "to_id": 3, "distance": 83 }, { "from_id": 5, "to_id": 4, "distance": 135 }, { "from_id": 5, "to_id": 6, "distance": 122 }, { "from_id": 5, "to_id": 7, "distance": 114 }, { "from_id": 5, "to_id": 8, "distance": 105 }, { "from_id": 5, "to_id": 9, "distance": 70 }, { "from_id": 5, "to_id": 10, "distance": 107 }, { "from_id": 5, "to_id": 11, "distance": 79 }, { "from_id": 5, "to_id": 12, "distance": 82 }, { "from_id": 5, "to_id": 13, "distance": 73 }, { "from_id": 5, "to_id": 14, "distance": 97 }, { "from_id": 5, "to_id": 15, "distance": 108 }, { "from_id": 5, "to_id": 16, "distance": 136 }, { "from_id": 5, "to_id": 17, "distance": 58 }, { "from_id": 5, "to_id": 18, "distance": 145 }, { "from_id": 5, "to_id": 19, "distance": 77 }, { "from_id": 5, "to_id": 20, "distance": 92 }, { "from_id": 6, "to_id": 0, "distance": 115 }, { "from_id": 6, "to_id": 1, "distance": 94 }, { "from_id": 6, "to_id": 2, "distance": 63 }, { "from_id": 6, "to_id": 3, "distance": 89 }, { "from_id": 6, "to_id": 4, "distance": 79 }, { "from_id": 6, "to_id": 5, "distance": 122 }, { "from_id": 6, "to_id": 7, "distance": 101 }, { "from_id": 6, "to_id": 8, "distance": 88 }, { "from_id": 6, "to_id": 9, "distance": 158 }, { "from_id": 6, "to_id": 10, "distance": 53 }, { "from_id": 6, "to_id": 11, "distance": 145 }, { "from_id": 6, "to_id": 12, "distance": 125 }, { "from_id": 6, "to_id": 13, "distance": 120 }, { "from_id": 6, "to_id": 14, "distance": 70 }, { "from_id": 6, "to_id": 15, "distance": 138 }, { "from_id": 6, "to_id": 16, "distance": 107 }, { "from_id": 6, "to_id": 17, "distance": 139 }, { "from_id": 6, "to_id": 18, "distance": 89 }, { "from_id": 6, "to_id": 19, "distance": 81 }, { "from_id": 6, "to_id": 20, "distance": 64 }, { "from_id": 7, "to_id": 0, "distance": 121 }, { "from_id": 7, "to_id": 1, "distance": 101 }, { "from_id": 7, "to_id": 2, "distance": 101 }, { "from_id": 7, "to_id": 3, "distance": 102 }, { "from_id": 7, "to_id": 4, "distance": 56 }, { "from_id": 7, "to_id": 5, "distance": 114 }, { "from_id": 7, "to_id": 6, "distance": 101 }, { "from_id": 7, "to_id": 8, "distance": 66 }, { "from_id": 7, "to_id": 9, "distance": 108 }, { "from_id": 7, "to_id": 10, "distance": 77 }, { "from_id": 7, "to_id": 11, "distance": 103 }, { "from_id": 7, "to_id": 12, "distance": 106 }, { "from_id": 7, "to_id": 13, "distance": 73 }, { "from_id": 7, "to_id": 14, "distance": 53 }, { "from_id": 7, "to_id": 15, "distance": 112 }, { "from_id": 7, "to_id": 16, 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{ "from_id": 8, "to_id": 18, "distance": 109 }, { "from_id": 8, "to_id": 19, "distance": 67 }, { "from_id": 8, "to_id": 20, "distance": 74 }, { "from_id": 9, "to_id": 0, "distance": 126 }, { "from_id": 9, "to_id": 1, "distance": 88 }, { "from_id": 9, "to_id": 2, "distance": 125 }, { "from_id": 9, "to_id": 3, "distance": 132 }, { "from_id": 9, "to_id": 4, "distance": 149 }, { "from_id": 9, "to_id": 5, "distance": 70 }, { "from_id": 9, "to_id": 6, "distance": 158 }, { "from_id": 9, "to_id": 7, "distance": 108 }, { "from_id": 9, "to_id": 8, "distance": 120 }, { "from_id": 9, "to_id": 10, "distance": 134 }, { "from_id": 9, "to_id": 11, "distance": 51 }, { "from_id": 9, "to_id": 12, "distance": 107 }, { "from_id": 9, "to_id": 13, "distance": 65 }, { "from_id": 9, "to_id": 14, "distance": 122 }, { "from_id": 9, "to_id": 15, "distance": 83 }, { "from_id": 9, "to_id": 16, "distance": 133 }, { "from_id": 9, "to_id": 17, "distance": 58 }, { "from_id": 9, "to_id": 18, "distance": 151 }, { 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{ "from_id": 10, "to_id": 20, "distance": 75 }, { "from_id": 11, "to_id": 0, "distance": 148 }, { "from_id": 11, "to_id": 1, "distance": 101 }, { "from_id": 11, "to_id": 2, "distance": 126 }, { "from_id": 11, "to_id": 3, "distance": 137 }, { "from_id": 11, "to_id": 4, "distance": 129 }, { "from_id": 11, "to_id": 5, "distance": 79 }, { "from_id": 11, "to_id": 6, "distance": 145 }, { "from_id": 11, "to_id": 7, "distance": 103 }, { "from_id": 11, "to_id": 8, "distance": 130 }, { "from_id": 11, "to_id": 9, "distance": 51 }, { "from_id": 11, "to_id": 10, "distance": 134 }, { "from_id": 11, "to_id": 12, "distance": 133 }, { "from_id": 11, "to_id": 13, "distance": 91 }, { "from_id": 11, "to_id": 14, "distance": 120 }, { "from_id": 11, "to_id": 15, "distance": 104 }, { "from_id": 11, "to_id": 16, "distance": 125 }, { "from_id": 11, "to_id": 17, "distance": 95 }, { "from_id": 11, "to_id": 18, "distance": 142 }, { "from_id": 11, "to_id": 19, "distance": 97 }, { "from_id": 11, "to_id": 20, "distance": 115 }, { "from_id": 12, "to_id": 0, "distance": 51 }, { "from_id": 12, "to_id": 1, "distance": 79 }, { "from_id": 12, "to_id": 2, "distance": 92 }, { "from_id": 12, "to_id": 3, "distance": 75 }, { "from_id": 12, "to_id": 4, "distance": 131 }, { "from_id": 12, "to_id": 5, "distance": 82 }, { "from_id": 12, "to_id": 6, "distance": 125 }, { "from_id": 12, "to_id": 7, "distance": 106 }, { "from_id": 12, "to_id": 8, "distance": 59 }, { "from_id": 12, "to_id": 9, "distance": 107 }, { "from_id": 12, "to_id": 10, "distance": 85 }, { "from_id": 12, "to_id": 11, "distance": 133 }, { "from_id": 12, "to_id": 13, "distance": 67 }, { "from_id": 12, "to_id": 14, "distance": 80 }, { "from_id": 12, "to_id": 15, "distance": 105 }, { "from_id": 12, "to_id": 16, "distance": 120 }, { "from_id": 12, "to_id": 17, "distance": 52 }, { "from_id": 12, "to_id": 18, "distance": 141 }, { "from_id": 12, "to_id": 19, "distance": 84 }, { "from_id": 12, "to_id": 20, "distance": 102 }, { "from_id": 13, "to_id": 0, "distance": 76 }, { "from_id": 13, "to_id": 1, "distance": 64 }, { "from_id": 13, "to_id": 2, "distance": 92 }, { "from_id": 13, "to_id": 3, "distance": 96 }, { "from_id": 13, "to_id": 4, "distance": 118 }, { "from_id": 13, "to_id": 5, "distance": 73 }, { "from_id": 13, "to_id": 6, "distance": 120 }, { "from_id": 13, "to_id": 7, "distance": 73 }, { "from_id": 13, "to_id": 8, "distance": 68 }, { "from_id": 13, "to_id": 9, "distance": 65 }, { "from_id": 13, "to_id": 10, "distance": 92 }, { "from_id": 13, "to_id": 11, "distance": 91 }, { "from_id": 13, "to_id": 12, "distance": 67 }, { "from_id": 13, "to_id": 14, "distance": 81 }, { "from_id": 13, "to_id": 15, "distance": 69 }, { "from_id": 13, "to_id": 16, "distance": 104 }, { "from_id": 13, "to_id": 17, "distance": 40 }, { "from_id": 13, "to_id": 18, "distance": 130 }, { "from_id": 13, "to_id": 19, "distance": 87 }, { "from_id": 13, "to_id": 20, "distance": 75 }, { "from_id": 14, "to_id": 0, "distance": 96 }, { "from_id": 14, "to_id": 1, "distance": 83 }, { "from_id": 14, "to_id": 2, "distance": 75 }, { "from_id": 14, "to_id": 3, "distance": 56 }, { "from_id": 14, "to_id": 4, "distance": 55 }, { "from_id": 14, "to_id": 5, "distance": 97 }, { "from_id": 14, "to_id": 6, "distance": 70 }, { "from_id": 14, "to_id": 7, "distance": 53 }, { "from_id": 14, "to_id": 8, "distance": 47 }, { "from_id": 14, "to_id": 9, "distance": 122 }, { "from_id": 14, "to_id": 10, "distance": 37 }, { "from_id": 14, "to_id": 11, "distance": 120 }, { "from_id": 14, "to_id": 12, "distance": 80 }, { "from_id": 14, "to_id": 13, "distance": 81 }, { "from_id": 14, "to_id": 15, "distance": 124 }, { "from_id": 14, "to_id": 16, "distance": 58 }, { "from_id": 14, "to_id": 17, "distance": 98 }, { "from_id": 14, "to_id": 18, "distance": 105 }, { "from_id": 14, "to_id": 19, "distance": 40 }, { "from_id": 14, "to_id": 20, "distance": 75 }, { "from_id": 15, "to_id": 0, "distance": 102 }, { "from_id": 15, "to_id": 1, "distance": 83 }, { "from_id": 15, "to_id": 2, "distance": 93 }, { "from_id": 15, "to_id": 3, "distance": 146 }, { "from_id": 15, "to_id": 4, "distance": 149 }, { "from_id": 15, "to_id": 5, "distance": 108 }, { "from_id": 15, "to_id": 6, "distance": 138 }, { "from_id": 15, "to_id": 7, "distance": 112 }, { "from_id": 15, "to_id": 8, "distance": 98 }, { "from_id": 15, "to_id": 9, "distance": 83 }, { "from_id": 15, "to_id": 10, "distance": 119 }, { "from_id": 15, "to_id": 11, "distance": 104 }, { "from_id": 15, "to_id": 12, "distance": 105 }, { "from_id": 15, "to_id": 13, "distance": 69 }, { "from_id": 15, "to_id": 14, "distance": 124 }, { "from_id": 15, "to_id": 16, "distance": 137 }, { "from_id": 15, "to_id": 17, "distance": 77 }, { "from_id": 15, "to_id": 18, "distance": 119 }, { "from_id": 15, "to_id": 19, "distance": 121 }, { "from_id": 15, "to_id": 20, "distance": 96 }, { "from_id": 16, "to_id": 0, "distance": 143 }, { "from_id": 16, "to_id": 1, "distance": 119 }, { "from_id": 16, "to_id": 2, "distance": 109 }, { "from_id": 16, "to_id": 3, "distance": 109 }, { "from_id": 16, "to_id": 4, "distance": 46 }, { "from_id": 16, "to_id": 5, "distance": 136 }, { "from_id": 16, "to_id": 6, "distance": 107 }, { "from_id": 16, "to_id": 7, "distance": 39 }, { "from_id": 16, "to_id": 8, "distance": 78 }, { "from_id": 16, "to_id": 9, "distance": 133 }, { "from_id": 16, "to_id": 10, "distance": 78 }, { "from_id": 16, "to_id": 11, "distance": 125 }, { "from_id": 16, "to_id": 12, "distance": 120 }, { "from_id": 16, "to_id": 13, "distance": 104 }, { "from_id": 16, "to_id": 14, "distance": 58 }, { "from_id": 16, "to_id": 15, "distance": 137 }, { "from_id": 16, "to_id": 17, "distance": 127 }, { "from_id": 16, "to_id": 18, "distance": 109 }, { "from_id": 16, "to_id": 19, "distance": 75 }, { "from_id": 16, "to_id": 20, "distance": 113 }, { "from_id": 17, "to_id": 0, "distance": 74 }, { "from_id": 17, "to_id": 1, "distance": 68 }, { "from_id": 17, "to_id": 2, "distance": 101 }, { "from_id": 17, "to_id": 3, "distance": 99 }, { "from_id": 17, "to_id": 4, "distance": 142 }, { "from_id": 17, "to_id": 5, "distance": 58 }, { "from_id": 17, "to_id": 6, "distance": 139 }, { "from_id": 17, "to_id": 7, "distance": 103 }, { "from_id": 17, "to_id": 8, "distance": 84 }, { "from_id": 17, "to_id": 9, "distance": 58 }, { "from_id": 17, "to_id": 10, "distance": 107 }, { "from_id": 17, "to_id": 11, "distance": 95 }, { "from_id": 17, "to_id": 12, "distance": 52 }, { "from_id": 17, "to_id": 13, "distance": 40 }, { "from_id": 17, "to_id": 14, "distance": 98 }, { "from_id": 17, "to_id": 15, "distance": 77 }, { "from_id": 17, "to_id": 16, "distance": 127 }, { "from_id": 17, "to_id": 18, "distance": 144 }, { "from_id": 17, "to_id": 19, "distance": 92 }, { "from_id": 17, "to_id": 20, "distance": 100 }, { "from_id": 18, "to_id": 0, "distance": 148 }, { "from_id": 18, "to_id": 1, "distance": 95 }, { "from_id": 18, "to_id": 2, "distance": 60 }, { "from_id": 18, "to_id": 3, "distance": 133 }, { "from_id": 18, "to_id": 4, "distance": 110 }, { "from_id": 18, "to_id": 5, "distance": 145 }, { "from_id": 18, "to_id": 6, "distance": 89 }, { "from_id": 18, "to_id": 7, "distance": 115 }, { "from_id": 18, "to_id": 8, "distance": 109 }, { "from_id": 18, "to_id": 9, "distance": 151 }, { "from_id": 18, "to_id": 10, "distance": 78 }, { "from_id": 18, "to_id": 11, "distance": 142 }, { "from_id": 18, "to_id": 12, "distance": 141 }, { "from_id": 18, "to_id": 13, "distance": 130 }, { "from_id": 18, "to_id": 14, "distance": 105 }, { "from_id": 18, "to_id": 15, "distance": 119 }, { "from_id": 18, "to_id": 16, "distance": 109 }, { "from_id": 18, "to_id": 17, "distance": 144 }, { "from_id": 18, "to_id": 19, "distance": 105 }, { "from_id": 18, "to_id": 20, "distance": 115 }, { "from_id": 19, "to_id": 0, "distance": 105 }, { "from_id": 19, "to_id": 1, "distance": 80 }, { "from_id": 19, "to_id": 2, "distance": 75 }, { "from_id": 19, "to_id": 3, "distance": 63 }, { "from_id": 19, "to_id": 4, "distance": 68 }, { "from_id": 19, "to_id": 5, "distance": 77 }, { "from_id": 19, "to_id": 6, "distance": 81 }, { "from_id": 19, "to_id": 7, "distance": 71 }, { "from_id": 19, "to_id": 8, "distance": 67 }, { "from_id": 19, "to_id": 9, "distance": 107 }, { "from_id": 19, "to_id": 10, "distance": 57 }, { "from_id": 19, "to_id": 11, "distance": 97 }, { "from_id": 19, "to_id": 12, "distance": 84 }, { "from_id": 19, "to_id": 13, "distance": 87 }, { "from_id": 19, "to_id": 14, "distance": 40 }, { "from_id": 19, "to_id": 15, "distance": 121 }, { "from_id": 19, "to_id": 16, "distance": 75 }, { "from_id": 19, "to_id": 17, "distance": 92 }, { "from_id": 19, "to_id": 18, "distance": 105 }, { "from_id": 19, "to_id": 20, "distance": 85 }, { "from_id": 20, "to_id": 0, "distance": 81 }, { "from_id": 20, "to_id": 1, "distance": 77 }, { "from_id": 20, "to_id": 2, "distance": 72 }, { "from_id": 20, "to_id": 3, "distance": 95 }, { "from_id": 20, "to_id": 4, "distance": 97 }, { "from_id": 20, "to_id": 5, "distance": 92 }, { "from_id": 20, "to_id": 6, "distance": 64 }, { "from_id": 20, "to_id": 7, "distance": 84 }, { "from_id": 20, "to_id": 8, "distance": 74 }, { "from_id": 20, "to_id": 9, "distance": 118 }, { "from_id": 20, "to_id": 10, "distance": 75 }, { "from_id": 20, "to_id": 11, "distance": 115 }, { "from_id": 20, "to_id": 12, "distance": 102 }, { "from_id": 20, "to_id": 13, "distance": 75 }, { "from_id": 20, "to_id": 14, "distance": 75 }, { "from_id": 20, "to_id": 15, "distance": 96 }, { "from_id": 20, "to_id": 16, "distance": 113 }, { "from_id": 20, "to_id": 17, "distance": 100 }, { "from_id": 20, "to_id": 18, "distance": 115 }, { "from_id": 20, "to_id": 19, "distance": 85 } ] }, "solution_variant": [ 0, 4, 9, 18 ], "context_index": 31, "input_format": "json", "input_index_base": 0 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "At the newsroom meeting the brief was short: assemble exactly the requested number of headline drafts, no repeats, and cover as many distinct tones and angles as possible. The tidy way to score a lineup is to measure how different each pair of drafts is and then add up those pairwise differences — the best collection is the one with the largest combined difference. The exact list of drafts and the number to pick follow below.\n\nThere are 22 drafts—IDs: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21—pick exactly 2 distinct headline drafts.\nPair 0 vs 1: tone dissimilarity 1.\nPair 0 vs 2: tone dissimilarity 4.\nPair 0 vs 3: tone dissimilarity 0.\nPair 0 vs 4: tone dissimilarity 6.\nPair 0 vs 5: tone dissimilarity 1.\nPair 0 vs 6: tone dissimilarity 7.\nPair 0 vs 7: tone dissimilarity 0.\nPair 0 vs 8: tone dissimilarity 8.\nPair 0 vs 9: tone dissimilarity 6.\nPair 0 vs 10: tone dissimilarity 5.\nPair 0 vs 11: tone dissimilarity 9.\nPair 0 vs 12: tone dissimilarity 8.\nPair 0 vs 13: tone dissimilarity 6.\nPair 0 vs 14: tone dissimilarity 8.\nPair 0 vs 15: tone dissimilarity 3.\nPair 0 vs 16: tone dissimilarity 2.\nPair 0 vs 17: tone dissimilarity 9.\nPair 0 vs 18: tone dissimilarity 3.\nPair 0 vs 19: tone dissimilarity 6.\nPair 0 vs 20: tone dissimilarity 8.\nPair 0 vs 21: tone dissimilarity 7.\nPair 1 vs 0: tone dissimilarity 1.\nPair 1 vs 2: tone dissimilarity 5.\nPair 1 vs 3: tone dissimilarity 2.\nPair 1 vs 4: tone dissimilarity 6.\nPair 1 vs 5: tone dissimilarity 9.\nPair 1 vs 6: tone dissimilarity 6.\nPair 1 vs 7: tone dissimilarity 4.\nPair 1 vs 8: tone dissimilarity 7.\nPair 1 vs 9: tone dissimilarity 7.\nPair 1 vs 10: tone dissimilarity 4.\nPair 1 vs 11: tone dissimilarity 5.\nPair 1 vs 12: tone dissimilarity 5.\nPair 1 vs 13: tone dissimilarity 5.\nPair 1 vs 14: tone dissimilarity 0.\nPair 1 vs 15: tone dissimilarity 1.\nPair 1 vs 16: tone dissimilarity 8.\nPair 1 vs 17: tone dissimilarity 6.\nPair 1 vs 18: tone dissimilarity 7.\nPair 1 vs 19: tone dissimilarity 9.\nPair 1 vs 20: tone dissimilarity 6.\nPair 1 vs 21: tone dissimilarity 9.\nPair 2 vs 0: tone dissimilarity 4.\nPair 2 vs 1: tone dissimilarity 5.\nPair 2 vs 3: tone dissimilarity 0.\nPair 2 vs 4: tone dissimilarity 2.\nPair 2 vs 5: tone dissimilarity 8.\nPair 2 vs 6: tone dissimilarity 5.\nPair 2 vs 7: tone dissimilarity 4.\nPair 2 vs 8: tone dissimilarity 0.\nPair 2 vs 9: tone dissimilarity 8.\nPair 2 vs 10: tone dissimilarity 0.\nPair 2 vs 11: tone dissimilarity 8.\nPair 2 vs 12: tone dissimilarity 8.\nPair 2 vs 13: tone dissimilarity 6.\nPair 2 vs 14: tone dissimilarity 8.\nPair 2 vs 15: tone dissimilarity 7.\nPair 2 vs 16: tone dissimilarity 3.\nPair 2 vs 17: tone dissimilarity 0.\nPair 2 vs 18: tone dissimilarity 3.\nPair 2 vs 19: tone dissimilarity 5.\nPair 2 vs 20: tone dissimilarity 0.\nPair 2 vs 21: tone dissimilarity 0.\nPair 3 vs 0: tone dissimilarity 0.\nPair 3 vs 1: tone dissimilarity 2.\nPair 3 vs 2: tone dissimilarity 0.\nPair 3 vs 4: tone dissimilarity 7.\nPair 3 vs 5: tone dissimilarity 1.\nPair 3 vs 6: tone dissimilarity 0.\nPair 3 vs 7: tone dissimilarity 5.\nPair 3 vs 8: tone dissimilarity 5.\nPair 3 vs 9: tone dissimilarity 5.\nPair 3 vs 10: tone dissimilarity 6.\nPair 3 vs 11: tone dissimilarity 8.\nPair 3 vs 12: tone dissimilarity 3.\nPair 3 vs 13: tone dissimilarity 1.\nPair 3 vs 14: tone dissimilarity 0.\nPair 3 vs 15: tone dissimilarity 5.\nPair 3 vs 16: tone dissimilarity 4.\nPair 3 vs 17: tone dissimilarity 9.\nPair 3 vs 18: tone dissimilarity 1.\nPair 3 vs 19: tone dissimilarity 8.\nPair 3 vs 20: tone dissimilarity 2.\nPair 3 vs 21: tone dissimilarity 7.\nPair 4 vs 0: tone dissimilarity 6.\nPair 4 vs 1: tone dissimilarity 6.\nPair 4 vs 2: tone dissimilarity 2.\nPair 4 vs 3: tone dissimilarity 7.\nPair 4 vs 5: tone dissimilarity 5.\nPair 4 vs 6: tone dissimilarity 0.\nPair 4 vs 7: tone dissimilarity 2.\nPair 4 vs 8: tone dissimilarity 0.\nPair 4 vs 9: tone dissimilarity 4.\nPair 4 vs 10: tone dissimilarity 4.\nPair 4 vs 11: tone dissimilarity 7.\nPair 4 vs 12: tone dissimilarity 5.\nPair 4 vs 13: tone dissimilarity 3.\nPair 4 vs 14: tone dissimilarity 3.\nPair 4 vs 15: tone dissimilarity 8.\nPair 4 vs 16: tone dissimilarity 6.\nPair 4 vs 17: tone dissimilarity 1.\nPair 4 vs 18: tone dissimilarity 3.\nPair 4 vs 19: tone dissimilarity 3.\nPair 4 vs 20: tone dissimilarity 2.\nPair 4 vs 21: tone dissimilarity 9.\nPair 5 vs 0: tone dissimilarity 1.\nPair 5 vs 1: tone dissimilarity 9.\nPair 5 vs 2: tone dissimilarity 8.\nPair 5 vs 3: tone dissimilarity 1.\nPair 5 vs 4: tone dissimilarity 5.\nPair 5 vs 6: tone dissimilarity 2.\nPair 5 vs 7: tone dissimilarity 8.\nPair 5 vs 8: tone dissimilarity 1.\nPair 5 vs 9: tone dissimilarity 1.\nPair 5 vs 10: tone dissimilarity 9.\nPair 5 vs 11: tone dissimilarity 5.\nPair 5 vs 12: tone dissimilarity 4.\nPair 5 vs 13: tone dissimilarity 2.\nPair 5 vs 14: tone dissimilarity 9.\nPair 5 vs 15: tone dissimilarity 3.\nPair 5 vs 16: tone dissimilarity 2.\nPair 5 vs 17: tone dissimilarity 5.\nPair 5 vs 18: tone dissimilarity 1.\nPair 5 vs 19: tone dissimilarity 1.\nPair 5 vs 20: tone dissimilarity 8.\nPair 5 vs 21: tone dissimilarity 7.\nPair 6 vs 0: tone dissimilarity 7.\nPair 6 vs 1: tone dissimilarity 6.\nPair 6 vs 2: tone dissimilarity 5.\nPair 6 vs 3: tone dissimilarity 0.\nPair 6 vs 4: tone dissimilarity 0.\nPair 6 vs 5: tone dissimilarity 2.\nPair 6 vs 7: tone dissimilarity 5.\nPair 6 vs 8: tone dissimilarity 2.\nPair 6 vs 9: tone dissimilarity 5.\nPair 6 vs 10: tone dissimilarity 5.\nPair 6 vs 11: tone dissimilarity 0.\nPair 6 vs 12: tone dissimilarity 4.\nPair 6 vs 13: tone dissimilarity 2.\nPair 6 vs 14: tone dissimilarity 6.\nPair 6 vs 15: tone dissimilarity 2.\nPair 6 vs 16: tone dissimilarity 0.\nPair 6 vs 17: tone dissimilarity 0.\nPair 6 vs 18: tone dissimilarity 5.\nPair 6 vs 19: tone dissimilarity 6.\nPair 6 vs 20: tone dissimilarity 2.\nPair 6 vs 21: tone dissimilarity 0.\nPair 7 vs 0: tone dissimilarity 0.\nPair 7 vs 1: tone dissimilarity 4.\nPair 7 vs 2: tone dissimilarity 4.\nPair 7 vs 3: tone dissimilarity 5.\nPair 7 vs 4: tone dissimilarity 2.\nPair 7 vs 5: tone dissimilarity 8.\nPair 7 vs 6: tone dissimilarity 5.\nPair 7 vs 8: tone dissimilarity 6.\nPair 7 vs 9: tone dissimilarity 9.\nPair 7 vs 10: tone dissimilarity 4.\nPair 7 vs 11: tone dissimilarity 6.\nPair 7 vs 12: tone dissimilarity 0.\nPair 7 vs 13: tone dissimilarity 2.\nPair 7 vs 14: tone dissimilarity 3.\nPair 7 vs 15: tone dissimilarity 9.\nPair 7 vs 16: tone dissimilarity 1.\nPair 7 vs 17: tone dissimilarity 6.\nPair 7 vs 18: tone dissimilarity 8.\nPair 7 vs 19: tone dissimilarity 6.\nPair 7 vs 20: tone dissimilarity 3.\nPair 7 vs 21: tone dissimilarity 8.\nPair 8 vs 0: tone dissimilarity 8.\nPair 8 vs 1: tone dissimilarity 7.\nPair 8 vs 2: tone dissimilarity 0.\nPair 8 vs 3: tone dissimilarity 5.\nPair 8 vs 4: tone dissimilarity 0.\nPair 8 vs 5: tone dissimilarity 1.\nPair 8 vs 6: tone dissimilarity 2.\nPair 8 vs 7: tone dissimilarity 6.\nPair 8 vs 9: tone dissimilarity 2.\nPair 8 vs 10: tone dissimilarity 8.\nPair 8 vs 11: tone dissimilarity 1.\nPair 8 vs 12: tone dissimilarity 4.\nPair 8 vs 13: tone dissimilarity 1.\nPair 8 vs 14: tone dissimilarity 9.\nPair 8 vs 15: tone dissimilarity 0.\nPair 8 vs 16: tone dissimilarity 3.\nPair 8 vs 17: tone dissimilarity 6.\nPair 8 vs 18: tone dissimilarity 5.\nPair 8 vs 19: tone dissimilarity 7.\nPair 8 vs 20: tone dissimilarity 3.\nPair 8 vs 21: tone dissimilarity 2.\nPair 9 vs 0: tone dissimilarity 6.\nPair 9 vs 1: tone dissimilarity 7.\nPair 9 vs 2: tone dissimilarity 8.\nPair 9 vs 3: tone dissimilarity 5.\nPair 9 vs 4: tone dissimilarity 4.\nPair 9 vs 5: tone dissimilarity 1.\nPair 9 vs 6: tone dissimilarity 5.\nPair 9 vs 7: tone dissimilarity 9.\nPair 9 vs 8: tone dissimilarity 2.\nPair 9 vs 10: tone dissimilarity 8.\nPair 9 vs 11: tone dissimilarity 0.\nPair 9 vs 12: tone dissimilarity 0.\nPair 9 vs 13: tone dissimilarity 2.\nPair 9 vs 14: tone dissimilarity 2.\nPair 9 vs 15: tone dissimilarity 6.\nPair 9 vs 16: tone dissimilarity 4.\nPair 9 vs 17: tone dissimilarity 1.\nPair 9 vs 18: tone dissimilarity 5.\nPair 9 vs 19: tone dissimilarity 0.\nPair 9 vs 20: tone dissimilarity 1.\nPair 9 vs 21: tone dissimilarity 3.\nPair 10 vs 0: tone dissimilarity 5.\nPair 10 vs 1: tone dissimilarity 4.\nPair 10 vs 2: tone dissimilarity 0.\nPair 10 vs 3: tone dissimilarity 6.\nPair 10 vs 4: tone dissimilarity 4.\nPair 10 vs 5: tone dissimilarity 9.\nPair 10 vs 6: tone dissimilarity 5.\nPair 10 vs 7: tone dissimilarity 4.\nPair 10 vs 8: tone dissimilarity 8.\nPair 10 vs 9: tone dissimilarity 8.\nPair 10 vs 11: tone dissimilarity 9.\nPair 10 vs 12: tone dissimilarity 4.\nPair 10 vs 13: tone dissimilarity 4.\nPair 10 vs 14: tone dissimilarity 9.\nPair 10 vs 15: tone dissimilarity 9.\nPair 10 vs 16: tone dissimilarity 1.\nPair 10 vs 17: tone dissimilarity 4.\nPair 10 vs 18: tone dissimilarity 7.\nPair 10 vs 19: tone dissimilarity 3.\nPair 10 vs 20: tone dissimilarity 8.\nPair 10 vs 21: tone dissimilarity 1.\nPair 11 vs 0: tone dissimilarity 9.\nPair 11 vs 1: tone dissimilarity 5.\nPair 11 vs 2: tone dissimilarity 8.\nPair 11 vs 3: tone dissimilarity 8.\nPair 11 vs 4: tone dissimilarity 7.\nPair 11 vs 5: tone dissimilarity 5.\nPair 11 vs 6: tone dissimilarity 0.\nPair 11 vs 7: tone dissimilarity 6.\nPair 11 vs 8: tone dissimilarity 1.\nPair 11 vs 9: tone dissimilarity 0.\nPair 11 vs 10: tone dissimilarity 9.\nPair 11 vs 12: tone dissimilarity 6.\nPair 11 vs 13: tone dissimilarity 5.\nPair 11 vs 14: tone dissimilarity 5.\nPair 11 vs 15: tone dissimilarity 8.\nPair 11 vs 16: tone dissimilarity 7.\nPair 11 vs 17: tone dissimilarity 3.\nPair 11 vs 18: tone dissimilarity 2.\nPair 11 vs 19: tone dissimilarity 1.\nPair 11 vs 20: tone dissimilarity 8.\nPair 11 vs 21: tone dissimilarity 3.\nPair 12 vs 0: tone dissimilarity 8.\nPair 12 vs 1: tone dissimilarity 5.\nPair 12 vs 2: tone dissimilarity 8.\nPair 12 vs 3: tone dissimilarity 3.\nPair 12 vs 4: tone dissimilarity 5.\nPair 12 vs 5: tone dissimilarity 4.\nPair 12 vs 6: tone dissimilarity 4.\nPair 12 vs 7: tone dissimilarity 0.\nPair 12 vs 8: tone dissimilarity 4.\nPair 12 vs 9: tone dissimilarity 0.\nPair 12 vs 10: tone dissimilarity 4.\nPair 12 vs 11: tone dissimilarity 6.\nPair 12 vs 13: tone dissimilarity 9.\nPair 12 vs 14: tone dissimilarity 1.\nPair 12 vs 15: tone dissimilarity 1.\nPair 12 vs 16: tone dissimilarity 8.\nPair 12 vs 17: tone dissimilarity 1.\nPair 12 vs 18: tone dissimilarity 0.\nPair 12 vs 19: tone dissimilarity 4.\nPair 12 vs 20: tone dissimilarity 8.\nPair 12 vs 21: tone dissimilarity 6.\nPair 13 vs 0: tone dissimilarity 6.\nPair 13 vs 1: tone dissimilarity 5.\nPair 13 vs 2: tone dissimilarity 6.\nPair 13 vs 3: tone dissimilarity 1.\nPair 13 vs 4: tone dissimilarity 3.\nPair 13 vs 5: tone dissimilarity 2.\nPair 13 vs 6: tone dissimilarity 2.\nPair 13 vs 7: tone dissimilarity 2.\nPair 13 vs 8: tone dissimilarity 1.\nPair 13 vs 9: tone dissimilarity 2.\nPair 13 vs 10: tone dissimilarity 4.\nPair 13 vs 11: tone dissimilarity 5.\nPair 13 vs 12: tone dissimilarity 9.\nPair 13 vs 14: tone dissimilarity 0.\nPair 13 vs 15: tone dissimilarity 4.\nPair 13 vs 16: tone dissimilarity 6.\nPair 13 vs 17: tone dissimilarity 4.\nPair 13 vs 18: tone dissimilarity 0.\nPair 13 vs 19: tone dissimilarity 0.\nPair 13 vs 20: tone dissimilarity 6.\nPair 13 vs 21: tone dissimilarity 0.\nPair 14 vs 0: tone dissimilarity 8.\nPair 14 vs 1: tone dissimilarity 0.\nPair 14 vs 2: tone dissimilarity 8.\nPair 14 vs 3: tone dissimilarity 0.\nPair 14 vs 4: tone dissimilarity 3.\nPair 14 vs 5: tone dissimilarity 9.\nPair 14 vs 6: tone dissimilarity 6.\nPair 14 vs 7: tone dissimilarity 3.\nPair 14 vs 8: tone dissimilarity 9.\nPair 14 vs 9: tone dissimilarity 2.\nPair 14 vs 10: tone dissimilarity 9.\nPair 14 vs 11: tone dissimilarity 5.\nPair 14 vs 12: tone dissimilarity 1.\nPair 14 vs 13: tone dissimilarity 0.\nPair 14 vs 15: tone dissimilarity 0.\nPair 14 vs 16: tone dissimilarity 2.\nPair 14 vs 17: tone dissimilarity 4.\nPair 14 vs 18: tone dissimilarity 8.\nPair 14 vs 19: tone dissimilarity 2.\nPair 14 vs 20: tone dissimilarity 3.\nPair 14 vs 21: tone dissimilarity 1.\nPair 15 vs 0: tone dissimilarity 3.\nPair 15 vs 1: tone dissimilarity 1.\nPair 15 vs 2: tone dissimilarity 7.\nPair 15 vs 3: tone dissimilarity 5.\nPair 15 vs 4: tone dissimilarity 8.\nPair 15 vs 5: tone dissimilarity 3.\nPair 15 vs 6: tone dissimilarity 2.\nPair 15 vs 7: tone dissimilarity 9.\nPair 15 vs 8: tone dissimilarity 0.\nPair 15 vs 9: tone dissimilarity 6.\nPair 15 vs 10: tone dissimilarity 9.\nPair 15 vs 11: tone dissimilarity 8.\nPair 15 vs 12: tone dissimilarity 1.\nPair 15 vs 13: tone dissimilarity 4.\nPair 15 vs 14: tone dissimilarity 0.\nPair 15 vs 16: tone dissimilarity 7.\nPair 15 vs 17: tone dissimilarity 7.\nPair 15 vs 18: tone dissimilarity 3.\nPair 15 vs 19: tone dissimilarity 4.\nPair 15 vs 20: tone dissimilarity 1.\nPair 15 vs 21: tone dissimilarity 2.\nPair 16 vs 0: tone dissimilarity 2.\nPair 16 vs 1: tone dissimilarity 8.\nPair 16 vs 2: tone dissimilarity 3.\nPair 16 vs 3: tone dissimilarity 4.\nPair 16 vs 4: tone dissimilarity 6.\nPair 16 vs 5: tone dissimilarity 2.\nPair 16 vs 6: tone dissimilarity 0.\nPair 16 vs 7: tone dissimilarity 1.\nPair 16 vs 8: tone dissimilarity 3.\nPair 16 vs 9: tone dissimilarity 4.\nPair 16 vs 10: tone dissimilarity 1.\nPair 16 vs 11: tone dissimilarity 7.\nPair 16 vs 12: tone dissimilarity 8.\nPair 16 vs 13: tone dissimilarity 6.\nPair 16 vs 14: tone dissimilarity 2.\nPair 16 vs 15: tone dissimilarity 7.\nPair 16 vs 17: tone dissimilarity 9.\nPair 16 vs 18: tone dissimilarity 7.\nPair 16 vs 19: tone dissimilarity 5.\nPair 16 vs 20: tone dissimilarity 8.\nPair 16 vs 21: tone dissimilarity 0.\nPair 17 vs 0: tone dissimilarity 9.\nPair 17 vs 1: tone dissimilarity 6.\nPair 17 vs 2: tone dissimilarity 0.\nPair 17 vs 3: tone dissimilarity 9.\nPair 17 vs 4: tone dissimilarity 1.\nPair 17 vs 5: tone dissimilarity 5.\nPair 17 vs 6: tone dissimilarity 0.\nPair 17 vs 7: tone dissimilarity 6.\nPair 17 vs 8: tone dissimilarity 6.\nPair 17 vs 9: tone dissimilarity 1.\nPair 17 vs 10: tone dissimilarity 4.\nPair 17 vs 11: tone dissimilarity 3.\nPair 17 vs 12: tone dissimilarity 1.\nPair 17 vs 13: tone dissimilarity 4.\nPair 17 vs 14: tone dissimilarity 4.\nPair 17 vs 15: tone dissimilarity 7.\nPair 17 vs 16: tone dissimilarity 9.\nPair 17 vs 18: tone dissimilarity 1.\nPair 17 vs 19: tone dissimilarity 3.\nPair 17 vs 20: tone dissimilarity 0.\nPair 17 vs 21: tone dissimilarity 4.\nPair 18 vs 0: tone dissimilarity 3.\nPair 18 vs 1: tone dissimilarity 7.\nPair 18 vs 2: tone dissimilarity 3.\nPair 18 vs 3: tone dissimilarity 1.\nPair 18 vs 4: tone dissimilarity 3.\nPair 18 vs 5: tone dissimilarity 1.\nPair 18 vs 6: tone dissimilarity 5.\nPair 18 vs 7: tone dissimilarity 8.\nPair 18 vs 8: tone dissimilarity 5.\nPair 18 vs 9: tone dissimilarity 5.\nPair 18 vs 10: tone dissimilarity 7.\nPair 18 vs 11: tone dissimilarity 2.\nPair 18 vs 12: tone dissimilarity 0.\nPair 18 vs 13: tone dissimilarity 0.\nPair 18 vs 14: tone dissimilarity 8.\nPair 18 vs 15: tone dissimilarity 3.\nPair 18 vs 16: tone dissimilarity 7.\nPair 18 vs 17: tone dissimilarity 1.\nPair 18 vs 19: tone dissimilarity 7.\nPair 18 vs 20: tone dissimilarity 3.\nPair 18 vs 21: tone dissimilarity 5.\nPair 19 vs 0: tone dissimilarity 6.\nPair 19 vs 1: tone dissimilarity 9.\nPair 19 vs 2: tone dissimilarity 5.\nPair 19 vs 3: tone dissimilarity 8.\nPair 19 vs 4: tone dissimilarity 3.\nPair 19 vs 5: tone dissimilarity 1.\nPair 19 vs 6: tone dissimilarity 6.\nPair 19 vs 7: tone dissimilarity 6.\nPair 19 vs 8: tone dissimilarity 7.\nPair 19 vs 9: tone dissimilarity 0.\nPair 19 vs 10: tone dissimilarity 3.\nPair 19 vs 11: tone dissimilarity 1.\nPair 19 vs 12: tone dissimilarity 4.\nPair 19 vs 13: tone dissimilarity 0.\nPair 19 vs 14: tone dissimilarity 2.\nPair 19 vs 15: tone dissimilarity 4.\nPair 19 vs 16: tone dissimilarity 5.\nPair 19 vs 17: tone dissimilarity 3.\nPair 19 vs 18: tone dissimilarity 7.\nPair 19 vs 20: tone dissimilarity 8.\nPair 19 vs 21: tone dissimilarity 2.\nPair 20 vs 0: tone dissimilarity 8.\nPair 20 vs 1: tone dissimilarity 6.\nPair 20 vs 2: tone dissimilarity 0.\nPair 20 vs 3: tone dissimilarity 2.\nPair 20 vs 4: tone dissimilarity 2.\nPair 20 vs 5: tone dissimilarity 8.\nPair 20 vs 6: tone dissimilarity 2.\nPair 20 vs 7: tone dissimilarity 3.\nPair 20 vs 8: tone dissimilarity 3.\nPair 20 vs 9: tone dissimilarity 1.\nPair 20 vs 10: tone dissimilarity 8.\nPair 20 vs 11: tone dissimilarity 8.\nPair 20 vs 12: tone dissimilarity 8.\nPair 20 vs 13: tone dissimilarity 6.\nPair 20 vs 14: tone dissimilarity 3.\nPair 20 vs 15: tone dissimilarity 1.\nPair 20 vs 16: tone dissimilarity 8.\nPair 20 vs 17: tone dissimilarity 0.\nPair 20 vs 18: tone dissimilarity 3.\nPair 20 vs 19: tone dissimilarity 8.\nPair 20 vs 21: tone dissimilarity 9.\nPair 21 vs 0: tone dissimilarity 7.\nPair 21 vs 1: tone dissimilarity 9.\nPair 21 vs 2: tone dissimilarity 0.\nPair 21 vs 3: tone dissimilarity 7.\nPair 21 vs 4: tone dissimilarity 9.\nPair 21 vs 5: tone dissimilarity 7.\nPair 21 vs 6: tone dissimilarity 0.\nPair 21 vs 7: tone dissimilarity 8.\nPair 21 vs 8: tone dissimilarity 2.\nPair 21 vs 9: tone dissimilarity 3.\nPair 21 vs 10: tone dissimilarity 1.\nPair 21 vs 11: tone dissimilarity 3.\nPair 21 vs 12: tone dissimilarity 6.\nPair 21 vs 13: tone dissimilarity 0.\nPair 21 vs 14: tone dissimilarity 1.\nPair 21 vs 15: tone dissimilarity 2.\nPair 21 vs 16: tone dissimilarity 0.\nPair 21 vs 17: tone dissimilarity 4.\nPair 21 vs 18: tone dissimilarity 5.\nPair 21 vs 19: tone dissimilarity 2.\nPair 21 vs 20: tone dissimilarity 9.\nScore lineups by summing the pairwise dissimilarities and select the 2 drafts that maximize the total.\n\nWhen you send the final pick, just drop it into this little JSON shape so it's easy to read and process:\n\n{\n \"solution\": [, , ...]\n}\n\nPretty simple — \"solution\" is the list of the headline drafts you picked (just their IDs, nothing else). This JSON is only a sketch of the expected shape, not the actual answer itself.\n\nPlease make sure to use the exact identifiers from the instance input — no renaming and no new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 1, 4, 0, 6, 1, 7, 0, 8, 6, 5, 9, 8, 6, 8, 3, 2, 9, 3, 6, 8, 7 ], [ 1, 0, 5, 2, 6, 9, 6, 4, 7, 7, 4, 5, 5, 5, 0, 1, 8, 6, 7, 9, 6, 9 ], [ 4, 5, 0, 0, 2, 8, 5, 4, 0, 8, 0, 8, 8, 6, 8, 7, 3, 0, 3, 5, 0, 0 ], [ 0, 2, 0, 0, 7, 1, 0, 5, 5, 5, 6, 8, 3, 1, 0, 5, 4, 9, 1, 8, 2, 7 ], [ 6, 6, 2, 7, 0, 5, 0, 2, 0, 4, 4, 7, 5, 3, 3, 8, 6, 1, 3, 3, 2, 9 ], [ 1, 9, 8, 1, 5, 0, 2, 8, 1, 1, 9, 5, 4, 2, 9, 3, 2, 5, 1, 1, 8, 7 ], [ 7, 6, 5, 0, 0, 2, 0, 5, 2, 5, 5, 0, 4, 2, 6, 2, 0, 0, 5, 6, 2, 0 ], [ 0, 4, 4, 5, 2, 8, 5, 0, 6, 9, 4, 6, 0, 2, 3, 9, 1, 6, 8, 6, 3, 8 ], [ 8, 7, 0, 5, 0, 1, 2, 6, 0, 2, 8, 1, 4, 1, 9, 0, 3, 6, 5, 7, 3, 2 ], [ 6, 7, 8, 5, 4, 1, 5, 9, 2, 0, 8, 0, 0, 2, 2, 6, 4, 1, 5, 0, 1, 3 ], [ 5, 4, 0, 6, 4, 9, 5, 4, 8, 8, 0, 9, 4, 4, 9, 9, 1, 4, 7, 3, 8, 1 ], [ 9, 5, 8, 8, 7, 5, 0, 6, 1, 0, 9, 0, 6, 5, 5, 8, 7, 3, 2, 1, 8, 3 ], [ 8, 5, 8, 3, 5, 4, 4, 0, 4, 0, 4, 6, 0, 9, 1, 1, 8, 1, 0, 4, 8, 6 ], [ 6, 5, 6, 1, 3, 2, 2, 2, 1, 2, 4, 5, 9, 0, 0, 4, 6, 4, 0, 0, 6, 0 ], [ 8, 0, 8, 0, 3, 9, 6, 3, 9, 2, 9, 5, 1, 0, 0, 0, 2, 4, 8, 2, 3, 1 ], [ 3, 1, 7, 5, 8, 3, 2, 9, 0, 6, 9, 8, 1, 4, 0, 0, 7, 7, 3, 4, 1, 2 ], [ 2, 8, 3, 4, 6, 2, 0, 1, 3, 4, 1, 7, 8, 6, 2, 7, 0, 9, 7, 5, 8, 0 ], [ 9, 6, 0, 9, 1, 5, 0, 6, 6, 1, 4, 3, 1, 4, 4, 7, 9, 0, 1, 3, 0, 4 ], [ 3, 7, 3, 1, 3, 1, 5, 8, 5, 5, 7, 2, 0, 0, 8, 3, 7, 1, 0, 7, 3, 5 ], [ 6, 9, 5, 8, 3, 1, 6, 6, 7, 0, 3, 1, 4, 0, 2, 4, 5, 3, 7, 0, 8, 2 ], [ 8, 6, 0, 2, 2, 8, 2, 3, 3, 1, 8, 8, 8, 6, 3, 1, 8, 0, 3, 8, 0, 9 ], [ 7, 9, 0, 7, 9, 7, 0, 8, 2, 3, 1, 3, 6, 0, 1, 2, 0, 4, 5, 2, 9, 0 ] ], "m": 2, "objective": 9.0 }, "solution": [ 12, 13 ], "obj": 9.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 22, "m": 2, "nodes": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 ], "distance_pairs": [ { "from_id": 0, "to_id": 1, "distance": 1 }, { "from_id": 0, "to_id": 2, "distance": 4 }, { "from_id": 0, "to_id": 3, "distance": 0 }, { "from_id": 0, "to_id": 4, "distance": 6 }, { "from_id": 0, "to_id": 5, "distance": 1 }, { "from_id": 0, "to_id": 6, "distance": 7 }, { "from_id": 0, "to_id": 7, "distance": 0 }, { "from_id": 0, "to_id": 8, "distance": 8 }, { "from_id": 0, "to_id": 9, "distance": 6 }, { "from_id": 0, "to_id": 10, "distance": 5 }, { "from_id": 0, "to_id": 11, "distance": 9 }, { "from_id": 0, "to_id": 12, "distance": 8 }, { "from_id": 0, "to_id": 13, "distance": 6 }, { "from_id": 0, "to_id": 14, "distance": 8 }, { "from_id": 0, "to_id": 15, "distance": 3 }, { "from_id": 0, "to_id": 16, "distance": 2 }, { "from_id": 0, "to_id": 17, "distance": 9 }, { "from_id": 0, "to_id": 18, "distance": 3 }, { "from_id": 0, "to_id": 19, "distance": 6 }, { "from_id": 0, "to_id": 20, "distance": 8 }, { "from_id": 0, "to_id": 21, "distance": 7 }, { "from_id": 1, "to_id": 0, "distance": 1 }, { "from_id": 1, "to_id": 2, "distance": 5 }, { "from_id": 1, "to_id": 3, "distance": 2 }, { "from_id": 1, "to_id": 4, "distance": 6 }, { "from_id": 1, "to_id": 5, "distance": 9 }, { "from_id": 1, "to_id": 6, "distance": 6 }, { "from_id": 1, "to_id": 7, "distance": 4 }, { "from_id": 1, "to_id": 8, "distance": 7 }, { "from_id": 1, "to_id": 9, "distance": 7 }, { "from_id": 1, "to_id": 10, "distance": 4 }, { "from_id": 1, "to_id": 11, "distance": 5 }, { "from_id": 1, "to_id": 12, "distance": 5 }, { "from_id": 1, "to_id": 13, "distance": 5 }, { "from_id": 1, "to_id": 14, "distance": 0 }, { "from_id": 1, "to_id": 15, "distance": 1 }, { "from_id": 1, "to_id": 16, "distance": 8 }, { "from_id": 1, "to_id": 17, "distance": 6 }, { "from_id": 1, "to_id": 18, "distance": 7 }, { "from_id": 1, "to_id": 19, "distance": 9 }, { "from_id": 1, "to_id": 20, "distance": 6 }, { "from_id": 1, "to_id": 21, "distance": 9 }, { "from_id": 2, "to_id": 0, "distance": 4 }, { "from_id": 2, "to_id": 1, "distance": 5 }, { "from_id": 2, "to_id": 3, "distance": 0 }, { "from_id": 2, "to_id": 4, "distance": 2 }, { "from_id": 2, "to_id": 5, "distance": 8 }, { "from_id": 2, "to_id": 6, "distance": 5 }, { "from_id": 2, "to_id": 7, "distance": 4 }, { "from_id": 2, "to_id": 8, "distance": 0 }, { "from_id": 2, "to_id": 9, "distance": 8 }, { "from_id": 2, "to_id": 10, "distance": 0 }, { "from_id": 2, "to_id": 11, "distance": 8 }, { "from_id": 2, "to_id": 12, "distance": 8 }, { "from_id": 2, "to_id": 13, "distance": 6 }, { "from_id": 2, "to_id": 14, "distance": 8 }, { "from_id": 2, "to_id": 15, "distance": 7 }, { "from_id": 2, "to_id": 16, "distance": 3 }, { "from_id": 2, "to_id": 17, "distance": 0 }, { "from_id": 2, "to_id": 18, "distance": 3 }, { "from_id": 2, "to_id": 19, "distance": 5 }, { "from_id": 2, "to_id": 20, "distance": 0 }, { "from_id": 2, "to_id": 21, "distance": 0 }, { "from_id": 3, "to_id": 0, "distance": 0 }, { "from_id": 3, "to_id": 1, "distance": 2 }, { "from_id": 3, "to_id": 2, "distance": 0 }, { "from_id": 3, "to_id": 4, "distance": 7 }, { "from_id": 3, "to_id": 5, "distance": 1 }, { "from_id": 3, "to_id": 6, "distance": 0 }, { "from_id": 3, "to_id": 7, "distance": 5 }, { "from_id": 3, "to_id": 8, "distance": 5 }, { "from_id": 3, "to_id": 9, "distance": 5 }, { "from_id": 3, "to_id": 10, "distance": 6 }, { "from_id": 3, "to_id": 11, "distance": 8 }, { "from_id": 3, "to_id": 12, "distance": 3 }, { "from_id": 3, "to_id": 13, "distance": 1 }, { "from_id": 3, "to_id": 14, "distance": 0 }, { "from_id": 3, "to_id": 15, "distance": 5 }, { "from_id": 3, "to_id": 16, "distance": 4 }, { "from_id": 3, "to_id": 17, "distance": 9 }, { "from_id": 3, "to_id": 18, "distance": 1 }, { "from_id": 3, "to_id": 19, "distance": 8 }, { "from_id": 3, "to_id": 20, "distance": 2 }, { "from_id": 3, "to_id": 21, "distance": 7 }, { "from_id": 4, "to_id": 0, "distance": 6 }, { "from_id": 4, "to_id": 1, "distance": 6 }, { "from_id": 4, "to_id": 2, "distance": 2 }, { "from_id": 4, "to_id": 3, "distance": 7 }, { "from_id": 4, "to_id": 5, "distance": 5 }, { "from_id": 4, "to_id": 6, "distance": 0 }, { "from_id": 4, "to_id": 7, "distance": 2 }, { "from_id": 4, "to_id": 8, "distance": 0 }, { "from_id": 4, "to_id": 9, "distance": 4 }, { "from_id": 4, "to_id": 10, "distance": 4 }, { "from_id": 4, "to_id": 11, "distance": 7 }, { "from_id": 4, "to_id": 12, "distance": 5 }, { "from_id": 4, "to_id": 13, "distance": 3 }, { "from_id": 4, "to_id": 14, "distance": 3 }, { "from_id": 4, "to_id": 15, "distance": 8 }, { "from_id": 4, "to_id": 16, "distance": 6 }, { "from_id": 4, "to_id": 17, "distance": 1 }, { "from_id": 4, "to_id": 18, "distance": 3 }, { "from_id": 4, "to_id": 19, "distance": 3 }, { "from_id": 4, "to_id": 20, "distance": 2 }, { "from_id": 4, "to_id": 21, "distance": 9 }, { "from_id": 5, "to_id": 0, "distance": 1 }, { "from_id": 5, "to_id": 1, "distance": 9 }, { "from_id": 5, "to_id": 2, "distance": 8 }, { "from_id": 5, "to_id": 3, "distance": 1 }, { "from_id": 5, "to_id": 4, "distance": 5 }, { "from_id": 5, "to_id": 6, "distance": 2 }, { "from_id": 5, "to_id": 7, "distance": 8 }, { "from_id": 5, "to_id": 8, "distance": 1 }, { "from_id": 5, "to_id": 9, "distance": 1 }, { "from_id": 5, "to_id": 10, "distance": 9 }, { "from_id": 5, "to_id": 11, "distance": 5 }, { "from_id": 5, "to_id": 12, "distance": 4 }, { "from_id": 5, "to_id": 13, "distance": 2 }, { "from_id": 5, "to_id": 14, "distance": 9 }, { "from_id": 5, "to_id": 15, "distance": 3 }, { "from_id": 5, "to_id": 16, "distance": 2 }, { "from_id": 5, "to_id": 17, "distance": 5 }, { "from_id": 5, "to_id": 18, "distance": 1 }, { "from_id": 5, "to_id": 19, "distance": 1 }, { "from_id": 5, "to_id": 20, "distance": 8 }, { "from_id": 5, "to_id": 21, "distance": 7 }, { "from_id": 6, "to_id": 0, "distance": 7 }, 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{\n \"location_a_id\": 19,\n \"location_b_id\": 17,\n \"dissimilarity_score\": 2\n },\n {\n \"location_a_id\": 19,\n \"location_b_id\": 18,\n \"dissimilarity_score\": 7\n },\n {\n \"location_a_id\": 19,\n \"location_b_id\": 20,\n \"dissimilarity_score\": 5\n },\n {\n \"location_a_id\": 19,\n \"location_b_id\": 21,\n \"dissimilarity_score\": 4\n },\n {\n \"location_a_id\": 19,\n \"location_b_id\": 22,\n \"dissimilarity_score\": 8\n },\n {\n \"location_a_id\": 19,\n \"location_b_id\": 23,\n \"dissimilarity_score\": 6\n },\n {\n \"location_a_id\": 19,\n \"location_b_id\": 24,\n \"dissimilarity_score\": 2\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 1,\n \"dissimilarity_score\": 6\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 2,\n \"dissimilarity_score\": 3\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 3,\n \"dissimilarity_score\": 6\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 4,\n \"dissimilarity_score\": 4\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 5,\n \"dissimilarity_score\": 0\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 6,\n \"dissimilarity_score\": 9\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 7,\n \"dissimilarity_score\": 9\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 8,\n \"dissimilarity_score\": 4\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 9,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 10,\n \"dissimilarity_score\": 3\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 11,\n \"dissimilarity_score\": 4\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 12,\n \"dissimilarity_score\": 5\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 13,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 14,\n \"dissimilarity_score\": 4\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 15,\n \"dissimilarity_score\": 0\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 16,\n \"dissimilarity_score\": 3\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 17,\n \"dissimilarity_score\": 6\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 18,\n \"dissimilarity_score\": 2\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 19,\n \"dissimilarity_score\": 5\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 21,\n \"dissimilarity_score\": 2\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 22,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 23,\n \"dissimilarity_score\": 5\n },\n {\n \"location_a_id\": 20,\n \"location_b_id\": 24,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 1,\n \"dissimilarity_score\": 4\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 2,\n \"dissimilarity_score\": 4\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 3,\n \"dissimilarity_score\": 6\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 4,\n \"dissimilarity_score\": 5\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 5,\n \"dissimilarity_score\": 3\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 6,\n \"dissimilarity_score\": 0\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 7,\n \"dissimilarity_score\": 6\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 8,\n \"dissimilarity_score\": 7\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 9,\n \"dissimilarity_score\": 7\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 10,\n \"dissimilarity_score\": 6\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 11,\n \"dissimilarity_score\": 8\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 12,\n \"dissimilarity_score\": 2\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 13,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 14,\n \"dissimilarity_score\": 8\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 15,\n \"dissimilarity_score\": 7\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 16,\n \"dissimilarity_score\": 5\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 17,\n \"dissimilarity_score\": 2\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 18,\n \"dissimilarity_score\": 8\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 19,\n \"dissimilarity_score\": 4\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 20,\n \"dissimilarity_score\": 2\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 22,\n \"dissimilarity_score\": 0\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 23,\n \"dissimilarity_score\": 3\n },\n {\n \"location_a_id\": 21,\n \"location_b_id\": 24,\n \"dissimilarity_score\": 9\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 1,\n \"dissimilarity_score\": 3\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 2,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 3,\n \"dissimilarity_score\": 9\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 4,\n \"dissimilarity_score\": 3\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 5,\n \"dissimilarity_score\": 5\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 6,\n \"dissimilarity_score\": 8\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 7,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 8,\n \"dissimilarity_score\": 3\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 9,\n \"dissimilarity_score\": 6\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 10,\n \"dissimilarity_score\": 2\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 11,\n \"dissimilarity_score\": 0\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 12,\n \"dissimilarity_score\": 8\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 13,\n \"dissimilarity_score\": 0\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 14,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 15,\n \"dissimilarity_score\": 3\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 16,\n \"dissimilarity_score\": 8\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 17,\n \"dissimilarity_score\": 6\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 18,\n \"dissimilarity_score\": 6\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 19,\n \"dissimilarity_score\": 8\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 20,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 21,\n \"dissimilarity_score\": 0\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 23,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 22,\n \"location_b_id\": 24,\n \"dissimilarity_score\": 3\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 1,\n \"dissimilarity_score\": 4\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 2,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 3,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 4,\n \"dissimilarity_score\": 6\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 5,\n \"dissimilarity_score\": 3\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 6,\n \"dissimilarity_score\": 9\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 7,\n \"dissimilarity_score\": 6\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 8,\n \"dissimilarity_score\": 0\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 9,\n \"dissimilarity_score\": 3\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 10,\n \"dissimilarity_score\": 2\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 11,\n \"dissimilarity_score\": 4\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 12,\n \"dissimilarity_score\": 6\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 13,\n \"dissimilarity_score\": 5\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 14,\n \"dissimilarity_score\": 2\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 15,\n \"dissimilarity_score\": 5\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 16,\n \"dissimilarity_score\": 0\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 17,\n \"dissimilarity_score\": 3\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 18,\n \"dissimilarity_score\": 6\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 19,\n \"dissimilarity_score\": 6\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 20,\n \"dissimilarity_score\": 5\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 21,\n \"dissimilarity_score\": 3\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 22,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 23,\n \"location_b_id\": 24,\n \"dissimilarity_score\": 3\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 1,\n \"dissimilarity_score\": 5\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 2,\n \"dissimilarity_score\": 2\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 3,\n \"dissimilarity_score\": 8\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 4,\n \"dissimilarity_score\": 2\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 5,\n \"dissimilarity_score\": 2\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 6,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 7,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 8,\n \"dissimilarity_score\": 5\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 9,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 10,\n \"dissimilarity_score\": 0\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 11,\n \"dissimilarity_score\": 4\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 12,\n \"dissimilarity_score\": 9\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 13,\n \"dissimilarity_score\": 9\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 14,\n \"dissimilarity_score\": 6\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 15,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 16,\n \"dissimilarity_score\": 2\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 17,\n \"dissimilarity_score\": 8\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 18,\n \"dissimilarity_score\": 0\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 19,\n \"dissimilarity_score\": 2\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 20,\n \"dissimilarity_score\": 1\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 21,\n \"dissimilarity_score\": 9\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 22,\n \"dissimilarity_score\": 3\n },\n {\n \"location_a_id\": 24,\n \"location_b_id\": 23,\n \"dissimilarity_score\": 3\n }\n ]\n}\n\nAlso, to keep things machine-friendly for whoever's collecting the picks, please drop the final selection into a tiny JSON object like this — super simple:\n\n{\n \"solution\": [, , ...]\n}\n\nThink of \"solution\" as just the list of picnic spots we pick — one identifier per spot. This JSON is just a sketch of the shape I expect (not the real answer yet), so when you fill it in use the actual IDs from the instance.\n\nPlease make sure you use the identifiers exactly as they appear in the input — no renaming, no made-up labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”. \"", "instance": { "distance_matrix": [ [ 0, 7, 5, 8, 7, 2, 0, 6, 2, 1, 8, 0, 2, 7, 5, 1, 2, 0, 2, 6, 4, 3, 4, 5 ], [ 7, 0, 0, 2, 8, 7, 7, 0, 4, 1, 7, 9, 3, 7, 7, 5, 6, 8, 2, 3, 4, 1, 1, 2 ], [ 5, 0, 0, 0, 9, 8, 9, 7, 8, 1, 6, 9, 4, 8, 6, 1, 6, 6, 5, 6, 6, 9, 1, 8 ], [ 8, 2, 0, 0, 0, 9, 2, 6, 0, 3, 4, 1, 5, 9, 4, 3, 6, 9, 0, 4, 5, 3, 6, 2 ], [ 7, 8, 9, 0, 0, 7, 0, 5, 3, 8, 3, 1, 7, 7, 1, 0, 5, 2, 4, 0, 3, 5, 3, 2 ], [ 2, 7, 8, 9, 7, 0, 1, 7, 9, 4, 7, 4, 3, 4, 7, 0, 3, 2, 9, 9, 0, 8, 9, 1 ], [ 0, 7, 9, 2, 0, 1, 0, 2, 6, 8, 2, 0, 7, 1, 7, 9, 3, 7, 2, 9, 6, 1, 6, 1 ], [ 6, 0, 7, 6, 5, 7, 2, 0, 5, 6, 5, 6, 9, 3, 1, 3, 9, 4, 6, 4, 7, 3, 0, 5 ], [ 2, 4, 8, 0, 3, 9, 6, 5, 0, 4, 6, 0, 2, 3, 5, 3, 8, 4, 6, 1, 7, 6, 3, 1 ], [ 1, 1, 1, 3, 8, 4, 8, 6, 4, 0, 6, 3, 6, 2, 3, 0, 1, 0, 2, 3, 6, 2, 2, 0 ], [ 8, 7, 6, 4, 3, 7, 2, 5, 6, 6, 0, 0, 6, 7, 7, 1, 7, 8, 6, 4, 8, 0, 4, 4 ], [ 0, 9, 9, 1, 1, 4, 0, 6, 0, 3, 0, 0, 2, 6, 5, 3, 9, 6, 6, 5, 2, 8, 6, 9 ], [ 2, 3, 4, 5, 7, 3, 7, 9, 2, 6, 6, 2, 0, 1, 8, 0, 6, 3, 3, 1, 1, 0, 5, 9 ], [ 7, 7, 8, 9, 7, 4, 1, 3, 3, 2, 7, 6, 1, 0, 2, 3, 5, 2, 5, 4, 8, 1, 2, 6 ], [ 5, 7, 6, 4, 1, 7, 7, 1, 5, 3, 7, 5, 8, 2, 0, 3, 5, 9, 5, 0, 7, 3, 5, 1 ], [ 1, 5, 1, 3, 0, 0, 9, 3, 3, 0, 1, 3, 0, 3, 3, 0, 6, 9, 8, 3, 5, 8, 0, 2 ], [ 2, 6, 6, 6, 5, 3, 3, 9, 8, 1, 7, 9, 6, 5, 5, 6, 0, 6, 2, 6, 2, 6, 3, 8 ], [ 0, 8, 6, 9, 2, 2, 7, 4, 4, 0, 8, 6, 3, 2, 9, 9, 6, 0, 7, 2, 8, 6, 6, 0 ], [ 2, 2, 5, 0, 4, 9, 2, 6, 6, 2, 6, 6, 3, 5, 5, 8, 2, 7, 0, 5, 4, 8, 6, 2 ], [ 6, 3, 6, 4, 0, 9, 9, 4, 1, 3, 4, 5, 1, 4, 0, 3, 6, 2, 5, 0, 2, 1, 5, 1 ], [ 4, 4, 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"to_id": 19, "distance": 2 }, { "from_id": 24, "to_id": 20, "distance": 1 }, { "from_id": 24, "to_id": 21, "distance": 9 }, { "from_id": 24, "to_id": 22, "distance": 3 }, { "from_id": 24, "to_id": 23, "distance": 3 } ] }, "solution_variant": [ 16, 18 ], "context_index": 33, "input_format": "json", "input_index_base": 1 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "One afternoon the counselor sketched out that the best group wouldn't be made of similar students but of ones who complement each other in contrasting ways. The instruction was to select exactly the fixed number of students, and to check a group's quality the counselor adds up how far apart every pair in that group is — the higher that summed amount, the more diverse the group feels. Every slot has to be filled by a different student and the group size can’t be changed. The concrete instance and detailed pairwise differences are listed below.\n\n{\n \"total_students_count\": 21,\n \"group_size\": 6,\n \"student_ids\": [\n 1,\n 2,\n 3,\n 4,\n 5,\n 6,\n 7,\n 8,\n 9,\n 10,\n 11,\n 12,\n 13,\n 14,\n 15,\n 16,\n 17,\n 18,\n 19,\n 20,\n 21\n ],\n \"data\": [\n {\n \"student_id_a\": 1,\n \"student_id_b\": 2,\n \"dissimilarity_score\": 1\n },\n {\n \"student_id_a\": 1,\n \"student_id_b\": 3,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 1,\n \"student_id_b\": 4,\n \"dissimilarity_score\": 2\n },\n {\n \"student_id_a\": 1,\n \"student_id_b\": 5,\n \"dissimilarity_score\": 7\n },\n {\n \"student_id_a\": 1,\n \"student_id_b\": 6,\n \"dissimilarity_score\": 1\n },\n {\n \"student_id_a\": 1,\n \"student_id_b\": 7,\n \"dissimilarity_score\": 0\n },\n {\n \"student_id_a\": 1,\n \"student_id_b\": 8,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 1,\n \"student_id_b\": 9,\n \"dissimilarity_score\": 9\n },\n {\n \"student_id_a\": 1,\n 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2\n },\n {\n \"student_id_a\": 2,\n \"student_id_b\": 14,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 2,\n \"student_id_b\": 15,\n \"dissimilarity_score\": 9\n },\n {\n \"student_id_a\": 2,\n \"student_id_b\": 16,\n \"dissimilarity_score\": 4\n },\n {\n \"student_id_a\": 2,\n \"student_id_b\": 17,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 2,\n \"student_id_b\": 18,\n \"dissimilarity_score\": 2\n },\n {\n \"student_id_a\": 2,\n \"student_id_b\": 19,\n \"dissimilarity_score\": 3\n },\n {\n \"student_id_a\": 2,\n \"student_id_b\": 20,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 2,\n \"student_id_b\": 21,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 3,\n \"student_id_b\": 1,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 3,\n \"student_id_b\": 2,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 3,\n \"student_id_b\": 4,\n \"dissimilarity_score\": 4\n },\n {\n \"student_id_a\": 3,\n \"student_id_b\": 5,\n 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{\n \"student_id_a\": 15,\n \"student_id_b\": 10,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 15,\n \"student_id_b\": 11,\n \"dissimilarity_score\": 7\n },\n {\n \"student_id_a\": 15,\n \"student_id_b\": 12,\n \"dissimilarity_score\": 4\n },\n {\n \"student_id_a\": 15,\n \"student_id_b\": 13,\n \"dissimilarity_score\": 7\n },\n {\n \"student_id_a\": 15,\n \"student_id_b\": 14,\n \"dissimilarity_score\": 7\n },\n {\n \"student_id_a\": 15,\n \"student_id_b\": 16,\n \"dissimilarity_score\": 7\n },\n {\n \"student_id_a\": 15,\n \"student_id_b\": 17,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 15,\n \"student_id_b\": 18,\n \"dissimilarity_score\": 2\n },\n {\n \"student_id_a\": 15,\n \"student_id_b\": 19,\n \"dissimilarity_score\": 0\n },\n {\n \"student_id_a\": 15,\n \"student_id_b\": 20,\n \"dissimilarity_score\": 1\n },\n {\n \"student_id_a\": 15,\n \"student_id_b\": 21,\n \"dissimilarity_score\": 2\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 1,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 2,\n \"dissimilarity_score\": 4\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 3,\n \"dissimilarity_score\": 3\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 4,\n \"dissimilarity_score\": 3\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 5,\n \"dissimilarity_score\": 9\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 6,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 7,\n \"dissimilarity_score\": 3\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 8,\n \"dissimilarity_score\": 9\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 9,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 10,\n \"dissimilarity_score\": 2\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 11,\n \"dissimilarity_score\": 0\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 12,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 13,\n \"dissimilarity_score\": 3\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 14,\n \"dissimilarity_score\": 2\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 15,\n \"dissimilarity_score\": 7\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 17,\n \"dissimilarity_score\": 9\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 18,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 19,\n \"dissimilarity_score\": 1\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 20,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 16,\n \"student_id_b\": 21,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 1,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 2,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 3,\n \"dissimilarity_score\": 1\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 4,\n \"dissimilarity_score\": 7\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 5,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 6,\n \"dissimilarity_score\": 1\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 7,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 8,\n \"dissimilarity_score\": 1\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 9,\n \"dissimilarity_score\": 7\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 10,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 11,\n \"dissimilarity_score\": 0\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 12,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 13,\n \"dissimilarity_score\": 7\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 14,\n \"dissimilarity_score\": 3\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 15,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 16,\n \"dissimilarity_score\": 9\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 18,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 19,\n \"dissimilarity_score\": 1\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 20,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 17,\n \"student_id_b\": 21,\n \"dissimilarity_score\": 4\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 1,\n \"dissimilarity_score\": 4\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 2,\n \"dissimilarity_score\": 2\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 3,\n \"dissimilarity_score\": 2\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 4,\n \"dissimilarity_score\": 1\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 5,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 6,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 7,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 8,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 9,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 10,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 11,\n \"dissimilarity_score\": 7\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 12,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 13,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 14,\n \"dissimilarity_score\": 3\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 15,\n \"dissimilarity_score\": 2\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 16,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 17,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 19,\n \"dissimilarity_score\": 9\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 20,\n \"dissimilarity_score\": 2\n },\n {\n \"student_id_a\": 18,\n \"student_id_b\": 21,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 1,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 2,\n \"dissimilarity_score\": 3\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 3,\n \"dissimilarity_score\": 3\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 4,\n \"dissimilarity_score\": 2\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 5,\n \"dissimilarity_score\": 9\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 6,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 7,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 8,\n \"dissimilarity_score\": 4\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 9,\n \"dissimilarity_score\": 4\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 10,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 11,\n \"dissimilarity_score\": 9\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 12,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 13,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 14,\n \"dissimilarity_score\": 1\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 15,\n \"dissimilarity_score\": 0\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 16,\n \"dissimilarity_score\": 1\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 17,\n \"dissimilarity_score\": 1\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 18,\n \"dissimilarity_score\": 9\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 20,\n \"dissimilarity_score\": 2\n },\n {\n \"student_id_a\": 19,\n \"student_id_b\": 21,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 1,\n \"dissimilarity_score\": 7\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 2,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 3,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 4,\n \"dissimilarity_score\": 0\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 5,\n \"dissimilarity_score\": 9\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 6,\n \"dissimilarity_score\": 2\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 7,\n \"dissimilarity_score\": 7\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 8,\n \"dissimilarity_score\": 3\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 9,\n \"dissimilarity_score\": 0\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 10,\n \"dissimilarity_score\": 2\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 11,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 12,\n \"dissimilarity_score\": 9\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 13,\n \"dissimilarity_score\": 1\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 14,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 15,\n \"dissimilarity_score\": 1\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 16,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 17,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 18,\n \"dissimilarity_score\": 2\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 19,\n \"dissimilarity_score\": 2\n },\n {\n \"student_id_a\": 20,\n \"student_id_b\": 21,\n \"dissimilarity_score\": 9\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 1,\n \"dissimilarity_score\": 7\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 2,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 3,\n \"dissimilarity_score\": 7\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 4,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 5,\n \"dissimilarity_score\": 0\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 6,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 7,\n \"dissimilarity_score\": 4\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 8,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 9,\n \"dissimilarity_score\": 7\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 10,\n \"dissimilarity_score\": 1\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 11,\n \"dissimilarity_score\": 4\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 12,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 13,\n \"dissimilarity_score\": 9\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 14,\n \"dissimilarity_score\": 0\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 15,\n \"dissimilarity_score\": 2\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 16,\n \"dissimilarity_score\": 5\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 17,\n \"dissimilarity_score\": 4\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 18,\n \"dissimilarity_score\": 6\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 19,\n \"dissimilarity_score\": 8\n },\n {\n \"student_id_a\": 21,\n \"student_id_b\": 20,\n \"dissimilarity_score\": 9\n }\n ]\n}\n\nIf you want to send the selected group back, just use a little JSON like this so I can read it reliably:\n\n{\n \"solution\": [, , ...]\n}\n\nThis just shows the shape I expect: the \"solution\" entry is an array where each placeholder like stands in for one chosen student's identifier. It's only a sketch of the format, not the actual selection — you'll replace each placeholder with the real identifiers from the instance.\n\nPlease make sure you use the exact identifiers given in the instance input — do not rename them or invent new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 1, 6, 2, 7, 1, 0, 8, 9, 3, 0, 8, 3, 3, 2, 8, 8, 4, 8, 7, 7 ], [ 1, 0, 8, 4, 2, 2, 5, 2, 6, 5, 6, 4, 2, 6, 9, 4, 6, 2, 3, 8, 5 ], [ 6, 8, 0, 4, 2, 7, 2, 1, 9, 9, 2, 1, 7, 9, 2, 3, 1, 2, 3, 6, 7 ], [ 2, 4, 4, 0, 2, 7, 7, 6, 3, 8, 9, 8, 4, 1, 2, 3, 7, 1, 2, 0, 6 ], [ 7, 2, 2, 2, 0, 4, 5, 9, 8, 6, 1, 8, 9, 7, 7, 9, 8, 6, 9, 9, 0 ], [ 1, 2, 7, 7, 4, 0, 3, 0, 4, 3, 1, 7, 8, 0, 3, 8, 1, 5, 5, 2, 5 ], [ 0, 5, 2, 7, 5, 3, 0, 0, 7, 6, 8, 6, 9, 2, 2, 3, 8, 6, 5, 7, 4 ], [ 8, 2, 1, 6, 9, 0, 0, 0, 6, 1, 2, 7, 6, 3, 2, 9, 1, 8, 4, 3, 5 ], [ 9, 6, 9, 3, 8, 4, 7, 6, 0, 1, 3, 3, 3, 8, 9, 5, 7, 5, 4, 0, 7 ], [ 3, 5, 9, 8, 6, 3, 6, 1, 1, 0, 0, 1, 3, 0, 8, 2, 5, 8, 6, 2, 1 ], [ 0, 6, 2, 9, 1, 1, 8, 2, 3, 0, 0, 9, 9, 7, 7, 0, 0, 7, 9, 8, 4 ], [ 8, 4, 1, 8, 8, 7, 6, 7, 3, 1, 9, 0, 4, 6, 4, 6, 5, 5, 6, 9, 6 ], [ 3, 2, 7, 4, 9, 8, 9, 6, 3, 3, 9, 4, 0, 5, 7, 3, 7, 6, 5, 1, 9 ], [ 3, 6, 9, 1, 7, 0, 2, 3, 8, 0, 7, 6, 5, 0, 7, 2, 3, 3, 1, 8, 0 ], [ 2, 9, 2, 2, 7, 3, 2, 2, 9, 8, 7, 4, 7, 7, 0, 7, 5, 2, 0, 1, 2 ], [ 8, 4, 3, 3, 9, 8, 3, 9, 5, 2, 0, 6, 3, 2, 7, 0, 9, 5, 1, 6, 5 ], [ 8, 6, 1, 7, 8, 1, 8, 1, 7, 5, 0, 5, 7, 3, 5, 9, 0, 5, 1, 8, 4 ], [ 4, 2, 2, 1, 6, 5, 6, 8, 5, 8, 7, 5, 6, 3, 2, 5, 5, 0, 9, 2, 6 ], [ 8, 3, 3, 2, 9, 5, 5, 4, 4, 6, 9, 6, 5, 1, 0, 1, 1, 9, 0, 2, 8 ], [ 7, 8, 6, 0, 9, 2, 7, 3, 0, 2, 8, 9, 1, 8, 1, 6, 8, 2, 2, 0, 9 ], [ 7, 5, 7, 6, 0, 5, 4, 5, 7, 1, 4, 6, 9, 0, 2, 5, 4, 6, 8, 9, 0 ] ], "m": 6, "objective": 115.0 }, "solution": [ 0, 4, 11, 15, 16, 19 ], "obj": 115.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 21, "m": 6, "nodes": [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 ], "distance_pairs": [ { "from_id": 1, "to_id": 2, "distance": 1 }, { "from_id": 1, "to_id": 3, "distance": 6 }, { "from_id": 1, "to_id": 4, "distance": 2 }, { "from_id": 1, "to_id": 5, "distance": 7 }, { "from_id": 1, "to_id": 6, "distance": 1 }, { "from_id": 1, "to_id": 7, "distance": 0 }, { "from_id": 1, "to_id": 8, "distance": 8 }, { "from_id": 1, "to_id": 9, "distance": 9 }, { "from_id": 1, "to_id": 10, "distance": 3 }, { "from_id": 1, "to_id": 11, "distance": 0 }, { "from_id": 1, "to_id": 12, "distance": 8 }, { "from_id": 1, "to_id": 13, "distance": 3 }, { "from_id": 1, "to_id": 14, "distance": 3 }, { "from_id": 1, "to_id": 15, "distance": 2 }, { "from_id": 1, "to_id": 16, "distance": 8 }, { "from_id": 1, "to_id": 17, "distance": 8 }, { "from_id": 1, "to_id": 18, "distance": 4 }, { "from_id": 1, "to_id": 19, "distance": 8 }, { "from_id": 1, "to_id": 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6, "distance": 2 }, { "from_id": 20, "to_id": 7, "distance": 7 }, { "from_id": 20, "to_id": 8, "distance": 3 }, { "from_id": 20, "to_id": 9, "distance": 0 }, { "from_id": 20, "to_id": 10, "distance": 2 }, { "from_id": 20, "to_id": 11, "distance": 8 }, { "from_id": 20, "to_id": 12, "distance": 9 }, { "from_id": 20, "to_id": 13, "distance": 1 }, { "from_id": 20, "to_id": 14, "distance": 8 }, { "from_id": 20, "to_id": 15, "distance": 1 }, { "from_id": 20, "to_id": 16, "distance": 6 }, { "from_id": 20, "to_id": 17, "distance": 8 }, { "from_id": 20, "to_id": 18, "distance": 2 }, { "from_id": 20, "to_id": 19, "distance": 2 }, { "from_id": 20, "to_id": 21, "distance": 9 }, { "from_id": 21, "to_id": 1, "distance": 7 }, { "from_id": 21, "to_id": 2, "distance": 5 }, { "from_id": 21, "to_id": 3, "distance": 7 }, { "from_id": 21, "to_id": 4, "distance": 6 }, { "from_id": 21, "to_id": 5, "distance": 0 }, { "from_id": 21, "to_id": 6, "distance": 5 }, { "from_id": 21, "to_id": 7, "distance": 4 }, { "from_id": 21, "to_id": 8, "distance": 5 }, { "from_id": 21, "to_id": 9, "distance": 7 }, { "from_id": 21, "to_id": 10, "distance": 1 }, { "from_id": 21, "to_id": 11, "distance": 4 }, { "from_id": 21, "to_id": 12, "distance": 6 }, { "from_id": 21, "to_id": 13, "distance": 9 }, { "from_id": 21, "to_id": 14, "distance": 0 }, { "from_id": 21, "to_id": 15, "distance": 2 }, { "from_id": 21, "to_id": 16, "distance": 5 }, { "from_id": 21, "to_id": 17, "distance": 4 }, { "from_id": 21, "to_id": 18, "distance": 6 }, { "from_id": 21, "to_id": 19, "distance": 8 }, { "from_id": 21, "to_id": 20, "distance": 9 } ] }, "solution_variant": [ 1, 5, 12, 16, 17, 20 ], "context_index": 34, "input_format": "json", "input_index_base": 1 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Across town the plant club is planning a demo bed and the brief was clear: plant exactly m species so the display shows lots of variety. For every pair of candidate plants there’s a difference rating; the way to pick the best group is to add every pair’s rating together — the biggest sum means the most varied selection. Each species can be used only once and the bed must have exactly m plants. The concrete list of species and their pairwise ratings are listed below.\n\nBelow are the 23 candidate species 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23; plant exactly 2 of them so the summed pairwise ratings are as large as possible.\n1 vs 2: difference rating 1.\n1 vs 3: difference rating 1.\n1 vs 4: difference rating 4.\n1 vs 5: difference rating 0.\n1 vs 6: difference rating 6.\n1 vs 7: difference rating 1.\n1 vs 8: difference rating 9.\n1 vs 9: difference rating 7.\n1 vs 10: difference rating 0.\n1 vs 11: difference rating 8.\n1 vs 12: difference rating 6.\n1 vs 13: difference rating 5.\n1 vs 14: difference rating 9.\n1 vs 15: difference rating 6.\n1 vs 16: difference rating 0.\n1 vs 17: difference rating 3.\n1 vs 18: difference rating 2.\n1 vs 19: difference rating 9.\n1 vs 20: difference rating 3.\n1 vs 21: difference rating 6.\n1 vs 22: difference rating 8.\n1 vs 23: difference rating 7.\n2 vs 1: difference rating 1.\n2 vs 3: difference rating 0.\n2 vs 4: difference rating 5.\n2 vs 5: difference rating 2.\n2 vs 6: difference rating 6.\n2 vs 7: difference rating 9.\n2 vs 8: difference rating 2.\n2 vs 9: difference rating 6.\n2 vs 10: difference rating 4.\n2 vs 11: difference rating 7.\n2 vs 12: difference rating 7.\n2 vs 13: difference rating 4.\n2 vs 14: difference rating 5.\n2 vs 15: difference rating 5.\n2 vs 16: difference rating 3.\n2 vs 17: difference rating 1.\n2 vs 18: difference rating 8.\n2 vs 19: difference rating 6.\n2 vs 20: difference rating 7.\n2 vs 21: difference rating 9.\n2 vs 22: difference rating 6.\n2 vs 23: difference rating 9.\n3 vs 1: difference rating 1.\n3 vs 2: difference rating 0.\n3 vs 4: difference rating 8.\n3 vs 5: difference rating 8.\n3 vs 6: difference rating 9.\n3 vs 7: difference rating 7.\n3 vs 8: difference rating 3.\n3 vs 9: difference rating 2.\n3 vs 10: difference rating 2.\n3 vs 11: difference rating 9.\n3 vs 12: difference rating 1.\n3 vs 13: difference rating 5.\n3 vs 14: difference rating 7.\n3 vs 15: difference rating 5.\n3 vs 16: difference rating 9.\n3 vs 17: difference rating 4.\n3 vs 18: difference rating 5.\n3 vs 19: difference rating 2.\n3 vs 20: difference rating 0.\n3 vs 21: difference rating 7.\n3 vs 22: difference rating 2.\n3 vs 23: difference rating 9.\n4 vs 1: difference rating 4.\n4 vs 2: difference rating 5.\n4 vs 3: difference rating 8.\n4 vs 5: difference rating 0.\n4 vs 6: difference rating 2.\n4 vs 7: difference rating 8.\n4 vs 8: difference rating 2.\n4 vs 9: difference rating 5.\n4 vs 10: difference rating 4.\n4 vs 11: difference rating 0.\n4 vs 12: difference rating 8.\n4 vs 13: difference rating 0.\n4 vs 14: difference rating 8.\n4 vs 15: difference rating 6.\n4 vs 16: difference rating 2.\n4 vs 17: difference rating 7.\n4 vs 18: difference rating 3.\n4 vs 19: difference rating 0.\n4 vs 20: difference rating 3.\n4 vs 21: difference rating 5.\n4 vs 22: difference rating 0.\n4 vs 23: difference rating 0.\n5 vs 1: difference rating 0.\n5 vs 2: difference rating 2.\n5 vs 3: difference rating 8.\n5 vs 4: difference rating 0.\n5 vs 6: difference rating 7.\n5 vs 7: difference rating 1.\n5 vs 8: difference rating 7.\n5 vs 9: difference rating 0.\n5 vs 10: difference rating 5.\n5 vs 11: difference rating 5.\n5 vs 12: difference rating 5.\n5 vs 13: difference rating 6.\n5 vs 14: difference rating 8.\n5 vs 15: difference rating 1.\n5 vs 16: difference rating 3.\n5 vs 17: difference rating 5.\n5 vs 18: difference rating 4.\n5 vs 19: difference rating 9.\n5 vs 20: difference rating 1.\n5 vs 21: difference rating 8.\n5 vs 22: difference rating 2.\n5 vs 23: difference rating 7.\n6 vs 1: difference rating 6.\n6 vs 2: difference rating 6.\n6 vs 3: difference rating 9.\n6 vs 4: difference rating 2.\n6 vs 5: difference rating 7.\n6 vs 7: difference rating 5.\n6 vs 8: difference rating 5.\n6 vs 9: difference rating 0.\n6 vs 10: difference rating 2.\n6 vs 11: difference rating 0.\n6 vs 12: difference rating 4.\n6 vs 13: difference rating 4.\n6 vs 14: difference rating 7.\n6 vs 15: difference rating 3.\n6 vs 16: difference rating 1.\n6 vs 17: difference rating 8.\n6 vs 18: difference rating 6.\n6 vs 19: difference rating 1.\n6 vs 20: difference rating 3.\n6 vs 21: difference rating 3.\n6 vs 22: difference rating 2.\n6 vs 23: difference rating 9.\n7 vs 1: difference rating 1.\n7 vs 2: difference rating 9.\n7 vs 3: difference rating 7.\n7 vs 4: difference rating 8.\n7 vs 5: difference rating 1.\n7 vs 6: difference rating 5.\n7 vs 8: difference rating 8.\n7 vs 9: difference rating 2.\n7 vs 10: difference rating 8.\n7 vs 11: difference rating 1.\n7 vs 12: difference rating 1.\n7 vs 13: difference rating 9.\n7 vs 14: difference rating 5.\n7 vs 15: difference rating 2.\n7 vs 16: difference rating 6.\n7 vs 17: difference rating 3.\n7 vs 18: difference rating 2.\n7 vs 19: difference rating 5.\n7 vs 20: difference rating 1.\n7 vs 21: difference rating 1.\n7 vs 22: difference rating 8.\n7 vs 23: difference rating 7.\n8 vs 1: difference rating 9.\n8 vs 2: difference rating 2.\n8 vs 3: difference rating 3.\n8 vs 4: difference rating 2.\n8 vs 5: difference rating 7.\n8 vs 6: difference rating 5.\n8 vs 7: difference rating 8.\n8 vs 9: difference rating 2.\n8 vs 10: difference rating 8.\n8 vs 11: difference rating 5.\n8 vs 12: difference rating 7.\n8 vs 13: difference rating 3.\n8 vs 14: difference rating 6.\n8 vs 15: difference rating 7.\n8 vs 16: difference rating 5.\n8 vs 17: difference rating 8.\n8 vs 18: difference rating 2.\n8 vs 19: difference rating 2.\n8 vs 20: difference rating 8.\n8 vs 21: difference rating 3.\n8 vs 22: difference rating 3.\n8 vs 23: difference rating 9.\n9 vs 1: difference rating 7.\n9 vs 2: difference rating 6.\n9 vs 3: difference rating 2.\n9 vs 4: difference rating 5.\n9 vs 5: difference rating 0.\n9 vs 6: difference rating 0.\n9 vs 7: difference rating 2.\n9 vs 8: difference rating 2.\n9 vs 10: difference rating 5.\n9 vs 11: difference rating 2.\n9 vs 12: difference rating 5.\n9 vs 13: difference rating 5.\n9 vs 14: difference rating 0.\n9 vs 15: difference rating 2.\n9 vs 16: difference rating 7.\n9 vs 17: difference rating 2.\n9 vs 18: difference rating 0.\n9 vs 19: difference rating 0.\n9 vs 20: difference rating 5.\n9 vs 21: difference rating 6.\n9 vs 22: difference rating 2.\n9 vs 23: difference rating 0.\n10 vs 1: difference rating 0.\n10 vs 2: difference rating 4.\n10 vs 3: difference rating 2.\n10 vs 4: difference rating 4.\n10 vs 5: difference rating 5.\n10 vs 6: difference rating 2.\n10 vs 7: difference rating 8.\n10 vs 8: difference rating 8.\n10 vs 9: difference rating 5.\n10 vs 11: difference rating 6.\n10 vs 12: difference rating 9.\n10 vs 13: difference rating 4.\n10 vs 14: difference rating 6.\n10 vs 15: difference rating 2.\n10 vs 16: difference rating 8.\n10 vs 17: difference rating 9.\n10 vs 18: difference rating 1.\n10 vs 19: difference rating 6.\n10 vs 20: difference rating 8.\n10 vs 21: difference rating 6.\n10 vs 22: difference rating 3.\n10 vs 23: difference rating 8.\n11 vs 1: difference rating 8.\n11 vs 2: difference rating 7.\n11 vs 3: difference rating 9.\n11 vs 4: difference rating 0.\n11 vs 5: difference rating 5.\n11 vs 6: difference rating 0.\n11 vs 7: difference rating 1.\n11 vs 8: difference rating 5.\n11 vs 9: difference rating 2.\n11 vs 10: difference rating 6.\n11 vs 12: difference rating 2.\n11 vs 13: difference rating 8.\n11 vs 14: difference rating 1.\n11 vs 15: difference rating 1.\n11 vs 16: difference rating 0.\n11 vs 17: difference rating 0.\n11 vs 18: difference rating 3.\n11 vs 19: difference rating 6.\n11 vs 20: difference rating 5.\n11 vs 21: difference rating 7.\n11 vs 22: difference rating 3.\n11 vs 23: difference rating 2.\n12 vs 1: difference rating 6.\n12 vs 2: difference rating 7.\n12 vs 3: difference rating 1.\n12 vs 4: difference rating 8.\n12 vs 5: difference rating 5.\n12 vs 6: difference rating 4.\n12 vs 7: difference rating 1.\n12 vs 8: difference rating 7.\n12 vs 9: difference rating 5.\n12 vs 10: difference rating 9.\n12 vs 11: difference rating 2.\n12 vs 13: difference rating 8.\n12 vs 14: difference rating 0.\n12 vs 15: difference rating 2.\n12 vs 16: difference rating 6.\n12 vs 17: difference rating 6.\n12 vs 18: difference rating 4.\n12 vs 19: difference rating 1.\n12 vs 20: difference rating 5.\n12 vs 21: difference rating 0.\n12 vs 22: difference rating 1.\n12 vs 23: difference rating 3.\n13 vs 1: difference rating 5.\n13 vs 2: difference rating 4.\n13 vs 3: difference rating 5.\n13 vs 4: difference rating 0.\n13 vs 5: difference rating 6.\n13 vs 6: difference rating 4.\n13 vs 7: difference rating 9.\n13 vs 8: difference rating 3.\n13 vs 9: difference rating 5.\n13 vs 10: difference rating 4.\n13 vs 11: difference rating 8.\n13 vs 12: difference rating 8.\n13 vs 14: difference rating 9.\n13 vs 15: difference rating 4.\n13 vs 16: difference rating 7.\n13 vs 17: difference rating 9.\n13 vs 18: difference rating 1.\n13 vs 19: difference rating 4.\n13 vs 20: difference rating 7.\n13 vs 21: difference rating 3.\n13 vs 22: difference rating 8.\n13 vs 23: difference rating 1.\n14 vs 1: difference rating 9.\n14 vs 2: difference rating 5.\n14 vs 3: difference rating 7.\n14 vs 4: difference rating 8.\n14 vs 5: difference rating 8.\n14 vs 6: difference rating 7.\n14 vs 7: difference rating 5.\n14 vs 8: difference rating 6.\n14 vs 9: difference rating 0.\n14 vs 10: difference rating 6.\n14 vs 11: difference rating 1.\n14 vs 12: difference rating 0.\n14 vs 13: difference rating 9.\n14 vs 15: difference rating 5.\n14 vs 16: difference rating 6.\n14 vs 17: difference rating 8.\n14 vs 18: difference rating 7.\n14 vs 19: difference rating 3.\n14 vs 20: difference rating 2.\n14 vs 21: difference rating 1.\n14 vs 22: difference rating 8.\n14 vs 23: difference rating 3.\n15 vs 1: difference rating 6.\n15 vs 2: difference rating 5.\n15 vs 3: difference rating 5.\n15 vs 4: difference rating 6.\n15 vs 5: difference rating 1.\n15 vs 6: difference rating 3.\n15 vs 7: difference rating 2.\n15 vs 8: difference rating 7.\n15 vs 9: difference rating 2.\n15 vs 10: difference rating 2.\n15 vs 11: difference rating 1.\n15 vs 12: difference rating 2.\n15 vs 13: difference rating 4.\n15 vs 14: difference rating 5.\n15 vs 16: difference rating 0.\n15 vs 17: difference rating 4.\n15 vs 18: difference rating 6.\n15 vs 19: difference rating 4.\n15 vs 20: difference rating 0.\n15 vs 21: difference rating 0.\n15 vs 22: difference rating 6.\n15 vs 23: difference rating 0.\n16 vs 1: difference rating 0.\n16 vs 2: difference rating 3.\n16 vs 3: difference rating 9.\n16 vs 4: difference rating 2.\n16 vs 5: difference rating 3.\n16 vs 6: difference rating 1.\n16 vs 7: difference rating 6.\n16 vs 8: difference rating 5.\n16 vs 9: difference rating 7.\n16 vs 10: difference rating 8.\n16 vs 11: difference rating 0.\n16 vs 12: difference rating 6.\n16 vs 13: difference rating 7.\n16 vs 14: difference rating 6.\n16 vs 15: difference rating 0.\n16 vs 17: difference rating 8.\n16 vs 18: difference rating 5.\n16 vs 19: difference rating 2.\n16 vs 20: difference rating 5.\n16 vs 21: difference rating 1.\n16 vs 22: difference rating 8.\n16 vs 23: difference rating 8.\n17 vs 1: difference rating 3.\n17 vs 2: difference rating 1.\n17 vs 3: difference rating 4.\n17 vs 4: difference rating 7.\n17 vs 5: difference rating 5.\n17 vs 6: difference rating 8.\n17 vs 7: difference rating 3.\n17 vs 8: difference rating 8.\n17 vs 9: difference rating 2.\n17 vs 10: difference rating 9.\n17 vs 11: difference rating 0.\n17 vs 12: difference rating 6.\n17 vs 13: difference rating 9.\n17 vs 14: difference rating 8.\n17 vs 15: difference rating 4.\n17 vs 16: difference rating 8.\n17 vs 18: difference rating 7.\n17 vs 19: difference rating 7.\n17 vs 20: difference rating 3.\n17 vs 21: difference rating 4.\n17 vs 22: difference rating 1.\n17 vs 23: difference rating 2.\n18 vs 1: difference rating 2.\n18 vs 2: difference rating 8.\n18 vs 3: difference rating 5.\n18 vs 4: difference rating 3.\n18 vs 5: difference rating 4.\n18 vs 6: difference rating 6.\n18 vs 7: difference rating 2.\n18 vs 8: difference rating 2.\n18 vs 9: difference rating 0.\n18 vs 10: difference rating 1.\n18 vs 11: difference rating 3.\n18 vs 12: difference rating 4.\n18 vs 13: difference rating 1.\n18 vs 14: difference rating 7.\n18 vs 15: difference rating 6.\n18 vs 16: difference rating 5.\n18 vs 17: difference rating 7.\n18 vs 19: difference rating 9.\n18 vs 20: difference rating 7.\n18 vs 21: difference rating 5.\n18 vs 22: difference rating 8.\n18 vs 23: difference rating 0.\n19 vs 1: difference rating 9.\n19 vs 2: difference rating 6.\n19 vs 3: difference rating 2.\n19 vs 4: difference rating 0.\n19 vs 5: difference rating 9.\n19 vs 6: difference rating 1.\n19 vs 7: difference rating 5.\n19 vs 8: difference rating 2.\n19 vs 9: difference rating 0.\n19 vs 10: difference rating 6.\n19 vs 11: difference rating 6.\n19 vs 12: difference rating 1.\n19 vs 13: difference rating 4.\n19 vs 14: difference rating 3.\n19 vs 15: difference rating 4.\n19 vs 16: difference rating 2.\n19 vs 17: difference rating 7.\n19 vs 18: difference rating 9.\n19 vs 20: difference rating 1.\n19 vs 21: difference rating 3.\n19 vs 22: difference rating 0.\n19 vs 23: difference rating 4.\n20 vs 1: difference rating 3.\n20 vs 2: difference rating 7.\n20 vs 3: difference rating 0.\n20 vs 4: difference rating 3.\n20 vs 5: difference rating 1.\n20 vs 6: difference rating 3.\n20 vs 7: difference rating 1.\n20 vs 8: difference rating 8.\n20 vs 9: difference rating 5.\n20 vs 10: difference rating 8.\n20 vs 11: difference rating 5.\n20 vs 12: difference rating 5.\n20 vs 13: difference rating 7.\n20 vs 14: difference rating 2.\n20 vs 15: difference rating 0.\n20 vs 16: difference rating 5.\n20 vs 17: difference rating 3.\n20 vs 18: difference rating 7.\n20 vs 19: difference rating 1.\n20 vs 21: difference rating 7.\n20 vs 22: difference rating 3.\n20 vs 23: difference rating 5.\n21 vs 1: difference rating 6.\n21 vs 2: difference rating 9.\n21 vs 3: difference rating 7.\n21 vs 4: difference rating 5.\n21 vs 5: difference rating 8.\n21 vs 6: difference rating 3.\n21 vs 7: difference rating 1.\n21 vs 8: difference rating 3.\n21 vs 9: difference rating 6.\n21 vs 10: difference rating 6.\n21 vs 11: difference rating 7.\n21 vs 12: difference rating 0.\n21 vs 13: difference rating 3.\n21 vs 14: difference rating 1.\n21 vs 15: difference rating 0.\n21 vs 16: difference rating 1.\n21 vs 17: difference rating 4.\n21 vs 18: difference rating 5.\n21 vs 19: difference rating 3.\n21 vs 20: difference rating 7.\n21 vs 22: difference rating 8.\n21 vs 23: difference rating 2.\n22 vs 1: difference rating 8.\n22 vs 2: difference rating 6.\n22 vs 3: difference rating 2.\n22 vs 4: difference rating 0.\n22 vs 5: difference rating 2.\n22 vs 6: difference rating 2.\n22 vs 7: difference rating 8.\n22 vs 8: difference rating 3.\n22 vs 9: difference rating 2.\n22 vs 10: difference rating 3.\n22 vs 11: difference rating 3.\n22 vs 12: difference rating 1.\n22 vs 13: difference rating 8.\n22 vs 14: difference rating 8.\n22 vs 15: difference rating 6.\n22 vs 16: difference rating 8.\n22 vs 17: difference rating 1.\n22 vs 18: difference rating 8.\n22 vs 19: difference rating 0.\n22 vs 20: difference rating 3.\n22 vs 21: difference rating 8.\n22 vs 23: difference rating 9.\n23 vs 1: difference rating 7.\n23 vs 2: difference rating 9.\n23 vs 3: difference rating 9.\n23 vs 4: difference rating 0.\n23 vs 5: difference rating 7.\n23 vs 6: difference rating 9.\n23 vs 7: difference rating 7.\n23 vs 8: difference rating 9.\n23 vs 9: difference rating 0.\n23 vs 10: difference rating 8.\n23 vs 11: difference rating 2.\n23 vs 12: difference rating 3.\n23 vs 13: difference rating 1.\n23 vs 14: difference rating 3.\n23 vs 15: difference rating 0.\n23 vs 16: difference rating 8.\n23 vs 17: difference rating 2.\n23 vs 18: difference rating 0.\n23 vs 19: difference rating 4.\n23 vs 20: difference rating 5.\n23 vs 21: difference rating 2.\n23 vs 22: difference rating 9.\nThe best demo bed is the 2-species selection whose pairwise ratings sum to the highest total.\n\nOh, and when you send the pick back, just use this little JSON layout so I can read it easily:\n\n{\n \"solution\": [, , ...]\n}\n\nHere \"solution\" is just the list of the m species you're choosing for the bed — the exact labels you picked from the candidate list. This block is only a sketch of the shape I expect, not the final selection itself.\n\nPlease be sure to use the exact identifiers from the instance input — don't rename them or invent new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 1, 1, 4, 0, 6, 1, 9, 7, 0, 8, 6, 5, 9, 6, 0, 3, 2, 9, 3, 6, 8, 7 ], [ 1, 0, 0, 5, 2, 6, 9, 2, 6, 4, 7, 7, 4, 5, 5, 3, 1, 8, 6, 7, 9, 6, 9 ], [ 1, 0, 0, 8, 8, 9, 7, 3, 2, 2, 9, 1, 5, 7, 5, 9, 4, 5, 2, 0, 7, 2, 9 ], [ 4, 5, 8, 0, 0, 2, 8, 2, 5, 4, 0, 8, 0, 8, 6, 2, 7, 3, 0, 3, 5, 0, 0 ], [ 0, 2, 8, 0, 0, 7, 1, 7, 0, 5, 5, 5, 6, 8, 1, 3, 5, 4, 9, 1, 8, 2, 7 ], [ 6, 6, 9, 2, 7, 0, 5, 5, 0, 2, 0, 4, 4, 7, 3, 1, 8, 6, 1, 3, 3, 2, 9 ], [ 1, 9, 7, 8, 1, 5, 0, 8, 2, 8, 1, 1, 9, 5, 2, 6, 3, 2, 5, 1, 1, 8, 7 ], [ 9, 2, 3, 2, 7, 5, 8, 0, 2, 8, 5, 7, 3, 6, 7, 5, 8, 2, 2, 8, 3, 3, 9 ], [ 7, 6, 2, 5, 0, 0, 2, 2, 0, 5, 2, 5, 5, 0, 2, 7, 2, 0, 0, 5, 6, 2, 0 ], [ 0, 4, 2, 4, 5, 2, 8, 8, 5, 0, 6, 9, 4, 6, 2, 8, 9, 1, 6, 8, 6, 3, 8 ], [ 8, 7, 9, 0, 5, 0, 1, 5, 2, 6, 0, 2, 8, 1, 1, 0, 0, 3, 6, 5, 7, 3, 2 ], [ 6, 7, 1, 8, 5, 4, 1, 7, 5, 9, 2, 0, 8, 0, 2, 6, 6, 4, 1, 5, 0, 1, 3 ], [ 5, 4, 5, 0, 6, 4, 9, 3, 5, 4, 8, 8, 0, 9, 4, 7, 9, 1, 4, 7, 3, 8, 1 ], [ 9, 5, 7, 8, 8, 7, 5, 6, 0, 6, 1, 0, 9, 0, 5, 6, 8, 7, 3, 2, 1, 8, 3 ], [ 6, 5, 5, 6, 1, 3, 2, 7, 2, 2, 1, 2, 4, 5, 0, 0, 4, 6, 4, 0, 0, 6, 0 ], [ 0, 3, 9, 2, 3, 1, 6, 5, 7, 8, 0, 6, 7, 6, 0, 0, 8, 5, 2, 5, 1, 8, 8 ], [ 3, 1, 4, 7, 5, 8, 3, 8, 2, 9, 0, 6, 9, 8, 4, 8, 0, 7, 7, 3, 4, 1, 2 ], [ 2, 8, 5, 3, 4, 6, 2, 2, 0, 1, 3, 4, 1, 7, 6, 5, 7, 0, 9, 7, 5, 8, 0 ], [ 9, 6, 2, 0, 9, 1, 5, 2, 0, 6, 6, 1, 4, 3, 4, 2, 7, 9, 0, 1, 3, 0, 4 ], [ 3, 7, 0, 3, 1, 3, 1, 8, 5, 8, 5, 5, 7, 2, 0, 5, 3, 7, 1, 0, 7, 3, 5 ], [ 6, 9, 7, 5, 8, 3, 1, 3, 6, 6, 7, 0, 3, 1, 0, 1, 4, 5, 3, 7, 0, 8, 2 ], [ 8, 6, 2, 0, 2, 2, 8, 3, 2, 3, 3, 1, 8, 8, 6, 8, 1, 8, 0, 3, 8, 0, 9 ], [ 7, 9, 9, 0, 7, 9, 7, 9, 0, 8, 2, 3, 1, 3, 0, 8, 2, 0, 4, 5, 2, 9, 0 ] ], "m": 2, "objective": 9.0 }, "solution": [ 1, 20 ], "obj": 9.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 23, "m": 2, "nodes": [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 ], "distance_pairs": [ { "from_id": 1, "to_id": 2, "distance": 1 }, { "from_id": 1, "to_id": 3, "distance": 1 }, { "from_id": 1, "to_id": 4, "distance": 4 }, { "from_id": 1, "to_id": 5, "distance": 0 }, { "from_id": 1, "to_id": 6, "distance": 6 }, { "from_id": 1, "to_id": 7, "distance": 1 }, { "from_id": 1, "to_id": 8, "distance": 9 }, { "from_id": 1, "to_id": 9, "distance": 7 }, { "from_id": 1, "to_id": 10, "distance": 0 }, { "from_id": 1, "to_id": 11, "distance": 8 }, { "from_id": 1, "to_id": 12, "distance": 6 }, { "from_id": 1, "to_id": 13, "distance": 5 }, { "from_id": 1, "to_id": 14, "distance": 9 }, { "from_id": 1, "to_id": 15, "distance": 6 }, { "from_id": 1, 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"distance": 5 }, { "from_id": 19, "to_id": 8, "distance": 2 }, { "from_id": 19, "to_id": 9, "distance": 0 }, { "from_id": 19, "to_id": 10, "distance": 6 }, { "from_id": 19, "to_id": 11, "distance": 6 }, { "from_id": 19, "to_id": 12, "distance": 1 }, { "from_id": 19, "to_id": 13, "distance": 4 }, { "from_id": 19, "to_id": 14, "distance": 3 }, { "from_id": 19, "to_id": 15, "distance": 4 }, { "from_id": 19, "to_id": 16, "distance": 2 }, { "from_id": 19, "to_id": 17, "distance": 7 }, { "from_id": 19, "to_id": 18, "distance": 9 }, { "from_id": 19, "to_id": 20, "distance": 1 }, { "from_id": 19, "to_id": 21, "distance": 3 }, { "from_id": 19, "to_id": 22, "distance": 0 }, { "from_id": 19, "to_id": 23, "distance": 4 }, { "from_id": 20, "to_id": 1, "distance": 3 }, { "from_id": 20, "to_id": 2, "distance": 7 }, { "from_id": 20, "to_id": 3, "distance": 0 }, { "from_id": 20, "to_id": 4, "distance": 3 }, { "from_id": 20, "to_id": 5, "distance": 1 }, { "from_id": 20, "to_id": 6, "distance": 3 }, { "from_id": 20, "to_id": 7, "distance": 1 }, { "from_id": 20, "to_id": 8, "distance": 8 }, { "from_id": 20, "to_id": 9, "distance": 5 }, { "from_id": 20, "to_id": 10, "distance": 8 }, { "from_id": 20, "to_id": 11, "distance": 5 }, { "from_id": 20, "to_id": 12, "distance": 5 }, { "from_id": 20, "to_id": 13, "distance": 7 }, { "from_id": 20, "to_id": 14, "distance": 2 }, { "from_id": 20, "to_id": 15, "distance": 0 }, { "from_id": 20, "to_id": 16, "distance": 5 }, { "from_id": 20, "to_id": 17, "distance": 3 }, { "from_id": 20, "to_id": 18, "distance": 7 }, { "from_id": 20, "to_id": 19, "distance": 1 }, { "from_id": 20, "to_id": 21, "distance": 7 }, { "from_id": 20, "to_id": 22, "distance": 3 }, { "from_id": 20, "to_id": 23, "distance": 5 }, { "from_id": 21, "to_id": 1, "distance": 6 }, { "from_id": 21, "to_id": 2, "distance": 9 }, { "from_id": 21, "to_id": 3, "distance": 7 }, { "from_id": 21, "to_id": 4, "distance": 5 }, { "from_id": 21, "to_id": 5, "distance": 8 }, { "from_id": 21, "to_id": 6, "distance": 3 }, { "from_id": 21, "to_id": 7, "distance": 1 }, { "from_id": 21, "to_id": 8, "distance": 3 }, { "from_id": 21, "to_id": 9, "distance": 6 }, { "from_id": 21, "to_id": 10, "distance": 6 }, { "from_id": 21, "to_id": 11, "distance": 7 }, { "from_id": 21, "to_id": 12, "distance": 0 }, { "from_id": 21, "to_id": 13, "distance": 3 }, { "from_id": 21, "to_id": 14, "distance": 1 }, { "from_id": 21, "to_id": 15, "distance": 0 }, { "from_id": 21, "to_id": 16, "distance": 1 }, { "from_id": 21, "to_id": 17, "distance": 4 }, { "from_id": 21, "to_id": 18, "distance": 5 }, { "from_id": 21, "to_id": 19, "distance": 3 }, { "from_id": 21, "to_id": 20, "distance": 7 }, { "from_id": 21, "to_id": 22, "distance": 8 }, { "from_id": 21, "to_id": 23, "distance": 2 }, { "from_id": 22, "to_id": 1, "distance": 8 }, { "from_id": 22, "to_id": 2, "distance": 6 }, { "from_id": 22, "to_id": 3, "distance": 2 }, { "from_id": 22, "to_id": 4, "distance": 0 }, { "from_id": 22, "to_id": 5, "distance": 2 }, { "from_id": 22, "to_id": 6, "distance": 2 }, { "from_id": 22, "to_id": 7, "distance": 8 }, { "from_id": 22, "to_id": 8, "distance": 3 }, { "from_id": 22, "to_id": 9, "distance": 2 }, { "from_id": 22, "to_id": 10, "distance": 3 }, { "from_id": 22, "to_id": 11, "distance": 3 }, { "from_id": 22, "to_id": 12, "distance": 1 }, { "from_id": 22, "to_id": 13, "distance": 8 }, { "from_id": 22, "to_id": 14, "distance": 8 }, { "from_id": 22, "to_id": 15, "distance": 6 }, { "from_id": 22, "to_id": 16, "distance": 8 }, { "from_id": 22, "to_id": 17, "distance": 1 }, { "from_id": 22, "to_id": 18, "distance": 8 }, { "from_id": 22, "to_id": 19, "distance": 0 }, { "from_id": 22, "to_id": 20, "distance": 3 }, { "from_id": 22, "to_id": 21, "distance": 8 }, { "from_id": 22, "to_id": 23, "distance": 9 }, { "from_id": 23, "to_id": 1, "distance": 7 }, { "from_id": 23, "to_id": 2, "distance": 9 }, { "from_id": 23, "to_id": 3, "distance": 9 }, { "from_id": 23, "to_id": 4, "distance": 0 }, { "from_id": 23, "to_id": 5, "distance": 7 }, { "from_id": 23, "to_id": 6, "distance": 9 }, { "from_id": 23, "to_id": 7, "distance": 7 }, { "from_id": 23, "to_id": 8, "distance": 9 }, { "from_id": 23, "to_id": 9, "distance": 0 }, { "from_id": 23, "to_id": 10, "distance": 8 }, { "from_id": 23, "to_id": 11, "distance": 2 }, { "from_id": 23, "to_id": 12, "distance": 3 }, { "from_id": 23, "to_id": 13, "distance": 1 }, { "from_id": 23, "to_id": 14, "distance": 3 }, { "from_id": 23, "to_id": 15, "distance": 0 }, { "from_id": 23, "to_id": 16, "distance": 8 }, { "from_id": 23, "to_id": 17, "distance": 2 }, { "from_id": 23, "to_id": 18, "distance": 0 }, { "from_id": 23, "to_id": 19, "distance": 4 }, { "from_id": 23, "to_id": 20, "distance": 5 }, { "from_id": 23, "to_id": 21, "distance": 2 }, { "from_id": 23, "to_id": 22, "distance": 9 } ] }, "solution_variant": [ 2, 21 ], "context_index": 35, "input_format": "nl", "input_index_base": 1 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "I’m putting together a variety night and have to pick exactly the number of performers the venue asked for. The goal is to make the bill feel fresh — pick acts that are as different from one another as possible. To judge that, there’s a simple rule: for every pair of chosen acts there’s a score for how different they are, and the better the lineup is, the bigger the total when all those pairwise differences are added up. Nobody can be picked twice and the count must be exact, and the specific acts and their pairwise difference scores are listed below.\n\nBelow are the 24 performers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23) and their pairwise difference scores — I must pick exactly 10 performers.\nFor 0 paired with 1, I record a difference of 112.\nFor 0 paired with 2, I record a difference of 80.\nFor 0 paired with 3, I record a difference of 82.\nFor 0 paired with 4, I record a difference of 91.\nFor 0 paired with 5, I record a difference of 97.\nFor 0 paired with 6, I record a difference of 110.\nFor 0 paired with 7, I record a difference of 87.\nFor 0 paired with 8, I record a difference of 95.\nFor 0 paired with 9, I record a difference of 102.\nFor 0 paired with 10, I record a difference of 127.\nFor 0 paired with 11, I record a difference of 86.\nFor 0 paired with 12, I record a difference of 143.\nFor 0 paired with 13, I record a difference of 117.\nFor 0 paired with 14, I record a difference of 112.\nFor 0 paired with 15, I record a difference of 114.\nFor 0 paired with 16, I record a difference of 136.\nFor 0 paired with 17, I record a difference of 89.\nFor 0 paired with 18, I record a difference of 139.\nFor 0 paired with 19, I record a difference of 92.\nFor 0 paired with 20, I record a difference of 89.\nFor 0 paired with 21, I record a difference of 113.\nFor 0 paired with 22, I record a difference of 103.\nFor 0 paired with 23, I record a difference of 130.\nFor 1 paired with 0, I record a difference of 112.\nFor 1 paired with 2, I record a difference of 89.\nFor 1 paired with 3, I record a difference of 91.\nFor 1 paired with 4, I record a difference of 164.\nFor 1 paired with 5, I record a difference of 95.\nFor 1 paired with 6, I record a difference of 134.\nFor 1 paired with 7, I record a difference of 59.\nFor 1 paired with 8, I record a difference of 149.\nFor 1 paired with 9, I record a difference of 148.\nFor 1 paired with 10, I record a difference of 139.\nFor 1 paired with 11, I record a difference of 139.\nFor 1 paired with 12, I record a difference of 134.\nFor 1 paired with 13, I record a difference of 138.\nFor 1 paired with 14, I record a difference of 74.\nFor 1 paired with 15, I record a difference of 146.\nFor 1 paired with 16, I record a difference of 153.\nFor 1 paired with 17, I record a difference of 118.\nFor 1 paired with 18, I record a difference of 122.\nFor 1 paired with 19, I record a difference of 132.\nFor 1 paired with 20, I record a difference of 115.\nFor 1 paired with 21, I record a difference of 162.\nFor 1 paired with 22, I record a difference of 134.\nFor 1 paired with 23, I record a difference of 112.\nFor 2 paired with 0, I record a difference of 80.\nFor 2 paired with 1, I record a difference of 89.\nFor 2 paired with 3, I record a difference of 102.\nFor 2 paired with 4, I record a difference of 116.\nFor 2 paired with 5, I record a difference of 101.\nFor 2 paired with 6, I record a difference of 147.\nFor 2 paired with 7, I record a difference of 69.\nFor 2 paired with 8, I record a difference of 138.\nFor 2 paired with 9, I record a difference of 106.\nFor 2 paired with 10, I record a difference of 159.\nFor 2 paired with 11, I record a difference of 128.\nFor 2 paired with 12, I record a difference of 159.\nFor 2 paired with 13, I record a difference of 133.\nFor 2 paired with 14, I record a difference of 79.\nFor 2 paired with 15, I record a difference of 151.\nFor 2 paired with 16, I record a difference of 144.\nFor 2 paired with 17, I record a difference of 120.\nFor 2 paired with 18, I record a difference of 149.\nFor 2 paired with 19, I record a difference of 105.\nFor 2 paired with 20, I record a difference of 113.\nFor 2 paired with 21, I record a difference of 140.\nFor 2 paired with 22, I record a difference of 120.\nFor 2 paired with 23, I record a difference of 130.\nFor 3 paired with 0, I record a difference of 82.\nFor 3 paired with 1, I record a difference of 91.\nFor 3 paired with 2, I record a difference of 102.\nFor 3 paired with 4, I record a difference of 131.\nFor 3 paired with 5, I record a difference of 100.\nFor 3 paired with 6, I record a difference of 105.\nFor 3 paired with 7, I record a difference of 105.\nFor 3 paired with 8, I record a difference of 99.\nFor 3 paired with 9, I record a difference of 120.\nFor 3 paired with 10, I record a difference of 104.\nFor 3 paired with 11, I record a difference of 109.\nFor 3 paired with 12, I record a difference of 130.\nFor 3 paired with 13, I record a difference of 125.\nFor 3 paired with 14, I record a difference of 94.\nFor 3 paired with 15, I record a difference of 112.\nFor 3 paired with 16, I record a difference of 157.\nFor 3 paired with 17, I record a difference of 73.\nFor 3 paired with 18, I record a difference of 97.\nFor 3 paired with 19, I record a difference of 111.\nFor 3 paired with 20, I record a difference of 129.\nFor 3 paired with 21, I record a difference of 153.\nFor 3 paired with 22, I record a difference of 103.\nFor 3 paired with 23, I record a difference of 144.\nFor 4 paired with 0, I record a difference of 91.\nFor 4 paired with 1, I record a difference of 164.\nFor 4 paired with 2, I record a difference of 116.\nFor 4 paired with 3, I record a difference of 131.\nFor 4 paired with 5, I record a difference of 105.\nFor 4 paired with 6, I record a difference of 111.\nFor 4 paired with 7, I record a difference of 146.\nFor 4 paired with 8, I record a difference of 124.\nFor 4 paired with 9, I record a difference of 86.\nFor 4 paired with 10, I record a difference of 161.\nFor 4 paired with 11, I record a difference of 92.\nFor 4 paired with 12, I record a difference of 132.\nFor 4 paired with 13, I record a difference of 134.\nFor 4 paired with 14, I record a difference of 153.\nFor 4 paired with 15, I record a difference of 119.\nFor 4 paired with 16, I record a difference of 116.\nFor 4 paired with 17, I record a difference of 107.\nFor 4 paired with 18, I record a difference of 142.\nFor 4 paired with 19, I record a difference of 68.\nFor 4 paired with 20, I record a difference of 126.\nFor 4 paired with 21, I record a difference of 134.\nFor 4 paired with 22, I record a difference of 137.\nFor 4 paired with 23, I record a difference of 143.\nFor 5 paired with 0, I record a difference of 97.\nFor 5 paired with 1, I record a difference of 95.\nFor 5 paired with 2, I record a difference of 101.\nFor 5 paired with 3, I record a difference of 100.\nFor 5 paired with 4, I record a difference of 105.\nFor 5 paired with 6, I record a difference of 86.\nFor 5 paired with 7, I record a difference of 98.\nFor 5 paired with 8, I record a difference of 119.\nFor 5 paired with 9, I record a difference of 111.\nFor 5 paired with 10, I record a difference of 135.\nFor 5 paired with 11, I record a difference of 104.\nFor 5 paired with 12, I record a difference of 108.\nFor 5 paired with 13, I record a difference of 128.\nFor 5 paired with 14, I record a difference of 115.\nFor 5 paired with 15, I record a difference of 89.\nFor 5 paired with 16, I record a difference of 113.\nFor 5 paired with 17, I record a difference of 90.\nFor 5 paired with 18, I record a difference of 79.\nFor 5 paired with 19, I record a difference of 90.\nFor 5 paired with 20, I record a difference of 124.\nFor 5 paired with 21, I record a difference of 137.\nFor 5 paired with 22, I record a difference of 111.\nFor 5 paired with 23, I record a difference of 74.\nFor 6 paired with 0, I record a difference of 110.\nFor 6 paired with 1, I record a difference of 134.\nFor 6 paired with 2, I record a difference of 147.\nFor 6 paired with 3, I record a difference of 105.\nFor 6 paired with 4, I record a difference of 111.\nFor 6 paired with 5, I record a difference of 86.\nFor 6 paired with 7, I record a difference of 141.\nFor 6 paired with 8, I record a difference of 79.\nFor 6 paired with 9, I record a difference of 107.\nFor 6 paired with 10, I record a difference of 93.\nFor 6 paired with 11, I record a difference of 66.\nFor 6 paired with 12, I record a difference of 97.\nFor 6 paired with 13, I record a difference of 112.\nFor 6 paired with 14, I record a difference of 131.\nFor 6 paired with 15, I record a difference of 76.\nFor 6 paired with 16, I record a difference of 105.\nFor 6 paired with 17, I record a difference of 82.\nFor 6 paired with 18, I record a difference of 81.\nFor 6 paired with 19, I record a difference of 87.\nFor 6 paired with 20, I record a difference of 121.\nFor 6 paired with 21, I record a difference of 123.\nFor 6 paired with 22, I record a difference of 104.\nFor 6 paired with 23, I record a difference of 122.\nFor 7 paired with 0, I record a difference of 87.\nFor 7 paired with 1, I record a difference of 59.\nFor 7 paired with 2, I record a difference of 69.\nFor 7 paired with 3, I record a difference of 105.\nFor 7 paired with 4, I record a difference of 146.\nFor 7 paired with 5, I record a difference of 98.\nFor 7 paired with 6, I record a difference of 141.\nFor 7 paired with 8, I record a difference of 144.\nFor 7 paired with 9, I record a difference of 128.\nFor 7 paired with 10, I record a difference of 152.\nFor 7 paired with 11, I record a difference of 122.\nFor 7 paired with 12, I record a difference of 154.\nFor 7 paired with 13, I record a difference of 149.\nFor 7 paired with 14, I record a difference of 77.\nFor 7 paired with 15, I record a difference of 135.\nFor 7 paired with 16, I record a difference of 143.\nFor 7 paired with 17, I record a difference of 136.\nFor 7 paired with 18, I record a difference of 151.\nFor 7 paired with 19, I record a difference of 115.\nFor 7 paired with 20, I record a difference of 103.\nFor 7 paired with 21, I record a difference of 148.\nFor 7 paired with 22, I record a difference of 123.\nFor 7 paired with 23, I record a difference of 112.\nFor 8 paired with 0, I record a difference of 95.\nFor 8 paired with 1, I record a difference of 149.\nFor 8 paired with 2, I record a difference of 138.\nFor 8 paired with 3, I record a difference of 99.\nFor 8 paired with 4, I record a difference of 124.\nFor 8 paired with 5, I record a difference of 119.\nFor 8 paired with 6, I record a difference of 79.\nFor 8 paired with 7, I record a difference of 144.\nFor 8 paired with 9, I record a difference of 95.\nFor 8 paired with 10, I record a difference of 92.\nFor 8 paired with 11, I record a difference of 89.\nFor 8 paired with 12, I record a difference of 163.\nFor 8 paired with 13, I record a difference of 129.\nFor 8 paired with 14, I record a difference of 126.\nFor 8 paired with 15, I record a difference of 101.\nFor 8 paired with 16, I record a difference of 145.\nFor 8 paired with 17, I record a difference of 97.\nFor 8 paired with 18, I record a difference of 120.\nFor 8 paired with 19, I record a difference of 111.\nFor 8 paired with 20, I record a difference of 139.\nFor 8 paired with 21, I record a difference of 123.\nFor 8 paired with 22, I record a difference of 65.\nFor 8 paired with 23, I record a difference of 166.\nFor 9 paired with 0, I record a difference of 102.\nFor 9 paired with 1, I record a difference of 148.\nFor 9 paired with 2, I record a difference of 106.\nFor 9 paired with 3, I record a difference of 120.\nFor 9 paired with 4, I record a difference of 86.\nFor 9 paired with 5, I record a difference of 111.\nFor 9 paired with 6, I record a difference of 107.\nFor 9 paired with 7, I record a difference of 128.\nFor 9 paired with 8, I record a difference of 95.\nFor 9 paired with 10, I record a difference of 140.\nFor 9 paired with 11, I record a difference of 84.\nFor 9 paired with 12, I record a difference of 160.\nFor 9 paired with 13, I record a difference of 153.\nFor 9 paired with 14, I record a difference of 104.\nFor 9 paired with 15, I record a difference of 101.\nFor 9 paired with 16, I record a difference of 117.\nFor 9 paired with 17, I record a difference of 128.\nFor 9 paired with 18, I record a difference of 137.\nFor 9 paired with 19, I record a difference of 56.\nFor 9 paired with 20, I record a difference of 140.\nFor 9 paired with 21, I record a difference of 143.\nFor 9 paired with 22, I record a difference of 88.\nFor 9 paired with 23, I record a difference of 159.\nFor 10 paired with 0, I record a difference of 127.\nFor 10 paired with 1, I record a difference of 139.\nFor 10 paired with 2, I record a difference of 159.\nFor 10 paired with 3, I record a difference of 104.\nFor 10 paired with 4, I record a difference of 161.\nFor 10 paired with 5, I record a difference of 135.\nFor 10 paired with 6, I record a difference of 93.\nFor 10 paired with 7, I record a difference of 152.\nFor 10 paired with 8, I record a difference of 92.\nFor 10 paired with 9, I record a difference of 140.\nFor 10 paired with 11, I record a difference of 123.\nFor 10 paired with 12, I record a difference of 153.\nFor 10 paired with 13, I record a difference of 117.\nFor 10 paired with 14, I record a difference of 134.\nFor 10 paired with 15, I record a difference of 120.\nFor 10 paired with 16, I record a difference of 178.\nFor 10 paired with 17, I record a difference of 108.\nFor 10 paired with 18, I record a difference of 131.\nFor 10 paired with 19, I record a difference of 137.\nFor 10 paired with 20, I record a difference of 157.\nFor 10 paired with 21, I record a difference of 154.\nFor 10 paired with 22, I record a difference of 128.\nFor 10 paired with 23, I record a difference of 160.\nFor 11 paired with 0, I record a difference of 86.\nFor 11 paired with 1, I record a difference of 139.\nFor 11 paired with 2, I record a difference of 128.\nFor 11 paired with 3, I record a difference of 109.\nFor 11 paired with 4, I record a difference of 92.\nFor 11 paired with 5, I record a difference of 104.\nFor 11 paired with 6, I record a difference of 66.\nFor 11 paired with 7, I record a difference of 122.\nFor 11 paired with 8, I record a difference of 89.\nFor 11 paired with 9, I record a difference of 84.\nFor 11 paired with 10, I record a difference of 123.\nFor 11 paired with 12, I record a difference of 113.\nFor 11 paired with 13, I record a difference of 117.\nFor 11 paired with 14, I record a difference of 119.\nFor 11 paired with 15, I record a difference of 88.\nFor 11 paired with 16, I record a difference of 78.\nFor 11 paired with 17, I record a difference of 109.\nFor 11 paired with 18, I record a difference of 123.\nFor 11 paired with 19, I record a difference of 54.\nFor 11 paired with 20, I record a difference of 92.\nFor 11 paired with 21, I record a difference of 101.\nFor 11 paired with 22, I record a difference of 95.\nFor 11 paired with 23, I record a difference of 130.\nFor 12 paired with 0, I record a difference of 143.\nFor 12 paired with 1, I record a difference of 134.\nFor 12 paired with 2, I record a difference of 159.\nFor 12 paired with 3, I record a difference of 130.\nFor 12 paired with 4, I record a difference of 132.\nFor 12 paired with 5, I record a difference of 108.\nFor 12 paired with 6, I record a difference of 97.\nFor 12 paired with 7, I record a difference of 154.\nFor 12 paired with 8, I record a difference of 163.\nFor 12 paired with 9, I record a difference of 160.\nFor 12 paired with 10, I record a difference of 153.\nFor 12 paired with 11, I record a difference of 113.\nFor 12 paired with 13, I record a difference of 117.\nFor 12 paired with 14, I record a difference of 155.\nFor 12 paired with 15, I record a difference of 141.\nFor 12 paired with 16, I record a difference of 108.\nFor 12 paired with 17, I record a difference of 103.\nFor 12 paired with 18, I record a difference of 101.\nFor 12 paired with 19, I record a difference of 114.\nFor 12 paired with 20, I record a difference of 117.\nFor 12 paired with 21, I record a difference of 154.\nFor 12 paired with 22, I record a difference of 173.\nFor 12 paired with 23, I record a difference of 116.\nFor 13 paired with 0, I record a difference of 117.\nFor 13 paired with 1, I record a difference of 138.\nFor 13 paired with 2, I record a difference of 133.\nFor 13 paired with 3, I record a difference of 125.\nFor 13 paired with 4, I record a difference of 134.\nFor 13 paired with 5, I record a difference of 128.\nFor 13 paired with 6, I record a difference of 112.\nFor 13 paired with 7, I record a difference of 149.\nFor 13 paired with 8, I record a difference of 129.\nFor 13 paired with 9, I record a difference of 153.\nFor 13 paired with 10, I record a difference of 117.\nFor 13 paired with 11, I record a difference of 117.\nFor 13 paired with 12, I record a difference of 117.\nFor 13 paired with 14, I record a difference of 142.\nFor 13 paired with 15, I record a difference of 165.\nFor 13 paired with 16, I record a difference of 133.\nFor 13 paired with 17, I record a difference of 100.\nFor 13 paired with 18, I record a difference of 139.\nFor 13 paired with 19, I record a difference of 129.\nFor 13 paired with 20, I record a difference of 105.\nFor 13 paired with 21, I record a difference of 82.\nFor 13 paired with 22, I record a difference of 146.\nFor 13 paired with 23, I record a difference of 124.\nFor 14 paired with 0, I record a difference of 112.\nFor 14 paired with 1, I record a difference of 74.\nFor 14 paired with 2, I record a difference of 79.\nFor 14 paired with 3, I record a difference of 94.\nFor 14 paired with 4, I record a difference of 153.\nFor 14 paired with 5, I record a difference of 115.\nFor 14 paired with 6, I record a difference of 131.\nFor 14 paired with 7, I record a difference of 77.\nFor 14 paired with 8, I record a difference of 126.\nFor 14 paired with 9, I record a difference of 104.\nFor 14 paired with 10, I record a difference of 134.\nFor 14 paired with 11, I record a difference of 119.\nFor 14 paired with 12, I record a difference of 155.\nFor 14 paired with 13, I record a difference of 142.\nFor 14 paired with 15, I record a difference of 140.\nFor 14 paired with 16, I record a difference of 138.\nFor 14 paired with 17, I record a difference of 132.\nFor 14 paired with 18, I record a difference of 134.\nFor 14 paired with 19, I record a difference of 107.\nFor 14 paired with 20, I record a difference of 123.\nFor 14 paired with 21, I record a difference of 153.\nFor 14 paired with 22, I record a difference of 99.\nFor 14 paired with 23, I record a difference of 146.\nFor 15 paired with 0, I record a difference of 114.\nFor 15 paired with 1, I record a difference of 146.\nFor 15 paired with 2, I record a difference of 151.\nFor 15 paired with 3, I record a difference of 112.\nFor 15 paired with 4, I record a difference of 119.\nFor 15 paired with 5, I record a difference of 89.\nFor 15 paired with 6, I record a difference of 76.\nFor 15 paired with 7, I record a difference of 135.\nFor 15 paired with 8, I record a difference of 101.\nFor 15 paired with 9, I record a difference of 101.\nFor 15 paired with 10, I record a difference of 120.\nFor 15 paired with 11, I record a difference of 88.\nFor 15 paired with 12, I record a difference of 141.\nFor 15 paired with 13, I record a difference of 165.\nFor 15 paired with 14, I record a difference of 140.\nFor 15 paired with 16, I record a difference of 136.\nFor 15 paired with 17, I record a difference of 123.\nFor 15 paired with 18, I record a difference of 109.\nFor 15 paired with 19, I record a difference of 89.\nFor 15 paired with 20, I record a difference of 157.\nFor 15 paired with 21, I record a difference of 161.\nFor 15 paired with 22, I record a difference of 105.\nFor 15 paired with 23, I record a difference of 131.\nFor 16 paired with 0, I record a difference of 136.\nFor 16 paired with 1, I record a difference of 153.\nFor 16 paired with 2, I record a difference of 144.\nFor 16 paired with 3, I record a difference of 157.\nFor 16 paired with 4, I record a difference of 116.\nFor 16 paired with 5, I record a difference of 113.\nFor 16 paired with 6, I record a difference of 105.\nFor 16 paired with 7, I record a difference of 143.\nFor 16 paired with 8, I record a difference of 145.\nFor 16 paired with 9, I record a difference of 117.\nFor 16 paired with 10, I record a difference of 178.\nFor 16 paired with 11, I record a difference of 78.\nFor 16 paired with 12, I record a difference of 108.\nFor 16 paired with 13, I record a difference of 133.\nFor 16 paired with 14, I record a difference of 138.\nFor 16 paired with 15, I record a difference of 136.\nFor 16 paired with 17, I record a difference of 144.\nFor 16 paired with 18, I record a difference of 135.\nFor 16 paired with 19, I record a difference of 82.\nFor 16 paired with 20, I record a difference of 102.\nFor 16 paired with 21, I record a difference of 109.\nFor 16 paired with 22, I record a difference of 132.\nFor 16 paired with 23, I record a difference of 121.\nFor 17 paired with 0, I record a difference of 89.\nFor 17 paired with 1, I record a difference of 118.\nFor 17 paired with 2, I record a difference of 120.\nFor 17 paired with 3, I record a difference of 73.\nFor 17 paired with 4, I record a difference of 107.\nFor 17 paired with 5, I record a difference of 90.\nFor 17 paired with 6, I record a difference of 82.\nFor 17 paired with 7, I record a difference of 136.\nFor 17 paired with 8, I record a difference of 97.\nFor 17 paired with 9, I record a difference of 128.\nFor 17 paired with 10, I record a difference of 108.\nFor 17 paired with 11, I record a difference of 109.\nFor 17 paired with 12, I record a difference of 103.\nFor 17 paired with 13, I record a difference of 100.\nFor 17 paired with 14, I record a difference of 132.\nFor 17 paired with 15, I record a difference of 123.\nFor 17 paired with 16, I record a difference of 144.\nFor 17 paired with 18, I record a difference of 83.\nFor 17 paired with 19, I record a difference of 113.\nFor 17 paired with 20, I record a difference of 118.\nFor 17 paired with 21, I record a difference of 138.\nFor 17 paired with 22, I record a difference of 125.\nFor 17 paired with 23, I record a difference of 137.\nFor 18 paired with 0, I record a difference of 139.\nFor 18 paired with 1, I record a difference of 122.\nFor 18 paired with 2, I record a difference of 149.\nFor 18 paired with 3, I record a difference of 97.\nFor 18 paired with 4, I record a difference of 142.\nFor 18 paired with 5, I record a difference of 79.\nFor 18 paired with 6, I record a difference of 81.\nFor 18 paired with 7, I record a difference of 151.\nFor 18 paired with 8, I record a difference of 120.\nFor 18 paired with 9, I record a difference of 137.\nFor 18 paired with 10, I record a difference of 131.\nFor 18 paired with 11, I record a difference of 123.\nFor 18 paired with 12, I record a difference of 101.\nFor 18 paired with 13, I record a difference of 139.\nFor 18 paired with 14, I record a difference of 134.\nFor 18 paired with 15, I record a difference of 109.\nFor 18 paired with 16, I record a difference of 135.\nFor 18 paired with 17, I record a difference of 83.\nFor 18 paired with 19, I record a difference of 121.\nFor 18 paired with 20, I record a difference of 157.\nFor 18 paired with 21, I record a difference of 165.\nFor 18 paired with 22, I record a difference of 121.\nFor 18 paired with 23, I record a difference of 129.\nFor 19 paired with 0, I record a difference of 92.\nFor 19 paired with 1, I record a difference of 132.\nFor 19 paired with 2, I record a difference of 105.\nFor 19 paired with 3, I record a difference of 111.\nFor 19 paired with 4, I record a difference of 68.\nFor 19 paired with 5, I record a difference of 90.\nFor 19 paired with 6, I record a difference of 87.\nFor 19 paired with 7, I record a difference of 115.\nFor 19 paired with 8, I record a difference of 111.\nFor 19 paired with 9, I record a difference of 56.\nFor 19 paired with 10, I record a difference of 137.\nFor 19 paired with 11, I record a difference of 54.\nFor 19 paired with 12, I record a difference of 114.\nFor 19 paired with 13, I record a difference of 129.\nFor 19 paired with 14, I record a difference of 107.\nFor 19 paired with 15, I record a difference of 89.\nFor 19 paired with 16, I record a difference of 82.\nFor 19 paired with 17, I record a difference of 113.\nFor 19 paired with 18, I record a difference of 121.\nFor 19 paired with 20, I record a difference of 111.\nFor 19 paired with 21, I record a difference of 127.\nFor 19 paired with 22, I record a difference of 107.\nFor 19 paired with 23, I record a difference of 124.\nFor 20 paired with 0, I record a difference of 89.\nFor 20 paired with 1, I record a difference of 115.\nFor 20 paired with 2, I record a difference of 113.\nFor 20 paired with 3, I record a difference of 129.\nFor 20 paired with 4, I record a difference of 126.\nFor 20 paired with 5, I record a difference of 124.\nFor 20 paired with 6, I record a difference of 121.\nFor 20 paired with 7, I record a difference of 103.\nFor 20 paired with 8, I record a difference of 139.\nFor 20 paired with 9, I record a difference of 140.\nFor 20 paired with 10, I record a difference of 157.\nFor 20 paired with 11, I record a difference of 92.\nFor 20 paired with 12, I record a difference of 117.\nFor 20 paired with 13, I record a difference of 105.\nFor 20 paired with 14, I record a difference of 123.\nFor 20 paired with 15, I record a difference of 157.\nFor 20 paired with 16, I record a difference of 102.\nFor 20 paired with 17, I record a difference of 118.\nFor 20 paired with 18, I record a difference of 157.\nFor 20 paired with 19, I record a difference of 111.\nFor 20 paired with 21, I record a difference of 100.\nFor 20 paired with 22, I record a difference of 142.\nFor 20 paired with 23, I record a difference of 128.\nFor 21 paired with 0, I record a difference of 113.\nFor 21 paired with 1, I record a difference of 162.\nFor 21 paired with 2, I record a difference of 140.\nFor 21 paired with 3, I record a difference of 153.\nFor 21 paired with 4, I record a difference of 134.\nFor 21 paired with 5, I record a difference of 137.\nFor 21 paired with 6, I record a difference of 123.\nFor 21 paired with 7, I record a difference of 148.\nFor 21 paired with 8, I record a difference of 123.\nFor 21 paired with 9, I record a difference of 143.\nFor 21 paired with 10, I record a difference of 154.\nFor 21 paired with 11, I record a difference of 101.\nFor 21 paired with 12, I record a difference of 154.\nFor 21 paired with 13, I record a difference of 82.\nFor 21 paired with 14, I record a difference of 153.\nFor 21 paired with 15, I record a difference of 161.\nFor 21 paired with 16, I record a difference of 109.\nFor 21 paired with 17, I record a difference of 138.\nFor 21 paired with 18, I record a difference of 165.\nFor 21 paired with 19, I record a difference of 127.\nFor 21 paired with 20, I record a difference of 100.\nFor 21 paired with 22, I record a difference of 122.\nFor 21 paired with 23, I record a difference of 130.\nFor 22 paired with 0, I record a difference of 103.\nFor 22 paired with 1, I record a difference of 134.\nFor 22 paired with 2, I record a difference of 120.\nFor 22 paired with 3, I record a difference of 103.\nFor 22 paired with 4, I record a difference of 137.\nFor 22 paired with 5, I record a difference of 111.\nFor 22 paired with 6, I record a difference of 104.\nFor 22 paired with 7, I record a difference of 123.\nFor 22 paired with 8, I record a difference of 65.\nFor 22 paired with 9, I record a difference of 88.\nFor 22 paired with 10, I record a difference of 128.\nFor 22 paired with 11, I record a difference of 95.\nFor 22 paired with 12, I record a difference of 173.\nFor 22 paired with 13, I record a difference of 146.\nFor 22 paired with 14, I record a difference of 99.\nFor 22 paired with 15, I record a difference of 105.\nFor 22 paired with 16, I record a difference of 132.\nFor 22 paired with 17, I record a difference of 125.\nFor 22 paired with 18, I record a difference of 121.\nFor 22 paired with 19, I record a difference of 107.\nFor 22 paired with 20, I record a difference of 142.\nFor 22 paired with 21, I record a difference of 122.\nFor 22 paired with 23, I record a difference of 154.\nFor 23 paired with 0, I record a difference of 130.\nFor 23 paired with 1, I record a difference of 112.\nFor 23 paired with 2, I record a difference of 130.\nFor 23 paired with 3, I record a difference of 144.\nFor 23 paired with 4, I record a difference of 143.\nFor 23 paired with 5, I record a difference of 74.\nFor 23 paired with 6, I record a difference of 122.\nFor 23 paired with 7, I record a difference of 112.\nFor 23 paired with 8, I record a difference of 166.\nFor 23 paired with 9, I record a difference of 159.\nFor 23 paired with 10, I record a difference of 160.\nFor 23 paired with 11, I record a difference of 130.\nFor 23 paired with 12, I record a difference of 116.\nFor 23 paired with 13, I record a difference of 124.\nFor 23 paired with 14, I record a difference of 146.\nFor 23 paired with 15, I record a difference of 131.\nFor 23 paired with 16, I record a difference of 121.\nFor 23 paired with 17, I record a difference of 137.\nFor 23 paired with 18, I record a difference of 129.\nFor 23 paired with 19, I record a difference of 124.\nFor 23 paired with 20, I record a difference of 128.\nFor 23 paired with 21, I record a difference of 130.\nFor 23 paired with 22, I record a difference of 154.\nI'll pick exactly 10 acts to maximize the summed pairwise differences.\n\nAlso, when you send back the chosen lineup, toss it into this simple JSON layout so it's easy to parse:\n\n{\n \"solution\": [, , ...]\n}\n\n\"solution\" is just the list of the performers you picked for the bill — put each performer's identifier (exactly as it appears in the instance) between the angle brackets, separated by commas. This JSON is only a sketch of the shape I expect, not the actual answer.\n\nPlease make sure to use the identifiers exactly as given in the input — don't rename them or invent new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 112, 80, 82, 91, 97, 110, 87, 95, 102, 127, 86, 143, 117, 112, 114, 136, 89, 139, 92, 89, 113, 103, 130 ], [ 112, 0, 89, 91, 164, 95, 134, 59, 149, 148, 139, 139, 134, 138, 74, 146, 153, 118, 122, 132, 115, 162, 134, 112 ], [ 80, 89, 0, 102, 116, 101, 147, 69, 138, 106, 159, 128, 159, 133, 79, 151, 144, 120, 149, 105, 113, 140, 120, 130 ], [ 82, 91, 102, 0, 131, 100, 105, 105, 99, 120, 104, 109, 130, 125, 94, 112, 157, 73, 97, 111, 129, 153, 103, 144 ], [ 91, 164, 116, 131, 0, 105, 111, 146, 124, 86, 161, 92, 132, 134, 153, 119, 116, 107, 142, 68, 126, 134, 137, 143 ], [ 97, 95, 101, 100, 105, 0, 86, 98, 119, 111, 135, 104, 108, 128, 115, 89, 113, 90, 79, 90, 124, 137, 111, 74 ], [ 110, 134, 147, 105, 111, 86, 0, 141, 79, 107, 93, 66, 97, 112, 131, 76, 105, 82, 81, 87, 121, 123, 104, 122 ], [ 87, 59, 69, 105, 146, 98, 141, 0, 144, 128, 152, 122, 154, 149, 77, 135, 143, 136, 151, 115, 103, 148, 123, 112 ], [ 95, 149, 138, 99, 124, 119, 79, 144, 0, 95, 92, 89, 163, 129, 126, 101, 145, 97, 120, 111, 139, 123, 65, 166 ], [ 102, 148, 106, 120, 86, 111, 107, 128, 95, 0, 140, 84, 160, 153, 104, 101, 117, 128, 137, 56, 140, 143, 88, 159 ], [ 127, 139, 159, 104, 161, 135, 93, 152, 92, 140, 0, 123, 153, 117, 134, 120, 178, 108, 131, 137, 157, 154, 128, 160 ], [ 86, 139, 128, 109, 92, 104, 66, 122, 89, 84, 123, 0, 113, 117, 119, 88, 78, 109, 123, 54, 92, 101, 95, 130 ], [ 143, 134, 159, 130, 132, 108, 97, 154, 163, 160, 153, 113, 0, 117, 155, 141, 108, 103, 101, 114, 117, 154, 173, 116 ], [ 117, 138, 133, 125, 134, 128, 112, 149, 129, 153, 117, 117, 117, 0, 142, 165, 133, 100, 139, 129, 105, 82, 146, 124 ], [ 112, 74, 79, 94, 153, 115, 131, 77, 126, 104, 134, 119, 155, 142, 0, 140, 138, 132, 134, 107, 123, 153, 99, 146 ], [ 114, 146, 151, 112, 119, 89, 76, 135, 101, 101, 120, 88, 141, 165, 140, 0, 136, 123, 109, 89, 157, 161, 105, 131 ], [ 136, 153, 144, 157, 116, 113, 105, 143, 145, 117, 178, 78, 108, 133, 138, 136, 0, 144, 135, 82, 102, 109, 132, 121 ], [ 89, 118, 120, 73, 107, 90, 82, 136, 97, 128, 108, 109, 103, 100, 132, 123, 144, 0, 83, 113, 118, 138, 125, 137 ], [ 139, 122, 149, 97, 142, 79, 81, 151, 120, 137, 131, 123, 101, 139, 134, 109, 135, 83, 0, 121, 157, 165, 121, 129 ], [ 92, 132, 105, 111, 68, 90, 87, 115, 111, 56, 137, 54, 114, 129, 107, 89, 82, 113, 121, 0, 111, 127, 107, 124 ], [ 89, 115, 113, 129, 126, 124, 121, 103, 139, 140, 157, 92, 117, 105, 123, 157, 102, 118, 157, 111, 0, 100, 142, 128 ], [ 113, 162, 140, 153, 134, 137, 123, 148, 123, 143, 154, 101, 154, 82, 153, 161, 109, 138, 165, 127, 100, 0, 122, 130 ], [ 103, 134, 120, 103, 137, 111, 104, 123, 65, 88, 128, 95, 173, 146, 99, 105, 132, 125, 121, 107, 142, 122, 0, 154 ], [ 130, 112, 130, 144, 143, 74, 122, 112, 166, 159, 160, 130, 116, 124, 146, 131, 121, 137, 129, 124, 128, 130, 154, 0 ] ], "m": 10, "objective": 6180.0 }, "solution": [ 1, 2, 4, 10, 12, 15, 16, 21, 22, 23 ], "obj": 6180.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 24, "m": 10, "nodes": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 ], "distance_pairs": [ { "from_id": 0, "to_id": 1, "distance": 112 }, { "from_id": 0, "to_id": 2, "distance": 80 }, { "from_id": 0, "to_id": 3, "distance": 82 }, { "from_id": 0, "to_id": 4, "distance": 91 }, { "from_id": 0, "to_id": 5, "distance": 97 }, { "from_id": 0, "to_id": 6, "distance": 110 }, { "from_id": 0, "to_id": 7, "distance": 87 }, { "from_id": 0, "to_id": 8, "distance": 95 }, { "from_id": 0, "to_id": 9, "distance": 102 }, { "from_id": 0, "to_id": 10, "distance": 127 }, { "from_id": 0, "to_id": 11, "distance": 86 }, { "from_id": 0, "to_id": 12, "distance": 143 }, { "from_id": 0, "to_id": 13, "distance": 117 }, { "from_id": 0, "to_id": 14, "distance": 112 }, { "from_id": 0, "to_id": 15, "distance": 114 }, { "from_id": 0, "to_id": 16, "distance": 136 }, { "from_id": 0, "to_id": 17, "distance": 89 }, { "from_id": 0, "to_id": 18, "distance": 139 }, { "from_id": 0, "to_id": 19, "distance": 92 }, { "from_id": 0, "to_id": 20, "distance": 89 }, { "from_id": 0, "to_id": 21, "distance": 113 }, { "from_id": 0, "to_id": 22, "distance": 103 }, { "from_id": 0, "to_id": 23, "distance": 130 }, { "from_id": 1, "to_id": 0, "distance": 112 }, { "from_id": 1, "to_id": 2, "distance": 89 }, { "from_id": 1, "to_id": 3, "distance": 91 }, { "from_id": 1, "to_id": 4, "distance": 164 }, { "from_id": 1, "to_id": 5, "distance": 95 }, { "from_id": 1, "to_id": 6, "distance": 134 }, { "from_id": 1, "to_id": 7, "distance": 59 }, { "from_id": 1, "to_id": 8, "distance": 149 }, { "from_id": 1, "to_id": 9, "distance": 148 }, { "from_id": 1, "to_id": 10, "distance": 139 }, { "from_id": 1, "to_id": 11, "distance": 139 }, { "from_id": 1, "to_id": 12, "distance": 134 }, { "from_id": 1, "to_id": 13, "distance": 138 }, { "from_id": 1, "to_id": 14, "distance": 74 }, { "from_id": 1, "to_id": 15, "distance": 146 }, { "from_id": 1, "to_id": 16, "distance": 153 }, { "from_id": 1, "to_id": 17, "distance": 118 }, { "from_id": 1, "to_id": 18, "distance": 122 }, { "from_id": 1, "to_id": 19, "distance": 132 }, { "from_id": 1, "to_id": 20, "distance": 115 }, { "from_id": 1, "to_id": 21, "distance": 162 }, { "from_id": 1, "to_id": 22, "distance": 134 }, { "from_id": 1, "to_id": 23, "distance": 112 }, { "from_id": 2, "to_id": 0, "distance": 80 }, { "from_id": 2, "to_id": 1, "distance": 89 }, { "from_id": 2, "to_id": 3, "distance": 102 }, { "from_id": 2, "to_id": 4, "distance": 116 }, { "from_id": 2, "to_id": 5, "distance": 101 }, { "from_id": 2, "to_id": 6, "distance": 147 }, { "from_id": 2, "to_id": 7, "distance": 69 }, { "from_id": 2, "to_id": 8, "distance": 138 }, { "from_id": 2, "to_id": 9, "distance": 106 }, { "from_id": 2, "to_id": 10, "distance": 159 }, { "from_id": 2, "to_id": 11, "distance": 128 }, { "from_id": 2, "to_id": 12, "distance": 159 }, { "from_id": 2, "to_id": 13, "distance": 133 }, { "from_id": 2, "to_id": 14, "distance": 79 }, { "from_id": 2, "to_id": 15, "distance": 151 }, { "from_id": 2, "to_id": 16, "distance": 144 }, { "from_id": 2, "to_id": 17, "distance": 120 }, { "from_id": 2, "to_id": 18, "distance": 149 }, { "from_id": 2, "to_id": 19, "distance": 105 }, { "from_id": 2, "to_id": 20, "distance": 113 }, { "from_id": 2, "to_id": 21, "distance": 140 }, { "from_id": 2, "to_id": 22, "distance": 120 }, { "from_id": 2, "to_id": 23, "distance": 130 }, { "from_id": 3, "to_id": 0, "distance": 82 }, { "from_id": 3, "to_id": 1, "distance": 91 }, { "from_id": 3, "to_id": 2, "distance": 102 }, { "from_id": 3, "to_id": 4, "distance": 131 }, { "from_id": 3, "to_id": 5, "distance": 100 }, { "from_id": 3, "to_id": 6, "distance": 105 }, { "from_id": 3, "to_id": 7, "distance": 105 }, { "from_id": 3, "to_id": 8, "distance": 99 }, { "from_id": 3, "to_id": 9, "distance": 120 }, { "from_id": 3, "to_id": 10, "distance": 104 }, { "from_id": 3, "to_id": 11, "distance": 109 }, { "from_id": 3, "to_id": 12, "distance": 130 }, { "from_id": 3, "to_id": 13, "distance": 125 }, { "from_id": 3, "to_id": 14, "distance": 94 }, { "from_id": 3, "to_id": 15, "distance": 112 }, { "from_id": 3, "to_id": 16, "distance": 157 }, { "from_id": 3, "to_id": 17, "distance": 73 }, { "from_id": 3, "to_id": 18, "distance": 97 }, { "from_id": 3, "to_id": 19, "distance": 111 }, { "from_id": 3, "to_id": 20, "distance": 129 }, { "from_id": 3, "to_id": 21, "distance": 153 }, { "from_id": 3, "to_id": 22, "distance": 103 }, { "from_id": 3, "to_id": 23, "distance": 144 }, { "from_id": 4, "to_id": 0, "distance": 91 }, { "from_id": 4, "to_id": 1, "distance": 164 }, { "from_id": 4, "to_id": 2, "distance": 116 }, { "from_id": 4, "to_id": 3, "distance": 131 }, { "from_id": 4, "to_id": 5, "distance": 105 }, { "from_id": 4, "to_id": 6, "distance": 111 }, { "from_id": 4, "to_id": 7, "distance": 146 }, { "from_id": 4, "to_id": 8, "distance": 124 }, { "from_id": 4, "to_id": 9, "distance": 86 }, { "from_id": 4, "to_id": 10, "distance": 161 }, { "from_id": 4, "to_id": 11, "distance": 92 }, { "from_id": 4, "to_id": 12, "distance": 132 }, { "from_id": 4, "to_id": 13, "distance": 134 }, { "from_id": 4, "to_id": 14, "distance": 153 }, { "from_id": 4, "to_id": 15, "distance": 119 }, { "from_id": 4, "to_id": 16, "distance": 116 }, { "from_id": 4, "to_id": 17, "distance": 107 }, { "from_id": 4, "to_id": 18, "distance": 142 }, { "from_id": 4, "to_id": 19, "distance": 68 }, { "from_id": 4, "to_id": 20, "distance": 126 }, { "from_id": 4, "to_id": 21, "distance": 134 }, { "from_id": 4, "to_id": 22, "distance": 137 }, { "from_id": 4, "to_id": 23, "distance": 143 }, { "from_id": 5, "to_id": 0, 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"from_id": 22, "to_id": 4, "distance": 137 }, { "from_id": 22, "to_id": 5, "distance": 111 }, { "from_id": 22, "to_id": 6, "distance": 104 }, { "from_id": 22, "to_id": 7, "distance": 123 }, { "from_id": 22, "to_id": 8, "distance": 65 }, { "from_id": 22, "to_id": 9, "distance": 88 }, { "from_id": 22, "to_id": 10, "distance": 128 }, { "from_id": 22, "to_id": 11, "distance": 95 }, { "from_id": 22, "to_id": 12, "distance": 173 }, { "from_id": 22, "to_id": 13, "distance": 146 }, { "from_id": 22, "to_id": 14, "distance": 99 }, { "from_id": 22, "to_id": 15, "distance": 105 }, { "from_id": 22, "to_id": 16, "distance": 132 }, { "from_id": 22, "to_id": 17, "distance": 125 }, { "from_id": 22, "to_id": 18, "distance": 121 }, { "from_id": 22, "to_id": 19, "distance": 107 }, { "from_id": 22, "to_id": 20, "distance": 142 }, { "from_id": 22, "to_id": 21, "distance": 122 }, { "from_id": 22, "to_id": 23, "distance": 154 }, { "from_id": 23, "to_id": 0, "distance": 130 }, { "from_id": 23, "to_id": 1, "distance": 112 }, { "from_id": 23, "to_id": 2, "distance": 130 }, { "from_id": 23, "to_id": 3, "distance": 144 }, { "from_id": 23, "to_id": 4, "distance": 143 }, { "from_id": 23, "to_id": 5, "distance": 74 }, { "from_id": 23, "to_id": 6, "distance": 122 }, { "from_id": 23, "to_id": 7, "distance": 112 }, { "from_id": 23, "to_id": 8, "distance": 166 }, { "from_id": 23, "to_id": 9, "distance": 159 }, { "from_id": 23, "to_id": 10, "distance": 160 }, { "from_id": 23, "to_id": 11, "distance": 130 }, { "from_id": 23, "to_id": 12, "distance": 116 }, { "from_id": 23, "to_id": 13, "distance": 124 }, { "from_id": 23, "to_id": 14, "distance": 146 }, { "from_id": 23, "to_id": 15, "distance": 131 }, { "from_id": 23, "to_id": 16, "distance": 121 }, { "from_id": 23, "to_id": 17, "distance": 137 }, { "from_id": 23, "to_id": 18, "distance": 129 }, { "from_id": 23, "to_id": 19, "distance": 124 }, { "from_id": 23, "to_id": 20, "distance": 128 }, { "from_id": 23, "to_id": 21, "distance": 130 }, { "from_id": 23, "to_id": 22, "distance": 154 } ] }, "solution_variant": [ 1, 2, 4, 10, 12, 15, 16, 21, 22, 23 ], "context_index": 36, "input_format": "nl", "input_index_base": 0 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Recently the programming team was asked to build a mini-retrospective with exactly the stated number of slots. For every pair of potential films there’s a number showing how dissimilar they are, and a stronger, more varied program comes from choosing films whose pair-by-pair difference numbers add up to a higher total. No duplicates are allowed and the count can’t be changed. See the specific titles and their pairwise differences below.\n\nThere are 24 candidate films, listed as A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, and the program must include exactly 19 films.\nPair A and B: dissimilarity 160.\nPair A and C: dissimilarity 193.\nPair A and D: dissimilarity 203.\nPair A and E: dissimilarity 165.\nPair A and F: dissimilarity 205.\nPair A and G: dissimilarity 184.\nPair A and H: dissimilarity 188.\nPair A and I: dissimilarity 147.\nPair A and J: dissimilarity 179.\nPair A and K: dissimilarity 170.\nPair A and L: dissimilarity 183.\nPair A and M: dissimilarity 202.\nPair A and N: dissimilarity 154.\nPair A and O: dissimilarity 128.\nPair A and P: dissimilarity 152.\nPair A and Q: dissimilarity 175.\nPair A and R: dissimilarity 157.\nPair A and S: dissimilarity 121.\nPair A and T: dissimilarity 165.\nPair A and U: dissimilarity 162.\nPair A and V: dissimilarity 170.\nPair A and W: dissimilarity 144.\nPair A and X: dissimilarity 148.\nPair B and A: dissimilarity 160.\nPair B and C: dissimilarity 195.\nPair B and D: dissimilarity 176.\nPair B and E: dissimilarity 153.\nPair B and F: dissimilarity 173.\nPair B and G: dissimilarity 155.\nPair B and H: dissimilarity 177.\nPair B and I: dissimilarity 159.\nPair B and J: dissimilarity 143.\nPair B and K: dissimilarity 136.\nPair B and L: dissimilarity 169.\nPair B and M: dissimilarity 178.\nPair B and N: dissimilarity 196.\nPair B and O: dissimilarity 130.\nPair B and P: dissimilarity 153.\nPair B and Q: dissimilarity 187.\nPair B and R: dissimilarity 137.\nPair B and S: dissimilarity 148.\nPair B and T: dissimilarity 154.\nPair B and U: dissimilarity 202.\nPair B and V: dissimilarity 180.\nPair B and W: dissimilarity 132.\nPair B and X: dissimilarity 174.\nPair C and A: dissimilarity 193.\nPair C and B: dissimilarity 195.\nPair C and D: dissimilarity 195.\nPair C and E: dissimilarity 182.\nPair C and F: dissimilarity 195.\nPair C and G: dissimilarity 157.\nPair C and H: dissimilarity 163.\nPair C and I: dissimilarity 231.\nPair C and J: dissimilarity 163.\nPair C and K: dissimilarity 185.\nPair C and L: dissimilarity 220.\nPair C and M: dissimilarity 137.\nPair C and N: dissimilarity 194.\nPair C and O: dissimilarity 199.\nPair C and P: dissimilarity 209.\nPair C and Q: dissimilarity 181.\nPair C and R: dissimilarity 184.\nPair C and S: dissimilarity 177.\nPair C and T: dissimilarity 182.\nPair C and U: dissimilarity 145.\nPair C and V: dissimilarity 210.\nPair C and W: dissimilarity 197.\nPair C and X: dissimilarity 198.\nPair D and A: dissimilarity 203.\nPair D and B: dissimilarity 176.\nPair D and C: dissimilarity 195.\nPair D and E: dissimilarity 147.\nPair D and F: dissimilarity 169.\nPair D and G: dissimilarity 152.\nPair D and H: dissimilarity 189.\nPair D and I: dissimilarity 190.\nPair D and J: dissimilarity 190.\nPair D and K: dissimilarity 204.\nPair D and L: dissimilarity 224.\nPair D and M: dissimilarity 149.\nPair D and N: dissimilarity 223.\nPair D and O: dissimilarity 171.\nPair D and P: dissimilarity 203.\nPair D and Q: dissimilarity 214.\nPair D and R: dissimilarity 154.\nPair D and S: dissimilarity 185.\nPair D and T: dissimilarity 182.\nPair D and U: dissimilarity 191.\nPair D and V: dissimilarity 173.\nPair D and W: dissimilarity 196.\nPair D and X: dissimilarity 199.\nPair E and A: dissimilarity 165.\nPair E and B: dissimilarity 153.\nPair E and C: dissimilarity 182.\nPair E and D: dissimilarity 147.\nPair E and F: dissimilarity 180.\nPair E and G: dissimilarity 124.\nPair E and H: dissimilarity 179.\nPair E and I: dissimilarity 203.\nPair E and J: dissimilarity 140.\nPair E and K: dissimilarity 185.\nPair E and L: dissimilarity 156.\nPair E and M: dissimilarity 145.\nPair E and N: dissimilarity 190.\nPair E and O: dissimilarity 129.\nPair E and P: dissimilarity 163.\nPair E and Q: dissimilarity 153.\nPair E and R: dissimilarity 167.\nPair E and S: dissimilarity 142.\nPair E and T: dissimilarity 167.\nPair E and U: dissimilarity 178.\nPair E and V: dissimilarity 179.\nPair E and W: dissimilarity 150.\nPair E and X: dissimilarity 165.\nPair F and A: dissimilarity 205.\nPair F and B: dissimilarity 173.\nPair F and C: dissimilarity 195.\nPair F and D: dissimilarity 169.\nPair F and E: dissimilarity 180.\nPair F and G: dissimilarity 182.\nPair F and H: dissimilarity 210.\nPair F and I: dissimilarity 154.\nPair F and J: dissimilarity 182.\nPair F and K: dissimilarity 173.\nPair F and L: dissimilarity 177.\nPair F and M: dissimilarity 152.\nPair F and N: dissimilarity 196.\nPair F and O: dissimilarity 159.\nPair F and P: dissimilarity 149.\nPair F and Q: dissimilarity 199.\nPair F and R: dissimilarity 153.\nPair F and S: dissimilarity 189.\nPair F and T: dissimilarity 166.\nPair F and U: dissimilarity 226.\nPair F and V: dissimilarity 179.\nPair F and W: dissimilarity 157.\nPair F and X: dissimilarity 178.\nPair G and A: dissimilarity 184.\nPair G and B: dissimilarity 155.\nPair G and C: dissimilarity 157.\nPair G and D: dissimilarity 152.\nPair G and E: dissimilarity 124.\nPair G and F: dissimilarity 182.\nPair G and H: dissimilarity 194.\nPair G and I: dissimilarity 205.\nPair G and J: dissimilarity 128.\nPair G and K: dissimilarity 200.\nPair G and L: dissimilarity 186.\nPair G and M: dissimilarity 144.\nPair G and N: dissimilarity 213.\nPair G and O: dissimilarity 180.\nPair G and P: dissimilarity 169.\nPair G and Q: dissimilarity 138.\nPair G and R: dissimilarity 156.\nPair G and S: dissimilarity 146.\nPair G and T: dissimilarity 153.\nPair G and U: dissimilarity 161.\nPair G and V: dissimilarity 178.\nPair G and W: dissimilarity 172.\nPair G and X: dissimilarity 176.\nPair H and A: dissimilarity 188.\nPair H and B: dissimilarity 177.\nPair H and C: dissimilarity 163.\nPair H and D: dissimilarity 189.\nPair H and E: dissimilarity 179.\nPair H and F: dissimilarity 210.\nPair H and G: dissimilarity 194.\nPair H and I: dissimilarity 196.\nPair H and J: dissimilarity 174.\nPair H and K: dissimilarity 171.\nPair H and L: dissimilarity 197.\nPair H and M: dissimilarity 177.\nPair H and N: dissimilarity 175.\nPair H and O: dissimilarity 182.\nPair H and P: dissimilarity 214.\nPair H and Q: dissimilarity 189.\nPair H and R: dissimilarity 132.\nPair H and S: dissimilarity 176.\nPair H and T: dissimilarity 183.\nPair H and U: dissimilarity 170.\nPair H and V: dissimilarity 194.\nPair H and W: dissimilarity 193.\nPair H and X: dissimilarity 164.\nPair I and A: dissimilarity 147.\nPair I and B: dissimilarity 159.\nPair I and C: dissimilarity 231.\nPair I and D: dissimilarity 190.\nPair I and E: dissimilarity 203.\nPair I and F: dissimilarity 154.\nPair I and G: dissimilarity 205.\nPair I and H: dissimilarity 196.\nPair I and J: dissimilarity 188.\nPair I and K: dissimilarity 182.\nPair I and L: dissimilarity 193.\nPair I and M: dissimilarity 196.\nPair I and N: dissimilarity 160.\nPair I and O: dissimilarity 134.\nPair I and P: dissimilarity 140.\nPair I and Q: dissimilarity 201.\nPair I and R: dissimilarity 99.\nPair I and S: dissimilarity 158.\nPair I and T: dissimilarity 181.\nPair I and U: dissimilarity 211.\nPair I and V: dissimilarity 166.\nPair I and W: dissimilarity 147.\nPair I and X: dissimilarity 156.\nPair J and A: dissimilarity 179.\nPair J and B: dissimilarity 143.\nPair J and C: dissimilarity 163.\nPair J and D: dissimilarity 190.\nPair J and E: dissimilarity 140.\nPair J and F: dissimilarity 182.\nPair J and G: dissimilarity 128.\nPair J and H: dissimilarity 174.\nPair J and I: dissimilarity 188.\nPair J and K: dissimilarity 162.\nPair J and L: dissimilarity 152.\nPair J and M: dissimilarity 145.\nPair J and N: dissimilarity 172.\nPair J and O: dissimilarity 155.\nPair J and P: dissimilarity 166.\nPair J and Q: dissimilarity 162.\nPair J and R: dissimilarity 129.\nPair J and S: dissimilarity 150.\nPair J and T: dissimilarity 131.\nPair J and U: dissimilarity 173.\nPair J and V: dissimilarity 161.\nPair J and W: dissimilarity 115.\nPair J and X: dissimilarity 158.\nPair K and A: dissimilarity 170.\nPair K and B: dissimilarity 136.\nPair K and C: dissimilarity 185.\nPair K and D: dissimilarity 204.\nPair K and E: dissimilarity 185.\nPair K and F: dissimilarity 173.\nPair K and G: dissimilarity 200.\nPair K and H: dissimilarity 171.\nPair K and I: dissimilarity 182.\nPair K and J: dissimilarity 162.\nPair K and L: dissimilarity 160.\nPair K and M: dissimilarity 171.\nPair K and N: dissimilarity 169.\nPair K and O: dissimilarity 172.\nPair K and P: dissimilarity 188.\nPair K and Q: dissimilarity 214.\nPair K and R: dissimilarity 165.\nPair K and S: dissimilarity 171.\nPair K and T: dissimilarity 181.\nPair K and U: dissimilarity 175.\nPair K and V: dissimilarity 180.\nPair K and W: dissimilarity 156.\nPair K and X: dissimilarity 201.\nPair L and A: dissimilarity 183.\nPair L and B: dissimilarity 169.\nPair L and C: dissimilarity 220.\nPair L and D: dissimilarity 224.\nPair L and E: dissimilarity 156.\nPair L and F: dissimilarity 177.\nPair L and G: dissimilarity 186.\nPair L and H: dissimilarity 197.\nPair L and I: dissimilarity 193.\nPair L and J: dissimilarity 152.\nPair L and K: dissimilarity 160.\nPair L and M: dissimilarity 191.\nPair L and N: dissimilarity 180.\nPair L and O: dissimilarity 177.\nPair L and P: dissimilarity 179.\nPair L and Q: dissimilarity 192.\nPair L and R: dissimilarity 175.\nPair L and S: dissimilarity 192.\nPair L and T: dissimilarity 169.\nPair L and U: dissimilarity 192.\nPair L and V: dissimilarity 168.\nPair L and W: dissimilarity 147.\nPair L and X: dissimilarity 145.\nPair M and A: dissimilarity 202.\nPair M and B: dissimilarity 178.\nPair M and C: dissimilarity 137.\nPair M and D: dissimilarity 149.\nPair M and E: dissimilarity 145.\nPair M and F: dissimilarity 152.\nPair M and G: dissimilarity 144.\nPair M and H: dissimilarity 177.\nPair M and I: dissimilarity 196.\nPair M and J: dissimilarity 145.\nPair M and K: dissimilarity 171.\nPair M and L: dissimilarity 191.\nPair M and N: dissimilarity 204.\nPair M and O: dissimilarity 183.\nPair M and P: dissimilarity 195.\nPair M and Q: dissimilarity 162.\nPair M and R: dissimilarity 146.\nPair M and S: dissimilarity 153.\nPair M and T: dissimilarity 182.\nPair M and U: dissimilarity 162.\nPair M and V: dissimilarity 175.\nPair M and W: dissimilarity 188.\nPair M and X: dissimilarity 168.\nPair N and A: dissimilarity 154.\nPair N and B: dissimilarity 196.\nPair N and C: dissimilarity 194.\nPair N and D: dissimilarity 223.\nPair N and E: dissimilarity 190.\nPair N and F: dissimilarity 196.\nPair N and G: dissimilarity 213.\nPair N and H: dissimilarity 175.\nPair N and I: dissimilarity 160.\nPair N and J: dissimilarity 172.\nPair N and K: dissimilarity 169.\nPair N and L: dissimilarity 180.\nPair N and M: dissimilarity 204.\nPair N and O: dissimilarity 147.\nPair N and P: dissimilarity 165.\nPair N and Q: dissimilarity 197.\nPair N and R: dissimilarity 160.\nPair N and S: dissimilarity 175.\nPair N and T: dissimilarity 170.\nPair N and U: dissimilarity 189.\nPair N and V: dissimilarity 189.\nPair N and W: dissimilarity 132.\nPair N and X: dissimilarity 190.\nPair O and A: dissimilarity 128.\nPair O and B: dissimilarity 130.\nPair O and C: dissimilarity 199.\nPair O and D: dissimilarity 171.\nPair O and E: dissimilarity 129.\nPair O and F: dissimilarity 159.\nPair O and G: dissimilarity 180.\nPair O and H: dissimilarity 182.\nPair O and I: dissimilarity 134.\nPair O and J: dissimilarity 155.\nPair O and K: dissimilarity 172.\nPair O and L: dissimilarity 177.\nPair O and M: dissimilarity 183.\nPair O and N: dissimilarity 147.\nPair O and P: dissimilarity 117.\nPair O and Q: dissimilarity 165.\nPair O and R: dissimilarity 139.\nPair O and S: dissimilarity 120.\nPair O and T: dissimilarity 155.\nPair O and U: dissimilarity 204.\nPair O and V: dissimilarity 176.\nPair O and W: dissimilarity 101.\nPair O and X: dissimilarity 174.\nPair P and A: dissimilarity 152.\nPair P and B: dissimilarity 153.\nPair P and C: dissimilarity 209.\nPair P and D: dissimilarity 203.\nPair P and E: dissimilarity 163.\nPair P and F: dissimilarity 149.\nPair P and G: dissimilarity 169.\nPair P and H: dissimilarity 214.\nPair P and I: dissimilarity 140.\nPair P and J: dissimilarity 166.\nPair P and K: dissimilarity 188.\nPair P and L: dissimilarity 179.\nPair P and M: dissimilarity 195.\nPair P and N: dissimilarity 165.\nPair P and O: dissimilarity 117.\nPair P and Q: dissimilarity 152.\nPair P and R: dissimilarity 147.\nPair P and S: dissimilarity 142.\nPair P and T: dissimilarity 124.\nPair P and U: dissimilarity 227.\nPair P and V: dissimilarity 180.\nPair P and W: dissimilarity 128.\nPair P and X: dissimilarity 175.\nPair Q and A: dissimilarity 175.\nPair Q and B: dissimilarity 187.\nPair Q and C: dissimilarity 181.\nPair Q and D: dissimilarity 214.\nPair Q and E: dissimilarity 153.\nPair Q and F: dissimilarity 199.\nPair Q and G: dissimilarity 138.\nPair Q and H: dissimilarity 189.\nPair Q and I: dissimilarity 201.\nPair Q and J: dissimilarity 162.\nPair Q and K: dissimilarity 214.\nPair Q and L: dissimilarity 192.\nPair Q and M: dissimilarity 162.\nPair Q and N: dissimilarity 197.\nPair Q and O: dissimilarity 165.\nPair Q and P: dissimilarity 152.\nPair Q and R: dissimilarity 157.\nPair Q and S: dissimilarity 113.\nPair Q and T: dissimilarity 177.\nPair Q and U: dissimilarity 207.\nPair Q and V: dissimilarity 213.\nPair Q and W: dissimilarity 171.\nPair Q and X: dissimilarity 176.\nPair R and A: dissimilarity 157.\nPair R and B: dissimilarity 137.\nPair R and C: dissimilarity 184.\nPair R and D: dissimilarity 154.\nPair R and E: dissimilarity 167.\nPair R and F: dissimilarity 153.\nPair R and G: dissimilarity 156.\nPair R and H: dissimilarity 132.\nPair R and I: dissimilarity 99.\nPair R and J: dissimilarity 129.\nPair R and K: dissimilarity 165.\nPair R and L: dissimilarity 175.\nPair R and M: dissimilarity 146.\nPair R and N: dissimilarity 160.\nPair R and O: dissimilarity 139.\nPair R and P: dissimilarity 147.\nPair R and Q: dissimilarity 157.\nPair R and S: dissimilarity 142.\nPair R and T: dissimilarity 140.\nPair R and U: dissimilarity 177.\nPair R and V: dissimilarity 148.\nPair R and W: dissimilarity 139.\nPair R and X: dissimilarity 134.\nPair S and A: dissimilarity 121.\nPair S and B: dissimilarity 148.\nPair S and C: dissimilarity 177.\nPair S and D: dissimilarity 185.\nPair S and E: dissimilarity 142.\nPair S and F: dissimilarity 189.\nPair S and G: dissimilarity 146.\nPair S and H: dissimilarity 176.\nPair S and I: dissimilarity 158.\nPair S and J: dissimilarity 150.\nPair S and K: dissimilarity 171.\nPair S and L: dissimilarity 192.\nPair S and M: dissimilarity 153.\nPair S and N: dissimilarity 175.\nPair S and O: dissimilarity 120.\nPair S and P: dissimilarity 142.\nPair S and Q: dissimilarity 113.\nPair S and R: dissimilarity 142.\nPair S and T: dissimilarity 160.\nPair S and U: dissimilarity 174.\nPair S and V: dissimilarity 160.\nPair S and W: dissimilarity 138.\nPair S and X: dissimilarity 155.\nPair T and A: dissimilarity 165.\nPair T and B: dissimilarity 154.\nPair T and C: dissimilarity 182.\nPair T and D: dissimilarity 182.\nPair T and E: dissimilarity 167.\nPair T and F: dissimilarity 166.\nPair T and G: dissimilarity 153.\nPair T and H: dissimilarity 183.\nPair T and I: dissimilarity 181.\nPair T and J: dissimilarity 131.\nPair T and K: dissimilarity 181.\nPair T and L: dissimilarity 169.\nPair T and M: dissimilarity 182.\nPair T and N: dissimilarity 170.\nPair T and O: dissimilarity 155.\nPair T and P: dissimilarity 124.\nPair T and Q: dissimilarity 177.\nPair T and R: dissimilarity 140.\nPair T and S: dissimilarity 160.\nPair T and U: dissimilarity 192.\nPair T and V: dissimilarity 125.\nPair T and W: dissimilarity 127.\nPair T and X: dissimilarity 157.\nPair U and A: dissimilarity 162.\nPair U and B: dissimilarity 202.\nPair U and C: dissimilarity 145.\nPair U and D: dissimilarity 191.\nPair U and E: dissimilarity 178.\nPair U and F: dissimilarity 226.\nPair U and G: dissimilarity 161.\nPair U and H: dissimilarity 170.\nPair U and I: dissimilarity 211.\nPair U and J: dissimilarity 173.\nPair U and K: dissimilarity 175.\nPair U and L: dissimilarity 192.\nPair U and M: dissimilarity 162.\nPair U and N: dissimilarity 189.\nPair U and O: dissimilarity 204.\nPair U and P: dissimilarity 227.\nPair U and Q: dissimilarity 207.\nPair U and R: dissimilarity 177.\nPair U and S: dissimilarity 174.\nPair U and T: dissimilarity 192.\nPair U and V: dissimilarity 164.\nPair U and W: dissimilarity 209.\nPair U and X: dissimilarity 190.\nPair V and A: dissimilarity 170.\nPair V and B: dissimilarity 180.\nPair V and C: dissimilarity 210.\nPair V and D: dissimilarity 173.\nPair V and E: dissimilarity 179.\nPair V and F: dissimilarity 179.\nPair V and G: dissimilarity 178.\nPair V and H: dissimilarity 194.\nPair V and I: dissimilarity 166.\nPair V and J: dissimilarity 161.\nPair V and K: dissimilarity 180.\nPair V and L: dissimilarity 168.\nPair V and M: dissimilarity 175.\nPair V and N: dissimilarity 189.\nPair V and O: dissimilarity 176.\nPair V and P: dissimilarity 180.\nPair V and Q: dissimilarity 213.\nPair V and R: dissimilarity 148.\nPair V and S: dissimilarity 160.\nPair V and T: dissimilarity 125.\nPair V and U: dissimilarity 164.\nPair V and W: dissimilarity 159.\nPair V and X: dissimilarity 153.\nPair W and A: dissimilarity 144.\nPair W and B: dissimilarity 132.\nPair W and C: dissimilarity 197.\nPair W and D: dissimilarity 196.\nPair W and E: dissimilarity 150.\nPair W and F: dissimilarity 157.\nPair W and G: dissimilarity 172.\nPair W and H: dissimilarity 193.\nPair W and I: dissimilarity 147.\nPair W and J: dissimilarity 115.\nPair W and K: dissimilarity 156.\nPair W and L: dissimilarity 147.\nPair W and M: dissimilarity 188.\nPair W and N: dissimilarity 132.\nPair W and O: dissimilarity 101.\nPair W and P: dissimilarity 128.\nPair W and Q: dissimilarity 171.\nPair W and R: dissimilarity 139.\nPair W and S: dissimilarity 138.\nPair W and T: dissimilarity 127.\nPair W and U: dissimilarity 209.\nPair W and V: dissimilarity 159.\nPair W and X: dissimilarity 162.\nPair X and A: dissimilarity 148.\nPair X and B: dissimilarity 174.\nPair X and C: dissimilarity 198.\nPair X and D: dissimilarity 199.\nPair X and E: dissimilarity 165.\nPair X and F: dissimilarity 178.\nPair X and G: dissimilarity 176.\nPair X and H: dissimilarity 164.\nPair X and I: dissimilarity 156.\nPair X and J: dissimilarity 158.\nPair X and K: dissimilarity 201.\nPair X and L: dissimilarity 145.\nPair X and M: dissimilarity 168.\nPair X and N: dissimilarity 190.\nPair X and O: dissimilarity 174.\nPair X and P: dissimilarity 175.\nPair X and Q: dissimilarity 176.\nPair X and R: dissimilarity 134.\nPair X and S: dissimilarity 155.\nPair X and T: dissimilarity 157.\nPair X and U: dissimilarity 190.\nPair X and V: dissimilarity 153.\nPair X and W: dissimilarity 162.\nThese dissimilarity values allow the programming team to select the 19 films that maximize the total pairwise dissimilarity.\n\nAlso, when you send back your chosen program, please use this simple JSON shape so it's easy to parse:\n\n{\n \"solution\": [\"\", \"\", ...]\n}\n\nHere \"solution\" is just the list of the film identifiers you picked for the slots — one identifier per chosen film, in any order. Think of it like filling out a short form: put the exact labels of the films you want inside that array. This JSON is just a sketch of the shape I expect, not the actual answer itself.\n\nPlease use the identifiers exactly as they appear in the instance input — do not rename them or invent new labels. \nValid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.", "instance": { "distance_matrix": [ [ 0, 160, 193, 203, 165, 205, 184, 188, 147, 179, 170, 183, 202, 154, 128, 152, 175, 157, 121, 165, 162, 170, 144, 148 ], [ 160, 0, 195, 176, 153, 173, 155, 177, 159, 143, 136, 169, 178, 196, 130, 153, 187, 137, 148, 154, 202, 180, 132, 174 ], [ 193, 195, 0, 195, 182, 195, 157, 163, 231, 163, 185, 220, 137, 194, 199, 209, 181, 184, 177, 182, 145, 210, 197, 198 ], [ 203, 176, 195, 0, 147, 169, 152, 189, 190, 190, 204, 224, 149, 223, 171, 203, 214, 154, 185, 182, 191, 173, 196, 199 ], [ 165, 153, 182, 147, 0, 180, 124, 179, 203, 140, 185, 156, 145, 190, 129, 163, 153, 167, 142, 167, 178, 179, 150, 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"context_index": 37, "input_format": "nl", "input_index_base": "names" }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "As coordinator of the fair, the task is simple to say and tricky to do: choose exactly the number of activities requested so the set of booths shows the broadest mix of styles. To judge any lineup, add up how dissimilar each pair of booths in that lineup is — the bigger that summed number, the better the variety. It’s mandatory to pick exactly that many booths and not repeat any. The specific candidates and the exact count to pick are shown below.\n\nThere are 19 candidate booths labeled 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18; pick exactly 15 distinct booths (no repeats).\nPair 0 vs 1: dissimilarity 62.\nPair 0 vs 2: dissimilarity 54.\nPair 0 vs 3: dissimilarity 71.\nPair 0 vs 4: dissimilarity 85.\nPair 0 vs 5: dissimilarity 49.\nPair 0 vs 6: dissimilarity 82.\nPair 0 vs 7: dissimilarity 10.\nPair 0 vs 8: dissimilarity 78.\nPair 0 vs 9: dissimilarity 89.\nPair 0 vs 10: dissimilarity 63.\nPair 0 vs 11: dissimilarity 73.\nPair 0 vs 12: dissimilarity 88.\nPair 0 vs 13: dissimilarity 77.\nPair 0 vs 14: dissimilarity 91.\nPair 0 vs 15: dissimilarity 70.\nPair 0 vs 16: dissimilarity 22.\nPair 0 vs 17: dissimilarity 39.\nPair 0 vs 18: dissimilarity 101.\nPair 1 vs 0: dissimilarity 62.\nPair 1 vs 2: dissimilarity 58.\nPair 1 vs 3: dissimilarity 55.\nPair 1 vs 4: dissimilarity 28.\nPair 1 vs 5: dissimilarity 55.\nPair 1 vs 6: dissimilarity 29.\nPair 1 vs 7: dissimilarity 63.\nPair 1 vs 8: dissimilarity 40.\nPair 1 vs 9: dissimilarity 37.\nPair 1 vs 10: dissimilarity 85.\nPair 1 vs 11: dissimilarity 55.\nPair 1 vs 12: dissimilarity 29.\nPair 1 vs 13: dissimilarity 55.\nPair 1 vs 14: dissimilarity 56.\nPair 1 vs 15: dissimilarity 62.\nPair 1 vs 16: dissimilarity 58.\nPair 1 vs 17: dissimilarity 75.\nPair 1 vs 18: dissimilarity 49.\nPair 2 vs 0: dissimilarity 54.\nPair 2 vs 1: dissimilarity 58.\nPair 2 vs 3: dissimilarity 89.\nPair 2 vs 4: dissimilarity 72.\nPair 2 vs 5: dissimilarity 7.\nPair 2 vs 6: dissimilarity 88.\nPair 2 vs 7: dissimilarity 63.\nPair 2 vs 8: dissimilarity 47.\nPair 2 vs 9: dissimilarity 80.\nPair 2 vs 10: dissimilarity 34.\nPair 2 vs 11: dissimilarity 59.\nPair 2 vs 12: dissimilarity 69.\nPair 2 vs 13: dissimilarity 57.\nPair 2 vs 14: dissimilarity 98.\nPair 2 vs 15: dissimilarity 48.\nPair 2 vs 16: dissimilarity 34.\nPair 2 vs 17: dissimilarity 30.\nPair 2 vs 18: dissimilarity 73.\nPair 3 vs 0: dissimilarity 71.\nPair 3 vs 1: dissimilarity 55.\nPair 3 vs 2: dissimilarity 89.\nPair 3 vs 4: dissimilarity 56.\nPair 3 vs 5: dissimilarity 88.\nPair 3 vs 6: dissimilarity 58.\nPair 3 vs 7: dissimilarity 67.\nPair 3 vs 8: dissimilarity 93.\nPair 3 vs 9: dissimilarity 49.\nPair 3 vs 10: dissimilarity 96.\nPair 3 vs 11: dissimilarity 108.\nPair 3 vs 12: dissimilarity 72.\nPair 3 vs 13: dissimilarity 109.\nPair 3 vs 14: dissimilarity 25.\nPair 3 vs 15: dissimilarity 63.\nPair 3 vs 16: dissimilarity 73.\nPair 3 vs 17: dissimilarity 94.\nPair 3 vs 18: dissimilarity 77.\nPair 4 vs 0: dissimilarity 85.\nPair 4 vs 1: dissimilarity 28.\nPair 4 vs 2: dissimilarity 72.\nPair 4 vs 3: dissimilarity 56.\nPair 4 vs 5: dissimilarity 71.\nPair 4 vs 6: dissimilarity 35.\nPair 4 vs 7: dissimilarity 86.\nPair 4 vs 8: dissimilarity 50.\nPair 4 vs 9: dissimilarity 13.\nPair 4 vs 10: dissimilarity 94.\nPair 4 vs 11: dissimilarity 76.\nPair 4 vs 12: dissimilarity 17.\nPair 4 vs 13: dissimilarity 74.\nPair 4 vs 14: dissimilarity 45.\nPair 4 vs 15: dissimilarity 60.\nPair 4 vs 16: dissimilarity 78.\nPair 4 vs 17: dissimilarity 94.\nPair 4 vs 18: dissimilarity 26.\nPair 5 vs 0: dissimilarity 49.\nPair 5 vs 1: dissimilarity 55.\nPair 5 vs 2: dissimilarity 7.\nPair 5 vs 3: dissimilarity 88.\nPair 5 vs 4: dissimilarity 71.\nPair 5 vs 6: dissimilarity 84.\nPair 5 vs 7: dissimilarity 58.\nPair 5 vs 8: dissimilarity 44.\nPair 5 vs 9: dissimilarity 79.\nPair 5 vs 10: dissimilarity 40.\nPair 5 vs 11: dissimilarity 52.\nPair 5 vs 12: dissimilarity 67.\nPair 5 vs 13: dissimilarity 51.\nPair 5 vs 14: dissimilarity 98.\nPair 5 vs 15: dissimilarity 53.\nPair 5 vs 16: dissimilarity 30.\nPair 5 vs 17: dissimilarity 29.\nPair 5 vs 18: dissimilarity 74.\nPair 6 vs 0: dissimilarity 82.\nPair 6 vs 1: dissimilarity 29.\nPair 6 vs 2: dissimilarity 88.\nPair 6 vs 3: dissimilarity 58.\nPair 6 vs 4: dissimilarity 35.\nPair 6 vs 5: dissimilarity 84.\nPair 6 vs 7: dissimilarity 80.\nPair 6 vs 8: dissimilarity 62.\nPair 6 vs 9: dissimilarity 39.\nPair 6 vs 10: dissimilarity 113.\nPair 6 vs 11: dissimilarity 73.\nPair 6 vs 12: dissimilarity 39.\nPair 6 vs 13: dissimilarity 74.\nPair 6 vs 14: dissimilarity 51.\nPair 6 vs 15: dissimilarity 86.\nPair 6 vs 16: dissimilarity 84.\nPair 6 vs 17: dissimilarity 103.\nPair 6 vs 18: dissimilarity 60.\nPair 7 vs 0: dissimilarity 10.\nPair 7 vs 1: dissimilarity 63.\nPair 7 vs 2: dissimilarity 63.\nPair 7 vs 3: dissimilarity 67.\nPair 7 vs 4: dissimilarity 86.\nPair 7 vs 5: dissimilarity 58.\nPair 7 vs 6: dissimilarity 80.\nPair 7 vs 8: dissimilarity 84.\nPair 7 vs 9: dissimilarity 90.\nPair 7 vs 10: dissimilarity 72.\nPair 7 vs 11: dissimilarity 78.\nPair 7 vs 12: dissimilarity 91.\nPair 7 vs 13: dissimilarity 83.\nPair 7 vs 14: dissimilarity 88.\nPair 7 vs 15: dissimilarity 75.\nPair 7 vs 16: dissimilarity 31.\nPair 7 vs 17: dissimilarity 49.\nPair 7 vs 18: dissimilarity 105.\nPair 8 vs 0: dissimilarity 78.\nPair 8 vs 1: dissimilarity 40.\nPair 8 vs 2: dissimilarity 47.\nPair 8 vs 3: dissimilarity 93.\nPair 8 vs 4: dissimilarity 50.\nPair 8 vs 5: dissimilarity 44.\nPair 8 vs 6: dissimilarity 62.\nPair 8 vs 7: dissimilarity 84.\nPair 8 vs 9: dissimilarity 63.\nPair 8 vs 10: dissimilarity 80.\nPair 8 vs 11: dissimilarity 35.\nPair 8 vs 12: dissimilarity 37.\nPair 8 vs 13: dissimilarity 29.\nPair 8 vs 14: dissimilarity 92.\nPair 8 vs 15: dissimilarity 72.\nPair 8 vs 16: dissimilarity 68.\nPair 8 vs 17: dissimilarity 74.\nPair 8 vs 18: dissimilarity 50.\nPair 9 vs 0: dissimilarity 89.\nPair 9 vs 1: dissimilarity 37.\nPair 9 vs 2: dissimilarity 80.\nPair 9 vs 3: dissimilarity 49.\nPair 9 vs 4: dissimilarity 13.\nPair 9 vs 5: dissimilarity 79.\nPair 9 vs 6: dissimilarity 39.\nPair 9 vs 7: dissimilarity 90.\nPair 9 vs 8: dissimilarity 63.\nPair 9 vs 10: dissimilarity 98.\nPair 9 vs 11: dissimilarity 89.\nPair 9 vs 12: dissimilarity 30.\nPair 9 vs 13: dissimilarity 87.\nPair 9 vs 14: dissimilarity 34.\nPair 9 vs 15: dissimilarity 59.\nPair 9 vs 16: dissimilarity 83.\nPair 9 vs 17: dissimilarity 99.\nPair 9 vs 18: dissimilarity 31.\nPair 10 vs 0: dissimilarity 63.\nPair 10 vs 1: dissimilarity 85.\nPair 10 vs 2: dissimilarity 34.\nPair 10 vs 3: dissimilarity 96.\nPair 10 vs 4: dissimilarity 94.\nPair 10 vs 5: dissimilarity 40.\nPair 10 vs 6: dissimilarity 113.\nPair 10 vs 7: dissimilarity 72.\nPair 10 vs 8: dissimilarity 80.\nPair 10 vs 9: dissimilarity 98.\nPair 10 vs 11: dissimilarity 92.\nPair 10 vs 12: dissimilarity 95.\nPair 10 vs 13: dissimilarity 91.\nPair 10 vs 14: dissimilarity 109.\nPair 10 vs 15: dissimilarity 45.\nPair 10 vs 16: dissimilarity 42.\nPair 10 vs 17: dissimilarity 31.\nPair 10 vs 18: dissimilarity 93.\nPair 11 vs 0: dissimilarity 73.\nPair 11 vs 1: dissimilarity 55.\nPair 11 vs 2: dissimilarity 59.\nPair 11 vs 3: dissimilarity 108.\nPair 11 vs 4: dissimilarity 76.\nPair 11 vs 5: dissimilarity 52.\nPair 11 vs 6: dissimilarity 73.\nPair 11 vs 7: dissimilarity 78.\nPair 11 vs 8: dissimilarity 35.\nPair 11 vs 9: dissimilarity 89.\nPair 11 vs 10: dissimilarity 92.\nPair 11 vs 12: dissimilarity 65.\nPair 11 vs 13: dissimilarity 8.\nPair 11 vs 14: dissimilarity 112.\nPair 11 vs 15: dissimilarity 96.\nPair 11 vs 16: dissimilarity 70.\nPair 11 vs 17: dissimilarity 74.\nPair 11 vs 18: dissimilarity 84.\nPair 12 vs 0: dissimilarity 88.\nPair 12 vs 1: dissimilarity 29.\nPair 12 vs 2: dissimilarity 69.\nPair 12 vs 3: dissimilarity 72.\nPair 12 vs 4: dissimilarity 17.\nPair 12 vs 5: dissimilarity 67.\nPair 12 vs 6: dissimilarity 39.\nPair 12 vs 7: dissimilarity 91.\nPair 12 vs 8: dissimilarity 37.\nPair 12 vs 9: dissimilarity 30.\nPair 12 vs 10: dissimilarity 95.\nPair 12 vs 11: dissimilarity 65.\nPair 12 vs 13: dissimilarity 61.\nPair 12 vs 14: dissimilarity 62.\nPair 12 vs 15: dissimilarity 67.\nPair 12 vs 16: dissimilarity 80.\nPair 12 vs 17: dissimilarity 93.\nPair 12 vs 18: dissimilarity 25.\nPair 13 vs 0: dissimilarity 77.\nPair 13 vs 1: dissimilarity 55.\nPair 13 vs 2: dissimilarity 57.\nPair 13 vs 3: dissimilarity 109.\nPair 13 vs 4: dissimilarity 74.\nPair 13 vs 5: dissimilarity 51.\nPair 13 vs 6: dissimilarity 74.\nPair 13 vs 7: dissimilarity 83.\nPair 13 vs 8: dissimilarity 29.\nPair 13 vs 9: dissimilarity 87.\nPair 13 vs 10: dissimilarity 91.\nPair 13 vs 11: dissimilarity 8.\nPair 13 vs 12: dissimilarity 61.\nPair 13 vs 14: dissimilarity 112.\nPair 13 vs 15: dissimilarity 93.\nPair 13 vs 16: dissimilarity 71.\nPair 13 vs 17: dissimilarity 75.\nPair 13 vs 18: dissimilarity 79.\nPair 14 vs 0: dissimilarity 91.\nPair 14 vs 1: dissimilarity 56.\nPair 14 vs 2: dissimilarity 98.\nPair 14 vs 3: dissimilarity 25.\nPair 14 vs 4: dissimilarity 45.\nPair 14 vs 5: dissimilarity 98.\nPair 14 vs 6: dissimilarity 51.\nPair 14 vs 7: dissimilarity 88.\nPair 14 vs 8: dissimilarity 92.\nPair 14 vs 9: dissimilarity 34.\nPair 14 vs 10: dissimilarity 109.\nPair 14 vs 11: dissimilarity 112.\nPair 14 vs 12: dissimilarity 62.\nPair 14 vs 13: dissimilarity 112.\nPair 14 vs 15: dissimilarity 69.\nPair 14 vs 16: dissimilarity 90.\nPair 14 vs 17: dissimilarity 110.\nPair 14 vs 18: dissimilarity 64.\nPair 15 vs 0: dissimilarity 70.\nPair 15 vs 1: dissimilarity 62.\nPair 15 vs 2: dissimilarity 48.\nPair 15 vs 3: dissimilarity 63.\nPair 15 vs 4: dissimilarity 60.\nPair 15 vs 5: dissimilarity 53.\nPair 15 vs 6: dissimilarity 86.\nPair 15 vs 7: dissimilarity 75.\nPair 15 vs 8: dissimilarity 72.\nPair 15 vs 9: dissimilarity 59.\nPair 15 vs 10: dissimilarity 45.\nPair 15 vs 11: dissimilarity 96.\nPair 15 vs 12: dissimilarity 67.\nPair 15 vs 13: dissimilarity 93.\nPair 15 vs 14: dissimilarity 69.\nPair 15 vs 16: dissimilarity 53.\nPair 15 vs 17: dissimilarity 61.\nPair 15 vs 18: dissimilarity 60.\nPair 16 vs 0: dissimilarity 22.\nPair 16 vs 1: dissimilarity 58.\nPair 16 vs 2: dissimilarity 34.\nPair 16 vs 3: dissimilarity 73.\nPair 16 vs 4: dissimilarity 78.\nPair 16 vs 5: dissimilarity 30.\nPair 16 vs 6: dissimilarity 84.\nPair 16 vs 7: dissimilarity 31.\nPair 16 vs 8: dissimilarity 68.\nPair 16 vs 9: dissimilarity 83.\nPair 16 vs 10: dissimilarity 42.\nPair 16 vs 11: dissimilarity 70.\nPair 16 vs 12: dissimilarity 80.\nPair 16 vs 13: dissimilarity 71.\nPair 16 vs 14: dissimilarity 90.\nPair 16 vs 15: dissimilarity 53.\nPair 16 vs 17: dissimilarity 22.\nPair 16 vs 18: dissimilarity 89.\nPair 17 vs 0: dissimilarity 39.\nPair 17 vs 1: dissimilarity 75.\nPair 17 vs 2: dissimilarity 30.\nPair 17 vs 3: dissimilarity 94.\nPair 17 vs 4: dissimilarity 94.\nPair 17 vs 5: dissimilarity 29.\nPair 17 vs 6: dissimilarity 103.\nPair 17 vs 7: dissimilarity 49.\nPair 17 vs 8: dissimilarity 74.\nPair 17 vs 9: dissimilarity 99.\nPair 17 vs 10: dissimilarity 31.\nPair 17 vs 11: dissimilarity 74.\nPair 17 vs 12: dissimilarity 93.\nPair 17 vs 13: dissimilarity 75.\nPair 17 vs 14: dissimilarity 110.\nPair 17 vs 15: dissimilarity 61.\nPair 17 vs 16: dissimilarity 22.\nPair 17 vs 18: dissimilarity 100.\nPair 18 vs 0: dissimilarity 101.\nPair 18 vs 1: dissimilarity 49.\nPair 18 vs 2: dissimilarity 73.\nPair 18 vs 3: dissimilarity 77.\nPair 18 vs 4: dissimilarity 26.\nPair 18 vs 5: dissimilarity 74.\nPair 18 vs 6: dissimilarity 60.\nPair 18 vs 7: dissimilarity 105.\nPair 18 vs 8: dissimilarity 50.\nPair 18 vs 9: dissimilarity 31.\nPair 18 vs 10: dissimilarity 93.\nPair 18 vs 11: dissimilarity 84.\nPair 18 vs 12: dissimilarity 25.\nPair 18 vs 13: dissimilarity 79.\nPair 18 vs 14: dissimilarity 64.\nPair 18 vs 15: dissimilarity 60.\nPair 18 vs 16: dissimilarity 89.\nPair 18 vs 17: dissimilarity 100.\nChoose the 15 booths that maximize the total summed dissimilarity across every pair in the lineup.\n\nAlso, when you hand me the final pick, a little JSON like this is the most convenient — nothing fancy, just a plain list of the booths you chose:\n\n{\n \"solution\": [, , ...]\n}\n\n\"solution\" is the list of chosen booths — put exactly the requested number of booth identifiers inside the array, one per chosen booth. The angled placeholders (like ) just show the shape I expect; they should be replaced by the actual identifiers from the instance.\n\nThis JSON is only a sketch of the expected shape, not the actual answer. Please use the identifiers exactly as they appear in the instance input — do not rename them or invent new labels.\n\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 62, 54, 71, 85, 49, 82, 10, 78, 89, 63, 73, 88, 77, 91, 70, 22, 39, 101 ], [ 62, 0, 58, 55, 28, 55, 29, 63, 40, 37, 85, 55, 29, 55, 56, 62, 58, 75, 49 ], [ 54, 58, 0, 89, 72, 7, 88, 63, 47, 80, 34, 59, 69, 57, 98, 48, 34, 30, 73 ], [ 71, 55, 89, 0, 56, 88, 58, 67, 93, 49, 96, 108, 72, 109, 25, 63, 73, 94, 77 ], [ 85, 28, 72, 56, 0, 71, 35, 86, 50, 13, 94, 76, 17, 74, 45, 60, 78, 94, 26 ], [ 49, 55, 7, 88, 71, 0, 84, 58, 44, 79, 40, 52, 67, 51, 98, 53, 30, 29, 74 ], [ 82, 29, 88, 58, 35, 84, 0, 80, 62, 39, 113, 73, 39, 74, 51, 86, 84, 103, 60 ], [ 10, 63, 63, 67, 86, 58, 80, 0, 84, 90, 72, 78, 91, 83, 88, 75, 31, 49, 105 ], [ 78, 40, 47, 93, 50, 44, 62, 84, 0, 63, 80, 35, 37, 29, 92, 72, 68, 74, 50 ], [ 89, 37, 80, 49, 13, 79, 39, 90, 63, 0, 98, 89, 30, 87, 34, 59, 83, 99, 31 ], [ 63, 85, 34, 96, 94, 40, 113, 72, 80, 98, 0, 92, 95, 91, 109, 45, 42, 31, 93 ], [ 73, 55, 59, 108, 76, 52, 73, 78, 35, 89, 92, 0, 65, 8, 112, 96, 70, 74, 84 ], [ 88, 29, 69, 72, 17, 67, 39, 91, 37, 30, 95, 65, 0, 61, 62, 67, 80, 93, 25 ], [ 77, 55, 57, 109, 74, 51, 74, 83, 29, 87, 91, 8, 61, 0, 112, 93, 71, 75, 79 ], [ 91, 56, 98, 25, 45, 98, 51, 88, 92, 34, 109, 112, 62, 112, 0, 69, 90, 110, 64 ], [ 70, 62, 48, 63, 60, 53, 86, 75, 72, 59, 45, 96, 67, 93, 69, 0, 53, 61, 60 ], [ 22, 58, 34, 73, 78, 30, 84, 31, 68, 83, 42, 70, 80, 71, 90, 53, 0, 22, 89 ], [ 39, 75, 30, 94, 94, 29, 103, 49, 74, 99, 31, 74, 93, 75, 110, 61, 22, 0, 100 ], [ 101, 49, 73, 77, 26, 74, 60, 105, 50, 31, 93, 84, 25, 79, 64, 60, 89, 100, 0 ] ], "m": 15, "objective": 7512.0 }, "solution": [ 0, 2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18 ], "obj": 7512.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 19, "m": 15, "nodes": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 ], "distance_pairs": [ { "from_id": 0, "to_id": 1, "distance": 62 }, { "from_id": 0, "to_id": 2, "distance": 54 }, { "from_id": 0, "to_id": 3, "distance": 71 }, { "from_id": 0, "to_id": 4, "distance": 85 }, { "from_id": 0, "to_id": 5, "distance": 49 }, { "from_id": 0, "to_id": 6, "distance": 82 }, { "from_id": 0, "to_id": 7, "distance": 10 }, { "from_id": 0, "to_id": 8, "distance": 78 }, { "from_id": 0, "to_id": 9, "distance": 89 }, { "from_id": 0, "to_id": 10, "distance": 63 }, { "from_id": 0, "to_id": 11, "distance": 73 }, { "from_id": 0, "to_id": 12, "distance": 88 }, { "from_id": 0, "to_id": 13, "distance": 77 }, { "from_id": 0, "to_id": 14, "distance": 91 }, { "from_id": 0, "to_id": 15, "distance": 70 }, { "from_id": 0, "to_id": 16, "distance": 22 }, { "from_id": 0, "to_id": 17, "distance": 39 }, { "from_id": 0, "to_id": 18, "distance": 101 }, { "from_id": 1, "to_id": 0, "distance": 62 }, { "from_id": 1, "to_id": 2, "distance": 58 }, { "from_id": 1, "to_id": 3, "distance": 55 }, { "from_id": 1, "to_id": 4, "distance": 28 }, { "from_id": 1, "to_id": 5, "distance": 55 }, { "from_id": 1, "to_id": 6, "distance": 29 }, { "from_id": 1, "to_id": 7, "distance": 63 }, { "from_id": 1, "to_id": 8, "distance": 40 }, { "from_id": 1, "to_id": 9, "distance": 37 }, { "from_id": 1, "to_id": 10, "distance": 85 }, { "from_id": 1, "to_id": 11, "distance": 55 }, { "from_id": 1, "to_id": 12, "distance": 29 }, { "from_id": 1, "to_id": 13, "distance": 55 }, { "from_id": 1, "to_id": 14, "distance": 56 }, { "from_id": 1, "to_id": 15, "distance": 62 }, { "from_id": 1, "to_id": 16, "distance": 58 }, { "from_id": 1, "to_id": 17, "distance": 75 }, { "from_id": 1, "to_id": 18, "distance": 49 }, { "from_id": 2, "to_id": 0, "distance": 54 }, { "from_id": 2, "to_id": 1, "distance": 58 }, { "from_id": 2, "to_id": 3, "distance": 89 }, { "from_id": 2, "to_id": 4, "distance": 72 }, { "from_id": 2, "to_id": 5, "distance": 7 }, { "from_id": 2, "to_id": 6, "distance": 88 }, { "from_id": 2, "to_id": 7, "distance": 63 }, { "from_id": 2, "to_id": 8, "distance": 47 }, { "from_id": 2, "to_id": 9, "distance": 80 }, { "from_id": 2, "to_id": 10, "distance": 34 }, { "from_id": 2, "to_id": 11, "distance": 59 }, { "from_id": 2, "to_id": 12, "distance": 69 }, { "from_id": 2, "to_id": 13, "distance": 57 }, { "from_id": 2, "to_id": 14, "distance": 98 }, { "from_id": 2, "to_id": 15, "distance": 48 }, { "from_id": 2, "to_id": 16, "distance": 34 }, { "from_id": 2, "to_id": 17, "distance": 30 }, { "from_id": 2, "to_id": 18, "distance": 73 }, { "from_id": 3, "to_id": 0, "distance": 71 }, { "from_id": 3, "to_id": 1, "distance": 55 }, { "from_id": 3, "to_id": 2, "distance": 89 }, { "from_id": 3, "to_id": 4, "distance": 56 }, { "from_id": 3, "to_id": 5, "distance": 88 }, { "from_id": 3, "to_id": 6, "distance": 58 }, { "from_id": 3, "to_id": 7, "distance": 67 }, { "from_id": 3, "to_id": 8, "distance": 93 }, { "from_id": 3, "to_id": 9, "distance": 49 }, { "from_id": 3, "to_id": 10, "distance": 96 }, { "from_id": 3, "to_id": 11, "distance": 108 }, { "from_id": 3, "to_id": 12, "distance": 72 }, { "from_id": 3, "to_id": 13, "distance": 109 }, { "from_id": 3, "to_id": 14, "distance": 25 }, { "from_id": 3, "to_id": 15, "distance": 63 }, { "from_id": 3, "to_id": 16, "distance": 73 }, { "from_id": 3, "to_id": 17, "distance": 94 }, { "from_id": 3, "to_id": 18, "distance": 77 }, { "from_id": 4, "to_id": 0, "distance": 85 }, { "from_id": 4, "to_id": 1, "distance": 28 }, { "from_id": 4, "to_id": 2, "distance": 72 }, { "from_id": 4, "to_id": 3, "distance": 56 }, { "from_id": 4, "to_id": 5, "distance": 71 }, { "from_id": 4, "to_id": 6, "distance": 35 }, { "from_id": 4, "to_id": 7, "distance": 86 }, { "from_id": 4, "to_id": 8, "distance": 50 }, { "from_id": 4, "to_id": 9, "distance": 13 }, { "from_id": 4, "to_id": 10, "distance": 94 }, { "from_id": 4, "to_id": 11, "distance": 76 }, { "from_id": 4, "to_id": 12, "distance": 17 }, { "from_id": 4, "to_id": 13, "distance": 74 }, { "from_id": 4, "to_id": 14, "distance": 45 }, { "from_id": 4, "to_id": 15, "distance": 60 }, { "from_id": 4, "to_id": 16, "distance": 78 }, { "from_id": 4, "to_id": 17, "distance": 94 }, { "from_id": 4, "to_id": 18, "distance": 26 }, { "from_id": 5, "to_id": 0, "distance": 49 }, { "from_id": 5, "to_id": 1, "distance": 55 }, { "from_id": 5, "to_id": 2, "distance": 7 }, { "from_id": 5, "to_id": 3, "distance": 88 }, { "from_id": 5, "to_id": 4, "distance": 71 }, { "from_id": 5, "to_id": 6, "distance": 84 }, { "from_id": 5, "to_id": 7, "distance": 58 }, { "from_id": 5, "to_id": 8, "distance": 44 }, { "from_id": 5, "to_id": 9, "distance": 79 }, { "from_id": 5, "to_id": 10, "distance": 40 }, { "from_id": 5, "to_id": 11, "distance": 52 }, { "from_id": 5, "to_id": 12, "distance": 67 }, { "from_id": 5, "to_id": 13, "distance": 51 }, { "from_id": 5, "to_id": 14, "distance": 98 }, { "from_id": 5, "to_id": 15, "distance": 53 }, { "from_id": 5, "to_id": 16, "distance": 30 }, { "from_id": 5, "to_id": 17, "distance": 29 }, { "from_id": 5, "to_id": 18, "distance": 74 }, { "from_id": 6, "to_id": 0, "distance": 82 }, { "from_id": 6, "to_id": 1, "distance": 29 }, { "from_id": 6, "to_id": 2, "distance": 88 }, { "from_id": 6, "to_id": 3, "distance": 58 }, { "from_id": 6, "to_id": 4, "distance": 35 }, { "from_id": 6, "to_id": 5, "distance": 84 }, { "from_id": 6, "to_id": 7, "distance": 80 }, { "from_id": 6, "to_id": 8, "distance": 62 }, { "from_id": 6, "to_id": 9, "distance": 39 }, { "from_id": 6, "to_id": 10, "distance": 113 }, { "from_id": 6, "to_id": 11, "distance": 73 }, { "from_id": 6, "to_id": 12, "distance": 39 }, { "from_id": 6, "to_id": 13, "distance": 74 }, { "from_id": 6, "to_id": 14, "distance": 51 }, { "from_id": 6, "to_id": 15, "distance": 86 }, { "from_id": 6, "to_id": 16, "distance": 84 }, { "from_id": 6, "to_id": 17, "distance": 103 }, { "from_id": 6, "to_id": 18, "distance": 60 }, { "from_id": 7, "to_id": 0, "distance": 10 }, { "from_id": 7, "to_id": 1, "distance": 63 }, { "from_id": 7, "to_id": 2, "distance": 63 }, { "from_id": 7, "to_id": 3, "distance": 67 }, { "from_id": 7, "to_id": 4, "distance": 86 }, { "from_id": 7, "to_id": 5, "distance": 58 }, { "from_id": 7, "to_id": 6, "distance": 80 }, { "from_id": 7, "to_id": 8, "distance": 84 }, { "from_id": 7, "to_id": 9, "distance": 90 }, { "from_id": 7, "to_id": 10, "distance": 72 }, { "from_id": 7, "to_id": 11, "distance": 78 }, { "from_id": 7, "to_id": 12, "distance": 91 }, { "from_id": 7, "to_id": 13, "distance": 83 }, { "from_id": 7, "to_id": 14, "distance": 88 }, { "from_id": 7, "to_id": 15, "distance": 75 }, { "from_id": 7, "to_id": 16, "distance": 31 }, { "from_id": 7, "to_id": 17, "distance": 49 }, { "from_id": 7, "to_id": 18, "distance": 105 }, { "from_id": 8, "to_id": 0, "distance": 78 }, { "from_id": 8, "to_id": 1, "distance": 40 }, { "from_id": 8, "to_id": 2, "distance": 47 }, { "from_id": 8, "to_id": 3, "distance": 93 }, { "from_id": 8, "to_id": 4, "distance": 50 }, { "from_id": 8, "to_id": 5, "distance": 44 }, { "from_id": 8, "to_id": 6, "distance": 62 }, { "from_id": 8, "to_id": 7, "distance": 84 }, { "from_id": 8, "to_id": 9, "distance": 63 }, { "from_id": 8, "to_id": 10, "distance": 80 }, { "from_id": 8, "to_id": 11, "distance": 35 }, { "from_id": 8, "to_id": 12, "distance": 37 }, { "from_id": 8, "to_id": 13, "distance": 29 }, { "from_id": 8, "to_id": 14, "distance": 92 }, { "from_id": 8, "to_id": 15, "distance": 72 }, { "from_id": 8, "to_id": 16, "distance": 68 }, { "from_id": 8, "to_id": 17, "distance": 74 }, { "from_id": 8, "to_id": 18, "distance": 50 }, { "from_id": 9, "to_id": 0, "distance": 89 }, { "from_id": 9, "to_id": 1, "distance": 37 }, { "from_id": 9, "to_id": 2, "distance": 80 }, { "from_id": 9, "to_id": 3, "distance": 49 }, { "from_id": 9, "to_id": 4, "distance": 13 }, { "from_id": 9, "to_id": 5, "distance": 79 }, { "from_id": 9, "to_id": 6, "distance": 39 }, { "from_id": 9, "to_id": 7, "distance": 90 }, { "from_id": 9, "to_id": 8, "distance": 63 }, { "from_id": 9, "to_id": 10, "distance": 98 }, { "from_id": 9, "to_id": 11, "distance": 89 }, { "from_id": 9, "to_id": 12, "distance": 30 }, { "from_id": 9, "to_id": 13, "distance": 87 }, { "from_id": 9, "to_id": 14, "distance": 34 }, { "from_id": 9, "to_id": 15, "distance": 59 }, { "from_id": 9, "to_id": 16, "distance": 83 }, { "from_id": 9, "to_id": 17, "distance": 99 }, { "from_id": 9, "to_id": 18, "distance": 31 }, { "from_id": 10, "to_id": 0, "distance": 63 }, { "from_id": 10, "to_id": 1, "distance": 85 }, { "from_id": 10, "to_id": 2, "distance": 34 }, { "from_id": 10, "to_id": 3, "distance": 96 }, { "from_id": 10, "to_id": 4, "distance": 94 }, { "from_id": 10, "to_id": 5, "distance": 40 }, { "from_id": 10, "to_id": 6, "distance": 113 }, { "from_id": 10, "to_id": 7, "distance": 72 }, { "from_id": 10, "to_id": 8, "distance": 80 }, { "from_id": 10, "to_id": 9, "distance": 98 }, { "from_id": 10, "to_id": 11, "distance": 92 }, { "from_id": 10, "to_id": 12, "distance": 95 }, { "from_id": 10, "to_id": 13, "distance": 91 }, { "from_id": 10, "to_id": 14, "distance": 109 }, { "from_id": 10, "to_id": 15, "distance": 45 }, { "from_id": 10, "to_id": 16, "distance": 42 }, { "from_id": 10, "to_id": 17, "distance": 31 }, { "from_id": 10, "to_id": 18, "distance": 93 }, { "from_id": 11, "to_id": 0, "distance": 73 }, { "from_id": 11, "to_id": 1, "distance": 55 }, { "from_id": 11, "to_id": 2, "distance": 59 }, { "from_id": 11, "to_id": 3, "distance": 108 }, { "from_id": 11, "to_id": 4, "distance": 76 }, { "from_id": 11, "to_id": 5, "distance": 52 }, { "from_id": 11, "to_id": 6, "distance": 73 }, { "from_id": 11, "to_id": 7, "distance": 78 }, { "from_id": 11, "to_id": 8, "distance": 35 }, { "from_id": 11, "to_id": 9, "distance": 89 }, { "from_id": 11, "to_id": 10, "distance": 92 }, { "from_id": 11, "to_id": 12, "distance": 65 }, { "from_id": 11, "to_id": 13, 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12, "distance": 25 }, { "from_id": 18, "to_id": 13, "distance": 79 }, { "from_id": 18, "to_id": 14, "distance": 64 }, { "from_id": 18, "to_id": 15, "distance": 60 }, { "from_id": 18, "to_id": 16, "distance": 89 }, { "from_id": 18, "to_id": 17, "distance": 100 } ] }, "solution_variant": [ 0, 2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18 ], "context_index": 38, "input_format": "nl", "input_index_base": 0 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Many teams face this: the marketing lead must nominate exactly the requested number of campaign concepts, and it’s better to present a spread of approaches rather than six clones of the same idea. The way to check is simple — for a given selection, add up the dissimilarity between every pair of chosen items; the selection with the largest total difference is preferred. No item can be picked twice and the number is fixed. 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\"strategy_audience_dissimilarity\": 115\n },\n {\n \"pitch_id_one\": 1,\n \"pitch_id_two\": 20,\n \"strategy_audience_dissimilarity\": 134\n },\n {\n \"pitch_id_one\": 1,\n \"pitch_id_two\": 21,\n \"strategy_audience_dissimilarity\": 155\n },\n {\n \"pitch_id_one\": 1,\n \"pitch_id_two\": 22,\n \"strategy_audience_dissimilarity\": 157\n },\n {\n \"pitch_id_one\": 2,\n \"pitch_id_two\": 1,\n \"strategy_audience_dissimilarity\": 128\n },\n {\n \"pitch_id_one\": 2,\n \"pitch_id_two\": 3,\n \"strategy_audience_dissimilarity\": 193\n },\n {\n \"pitch_id_one\": 2,\n \"pitch_id_two\": 4,\n \"strategy_audience_dissimilarity\": 166\n },\n {\n \"pitch_id_one\": 2,\n \"pitch_id_two\": 5,\n \"strategy_audience_dissimilarity\": 133\n },\n {\n \"pitch_id_one\": 2,\n \"pitch_id_two\": 6,\n \"strategy_audience_dissimilarity\": 165\n },\n {\n \"pitch_id_one\": 2,\n \"pitch_id_two\": 7,\n \"strategy_audience_dissimilarity\": 164\n },\n {\n \"pitch_id_one\": 2,\n \"pitch_id_two\": 8,\n 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\"strategy_audience_dissimilarity\": 137\n },\n {\n \"pitch_id_one\": 3,\n \"pitch_id_two\": 8,\n \"strategy_audience_dissimilarity\": 165\n },\n {\n \"pitch_id_one\": 3,\n \"pitch_id_two\": 9,\n \"strategy_audience_dissimilarity\": 164\n },\n {\n \"pitch_id_one\": 3,\n \"pitch_id_two\": 10,\n \"strategy_audience_dissimilarity\": 136\n },\n {\n \"pitch_id_one\": 3,\n \"pitch_id_two\": 11,\n \"strategy_audience_dissimilarity\": 125\n },\n {\n \"pitch_id_one\": 3,\n \"pitch_id_two\": 12,\n \"strategy_audience_dissimilarity\": 148\n },\n {\n \"pitch_id_one\": 3,\n \"pitch_id_two\": 13,\n \"strategy_audience_dissimilarity\": 133\n },\n {\n \"pitch_id_one\": 3,\n \"pitch_id_two\": 14,\n \"strategy_audience_dissimilarity\": 164\n },\n {\n \"pitch_id_one\": 3,\n \"pitch_id_two\": 15,\n \"strategy_audience_dissimilarity\": 190\n },\n {\n \"pitch_id_one\": 3,\n \"pitch_id_two\": 16,\n \"strategy_audience_dissimilarity\": 151\n },\n {\n \"pitch_id_one\": 3,\n \"pitch_id_two\": 17,\n 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\"strategy_audience_dissimilarity\": 149\n },\n {\n \"pitch_id_one\": 14,\n \"pitch_id_two\": 17,\n \"strategy_audience_dissimilarity\": 140\n },\n {\n \"pitch_id_one\": 14,\n \"pitch_id_two\": 18,\n \"strategy_audience_dissimilarity\": 158\n },\n {\n \"pitch_id_one\": 14,\n \"pitch_id_two\": 19,\n \"strategy_audience_dissimilarity\": 103\n },\n {\n \"pitch_id_one\": 14,\n \"pitch_id_two\": 20,\n \"strategy_audience_dissimilarity\": 144\n },\n {\n \"pitch_id_one\": 14,\n \"pitch_id_two\": 21,\n \"strategy_audience_dissimilarity\": 133\n },\n {\n \"pitch_id_one\": 14,\n \"pitch_id_two\": 22,\n \"strategy_audience_dissimilarity\": 131\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 1,\n \"strategy_audience_dissimilarity\": 133\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 2,\n \"strategy_audience_dissimilarity\": 144\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 3,\n \"strategy_audience_dissimilarity\": 190\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 4,\n \"strategy_audience_dissimilarity\": 174\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 5,\n \"strategy_audience_dissimilarity\": 155\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 6,\n \"strategy_audience_dissimilarity\": 181\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 7,\n \"strategy_audience_dissimilarity\": 129\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 8,\n \"strategy_audience_dissimilarity\": 137\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 9,\n \"strategy_audience_dissimilarity\": 95\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 10,\n \"strategy_audience_dissimilarity\": 139\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 11,\n \"strategy_audience_dissimilarity\": 184\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 12,\n \"strategy_audience_dissimilarity\": 126\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 13,\n \"strategy_audience_dissimilarity\": 203\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 14,\n \"strategy_audience_dissimilarity\": 161\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 16,\n \"strategy_audience_dissimilarity\": 148\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 17,\n \"strategy_audience_dissimilarity\": 147\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 18,\n \"strategy_audience_dissimilarity\": 140\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 19,\n \"strategy_audience_dissimilarity\": 173\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 20,\n \"strategy_audience_dissimilarity\": 134\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 21,\n \"strategy_audience_dissimilarity\": 151\n },\n {\n \"pitch_id_one\": 15,\n \"pitch_id_two\": 22,\n \"strategy_audience_dissimilarity\": 181\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 1,\n \"strategy_audience_dissimilarity\": 115\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 2,\n \"strategy_audience_dissimilarity\": 151\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 3,\n \"strategy_audience_dissimilarity\": 151\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 4,\n \"strategy_audience_dissimilarity\": 144\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 5,\n \"strategy_audience_dissimilarity\": 191\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 6,\n \"strategy_audience_dissimilarity\": 159\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 7,\n \"strategy_audience_dissimilarity\": 120\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 8,\n \"strategy_audience_dissimilarity\": 147\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 9,\n \"strategy_audience_dissimilarity\": 150\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 10,\n \"strategy_audience_dissimilarity\": 126\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 11,\n \"strategy_audience_dissimilarity\": 120\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 12,\n \"strategy_audience_dissimilarity\": 132\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 13,\n \"strategy_audience_dissimilarity\": 167\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 14,\n \"strategy_audience_dissimilarity\": 149\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 15,\n \"strategy_audience_dissimilarity\": 148\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 17,\n \"strategy_audience_dissimilarity\": 121\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 18,\n \"strategy_audience_dissimilarity\": 131\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 19,\n \"strategy_audience_dissimilarity\": 113\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 20,\n \"strategy_audience_dissimilarity\": 162\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 21,\n \"strategy_audience_dissimilarity\": 107\n },\n {\n \"pitch_id_one\": 16,\n \"pitch_id_two\": 22,\n \"strategy_audience_dissimilarity\": 142\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 1,\n \"strategy_audience_dissimilarity\": 112\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 2,\n \"strategy_audience_dissimilarity\": 141\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 3,\n \"strategy_audience_dissimilarity\": 160\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 4,\n \"strategy_audience_dissimilarity\": 158\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 5,\n \"strategy_audience_dissimilarity\": 156\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 6,\n \"strategy_audience_dissimilarity\": 161\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 7,\n \"strategy_audience_dissimilarity\": 142\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 8,\n \"strategy_audience_dissimilarity\": 168\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 9,\n \"strategy_audience_dissimilarity\": 152\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 10,\n \"strategy_audience_dissimilarity\": 136\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 11,\n \"strategy_audience_dissimilarity\": 131\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 12,\n \"strategy_audience_dissimilarity\": 109\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 13,\n \"strategy_audience_dissimilarity\": 181\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 14,\n \"strategy_audience_dissimilarity\": 140\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 15,\n \"strategy_audience_dissimilarity\": 147\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 16,\n \"strategy_audience_dissimilarity\": 121\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 18,\n \"strategy_audience_dissimilarity\": 116\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 19,\n \"strategy_audience_dissimilarity\": 139\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 20,\n \"strategy_audience_dissimilarity\": 158\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 21,\n \"strategy_audience_dissimilarity\": 157\n },\n {\n \"pitch_id_one\": 17,\n \"pitch_id_two\": 22,\n \"strategy_audience_dissimilarity\": 152\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 1,\n \"strategy_audience_dissimilarity\": 127\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 2,\n \"strategy_audience_dissimilarity\": 133\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 3,\n \"strategy_audience_dissimilarity\": 126\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 4,\n \"strategy_audience_dissimilarity\": 108\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 5,\n \"strategy_audience_dissimilarity\": 151\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 6,\n \"strategy_audience_dissimilarity\": 125\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 7,\n \"strategy_audience_dissimilarity\": 142\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 8,\n \"strategy_audience_dissimilarity\": 129\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 9,\n \"strategy_audience_dissimilarity\": 133\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 10,\n \"strategy_audience_dissimilarity\": 128\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 11,\n \"strategy_audience_dissimilarity\": 129\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 12,\n \"strategy_audience_dissimilarity\": 118\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 13,\n \"strategy_audience_dissimilarity\": 160\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 14,\n \"strategy_audience_dissimilarity\": 158\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 15,\n \"strategy_audience_dissimilarity\": 140\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 16,\n \"strategy_audience_dissimilarity\": 131\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 17,\n \"strategy_audience_dissimilarity\": 116\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 19,\n \"strategy_audience_dissimilarity\": 175\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 20,\n \"strategy_audience_dissimilarity\": 139\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 21,\n \"strategy_audience_dissimilarity\": 148\n },\n {\n \"pitch_id_one\": 18,\n \"pitch_id_two\": 22,\n \"strategy_audience_dissimilarity\": 155\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 1,\n \"strategy_audience_dissimilarity\": 115\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 2,\n \"strategy_audience_dissimilarity\": 143\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 3,\n \"strategy_audience_dissimilarity\": 180\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 4,\n \"strategy_audience_dissimilarity\": 153\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 5,\n \"strategy_audience_dissimilarity\": 181\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 6,\n \"strategy_audience_dissimilarity\": 178\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 7,\n \"strategy_audience_dissimilarity\": 130\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 8,\n \"strategy_audience_dissimilarity\": 159\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 9,\n \"strategy_audience_dissimilarity\": 161\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 10,\n \"strategy_audience_dissimilarity\": 104\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 11,\n \"strategy_audience_dissimilarity\": 155\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 12,\n \"strategy_audience_dissimilarity\": 132\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 13,\n \"strategy_audience_dissimilarity\": 181\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 14,\n \"strategy_audience_dissimilarity\": 103\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 15,\n \"strategy_audience_dissimilarity\": 173\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 16,\n \"strategy_audience_dissimilarity\": 113\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 17,\n \"strategy_audience_dissimilarity\": 139\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 18,\n \"strategy_audience_dissimilarity\": 175\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 20,\n \"strategy_audience_dissimilarity\": 160\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 21,\n \"strategy_audience_dissimilarity\": 129\n },\n {\n \"pitch_id_one\": 19,\n \"pitch_id_two\": 22,\n \"strategy_audience_dissimilarity\": 131\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 1,\n \"strategy_audience_dissimilarity\": 134\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 2,\n \"strategy_audience_dissimilarity\": 148\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 3,\n \"strategy_audience_dissimilarity\": 127\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 4,\n \"strategy_audience_dissimilarity\": 146\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 5,\n \"strategy_audience_dissimilarity\": 118\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 6,\n \"strategy_audience_dissimilarity\": 110\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 7,\n \"strategy_audience_dissimilarity\": 126\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 8,\n \"strategy_audience_dissimilarity\": 153\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 9,\n \"strategy_audience_dissimilarity\": 102\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 10,\n \"strategy_audience_dissimilarity\": 138\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 11,\n \"strategy_audience_dissimilarity\": 143\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 12,\n \"strategy_audience_dissimilarity\": 116\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 13,\n \"strategy_audience_dissimilarity\": 130\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 14,\n \"strategy_audience_dissimilarity\": 144\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 15,\n \"strategy_audience_dissimilarity\": 134\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 16,\n \"strategy_audience_dissimilarity\": 162\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 17,\n \"strategy_audience_dissimilarity\": 158\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 18,\n \"strategy_audience_dissimilarity\": 139\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 19,\n \"strategy_audience_dissimilarity\": 160\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 21,\n \"strategy_audience_dissimilarity\": 129\n },\n {\n \"pitch_id_one\": 20,\n \"pitch_id_two\": 22,\n \"strategy_audience_dissimilarity\": 141\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 1,\n \"strategy_audience_dissimilarity\": 155\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 2,\n \"strategy_audience_dissimilarity\": 185\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 3,\n \"strategy_audience_dissimilarity\": 131\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 4,\n \"strategy_audience_dissimilarity\": 110\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 5,\n \"strategy_audience_dissimilarity\": 206\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 6,\n \"strategy_audience_dissimilarity\": 169\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 7,\n \"strategy_audience_dissimilarity\": 85\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 8,\n \"strategy_audience_dissimilarity\": 159\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 9,\n \"strategy_audience_dissimilarity\": 144\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 10,\n \"strategy_audience_dissimilarity\": 137\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 11,\n \"strategy_audience_dissimilarity\": 130\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 12,\n \"strategy_audience_dissimilarity\": 119\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 13,\n \"strategy_audience_dissimilarity\": 173\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 14,\n \"strategy_audience_dissimilarity\": 133\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 15,\n \"strategy_audience_dissimilarity\": 151\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 16,\n \"strategy_audience_dissimilarity\": 107\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 17,\n \"strategy_audience_dissimilarity\": 157\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 18,\n \"strategy_audience_dissimilarity\": 148\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 19,\n \"strategy_audience_dissimilarity\": 129\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 20,\n \"strategy_audience_dissimilarity\": 129\n },\n {\n \"pitch_id_one\": 21,\n \"pitch_id_two\": 22,\n \"strategy_audience_dissimilarity\": 155\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 1,\n \"strategy_audience_dissimilarity\": 157\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 2,\n \"strategy_audience_dissimilarity\": 116\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 3,\n \"strategy_audience_dissimilarity\": 182\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 4,\n \"strategy_audience_dissimilarity\": 146\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 5,\n \"strategy_audience_dissimilarity\": 151\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 6,\n \"strategy_audience_dissimilarity\": 161\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 7,\n \"strategy_audience_dissimilarity\": 167\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 8,\n \"strategy_audience_dissimilarity\": 186\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 9,\n \"strategy_audience_dissimilarity\": 181\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 10,\n \"strategy_audience_dissimilarity\": 174\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 11,\n \"strategy_audience_dissimilarity\": 155\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 12,\n \"strategy_audience_dissimilarity\": 120\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 13,\n \"strategy_audience_dissimilarity\": 176\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 14,\n \"strategy_audience_dissimilarity\": 131\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 15,\n \"strategy_audience_dissimilarity\": 181\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 16,\n \"strategy_audience_dissimilarity\": 142\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 17,\n \"strategy_audience_dissimilarity\": 152\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 18,\n \"strategy_audience_dissimilarity\": 155\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 19,\n \"strategy_audience_dissimilarity\": 131\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 20,\n \"strategy_audience_dissimilarity\": 141\n },\n {\n \"pitch_id_one\": 22,\n \"pitch_id_two\": 21,\n \"strategy_audience_dissimilarity\": 155\n }\n ]\n}\n\nIf you want to send the answer back in a tidy way, just stick to this little JSON shape so it's easy to read and parse.\n\n{\n \"solution\": [, , ...]\n}\n\nThink of \"solution\" as the list of the campaign concept IDs you pick — just the exact IDs (no extra text), listed in the array. This is just a sketch of the shape I expect, not the actual selection.\n\nPlease make sure to use the identifiers exactly as they appear in the instance input — no renaming and no new labels. 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124, 145, 165, 138, 191, 154, 103, 0, 119, 120, 163, 146, 192, 164, 137, 147, 168, 129, 159, 153, 159, 186 ], [ 131, 149, 164, 165, 147, 164, 113, 119, 0, 119, 148, 144, 186, 177, 95, 150, 152, 133, 161, 102, 144, 181 ], [ 92, 137, 136, 133, 155, 142, 124, 120, 119, 0, 155, 133, 157, 123, 139, 126, 136, 128, 104, 138, 137, 174 ], [ 135, 173, 125, 127, 191, 155, 123, 163, 148, 155, 0, 130, 183, 167, 184, 120, 131, 129, 155, 143, 130, 155 ], [ 114, 122, 148, 109, 138, 143, 108, 146, 144, 133, 130, 0, 164, 82, 126, 132, 109, 118, 132, 116, 119, 120 ], [ 142, 191, 133, 182, 141, 64, 189, 192, 186, 157, 183, 164, 0, 161, 203, 167, 181, 160, 181, 130, 173, 176 ], [ 116, 153, 164, 122, 154, 150, 136, 164, 177, 123, 167, 82, 161, 0, 161, 149, 140, 158, 103, 144, 133, 131 ], [ 133, 144, 190, 174, 155, 181, 129, 137, 95, 139, 184, 126, 203, 161, 0, 148, 147, 140, 173, 134, 151, 181 ], [ 115, 151, 151, 144, 191, 159, 120, 147, 150, 126, 120, 132, 167, 149, 148, 0, 121, 131, 113, 162, 107, 142 ], [ 112, 141, 160, 158, 156, 161, 142, 168, 152, 136, 131, 109, 181, 140, 147, 121, 0, 116, 139, 158, 157, 152 ], [ 127, 133, 126, 108, 151, 125, 142, 129, 133, 128, 129, 118, 160, 158, 140, 131, 116, 0, 175, 139, 148, 155 ], [ 115, 143, 180, 153, 181, 178, 130, 159, 161, 104, 155, 132, 181, 103, 173, 113, 139, 175, 0, 160, 129, 131 ], [ 134, 148, 127, 146, 118, 110, 126, 153, 102, 138, 143, 116, 130, 144, 134, 162, 158, 139, 160, 0, 129, 141 ], [ 155, 185, 131, 110, 206, 169, 85, 159, 144, 137, 130, 119, 173, 133, 151, 107, 157, 148, 129, 129, 0, 155 ], [ 157, 116, 182, 146, 151, 161, 167, 186, 181, 174, 155, 120, 176, 131, 181, 142, 152, 155, 131, 141, 155, 0 ] ], "m": 13, "objective": 12389.0 }, "solution": [ 1, 2, 3, 4, 7, 8, 10, 12, 14, 16, 18, 20, 21 ], "obj": 12389.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 22, "m": 13, "nodes": [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 ], "distance_pairs": [ { "from_id": 1, "to_id": 2, 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"distance": 173 }, { "from_id": 13, "to_id": 22, "distance": 176 }, { "from_id": 14, "to_id": 1, "distance": 116 }, { "from_id": 14, "to_id": 2, "distance": 153 }, { "from_id": 14, "to_id": 3, "distance": 164 }, { "from_id": 14, "to_id": 4, "distance": 122 }, { "from_id": 14, "to_id": 5, "distance": 154 }, { "from_id": 14, "to_id": 6, "distance": 150 }, { "from_id": 14, "to_id": 7, "distance": 136 }, { "from_id": 14, "to_id": 8, "distance": 164 }, { "from_id": 14, "to_id": 9, "distance": 177 }, { "from_id": 14, "to_id": 10, "distance": 123 }, { "from_id": 14, "to_id": 11, "distance": 167 }, { "from_id": 14, "to_id": 12, "distance": 82 }, { "from_id": 14, "to_id": 13, "distance": 161 }, { "from_id": 14, "to_id": 15, "distance": 161 }, { "from_id": 14, "to_id": 16, "distance": 149 }, { "from_id": 14, "to_id": 17, "distance": 140 }, { "from_id": 14, "to_id": 18, "distance": 158 }, { "from_id": 14, "to_id": 19, "distance": 103 }, { "from_id": 14, "to_id": 20, "distance": 144 }, { 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"distance": 134 }, { "from_id": 15, "to_id": 21, "distance": 151 }, { "from_id": 15, "to_id": 22, "distance": 181 }, { "from_id": 16, "to_id": 1, "distance": 115 }, { "from_id": 16, "to_id": 2, "distance": 151 }, { "from_id": 16, "to_id": 3, "distance": 151 }, { "from_id": 16, "to_id": 4, "distance": 144 }, { "from_id": 16, "to_id": 5, "distance": 191 }, { "from_id": 16, "to_id": 6, "distance": 159 }, { "from_id": 16, "to_id": 7, "distance": 120 }, { "from_id": 16, "to_id": 8, "distance": 147 }, { "from_id": 16, "to_id": 9, "distance": 150 }, { "from_id": 16, "to_id": 10, "distance": 126 }, { "from_id": 16, "to_id": 11, "distance": 120 }, { "from_id": 16, "to_id": 12, "distance": 132 }, { "from_id": 16, "to_id": 13, "distance": 167 }, { "from_id": 16, "to_id": 14, "distance": 149 }, { "from_id": 16, "to_id": 15, "distance": 148 }, { "from_id": 16, "to_id": 17, "distance": 121 }, { "from_id": 16, "to_id": 18, "distance": 131 }, { "from_id": 16, "to_id": 19, "distance": 113 }, { "from_id": 16, "to_id": 20, "distance": 162 }, { "from_id": 16, "to_id": 21, "distance": 107 }, { "from_id": 16, "to_id": 22, "distance": 142 }, { "from_id": 17, "to_id": 1, "distance": 112 }, { "from_id": 17, "to_id": 2, "distance": 141 }, { "from_id": 17, "to_id": 3, "distance": 160 }, { "from_id": 17, "to_id": 4, "distance": 158 }, { "from_id": 17, "to_id": 5, "distance": 156 }, { "from_id": 17, "to_id": 6, "distance": 161 }, { "from_id": 17, "to_id": 7, "distance": 142 }, { "from_id": 17, "to_id": 8, "distance": 168 }, { "from_id": 17, "to_id": 9, "distance": 152 }, { "from_id": 17, "to_id": 10, "distance": 136 }, { "from_id": 17, "to_id": 11, "distance": 131 }, { "from_id": 17, "to_id": 12, "distance": 109 }, { "from_id": 17, "to_id": 13, "distance": 181 }, { "from_id": 17, "to_id": 14, "distance": 140 }, { "from_id": 17, "to_id": 15, "distance": 147 }, { "from_id": 17, "to_id": 16, "distance": 121 }, { "from_id": 17, "to_id": 18, "distance": 116 }, { "from_id": 17, "to_id": 19, "distance": 139 }, { "from_id": 17, "to_id": 20, "distance": 158 }, { "from_id": 17, "to_id": 21, "distance": 157 }, { "from_id": 17, "to_id": 22, "distance": 152 }, { "from_id": 18, "to_id": 1, "distance": 127 }, { "from_id": 18, "to_id": 2, "distance": 133 }, { "from_id": 18, "to_id": 3, "distance": 126 }, { "from_id": 18, "to_id": 4, "distance": 108 }, { "from_id": 18, "to_id": 5, "distance": 151 }, { "from_id": 18, "to_id": 6, "distance": 125 }, { "from_id": 18, "to_id": 7, "distance": 142 }, { "from_id": 18, "to_id": 8, "distance": 129 }, { "from_id": 18, "to_id": 9, "distance": 133 }, { "from_id": 18, "to_id": 10, "distance": 128 }, { "from_id": 18, "to_id": 11, "distance": 129 }, { "from_id": 18, "to_id": 12, "distance": 118 }, { "from_id": 18, "to_id": 13, "distance": 160 }, { "from_id": 18, "to_id": 14, "distance": 158 }, { "from_id": 18, "to_id": 15, "distance": 140 }, { "from_id": 18, "to_id": 16, "distance": 131 }, { "from_id": 18, "to_id": 17, "distance": 116 }, { "from_id": 18, "to_id": 19, "distance": 175 }, { "from_id": 18, "to_id": 20, "distance": 139 }, { "from_id": 18, "to_id": 21, "distance": 148 }, { "from_id": 18, "to_id": 22, "distance": 155 }, { "from_id": 19, "to_id": 1, "distance": 115 }, { "from_id": 19, "to_id": 2, "distance": 143 }, { "from_id": 19, "to_id": 3, "distance": 180 }, { "from_id": 19, "to_id": 4, "distance": 153 }, { "from_id": 19, "to_id": 5, "distance": 181 }, { "from_id": 19, "to_id": 6, "distance": 178 }, { "from_id": 19, "to_id": 7, "distance": 130 }, { "from_id": 19, "to_id": 8, "distance": 159 }, { "from_id": 19, "to_id": 9, "distance": 161 }, { "from_id": 19, "to_id": 10, "distance": 104 }, { "from_id": 19, "to_id": 11, "distance": 155 }, { "from_id": 19, "to_id": 12, "distance": 132 }, { "from_id": 19, "to_id": 13, "distance": 181 }, { "from_id": 19, "to_id": 14, "distance": 103 }, { "from_id": 19, "to_id": 15, "distance": 173 }, { "from_id": 19, "to_id": 16, "distance": 113 }, { "from_id": 19, "to_id": 17, "distance": 139 }, { "from_id": 19, "to_id": 18, "distance": 175 }, { "from_id": 19, "to_id": 20, "distance": 160 }, { "from_id": 19, "to_id": 21, "distance": 129 }, { "from_id": 19, "to_id": 22, "distance": 131 }, { "from_id": 20, "to_id": 1, "distance": 134 }, { "from_id": 20, "to_id": 2, "distance": 148 }, { "from_id": 20, "to_id": 3, "distance": 127 }, { "from_id": 20, "to_id": 4, "distance": 146 }, { "from_id": 20, "to_id": 5, "distance": 118 }, { "from_id": 20, "to_id": 6, "distance": 110 }, { "from_id": 20, "to_id": 7, "distance": 126 }, { "from_id": 20, "to_id": 8, "distance": 153 }, { "from_id": 20, "to_id": 9, "distance": 102 }, { "from_id": 20, "to_id": 10, "distance": 138 }, { "from_id": 20, "to_id": 11, "distance": 143 }, { "from_id": 20, "to_id": 12, "distance": 116 }, { "from_id": 20, "to_id": 13, "distance": 130 }, { "from_id": 20, "to_id": 14, "distance": 144 }, { "from_id": 20, "to_id": 15, "distance": 134 }, { "from_id": 20, "to_id": 16, "distance": 162 }, { "from_id": 20, "to_id": 17, "distance": 158 }, { "from_id": 20, "to_id": 18, "distance": 139 }, { "from_id": 20, "to_id": 19, "distance": 160 }, { "from_id": 20, "to_id": 21, "distance": 129 }, { "from_id": 20, "to_id": 22, "distance": 141 }, { "from_id": 21, "to_id": 1, "distance": 155 }, { "from_id": 21, "to_id": 2, "distance": 185 }, { "from_id": 21, "to_id": 3, "distance": 131 }, { "from_id": 21, "to_id": 4, "distance": 110 }, { "from_id": 21, "to_id": 5, "distance": 206 }, { "from_id": 21, "to_id": 6, "distance": 169 }, { "from_id": 21, "to_id": 7, "distance": 85 }, { "from_id": 21, "to_id": 8, "distance": 159 }, { "from_id": 21, "to_id": 9, "distance": 144 }, { "from_id": 21, "to_id": 10, "distance": 137 }, { "from_id": 21, "to_id": 11, "distance": 130 }, { "from_id": 21, "to_id": 12, "distance": 119 }, { "from_id": 21, "to_id": 13, "distance": 173 }, { "from_id": 21, "to_id": 14, "distance": 133 }, { "from_id": 21, "to_id": 15, "distance": 151 }, { "from_id": 21, "to_id": 16, "distance": 107 }, { "from_id": 21, "to_id": 17, "distance": 157 }, { "from_id": 21, "to_id": 18, "distance": 148 }, { "from_id": 21, "to_id": 19, "distance": 129 }, { "from_id": 21, "to_id": 20, "distance": 129 }, { "from_id": 21, "to_id": 22, "distance": 155 }, { "from_id": 22, "to_id": 1, "distance": 157 }, { "from_id": 22, "to_id": 2, "distance": 116 }, { "from_id": 22, "to_id": 3, "distance": 182 }, { "from_id": 22, "to_id": 4, "distance": 146 }, { "from_id": 22, "to_id": 5, "distance": 151 }, { "from_id": 22, "to_id": 6, "distance": 161 }, { "from_id": 22, "to_id": 7, "distance": 167 }, { "from_id": 22, "to_id": 8, "distance": 186 }, { "from_id": 22, "to_id": 9, "distance": 181 }, { "from_id": 22, "to_id": 10, "distance": 174 }, { "from_id": 22, "to_id": 11, "distance": 155 }, { "from_id": 22, "to_id": 12, "distance": 120 }, { "from_id": 22, "to_id": 13, "distance": 176 }, { "from_id": 22, "to_id": 14, "distance": 131 }, { "from_id": 22, "to_id": 15, "distance": 181 }, { "from_id": 22, "to_id": 16, "distance": 142 }, { "from_id": 22, "to_id": 17, "distance": 152 }, { "from_id": 22, "to_id": 18, "distance": 155 }, { "from_id": 22, "to_id": 19, "distance": 131 }, { "from_id": 22, "to_id": 20, "distance": 141 }, { "from_id": 22, "to_id": 21, "distance": 155 } ] }, "solution_variant": [ 2, 3, 4, 5, 8, 9, 11, 13, 15, 17, 19, 21, 22 ], "context_index": 39, "input_format": "json", "input_index_base": 1 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Many people on the team wanted a mini-series that felt wide-ranging, so the job was to pick precisely the given number of episodes and avoid overlap in perspective. For each pair we already have a difference value, and the lineup’s diversity is measured by adding all those pairwise values together — the bigger that sum, the better the spread. Every slot needs to be filled and repeats aren’t allowed. The specific episode options and their pairwise difference figures are listed below.\n\n# total_candidate_episodes=23\n# episodes_to_select=6\n# episode_ids=A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W\nepisode_id_a,episode_id_b,difference_score\nA,B,4\nA,C,8\nA,D,8\nA,E,4\nA,F,3\nA,G,5\nA,H,1\nA,I,9\nA,J,5\nA,K,7\nA,L,9\nA,M,9\nA,N,5\nA,O,4\nA,P,8\nA,Q,4\nA,R,7\nA,S,9\nA,T,8\nA,U,6\nA,V,8\nA,W,4\nB,A,4\nB,C,7\nB,D,2\nB,E,0\nB,F,9\nB,G,1\nB,H,6\nB,I,1\nB,J,4\nB,K,6\nB,L,7\nB,M,4\nB,N,0\nB,O,5\nB,P,2\nB,Q,4\nB,R,4\nB,S,3\nB,T,7\nB,U,9\nB,V,9\nB,W,8\nC,A,8\nC,B,7\nC,D,8\nC,E,8\nC,F,5\nC,G,1\nC,H,5\nC,I,2\nC,J,4\nC,K,2\nC,L,3\nC,M,4\nC,N,2\nC,O,6\nC,P,0\nC,Q,0\nC,R,7\nC,S,2\nC,T,3\nC,U,4\nC,V,7\nC,W,8\nD,A,8\nD,B,2\nD,C,8\nD,E,7\nD,F,8\nD,G,8\nD,H,5\nD,I,4\nD,J,0\nD,K,5\nD,L,5\nD,M,9\nD,N,2\nD,O,0\nD,P,6\nD,Q,6\nD,R,4\nD,S,1\nD,T,1\nD,U,6\nD,V,6\nD,W,8\nE,A,4\nE,B,0\nE,C,8\nE,D,7\nE,F,3\nE,G,7\nE,H,3\nE,I,1\nE,J,4\nE,K,6\nE,L,4\nE,M,1\nE,N,4\nE,O,1\nE,P,4\nE,Q,9\nE,R,3\nE,S,1\nE,T,8\nE,U,7\nE,V,4\nE,W,4\nF,A,3\nF,B,9\nF,C,5\nF,D,8\nF,E,3\nF,G,6\nF,H,3\nF,I,0\nF,J,0\nF,K,4\nF,L,0\nF,M,7\nF,N,6\nF,O,3\nF,P,4\nF,Q,0\nF,R,6\nF,S,2\nF,T,7\nF,U,2\nF,V,7\nF,W,3\nG,A,5\nG,B,1\nG,C,1\nG,D,8\nG,E,7\nG,F,6\nG,H,9\nG,I,3\nG,J,3\nG,K,8\nG,L,7\nG,M,7\nG,N,3\nG,O,2\nG,P,9\nG,Q,9\nG,R,2\nG,S,5\nG,T,7\nG,U,4\nG,V,8\nG,W,9\nH,A,1\nH,B,6\nH,C,5\nH,D,5\nH,E,3\nH,F,3\nH,G,9\nH,I,1\nH,J,6\nH,K,2\nH,L,2\nH,M,7\nH,N,8\nH,O,8\nH,P,8\nH,Q,1\nH,R,6\nH,S,3\nH,T,9\nH,U,3\nH,V,5\nH,W,5\nI,A,9\nI,B,1\nI,C,2\nI,D,4\nI,E,1\nI,F,0\nI,G,3\nI,H,1\nI,J,7\nI,K,3\nI,L,9\nI,M,6\nI,N,6\nI,O,4\nI,P,6\nI,Q,2\nI,R,3\nI,S,0\nI,T,8\nI,U,2\nI,V,5\nI,W,6\nJ,A,5\nJ,B,4\nJ,C,4\nJ,D,0\nJ,E,4\nJ,F,0\nJ,G,3\nJ,H,6\nJ,I,7\nJ,K,5\nJ,L,2\nJ,M,2\nJ,N,4\nJ,O,8\nJ,P,9\nJ,Q,2\nJ,R,6\nJ,S,2\nJ,T,4\nJ,U,8\nJ,V,7\nJ,W,8\nK,A,7\nK,B,6\nK,C,2\nK,D,5\nK,E,6\nK,F,4\nK,G,8\nK,H,2\nK,I,3\nK,J,5\nK,L,3\nK,M,5\nK,N,4\nK,O,3\nK,P,0\nK,Q,7\nK,R,4\nK,S,8\nK,T,1\nK,U,4\nK,V,8\nK,W,9\nL,A,9\nL,B,7\nL,C,3\nL,D,5\nL,E,4\nL,F,0\nL,G,7\nL,H,2\nL,I,9\nL,J,2\nL,K,3\nL,M,4\nL,N,0\nL,O,1\nL,P,1\nL,Q,1\nL,R,9\nL,S,0\nL,T,9\nL,U,6\nL,V,1\nL,W,0\nM,A,9\nM,B,4\nM,C,4\nM,D,9\nM,E,1\nM,F,7\nM,G,7\nM,H,7\nM,I,6\nM,J,2\nM,K,5\nM,L,4\nM,N,5\nM,O,2\nM,P,1\nM,Q,4\nM,R,2\nM,S,7\nM,T,5\nM,U,2\nM,V,0\nM,W,0\nN,A,5\nN,B,0\nN,C,2\nN,D,2\nN,E,4\nN,F,6\nN,G,3\nN,H,8\nN,I,6\nN,J,4\nN,K,4\nN,L,0\nN,M,5\nN,O,0\nN,P,9\nN,Q,3\nN,R,4\nN,S,9\nN,T,0\nN,U,9\nN,V,9\nN,W,6\nO,A,4\nO,B,5\nO,C,6\nO,D,0\nO,E,1\nO,F,3\nO,G,2\nO,H,8\nO,I,4\nO,J,8\nO,K,3\nO,L,1\nO,M,2\nO,N,0\nO,P,2\nO,Q,3\nO,R,5\nO,S,0\nO,T,4\nO,U,4\nO,V,4\nO,W,2\nP,A,8\nP,B,2\nP,C,0\nP,D,6\nP,E,4\nP,F,4\nP,G,9\nP,H,8\nP,I,6\nP,J,9\nP,K,0\nP,L,1\nP,M,1\nP,N,9\nP,O,2\nP,Q,2\nP,R,4\nP,S,1\nP,T,8\nP,U,5\nP,V,1\nP,W,6\nQ,A,4\nQ,B,4\nQ,C,0\nQ,D,6\nQ,E,9\nQ,F,0\nQ,G,9\nQ,H,1\nQ,I,2\nQ,J,2\nQ,K,7\nQ,L,1\nQ,M,4\nQ,N,3\nQ,O,3\nQ,P,2\nQ,R,5\nQ,S,9\nQ,T,1\nQ,U,7\nQ,V,1\nQ,W,0\nR,A,7\nR,B,4\nR,C,7\nR,D,4\nR,E,3\nR,F,6\nR,G,2\nR,H,6\nR,I,3\nR,J,6\nR,K,4\nR,L,9\nR,M,2\nR,N,4\nR,O,5\nR,P,4\nR,Q,5\nR,S,7\nR,T,2\nR,U,8\nR,V,2\nR,W,6\nS,A,9\nS,B,3\nS,C,2\nS,D,1\nS,E,1\nS,F,2\nS,G,5\nS,H,3\nS,I,0\nS,J,2\nS,K,8\nS,L,0\nS,M,7\nS,N,9\nS,O,0\nS,P,1\nS,Q,9\nS,R,7\nS,T,0\nS,U,2\nS,V,9\nS,W,2\nT,A,8\nT,B,7\nT,C,3\nT,D,1\nT,E,8\nT,F,7\nT,G,7\nT,H,9\nT,I,8\nT,J,4\nT,K,1\nT,L,9\nT,M,5\nT,N,0\nT,O,4\nT,P,8\nT,Q,1\nT,R,2\nT,S,0\nT,U,3\nT,V,9\nT,W,9\nU,A,6\nU,B,9\nU,C,4\nU,D,6\nU,E,7\nU,F,2\nU,G,4\nU,H,3\nU,I,2\nU,J,8\nU,K,4\nU,L,6\nU,M,2\nU,N,9\nU,O,4\nU,P,5\nU,Q,7\nU,R,8\nU,S,2\nU,T,3\nU,V,5\nU,W,0\nV,A,8\nV,B,9\nV,C,7\nV,D,6\nV,E,4\nV,F,7\nV,G,8\nV,H,5\nV,I,5\nV,J,7\nV,K,8\nV,L,1\nV,M,0\nV,N,9\nV,O,4\nV,P,1\nV,Q,1\nV,R,2\nV,S,9\nV,T,9\nV,U,5\nV,W,4\nW,A,4\nW,B,8\nW,C,8\nW,D,8\nW,E,4\nW,F,3\nW,G,9\nW,H,5\nW,I,6\nW,J,8\nW,K,9\nW,L,0\nW,M,0\nW,N,6\nW,O,2\nW,P,6\nW,Q,0\nW,R,6\nW,S,2\nW,T,9\nW,U,0\nW,V,4\n\nYou can just hand the final lineup over in a tiny JSON snippet like this so it's easy to parse:\n\n{\n \"solution\": [, , ...]\n}\n\nHere \"solution\" is where you list the chosen episodes to fill the slots — just the identifiers from the instance, nothing else. Think of it like filling out a very short form: the array holds the exactly m items you picked, order doesn't matter. This JSON is only a sketch of the expected shape, not the actual answer — replace the placeholders with the real identifiers from the instance.\n\nPlease use the identifiers exactly as they appear in the input — no renaming, no made-up labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 4, 8, 8, 4, 3, 5, 1, 9, 5, 7, 9, 9, 5, 4, 8, 4, 7, 9, 8, 6, 8, 4 ], [ 4, 0, 7, 2, 0, 9, 1, 6, 1, 4, 6, 7, 4, 0, 5, 2, 4, 4, 3, 7, 9, 9, 8 ], [ 8, 7, 0, 8, 8, 5, 1, 5, 2, 4, 2, 3, 4, 2, 6, 0, 0, 7, 2, 3, 4, 7, 8 ], [ 8, 2, 8, 0, 7, 8, 8, 5, 4, 0, 5, 5, 9, 2, 0, 6, 6, 4, 1, 1, 6, 6, 8 ], [ 4, 0, 8, 7, 0, 3, 7, 3, 1, 4, 6, 4, 1, 4, 1, 4, 9, 3, 1, 8, 7, 4, 4 ], [ 3, 9, 5, 8, 3, 0, 6, 3, 0, 0, 4, 0, 7, 6, 3, 4, 0, 6, 2, 7, 2, 7, 3 ], [ 5, 1, 1, 8, 7, 6, 0, 9, 3, 3, 8, 7, 7, 3, 2, 9, 9, 2, 5, 7, 4, 8, 9 ], [ 1, 6, 5, 5, 3, 3, 9, 0, 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9, 7, 6, 4, 7, 8, 5, 5, 7, 8, 1, 0, 9, 4, 1, 1, 2, 9, 9, 5, 0, 4 ], [ 4, 8, 8, 8, 4, 3, 9, 5, 6, 8, 9, 0, 0, 6, 2, 6, 0, 6, 2, 9, 0, 4, 0 ] ], "m": 6, "objective": 109.0 }, "solution": [ 6, 7, 9, 15, 19, 22 ], "obj": 109.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 23, "m": 6, "nodes": [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S", "T", "U", "V", "W" ], "distance_pairs": [ { "from_id": "A", "to_id": "B", "distance": 4 }, { "from_id": "A", "to_id": "C", "distance": 8 }, { "from_id": "A", "to_id": "D", "distance": 8 }, { "from_id": "A", "to_id": "E", "distance": 4 }, { "from_id": "A", "to_id": "F", "distance": 3 }, { "from_id": "A", "to_id": "G", "distance": 5 }, { "from_id": "A", "to_id": "H", "distance": 1 }, { "from_id": "A", "to_id": "I", "distance": 9 }, { "from_id": "A", "to_id": "J", "distance": 5 }, { "from_id": "A", "to_id": "K", "distance": 7 }, { "from_id": "A", "to_id": "L", "distance": 9 }, { "from_id": "A", 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The way to tell which grouping wins is to look at every pair of chosen items, score how different they are from one another, and add all those pairwise scores together — the larger that sum, the more diverse the selection. Duplicates aren’t allowed, and the concrete choices and the number required appear below.\n\n# total_elements=25\n# selection_size=10\n# element_ids=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24\nelement_a_id,element_b_id,dissimilarity_score\n0,1,179\n0,2,187\n0,3,168\n0,4,155\n0,5,179\n0,6,172\n0,7,202\n0,8,141\n0,9,161\n0,10,222\n0,11,215\n0,12,163\n0,13,200\n0,14,161\n0,15,156\n0,16,195\n0,17,186\n0,18,175\n0,19,159\n0,20,197\n0,21,166\n0,22,171\n0,23,206\n0,24,145\n1,0,179\n1,2,181\n1,3,212\n1,4,205\n1,5,143\n1,6,176\n1,7,190\n1,8,178\n1,9,186\n1,10,219\n1,11,221\n1,12,168\n1,13,215\n1,14,235\n1,15,219\n1,16,253\n1,17,202\n1,18,222\n1,19,205\n1,20,226\n1,21,186\n1,22,204\n1,23,210\n1,24,196\n2,0,187\n2,1,181\n2,3,167\n2,4,166\n2,5,210\n2,6,181\n2,7,153\n2,8,222\n2,9,204\n2,10,135\n2,11,182\n2,12,154\n2,13,135\n2,14,195\n2,15,171\n2,16,180\n2,17,163\n2,18,201\n2,19,191\n2,20,204\n2,21,172\n2,22,187\n2,23,193\n2,24,183\n3,0,168\n3,1,212\n3,2,167\n3,4,176\n3,5,187\n3,6,179\n3,7,172\n3,8,184\n3,9,188\n3,10,195\n3,11,208\n3,12,191\n3,13,145\n3,14,172\n3,15,186\n3,16,148\n3,17,203\n3,18,226\n3,19,199\n3,20,164\n3,21,123\n3,22,156\n3,23,186\n3,24,171\n4,0,155\n4,1,205\n4,2,166\n4,3,176\n4,5,189\n4,6,177\n4,7,153\n4,8,196\n4,9,147\n4,10,189\n4,11,167\n4,12,174\n4,13,179\n4,14,188\n4,15,169\n4,16,212\n4,17,203\n4,18,171\n4,19,175\n4,20,162\n4,21,179\n4,22,150\n4,23,174\n4,24,198\n5,0,179\n5,1,143\n5,2,210\n5,3,187\n5,4,189\n5,6,207\n5,7,172\n5,8,144\n5,9,175\n5,10,214\n5,11,189\n5,12,158\n5,13,195\n5,14,224\n5,15,225\n5,16,204\n5,17,183\n5,18,218\n5,19,186\n5,20,173\n5,21,155\n5,22,181\n5,23,196\n5,24,194\n6,0,172\n6,1,176\n6,2,181\n6,3,179\n6,4,177\n6,5,207\n6,7,199\n6,8,192\n6,9,200\n6,10,214\n6,11,245\n6,12,194\n6,13,219\n6,14,162\n6,15,175\n6,16,218\n6,17,176\n6,18,174\n6,19,202\n6,20,194\n6,21,170\n6,22,209\n6,23,218\n6,24,175\n7,0,202\n7,1,190\n7,2,153\n7,3,172\n7,4,153\n7,5,172\n7,6,199\n7,8,208\n7,9,193\n7,10,186\n7,11,167\n7,12,182\n7,13,149\n7,14,214\n7,15,169\n7,16,181\n7,17,181\n7,18,206\n7,19,234\n7,20,176\n7,21,173\n7,22,197\n7,23,201\n7,24,216\n8,0,141\n8,1,178\n8,2,222\n8,3,184\n8,4,196\n8,5,144\n8,6,192\n8,7,208\n8,9,170\n8,10,237\n8,11,229\n8,12,180\n8,13,206\n8,14,188\n8,15,209\n8,16,183\n8,17,166\n8,18,207\n8,19,183\n8,20,182\n8,21,180\n8,22,189\n8,23,186\n8,24,182\n9,0,161\n9,1,186\n9,2,204\n9,3,188\n9,4,147\n9,5,175\n9,6,200\n9,7,193\n9,8,170\n9,10,226\n9,11,205\n9,12,173\n9,13,205\n9,14,169\n9,15,201\n9,16,233\n9,17,206\n9,18,176\n9,19,164\n9,20,177\n9,21,178\n9,22,184\n9,23,191\n9,24,196\n10,0,222\n10,1,219\n10,2,135\n10,3,195\n10,4,189\n10,5,214\n10,6,214\n10,7,186\n10,8,237\n10,9,226\n10,11,165\n10,12,169\n10,13,137\n10,14,240\n10,15,212\n10,16,204\n10,17,208\n10,18,221\n10,19,225\n10,20,201\n10,21,199\n10,22,202\n10,23,228\n10,24,245\n11,0,215\n11,1,221\n11,2,182\n11,3,208\n11,4,167\n11,5,189\n11,6,245\n11,7,167\n11,8,229\n11,9,205\n11,10,165\n11,12,177\n11,13,133\n11,14,270\n11,15,183\n11,16,215\n11,17,210\n11,18,213\n11,19,184\n11,20,199\n11,21,196\n11,22,136\n11,23,178\n11,24,226\n12,0,163\n12,1,168\n12,2,154\n12,3,191\n12,4,174\n12,5,158\n12,6,194\n12,7,182\n12,8,180\n12,9,173\n12,10,169\n12,11,177\n12,13,152\n12,14,200\n12,15,190\n12,16,170\n12,17,180\n12,18,176\n12,19,149\n12,20,138\n12,21,132\n12,22,188\n12,23,186\n12,24,174\n13,0,200\n13,1,215\n13,2,135\n13,3,145\n13,4,179\n13,5,195\n13,6,219\n13,7,149\n13,8,206\n13,9,205\n13,10,137\n13,11,133\n13,12,152\n13,14,212\n13,15,177\n13,16,147\n13,17,192\n13,18,225\n13,19,192\n13,20,160\n13,21,167\n13,22,167\n13,23,178\n13,24,202\n14,0,161\n14,1,235\n14,2,195\n14,3,172\n14,4,188\n14,5,224\n14,6,162\n14,7,214\n14,8,188\n14,9,169\n14,10,240\n14,11,270\n14,12,200\n14,13,212\n14,15,198\n14,16,203\n14,17,203\n14,18,205\n14,19,207\n14,20,197\n14,21,177\n14,22,235\n14,23,226\n14,24,193\n15,0,156\n15,1,219\n15,2,171\n15,3,186\n15,4,169\n15,5,225\n15,6,175\n15,7,169\n15,8,209\n15,9,201\n15,10,212\n15,11,183\n15,12,190\n15,13,177\n15,14,198\n15,16,209\n15,17,189\n15,18,187\n15,19,183\n15,20,218\n15,21,193\n15,22,169\n15,23,218\n15,24,198\n16,0,195\n16,1,253\n16,2,180\n16,3,148\n16,4,212\n16,5,204\n16,6,218\n16,7,181\n16,8,183\n16,9,233\n16,10,204\n16,11,215\n16,12,170\n16,13,147\n16,14,203\n16,15,209\n16,17,163\n16,18,224\n16,19,208\n16,20,148\n16,21,157\n16,22,212\n16,23,198\n16,24,184\n17,0,186\n17,1,202\n17,2,163\n17,3,203\n17,4,203\n17,5,183\n17,6,176\n17,7,181\n17,8,166\n17,9,206\n17,10,208\n17,11,210\n17,12,180\n17,13,192\n17,14,203\n17,15,189\n17,16,163\n17,18,171\n17,19,188\n17,20,193\n17,21,193\n17,22,216\n17,23,178\n17,24,191\n18,0,175\n18,1,222\n18,2,201\n18,3,226\n18,4,171\n18,5,218\n18,6,174\n18,7,206\n18,8,207\n18,9,176\n18,10,221\n18,11,213\n18,12,176\n18,13,225\n18,14,205\n18,15,187\n18,16,224\n18,17,171\n18,19,178\n18,20,192\n18,21,180\n18,22,197\n18,23,212\n18,24,186\n19,0,159\n19,1,205\n19,2,191\n19,3,199\n19,4,175\n19,5,186\n19,6,202\n19,7,234\n19,8,183\n19,9,164\n19,10,225\n19,11,184\n19,12,149\n19,13,192\n19,14,207\n19,15,183\n19,16,208\n19,17,188\n19,18,178\n19,20,170\n19,21,161\n19,22,134\n19,23,178\n19,24,165\n20,0,197\n20,1,226\n20,2,204\n20,3,164\n20,4,162\n20,5,173\n20,6,194\n20,7,176\n20,8,182\n20,9,177\n20,10,201\n20,11,199\n20,12,138\n20,13,160\n20,14,197\n20,15,218\n20,16,148\n20,17,193\n20,18,192\n20,19,170\n20,21,133\n20,22,190\n20,23,182\n20,24,200\n21,0,166\n21,1,186\n21,2,172\n21,3,123\n21,4,179\n21,5,155\n21,6,170\n21,7,173\n21,8,180\n21,9,178\n21,10,199\n21,11,196\n21,12,132\n21,13,167\n21,14,177\n21,15,193\n21,16,157\n21,17,193\n21,18,180\n21,19,161\n21,20,133\n21,22,154\n21,23,179\n21,24,147\n22,0,171\n22,1,204\n22,2,187\n22,3,156\n22,4,150\n22,5,181\n22,6,209\n22,7,197\n22,8,189\n22,9,184\n22,10,202\n22,11,136\n22,12,188\n22,13,167\n22,14,235\n22,15,169\n22,16,212\n22,17,216\n22,18,197\n22,19,134\n22,20,190\n22,21,154\n22,23,170\n22,24,184\n23,0,206\n23,1,210\n23,2,193\n23,3,186\n23,4,174\n23,5,196\n23,6,218\n23,7,201\n23,8,186\n23,9,191\n23,10,228\n23,11,178\n23,12,186\n23,13,178\n23,14,226\n23,15,218\n23,16,198\n23,17,178\n23,18,212\n23,19,178\n23,20,182\n23,21,179\n23,22,170\n23,24,226\n24,0,145\n24,1,196\n24,2,183\n24,3,171\n24,4,198\n24,5,194\n24,6,175\n24,7,216\n24,8,182\n24,9,196\n24,10,245\n24,11,226\n24,12,174\n24,13,202\n24,14,193\n24,15,198\n24,16,184\n24,17,191\n24,18,186\n24,19,165\n24,20,200\n24,21,147\n24,22,184\n24,23,226\n\nWhen you’re ready, just drop your chosen items into a tiny JSON sketch so everything’s neat and unambiguous — something like this:\n\n{\n \"solution\": [, , ...]\n}\n\nThe \"solution\" array should list the exact hardscape elements you picked for the courtyard, one identifier per entry. Swap each for the identifier given in the instance input — this block is just the shape I’m expecting, not the filled-in final.\n\nPlease 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”.", "instance": { "distance_matrix": [ [ 0, 179, 187, 168, 155, 179, 172, 202, 141, 161, 222, 215, 163, 200, 161, 156, 195, 186, 175, 159, 197, 166, 171, 206, 145 ], [ 179, 0, 181, 212, 205, 143, 176, 190, 178, 186, 219, 221, 168, 215, 235, 219, 253, 202, 222, 205, 226, 186, 204, 210, 196 ], [ 187, 181, 0, 167, 166, 210, 181, 153, 222, 204, 135, 182, 154, 135, 195, 171, 180, 163, 201, 191, 204, 172, 187, 193, 183 ], [ 168, 212, 167, 0, 176, 187, 179, 172, 184, 188, 195, 208, 191, 145, 172, 186, 148, 203, 226, 199, 164, 123, 156, 186, 171 ], [ 155, 205, 166, 176, 0, 189, 177, 153, 196, 147, 189, 167, 174, 179, 188, 169, 212, 203, 171, 175, 162, 179, 150, 174, 198 ], [ 179, 143, 210, 187, 189, 0, 207, 172, 144, 175, 214, 189, 158, 195, 224, 225, 204, 183, 218, 186, 173, 155, 181, 196, 194 ], [ 172, 176, 181, 179, 177, 207, 0, 199, 192, 200, 214, 245, 194, 219, 162, 175, 218, 176, 174, 202, 194, 170, 209, 218, 175 ], [ 202, 190, 153, 172, 153, 172, 199, 0, 208, 193, 186, 167, 182, 149, 214, 169, 181, 181, 206, 234, 176, 173, 197, 201, 216 ], [ 141, 178, 222, 184, 196, 144, 192, 208, 0, 170, 237, 229, 180, 206, 188, 209, 183, 166, 207, 183, 182, 180, 189, 186, 182 ], [ 161, 186, 204, 188, 147, 175, 200, 193, 170, 0, 226, 205, 173, 205, 169, 201, 233, 206, 176, 164, 177, 178, 184, 191, 196 ], [ 222, 219, 135, 195, 189, 214, 214, 186, 237, 226, 0, 165, 169, 137, 240, 212, 204, 208, 221, 225, 201, 199, 202, 228, 245 ], [ 215, 221, 182, 208, 167, 189, 245, 167, 229, 205, 165, 0, 177, 133, 270, 183, 215, 210, 213, 184, 199, 196, 136, 178, 226 ], [ 163, 168, 154, 191, 174, 158, 194, 182, 180, 173, 169, 177, 0, 152, 200, 190, 170, 180, 176, 149, 138, 132, 188, 186, 174 ], [ 200, 215, 135, 145, 179, 195, 219, 149, 206, 205, 137, 133, 152, 0, 212, 177, 147, 192, 225, 192, 160, 167, 167, 178, 202 ], [ 161, 235, 195, 172, 188, 224, 162, 214, 188, 169, 240, 270, 200, 212, 0, 198, 203, 203, 205, 207, 197, 177, 235, 226, 193 ], [ 156, 219, 171, 186, 169, 225, 175, 169, 209, 201, 212, 183, 190, 177, 198, 0, 209, 189, 187, 183, 218, 193, 169, 218, 198 ], [ 195, 253, 180, 148, 212, 204, 218, 181, 183, 233, 204, 215, 170, 147, 203, 209, 0, 163, 224, 208, 148, 157, 212, 198, 184 ], [ 186, 202, 163, 203, 203, 183, 176, 181, 166, 206, 208, 210, 180, 192, 203, 189, 163, 0, 171, 188, 193, 193, 216, 178, 191 ], [ 175, 222, 201, 226, 171, 218, 174, 206, 207, 176, 221, 213, 176, 225, 205, 187, 224, 171, 0, 178, 192, 180, 197, 212, 186 ], [ 159, 205, 191, 199, 175, 186, 202, 234, 183, 164, 225, 184, 149, 192, 207, 183, 208, 188, 178, 0, 170, 161, 134, 178, 165 ], [ 197, 226, 204, 164, 162, 173, 194, 176, 182, 177, 201, 199, 138, 160, 197, 218, 148, 193, 192, 170, 0, 133, 190, 182, 200 ], [ 166, 186, 172, 123, 179, 155, 170, 173, 180, 178, 199, 196, 132, 167, 177, 193, 157, 193, 180, 161, 133, 0, 154, 179, 147 ], [ 171, 204, 187, 156, 150, 181, 209, 197, 189, 184, 202, 136, 188, 167, 235, 169, 212, 216, 197, 134, 190, 154, 0, 170, 184 ], [ 206, 210, 193, 186, 174, 196, 218, 201, 186, 191, 228, 178, 186, 178, 226, 218, 198, 178, 212, 178, 182, 179, 170, 0, 226 ], [ 145, 196, 183, 171, 198, 194, 175, 216, 182, 196, 245, 226, 174, 202, 193, 198, 184, 191, 186, 165, 200, 147, 184, 226, 0 ] ], "m": 10, "objective": 9449.0 }, "solution": [ 1, 5, 10, 11, 14, 15, 16, 18, 23, 24 ], "obj": 9449.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 25, "m": 10, "nodes": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 ], "distance_pairs": [ { "from_id": 0, "to_id": 1, "distance": 179 }, { "from_id": 0, "to_id": 2, "distance": 187 }, { "from_id": 0, "to_id": 3, "distance": 168 }, { "from_id": 0, "to_id": 4, "distance": 155 }, { "from_id": 0, "to_id": 5, "distance": 179 }, { "from_id": 0, "to_id": 6, "distance": 172 }, { "from_id": 0, "to_id": 7, "distance": 202 }, { "from_id": 0, "to_id": 8, "distance": 141 }, { "from_id": 0, "to_id": 9, "distance": 161 }, { "from_id": 0, "to_id": 10, "distance": 222 }, { "from_id": 0, "to_id": 11, "distance": 215 }, { "from_id": 0, "to_id": 12, "distance": 163 }, { "from_id": 0, "to_id": 13, "distance": 200 }, { "from_id": 0, "to_id": 14, "distance": 161 }, { "from_id": 0, "to_id": 15, "distance": 156 }, { "from_id": 0, "to_id": 16, "distance": 195 }, { "from_id": 0, "to_id": 17, "distance": 186 }, { "from_id": 0, "to_id": 18, "distance": 175 }, { "from_id": 0, "to_id": 19, "distance": 159 }, { "from_id": 0, "to_id": 20, "distance": 197 }, { "from_id": 0, "to_id": 21, "distance": 166 }, { "from_id": 0, "to_id": 22, "distance": 171 }, { "from_id": 0, "to_id": 23, "distance": 206 }, { "from_id": 0, "to_id": 24, "distance": 145 }, { "from_id": 1, "to_id": 0, "distance": 179 }, { "from_id": 1, "to_id": 2, "distance": 181 }, { "from_id": 1, "to_id": 3, "distance": 212 }, { "from_id": 1, "to_id": 4, "distance": 205 }, { "from_id": 1, "to_id": 5, "distance": 143 }, { "from_id": 1, "to_id": 6, "distance": 176 }, { "from_id": 1, "to_id": 7, "distance": 190 }, { "from_id": 1, "to_id": 8, "distance": 178 }, { "from_id": 1, "to_id": 9, "distance": 186 }, { "from_id": 1, "to_id": 10, "distance": 219 }, { "from_id": 1, "to_id": 11, "distance": 221 }, { "from_id": 1, "to_id": 12, "distance": 168 }, { "from_id": 1, "to_id": 13, "distance": 215 }, { "from_id": 1, "to_id": 14, "distance": 235 }, { "from_id": 1, "to_id": 15, "distance": 219 }, { "from_id": 1, "to_id": 16, "distance": 253 }, { "from_id": 1, "to_id": 17, "distance": 202 }, { "from_id": 1, "to_id": 18, "distance": 222 }, { "from_id": 1, "to_id": 19, "distance": 205 }, { "from_id": 1, "to_id": 20, "distance": 226 }, { "from_id": 1, "to_id": 21, "distance": 186 }, { "from_id": 1, "to_id": 22, "distance": 204 }, { "from_id": 1, "to_id": 23, "distance": 210 }, { "from_id": 1, "to_id": 24, "distance": 196 }, { "from_id": 2, 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"distance": 196 }, { "from_id": 24, "to_id": 10, "distance": 245 }, { "from_id": 24, "to_id": 11, "distance": 226 }, { "from_id": 24, "to_id": 12, "distance": 174 }, { "from_id": 24, "to_id": 13, "distance": 202 }, { "from_id": 24, "to_id": 14, "distance": 193 }, { "from_id": 24, "to_id": 15, "distance": 198 }, { "from_id": 24, "to_id": 16, "distance": 184 }, { "from_id": 24, "to_id": 17, "distance": 191 }, { "from_id": 24, "to_id": 18, "distance": 186 }, { "from_id": 24, "to_id": 19, "distance": 165 }, { "from_id": 24, "to_id": 20, "distance": 200 }, { "from_id": 24, "to_id": 21, "distance": 147 }, { "from_id": 24, "to_id": 22, "distance": 184 }, { "from_id": 24, "to_id": 23, "distance": 226 } ] }, "solution_variant": [ 1, 5, 10, 11, 14, 15, 16, 18, 23, 24 ], "context_index": 41, "input_format": "csv", "input_index_base": 0 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "I run a little bookshop and need to put together a recommendation shelf of exactly a set number of titles; the idea is to pick that many books so the whole shelf feels wide-ranging in genre and viewpoint. To judge a selection, I look at how different each pair of chosen books is (there’s a distance number for every pair) and add up all those pairwise differences — the better shelf is the one with the biggest total. I can’t pick more or fewer than asked, and each title counts once, not twice. Concrete details are shown below.\n\nThere are 24 titles available, listed as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23; I must pick exactly 7 to place on the recommendation shelf.\n\n| title_id_first | title_id_second | genre_viewpoint_dissimilarity |\n|---|---|---|\n| 0 | 1 | 175 |\n| 0 | 2 | 185 |\n| 0 | 3 | 172 |\n| 0 | 4 | 170 |\n| 0 | 5 | 141 |\n| 0 | 6 | 156 |\n| 0 | 7 | 178 |\n| 0 | 8 | 186 |\n| 0 | 9 | 156 |\n| 0 | 10 | 185 |\n| 0 | 11 | 189 |\n| 0 | 12 | 170 |\n| 0 | 13 | 210 |\n| 0 | 14 | 173 |\n| 0 | 15 | 149 |\n| 0 | 16 | 186 |\n| 0 | 17 | 170 |\n| 0 | 18 | 192 |\n| 0 | 19 | 146 |\n| 0 | 20 | 186 |\n| 0 | 21 | 201 |\n| 0 | 22 | 194 |\n| 0 | 23 | 144 |\n| 1 | 0 | 175 |\n| 1 | 2 | 214 |\n| 1 | 3 | 185 |\n| 1 | 4 | 166 |\n| 1 | 5 | 148 |\n| 1 | 6 | 180 |\n| 1 | 7 | 185 |\n| 1 | 8 | 185 |\n| 1 | 9 | 216 |\n| 1 | 10 | 179 |\n| 1 | 11 | 194 |\n| 1 | 12 | 184 |\n| 1 | 13 | 215 |\n| 1 | 14 | 185 |\n| 1 | 15 | 218 |\n| 1 | 16 | 179 |\n| 1 | 17 | 167 |\n| 1 | 18 | 196 |\n| 1 | 19 | 180 |\n| 1 | 20 | 223 |\n| 1 | 21 | 215 |\n| 1 | 22 | 217 |\n| 1 | 23 | 196 |\n| 2 | 0 | 185 |\n| 2 | 1 | 214 |\n| 2 | 3 | 192 |\n| 2 | 4 | 205 |\n| 2 | 5 | 157 |\n| 2 | 6 | 223 |\n| 2 | 7 | 223 |\n| 2 | 8 | 220 |\n| 2 | 9 | 190 |\n| 2 | 10 | 234 |\n| 2 | 11 | 236 |\n| 2 | 12 | 187 |\n| 2 | 13 | 233 |\n| 2 | 14 | 197 |\n| 2 | 15 | 201 |\n| 2 | 16 | 198 |\n| 2 | 17 | 243 |\n| 2 | 18 | 221 |\n| 2 | 19 | 135 |\n| 2 | 20 | 181 |\n| 2 | 21 | 218 |\n| 2 | 22 | 212 |\n| 2 | 23 | 173 |\n| 3 | 0 | 172 |\n| 3 | 1 | 185 |\n| 3 | 2 | 192 |\n| 3 | 4 | 191 |\n| 3 | 5 | 182 |\n| 3 | 6 | 235 |\n| 3 | 7 | 191 |\n| 3 | 8 | 163 |\n| 3 | 9 | 190 |\n| 3 | 10 | 196 |\n| 3 | 11 | 140 |\n| 3 | 12 | 189 |\n| 3 | 13 | 167 |\n| 3 | 14 | 198 |\n| 3 | 15 | 182 |\n| 3 | 16 | 187 |\n| 3 | 17 | 200 |\n| 3 | 18 | 200 |\n| 3 | 19 | 179 |\n| 3 | 20 | 190 |\n| 3 | 21 | 181 |\n| 3 | 22 | 184 |\n| 3 | 23 | 178 |\n| 4 | 0 | 170 |\n| 4 | 1 | 166 |\n| 4 | 2 | 205 |\n| 4 | 3 | 191 |\n| 4 | 5 | 194 |\n| 4 | 6 | 167 |\n| 4 | 7 | 181 |\n| 4 | 8 | 210 |\n| 4 | 9 | 207 |\n| 4 | 10 | 164 |\n| 4 | 11 | 190 |\n| 4 | 12 | 193 |\n| 4 | 13 | 192 |\n| 4 | 14 | 187 |\n| 4 | 15 | 224 |\n| 4 | 16 | 183 |\n| 4 | 17 | 204 |\n| 4 | 18 | 171 |\n| 4 | 19 | 176 |\n| 4 | 20 | 195 |\n| 4 | 21 | 229 |\n| 4 | 22 | 217 |\n| 4 | 23 | 191 |\n| 5 | 0 | 141 |\n| 5 | 1 | 148 |\n| 5 | 2 | 157 |\n| 5 | 3 | 182 |\n| 5 | 4 | 194 |\n| 5 | 6 | 172 |\n| 5 | 7 | 186 |\n| 5 | 8 | 184 |\n| 5 | 9 | 173 |\n| 5 | 10 | 169 |\n| 5 | 11 | 189 |\n| 5 | 12 | 135 |\n| 5 | 13 | 192 |\n| 5 | 14 | 149 |\n| 5 | 15 | 181 |\n| 5 | 16 | 137 |\n| 5 | 17 | 150 |\n| 5 | 18 | 166 |\n| 5 | 19 | 158 |\n| 5 | 20 | 204 |\n| 5 | 21 | 198 |\n| 5 | 22 | 207 |\n| 5 | 23 | 146 |\n| 6 | 0 | 156 |\n| 6 | 1 | 180 |\n| 6 | 2 | 223 |\n| 6 | 3 | 235 |\n| 6 | 4 | 167 |\n| 6 | 5 | 172 |\n| 6 | 7 | 191 |\n| 6 | 8 | 194 |\n| 6 | 9 | 158 |\n| 6 | 10 | 185 |\n| 6 | 11 | 216 |\n| 6 | 12 | 158 |\n| 6 | 13 | 244 |\n| 6 | 14 | 193 |\n| 6 | 15 | 190 |\n| 6 | 16 | 199 |\n| 6 | 17 | 172 |\n| 6 | 18 | 182 |\n| 6 | 19 | 174 |\n| 6 | 20 | 215 |\n| 6 | 21 | 210 |\n| 6 | 22 | 209 |\n| 6 | 23 | 195 |\n| 7 | 0 | 178 |\n| 7 | 1 | 185 |\n| 7 | 2 | 223 |\n| 7 | 3 | 191 |\n| 7 | 4 | 181 |\n| 7 | 5 | 186 |\n| 7 | 6 | 191 |\n| 7 | 8 | 203 |\n| 7 | 9 | 196 |\n| 7 | 10 | 169 |\n| 7 | 11 | 234 |\n| 7 | 12 | 209 |\n| 7 | 13 | 195 |\n| 7 | 14 | 238 |\n| 7 | 15 | 234 |\n| 7 | 16 | 198 |\n| 7 | 17 | 193 |\n| 7 | 18 | 223 |\n| 7 | 19 | 216 |\n| 7 | 20 | 251 |\n| 7 | 21 | 216 |\n| 7 | 22 | 264 |\n| 7 | 23 | 198 |\n| 8 | 0 | 186 |\n| 8 | 1 | 185 |\n| 8 | 2 | 220 |\n| 8 | 3 | 163 |\n| 8 | 4 | 210 |\n| 8 | 5 | 184 |\n| 8 | 6 | 194 |\n| 8 | 7 | 203 |\n| 8 | 9 | 209 |\n| 8 | 10 | 191 |\n| 8 | 11 | 206 |\n| 8 | 12 | 178 |\n| 8 | 13 | 218 |\n| 8 | 14 | 193 |\n| 8 | 15 | 201 |\n| 8 | 16 | 164 |\n| 8 | 17 | 211 |\n| 8 | 18 | 207 |\n| 8 | 19 | 171 |\n| 8 | 20 | 247 |\n| 8 | 21 | 231 |\n| 8 | 22 | 170 |\n| 8 | 23 | 199 |\n| 9 | 0 | 156 |\n| 9 | 1 | 216 |\n| 9 | 2 | 190 |\n| 9 | 3 | 190 |\n| 9 | 4 | 207 |\n| 9 | 5 | 173 |\n| 9 | 6 | 158 |\n| 9 | 7 | 196 |\n| 9 | 8 | 209 |\n| 9 | 10 | 185 |\n| 9 | 11 | 187 |\n| 9 | 12 | 147 |\n| 9 | 13 | 217 |\n| 9 | 14 | 197 |\n| 9 | 15 | 156 |\n| 9 | 16 | 206 |\n| 9 | 17 | 167 |\n| 9 | 18 | 219 |\n| 9 | 19 | 206 |\n| 9 | 20 | 169 |\n| 9 | 21 | 167 |\n| 9 | 22 | 199 |\n| 9 | 23 | 200 |\n| 10 | 0 | 185 |\n| 10 | 1 | 179 |\n| 10 | 2 | 234 |\n| 10 | 3 | 196 |\n| 10 | 4 | 164 |\n| 10 | 5 | 169 |\n| 10 | 6 | 185 |\n| 10 | 7 | 169 |\n| 10 | 8 | 191 |\n| 10 | 9 | 185 |\n| 10 | 11 | 184 |\n| 10 | 12 | 171 |\n| 10 | 13 | 181 |\n| 10 | 14 | 161 |\n| 10 | 15 | 231 |\n| 10 | 16 | 169 |\n| 10 | 17 | 182 |\n| 10 | 18 | 181 |\n| 10 | 19 | 214 |\n| 10 | 20 | 229 |\n| 10 | 21 | 215 |\n| 10 | 22 | 204 |\n| 10 | 23 | 179 |\n| 11 | 0 | 189 |\n| 11 | 1 | 194 |\n| 11 | 2 | 236 |\n| 11 | 3 | 140 |\n| 11 | 4 | 190 |\n| 11 | 5 | 189 |\n| 11 | 6 | 216 |\n| 11 | 7 | 234 |\n| 11 | 8 | 206 |\n| 11 | 9 | 187 |\n| 11 | 10 | 184 |\n| 11 | 12 | 204 |\n| 11 | 13 | 154 |\n| 11 | 14 | 158 |\n| 11 | 15 | 198 |\n| 11 | 16 | 194 |\n| 11 | 17 | 178 |\n| 11 | 18 | 181 |\n| 11 | 19 | 230 |\n| 11 | 20 | 187 |\n| 11 | 21 | 170 |\n| 11 | 22 | 165 |\n| 11 | 23 | 181 |\n| 12 | 0 | 170 |\n| 12 | 1 | 184 |\n| 12 | 2 | 187 |\n| 12 | 3 | 189 |\n| 12 | 4 | 193 |\n| 12 | 5 | 135 |\n| 12 | 6 | 158 |\n| 12 | 7 | 209 |\n| 12 | 8 | 178 |\n| 12 | 9 | 147 |\n| 12 | 10 | 171 |\n| 12 | 11 | 204 |\n| 12 | 13 | 207 |\n| 12 | 14 | 164 |\n| 12 | 15 | 194 |\n| 12 | 16 | 158 |\n| 12 | 17 | 166 |\n| 12 | 18 | 174 |\n| 12 | 19 | 170 |\n| 12 | 20 | 207 |\n| 12 | 21 | 226 |\n| 12 | 22 | 215 |\n| 12 | 23 | 177 |\n| 13 | 0 | 210 |\n| 13 | 1 | 215 |\n| 13 | 2 | 233 |\n| 13 | 3 | 167 |\n| 13 | 4 | 192 |\n| 13 | 5 | 192 |\n| 13 | 6 | 244 |\n| 13 | 7 | 195 |\n| 13 | 8 | 218 |\n| 13 | 9 | 217 |\n| 13 | 10 | 181 |\n| 13 | 11 | 154 |\n| 13 | 12 | 207 |\n| 13 | 14 | 177 |\n| 13 | 15 | 247 |\n| 13 | 16 | 166 |\n| 13 | 17 | 177 |\n| 13 | 18 | 184 |\n| 13 | 19 | 229 |\n| 13 | 20 | 204 |\n| 13 | 21 | 195 |\n| 13 | 22 | 207 |\n| 13 | 23 | 168 |\n| 14 | 0 | 173 |\n| 14 | 1 | 185 |\n| 14 | 2 | 197 |\n| 14 | 3 | 198 |\n| 14 | 4 | 187 |\n| 14 | 5 | 149 |\n| 14 | 6 | 193 |\n| 14 | 7 | 238 |\n| 14 | 8 | 193 |\n| 14 | 9 | 197 |\n| 14 | 10 | 161 |\n| 14 | 11 | 158 |\n| 14 | 12 | 164 |\n| 14 | 13 | 177 |\n| 14 | 15 | 206 |\n| 14 | 16 | 149 |\n| 14 | 17 | 176 |\n| 14 | 18 | 179 |\n| 14 | 19 | 179 |\n| 14 | 20 | 188 |\n| 14 | 21 | 205 |\n| 14 | 22 | 160 |\n| 14 | 23 | 148 |\n| 15 | 0 | 149 |\n| 15 | 1 | 218 |\n| 15 | 2 | 201 |\n| 15 | 3 | 182 |\n| 15 | 4 | 224 |\n| 15 | 5 | 181 |\n| 15 | 6 | 190 |\n| 15 | 7 | 234 |\n| 15 | 8 | 201 |\n| 15 | 9 | 156 |\n| 15 | 10 | 231 |\n| 15 | 11 | 198 |\n| 15 | 12 | 194 |\n| 15 | 13 | 247 |\n| 15 | 14 | 206 |\n| 15 | 16 | 225 |\n| 15 | 17 | 178 |\n| 15 | 18 | 217 |\n| 15 | 19 | 176 |\n| 15 | 20 | 213 |\n| 15 | 21 | 217 |\n| 15 | 22 | 174 |\n| 15 | 23 | 195 |\n| 16 | 0 | 186 |\n| 16 | 1 | 179 |\n| 16 | 2 | 198 |\n| 16 | 3 | 187 |\n| 16 | 4 | 183 |\n| 16 | 5 | 137 |\n| 16 | 6 | 199 |\n| 16 | 7 | 198 |\n| 16 | 8 | 164 |\n| 16 | 9 | 206 |\n| 16 | 10 | 169 |\n| 16 | 11 | 194 |\n| 16 | 12 | 158 |\n| 16 | 13 | 166 |\n| 16 | 14 | 149 |\n| 16 | 15 | 225 |\n| 16 | 17 | 191 |\n| 16 | 18 | 208 |\n| 16 | 19 | 186 |\n| 16 | 20 | 199 |\n| 16 | 21 | 222 |\n| 16 | 22 | 219 |\n| 16 | 23 | 128 |\n| 17 | 0 | 170 |\n| 17 | 1 | 167 |\n| 17 | 2 | 243 |\n| 17 | 3 | 200 |\n| 17 | 4 | 204 |\n| 17 | 5 | 150 |\n| 17 | 6 | 172 |\n| 17 | 7 | 193 |\n| 17 | 8 | 211 |\n| 17 | 9 | 167 |\n| 17 | 10 | 182 |\n| 17 | 11 | 178 |\n| 17 | 12 | 166 |\n| 17 | 13 | 177 |\n| 17 | 14 | 176 |\n| 17 | 15 | 178 |\n| 17 | 16 | 191 |\n| 17 | 18 | 182 |\n| 17 | 19 | 208 |\n| 17 | 20 | 223 |\n| 17 | 21 | 174 |\n| 17 | 22 | 199 |\n| 17 | 23 | 187 |\n| 18 | 0 | 192 |\n| 18 | 1 | 196 |\n| 18 | 2 | 221 |\n| 18 | 3 | 200 |\n| 18 | 4 | 171 |\n| 18 | 5 | 166 |\n| 18 | 6 | 182 |\n| 18 | 7 | 223 |\n| 18 | 8 | 207 |\n| 18 | 9 | 219 |\n| 18 | 10 | 181 |\n| 18 | 11 | 181 |\n| 18 | 12 | 174 |\n| 18 | 13 | 184 |\n| 18 | 14 | 179 |\n| 18 | 15 | 217 |\n| 18 | 16 | 208 |\n| 18 | 17 | 182 |\n| 18 | 19 | 177 |\n| 18 | 20 | 224 |\n| 18 | 21 | 212 |\n| 18 | 22 | 187 |\n| 18 | 23 | 190 |\n| 19 | 0 | 146 |\n| 19 | 1 | 180 |\n| 19 | 2 | 135 |\n| 19 | 3 | 179 |\n| 19 | 4 | 176 |\n| 19 | 5 | 158 |\n| 19 | 6 | 174 |\n| 19 | 7 | 216 |\n| 19 | 8 | 171 |\n| 19 | 9 | 206 |\n| 19 | 10 | 214 |\n| 19 | 11 | 230 |\n| 19 | 12 | 170 |\n| 19 | 13 | 229 |\n| 19 | 14 | 179 |\n| 19 | 15 | 176 |\n| 19 | 16 | 186 |\n| 19 | 17 | 208 |\n| 19 | 18 | 177 |\n| 19 | 20 | 182 |\n| 19 | 21 | 207 |\n| 19 | 22 | 164 |\n| 19 | 23 | 151 |\n| 20 | 0 | 186 |\n| 20 | 1 | 223 |\n| 20 | 2 | 181 |\n| 20 | 3 | 190 |\n| 20 | 4 | 195 |\n| 20 | 5 | 204 |\n| 20 | 6 | 215 |\n| 20 | 7 | 251 |\n| 20 | 8 | 247 |\n| 20 | 9 | 169 |\n| 20 | 10 | 229 |\n| 20 | 11 | 187 |\n| 20 | 12 | 207 |\n| 20 | 13 | 204 |\n| 20 | 14 | 188 |\n| 20 | 15 | 213 |\n| 20 | 16 | 199 |\n| 20 | 17 | 223 |\n| 20 | 18 | 224 |\n| 20 | 19 | 182 |\n| 20 | 21 | 169 |\n| 20 | 22 | 195 |\n| 20 | 23 | 166 |\n| 21 | 0 | 201 |\n| 21 | 1 | 215 |\n| 21 | 2 | 218 |\n| 21 | 3 | 181 |\n| 21 | 4 | 229 |\n| 21 | 5 | 198 |\n| 21 | 6 | 210 |\n| 21 | 7 | 216 |\n| 21 | 8 | 231 |\n| 21 | 9 | 167 |\n| 21 | 10 | 215 |\n| 21 | 11 | 170 |\n| 21 | 12 | 226 |\n| 21 | 13 | 195 |\n| 21 | 14 | 205 |\n| 21 | 15 | 217 |\n| 21 | 16 | 222 |\n| 21 | 17 | 174 |\n| 21 | 18 | 212 |\n| 21 | 19 | 207 |\n| 21 | 20 | 169 |\n| 21 | 22 | 175 |\n| 21 | 23 | 196 |\n| 22 | 0 | 194 |\n| 22 | 1 | 217 |\n| 22 | 2 | 212 |\n| 22 | 3 | 184 |\n| 22 | 4 | 217 |\n| 22 | 5 | 207 |\n| 22 | 6 | 209 |\n| 22 | 7 | 264 |\n| 22 | 8 | 170 |\n| 22 | 9 | 199 |\n| 22 | 10 | 204 |\n| 22 | 11 | 165 |\n| 22 | 12 | 215 |\n| 22 | 13 | 207 |\n| 22 | 14 | 160 |\n| 22 | 15 | 174 |\n| 22 | 16 | 219 |\n| 22 | 17 | 199 |\n| 22 | 18 | 187 |\n| 22 | 19 | 164 |\n| 22 | 20 | 195 |\n| 22 | 21 | 175 |\n| 22 | 23 | 189 |\n| 23 | 0 | 144 |\n| 23 | 1 | 196 |\n| 23 | 2 | 173 |\n| 23 | 3 | 178 |\n| 23 | 4 | 191 |\n| 23 | 5 | 146 |\n| 23 | 6 | 195 |\n| 23 | 7 | 198 |\n| 23 | 8 | 199 |\n| 23 | 9 | 200 |\n| 23 | 10 | 179 |\n| 23 | 11 | 181 |\n| 23 | 12 | 177 |\n| 23 | 13 | 168 |\n| 23 | 14 | 148 |\n| 23 | 15 | 195 |\n| 23 | 16 | 128 |\n| 23 | 17 | 187 |\n| 23 | 18 | 190 |\n| 23 | 19 | 151 |\n| 23 | 20 | 166 |\n| 23 | 21 | 196 |\n| 23 | 22 | 189 |\n\nI'll add up the relevant pairwise dissimilarities to score any selection of 7 titles from the 24 candidates.\n\nAlso, I’ll give the final picks in a tiny JSON snippet so it’s easy to read or drop into something else. It’ll follow this shape:\n\n{\n \"solution\": [, , ...]\n}\n\nHere \"solution\" is just the list of book identifiers you’re selecting for the recommendation shelf — one identifier per chosen title. This JSON is only a sketch of the expected shape, not the actual answer.\n\nPlease use 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”.\n\nI’ll return the chosen m book IDs in that exact format.", "instance": { "distance_matrix": [ [ 0, 175, 185, 172, 170, 141, 156, 178, 186, 156, 185, 189, 170, 210, 173, 149, 186, 170, 192, 146, 186, 201, 194, 144 ], [ 175, 0, 214, 185, 166, 148, 180, 185, 185, 216, 179, 194, 184, 215, 185, 218, 179, 167, 196, 180, 223, 215, 217, 196 ], [ 185, 214, 0, 192, 205, 157, 223, 223, 220, 190, 234, 236, 187, 233, 197, 201, 198, 243, 221, 135, 181, 218, 212, 173 ], [ 172, 185, 192, 0, 191, 182, 235, 191, 163, 190, 196, 140, 189, 167, 198, 182, 187, 200, 200, 179, 190, 181, 184, 178 ], [ 170, 166, 205, 191, 0, 194, 167, 181, 210, 207, 164, 190, 193, 192, 187, 224, 183, 204, 171, 176, 195, 229, 217, 191 ], [ 141, 148, 157, 182, 194, 0, 172, 186, 184, 173, 169, 189, 135, 192, 149, 181, 137, 150, 166, 158, 204, 198, 207, 146 ], [ 156, 180, 223, 235, 167, 172, 0, 191, 194, 158, 185, 216, 158, 244, 193, 190, 199, 172, 182, 174, 215, 210, 209, 195 ], [ 178, 185, 223, 191, 181, 186, 191, 0, 203, 196, 169, 234, 209, 195, 238, 234, 198, 193, 223, 216, 251, 216, 264, 198 ], [ 186, 185, 220, 163, 210, 184, 194, 203, 0, 209, 191, 206, 178, 218, 193, 201, 164, 211, 207, 171, 247, 231, 170, 199 ], [ 156, 216, 190, 190, 207, 173, 158, 196, 209, 0, 185, 187, 147, 217, 197, 156, 206, 167, 219, 206, 169, 167, 199, 200 ], [ 185, 179, 234, 196, 164, 169, 185, 169, 191, 185, 0, 184, 171, 181, 161, 231, 169, 182, 181, 214, 229, 215, 204, 179 ], [ 189, 194, 236, 140, 190, 189, 216, 234, 206, 187, 184, 0, 204, 154, 158, 198, 194, 178, 181, 230, 187, 170, 165, 181 ], [ 170, 184, 187, 189, 193, 135, 158, 209, 178, 147, 171, 204, 0, 207, 164, 194, 158, 166, 174, 170, 207, 226, 215, 177 ], [ 210, 215, 233, 167, 192, 192, 244, 195, 218, 217, 181, 154, 207, 0, 177, 247, 166, 177, 184, 229, 204, 195, 207, 168 ], [ 173, 185, 197, 198, 187, 149, 193, 238, 193, 197, 161, 158, 164, 177, 0, 206, 149, 176, 179, 179, 188, 205, 160, 148 ], [ 149, 218, 201, 182, 224, 181, 190, 234, 201, 156, 231, 198, 194, 247, 206, 0, 225, 178, 217, 176, 213, 217, 174, 195 ], [ 186, 179, 198, 187, 183, 137, 199, 198, 164, 206, 169, 194, 158, 166, 149, 225, 0, 191, 208, 186, 199, 222, 219, 128 ], [ 170, 167, 243, 200, 204, 150, 172, 193, 211, 167, 182, 178, 166, 177, 176, 178, 191, 0, 182, 208, 223, 174, 199, 187 ], [ 192, 196, 221, 200, 171, 166, 182, 223, 207, 219, 181, 181, 174, 184, 179, 217, 208, 182, 0, 177, 224, 212, 187, 190 ], [ 146, 180, 135, 179, 176, 158, 174, 216, 171, 206, 214, 230, 170, 229, 179, 176, 186, 208, 177, 0, 182, 207, 164, 151 ], [ 186, 223, 181, 190, 195, 204, 215, 251, 247, 169, 229, 187, 207, 204, 188, 213, 199, 223, 224, 182, 0, 169, 195, 166 ], [ 201, 215, 218, 181, 229, 198, 210, 216, 231, 167, 215, 170, 226, 195, 205, 217, 222, 174, 212, 207, 169, 0, 175, 196 ], [ 194, 217, 212, 184, 217, 207, 209, 264, 170, 199, 204, 165, 215, 207, 160, 174, 219, 199, 187, 164, 195, 175, 0, 189 ], [ 144, 196, 173, 178, 191, 146, 195, 198, 199, 200, 179, 181, 177, 168, 148, 195, 128, 187, 190, 151, 166, 196, 189, 0 ] ], "m": 7, "objective": 4547.0 }, "solution": [ 2, 7, 8, 13, 15, 18, 20 ], "obj": 4547.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 24, "m": 7, "nodes": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 ], "distance_pairs": [ { "from_id": 0, "to_id": 1, "distance": 175 }, { "from_id": 0, "to_id": 2, "distance": 185 }, { "from_id": 0, "to_id": 3, "distance": 172 }, { "from_id": 0, "to_id": 4, "distance": 170 }, { "from_id": 0, "to_id": 5, "distance": 141 }, { "from_id": 0, "to_id": 6, "distance": 156 }, { "from_id": 0, "to_id": 7, "distance": 178 }, { "from_id": 0, "to_id": 8, "distance": 186 }, { "from_id": 0, "to_id": 9, "distance": 156 }, { "from_id": 0, "to_id": 10, "distance": 185 }, { "from_id": 0, "to_id": 11, "distance": 189 }, { "from_id": 0, "to_id": 12, "distance": 170 }, { "from_id": 0, "to_id": 13, "distance": 210 }, { "from_id": 0, "to_id": 14, "distance": 173 }, { "from_id": 0, "to_id": 15, "distance": 149 }, { "from_id": 0, "to_id": 16, "distance": 186 }, { "from_id": 0, "to_id": 17, "distance": 170 }, { "from_id": 0, "to_id": 18, "distance": 192 }, { "from_id": 0, "to_id": 19, "distance": 146 }, { "from_id": 0, "to_id": 20, "distance": 186 }, { "from_id": 0, "to_id": 21, "distance": 201 }, { "from_id": 0, "to_id": 22, "distance": 194 }, { "from_id": 0, "to_id": 23, "distance": 144 }, { "from_id": 1, "to_id": 0, "distance": 175 }, { "from_id": 1, "to_id": 2, "distance": 214 }, { "from_id": 1, "to_id": 3, "distance": 185 }, { "from_id": 1, "to_id": 4, "distance": 166 }, { "from_id": 1, "to_id": 5, "distance": 148 }, { "from_id": 1, "to_id": 6, "distance": 180 }, { "from_id": 1, "to_id": 7, "distance": 185 }, { "from_id": 1, "to_id": 8, "distance": 185 }, { "from_id": 1, "to_id": 9, "distance": 216 }, { "from_id": 1, "to_id": 10, "distance": 179 }, { "from_id": 1, "to_id": 11, "distance": 194 }, { "from_id": 1, "to_id": 12, "distance": 184 }, { "from_id": 1, "to_id": 13, "distance": 215 }, { "from_id": 1, "to_id": 14, "distance": 185 }, { "from_id": 1, "to_id": 15, "distance": 218 }, { "from_id": 1, "to_id": 16, "distance": 179 }, { "from_id": 1, "to_id": 17, "distance": 167 }, { "from_id": 1, "to_id": 18, "distance": 196 }, { "from_id": 1, "to_id": 19, "distance": 180 }, { "from_id": 1, "to_id": 20, "distance": 223 }, { "from_id": 1, "to_id": 21, "distance": 215 }, { "from_id": 1, "to_id": 22, "distance": 217 }, { "from_id": 1, "to_id": 23, "distance": 196 }, { "from_id": 2, "to_id": 0, "distance": 185 }, { "from_id": 2, "to_id": 1, "distance": 214 }, { "from_id": 2, "to_id": 3, "distance": 192 }, { "from_id": 2, "to_id": 4, "distance": 205 }, { "from_id": 2, "to_id": 5, "distance": 157 }, { "from_id": 2, "to_id": 6, "distance": 223 }, { "from_id": 2, "to_id": 7, "distance": 223 }, { 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"from_id": 5, "to_id": 1, "distance": 148 }, { "from_id": 5, "to_id": 2, "distance": 157 }, { "from_id": 5, "to_id": 3, "distance": 182 }, { "from_id": 5, "to_id": 4, "distance": 194 }, { "from_id": 5, "to_id": 6, "distance": 172 }, { "from_id": 5, "to_id": 7, "distance": 186 }, { "from_id": 5, "to_id": 8, "distance": 184 }, { "from_id": 5, "to_id": 9, "distance": 173 }, { "from_id": 5, "to_id": 10, "distance": 169 }, { "from_id": 5, "to_id": 11, "distance": 189 }, { "from_id": 5, "to_id": 12, "distance": 135 }, { "from_id": 5, "to_id": 13, "distance": 192 }, { "from_id": 5, "to_id": 14, "distance": 149 }, { "from_id": 5, "to_id": 15, "distance": 181 }, { "from_id": 5, "to_id": 16, "distance": 137 }, { "from_id": 5, "to_id": 17, "distance": 150 }, { "from_id": 5, "to_id": 18, "distance": 166 }, { "from_id": 5, "to_id": 19, "distance": 158 }, { "from_id": 5, "to_id": 20, "distance": 204 }, { "from_id": 5, "to_id": 21, "distance": 198 }, { "from_id": 5, "to_id": 22, "distance": 207 }, { 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"distance": 166 }, { "from_id": 23, "to_id": 21, "distance": 196 }, { "from_id": 23, "to_id": 22, "distance": 189 } ] }, "solution_variant": [ 2, 7, 8, 13, 15, 18, 20 ], "context_index": 42, "input_format": "markdown_table", "input_index_base": 0 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "I’m putting together the entertainment lineup for a reception and need to pick exactly m acts so the party feels fresh and varied. The idea is simple: every pair of options has a score for how different they are, and the stronger the overall variety the better — so add up the difference between each pair in the chosen group and aim for the highest total. There can’t be repeats and the list must hit exactly m slots, no more and no less. The exact details and the distance numbers are shown below.\n\n# total_entertainment_options=23\n# acts_to_select=7\n# available_act_ids=A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W\nact_id_first,act_id_second,variety_score_between_acts\nA,B,2\nA,C,7\nA,D,8\nA,E,4\nA,F,9\nA,G,5\nA,H,9\nA,I,1\nA,J,6\nA,K,5\nA,L,2\nA,M,0\nA,N,1\nA,O,4\nA,P,8\nA,Q,4\nA,R,8\nA,S,4\nA,T,5\nA,U,1\nA,V,4\nA,W,4\nB,A,2\nB,C,0\nB,D,2\nB,E,0\nB,F,0\nB,G,8\nB,H,0\nB,I,3\nB,J,6\nB,K,9\nB,L,1\nB,M,1\nB,N,0\nB,O,6\nB,P,3\nB,Q,8\nB,R,1\nB,S,3\nB,T,2\nB,U,9\nB,V,5\nB,W,3\nC,A,7\nC,B,0\nC,D,8\nC,E,7\nC,F,5\nC,G,7\nC,H,5\nC,I,4\nC,J,3\nC,K,1\nC,L,0\nC,M,4\nC,N,4\nC,O,7\nC,P,8\nC,Q,0\nC,R,4\nC,S,8\nC,T,3\nC,U,9\nC,V,0\nC,W,1\nD,A,8\nD,B,2\nD,C,8\nD,E,9\nD,F,5\nD,G,9\nD,H,1\nD,I,2\nD,J,0\nD,K,6\nD,L,1\nD,M,3\nD,N,3\nD,O,4\nD,P,6\nD,Q,5\nD,R,4\nD,S,3\nD,T,6\nD,U,8\nD,V,0\nD,W,9\nE,A,4\nE,B,0\nE,C,7\nE,D,9\nE,F,6\nE,G,8\nE,H,3\nE,I,2\nE,J,4\nE,K,9\nE,L,5\nE,M,2\nE,N,8\nE,O,9\nE,P,8\nE,Q,1\nE,R,8\nE,S,9\nE,T,2\nE,U,7\nE,V,1\nE,W,4\nF,A,9\nF,B,0\nF,C,5\nF,D,5\nF,E,6\nF,G,8\nF,H,0\nF,I,4\nF,J,6\nF,K,3\nF,L,3\nF,M,3\nF,N,5\nF,O,0\nF,P,0\nF,Q,6\nF,R,5\nF,S,4\nF,T,4\nF,U,5\nF,V,0\nF,W,0\nG,A,5\nG,B,8\nG,C,7\nG,D,9\nG,E,8\nG,F,8\nG,H,1\nG,I,6\nG,J,9\nG,K,4\nG,L,3\nG,M,0\nG,N,0\nG,O,1\nG,P,3\nG,Q,0\nG,R,0\nG,S,9\nG,T,7\nG,U,5\nG,V,1\nG,W,7\nH,A,9\nH,B,0\nH,C,5\nH,D,1\nH,E,3\nH,F,0\nH,G,1\nH,I,3\nH,J,7\nH,K,5\nH,L,3\nH,M,5\nH,N,4\nH,O,6\nH,P,3\nH,Q,3\nH,R,6\nH,S,7\nH,T,5\nH,U,6\nH,V,8\nH,W,7\nI,A,1\nI,B,3\nI,C,4\nI,D,2\nI,E,2\nI,F,4\nI,G,6\nI,H,3\nI,J,6\nI,K,7\nI,L,1\nI,M,6\nI,N,1\nI,O,3\nI,P,2\nI,Q,7\nI,R,2\nI,S,9\nI,T,0\nI,U,5\nI,V,3\nI,W,1\nJ,A,6\nJ,B,6\nJ,C,3\nJ,D,0\nJ,E,4\nJ,F,6\nJ,G,9\nJ,H,7\nJ,I,6\nJ,K,3\nJ,L,7\nJ,M,7\nJ,N,8\nJ,O,4\nJ,P,9\nJ,Q,0\nJ,R,8\nJ,S,6\nJ,T,3\nJ,U,1\nJ,V,2\nJ,W,7\nK,A,5\nK,B,9\nK,C,1\nK,D,6\nK,E,9\nK,F,3\nK,G,4\nK,H,5\nK,I,7\nK,J,3\nK,L,2\nK,M,9\nK,N,8\nK,O,1\nK,P,5\nK,Q,9\nK,R,6\nK,S,3\nK,T,7\nK,U,4\nK,V,0\nK,W,2\nL,A,2\nL,B,1\nL,C,0\nL,D,1\nL,E,5\nL,F,3\nL,G,3\nL,H,3\nL,I,1\nL,J,7\nL,K,2\nL,M,5\nL,N,9\nL,O,2\nL,P,1\nL,Q,9\nL,R,0\nL,S,1\nL,T,8\nL,U,9\nL,V,9\nL,W,0\nM,A,0\nM,B,1\nM,C,4\nM,D,3\nM,E,2\nM,F,3\nM,G,0\nM,H,5\nM,I,6\nM,J,7\nM,K,9\nM,L,5\nM,N,8\nM,O,8\nM,P,5\nM,Q,0\nM,R,1\nM,S,8\nM,T,6\nM,U,4\nM,V,3\nM,W,4\nN,A,1\nN,B,0\nN,C,4\nN,D,3\nN,E,8\nN,F,5\nN,G,0\nN,H,4\nN,I,1\nN,J,8\nN,K,8\nN,L,9\nN,M,8\nN,O,2\nN,P,8\nN,Q,2\nN,R,1\nN,S,7\nN,T,9\nN,U,7\nN,V,4\nN,W,1\nO,A,4\nO,B,6\nO,C,7\nO,D,4\nO,E,9\nO,F,0\nO,G,1\nO,H,6\nO,I,3\nO,J,4\nO,K,1\nO,L,2\nO,M,8\nO,N,2\nO,P,3\nO,Q,6\nO,R,9\nO,S,9\nO,T,3\nO,U,8\nO,V,5\nO,W,4\nP,A,8\nP,B,3\nP,C,8\nP,D,6\nP,E,8\nP,F,0\nP,G,3\nP,H,3\nP,I,2\nP,J,9\nP,K,5\nP,L,1\nP,M,5\nP,N,8\nP,O,3\nP,Q,9\nP,R,2\nP,S,0\nP,T,1\nP,U,3\nP,V,8\nP,W,3\nQ,A,4\nQ,B,8\nQ,C,0\nQ,D,5\nQ,E,1\nQ,F,6\nQ,G,0\nQ,H,3\nQ,I,7\nQ,J,0\nQ,K,9\nQ,L,9\nQ,M,0\nQ,N,2\nQ,O,6\nQ,P,9\nQ,R,4\nQ,S,6\nQ,T,1\nQ,U,8\nQ,V,2\nQ,W,8\nR,A,8\nR,B,1\nR,C,4\nR,D,4\nR,E,8\nR,F,5\nR,G,0\nR,H,6\nR,I,2\nR,J,8\nR,K,6\nR,L,0\nR,M,1\nR,N,1\nR,O,9\nR,P,2\nR,Q,4\nR,S,2\nR,T,5\nR,U,1\nR,V,8\nR,W,9\nS,A,4\nS,B,3\nS,C,8\nS,D,3\nS,E,9\nS,F,4\nS,G,9\nS,H,7\nS,I,9\nS,J,6\nS,K,3\nS,L,1\nS,M,8\nS,N,7\nS,O,9\nS,P,0\nS,Q,6\nS,R,2\nS,T,3\nS,U,4\nS,V,1\nS,W,6\nT,A,5\nT,B,2\nT,C,3\nT,D,6\nT,E,2\nT,F,4\nT,G,7\nT,H,5\nT,I,0\nT,J,3\nT,K,7\nT,L,8\nT,M,6\nT,N,9\nT,O,3\nT,P,1\nT,Q,1\nT,R,5\nT,S,3\nT,U,2\nT,V,9\nT,W,8\nU,A,1\nU,B,9\nU,C,9\nU,D,8\nU,E,7\nU,F,5\nU,G,5\nU,H,6\nU,I,5\nU,J,1\nU,K,4\nU,L,9\nU,M,4\nU,N,7\nU,O,8\nU,P,3\nU,Q,8\nU,R,1\nU,S,4\nU,T,2\nU,V,9\nU,W,5\nV,A,4\nV,B,5\nV,C,0\nV,D,0\nV,E,1\nV,F,0\nV,G,1\nV,H,8\nV,I,3\nV,J,2\nV,K,0\nV,L,9\nV,M,3\nV,N,4\nV,O,5\nV,P,8\nV,Q,2\nV,R,8\nV,S,1\nV,T,9\nV,U,9\nV,W,8\nW,A,4\nW,B,3\nW,C,1\nW,D,9\nW,E,4\nW,F,0\nW,G,7\nW,H,7\nW,I,1\nW,J,7\nW,K,2\nW,L,0\nW,M,4\nW,N,1\nW,O,4\nW,P,3\nW,Q,8\nW,R,9\nW,S,6\nW,T,8\nW,U,5\nW,V,8\n\nOh, and when you reply with your chosen lineup, please use this simple JSON layout so it's easy to parse:\n\n{\n \"solution\": [, , ...]\n}\n\n\"solution\" is just the list of the m acts you're picking for the reception — put each act's identifier in the array (no repeats), in whatever order you like. This JSON is just a sketch of the shape I expect, not the actual answer itself.\n\nPlease 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”.\"", "instance": { "distance_matrix": [ [ 0, 2, 7, 8, 4, 9, 5, 9, 1, 6, 5, 2, 0, 1, 4, 8, 4, 8, 4, 5, 1, 4, 4 ], [ 2, 0, 0, 2, 0, 0, 8, 0, 3, 6, 9, 1, 1, 0, 6, 3, 8, 1, 3, 2, 9, 5, 3 ], [ 7, 0, 0, 8, 7, 5, 7, 5, 4, 3, 1, 0, 4, 4, 7, 8, 0, 4, 8, 3, 9, 0, 1 ], [ 8, 2, 8, 0, 9, 5, 9, 1, 2, 0, 6, 1, 3, 3, 4, 6, 5, 4, 3, 6, 8, 0, 9 ], [ 4, 0, 7, 9, 0, 6, 8, 3, 2, 4, 9, 5, 2, 8, 9, 8, 1, 8, 9, 2, 7, 1, 4 ], [ 9, 0, 5, 5, 6, 0, 8, 0, 4, 6, 3, 3, 3, 5, 0, 0, 6, 5, 4, 4, 5, 0, 0 ], [ 5, 8, 7, 9, 8, 8, 0, 1, 6, 9, 4, 3, 0, 0, 1, 3, 0, 0, 9, 7, 5, 1, 7 ], [ 9, 0, 5, 1, 3, 0, 1, 0, 3, 7, 5, 3, 5, 4, 6, 3, 3, 6, 7, 5, 6, 8, 7 ], [ 1, 3, 4, 2, 2, 4, 6, 3, 0, 6, 7, 1, 6, 1, 3, 2, 7, 2, 9, 0, 5, 3, 1 ], [ 6, 6, 3, 0, 4, 6, 9, 7, 6, 0, 3, 7, 7, 8, 4, 9, 0, 8, 6, 3, 1, 2, 7 ], [ 5, 9, 1, 6, 9, 3, 4, 5, 7, 3, 0, 2, 9, 8, 1, 5, 9, 6, 3, 7, 4, 0, 2 ], [ 2, 1, 0, 1, 5, 3, 3, 3, 1, 7, 2, 0, 5, 9, 2, 1, 9, 0, 1, 8, 9, 9, 0 ], [ 0, 1, 4, 3, 2, 3, 0, 5, 6, 7, 9, 5, 0, 8, 8, 5, 0, 1, 8, 6, 4, 3, 4 ], [ 1, 0, 4, 3, 8, 5, 0, 4, 1, 8, 8, 9, 8, 0, 2, 8, 2, 1, 7, 9, 7, 4, 1 ], [ 4, 6, 7, 4, 9, 0, 1, 6, 3, 4, 1, 2, 8, 2, 0, 3, 6, 9, 9, 3, 8, 5, 4 ], [ 8, 3, 8, 6, 8, 0, 3, 3, 2, 9, 5, 1, 5, 8, 3, 0, 9, 2, 0, 1, 3, 8, 3 ], [ 4, 8, 0, 5, 1, 6, 0, 3, 7, 0, 9, 9, 0, 2, 6, 9, 0, 4, 6, 1, 8, 2, 8 ], [ 8, 1, 4, 4, 8, 5, 0, 6, 2, 8, 6, 0, 1, 1, 9, 2, 4, 0, 2, 5, 1, 8, 9 ], [ 4, 3, 8, 3, 9, 4, 9, 7, 9, 6, 3, 1, 8, 7, 9, 0, 6, 2, 0, 3, 4, 1, 6 ], [ 5, 2, 3, 6, 2, 4, 7, 5, 0, 3, 7, 8, 6, 9, 3, 1, 1, 5, 3, 0, 2, 9, 8 ], [ 1, 9, 9, 8, 7, 5, 5, 6, 5, 1, 4, 9, 4, 7, 8, 3, 8, 1, 4, 2, 0, 9, 5 ], [ 4, 5, 0, 0, 1, 0, 1, 8, 3, 2, 0, 9, 3, 4, 5, 8, 2, 8, 1, 9, 9, 0, 8 ], [ 4, 3, 1, 9, 4, 0, 7, 7, 1, 7, 2, 0, 4, 1, 4, 3, 8, 9, 6, 8, 5, 8, 0 ] ], "m": 7, "objective": 148.0 }, "solution": [ 2, 3, 4, 6, 14, 18, 20 ], "obj": 148.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 23, "m": 7, "nodes": [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S", "T", "U", "V", "W" ], "distance_pairs": [ { "from_id": "A", "to_id": "B", "distance": 2 }, { "from_id": "A", "to_id": "C", "distance": 7 }, { "from_id": "A", "to_id": "D", "distance": 8 }, { "from_id": "A", "to_id": "E", "distance": 4 }, { "from_id": "A", "to_id": "F", "distance": 9 }, { "from_id": "A", "to_id": "G", "distance": 5 }, { "from_id": "A", "to_id": "H", "distance": 9 }, { "from_id": "A", "to_id": "I", "distance": 1 }, { "from_id": "A", "to_id": "J", "distance": 6 }, { "from_id": "A", "to_id": "K", "distance": 5 }, { "from_id": "A", "to_id": "L", "distance": 2 }, { "from_id": "A", "to_id": "M", "distance": 0 }, { "from_id": "A", "to_id": "N", "distance": 1 }, { "from_id": "A", "to_id": "O", "distance": 4 }, { "from_id": "A", "to_id": "P", "distance": 8 }, { "from_id": "A", "to_id": "Q", "distance": 4 }, { "from_id": "A", "to_id": "R", "distance": 8 }, { "from_id": "A", "to_id": "S", "distance": 4 }, { "from_id": "A", "to_id": "T", "distance": 5 }, { "from_id": "A", "to_id": "U", "distance": 1 }, { "from_id": "A", "to_id": "V", "distance": 4 }, { "from_id": "A", "to_id": "W", "distance": 4 }, { "from_id": "B", "to_id": "A", "distance": 2 }, { "from_id": "B", "to_id": "C", "distance": 0 }, { "from_id": "B", "to_id": "D", "distance": 2 }, { "from_id": "B", "to_id": "E", "distance": 0 }, { "from_id": "B", "to_id": "F", "distance": 0 }, { "from_id": "B", "to_id": "G", "distance": 8 }, { "from_id": "B", "to_id": "H", "distance": 0 }, { "from_id": "B", "to_id": "I", "distance": 3 }, { "from_id": "B", "to_id": "J", "distance": 6 }, { "from_id": "B", "to_id": "K", "distance": 9 }, { "from_id": "B", "to_id": "L", "distance": 1 }, { "from_id": "B", "to_id": "M", "distance": 1 }, { "from_id": "B", "to_id": "N", "distance": 0 }, { "from_id": "B", "to_id": "O", "distance": 6 }, { "from_id": "B", "to_id": "P", "distance": 3 }, { "from_id": "B", "to_id": "Q", "distance": 8 }, { "from_id": "B", "to_id": "R", "distance": 1 }, { "from_id": "B", "to_id": "S", "distance": 3 }, { "from_id": "B", "to_id": "T", "distance": 2 }, { "from_id": "B", "to_id": "U", "distance": 9 }, { "from_id": "B", "to_id": "V", "distance": 5 }, { "from_id": "B", "to_id": "W", "distance": 3 }, { "from_id": "C", "to_id": "A", "distance": 7 }, { "from_id": "C", "to_id": "B", "distance": 0 }, { "from_id": "C", "to_id": "D", "distance": 8 }, { "from_id": "C", "to_id": "E", "distance": 7 }, { "from_id": "C", "to_id": "F", "distance": 5 }, { "from_id": "C", "to_id": "G", "distance": 7 }, { "from_id": "C", "to_id": "H", "distance": 5 }, { "from_id": "C", "to_id": "I", "distance": 4 }, { "from_id": "C", "to_id": "J", "distance": 3 }, { "from_id": "C", "to_id": "K", "distance": 1 }, { "from_id": "C", "to_id": "L", "distance": 0 }, { "from_id": "C", "to_id": "M", "distance": 4 }, { "from_id": "C", "to_id": "N", "distance": 4 }, { "from_id": "C", "to_id": "O", "distance": 7 }, { "from_id": "C", "to_id": "P", "distance": 8 }, { "from_id": "C", "to_id": "Q", "distance": 0 }, { "from_id": "C", "to_id": "R", "distance": 4 }, { "from_id": "C", "to_id": "S", "distance": 8 }, { "from_id": "C", "to_id": "T", "distance": 3 }, { "from_id": "C", "to_id": "U", "distance": 9 }, { "from_id": "C", "to_id": "V", "distance": 0 }, { "from_id": "C", "to_id": "W", "distance": 1 }, { "from_id": "D", "to_id": "A", "distance": 8 }, { "from_id": "D", "to_id": "B", "distance": 2 }, { "from_id": "D", "to_id": "C", "distance": 8 }, { "from_id": "D", "to_id": "E", "distance": 9 }, { "from_id": "D", "to_id": "F", "distance": 5 }, { "from_id": "D", "to_id": "G", "distance": 9 }, { "from_id": "D", "to_id": "H", "distance": 1 }, { "from_id": "D", "to_id": "I", "distance": 2 }, { "from_id": "D", "to_id": "J", "distance": 0 }, { "from_id": "D", "to_id": "K", "distance": 6 }, { "from_id": "D", "to_id": "L", "distance": 1 }, { "from_id": "D", 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"to_id": "D", "distance": 8 }, { "from_id": "U", "to_id": "E", "distance": 7 }, { "from_id": "U", "to_id": "F", "distance": 5 }, { "from_id": "U", "to_id": "G", "distance": 5 }, { "from_id": "U", "to_id": "H", "distance": 6 }, { "from_id": "U", "to_id": "I", "distance": 5 }, { "from_id": "U", "to_id": "J", "distance": 1 }, { "from_id": "U", "to_id": "K", "distance": 4 }, { "from_id": "U", "to_id": "L", "distance": 9 }, { "from_id": "U", "to_id": "M", "distance": 4 }, { "from_id": "U", "to_id": "N", "distance": 7 }, { "from_id": "U", "to_id": "O", "distance": 8 }, { "from_id": "U", "to_id": "P", "distance": 3 }, { "from_id": "U", "to_id": "Q", "distance": 8 }, { "from_id": "U", "to_id": "R", "distance": 1 }, { "from_id": "U", "to_id": "S", "distance": 4 }, { "from_id": "U", "to_id": "T", "distance": 2 }, { "from_id": "U", "to_id": "V", "distance": 9 }, { "from_id": "U", "to_id": "W", "distance": 5 }, { "from_id": "V", "to_id": "A", "distance": 4 }, { "from_id": "V", "to_id": "B", "distance": 5 }, { "from_id": "V", "to_id": "C", "distance": 0 }, { "from_id": "V", "to_id": "D", "distance": 0 }, { "from_id": "V", "to_id": "E", "distance": 1 }, { "from_id": "V", "to_id": "F", "distance": 0 }, { "from_id": "V", "to_id": "G", "distance": 1 }, { "from_id": "V", "to_id": "H", "distance": 8 }, { "from_id": "V", "to_id": "I", "distance": 3 }, { "from_id": "V", "to_id": "J", "distance": 2 }, { "from_id": "V", "to_id": "K", "distance": 0 }, { "from_id": "V", "to_id": "L", "distance": 9 }, { "from_id": "V", "to_id": "M", "distance": 3 }, { "from_id": "V", "to_id": "N", "distance": 4 }, { "from_id": "V", "to_id": "O", "distance": 5 }, { "from_id": "V", "to_id": "P", "distance": 8 }, { "from_id": "V", "to_id": "Q", "distance": 2 }, { "from_id": "V", "to_id": "R", "distance": 8 }, { "from_id": "V", "to_id": "S", "distance": 1 }, { "from_id": "V", "to_id": "T", "distance": 9 }, { "from_id": "V", "to_id": "U", "distance": 9 }, { "from_id": "V", "to_id": "W", "distance": 8 }, { "from_id": "W", "to_id": "A", "distance": 4 }, { "from_id": "W", "to_id": "B", "distance": 3 }, { "from_id": "W", "to_id": "C", "distance": 1 }, { "from_id": "W", "to_id": "D", "distance": 9 }, { "from_id": "W", "to_id": "E", "distance": 4 }, { "from_id": "W", "to_id": "F", "distance": 0 }, { "from_id": "W", "to_id": "G", "distance": 7 }, { "from_id": "W", "to_id": "H", "distance": 7 }, { "from_id": "W", "to_id": "I", "distance": 1 }, { "from_id": "W", "to_id": "J", "distance": 7 }, { "from_id": "W", "to_id": "K", "distance": 2 }, { "from_id": "W", "to_id": "L", "distance": 0 }, { "from_id": "W", "to_id": "M", "distance": 4 }, { "from_id": "W", "to_id": "N", "distance": 1 }, { "from_id": "W", "to_id": "O", "distance": 4 }, { "from_id": "W", "to_id": "P", "distance": 3 }, { "from_id": "W", "to_id": "Q", "distance": 8 }, { "from_id": "W", "to_id": "R", "distance": 9 }, { "from_id": "W", "to_id": "S", "distance": 6 }, { "from_id": "W", "to_id": "T", "distance": 8 }, { "from_id": "W", "to_id": "U", "distance": 5 }, { "from_id": "W", "to_id": "V", "distance": 8 } ] }, "solution_variant": [ "C", "D", "E", "G", "O", "S", "U" ], "context_index": 43, "input_format": "csv", "input_index_base": "names" }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Someone at the bar is curating a flight and wants every sip to be distinct — different regions, different flavor signatures. The way they decide is by looking at every pair of bottles in the chosen flight, seeing how different that pair is, and adding up those differences; the fuller that combined difference number, the better the flight. The curator must pick exactly the given number of unique drinks. The specific drink list and required count follow below.\n\nThere are 25 candidate bottles — 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24; they must select exactly 8.\n\n| drink_id_a | drink_id_b | dissimilarity_score |\n|---|---|---|\n| 0 | 1 | 175 |\n| 0 | 2 | 185 |\n| 0 | 3 | 172 |\n| 0 | 4 | 170 |\n| 0 | 5 | 149 |\n| 0 | 6 | 221 |\n| 0 | 7 | 141 |\n| 0 | 8 | 156 |\n| 0 | 9 | 178 |\n| 0 | 10 | 180 |\n| 0 | 11 | 185 |\n| 0 | 12 | 189 |\n| 0 | 13 | 170 |\n| 0 | 14 | 210 |\n| 0 | 15 | 173 |\n| 0 | 16 | 170 |\n| 0 | 17 | 192 |\n| 0 | 18 | 192 |\n| 0 | 19 | 146 |\n| 0 | 20 | 186 |\n| 0 | 21 | 194 |\n| 0 | 22 | 201 |\n| 0 | 23 | 194 |\n| 0 | 24 | 144 |\n| 1 | 0 | 175 |\n| 1 | 2 | 214 |\n| 1 | 3 | 185 |\n| 1 | 4 | 166 |\n| 1 | 5 | 179 |\n| 1 | 6 | 229 |\n| 1 | 7 | 148 |\n| 1 | 8 | 180 |\n| 1 | 9 | 185 |\n| 1 | 10 | 213 |\n| 1 | 11 | 179 |\n| 1 | 12 | 194 |\n| 1 | 13 | 184 |\n| 1 | 14 | 215 |\n| 1 | 15 | 185 |\n| 1 | 16 | 167 |\n| 1 | 17 | 169 |\n| 1 | 18 | 196 |\n| 1 | 19 | 180 |\n| 1 | 20 | 223 |\n| 1 | 21 | 197 |\n| 1 | 22 | 215 |\n| 1 | 23 | 217 |\n| 1 | 24 | 196 |\n| 2 | 0 | 185 |\n| 2 | 1 | 214 |\n| 2 | 3 | 192 |\n| 2 | 4 | 205 |\n| 2 | 5 | 134 |\n| 2 | 6 | 151 |\n| 2 | 7 | 157 |\n| 2 | 8 | 223 |\n| 2 | 9 | 223 |\n| 2 | 10 | 165 |\n| 2 | 11 | 234 |\n| 2 | 12 | 236 |\n| 2 | 13 | 187 |\n| 2 | 14 | 233 |\n| 2 | 15 | 197 |\n| 2 | 16 | 243 |\n| 2 | 17 | 224 |\n| 2 | 18 | 221 |\n| 2 | 19 | 135 |\n| 2 | 20 | 181 |\n| 2 | 21 | 213 |\n| 2 | 22 | 218 |\n| 2 | 23 | 212 |\n| 2 | 24 | 173 |\n| 3 | 0 | 172 |\n| 3 | 1 | 185 |\n| 3 | 2 | 192 |\n| 3 | 4 | 191 |\n| 3 | 5 | 158 |\n| 3 | 6 | 169 |\n| 3 | 7 | 182 |\n| 3 | 8 | 235 |\n| 3 | 9 | 191 |\n| 3 | 10 | 160 |\n| 3 | 11 | 196 |\n| 3 | 12 | 140 |\n| 3 | 13 | 189 |\n| 3 | 14 | 167 |\n| 3 | 15 | 198 |\n| 3 | 16 | 200 |\n| 3 | 17 | 182 |\n| 3 | 18 | 200 |\n| 3 | 19 | 179 |\n| 3 | 20 | 190 |\n| 3 | 21 | 224 |\n| 3 | 22 | 181 |\n| 3 | 23 | 184 |\n| 3 | 24 | 178 |\n| 4 | 0 | 170 |\n| 4 | 1 | 166 |\n| 4 | 2 | 205 |\n| 4 | 3 | 191 |\n| 4 | 5 | 193 |\n| 4 | 6 | 201 |\n| 4 | 7 | 194 |\n| 4 | 8 | 167 |\n| 4 | 9 | 181 |\n| 4 | 10 | 153 |\n| 4 | 11 | 164 |\n| 4 | 12 | 190 |\n| 4 | 13 | 193 |\n| 4 | 14 | 192 |\n| 4 | 15 | 187 |\n| 4 | 16 | 204 |\n| 4 | 17 | 173 |\n| 4 | 18 | 171 |\n| 4 | 19 | 176 |\n| 4 | 20 | 195 |\n| 4 | 21 | 211 |\n| 4 | 22 | 229 |\n| 4 | 23 | 217 |\n| 4 | 24 | 191 |\n| 5 | 0 | 149 |\n| 5 | 1 | 179 |\n| 5 | 2 | 134 |\n| 5 | 3 | 158 |\n| 5 | 4 | 193 |\n| 5 | 6 | 154 |\n| 5 | 7 | 164 |\n| 5 | 8 | 203 |\n| 5 | 9 | 180 |\n| 5 | 10 | 184 |\n| 5 | 11 | 225 |\n| 5 | 12 | 200 |\n| 5 | 13 | 204 |\n| 5 | 14 | 187 |\n| 5 | 15 | 187 |\n| 5 | 16 | 188 |\n| 5 | 17 | 196 |\n| 5 | 18 | 195 |\n| 5 | 19 | 132 |\n| 5 | 20 | 187 |\n| 5 | 21 | 224 |\n| 5 | 22 | 183 |\n| 5 | 23 | 178 |\n| 5 | 24 | 158 |\n| 6 | 0 | 221 |\n| 6 | 1 | 229 |\n| 6 | 2 | 151 |\n| 6 | 3 | 169 |\n| 6 | 4 | 201 |\n| 6 | 5 | 154 |\n| 6 | 7 | 202 |\n| 6 | 8 | 222 |\n| 6 | 9 | 221 |\n| 6 | 10 | 160 |\n| 6 | 11 | 200 |\n| 6 | 12 | 184 |\n| 6 | 13 | 189 |\n| 6 | 14 | 171 |\n| 6 | 15 | 170 |\n| 6 | 16 | 234 |\n| 6 | 17 | 198 |\n| 6 | 18 | 183 |\n| 6 | 19 | 179 |\n| 6 | 20 | 180 |\n| 6 | 21 | 220 |\n| 6 | 22 | 180 |\n| 6 | 23 | 172 |\n| 6 | 24 | 185 |\n| 7 | 0 | 141 |\n| 7 | 1 | 148 |\n| 7 | 2 | 157 |\n| 7 | 3 | 182 |\n| 7 | 4 | 194 |\n| 7 | 5 | 164 |\n| 7 | 6 | 202 |\n| 7 | 8 | 172 |\n| 7 | 9 | 186 |\n| 7 | 10 | 176 |\n| 7 | 11 | 169 |\n| 7 | 12 | 189 |\n| 7 | 13 | 135 |\n| 7 | 14 | 192 |\n| 7 | 15 | 149 |\n| 7 | 16 | 150 |\n| 7 | 17 | 174 |\n| 7 | 18 | 166 |\n| 7 | 19 | 158 |\n| 7 | 20 | 204 |\n| 7 | 21 | 165 |\n| 7 | 22 | 198 |\n| 7 | 23 | 207 |\n| 7 | 24 | 146 |\n| 8 | 0 | 156 |\n| 8 | 1 | 180 |\n| 8 | 2 | 223 |\n| 8 | 3 | 235 |\n| 8 | 4 | 167 |\n| 8 | 5 | 203 |\n| 8 | 6 | 222 |\n| 8 | 7 | 172 |\n| 8 | 9 | 191 |\n| 8 | 10 | 192 |\n| 8 | 11 | 185 |\n| 8 | 12 | 216 |\n| 8 | 13 | 158 |\n| 8 | 14 | 244 |\n| 8 | 15 | 193 |\n| 8 | 16 | 172 |\n| 8 | 17 | 196 |\n| 8 | 18 | 182 |\n| 8 | 19 | 174 |\n| 8 | 20 | 215 |\n| 8 | 21 | 201 |\n| 8 | 22 | 210 |\n| 8 | 23 | 209 |\n| 8 | 24 | 195 |\n| 9 | 0 | 178 |\n| 9 | 1 | 185 |\n| 9 | 2 | 223 |\n| 9 | 3 | 191 |\n| 9 | 4 | 181 |\n| 9 | 5 | 180 |\n| 9 | 6 | 221 |\n| 9 | 7 | 186 |\n| 9 | 8 | 191 |\n| 9 | 10 | 212 |\n| 9 | 11 | 169 |\n| 9 | 12 | 234 |\n| 9 | 13 | 209 |\n| 9 | 14 | 195 |\n| 9 | 15 | 238 |\n| 9 | 16 | 193 |\n| 9 | 17 | 166 |\n| 9 | 18 | 223 |\n| 9 | 19 | 216 |\n| 9 | 20 | 251 |\n| 9 | 21 | 272 |\n| 9 | 22 | 216 |\n| 9 | 23 | 264 |\n| 9 | 24 | 198 |\n| 10 | 0 | 180 |\n| 10 | 1 | 213 |\n| 10 | 2 | 165 |\n| 10 | 3 | 160 |\n| 10 | 4 | 153 |\n| 10 | 5 | 184 |\n| 10 | 6 | 160 |\n| 10 | 7 | 176 |\n| 10 | 8 | 192 |\n| 10 | 9 | 212 |\n| 10 | 11 | 180 |\n| 10 | 12 | 173 |\n| 10 | 13 | 181 |\n| 10 | 14 | 174 |\n| 10 | 15 | 177 |\n| 10 | 16 | 222 |\n| 10 | 17 | 204 |\n| 10 | 18 | 194 |\n| 10 | 19 | 166 |\n| 10 | 20 | 178 |\n| 10 | 21 | 196 |\n| 10 | 22 | 220 |\n| 10 | 23 | 184 |\n| 10 | 24 | 146 |\n| 11 | 0 | 185 |\n| 11 | 1 | 179 |\n| 11 | 2 | 234 |\n| 11 | 3 | 196 |\n| 11 | 4 | 164 |\n| 11 | 5 | 225 |\n| 11 | 6 | 200 |\n| 11 | 7 | 169 |\n| 11 | 8 | 185 |\n| 11 | 9 | 169 |\n| 11 | 10 | 180 |\n| 11 | 12 | 184 |\n| 11 | 13 | 171 |\n| 11 | 14 | 181 |\n| 11 | 15 | 161 |\n| 11 | 16 | 182 |\n| 11 | 17 | 177 |\n| 11 | 18 | 181 |\n| 11 | 19 | 214 |\n| 11 | 20 | 229 |\n| 11 | 21 | 201 |\n| 11 | 22 | 215 |\n| 11 | 23 | 204 |\n| 11 | 24 | 179 |\n| 12 | 0 | 189 |\n| 12 | 1 | 194 |\n| 12 | 2 | 236 |\n| 12 | 3 | 140 |\n| 12 | 4 | 190 |\n| 12 | 5 | 200 |\n| 12 | 6 | 184 |\n| 12 | 7 | 189 |\n| 12 | 8 | 216 |\n| 12 | 9 | 234 |\n| 12 | 10 | 173 |\n| 12 | 11 | 184 |\n| 12 | 13 | 204 |\n| 12 | 14 | 154 |\n| 12 | 15 | 158 |\n| 12 | 16 | 178 |\n| 12 | 17 | 196 |\n| 12 | 18 | 181 |\n| 12 | 19 | 230 |\n| 12 | 20 | 187 |\n| 12 | 21 | 185 |\n| 12 | 22 | 170 |\n| 12 | 23 | 165 |\n| 12 | 24 | 181 |\n| 13 | 0 | 170 |\n| 13 | 1 | 184 |\n| 13 | 2 | 187 |\n| 13 | 3 | 189 |\n| 13 | 4 | 193 |\n| 13 | 5 | 204 |\n| 13 | 6 | 189 |\n| 13 | 7 | 135 |\n| 13 | 8 | 158 |\n| 13 | 9 | 209 |\n| 13 | 10 | 181 |\n| 13 | 11 | 171 |\n| 13 | 12 | 204 |\n| 13 | 14 | 207 |\n| 13 | 15 | 164 |\n| 13 | 16 | 166 |\n| 13 | 17 | 200 |\n| 13 | 18 | 174 |\n| 13 | 19 | 170 |\n| 13 | 20 | 207 |\n| 13 | 21 | 170 |\n| 13 | 22 | 226 |\n| 13 | 23 | 215 |\n| 13 | 24 | 177 |\n| 14 | 0 | 210 |\n| 14 | 1 | 215 |\n| 14 | 2 | 233 |\n| 14 | 3 | 167 |\n| 14 | 4 | 192 |\n| 14 | 5 | 187 |\n| 14 | 6 | 171 |\n| 14 | 7 | 192 |\n| 14 | 8 | 244 |\n| 14 | 9 | 195 |\n| 14 | 10 | 174 |\n| 14 | 11 | 181 |\n| 14 | 12 | 154 |\n| 14 | 13 | 207 |\n| 14 | 15 | 177 |\n| 14 | 16 | 177 |\n| 14 | 17 | 184 |\n| 14 | 18 | 184 |\n| 14 | 19 | 229 |\n| 14 | 20 | 204 |\n| 14 | 21 | 223 |\n| 14 | 22 | 195 |\n| 14 | 23 | 207 |\n| 14 | 24 | 168 |\n| 15 | 0 | 173 |\n| 15 | 1 | 185 |\n| 15 | 2 | 197 |\n| 15 | 3 | 198 |\n| 15 | 4 | 187 |\n| 15 | 5 | 187 |\n| 15 | 6 | 170 |\n| 15 | 7 | 149 |\n| 15 | 8 | 193 |\n| 15 | 9 | 238 |\n| 15 | 10 | 177 |\n| 15 | 11 | 161 |\n| 15 | 12 | 158 |\n| 15 | 13 | 164 |\n| 15 | 14 | 177 |\n| 15 | 16 | 176 |\n| 15 | 17 | 162 |\n| 15 | 18 | 179 |\n| 15 | 19 | 179 |\n| 15 | 20 | 188 |\n| 15 | 21 | 141 |\n| 15 | 22 | 205 |\n| 15 | 23 | 160 |\n| 15 | 24 | 148 |\n| 16 | 0 | 170 |\n| 16 | 1 | 167 |\n| 16 | 2 | 243 |\n| 16 | 3 | 200 |\n| 16 | 4 | 204 |\n| 16 | 5 | 188 |\n| 16 | 6 | 234 |\n| 16 | 7 | 150 |\n| 16 | 8 | 172 |\n| 16 | 9 | 193 |\n| 16 | 10 | 222 |\n| 16 | 11 | 182 |\n| 16 | 12 | 178 |\n| 16 | 13 | 166 |\n| 16 | 14 | 177 |\n| 16 | 15 | 176 |\n| 16 | 17 | 180 |\n| 16 | 18 | 182 |\n| 16 | 19 | 208 |\n| 16 | 20 | 223 |\n| 16 | 21 | 198 |\n| 16 | 22 | 174 |\n| 16 | 23 | 199 |\n| 16 | 24 | 187 |\n| 17 | 0 | 192 |\n| 17 | 1 | 169 |\n| 17 | 2 | 224 |\n| 17 | 3 | 182 |\n| 17 | 4 | 173 |\n| 17 | 5 | 196 |\n| 17 | 6 | 198 |\n| 17 | 7 | 174 |\n| 17 | 8 | 196 |\n| 17 | 9 | 166 |\n| 17 | 10 | 204 |\n| 17 | 11 | 177 |\n| 17 | 12 | 196 |\n| 17 | 13 | 200 |\n| 17 | 14 | 184 |\n| 17 | 15 | 162 |\n| 17 | 16 | 180 |\n| 17 | 18 | 210 |\n| 17 | 19 | 197 |\n| 17 | 20 | 203 |\n| 17 | 21 | 208 |\n| 17 | 22 | 172 |\n| 17 | 23 | 238 |\n| 17 | 24 | 179 |\n| 18 | 0 | 192 |\n| 18 | 1 | 196 |\n| 18 | 2 | 221 |\n| 18 | 3 | 200 |\n| 18 | 4 | 171 |\n| 18 | 5 | 195 |\n| 18 | 6 | 183 |\n| 18 | 7 | 166 |\n| 18 | 8 | 182 |\n| 18 | 9 | 223 |\n| 18 | 10 | 194 |\n| 18 | 11 | 181 |\n| 18 | 12 | 181 |\n| 18 | 13 | 174 |\n| 18 | 14 | 184 |\n| 18 | 15 | 179 |\n| 18 | 16 | 182 |\n| 18 | 17 | 210 |\n| 18 | 19 | 177 |\n| 18 | 20 | 224 |\n| 18 | 21 | 236 |\n| 18 | 22 | 212 |\n| 18 | 23 | 187 |\n| 18 | 24 | 190 |\n| 19 | 0 | 146 |\n| 19 | 1 | 180 |\n| 19 | 2 | 135 |\n| 19 | 3 | 179 |\n| 19 | 4 | 176 |\n| 19 | 5 | 132 |\n| 19 | 6 | 179 |\n| 19 | 7 | 158 |\n| 19 | 8 | 174 |\n| 19 | 9 | 216 |\n| 19 | 10 | 166 |\n| 19 | 11 | 214 |\n| 19 | 12 | 230 |\n| 19 | 13 | 170 |\n| 19 | 14 | 229 |\n| 19 | 15 | 179 |\n| 19 | 16 | 208 |\n| 19 | 17 | 197 |\n| 19 | 18 | 177 |\n| 19 | 20 | 182 |\n| 19 | 21 | 212 |\n| 19 | 22 | 207 |\n| 19 | 23 | 164 |\n| 19 | 24 | 151 |\n| 20 | 0 | 186 |\n| 20 | 1 | 223 |\n| 20 | 2 | 181 |\n| 20 | 3 | 190 |\n| 20 | 4 | 195 |\n| 20 | 5 | 187 |\n| 20 | 6 | 180 |\n| 20 | 7 | 204 |\n| 20 | 8 | 215 |\n| 20 | 9 | 251 |\n| 20 | 10 | 178 |\n| 20 | 11 | 229 |\n| 20 | 12 | 187 |\n| 20 | 13 | 207 |\n| 20 | 14 | 204 |\n| 20 | 15 | 188 |\n| 20 | 16 | 223 |\n| 20 | 17 | 203 |\n| 20 | 18 | 224 |\n| 20 | 19 | 182 |\n| 20 | 21 | 156 |\n| 20 | 22 | 169 |\n| 20 | 23 | 195 |\n| 20 | 24 | 166 |\n| 21 | 0 | 194 |\n| 21 | 1 | 197 |\n| 21 | 2 | 213 |\n| 21 | 3 | 224 |\n| 21 | 4 | 211 |\n| 21 | 5 | 224 |\n| 21 | 6 | 220 |\n| 21 | 7 | 165 |\n| 21 | 8 | 201 |\n| 21 | 9 | 272 |\n| 21 | 10 | 196 |\n| 21 | 11 | 201 |\n| 21 | 12 | 185 |\n| 21 | 13 | 170 |\n| 21 | 14 | 223 |\n| 21 | 15 | 141 |\n| 21 | 16 | 198 |\n| 21 | 17 | 208 |\n| 21 | 18 | 236 |\n| 21 | 19 | 212 |\n| 21 | 20 | 156 |\n| 21 | 22 | 222 |\n| 21 | 23 | 214 |\n| 21 | 24 | 191 |\n| 22 | 0 | 201 |\n| 22 | 1 | 215 |\n| 22 | 2 | 218 |\n| 22 | 3 | 181 |\n| 22 | 4 | 229 |\n| 22 | 5 | 183 |\n| 22 | 6 | 180 |\n| 22 | 7 | 198 |\n| 22 | 8 | 210 |\n| 22 | 9 | 216 |\n| 22 | 10 | 220 |\n| 22 | 11 | 215 |\n| 22 | 12 | 170 |\n| 22 | 13 | 226 |\n| 22 | 14 | 195 |\n| 22 | 15 | 205 |\n| 22 | 16 | 174 |\n| 22 | 17 | 172 |\n| 22 | 18 | 212 |\n| 22 | 19 | 207 |\n| 22 | 20 | 169 |\n| 22 | 21 | 222 |\n| 22 | 23 | 175 |\n| 22 | 24 | 196 |\n| 23 | 0 | 194 |\n| 23 | 1 | 217 |\n| 23 | 2 | 212 |\n| 23 | 3 | 184 |\n| 23 | 4 | 217 |\n| 23 | 5 | 178 |\n| 23 | 6 | 172 |\n| 23 | 7 | 207 |\n| 23 | 8 | 209 |\n| 23 | 9 | 264 |\n| 23 | 10 | 184 |\n| 23 | 11 | 204 |\n| 23 | 12 | 165 |\n| 23 | 13 | 215 |\n| 23 | 14 | 207 |\n| 23 | 15 | 160 |\n| 23 | 16 | 199 |\n| 23 | 17 | 238 |\n| 23 | 18 | 187 |\n| 23 | 19 | 164 |\n| 23 | 20 | 195 |\n| 23 | 21 | 214 |\n| 23 | 22 | 175 |\n| 23 | 24 | 189 |\n| 24 | 0 | 144 |\n| 24 | 1 | 196 |\n| 24 | 2 | 173 |\n| 24 | 3 | 178 |\n| 24 | 4 | 191 |\n| 24 | 5 | 158 |\n| 24 | 6 | 185 |\n| 24 | 7 | 146 |\n| 24 | 8 | 195 |\n| 24 | 9 | 198 |\n| 24 | 10 | 146 |\n| 24 | 11 | 179 |\n| 24 | 12 | 181 |\n| 24 | 13 | 177 |\n| 24 | 14 | 168 |\n| 24 | 15 | 148 |\n| 24 | 16 | 187 |\n| 24 | 17 | 179 |\n| 24 | 18 | 190 |\n| 24 | 19 | 151 |\n| 24 | 20 | 166 |\n| 24 | 21 | 191 |\n| 24 | 22 | 196 |\n| 24 | 23 | 189 |\n\nThey will sum every pair's dissimilarity scores to find the most diverse 8-bottle flight.\n\nWhen you're ready to hand over the chosen bottles, just drop them into this tiny JSON shape so it's easy to read and parse:\n\n{\n \"solution\": [, , ...]\n}\n\nThink of \"solution\" as the list of picks for the flight, and each is the identifier for one bottle from the list — just the ID, nothing fancy. This is only a sketch of the format I expect, not the actual answer itself.\n\nPlease use the exact identifiers from the instance input — do not rename them or invent new labels. \nValid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.", "instance": { "distance_matrix": [ [ 0, 175, 185, 172, 170, 149, 221, 141, 156, 178, 180, 185, 189, 170, 210, 173, 170, 192, 192, 146, 186, 194, 201, 194, 144 ], [ 175, 0, 214, 185, 166, 179, 229, 148, 180, 185, 213, 179, 194, 184, 215, 185, 167, 169, 196, 180, 223, 197, 215, 217, 196 ], [ 185, 214, 0, 192, 205, 134, 151, 157, 223, 223, 165, 234, 236, 187, 233, 197, 243, 224, 221, 135, 181, 213, 218, 212, 173 ], [ 172, 185, 192, 0, 191, 158, 169, 182, 235, 191, 160, 196, 140, 189, 167, 198, 200, 182, 200, 179, 190, 224, 181, 184, 178 ], [ 170, 166, 205, 191, 0, 193, 201, 194, 167, 181, 153, 164, 190, 193, 192, 187, 204, 173, 171, 176, 195, 211, 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"MDP", "problem_type": "MDP", "instruction": "My team has to fill a set number of instructor slots for an upcoming series, and the point is to make the schedule feel rich and diverse. That means choosing exactly the requested number of teachers (no duplicates, no extra slots) so that when you look at how different each pair is and add up all those pairwise differences, the sum is as big as possible — that sum is the simple way we measure variety. The concrete candidates and their pairwise difference table appear below.\n\n# num_instructor_candidates=23\n# slots_to_fill=2\n# instructor_ids=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22\ninstructor_a_id,instructor_b_id,topic_style_difference\n0,1,9\n0,2,3\n0,3,8\n0,4,0\n0,5,8\n0,6,7\n0,7,5\n0,8,4\n0,9,2\n0,10,7\n0,11,0\n0,12,9\n0,13,9\n0,14,1\n0,15,1\n0,16,8\n0,17,2\n0,18,9\n0,19,5\n0,20,4\n0,21,2\n0,22,6\n1,0,9\n1,2,8\n1,3,3\n1,4,3\n1,5,3\n1,6,4\n1,7,0\n1,8,7\n1,9,4\n1,10,7\n1,11,3\n1,12,9\n1,13,7\n1,14,2\n1,15,0\n1,16,1\n1,17,3\n1,18,4\n1,19,4\n1,20,9\n1,21,0\n1,22,4\n2,0,3\n2,1,8\n2,3,3\n2,4,6\n2,5,4\n2,6,9\n2,7,9\n2,8,2\n2,9,9\n2,10,8\n2,11,9\n2,12,9\n2,13,0\n2,14,1\n2,15,9\n2,16,0\n2,17,2\n2,18,8\n2,19,0\n2,20,5\n2,21,1\n2,22,9\n3,0,8\n3,1,3\n3,2,3\n3,4,0\n3,5,1\n3,6,4\n3,7,1\n3,8,0\n3,9,1\n3,10,1\n3,11,0\n3,12,3\n3,13,3\n3,14,6\n3,15,1\n3,16,3\n3,17,6\n3,18,1\n3,19,8\n3,20,0\n3,21,4\n3,22,5\n4,0,0\n4,1,3\n4,2,6\n4,3,0\n4,5,1\n4,6,1\n4,7,3\n4,8,5\n4,9,6\n4,10,9\n4,11,5\n4,12,7\n4,13,8\n4,14,3\n4,15,8\n4,16,1\n4,17,2\n4,18,1\n4,19,5\n4,20,0\n4,21,4\n4,22,4\n5,0,8\n5,1,3\n5,2,4\n5,3,1\n5,4,1\n5,6,3\n5,7,9\n5,8,1\n5,9,3\n5,10,1\n5,11,1\n5,12,1\n5,13,7\n5,14,7\n5,15,3\n5,16,8\n5,17,0\n5,18,2\n5,19,6\n5,20,1\n5,21,2\n5,22,5\n6,0,7\n6,1,4\n6,2,9\n6,3,4\n6,4,1\n6,5,3\n6,7,2\n6,8,4\n6,9,9\n6,10,0\n6,11,2\n6,12,4\n6,13,7\n6,14,7\n6,15,6\n6,16,9\n6,17,1\n6,18,5\n6,19,5\n6,20,0\n6,21,0\n6,22,2\n7,0,5\n7,1,0\n7,2,9\n7,3,1\n7,4,3\n7,5,9\n7,6,2\n7,8,4\n7,9,1\n7,10,8\n7,11,4\n7,12,5\n7,13,7\n7,14,0\n7,15,2\n7,16,3\n7,17,2\n7,18,0\n7,19,9\n7,20,9\n7,21,4\n7,22,1\n8,0,4\n8,1,7\n8,2,2\n8,3,0\n8,4,5\n8,5,1\n8,6,4\n8,7,4\n8,9,6\n8,10,1\n8,11,7\n8,12,7\n8,13,4\n8,14,3\n8,15,8\n8,16,3\n8,17,2\n8,18,4\n8,19,0\n8,20,3\n8,21,2\n8,22,1\n9,0,2\n9,1,4\n9,2,9\n9,3,1\n9,4,6\n9,5,3\n9,6,9\n9,7,1\n9,8,6\n9,10,1\n9,11,1\n9,12,5\n9,13,2\n9,14,3\n9,15,8\n9,16,5\n9,17,2\n9,18,1\n9,19,1\n9,20,6\n9,21,3\n9,22,7\n10,0,7\n10,1,7\n10,2,8\n10,3,1\n10,4,9\n10,5,1\n10,6,0\n10,7,8\n10,8,1\n10,9,1\n10,11,3\n10,12,4\n10,13,6\n10,14,1\n10,15,6\n10,16,5\n10,17,5\n10,18,9\n10,19,5\n10,20,2\n10,21,7\n10,22,2\n11,0,0\n11,1,3\n11,2,9\n11,3,0\n11,4,5\n11,5,1\n11,6,2\n11,7,4\n11,8,7\n11,9,1\n11,10,3\n11,12,3\n11,13,5\n11,14,7\n11,15,7\n11,16,3\n11,17,2\n11,18,3\n11,19,7\n11,20,1\n11,21,5\n11,22,4\n12,0,9\n12,1,9\n12,2,9\n12,3,3\n12,4,7\n12,5,1\n12,6,4\n12,7,5\n12,8,7\n12,9,5\n12,10,4\n12,11,3\n12,13,6\n12,14,2\n12,15,4\n12,16,8\n12,17,5\n12,18,9\n12,19,2\n12,20,7\n12,21,8\n12,22,5\n13,0,9\n13,1,7\n13,2,0\n13,3,3\n13,4,8\n13,5,7\n13,6,7\n13,7,7\n13,8,4\n13,9,2\n13,10,6\n13,11,5\n13,12,6\n13,14,0\n13,15,0\n13,16,8\n13,17,3\n13,18,2\n13,19,2\n13,20,5\n13,21,6\n13,22,5\n14,0,1\n14,1,2\n14,2,1\n14,3,6\n14,4,3\n14,5,7\n14,6,7\n14,7,0\n14,8,3\n14,9,3\n14,10,1\n14,11,7\n14,12,2\n14,13,0\n14,15,8\n14,16,1\n14,17,3\n14,18,8\n14,19,9\n14,20,2\n14,21,6\n14,22,4\n15,0,1\n15,1,0\n15,2,9\n15,3,1\n15,4,8\n15,5,3\n15,6,6\n15,7,2\n15,8,8\n15,9,8\n15,10,6\n15,11,7\n15,12,4\n15,13,0\n15,14,8\n15,16,8\n15,17,9\n15,18,2\n15,19,7\n15,20,6\n15,21,1\n15,22,0\n16,0,8\n16,1,1\n16,2,0\n16,3,3\n16,4,1\n16,5,8\n16,6,9\n16,7,3\n16,8,3\n16,9,5\n16,10,5\n16,11,3\n16,12,8\n16,13,8\n16,14,1\n16,15,8\n16,17,5\n16,18,0\n16,19,7\n16,20,7\n16,21,6\n16,22,8\n17,0,2\n17,1,3\n17,2,2\n17,3,6\n17,4,2\n17,5,0\n17,6,1\n17,7,2\n17,8,2\n17,9,2\n17,10,5\n17,11,2\n17,12,5\n17,13,3\n17,14,3\n17,15,9\n17,16,5\n17,18,7\n17,19,4\n17,20,6\n17,21,7\n17,22,3\n18,0,9\n18,1,4\n18,2,8\n18,3,1\n18,4,1\n18,5,2\n18,6,5\n18,7,0\n18,8,4\n18,9,1\n18,10,9\n18,11,3\n18,12,9\n18,13,2\n18,14,8\n18,15,2\n18,16,0\n18,17,7\n18,19,1\n18,20,3\n18,21,9\n18,22,1\n19,0,5\n19,1,4\n19,2,0\n19,3,8\n19,4,5\n19,5,6\n19,6,5\n19,7,9\n19,8,0\n19,9,1\n19,10,5\n19,11,7\n19,12,2\n19,13,2\n19,14,9\n19,15,7\n19,16,7\n19,17,4\n19,18,1\n19,20,7\n19,21,4\n19,22,8\n20,0,4\n20,1,9\n20,2,5\n20,3,0\n20,4,0\n20,5,1\n20,6,0\n20,7,9\n20,8,3\n20,9,6\n20,10,2\n20,11,1\n20,12,7\n20,13,5\n20,14,2\n20,15,6\n20,16,7\n20,17,6\n20,18,3\n20,19,7\n20,21,0\n20,22,5\n21,0,2\n21,1,0\n21,2,1\n21,3,4\n21,4,4\n21,5,2\n21,6,0\n21,7,4\n21,8,2\n21,9,3\n21,10,7\n21,11,5\n21,12,8\n21,13,6\n21,14,6\n21,15,1\n21,16,6\n21,17,7\n21,18,9\n21,19,4\n21,20,0\n21,22,3\n22,0,6\n22,1,4\n22,2,9\n22,3,5\n22,4,4\n22,5,5\n22,6,2\n22,7,1\n22,8,1\n22,9,7\n22,10,2\n22,11,4\n22,12,5\n22,13,5\n22,14,4\n22,15,0\n22,16,8\n22,17,3\n22,18,1\n22,19,8\n22,20,5\n22,21,3\n\nWhen you send back the actual picks, please put them in a tiny JSON object so our little schedule parser can read it easily — something like this:\n\n{\n \"solution\": [, , ...]\n}\n\nThat \"solution\" array should just list the exact IDs of the instructors you choose (one entry per chosen slot). Think of this snippet as a blank form showing the shape we need — it's not the real answer yet, just the template.\n\nAnd please use the identifiers exactly as they appear in the instance input — no renaming, no shorthand, and don't invent new labels.\n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 9, 3, 8, 0, 8, 7, 5, 4, 2, 7, 0, 9, 9, 1, 1, 8, 2, 9, 5, 4, 2, 6 ], [ 9, 0, 8, 3, 3, 3, 4, 0, 7, 4, 7, 3, 9, 7, 2, 0, 1, 3, 4, 4, 9, 0, 4 ], [ 3, 8, 0, 3, 6, 4, 9, 9, 2, 9, 8, 9, 9, 0, 1, 9, 0, 2, 8, 0, 5, 1, 9 ], [ 8, 3, 3, 0, 0, 1, 4, 1, 0, 1, 1, 0, 3, 3, 6, 1, 3, 6, 1, 8, 0, 4, 5 ], [ 0, 3, 6, 0, 0, 1, 1, 3, 5, 6, 9, 5, 7, 8, 3, 8, 1, 2, 1, 5, 0, 4, 4 ], [ 8, 3, 4, 1, 1, 0, 3, 9, 1, 3, 1, 1, 1, 7, 7, 3, 8, 0, 2, 6, 1, 2, 5 ], [ 7, 4, 9, 4, 1, 3, 0, 2, 4, 9, 0, 2, 4, 7, 7, 6, 9, 1, 5, 5, 0, 0, 2 ], [ 5, 0, 9, 1, 3, 9, 2, 0, 4, 1, 8, 4, 5, 7, 0, 2, 3, 2, 0, 9, 9, 4, 1 ], [ 4, 7, 2, 0, 5, 1, 4, 4, 0, 6, 1, 7, 7, 4, 3, 8, 3, 2, 4, 0, 3, 2, 1 ], [ 2, 4, 9, 1, 6, 3, 9, 1, 6, 0, 1, 1, 5, 2, 3, 8, 5, 2, 1, 1, 6, 3, 7 ], [ 7, 7, 8, 1, 9, 1, 0, 8, 1, 1, 0, 3, 4, 6, 1, 6, 5, 5, 9, 5, 2, 7, 2 ], [ 0, 3, 9, 0, 5, 1, 2, 4, 7, 1, 3, 0, 3, 5, 7, 7, 3, 2, 3, 7, 1, 5, 4 ], [ 9, 9, 9, 3, 7, 1, 4, 5, 7, 5, 4, 3, 0, 6, 2, 4, 8, 5, 9, 2, 7, 8, 5 ], [ 9, 7, 0, 3, 8, 7, 7, 7, 4, 2, 6, 5, 6, 0, 0, 0, 8, 3, 2, 2, 5, 6, 5 ], [ 1, 2, 1, 6, 3, 7, 7, 0, 3, 3, 1, 7, 2, 0, 0, 8, 1, 3, 8, 9, 2, 6, 4 ], [ 1, 0, 9, 1, 8, 3, 6, 2, 8, 8, 6, 7, 4, 0, 8, 0, 8, 9, 2, 7, 6, 1, 0 ], [ 8, 1, 0, 3, 1, 8, 9, 3, 3, 5, 5, 3, 8, 8, 1, 8, 0, 5, 0, 7, 7, 6, 8 ], [ 2, 3, 2, 6, 2, 0, 1, 2, 2, 2, 5, 2, 5, 3, 3, 9, 5, 0, 7, 4, 6, 7, 3 ], [ 9, 4, 8, 1, 1, 2, 5, 0, 4, 1, 9, 3, 9, 2, 8, 2, 0, 7, 0, 1, 3, 9, 1 ], [ 5, 4, 0, 8, 5, 6, 5, 9, 0, 1, 5, 7, 2, 2, 9, 7, 7, 4, 1, 0, 7, 4, 8 ], [ 4, 9, 5, 0, 0, 1, 0, 9, 3, 6, 2, 1, 7, 5, 2, 6, 7, 6, 3, 7, 0, 0, 5 ], [ 2, 0, 1, 4, 4, 2, 0, 4, 2, 3, 7, 5, 8, 6, 6, 1, 6, 7, 9, 4, 0, 0, 3 ], [ 6, 4, 9, 5, 4, 5, 2, 1, 1, 7, 2, 4, 5, 5, 4, 0, 8, 3, 1, 8, 5, 3, 0 ] ], "m": 2, "objective": 9.0 }, "solution": [ 18, 21 ], "obj": 9.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 23, "m": 2, "nodes": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 ], "distance_pairs": [ { "from_id": 0, "to_id": 1, "distance": 9 }, { "from_id": 0, "to_id": 2, "distance": 3 }, { "from_id": 0, "to_id": 3, "distance": 8 }, { "from_id": 0, "to_id": 4, "distance": 0 }, { "from_id": 0, "to_id": 5, "distance": 8 }, { "from_id": 0, "to_id": 6, "distance": 7 }, { "from_id": 0, "to_id": 7, "distance": 5 }, { "from_id": 0, "to_id": 8, "distance": 4 }, { "from_id": 0, "to_id": 9, "distance": 2 }, { "from_id": 0, "to_id": 10, "distance": 7 }, { "from_id": 0, "to_id": 11, "distance": 0 }, { "from_id": 0, "to_id": 12, "distance": 9 }, { "from_id": 0, "to_id": 13, "distance": 9 }, { "from_id": 0, "to_id": 14, "distance": 1 }, { "from_id": 0, "to_id": 15, "distance": 1 }, { "from_id": 0, "to_id": 16, "distance": 8 }, { "from_id": 0, "to_id": 17, "distance": 2 }, { "from_id": 0, "to_id": 18, "distance": 9 }, { "from_id": 0, "to_id": 19, "distance": 5 }, { "from_id": 0, "to_id": 20, "distance": 4 }, { "from_id": 0, "to_id": 21, "distance": 2 }, { "from_id": 0, "to_id": 22, "distance": 6 }, { "from_id": 1, "to_id": 0, "distance": 9 }, { "from_id": 1, "to_id": 2, "distance": 8 }, { "from_id": 1, "to_id": 3, "distance": 3 }, { "from_id": 1, "to_id": 4, "distance": 3 }, { "from_id": 1, "to_id": 5, "distance": 3 }, { "from_id": 1, "to_id": 6, "distance": 4 }, { "from_id": 1, "to_id": 7, "distance": 0 }, { "from_id": 1, "to_id": 8, "distance": 7 }, { "from_id": 1, "to_id": 9, "distance": 4 }, { "from_id": 1, "to_id": 10, "distance": 7 }, { "from_id": 1, "to_id": 11, "distance": 3 }, { "from_id": 1, "to_id": 12, "distance": 9 }, { "from_id": 1, "to_id": 13, "distance": 7 }, { "from_id": 1, "to_id": 14, "distance": 2 }, { "from_id": 1, "to_id": 15, "distance": 0 }, { "from_id": 1, "to_id": 16, "distance": 1 }, { "from_id": 1, "to_id": 17, "distance": 3 }, { "from_id": 1, "to_id": 18, "distance": 4 }, { "from_id": 1, "to_id": 19, "distance": 4 }, { "from_id": 1, "to_id": 20, "distance": 9 }, { "from_id": 1, "to_id": 21, "distance": 0 }, { "from_id": 1, "to_id": 22, "distance": 4 }, { "from_id": 2, "to_id": 0, "distance": 3 }, { "from_id": 2, "to_id": 1, "distance": 8 }, { "from_id": 2, "to_id": 3, "distance": 3 }, { "from_id": 2, "to_id": 4, "distance": 6 }, { "from_id": 2, "to_id": 5, "distance": 4 }, { "from_id": 2, "to_id": 6, "distance": 9 }, { "from_id": 2, "to_id": 7, "distance": 9 }, { "from_id": 2, "to_id": 8, "distance": 2 }, { "from_id": 2, "to_id": 9, "distance": 9 }, { "from_id": 2, "to_id": 10, "distance": 8 }, { "from_id": 2, "to_id": 11, "distance": 9 }, { "from_id": 2, "to_id": 12, "distance": 9 }, { "from_id": 2, "to_id": 13, "distance": 0 }, { "from_id": 2, "to_id": 14, "distance": 1 }, { "from_id": 2, "to_id": 15, "distance": 9 }, { "from_id": 2, "to_id": 16, "distance": 0 }, { "from_id": 2, "to_id": 17, "distance": 2 }, { "from_id": 2, "to_id": 18, "distance": 8 }, { "from_id": 2, "to_id": 19, "distance": 0 }, { "from_id": 2, "to_id": 20, "distance": 5 }, { "from_id": 2, "to_id": 21, "distance": 1 }, { "from_id": 2, "to_id": 22, "distance": 9 }, { "from_id": 3, "to_id": 0, "distance": 8 }, { "from_id": 3, "to_id": 1, "distance": 3 }, { "from_id": 3, "to_id": 2, "distance": 3 }, { "from_id": 3, "to_id": 4, "distance": 0 }, { "from_id": 3, "to_id": 5, "distance": 1 }, { "from_id": 3, "to_id": 6, "distance": 4 }, { "from_id": 3, "to_id": 7, "distance": 1 }, { "from_id": 3, "to_id": 8, "distance": 0 }, { "from_id": 3, "to_id": 9, "distance": 1 }, { "from_id": 3, "to_id": 10, "distance": 1 }, { "from_id": 3, "to_id": 11, "distance": 0 }, { "from_id": 3, "to_id": 12, "distance": 3 }, { "from_id": 3, "to_id": 13, "distance": 3 }, { "from_id": 3, "to_id": 14, "distance": 6 }, { "from_id": 3, "to_id": 15, "distance": 1 }, { "from_id": 3, "to_id": 16, "distance": 3 }, { "from_id": 3, "to_id": 17, "distance": 6 }, { "from_id": 3, "to_id": 18, "distance": 1 }, { "from_id": 3, "to_id": 19, "distance": 8 }, { "from_id": 3, "to_id": 20, "distance": 0 }, { "from_id": 3, "to_id": 21, "distance": 4 }, { "from_id": 3, "to_id": 22, "distance": 5 }, { "from_id": 4, "to_id": 0, "distance": 0 }, { "from_id": 4, "to_id": 1, "distance": 3 }, { "from_id": 4, "to_id": 2, "distance": 6 }, { "from_id": 4, "to_id": 3, "distance": 0 }, { "from_id": 4, "to_id": 5, "distance": 1 }, { "from_id": 4, "to_id": 6, "distance": 1 }, { "from_id": 4, "to_id": 7, "distance": 3 }, { "from_id": 4, "to_id": 8, "distance": 5 }, { "from_id": 4, "to_id": 9, "distance": 6 }, { "from_id": 4, "to_id": 10, "distance": 9 }, { "from_id": 4, "to_id": 11, "distance": 5 }, { "from_id": 4, "to_id": 12, "distance": 7 }, { "from_id": 4, "to_id": 13, 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"from_id": 22, "to_id": 4, "distance": 4 }, { "from_id": 22, "to_id": 5, "distance": 5 }, { "from_id": 22, "to_id": 6, "distance": 2 }, { "from_id": 22, "to_id": 7, "distance": 1 }, { "from_id": 22, "to_id": 8, "distance": 1 }, { "from_id": 22, "to_id": 9, "distance": 7 }, { "from_id": 22, "to_id": 10, "distance": 2 }, { "from_id": 22, "to_id": 11, "distance": 4 }, { "from_id": 22, "to_id": 12, "distance": 5 }, { "from_id": 22, "to_id": 13, "distance": 5 }, { "from_id": 22, "to_id": 14, "distance": 4 }, { "from_id": 22, "to_id": 15, "distance": 0 }, { "from_id": 22, "to_id": 16, "distance": 8 }, { "from_id": 22, "to_id": 17, "distance": 3 }, { "from_id": 22, "to_id": 18, "distance": 1 }, { "from_id": 22, "to_id": 19, "distance": 8 }, { "from_id": 22, "to_id": 20, "distance": 5 }, { "from_id": 22, "to_id": 21, "distance": 3 } ] }, "solution_variant": [ 18, 21 ], "context_index": 45, "input_format": "csv", "input_index_base": 0 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Imagine being in charge of choosing exactly m parks to renovate and trying to scatter them so they don’t all sit next to each other. For any choice, check every pair of parks in it, note how far apart they are, sum up those distances, and the choice with the biggest sum is the one that’s most spread out. The plan calls for exactly m unique parks — can’t pick the same park twice — and the concrete details are listed below.\n\nThere are 22 parks available, identified as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22; you must select exactly 7 distinct parks to renovate.\n\n| park_id_a | park_id_b | distance_between_parks |\n|---|---|---|\n| 1 | 2 | 7 |\n| 1 | 3 | 4 |\n| 1 | 4 | 7 |\n| 1 | 5 | 1 |\n| 1 | 6 | 1 |\n| 1 | 7 | 6 |\n| 1 | 8 | 3 |\n| 1 | 9 | 1 |\n| 1 | 10 | 4 |\n| 1 | 11 | 8 |\n| 1 | 12 | 9 |\n| 1 | 13 | 9 |\n| 1 | 14 | 7 |\n| 1 | 15 | 5 |\n| 1 | 16 | 9 |\n| 1 | 17 | 9 |\n| 1 | 18 | 2 |\n| 1 | 19 | 2 |\n| 1 | 20 | 7 |\n| 1 | 21 | 7 |\n| 1 | 22 | 7 |\n| 2 | 1 | 7 |\n| 2 | 3 | 3 |\n| 2 | 4 | 8 |\n| 2 | 5 | 8 |\n| 2 | 6 | 6 |\n| 2 | 7 | 2 |\n| 2 | 8 | 3 |\n| 2 | 9 | 0 |\n| 2 | 10 | 7 |\n| 2 | 11 | 4 |\n| 2 | 12 | 9 |\n| 2 | 13 | 5 |\n| 2 | 14 | 8 |\n| 2 | 15 | 3 |\n| 2 | 16 | 3 |\n| 2 | 17 | 5 |\n| 2 | 18 | 7 |\n| 2 | 19 | 7 |\n| 2 | 20 | 1 |\n| 2 | 21 | 6 |\n| 2 | 22 | 4 |\n| 3 | 1 | 4 |\n| 3 | 2 | 3 |\n| 3 | 4 | 2 |\n| 3 | 5 | 3 |\n| 3 | 6 | 3 |\n| 3 | 7 | 0 |\n| 3 | 8 | 7 |\n| 3 | 9 | 2 |\n| 3 | 10 | 9 |\n| 3 | 11 | 2 |\n| 3 | 12 | 2 |\n| 3 | 13 | 6 |\n| 3 | 14 | 5 |\n| 3 | 15 | 7 |\n| 3 | 16 | 7 |\n| 3 | 17 | 5 |\n| 3 | 18 | 9 |\n| 3 | 19 | 5 |\n| 3 | 20 | 7 |\n| 3 | 21 | 9 |\n| 3 | 22 | 2 |\n| 4 | 1 | 7 |\n| 4 | 2 | 8 |\n| 4 | 3 | 2 |\n| 4 | 5 | 5 |\n| 4 | 6 | 0 |\n| 4 | 7 | 8 |\n| 4 | 8 | 9 |\n| 4 | 9 | 8 |\n| 4 | 10 | 2 |\n| 4 | 11 | 7 |\n| 4 | 12 | 9 |\n| 4 | 13 | 4 |\n| 4 | 14 | 6 |\n| 4 | 15 | 6 |\n| 4 | 16 | 7 |\n| 4 | 17 | 1 |\n| 4 | 18 | 8 |\n| 4 | 19 | 6 |\n| 4 | 20 | 1 |\n| 4 | 21 | 1 |\n| 4 | 22 | 3 |\n| 5 | 1 | 1 |\n| 5 | 2 | 8 |\n| 5 | 3 | 3 |\n| 5 | 4 | 5 |\n| 5 | 6 | 4 |\n| 5 | 7 | 4 |\n| 5 | 8 | 7 |\n| 5 | 9 | 5 |\n| 5 | 10 | 5 |\n| 5 | 11 | 3 |\n| 5 | 12 | 0 |\n| 5 | 13 | 3 |\n| 5 | 14 | 6 |\n| 5 | 15 | 9 |\n| 5 | 16 | 4 |\n| 5 | 17 | 7 |\n| 5 | 18 | 8 |\n| 5 | 19 | 3 |\n| 5 | 20 | 8 |\n| 5 | 21 | 8 |\n| 5 | 22 | 3 |\n| 6 | 1 | 1 |\n| 6 | 2 | 6 |\n| 6 | 3 | 3 |\n| 6 | 4 | 0 |\n| 6 | 5 | 4 |\n| 6 | 7 | 7 |\n| 6 | 8 | 2 |\n| 6 | 9 | 5 |\n| 6 | 10 | 0 |\n| 6 | 11 | 0 |\n| 6 | 12 | 3 |\n| 6 | 13 | 6 |\n| 6 | 14 | 2 |\n| 6 | 15 | 1 |\n| 6 | 16 | 5 |\n| 6 | 17 | 4 |\n| 6 | 18 | 5 |\n| 6 | 19 | 3 |\n| 6 | 20 | 7 |\n| 6 | 21 | 9 |\n| 6 | 22 | 3 |\n| 7 | 1 | 6 |\n| 7 | 2 | 2 |\n| 7 | 3 | 0 |\n| 7 | 4 | 8 |\n| 7 | 5 | 4 |\n| 7 | 6 | 7 |\n| 7 | 8 | 9 |\n| 7 | 9 | 5 |\n| 7 | 10 | 4 |\n| 7 | 11 | 6 |\n| 7 | 12 | 3 |\n| 7 | 13 | 4 |\n| 7 | 14 | 3 |\n| 7 | 15 | 0 |\n| 7 | 16 | 0 |\n| 7 | 17 | 9 |\n| 7 | 18 | 0 |\n| 7 | 19 | 6 |\n| 7 | 20 | 1 |\n| 7 | 21 | 9 |\n| 7 | 22 | 8 |\n| 8 | 1 | 3 |\n| 8 | 2 | 3 |\n| 8 | 3 | 7 |\n| 8 | 4 | 9 |\n| 8 | 5 | 7 |\n| 8 | 6 | 2 |\n| 8 | 7 | 9 |\n| 8 | 9 | 8 |\n| 8 | 10 | 4 |\n| 8 | 11 | 8 |\n| 8 | 12 | 7 |\n| 8 | 13 | 8 |\n| 8 | 14 | 8 |\n| 8 | 15 | 0 |\n| 8 | 16 | 2 |\n| 8 | 17 | 5 |\n| 8 | 18 | 7 |\n| 8 | 19 | 2 |\n| 8 | 20 | 2 |\n| 8 | 21 | 1 |\n| 8 | 22 | 7 |\n| 9 | 1 | 1 |\n| 9 | 2 | 0 |\n| 9 | 3 | 2 |\n| 9 | 4 | 8 |\n| 9 | 5 | 5 |\n| 9 | 6 | 5 |\n| 9 | 7 | 5 |\n| 9 | 8 | 8 |\n| 9 | 10 | 7 |\n| 9 | 11 | 5 |\n| 9 | 12 | 5 |\n| 9 | 13 | 2 |\n| 9 | 14 | 7 |\n| 9 | 15 | 2 |\n| 9 | 16 | 2 |\n| 9 | 17 | 9 |\n| 9 | 18 | 1 |\n| 9 | 19 | 4 |\n| 9 | 20 | 6 |\n| 9 | 21 | 5 |\n| 9 | 22 | 8 |\n| 10 | 1 | 4 |\n| 10 | 2 | 7 |\n| 10 | 3 | 9 |\n| 10 | 4 | 2 |\n| 10 | 5 | 5 |\n| 10 | 6 | 0 |\n| 10 | 7 | 4 |\n| 10 | 8 | 4 |\n| 10 | 9 | 7 |\n| 10 | 11 | 4 |\n| 10 | 12 | 6 |\n| 10 | 13 | 5 |\n| 10 | 14 | 8 |\n| 10 | 15 | 6 |\n| 10 | 16 | 1 |\n| 10 | 17 | 1 |\n| 10 | 18 | 2 |\n| 10 | 19 | 4 |\n| 10 | 20 | 9 |\n| 10 | 21 | 7 |\n| 10 | 22 | 9 |\n| 11 | 1 | 8 |\n| 11 | 2 | 4 |\n| 11 | 3 | 2 |\n| 11 | 4 | 7 |\n| 11 | 5 | 3 |\n| 11 | 6 | 0 |\n| 11 | 7 | 6 |\n| 11 | 8 | 8 |\n| 11 | 9 | 5 |\n| 11 | 10 | 4 |\n| 11 | 12 | 8 |\n| 11 | 13 | 9 |\n| 11 | 14 | 9 |\n| 11 | 15 | 5 |\n| 11 | 16 | 3 |\n| 11 | 17 | 1 |\n| 11 | 18 | 7 |\n| 11 | 19 | 4 |\n| 11 | 20 | 6 |\n| 11 | 21 | 7 |\n| 11 | 22 | 1 |\n| 12 | 1 | 9 |\n| 12 | 2 | 9 |\n| 12 | 3 | 2 |\n| 12 | 4 | 9 |\n| 12 | 5 | 0 |\n| 12 | 6 | 3 |\n| 12 | 7 | 3 |\n| 12 | 8 | 7 |\n| 12 | 9 | 5 |\n| 12 | 10 | 6 |\n| 12 | 11 | 8 |\n| 12 | 13 | 7 |\n| 12 | 14 | 6 |\n| 12 | 15 | 0 |\n| 12 | 16 | 6 |\n| 12 | 17 | 1 |\n| 12 | 18 | 8 |\n| 12 | 19 | 5 |\n| 12 | 20 | 3 |\n| 12 | 21 | 3 |\n| 12 | 22 | 1 |\n| 13 | 1 | 9 |\n| 13 | 2 | 5 |\n| 13 | 3 | 6 |\n| 13 | 4 | 4 |\n| 13 | 5 | 3 |\n| 13 | 6 | 6 |\n| 13 | 7 | 4 |\n| 13 | 8 | 8 |\n| 13 | 9 | 2 |\n| 13 | 10 | 5 |\n| 13 | 11 | 9 |\n| 13 | 12 | 7 |\n| 13 | 14 | 8 |\n| 13 | 15 | 4 |\n| 13 | 16 | 9 |\n| 13 | 17 | 0 |\n| 13 | 18 | 0 |\n| 13 | 19 | 7 |\n| 13 | 20 | 0 |\n| 13 | 21 | 4 |\n| 13 | 22 | 6 |\n| 14 | 1 | 7 |\n| 14 | 2 | 8 |\n| 14 | 3 | 5 |\n| 14 | 4 | 6 |\n| 14 | 5 | 6 |\n| 14 | 6 | 2 |\n| 14 | 7 | 3 |\n| 14 | 8 | 8 |\n| 14 | 9 | 7 |\n| 14 | 10 | 8 |\n| 14 | 11 | 9 |\n| 14 | 12 | 6 |\n| 14 | 13 | 8 |\n| 14 | 15 | 6 |\n| 14 | 16 | 0 |\n| 14 | 17 | 0 |\n| 14 | 18 | 3 |\n| 14 | 19 | 2 |\n| 14 | 20 | 0 |\n| 14 | 21 | 8 |\n| 14 | 22 | 4 |\n| 15 | 1 | 5 |\n| 15 | 2 | 3 |\n| 15 | 3 | 7 |\n| 15 | 4 | 6 |\n| 15 | 5 | 9 |\n| 15 | 6 | 1 |\n| 15 | 7 | 0 |\n| 15 | 8 | 0 |\n| 15 | 9 | 2 |\n| 15 | 10 | 6 |\n| 15 | 11 | 5 |\n| 15 | 12 | 0 |\n| 15 | 13 | 4 |\n| 15 | 14 | 6 |\n| 15 | 16 | 4 |\n| 15 | 17 | 6 |\n| 15 | 18 | 1 |\n| 15 | 19 | 2 |\n| 15 | 20 | 4 |\n| 15 | 21 | 2 |\n| 15 | 22 | 0 |\n| 16 | 1 | 9 |\n| 16 | 2 | 3 |\n| 16 | 3 | 7 |\n| 16 | 4 | 7 |\n| 16 | 5 | 4 |\n| 16 | 6 | 5 |\n| 16 | 7 | 0 |\n| 16 | 8 | 2 |\n| 16 | 9 | 2 |\n| 16 | 10 | 1 |\n| 16 | 11 | 3 |\n| 16 | 12 | 6 |\n| 16 | 13 | 9 |\n| 16 | 14 | 0 |\n| 16 | 15 | 4 |\n| 16 | 17 | 4 |\n| 16 | 18 | 6 |\n| 16 | 19 | 7 |\n| 16 | 20 | 0 |\n| 16 | 21 | 3 |\n| 16 | 22 | 6 |\n| 17 | 1 | 9 |\n| 17 | 2 | 5 |\n| 17 | 3 | 5 |\n| 17 | 4 | 1 |\n| 17 | 5 | 7 |\n| 17 | 6 | 4 |\n| 17 | 7 | 9 |\n| 17 | 8 | 5 |\n| 17 | 9 | 9 |\n| 17 | 10 | 1 |\n| 17 | 11 | 1 |\n| 17 | 12 | 1 |\n| 17 | 13 | 0 |\n| 17 | 14 | 0 |\n| 17 | 15 | 6 |\n| 17 | 16 | 4 |\n| 17 | 18 | 4 |\n| 17 | 19 | 5 |\n| 17 | 20 | 7 |\n| 17 | 21 | 1 |\n| 17 | 22 | 3 |\n| 18 | 1 | 2 |\n| 18 | 2 | 7 |\n| 18 | 3 | 9 |\n| 18 | 4 | 8 |\n| 18 | 5 | 8 |\n| 18 | 6 | 5 |\n| 18 | 7 | 0 |\n| 18 | 8 | 7 |\n| 18 | 9 | 1 |\n| 18 | 10 | 2 |\n| 18 | 11 | 7 |\n| 18 | 12 | 8 |\n| 18 | 13 | 0 |\n| 18 | 14 | 3 |\n| 18 | 15 | 1 |\n| 18 | 16 | 6 |\n| 18 | 17 | 4 |\n| 18 | 19 | 1 |\n| 18 | 20 | 9 |\n| 18 | 21 | 5 |\n| 18 | 22 | 2 |\n| 19 | 1 | 2 |\n| 19 | 2 | 7 |\n| 19 | 3 | 5 |\n| 19 | 4 | 6 |\n| 19 | 5 | 3 |\n| 19 | 6 | 3 |\n| 19 | 7 | 6 |\n| 19 | 8 | 2 |\n| 19 | 9 | 4 |\n| 19 | 10 | 4 |\n| 19 | 11 | 4 |\n| 19 | 12 | 5 |\n| 19 | 13 | 7 |\n| 19 | 14 | 2 |\n| 19 | 15 | 2 |\n| 19 | 16 | 7 |\n| 19 | 17 | 5 |\n| 19 | 18 | 1 |\n| 19 | 20 | 0 |\n| 19 | 21 | 0 |\n| 19 | 22 | 6 |\n| 20 | 1 | 7 |\n| 20 | 2 | 1 |\n| 20 | 3 | 7 |\n| 20 | 4 | 1 |\n| 20 | 5 | 8 |\n| 20 | 6 | 7 |\n| 20 | 7 | 1 |\n| 20 | 8 | 2 |\n| 20 | 9 | 6 |\n| 20 | 10 | 9 |\n| 20 | 11 | 6 |\n| 20 | 12 | 3 |\n| 20 | 13 | 0 |\n| 20 | 14 | 0 |\n| 20 | 15 | 4 |\n| 20 | 16 | 0 |\n| 20 | 17 | 7 |\n| 20 | 18 | 9 |\n| 20 | 19 | 0 |\n| 20 | 21 | 6 |\n| 20 | 22 | 0 |\n| 21 | 1 | 7 |\n| 21 | 2 | 6 |\n| 21 | 3 | 9 |\n| 21 | 4 | 1 |\n| 21 | 5 | 8 |\n| 21 | 6 | 9 |\n| 21 | 7 | 9 |\n| 21 | 8 | 1 |\n| 21 | 9 | 5 |\n| 21 | 10 | 7 |\n| 21 | 11 | 7 |\n| 21 | 12 | 3 |\n| 21 | 13 | 4 |\n| 21 | 14 | 8 |\n| 21 | 15 | 2 |\n| 21 | 16 | 3 |\n| 21 | 17 | 1 |\n| 21 | 18 | 5 |\n| 21 | 19 | 0 |\n| 21 | 20 | 6 |\n| 21 | 22 | 2 |\n| 22 | 1 | 7 |\n| 22 | 2 | 4 |\n| 22 | 3 | 2 |\n| 22 | 4 | 3 |\n| 22 | 5 | 3 |\n| 22 | 6 | 3 |\n| 22 | 7 | 8 |\n| 22 | 8 | 7 |\n| 22 | 9 | 8 |\n| 22 | 10 | 9 |\n| 22 | 11 | 1 |\n| 22 | 12 | 1 |\n| 22 | 13 | 6 |\n| 22 | 14 | 4 |\n| 22 | 15 | 0 |\n| 22 | 16 | 6 |\n| 22 | 17 | 3 |\n| 22 | 18 | 2 |\n| 22 | 19 | 6 |\n| 22 | 20 | 0 |\n| 22 | 21 | 2 |\n\nWhen you evaluate each candidate set, sum the distances for every unordered pair; the set with the largest total is the most spread-out.\n\nOh, and when you send the final selection back, please use this simple JSON layout so it's easy to read by whatever's collecting the answers:\n\n{\n \"solution\": [, , ...]\n}\n\nHere \"solution\" is just the list/array of the m parks you're choosing to renovate — one identifier per park, in that same placeholder style. This is only a sketch of the shape I expect, not the actual answer you should submit.\n\nPlease make sure to use the exact identifiers from the instance input — do not rename them or invent new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 7, 4, 7, 1, 1, 6, 3, 1, 4, 8, 9, 9, 7, 5, 9, 9, 2, 2, 7, 7, 7 ], [ 7, 0, 3, 8, 8, 6, 2, 3, 0, 7, 4, 9, 5, 8, 3, 3, 5, 7, 7, 1, 6, 4 ], [ 4, 3, 0, 2, 3, 3, 0, 7, 2, 9, 2, 2, 6, 5, 7, 7, 5, 9, 5, 7, 9, 2 ], [ 7, 8, 2, 0, 5, 0, 8, 9, 8, 2, 7, 9, 4, 6, 6, 7, 1, 8, 6, 1, 1, 3 ], [ 1, 8, 3, 5, 0, 4, 4, 7, 5, 5, 3, 0, 3, 6, 9, 4, 7, 8, 3, 8, 8, 3 ], [ 1, 6, 3, 0, 4, 0, 7, 2, 5, 0, 0, 3, 6, 2, 1, 5, 4, 5, 3, 7, 9, 3 ], [ 6, 2, 0, 8, 4, 7, 0, 9, 5, 4, 6, 3, 4, 3, 0, 0, 9, 0, 6, 1, 9, 8 ], [ 3, 3, 7, 9, 7, 2, 9, 0, 8, 4, 8, 7, 8, 8, 0, 2, 5, 7, 2, 2, 1, 7 ], [ 1, 0, 2, 8, 5, 5, 5, 8, 0, 7, 5, 5, 2, 7, 2, 2, 9, 1, 4, 6, 5, 8 ], [ 4, 7, 9, 2, 5, 0, 4, 4, 7, 0, 4, 6, 5, 8, 6, 1, 1, 2, 4, 9, 7, 9 ], [ 8, 4, 2, 7, 3, 0, 6, 8, 5, 4, 0, 8, 9, 9, 5, 3, 1, 7, 4, 6, 7, 1 ], [ 9, 9, 2, 9, 0, 3, 3, 7, 5, 6, 8, 0, 7, 6, 0, 6, 1, 8, 5, 3, 3, 1 ], [ 9, 5, 6, 4, 3, 6, 4, 8, 2, 5, 9, 7, 0, 8, 4, 9, 0, 0, 7, 0, 4, 6 ], [ 7, 8, 5, 6, 6, 2, 3, 8, 7, 8, 9, 6, 8, 0, 6, 0, 0, 3, 2, 0, 8, 4 ], [ 5, 3, 7, 6, 9, 1, 0, 0, 2, 6, 5, 0, 4, 6, 0, 4, 6, 1, 2, 4, 2, 0 ], [ 9, 3, 7, 7, 4, 5, 0, 2, 2, 1, 3, 6, 9, 0, 4, 0, 4, 6, 7, 0, 3, 6 ], [ 9, 5, 5, 1, 7, 4, 9, 5, 9, 1, 1, 1, 0, 0, 6, 4, 0, 4, 5, 7, 1, 3 ], [ 2, 7, 9, 8, 8, 5, 0, 7, 1, 2, 7, 8, 0, 3, 1, 6, 4, 0, 1, 9, 5, 2 ], [ 2, 7, 5, 6, 3, 3, 6, 2, 4, 4, 4, 5, 7, 2, 2, 7, 5, 1, 0, 0, 0, 6 ], [ 7, 1, 7, 1, 8, 7, 1, 2, 6, 9, 6, 3, 0, 0, 4, 0, 7, 9, 0, 0, 6, 0 ], [ 7, 6, 9, 1, 8, 9, 9, 1, 5, 7, 7, 3, 4, 8, 2, 3, 1, 5, 0, 6, 0, 2 ], [ 7, 4, 2, 3, 3, 3, 8, 7, 8, 9, 1, 1, 6, 4, 0, 6, 3, 2, 6, 0, 2, 0 ] ], "m": 7, "objective": 156.0 }, "solution": [ 0, 3, 7, 10, 11, 12, 13 ], "obj": 156.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 22, "m": 7, "nodes": [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 ], "distance_pairs": [ { "from_id": 1, "to_id": 2, "distance": 7 }, { "from_id": 1, "to_id": 3, "distance": 4 }, { "from_id": 1, "to_id": 4, "distance": 7 }, { "from_id": 1, "to_id": 5, "distance": 1 }, { "from_id": 1, "to_id": 6, "distance": 1 }, { "from_id": 1, "to_id": 7, "distance": 6 }, { "from_id": 1, "to_id": 8, "distance": 3 }, { "from_id": 1, "to_id": 9, "distance": 1 }, { "from_id": 1, "to_id": 10, "distance": 4 }, { "from_id": 1, "to_id": 11, "distance": 8 }, { "from_id": 1, "to_id": 12, "distance": 9 }, { "from_id": 1, "to_id": 13, "distance": 9 }, { "from_id": 1, "to_id": 14, "distance": 7 }, { "from_id": 1, "to_id": 15, "distance": 5 }, { "from_id": 1, "to_id": 16, "distance": 9 }, { "from_id": 1, "to_id": 17, "distance": 9 }, { "from_id": 1, "to_id": 18, "distance": 2 }, { "from_id": 1, "to_id": 19, "distance": 2 }, { "from_id": 1, "to_id": 20, "distance": 7 }, { "from_id": 1, "to_id": 21, "distance": 7 }, { "from_id": 1, "to_id": 22, "distance": 7 }, { "from_id": 2, "to_id": 1, "distance": 7 }, { "from_id": 2, "to_id": 3, "distance": 3 }, { "from_id": 2, "to_id": 4, "distance": 8 }, { "from_id": 2, "to_id": 5, "distance": 8 }, { "from_id": 2, "to_id": 6, "distance": 6 }, { "from_id": 2, "to_id": 7, "distance": 2 }, { "from_id": 2, "to_id": 8, "distance": 3 }, { "from_id": 2, "to_id": 9, "distance": 0 }, { "from_id": 2, "to_id": 10, "distance": 7 }, { "from_id": 2, "to_id": 11, "distance": 4 }, { "from_id": 2, "to_id": 12, "distance": 9 }, { "from_id": 2, "to_id": 13, "distance": 5 }, { "from_id": 2, "to_id": 14, "distance": 8 }, { "from_id": 2, "to_id": 15, "distance": 3 }, { "from_id": 2, "to_id": 16, "distance": 3 }, { "from_id": 2, "to_id": 17, "distance": 5 }, { "from_id": 2, "to_id": 18, "distance": 7 }, { "from_id": 2, "to_id": 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"from_id": 21, "to_id": 7, "distance": 9 }, { "from_id": 21, "to_id": 8, "distance": 1 }, { "from_id": 21, "to_id": 9, "distance": 5 }, { "from_id": 21, "to_id": 10, "distance": 7 }, { "from_id": 21, "to_id": 11, "distance": 7 }, { "from_id": 21, "to_id": 12, "distance": 3 }, { "from_id": 21, "to_id": 13, "distance": 4 }, { "from_id": 21, "to_id": 14, "distance": 8 }, { "from_id": 21, "to_id": 15, "distance": 2 }, { "from_id": 21, "to_id": 16, "distance": 3 }, { "from_id": 21, "to_id": 17, "distance": 1 }, { "from_id": 21, "to_id": 18, "distance": 5 }, { "from_id": 21, "to_id": 19, "distance": 0 }, { "from_id": 21, "to_id": 20, "distance": 6 }, { "from_id": 21, "to_id": 22, "distance": 2 }, { "from_id": 22, "to_id": 1, "distance": 7 }, { "from_id": 22, "to_id": 2, "distance": 4 }, { "from_id": 22, "to_id": 3, "distance": 2 }, { "from_id": 22, "to_id": 4, "distance": 3 }, { "from_id": 22, "to_id": 5, "distance": 3 }, { "from_id": 22, "to_id": 6, "distance": 3 }, { "from_id": 22, "to_id": 7, "distance": 8 }, { "from_id": 22, "to_id": 8, "distance": 7 }, { "from_id": 22, "to_id": 9, "distance": 8 }, { "from_id": 22, "to_id": 10, "distance": 9 }, { "from_id": 22, "to_id": 11, "distance": 1 }, { "from_id": 22, "to_id": 12, "distance": 1 }, { "from_id": 22, "to_id": 13, "distance": 6 }, { "from_id": 22, "to_id": 14, "distance": 4 }, { "from_id": 22, "to_id": 15, "distance": 0 }, { "from_id": 22, "to_id": 16, "distance": 6 }, { "from_id": 22, "to_id": 17, "distance": 3 }, { "from_id": 22, "to_id": 18, "distance": 2 }, { "from_id": 22, "to_id": 19, "distance": 6 }, { "from_id": 22, "to_id": 20, "distance": 0 }, { "from_id": 22, "to_id": 21, "distance": 2 } ] }, "solution_variant": [ 1, 4, 8, 11, 12, 13, 14 ], "context_index": 46, "input_format": "markdown_table", "input_index_base": 1 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "A teacher put together a list of potential trips and a target of m outings, hoping the kids get a range of experiences. Between each pair of sites there’s a “difference” score; pick m distinct sites, sum the difference scores for all pairs among them, and the set with the highest sum will be the most varied schedule. There’s no swapping in duplicates — it has to be exactly m different locations. The detailed scores appear below.\n\nThey listed 21 potential sites (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20) and must pick exactly 6 different locations.\n\n| site_id_first | site_id_second | difference_score |\n|---|---|---|\n| 0 | 1 | 1 |\n| 0 | 2 | 4 |\n| 0 | 3 | 0 |\n| 0 | 4 | 8 |\n| 0 | 5 | 6 |\n| 0 | 6 | 1 |\n| 0 | 7 | 2 |\n| 0 | 8 | 9 |\n| 0 | 9 | 2 |\n| 0 | 10 | 2 |\n| 0 | 11 | 3 |\n| 0 | 12 | 7 |\n| 0 | 13 | 9 |\n| 0 | 14 | 8 |\n| 0 | 15 | 0 |\n| 0 | 16 | 9 |\n| 0 | 17 | 8 |\n| 0 | 18 | 2 |\n| 0 | 19 | 6 |\n| 0 | 20 | 9 |\n| 1 | 0 | 1 |\n| 1 | 2 | 2 |\n| 1 | 3 | 9 |\n| 1 | 4 | 9 |\n| 1 | 5 | 1 |\n| 1 | 6 | 3 |\n| 1 | 7 | 1 |\n| 1 | 8 | 2 |\n| 1 | 9 | 7 |\n| 1 | 10 | 8 |\n| 1 | 11 | 2 |\n| 1 | 12 | 6 |\n| 1 | 13 | 1 |\n| 1 | 14 | 2 |\n| 1 | 15 | 5 |\n| 1 | 16 | 2 |\n| 1 | 17 | 1 |\n| 1 | 18 | 5 |\n| 1 | 19 | 6 |\n| 1 | 20 | 8 |\n| 2 | 0 | 4 |\n| 2 | 1 | 2 |\n| 2 | 3 | 8 |\n| 2 | 4 | 0 |\n| 2 | 5 | 4 |\n| 2 | 6 | 2 |\n| 2 | 7 | 6 |\n| 2 | 8 | 8 |\n| 2 | 9 | 3 |\n| 2 | 10 | 4 |\n| 2 | 11 | 3 |\n| 2 | 12 | 1 |\n| 2 | 13 | 8 |\n| 2 | 14 | 7 |\n| 2 | 15 | 6 |\n| 2 | 16 | 4 |\n| 2 | 17 | 2 |\n| 2 | 18 | 2 |\n| 2 | 19 | 3 |\n| 2 | 20 | 5 |\n| 3 | 0 | 0 |\n| 3 | 1 | 9 |\n| 3 | 2 | 8 |\n| 3 | 4 | 4 |\n| 3 | 5 | 1 |\n| 3 | 6 | 7 |\n| 3 | 7 | 4 |\n| 3 | 8 | 1 |\n| 3 | 9 | 7 |\n| 3 | 10 | 8 |\n| 3 | 11 | 1 |\n| 3 | 12 | 2 |\n| 3 | 13 | 0 |\n| 3 | 14 | 3 |\n| 3 | 15 | 2 |\n| 3 | 16 | 4 |\n| 3 | 17 | 1 |\n| 3 | 18 | 5 |\n| 3 | 19 | 1 |\n| 3 | 20 | 6 |\n| 4 | 0 | 8 |\n| 4 | 1 | 9 |\n| 4 | 2 | 0 |\n| 4 | 3 | 4 |\n| 4 | 5 | 0 |\n| 4 | 6 | 5 |\n| 4 | 7 | 1 |\n| 4 | 8 | 9 |\n| 4 | 9 | 9 |\n| 4 | 10 | 0 |\n| 4 | 11 | 3 |\n| 4 | 12 | 5 |\n| 4 | 13 | 6 |\n| 4 | 14 | 3 |\n| 4 | 15 | 6 |\n| 4 | 16 | 5 |\n| 4 | 17 | 4 |\n| 4 | 18 | 9 |\n| 4 | 19 | 3 |\n| 4 | 20 | 4 |\n| 5 | 0 | 6 |\n| 5 | 1 | 1 |\n| 5 | 2 | 4 |\n| 5 | 3 | 1 |\n| 5 | 4 | 0 |\n| 5 | 6 | 3 |\n| 5 | 7 | 8 |\n| 5 | 8 | 9 |\n| 5 | 9 | 4 |\n| 5 | 10 | 5 |\n| 5 | 11 | 3 |\n| 5 | 12 | 7 |\n| 5 | 13 | 2 |\n| 5 | 14 | 9 |\n| 5 | 15 | 7 |\n| 5 | 16 | 1 |\n| 5 | 17 | 9 |\n| 5 | 18 | 1 |\n| 5 | 19 | 4 |\n| 5 | 20 | 1 |\n| 6 | 0 | 1 |\n| 6 | 1 | 3 |\n| 6 | 2 | 2 |\n| 6 | 3 | 7 |\n| 6 | 4 | 5 |\n| 6 | 5 | 3 |\n| 6 | 7 | 8 |\n| 6 | 8 | 4 |\n| 6 | 9 | 2 |\n| 6 | 10 | 1 |\n| 6 | 11 | 2 |\n| 6 | 12 | 3 |\n| 6 | 13 | 8 |\n| 6 | 14 | 1 |\n| 6 | 15 | 9 |\n| 6 | 16 | 6 |\n| 6 | 17 | 0 |\n| 6 | 18 | 1 |\n| 6 | 19 | 3 |\n| 6 | 20 | 1 |\n| 7 | 0 | 2 |\n| 7 | 1 | 1 |\n| 7 | 2 | 6 |\n| 7 | 3 | 4 |\n| 7 | 4 | 1 |\n| 7 | 5 | 8 |\n| 7 | 6 | 8 |\n| 7 | 8 | 3 |\n| 7 | 9 | 4 |\n| 7 | 10 | 8 |\n| 7 | 11 | 8 |\n| 7 | 12 | 8 |\n| 7 | 13 | 9 |\n| 7 | 14 | 8 |\n| 7 | 15 | 0 |\n| 7 | 16 | 6 |\n| 7 | 17 | 1 |\n| 7 | 18 | 1 |\n| 7 | 19 | 3 |\n| 7 | 20 | 8 |\n| 8 | 0 | 9 |\n| 8 | 1 | 2 |\n| 8 | 2 | 8 |\n| 8 | 3 | 1 |\n| 8 | 4 | 9 |\n| 8 | 5 | 9 |\n| 8 | 6 | 4 |\n| 8 | 7 | 3 |\n| 8 | 9 | 6 |\n| 8 | 10 | 4 |\n| 8 | 11 | 3 |\n| 8 | 12 | 9 |\n| 8 | 13 | 3 |\n| 8 | 14 | 8 |\n| 8 | 15 | 7 |\n| 8 | 16 | 2 |\n| 8 | 17 | 0 |\n| 8 | 18 | 9 |\n| 8 | 19 | 4 |\n| 8 | 20 | 1 |\n| 9 | 0 | 2 |\n| 9 | 1 | 7 |\n| 9 | 2 | 3 |\n| 9 | 3 | 7 |\n| 9 | 4 | 9 |\n| 9 | 5 | 4 |\n| 9 | 6 | 2 |\n| 9 | 7 | 4 |\n| 9 | 8 | 6 |\n| 9 | 10 | 4 |\n| 9 | 11 | 7 |\n| 9 | 12 | 9 |\n| 9 | 13 | 7 |\n| 9 | 14 | 8 |\n| 9 | 15 | 2 |\n| 9 | 16 | 5 |\n| 9 | 17 | 0 |\n| 9 | 18 | 6 |\n| 9 | 19 | 6 |\n| 9 | 20 | 8 |\n| 10 | 0 | 2 |\n| 10 | 1 | 8 |\n| 10 | 2 | 4 |\n| 10 | 3 | 8 |\n| 10 | 4 | 0 |\n| 10 | 5 | 5 |\n| 10 | 6 | 1 |\n| 10 | 7 | 8 |\n| 10 | 8 | 4 |\n| 10 | 9 | 4 |\n| 10 | 11 | 8 |\n| 10 | 12 | 4 |\n| 10 | 13 | 5 |\n| 10 | 14 | 9 |\n| 10 | 15 | 3 |\n| 10 | 16 | 4 |\n| 10 | 17 | 8 |\n| 10 | 18 | 7 |\n| 10 | 19 | 1 |\n| 10 | 20 | 3 |\n| 11 | 0 | 3 |\n| 11 | 1 | 2 |\n| 11 | 2 | 3 |\n| 11 | 3 | 1 |\n| 11 | 4 | 3 |\n| 11 | 5 | 3 |\n| 11 | 6 | 2 |\n| 11 | 7 | 8 |\n| 11 | 8 | 3 |\n| 11 | 9 | 7 |\n| 11 | 10 | 8 |\n| 11 | 12 | 3 |\n| 11 | 13 | 3 |\n| 11 | 14 | 7 |\n| 11 | 15 | 7 |\n| 11 | 16 | 0 |\n| 11 | 17 | 2 |\n| 11 | 18 | 5 |\n| 11 | 19 | 4 |\n| 11 | 20 | 5 |\n| 12 | 0 | 7 |\n| 12 | 1 | 6 |\n| 12 | 2 | 1 |\n| 12 | 3 | 2 |\n| 12 | 4 | 5 |\n| 12 | 5 | 7 |\n| 12 | 6 | 3 |\n| 12 | 7 | 8 |\n| 12 | 8 | 9 |\n| 12 | 9 | 9 |\n| 12 | 10 | 4 |\n| 12 | 11 | 3 |\n| 12 | 13 | 3 |\n| 12 | 14 | 7 |\n| 12 | 15 | 6 |\n| 12 | 16 | 4 |\n| 12 | 17 | 1 |\n| 12 | 18 | 3 |\n| 12 | 19 | 5 |\n| 12 | 20 | 1 |\n| 13 | 0 | 9 |\n| 13 | 1 | 1 |\n| 13 | 2 | 8 |\n| 13 | 3 | 0 |\n| 13 | 4 | 6 |\n| 13 | 5 | 2 |\n| 13 | 6 | 8 |\n| 13 | 7 | 9 |\n| 13 | 8 | 3 |\n| 13 | 9 | 7 |\n| 13 | 10 | 5 |\n| 13 | 11 | 3 |\n| 13 | 12 | 3 |\n| 13 | 14 | 4 |\n| 13 | 15 | 5 |\n| 13 | 16 | 0 |\n| 13 | 17 | 6 |\n| 13 | 18 | 4 |\n| 13 | 19 | 8 |\n| 13 | 20 | 1 |\n| 14 | 0 | 8 |\n| 14 | 1 | 2 |\n| 14 | 2 | 7 |\n| 14 | 3 | 3 |\n| 14 | 4 | 3 |\n| 14 | 5 | 9 |\n| 14 | 6 | 1 |\n| 14 | 7 | 8 |\n| 14 | 8 | 8 |\n| 14 | 9 | 8 |\n| 14 | 10 | 9 |\n| 14 | 11 | 7 |\n| 14 | 12 | 7 |\n| 14 | 13 | 4 |\n| 14 | 15 | 6 |\n| 14 | 16 | 2 |\n| 14 | 17 | 4 |\n| 14 | 18 | 9 |\n| 14 | 19 | 1 |\n| 14 | 20 | 9 |\n| 15 | 0 | 0 |\n| 15 | 1 | 5 |\n| 15 | 2 | 6 |\n| 15 | 3 | 2 |\n| 15 | 4 | 6 |\n| 15 | 5 | 7 |\n| 15 | 6 | 9 |\n| 15 | 7 | 0 |\n| 15 | 8 | 7 |\n| 15 | 9 | 2 |\n| 15 | 10 | 3 |\n| 15 | 11 | 7 |\n| 15 | 12 | 6 |\n| 15 | 13 | 5 |\n| 15 | 14 | 6 |\n| 15 | 16 | 4 |\n| 15 | 17 | 8 |\n| 15 | 18 | 2 |\n| 15 | 19 | 9 |\n| 15 | 20 | 6 |\n| 16 | 0 | 9 |\n| 16 | 1 | 2 |\n| 16 | 2 | 4 |\n| 16 | 3 | 4 |\n| 16 | 4 | 5 |\n| 16 | 5 | 1 |\n| 16 | 6 | 6 |\n| 16 | 7 | 6 |\n| 16 | 8 | 2 |\n| 16 | 9 | 5 |\n| 16 | 10 | 4 |\n| 16 | 11 | 0 |\n| 16 | 12 | 4 |\n| 16 | 13 | 0 |\n| 16 | 14 | 2 |\n| 16 | 15 | 4 |\n| 16 | 17 | 1 |\n| 16 | 18 | 2 |\n| 16 | 19 | 4 |\n| 16 | 20 | 9 |\n| 17 | 0 | 8 |\n| 17 | 1 | 1 |\n| 17 | 2 | 2 |\n| 17 | 3 | 1 |\n| 17 | 4 | 4 |\n| 17 | 5 | 9 |\n| 17 | 6 | 0 |\n| 17 | 7 | 1 |\n| 17 | 8 | 0 |\n| 17 | 9 | 0 |\n| 17 | 10 | 8 |\n| 17 | 11 | 2 |\n| 17 | 12 | 1 |\n| 17 | 13 | 6 |\n| 17 | 14 | 4 |\n| 17 | 15 | 8 |\n| 17 | 16 | 1 |\n| 17 | 18 | 5 |\n| 17 | 19 | 9 |\n| 17 | 20 | 3 |\n| 18 | 0 | 2 |\n| 18 | 1 | 5 |\n| 18 | 2 | 2 |\n| 18 | 3 | 5 |\n| 18 | 4 | 9 |\n| 18 | 5 | 1 |\n| 18 | 6 | 1 |\n| 18 | 7 | 1 |\n| 18 | 8 | 9 |\n| 18 | 9 | 6 |\n| 18 | 10 | 7 |\n| 18 | 11 | 5 |\n| 18 | 12 | 3 |\n| 18 | 13 | 4 |\n| 18 | 14 | 9 |\n| 18 | 15 | 2 |\n| 18 | 16 | 2 |\n| 18 | 17 | 5 |\n| 18 | 19 | 5 |\n| 18 | 20 | 5 |\n| 19 | 0 | 6 |\n| 19 | 1 | 6 |\n| 19 | 2 | 3 |\n| 19 | 3 | 1 |\n| 19 | 4 | 3 |\n| 19 | 5 | 4 |\n| 19 | 6 | 3 |\n| 19 | 7 | 3 |\n| 19 | 8 | 4 |\n| 19 | 9 | 6 |\n| 19 | 10 | 1 |\n| 19 | 11 | 4 |\n| 19 | 12 | 5 |\n| 19 | 13 | 8 |\n| 19 | 14 | 1 |\n| 19 | 15 | 9 |\n| 19 | 16 | 4 |\n| 19 | 17 | 9 |\n| 19 | 18 | 5 |\n| 19 | 20 | 5 |\n| 20 | 0 | 9 |\n| 20 | 1 | 8 |\n| 20 | 2 | 5 |\n| 20 | 3 | 6 |\n| 20 | 4 | 4 |\n| 20 | 5 | 1 |\n| 20 | 6 | 1 |\n| 20 | 7 | 8 |\n| 20 | 8 | 1 |\n| 20 | 9 | 8 |\n| 20 | 10 | 3 |\n| 20 | 11 | 5 |\n| 20 | 12 | 1 |\n| 20 | 13 | 1 |\n| 20 | 14 | 9 |\n| 20 | 15 | 6 |\n| 20 | 16 | 9 |\n| 20 | 17 | 3 |\n| 20 | 18 | 5 |\n| 20 | 19 | 5 |\n\nThe teacher will use these pairwise scores to pick the 6 most diverse outings.\n\nOh, and when you send back your chosen set, please use this simple JSON layout so it's easy to parse:\n\n{\n \"solution\": [, , ...]\n}\n\nThis just means \"solution\" should be a list of the m sites you picked. Treat each placeholder as a single site identifier — it's just a sketch of the shape I expect, not the actual answer.\n\nPlease use the exact identifiers from the instance input (don't rename them or invent new labels). Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.", "instance": { "distance_matrix": [ [ 0, 1, 4, 0, 8, 6, 1, 2, 9, 2, 2, 3, 7, 9, 8, 0, 9, 8, 2, 6, 9 ], [ 1, 0, 2, 9, 9, 1, 3, 1, 2, 7, 8, 2, 6, 1, 2, 5, 2, 1, 5, 6, 8 ], [ 4, 2, 0, 8, 0, 4, 2, 6, 8, 3, 4, 3, 1, 8, 7, 6, 4, 2, 2, 3, 5 ], [ 0, 9, 8, 0, 4, 1, 7, 4, 1, 7, 8, 1, 2, 0, 3, 2, 4, 1, 5, 1, 6 ], [ 8, 9, 0, 4, 0, 0, 5, 1, 9, 9, 0, 3, 5, 6, 3, 6, 5, 4, 9, 3, 4 ], [ 6, 1, 4, 1, 0, 0, 3, 8, 9, 4, 5, 3, 7, 2, 9, 7, 1, 9, 1, 4, 1 ], [ 1, 3, 2, 7, 5, 3, 0, 8, 4, 2, 1, 2, 3, 8, 1, 9, 6, 0, 1, 3, 1 ], [ 2, 1, 6, 4, 1, 8, 8, 0, 3, 4, 8, 8, 8, 9, 8, 0, 6, 1, 1, 3, 8 ], [ 9, 2, 8, 1, 9, 9, 4, 3, 0, 6, 4, 3, 9, 3, 8, 7, 2, 0, 9, 4, 1 ], [ 2, 7, 3, 7, 9, 4, 2, 4, 6, 0, 4, 7, 9, 7, 8, 2, 5, 0, 6, 6, 8 ], [ 2, 8, 4, 8, 0, 5, 1, 8, 4, 4, 0, 8, 4, 5, 9, 3, 4, 8, 7, 1, 3 ], [ 3, 2, 3, 1, 3, 3, 2, 8, 3, 7, 8, 0, 3, 3, 7, 7, 0, 2, 5, 4, 5 ], [ 7, 6, 1, 2, 5, 7, 3, 8, 9, 9, 4, 3, 0, 3, 7, 6, 4, 1, 3, 5, 1 ], [ 9, 1, 8, 0, 6, 2, 8, 9, 3, 7, 5, 3, 3, 0, 4, 5, 0, 6, 4, 8, 1 ], [ 8, 2, 7, 3, 3, 9, 1, 8, 8, 8, 9, 7, 7, 4, 0, 6, 2, 4, 9, 1, 9 ], [ 0, 5, 6, 2, 6, 7, 9, 0, 7, 2, 3, 7, 6, 5, 6, 0, 4, 8, 2, 9, 6 ], [ 9, 2, 4, 4, 5, 1, 6, 6, 2, 5, 4, 0, 4, 0, 2, 4, 0, 1, 2, 4, 9 ], [ 8, 1, 2, 1, 4, 9, 0, 1, 0, 0, 8, 2, 1, 6, 4, 8, 1, 0, 5, 9, 3 ], [ 2, 5, 2, 5, 9, 1, 1, 1, 9, 6, 7, 5, 3, 4, 9, 2, 2, 5, 0, 5, 5 ], [ 6, 6, 3, 1, 3, 4, 3, 3, 4, 6, 1, 4, 5, 8, 1, 9, 4, 9, 5, 0, 5 ], [ 9, 8, 5, 6, 4, 1, 1, 8, 1, 8, 3, 5, 1, 1, 9, 6, 9, 3, 5, 5, 0 ] ], "m": 6, "objective": 109.0 }, "solution": [ 4, 8, 9, 12, 14, 18 ], "obj": 109.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 21, "m": 6, "nodes": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 ], "distance_pairs": [ { "from_id": 0, "to_id": 1, "distance": 1 }, { "from_id": 0, "to_id": 2, "distance": 4 }, { "from_id": 0, "to_id": 3, "distance": 0 }, { "from_id": 0, "to_id": 4, "distance": 8 }, { "from_id": 0, "to_id": 5, "distance": 6 }, { "from_id": 0, "to_id": 6, "distance": 1 }, { "from_id": 0, "to_id": 7, "distance": 2 }, { "from_id": 0, "to_id": 8, "distance": 9 }, { "from_id": 0, "to_id": 9, "distance": 2 }, { "from_id": 0, "to_id": 10, "distance": 2 }, { "from_id": 0, "to_id": 11, "distance": 3 }, { "from_id": 0, "to_id": 12, "distance": 7 }, { "from_id": 0, "to_id": 13, "distance": 9 }, { "from_id": 0, "to_id": 14, "distance": 8 }, { "from_id": 0, "to_id": 15, "distance": 0 }, { "from_id": 0, "to_id": 16, "distance": 9 }, { "from_id": 0, "to_id": 17, "distance": 8 }, { "from_id": 0, "to_id": 18, "distance": 2 }, { "from_id": 0, "to_id": 19, "distance": 6 }, { "from_id": 0, "to_id": 20, "distance": 9 }, { "from_id": 1, "to_id": 0, "distance": 1 }, { "from_id": 1, "to_id": 2, "distance": 2 }, { "from_id": 1, "to_id": 3, "distance": 9 }, { "from_id": 1, "to_id": 4, "distance": 9 }, { "from_id": 1, "to_id": 5, "distance": 1 }, 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"from_id": 20, "to_id": 15, "distance": 6 }, { "from_id": 20, "to_id": 16, "distance": 9 }, { "from_id": 20, "to_id": 17, "distance": 3 }, { "from_id": 20, "to_id": 18, "distance": 5 }, { "from_id": 20, "to_id": 19, "distance": 5 } ] }, "solution_variant": [ 4, 8, 9, 12, 14, 18 ], "context_index": 47, "input_format": "markdown_table", "input_index_base": 0 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Many people on the crew pitched in ideas for a spice sampler, but the task boiled down to picking exactly m unique spices that together give the most distinctive mix. The rule: fill exactly m slots with different spices, and prefer combinations where members are more distant in flavor from one another. To compare boxes, total up the flavor-distance between every pair inside each selection — the larger that total, the more contrasting the sampler. The concrete spice list and all pairwise distances are provided below.\n\n# num_spices=21\n# sampler_size=2\n# spice_ids=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\nspice_id_a,spice_id_b,flavor_distance\n0,1,1\n0,2,6\n0,3,3\n0,4,7\n0,5,9\n0,6,4\n0,7,2\n0,8,7\n0,9,8\n0,10,1\n0,11,4\n0,12,5\n0,13,2\n0,14,5\n0,15,5\n0,16,8\n0,17,4\n0,18,6\n0,19,0\n0,20,9\n1,0,1\n1,2,7\n1,3,0\n1,4,3\n1,5,6\n1,6,2\n1,7,7\n1,8,9\n1,9,7\n1,10,5\n1,11,8\n1,12,3\n1,13,8\n1,14,4\n1,15,4\n1,16,1\n1,17,9\n1,18,1\n1,19,7\n1,20,9\n2,0,6\n2,1,7\n2,3,0\n2,4,9\n2,5,4\n2,6,7\n2,7,2\n2,8,2\n2,9,6\n2,10,8\n2,11,8\n2,12,5\n2,13,7\n2,14,2\n2,15,7\n2,16,0\n2,17,3\n2,18,8\n2,19,9\n2,20,6\n3,0,3\n3,1,0\n3,2,0\n3,4,6\n3,5,9\n3,6,9\n3,7,2\n3,8,6\n3,9,3\n3,10,1\n3,11,4\n3,12,5\n3,13,7\n3,14,7\n3,15,8\n3,16,6\n3,17,4\n3,18,8\n3,19,1\n3,20,6\n4,0,7\n4,1,3\n4,2,9\n4,3,6\n4,5,6\n4,6,4\n4,7,4\n4,8,6\n4,9,7\n4,10,4\n4,11,4\n4,12,8\n4,13,9\n4,14,8\n4,15,4\n4,16,7\n4,17,9\n4,18,7\n4,19,3\n4,20,7\n5,0,9\n5,1,6\n5,2,4\n5,3,9\n5,4,6\n5,6,6\n5,7,4\n5,8,8\n5,9,2\n5,10,0\n5,11,5\n5,12,6\n5,13,9\n5,14,8\n5,15,3\n5,16,4\n5,17,5\n5,18,2\n5,19,2\n5,20,6\n6,0,4\n6,1,2\n6,2,7\n6,3,9\n6,4,4\n6,5,6\n6,7,0\n6,8,6\n6,9,6\n6,10,0\n6,11,2\n6,12,1\n6,13,4\n6,14,4\n6,15,8\n6,16,5\n6,17,6\n6,18,3\n6,19,1\n6,20,2\n7,0,2\n7,1,7\n7,2,2\n7,3,2\n7,4,4\n7,5,4\n7,6,0\n7,8,2\n7,9,2\n7,10,0\n7,11,5\n7,12,0\n7,13,0\n7,14,5\n7,15,6\n7,16,0\n7,17,3\n7,18,9\n7,19,9\n7,20,8\n8,0,7\n8,1,9\n8,2,2\n8,3,6\n8,4,6\n8,5,8\n8,6,6\n8,7,2\n8,9,9\n8,10,4\n8,11,7\n8,12,5\n8,13,9\n8,14,4\n8,15,9\n8,16,4\n8,17,7\n8,18,8\n8,19,3\n8,20,7\n9,0,8\n9,1,7\n9,2,6\n9,3,3\n9,4,7\n9,5,2\n9,6,6\n9,7,2\n9,8,9\n9,10,5\n9,11,1\n9,12,4\n9,13,4\n9,14,9\n9,15,8\n9,16,2\n9,17,6\n9,18,6\n9,19,5\n9,20,5\n10,0,1\n10,1,5\n10,2,8\n10,3,1\n10,4,4\n10,5,0\n10,6,0\n10,7,0\n10,8,4\n10,9,5\n10,11,6\n10,12,9\n10,13,4\n10,14,9\n10,15,1\n10,16,1\n10,17,7\n10,18,1\n10,19,8\n10,20,6\n11,0,4\n11,1,8\n11,2,8\n11,3,4\n11,4,4\n11,5,5\n11,6,2\n11,7,5\n11,8,7\n11,9,1\n11,10,6\n11,12,7\n11,13,7\n11,14,3\n11,15,9\n11,16,9\n11,17,5\n11,18,4\n11,19,6\n11,20,3\n12,0,5\n12,1,3\n12,2,5\n12,3,5\n12,4,8\n12,5,6\n12,6,1\n12,7,0\n12,8,5\n12,9,4\n12,10,9\n12,11,7\n12,13,0\n12,14,9\n12,15,7\n12,16,9\n12,17,9\n12,18,3\n12,19,2\n12,20,5\n13,0,2\n13,1,8\n13,2,7\n13,3,7\n13,4,9\n13,5,9\n13,6,4\n13,7,0\n13,8,9\n13,9,4\n13,10,4\n13,11,7\n13,12,0\n13,14,4\n13,15,7\n13,16,7\n13,17,1\n13,18,5\n13,19,5\n13,20,9\n14,0,5\n14,1,4\n14,2,2\n14,3,7\n14,4,8\n14,5,8\n14,6,4\n14,7,5\n14,8,4\n14,9,9\n14,10,9\n14,11,3\n14,12,9\n14,13,4\n14,15,5\n14,16,2\n14,17,3\n14,18,0\n14,19,1\n14,20,8\n15,0,5\n15,1,4\n15,2,7\n15,3,8\n15,4,4\n15,5,3\n15,6,8\n15,7,6\n15,8,9\n15,9,8\n15,10,1\n15,11,9\n15,12,7\n15,13,7\n15,14,5\n15,16,8\n15,17,0\n15,18,0\n15,19,2\n15,20,1\n16,0,8\n16,1,1\n16,2,0\n16,3,6\n16,4,7\n16,5,4\n16,6,5\n16,7,0\n16,8,4\n16,9,2\n16,10,1\n16,11,9\n16,12,9\n16,13,7\n16,14,2\n16,15,8\n16,17,3\n16,18,2\n16,19,8\n16,20,6\n17,0,4\n17,1,9\n17,2,3\n17,3,4\n17,4,9\n17,5,5\n17,6,6\n17,7,3\n17,8,7\n17,9,6\n17,10,7\n17,11,5\n17,12,9\n17,13,1\n17,14,3\n17,15,0\n17,16,3\n17,18,2\n17,19,5\n17,20,8\n18,0,6\n18,1,1\n18,2,8\n18,3,8\n18,4,7\n18,5,2\n18,6,3\n18,7,9\n18,8,8\n18,9,6\n18,10,1\n18,11,4\n18,12,3\n18,13,5\n18,14,0\n18,15,0\n18,16,2\n18,17,2\n18,19,3\n18,20,5\n19,0,0\n19,1,7\n19,2,9\n19,3,1\n19,4,3\n19,5,2\n19,6,1\n19,7,9\n19,8,3\n19,9,5\n19,10,8\n19,11,6\n19,12,2\n19,13,5\n19,14,1\n19,15,2\n19,16,8\n19,17,5\n19,18,3\n19,20,7\n20,0,9\n20,1,9\n20,2,6\n20,3,6\n20,4,7\n20,5,6\n20,6,2\n20,7,8\n20,8,7\n20,9,5\n20,10,6\n20,11,3\n20,12,5\n20,13,9\n20,14,8\n20,15,1\n20,16,6\n20,17,8\n20,18,5\n20,19,7\n\nOh, and when you share the final spice picks, please drop them into a tiny JSON snippet like this so it's easy to read and reuse:\n\n{\n \"solution\": [, , ...]\n}\n\nThink of that \"solution\" array as the box list — just put the exact m spice identifiers from the instance inside it. This little sketch is just the expected shape, not the actual answer.\n\nPlease use the identifiers exactly as they appear in the input — no renaming or new labels. \n- for example: \"Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.\"", "instance": { "distance_matrix": [ [ 0, 1, 6, 3, 7, 9, 4, 2, 7, 8, 1, 4, 5, 2, 5, 5, 8, 4, 6, 0, 9 ], [ 1, 0, 7, 0, 3, 6, 2, 7, 9, 7, 5, 8, 3, 8, 4, 4, 1, 9, 1, 7, 9 ], [ 6, 7, 0, 0, 9, 4, 7, 2, 2, 6, 8, 8, 5, 7, 2, 7, 0, 3, 8, 9, 6 ], [ 3, 0, 0, 0, 6, 9, 9, 2, 6, 3, 1, 4, 5, 7, 7, 8, 6, 4, 8, 1, 6 ], [ 7, 3, 9, 6, 0, 6, 4, 4, 6, 7, 4, 4, 8, 9, 8, 4, 7, 9, 7, 3, 7 ], [ 9, 6, 4, 9, 6, 0, 6, 4, 8, 2, 0, 5, 6, 9, 8, 3, 4, 5, 2, 2, 6 ], [ 4, 2, 7, 9, 4, 6, 0, 0, 6, 6, 0, 2, 1, 4, 4, 8, 5, 6, 3, 1, 2 ], [ 2, 7, 2, 2, 4, 4, 0, 0, 2, 2, 0, 5, 0, 0, 5, 6, 0, 3, 9, 9, 8 ], [ 7, 9, 2, 6, 6, 8, 6, 2, 0, 9, 4, 7, 5, 9, 4, 9, 4, 7, 8, 3, 7 ], [ 8, 7, 6, 3, 7, 2, 6, 2, 9, 0, 5, 1, 4, 4, 9, 8, 2, 6, 6, 5, 5 ], [ 1, 5, 8, 1, 4, 0, 0, 0, 4, 5, 0, 6, 9, 4, 9, 1, 1, 7, 1, 8, 6 ], [ 4, 8, 8, 4, 4, 5, 2, 5, 7, 1, 6, 0, 7, 7, 3, 9, 9, 5, 4, 6, 3 ], [ 5, 3, 5, 5, 8, 6, 1, 0, 5, 4, 9, 7, 0, 0, 9, 7, 9, 9, 3, 2, 5 ], [ 2, 8, 7, 7, 9, 9, 4, 0, 9, 4, 4, 7, 0, 0, 4, 7, 7, 1, 5, 5, 9 ], [ 5, 4, 2, 7, 8, 8, 4, 5, 4, 9, 9, 3, 9, 4, 0, 5, 2, 3, 0, 1, 8 ], [ 5, 4, 7, 8, 4, 3, 8, 6, 9, 8, 1, 9, 7, 7, 5, 0, 8, 0, 0, 2, 1 ], [ 8, 1, 0, 6, 7, 4, 5, 0, 4, 2, 1, 9, 9, 7, 2, 8, 0, 3, 2, 8, 6 ], [ 4, 9, 3, 4, 9, 5, 6, 3, 7, 6, 7, 5, 9, 1, 3, 0, 3, 0, 2, 5, 8 ], [ 6, 1, 8, 8, 7, 2, 3, 9, 8, 6, 1, 4, 3, 5, 0, 0, 2, 2, 0, 3, 5 ], [ 0, 7, 9, 1, 3, 2, 1, 9, 3, 5, 8, 6, 2, 5, 1, 2, 8, 5, 3, 0, 7 ], [ 9, 9, 6, 6, 7, 6, 2, 8, 7, 5, 6, 3, 5, 9, 8, 1, 6, 8, 5, 7, 0 ] ], "m": 2, "objective": 9.0 }, "solution": [ 7, 18 ], "obj": 9.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 21, "m": 2, "nodes": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 ], "distance_pairs": [ { "from_id": 0, "to_id": 1, "distance": 1 }, { "from_id": 0, "to_id": 2, "distance": 6 }, { "from_id": 0, "to_id": 3, "distance": 3 }, { "from_id": 0, "to_id": 4, "distance": 7 }, { "from_id": 0, "to_id": 5, "distance": 9 }, { "from_id": 0, "to_id": 6, "distance": 4 }, { "from_id": 0, "to_id": 7, "distance": 2 }, { "from_id": 0, "to_id": 8, "distance": 7 }, { "from_id": 0, "to_id": 9, "distance": 8 }, { "from_id": 0, "to_id": 10, "distance": 1 }, { "from_id": 0, "to_id": 11, "distance": 4 }, { "from_id": 0, "to_id": 12, "distance": 5 }, { "from_id": 0, "to_id": 13, "distance": 2 }, { "from_id": 0, "to_id": 14, "distance": 5 }, { "from_id": 0, "to_id": 15, "distance": 5 }, { "from_id": 0, "to_id": 16, "distance": 8 }, { "from_id": 0, "to_id": 17, "distance": 4 }, { "from_id": 0, "to_id": 18, "distance": 6 }, { "from_id": 0, "to_id": 19, "distance": 0 }, { "from_id": 0, "to_id": 20, "distance": 9 }, { "from_id": 1, "to_id": 0, "distance": 1 }, { "from_id": 1, "to_id": 2, "distance": 7 }, { "from_id": 1, "to_id": 3, "distance": 0 }, { "from_id": 1, "to_id": 4, "distance": 3 }, { "from_id": 1, "to_id": 5, "distance": 6 }, { "from_id": 1, "to_id": 6, "distance": 2 }, { "from_id": 1, "to_id": 7, "distance": 7 }, { "from_id": 1, "to_id": 8, "distance": 9 }, { "from_id": 1, "to_id": 9, "distance": 7 }, { "from_id": 1, "to_id": 10, "distance": 5 }, { "from_id": 1, "to_id": 11, "distance": 8 }, { "from_id": 1, "to_id": 12, "distance": 3 }, { "from_id": 1, "to_id": 13, "distance": 8 }, { "from_id": 1, "to_id": 14, "distance": 4 }, { "from_id": 1, "to_id": 15, "distance": 4 }, { "from_id": 1, "to_id": 16, "distance": 1 }, { "from_id": 1, "to_id": 17, "distance": 9 }, { "from_id": 1, "to_id": 18, "distance": 1 }, { "from_id": 1, "to_id": 19, "distance": 7 }, { "from_id": 1, "to_id": 20, "distance": 9 }, { "from_id": 2, "to_id": 0, "distance": 6 }, { "from_id": 2, "to_id": 1, "distance": 7 }, { "from_id": 2, "to_id": 3, "distance": 0 }, { "from_id": 2, "to_id": 4, "distance": 9 }, { "from_id": 2, "to_id": 5, "distance": 4 }, { "from_id": 2, "to_id": 6, "distance": 7 }, { "from_id": 2, "to_id": 7, "distance": 2 }, { "from_id": 2, "to_id": 8, "distance": 2 }, { "from_id": 2, "to_id": 9, "distance": 6 }, { "from_id": 2, "to_id": 10, "distance": 8 }, { "from_id": 2, "to_id": 11, "distance": 8 }, { "from_id": 2, "to_id": 12, "distance": 5 }, { "from_id": 2, "to_id": 13, "distance": 7 }, { "from_id": 2, "to_id": 14, "distance": 2 }, { "from_id": 2, "to_id": 15, "distance": 7 }, { "from_id": 2, "to_id": 16, "distance": 0 }, { "from_id": 2, "to_id": 17, "distance": 3 }, { "from_id": 2, "to_id": 18, "distance": 8 }, { "from_id": 2, "to_id": 19, "distance": 9 }, { "from_id": 2, "to_id": 20, "distance": 6 }, { "from_id": 3, "to_id": 0, "distance": 3 }, { "from_id": 3, "to_id": 1, "distance": 0 }, { "from_id": 3, "to_id": 2, "distance": 0 }, { "from_id": 3, "to_id": 4, "distance": 6 }, { "from_id": 3, "to_id": 5, "distance": 9 }, { "from_id": 3, "to_id": 6, "distance": 9 }, { "from_id": 3, "to_id": 7, "distance": 2 }, { "from_id": 3, "to_id": 8, "distance": 6 }, { "from_id": 3, "to_id": 9, "distance": 3 }, { "from_id": 3, "to_id": 10, "distance": 1 }, { "from_id": 3, "to_id": 11, "distance": 4 }, { "from_id": 3, "to_id": 12, "distance": 5 }, { "from_id": 3, "to_id": 13, "distance": 7 }, { "from_id": 3, "to_id": 14, "distance": 7 }, { "from_id": 3, "to_id": 15, "distance": 8 }, { "from_id": 3, "to_id": 16, "distance": 6 }, { "from_id": 3, "to_id": 17, "distance": 4 }, { "from_id": 3, "to_id": 18, "distance": 8 }, { "from_id": 3, "to_id": 19, "distance": 1 }, { "from_id": 3, "to_id": 20, "distance": 6 }, { "from_id": 4, "to_id": 0, "distance": 7 }, { "from_id": 4, "to_id": 1, "distance": 3 }, { "from_id": 4, "to_id": 2, "distance": 9 }, { "from_id": 4, "to_id": 3, "distance": 6 }, { "from_id": 4, "to_id": 5, "distance": 6 }, { "from_id": 4, "to_id": 6, "distance": 4 }, { "from_id": 4, "to_id": 7, "distance": 4 }, { "from_id": 4, "to_id": 8, "distance": 6 }, { "from_id": 4, "to_id": 9, "distance": 7 }, { "from_id": 4, "to_id": 10, "distance": 4 }, { "from_id": 4, "to_id": 11, "distance": 4 }, { "from_id": 4, "to_id": 12, "distance": 8 }, { "from_id": 4, "to_id": 13, "distance": 9 }, { "from_id": 4, "to_id": 14, "distance": 8 }, { "from_id": 4, "to_id": 15, "distance": 4 }, { "from_id": 4, "to_id": 16, "distance": 7 }, { "from_id": 4, "to_id": 17, "distance": 9 }, { "from_id": 4, "to_id": 18, "distance": 7 }, { "from_id": 4, "to_id": 19, "distance": 3 }, { "from_id": 4, "to_id": 20, "distance": 7 }, { "from_id": 5, "to_id": 0, "distance": 9 }, { "from_id": 5, "to_id": 1, "distance": 6 }, { "from_id": 5, "to_id": 2, "distance": 4 }, { "from_id": 5, "to_id": 3, "distance": 9 }, { "from_id": 5, "to_id": 4, "distance": 6 }, { "from_id": 5, "to_id": 6, "distance": 6 }, { "from_id": 5, "to_id": 7, "distance": 4 }, { "from_id": 5, "to_id": 8, "distance": 8 }, { "from_id": 5, "to_id": 9, "distance": 2 }, { "from_id": 5, "to_id": 10, "distance": 0 }, { "from_id": 5, "to_id": 11, "distance": 5 }, { "from_id": 5, "to_id": 12, "distance": 6 }, { "from_id": 5, "to_id": 13, "distance": 9 }, { "from_id": 5, "to_id": 14, "distance": 8 }, { "from_id": 5, "to_id": 15, "distance": 3 }, { "from_id": 5, "to_id": 16, "distance": 4 }, { "from_id": 5, "to_id": 17, "distance": 5 }, { "from_id": 5, "to_id": 18, "distance": 2 }, { "from_id": 5, "to_id": 19, "distance": 2 }, { "from_id": 5, "to_id": 20, "distance": 6 }, { "from_id": 6, "to_id": 0, "distance": 4 }, { "from_id": 6, "to_id": 1, "distance": 2 }, { "from_id": 6, "to_id": 2, "distance": 7 }, { "from_id": 6, "to_id": 3, "distance": 9 }, { "from_id": 6, "to_id": 4, "distance": 4 }, { "from_id": 6, "to_id": 5, "distance": 6 }, { "from_id": 6, "to_id": 7, "distance": 0 }, { "from_id": 6, "to_id": 8, "distance": 6 }, { "from_id": 6, "to_id": 9, "distance": 6 }, { "from_id": 6, "to_id": 10, "distance": 0 }, { "from_id": 6, "to_id": 11, "distance": 2 }, { "from_id": 6, "to_id": 12, "distance": 1 }, { "from_id": 6, "to_id": 13, "distance": 4 }, { "from_id": 6, "to_id": 14, "distance": 4 }, { "from_id": 6, "to_id": 15, "distance": 8 }, { "from_id": 6, "to_id": 16, "distance": 5 }, { "from_id": 6, "to_id": 17, "distance": 6 }, { "from_id": 6, "to_id": 18, "distance": 3 }, { "from_id": 6, "to_id": 19, "distance": 1 }, { "from_id": 6, "to_id": 20, "distance": 2 }, { "from_id": 7, "to_id": 0, "distance": 2 }, { "from_id": 7, "to_id": 1, "distance": 7 }, { "from_id": 7, "to_id": 2, "distance": 2 }, { "from_id": 7, "to_id": 3, "distance": 2 }, { "from_id": 7, "to_id": 4, "distance": 4 }, { "from_id": 7, "to_id": 5, "distance": 4 }, { "from_id": 7, "to_id": 6, "distance": 0 }, { "from_id": 7, "to_id": 8, "distance": 2 }, { "from_id": 7, "to_id": 9, "distance": 2 }, { "from_id": 7, "to_id": 10, "distance": 0 }, { "from_id": 7, "to_id": 11, "distance": 5 }, { "from_id": 7, "to_id": 12, "distance": 0 }, { "from_id": 7, "to_id": 13, "distance": 0 }, { "from_id": 7, "to_id": 14, "distance": 5 }, { "from_id": 7, "to_id": 15, "distance": 6 }, { "from_id": 7, "to_id": 16, "distance": 0 }, { "from_id": 7, "to_id": 17, "distance": 3 }, { "from_id": 7, "to_id": 18, "distance": 9 }, { "from_id": 7, "to_id": 19, "distance": 9 }, { "from_id": 7, "to_id": 20, "distance": 8 }, { "from_id": 8, "to_id": 0, "distance": 7 }, { "from_id": 8, "to_id": 1, "distance": 9 }, { "from_id": 8, "to_id": 2, "distance": 2 }, { "from_id": 8, "to_id": 3, "distance": 6 }, { "from_id": 8, "to_id": 4, "distance": 6 }, { "from_id": 8, "to_id": 5, "distance": 8 }, { "from_id": 8, "to_id": 6, "distance": 6 }, { "from_id": 8, "to_id": 7, "distance": 2 }, { "from_id": 8, "to_id": 9, "distance": 9 }, { "from_id": 8, "to_id": 10, "distance": 4 }, { "from_id": 8, "to_id": 11, "distance": 7 }, { "from_id": 8, "to_id": 12, "distance": 5 }, { "from_id": 8, "to_id": 13, "distance": 9 }, { "from_id": 8, "to_id": 14, "distance": 4 }, { "from_id": 8, "to_id": 15, "distance": 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"from_id": 19, "to_id": 10, "distance": 8 }, { "from_id": 19, "to_id": 11, "distance": 6 }, { "from_id": 19, "to_id": 12, "distance": 2 }, { "from_id": 19, "to_id": 13, "distance": 5 }, { "from_id": 19, "to_id": 14, "distance": 1 }, { "from_id": 19, "to_id": 15, "distance": 2 }, { "from_id": 19, "to_id": 16, "distance": 8 }, { "from_id": 19, "to_id": 17, "distance": 5 }, { "from_id": 19, "to_id": 18, "distance": 3 }, { "from_id": 19, "to_id": 20, "distance": 7 }, { "from_id": 20, "to_id": 0, "distance": 9 }, { "from_id": 20, "to_id": 1, "distance": 9 }, { "from_id": 20, "to_id": 2, "distance": 6 }, { "from_id": 20, "to_id": 3, "distance": 6 }, { "from_id": 20, "to_id": 4, "distance": 7 }, { "from_id": 20, "to_id": 5, "distance": 6 }, { "from_id": 20, "to_id": 6, "distance": 2 }, { "from_id": 20, "to_id": 7, "distance": 8 }, { "from_id": 20, "to_id": 8, "distance": 7 }, { "from_id": 20, "to_id": 9, "distance": 5 }, { "from_id": 20, "to_id": 10, "distance": 6 }, { "from_id": 20, "to_id": 11, "distance": 3 }, { "from_id": 20, "to_id": 12, "distance": 5 }, { "from_id": 20, "to_id": 13, "distance": 9 }, { "from_id": 20, "to_id": 14, "distance": 8 }, { "from_id": 20, "to_id": 15, "distance": 1 }, { "from_id": 20, "to_id": 16, "distance": 6 }, { "from_id": 20, "to_id": 17, "distance": 8 }, { "from_id": 20, "to_id": 18, "distance": 5 }, { "from_id": 20, "to_id": 19, "distance": 7 } ] }, "solution_variant": [ 7, 18 ], "context_index": 48, "input_format": "csv", "input_index_base": 0 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "I’m picturing a shop owner trying to set up the seasonal display and deciding which items to put out. The owner has to pick exactly m products to fill the display, no repeats and no empty spots, and wants the whole arrangement to feel as varied as possible. For any two products there’s a number that says how different they are, so the owner looks at every pair among the chosen m and adds up those difference scores — the higher that total, the more interesting and spread-out the display. The exact product list and the difference numbers are shown below.\n\nI see 22 distinct products listed as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, and the owner must pick exactly 2 to fill the display.\n\n| product_a_id | product_b_id | dissimilarity_score |\n|---|---|---|\n| 0 | 1 | 1 |\n| 0 | 2 | 9 |\n| 0 | 3 | 2 |\n| 0 | 4 | 0 |\n| 0 | 5 | 0 |\n| 0 | 6 | 7 |\n| 0 | 7 | 9 |\n| 0 | 8 | 2 |\n| 0 | 9 | 0 |\n| 0 | 10 | 8 |\n| 0 | 11 | 7 |\n| 0 | 12 | 5 |\n| 0 | 13 | 1 |\n| 0 | 14 | 8 |\n| 0 | 15 | 7 |\n| 0 | 16 | 4 |\n| 0 | 17 | 2 |\n| 0 | 18 | 7 |\n| 0 | 19 | 3 |\n| 0 | 20 | 0 |\n| 0 | 21 | 9 |\n| 1 | 0 | 1 |\n| 1 | 2 | 4 |\n| 1 | 3 | 2 |\n| 1 | 4 | 9 |\n| 1 | 5 | 5 |\n| 1 | 6 | 3 |\n| 1 | 7 | 0 |\n| 1 | 8 | 3 |\n| 1 | 9 | 0 |\n| 1 | 10 | 9 |\n| 1 | 11 | 5 |\n| 1 | 12 | 2 |\n| 1 | 13 | 1 |\n| 1 | 14 | 1 |\n| 1 | 15 | 4 |\n| 1 | 16 | 2 |\n| 1 | 17 | 5 |\n| 1 | 18 | 3 |\n| 1 | 19 | 6 |\n| 1 | 20 | 0 |\n| 1 | 21 | 1 |\n| 2 | 0 | 9 |\n| 2 | 1 | 4 |\n| 2 | 3 | 6 |\n| 2 | 4 | 5 |\n| 2 | 5 | 2 |\n| 2 | 6 | 6 |\n| 2 | 7 | 1 |\n| 2 | 8 | 2 |\n| 2 | 9 | 9 |\n| 2 | 10 | 9 |\n| 2 | 11 | 4 |\n| 2 | 12 | 9 |\n| 2 | 13 | 0 |\n| 2 | 14 | 3 |\n| 2 | 15 | 8 |\n| 2 | 16 | 3 |\n| 2 | 17 | 9 |\n| 2 | 18 | 1 |\n| 2 | 19 | 3 |\n| 2 | 20 | 0 |\n| 2 | 21 | 4 |\n| 3 | 0 | 2 |\n| 3 | 1 | 2 |\n| 3 | 2 | 6 |\n| 3 | 4 | 7 |\n| 3 | 5 | 9 |\n| 3 | 6 | 8 |\n| 3 | 7 | 8 |\n| 3 | 8 | 5 |\n| 3 | 9 | 3 |\n| 3 | 10 | 5 |\n| 3 | 11 | 1 |\n| 3 | 12 | 9 |\n| 3 | 13 | 1 |\n| 3 | 14 | 1 |\n| 3 | 15 | 8 |\n| 3 | 16 | 7 |\n| 3 | 17 | 1 |\n| 3 | 18 | 9 |\n| 3 | 19 | 6 |\n| 3 | 20 | 4 |\n| 3 | 21 | 6 |\n| 4 | 0 | 0 |\n| 4 | 1 | 9 |\n| 4 | 2 | 5 |\n| 4 | 3 | 7 |\n| 4 | 5 | 2 |\n| 4 | 6 | 5 |\n| 4 | 7 | 1 |\n| 4 | 8 | 5 |\n| 4 | 9 | 9 |\n| 4 | 10 | 7 |\n| 4 | 11 | 4 |\n| 4 | 12 | 5 |\n| 4 | 13 | 1 |\n| 4 | 14 | 7 |\n| 4 | 15 | 9 |\n| 4 | 16 | 0 |\n| 4 | 17 | 0 |\n| 4 | 18 | 7 |\n| 4 | 19 | 7 |\n| 4 | 20 | 1 |\n| 4 | 21 | 4 |\n| 5 | 0 | 0 |\n| 5 | 1 | 5 |\n| 5 | 2 | 2 |\n| 5 | 3 | 9 |\n| 5 | 4 | 2 |\n| 5 | 6 | 4 |\n| 5 | 7 | 3 |\n| 5 | 8 | 2 |\n| 5 | 9 | 6 |\n| 5 | 10 | 4 |\n| 5 | 11 | 3 |\n| 5 | 12 | 1 |\n| 5 | 13 | 4 |\n| 5 | 14 | 4 |\n| 5 | 15 | 9 |\n| 5 | 16 | 0 |\n| 5 | 17 | 8 |\n| 5 | 18 | 9 |\n| 5 | 19 | 3 |\n| 5 | 20 | 6 |\n| 5 | 21 | 0 |\n| 6 | 0 | 7 |\n| 6 | 1 | 3 |\n| 6 | 2 | 6 |\n| 6 | 3 | 8 |\n| 6 | 4 | 5 |\n| 6 | 5 | 4 |\n| 6 | 7 | 7 |\n| 6 | 8 | 4 |\n| 6 | 9 | 9 |\n| 6 | 10 | 6 |\n| 6 | 11 | 6 |\n| 6 | 12 | 8 |\n| 6 | 13 | 9 |\n| 6 | 14 | 3 |\n| 6 | 15 | 0 |\n| 6 | 16 | 9 |\n| 6 | 17 | 7 |\n| 6 | 18 | 0 |\n| 6 | 19 | 2 |\n| 6 | 20 | 8 |\n| 6 | 21 | 3 |\n| 7 | 0 | 9 |\n| 7 | 1 | 0 |\n| 7 | 2 | 1 |\n| 7 | 3 | 8 |\n| 7 | 4 | 1 |\n| 7 | 5 | 3 |\n| 7 | 6 | 7 |\n| 7 | 8 | 5 |\n| 7 | 9 | 4 |\n| 7 | 10 | 2 |\n| 7 | 11 | 9 |\n| 7 | 12 | 1 |\n| 7 | 13 | 8 |\n| 7 | 14 | 3 |\n| 7 | 15 | 6 |\n| 7 | 16 | 7 |\n| 7 | 17 | 0 |\n| 7 | 18 | 8 |\n| 7 | 19 | 1 |\n| 7 | 20 | 1 |\n| 7 | 21 | 6 |\n| 8 | 0 | 2 |\n| 8 | 1 | 3 |\n| 8 | 2 | 2 |\n| 8 | 3 | 5 |\n| 8 | 4 | 5 |\n| 8 | 5 | 2 |\n| 8 | 6 | 4 |\n| 8 | 7 | 5 |\n| 8 | 9 | 1 |\n| 8 | 10 | 8 |\n| 8 | 11 | 0 |\n| 8 | 12 | 9 |\n| 8 | 13 | 1 |\n| 8 | 14 | 4 |\n| 8 | 15 | 2 |\n| 8 | 16 | 4 |\n| 8 | 17 | 9 |\n| 8 | 18 | 1 |\n| 8 | 19 | 8 |\n| 8 | 20 | 8 |\n| 8 | 21 | 7 |\n| 9 | 0 | 0 |\n| 9 | 1 | 0 |\n| 9 | 2 | 9 |\n| 9 | 3 | 3 |\n| 9 | 4 | 9 |\n| 9 | 5 | 6 |\n| 9 | 6 | 9 |\n| 9 | 7 | 4 |\n| 9 | 8 | 1 |\n| 9 | 10 | 3 |\n| 9 | 11 | 9 |\n| 9 | 12 | 0 |\n| 9 | 13 | 1 |\n| 9 | 14 | 8 |\n| 9 | 15 | 0 |\n| 9 | 16 | 9 |\n| 9 | 17 | 9 |\n| 9 | 18 | 4 |\n| 9 | 19 | 5 |\n| 9 | 20 | 6 |\n| 9 | 21 | 7 |\n| 10 | 0 | 8 |\n| 10 | 1 | 9 |\n| 10 | 2 | 9 |\n| 10 | 3 | 5 |\n| 10 | 4 | 7 |\n| 10 | 5 | 4 |\n| 10 | 6 | 6 |\n| 10 | 7 | 2 |\n| 10 | 8 | 8 |\n| 10 | 9 | 3 |\n| 10 | 11 | 2 |\n| 10 | 12 | 8 |\n| 10 | 13 | 3 |\n| 10 | 14 | 3 |\n| 10 | 15 | 5 |\n| 10 | 16 | 3 |\n| 10 | 17 | 2 |\n| 10 | 18 | 2 |\n| 10 | 19 | 9 |\n| 10 | 20 | 8 |\n| 10 | 21 | 1 |\n| 11 | 0 | 7 |\n| 11 | 1 | 5 |\n| 11 | 2 | 4 |\n| 11 | 3 | 1 |\n| 11 | 4 | 4 |\n| 11 | 5 | 3 |\n| 11 | 6 | 6 |\n| 11 | 7 | 9 |\n| 11 | 8 | 0 |\n| 11 | 9 | 9 |\n| 11 | 10 | 2 |\n| 11 | 12 | 5 |\n| 11 | 13 | 1 |\n| 11 | 14 | 1 |\n| 11 | 15 | 8 |\n| 11 | 16 | 2 |\n| 11 | 17 | 3 |\n| 11 | 18 | 6 |\n| 11 | 19 | 4 |\n| 11 | 20 | 7 |\n| 11 | 21 | 2 |\n| 12 | 0 | 5 |\n| 12 | 1 | 2 |\n| 12 | 2 | 9 |\n| 12 | 3 | 9 |\n| 12 | 4 | 5 |\n| 12 | 5 | 1 |\n| 12 | 6 | 8 |\n| 12 | 7 | 1 |\n| 12 | 8 | 9 |\n| 12 | 9 | 0 |\n| 12 | 10 | 8 |\n| 12 | 11 | 5 |\n| 12 | 13 | 5 |\n| 12 | 14 | 6 |\n| 12 | 15 | 3 |\n| 12 | 16 | 5 |\n| 12 | 17 | 9 |\n| 12 | 18 | 5 |\n| 12 | 19 | 6 |\n| 12 | 20 | 5 |\n| 12 | 21 | 2 |\n| 13 | 0 | 1 |\n| 13 | 1 | 1 |\n| 13 | 2 | 0 |\n| 13 | 3 | 1 |\n| 13 | 4 | 1 |\n| 13 | 5 | 4 |\n| 13 | 6 | 9 |\n| 13 | 7 | 8 |\n| 13 | 8 | 1 |\n| 13 | 9 | 1 |\n| 13 | 10 | 3 |\n| 13 | 11 | 1 |\n| 13 | 12 | 5 |\n| 13 | 14 | 9 |\n| 13 | 15 | 1 |\n| 13 | 16 | 1 |\n| 13 | 17 | 5 |\n| 13 | 18 | 4 |\n| 13 | 19 | 0 |\n| 13 | 20 | 8 |\n| 13 | 21 | 1 |\n| 14 | 0 | 8 |\n| 14 | 1 | 1 |\n| 14 | 2 | 3 |\n| 14 | 3 | 1 |\n| 14 | 4 | 7 |\n| 14 | 5 | 4 |\n| 14 | 6 | 3 |\n| 14 | 7 | 3 |\n| 14 | 8 | 4 |\n| 14 | 9 | 8 |\n| 14 | 10 | 3 |\n| 14 | 11 | 1 |\n| 14 | 12 | 6 |\n| 14 | 13 | 9 |\n| 14 | 15 | 2 |\n| 14 | 16 | 2 |\n| 14 | 17 | 9 |\n| 14 | 18 | 8 |\n| 14 | 19 | 1 |\n| 14 | 20 | 2 |\n| 14 | 21 | 7 |\n| 15 | 0 | 7 |\n| 15 | 1 | 4 |\n| 15 | 2 | 8 |\n| 15 | 3 | 8 |\n| 15 | 4 | 9 |\n| 15 | 5 | 9 |\n| 15 | 6 | 0 |\n| 15 | 7 | 6 |\n| 15 | 8 | 2 |\n| 15 | 9 | 0 |\n| 15 | 10 | 5 |\n| 15 | 11 | 8 |\n| 15 | 12 | 3 |\n| 15 | 13 | 1 |\n| 15 | 14 | 2 |\n| 15 | 16 | 0 |\n| 15 | 17 | 3 |\n| 15 | 18 | 1 |\n| 15 | 19 | 7 |\n| 15 | 20 | 3 |\n| 15 | 21 | 5 |\n| 16 | 0 | 4 |\n| 16 | 1 | 2 |\n| 16 | 2 | 3 |\n| 16 | 3 | 7 |\n| 16 | 4 | 0 |\n| 16 | 5 | 0 |\n| 16 | 6 | 9 |\n| 16 | 7 | 7 |\n| 16 | 8 | 4 |\n| 16 | 9 | 9 |\n| 16 | 10 | 3 |\n| 16 | 11 | 2 |\n| 16 | 12 | 5 |\n| 16 | 13 | 1 |\n| 16 | 14 | 2 |\n| 16 | 15 | 0 |\n| 16 | 17 | 4 |\n| 16 | 18 | 5 |\n| 16 | 19 | 9 |\n| 16 | 20 | 8 |\n| 16 | 21 | 6 |\n| 17 | 0 | 2 |\n| 17 | 1 | 5 |\n| 17 | 2 | 9 |\n| 17 | 3 | 1 |\n| 17 | 4 | 0 |\n| 17 | 5 | 8 |\n| 17 | 6 | 7 |\n| 17 | 7 | 0 |\n| 17 | 8 | 9 |\n| 17 | 9 | 9 |\n| 17 | 10 | 2 |\n| 17 | 11 | 3 |\n| 17 | 12 | 9 |\n| 17 | 13 | 5 |\n| 17 | 14 | 9 |\n| 17 | 15 | 3 |\n| 17 | 16 | 4 |\n| 17 | 18 | 3 |\n| 17 | 19 | 2 |\n| 17 | 20 | 4 |\n| 17 | 21 | 9 |\n| 18 | 0 | 7 |\n| 18 | 1 | 3 |\n| 18 | 2 | 1 |\n| 18 | 3 | 9 |\n| 18 | 4 | 7 |\n| 18 | 5 | 9 |\n| 18 | 6 | 0 |\n| 18 | 7 | 8 |\n| 18 | 8 | 1 |\n| 18 | 9 | 4 |\n| 18 | 10 | 2 |\n| 18 | 11 | 6 |\n| 18 | 12 | 5 |\n| 18 | 13 | 4 |\n| 18 | 14 | 8 |\n| 18 | 15 | 1 |\n| 18 | 16 | 5 |\n| 18 | 17 | 3 |\n| 18 | 19 | 7 |\n| 18 | 20 | 4 |\n| 18 | 21 | 9 |\n| 19 | 0 | 3 |\n| 19 | 1 | 6 |\n| 19 | 2 | 3 |\n| 19 | 3 | 6 |\n| 19 | 4 | 7 |\n| 19 | 5 | 3 |\n| 19 | 6 | 2 |\n| 19 | 7 | 1 |\n| 19 | 8 | 8 |\n| 19 | 9 | 5 |\n| 19 | 10 | 9 |\n| 19 | 11 | 4 |\n| 19 | 12 | 6 |\n| 19 | 13 | 0 |\n| 19 | 14 | 1 |\n| 19 | 15 | 7 |\n| 19 | 16 | 9 |\n| 19 | 17 | 2 |\n| 19 | 18 | 7 |\n| 19 | 20 | 2 |\n| 19 | 21 | 0 |\n| 20 | 0 | 0 |\n| 20 | 1 | 0 |\n| 20 | 2 | 0 |\n| 20 | 3 | 4 |\n| 20 | 4 | 1 |\n| 20 | 5 | 6 |\n| 20 | 6 | 8 |\n| 20 | 7 | 1 |\n| 20 | 8 | 8 |\n| 20 | 9 | 6 |\n| 20 | 10 | 8 |\n| 20 | 11 | 7 |\n| 20 | 12 | 5 |\n| 20 | 13 | 8 |\n| 20 | 14 | 2 |\n| 20 | 15 | 3 |\n| 20 | 16 | 8 |\n| 20 | 17 | 4 |\n| 20 | 18 | 4 |\n| 20 | 19 | 2 |\n| 20 | 21 | 7 |\n| 21 | 0 | 9 |\n| 21 | 1 | 1 |\n| 21 | 2 | 4 |\n| 21 | 3 | 6 |\n| 21 | 4 | 4 |\n| 21 | 5 | 0 |\n| 21 | 6 | 3 |\n| 21 | 7 | 6 |\n| 21 | 8 | 7 |\n| 21 | 9 | 7 |\n| 21 | 10 | 1 |\n| 21 | 11 | 2 |\n| 21 | 12 | 2 |\n| 21 | 13 | 1 |\n| 21 | 14 | 7 |\n| 21 | 15 | 5 |\n| 21 | 16 | 6 |\n| 21 | 17 | 9 |\n| 21 | 18 | 9 |\n| 21 | 19 | 0 |\n| 21 | 20 | 7 |\n\nThese pairwise dissimilarities will be summed for every pair in any chosen set of 2 to judge display variety.\n\nIf you want the picks returned in a simple, consistent layout, just use this little JSON shape in your reply:\n\n{\n \"solution\": [, , ...]\n}\n\nThink of \"solution\" as the list of exactly m products to put in the display — each entry is a product identifier from the instance. This block is just a sketch of the shape I expect, not the final answer itself.\n\nAll identifiers must be used exactly as they appear in the instance input — no renaming and no new labels.\n\nValid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.", "instance": { "distance_matrix": [ [ 0, 1, 9, 2, 0, 0, 7, 9, 2, 0, 8, 7, 5, 1, 8, 7, 4, 2, 7, 3, 0, 9 ], [ 1, 0, 4, 2, 9, 5, 3, 0, 3, 0, 9, 5, 2, 1, 1, 4, 2, 5, 3, 6, 0, 1 ], [ 9, 4, 0, 6, 5, 2, 6, 1, 2, 9, 9, 4, 9, 0, 3, 8, 3, 9, 1, 3, 0, 4 ], [ 2, 2, 6, 0, 7, 9, 8, 8, 5, 3, 5, 1, 9, 1, 1, 8, 7, 1, 9, 6, 4, 6 ], [ 0, 9, 5, 7, 0, 2, 5, 1, 5, 9, 7, 4, 5, 1, 7, 9, 0, 0, 7, 7, 1, 4 ], [ 0, 5, 2, 9, 2, 0, 4, 3, 2, 6, 4, 3, 1, 4, 4, 9, 0, 8, 9, 3, 6, 0 ], [ 7, 3, 6, 8, 5, 4, 0, 7, 4, 9, 6, 6, 8, 9, 3, 0, 9, 7, 0, 2, 8, 3 ], [ 9, 0, 1, 8, 1, 3, 7, 0, 5, 4, 2, 9, 1, 8, 3, 6, 7, 0, 8, 1, 1, 6 ], [ 2, 3, 2, 5, 5, 2, 4, 5, 0, 1, 8, 0, 9, 1, 4, 2, 4, 9, 1, 8, 8, 7 ], [ 0, 0, 9, 3, 9, 6, 9, 4, 1, 0, 3, 9, 0, 1, 8, 0, 9, 9, 4, 5, 6, 7 ], [ 8, 9, 9, 5, 7, 4, 6, 2, 8, 3, 0, 2, 8, 3, 3, 5, 3, 2, 2, 9, 8, 1 ], [ 7, 5, 4, 1, 4, 3, 6, 9, 0, 9, 2, 0, 5, 1, 1, 8, 2, 3, 6, 4, 7, 2 ], [ 5, 2, 9, 9, 5, 1, 8, 1, 9, 0, 8, 5, 0, 5, 6, 3, 5, 9, 5, 6, 5, 2 ], [ 1, 1, 0, 1, 1, 4, 9, 8, 1, 1, 3, 1, 5, 0, 9, 1, 1, 5, 4, 0, 8, 1 ], [ 8, 1, 3, 1, 7, 4, 3, 3, 4, 8, 3, 1, 6, 9, 0, 2, 2, 9, 8, 1, 2, 7 ], [ 7, 4, 8, 8, 9, 9, 0, 6, 2, 0, 5, 8, 3, 1, 2, 0, 0, 3, 1, 7, 3, 5 ], [ 4, 2, 3, 7, 0, 0, 9, 7, 4, 9, 3, 2, 5, 1, 2, 0, 0, 4, 5, 9, 8, 6 ], [ 2, 5, 9, 1, 0, 8, 7, 0, 9, 9, 2, 3, 9, 5, 9, 3, 4, 0, 3, 2, 4, 9 ], [ 7, 3, 1, 9, 7, 9, 0, 8, 1, 4, 2, 6, 5, 4, 8, 1, 5, 3, 0, 7, 4, 9 ], [ 3, 6, 3, 6, 7, 3, 2, 1, 8, 5, 9, 4, 6, 0, 1, 7, 9, 2, 7, 0, 2, 0 ], [ 0, 0, 0, 4, 1, 6, 8, 1, 8, 6, 8, 7, 5, 8, 2, 3, 8, 4, 4, 2, 0, 7 ], [ 9, 1, 4, 6, 4, 0, 3, 6, 7, 7, 1, 2, 2, 1, 7, 5, 6, 9, 9, 0, 7, 0 ] ], "m": 2, "objective": 9.0 }, "solution": [ 8, 12 ], "obj": 9.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 22, "m": 2, "nodes": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 ], "distance_pairs": [ { "from_id": 0, "to_id": 1, "distance": 1 }, { "from_id": 0, "to_id": 2, "distance": 9 }, { "from_id": 0, "to_id": 3, "distance": 2 }, { "from_id": 0, "to_id": 4, "distance": 0 }, { "from_id": 0, "to_id": 5, "distance": 0 }, { "from_id": 0, "to_id": 6, "distance": 7 }, { "from_id": 0, "to_id": 7, "distance": 9 }, { "from_id": 0, "to_id": 8, "distance": 2 }, { "from_id": 0, "to_id": 9, "distance": 0 }, { "from_id": 0, "to_id": 10, "distance": 8 }, { "from_id": 0, "to_id": 11, "distance": 7 }, { "from_id": 0, "to_id": 12, "distance": 5 }, { "from_id": 0, "to_id": 13, "distance": 1 }, { "from_id": 0, "to_id": 14, "distance": 8 }, { "from_id": 0, "to_id": 15, "distance": 7 }, { "from_id": 0, "to_id": 16, "distance": 4 }, { "from_id": 0, "to_id": 17, "distance": 2 }, { "from_id": 0, "to_id": 18, "distance": 7 }, { "from_id": 0, "to_id": 19, "distance": 3 }, { "from_id": 0, "to_id": 20, "distance": 0 }, { "from_id": 0, "to_id": 21, "distance": 9 }, { "from_id": 1, "to_id": 0, "distance": 1 }, { "from_id": 1, "to_id": 2, "distance": 4 }, { "from_id": 1, "to_id": 3, "distance": 2 }, { "from_id": 1, "to_id": 4, "distance": 9 }, { "from_id": 1, "to_id": 5, "distance": 5 }, { "from_id": 1, "to_id": 6, "distance": 3 }, { "from_id": 1, "to_id": 7, "distance": 0 }, { "from_id": 1, "to_id": 8, "distance": 3 }, { "from_id": 1, "to_id": 9, "distance": 0 }, { "from_id": 1, "to_id": 10, "distance": 9 }, { "from_id": 1, "to_id": 11, "distance": 5 }, { "from_id": 1, "to_id": 12, "distance": 2 }, { "from_id": 1, "to_id": 13, "distance": 1 }, { "from_id": 1, "to_id": 14, "distance": 1 }, { "from_id": 1, "to_id": 15, "distance": 4 }, { "from_id": 1, "to_id": 16, "distance": 2 }, { "from_id": 1, "to_id": 17, "distance": 5 }, { "from_id": 1, "to_id": 18, "distance": 3 }, { "from_id": 1, "to_id": 19, "distance": 6 }, { "from_id": 1, "to_id": 20, "distance": 0 }, { "from_id": 1, "to_id": 21, "distance": 1 }, { "from_id": 2, "to_id": 0, "distance": 9 }, { "from_id": 2, "to_id": 1, "distance": 4 }, { "from_id": 2, "to_id": 3, "distance": 6 }, { "from_id": 2, "to_id": 4, "distance": 5 }, { "from_id": 2, "to_id": 5, "distance": 2 }, { "from_id": 2, "to_id": 6, "distance": 6 }, { "from_id": 2, "to_id": 7, "distance": 1 }, { "from_id": 2, "to_id": 8, "distance": 2 }, { "from_id": 2, "to_id": 9, "distance": 9 }, { "from_id": 2, "to_id": 10, "distance": 9 }, { "from_id": 2, "to_id": 11, "distance": 4 }, { "from_id": 2, "to_id": 12, "distance": 9 }, { "from_id": 2, "to_id": 13, "distance": 0 }, { "from_id": 2, "to_id": 14, "distance": 3 }, { "from_id": 2, "to_id": 15, "distance": 8 }, { "from_id": 2, "to_id": 16, "distance": 3 }, { "from_id": 2, "to_id": 17, "distance": 9 }, { "from_id": 2, "to_id": 18, "distance": 1 }, { "from_id": 2, "to_id": 19, "distance": 3 }, { "from_id": 2, "to_id": 20, "distance": 0 }, { "from_id": 2, "to_id": 21, "distance": 4 }, { "from_id": 3, "to_id": 0, "distance": 2 }, { "from_id": 3, "to_id": 1, "distance": 2 }, { "from_id": 3, "to_id": 2, "distance": 6 }, { "from_id": 3, "to_id": 4, "distance": 7 }, { "from_id": 3, "to_id": 5, "distance": 9 }, { "from_id": 3, "to_id": 6, "distance": 8 }, { "from_id": 3, "to_id": 7, "distance": 8 }, { "from_id": 3, "to_id": 8, "distance": 5 }, { "from_id": 3, "to_id": 9, "distance": 3 }, { "from_id": 3, "to_id": 10, "distance": 5 }, { "from_id": 3, "to_id": 11, "distance": 1 }, { "from_id": 3, "to_id": 12, "distance": 9 }, { "from_id": 3, "to_id": 13, "distance": 1 }, { "from_id": 3, "to_id": 14, "distance": 1 }, { "from_id": 3, "to_id": 15, "distance": 8 }, { "from_id": 3, "to_id": 16, "distance": 7 }, { "from_id": 3, "to_id": 17, "distance": 1 }, { "from_id": 3, "to_id": 18, "distance": 9 }, { "from_id": 3, "to_id": 19, "distance": 6 }, { "from_id": 3, "to_id": 20, "distance": 4 }, { "from_id": 3, "to_id": 21, "distance": 6 }, { "from_id": 4, "to_id": 0, "distance": 0 }, { "from_id": 4, "to_id": 1, "distance": 9 }, { "from_id": 4, "to_id": 2, "distance": 5 }, { "from_id": 4, "to_id": 3, "distance": 7 }, { "from_id": 4, "to_id": 5, "distance": 2 }, { "from_id": 4, "to_id": 6, "distance": 5 }, { "from_id": 4, "to_id": 7, "distance": 1 }, { "from_id": 4, "to_id": 8, "distance": 5 }, { "from_id": 4, "to_id": 9, "distance": 9 }, { "from_id": 4, "to_id": 10, "distance": 7 }, { "from_id": 4, "to_id": 11, "distance": 4 }, { "from_id": 4, "to_id": 12, "distance": 5 }, { "from_id": 4, "to_id": 13, "distance": 1 }, { "from_id": 4, "to_id": 14, "distance": 7 }, { "from_id": 4, "to_id": 15, "distance": 9 }, { "from_id": 4, "to_id": 16, "distance": 0 }, { "from_id": 4, "to_id": 17, "distance": 0 }, { "from_id": 4, "to_id": 18, "distance": 7 }, { "from_id": 4, "to_id": 19, "distance": 7 }, { "from_id": 4, "to_id": 20, "distance": 1 }, { "from_id": 4, "to_id": 21, "distance": 4 }, { "from_id": 5, "to_id": 0, "distance": 0 }, { "from_id": 5, "to_id": 1, "distance": 5 }, { "from_id": 5, "to_id": 2, "distance": 2 }, { "from_id": 5, "to_id": 3, "distance": 9 }, { "from_id": 5, "to_id": 4, "distance": 2 }, { "from_id": 5, "to_id": 6, "distance": 4 }, { "from_id": 5, "to_id": 7, "distance": 3 }, { "from_id": 5, "to_id": 8, "distance": 2 }, { "from_id": 5, "to_id": 9, "distance": 6 }, { "from_id": 5, "to_id": 10, "distance": 4 }, { "from_id": 5, "to_id": 11, "distance": 3 }, { "from_id": 5, "to_id": 12, "distance": 1 }, { "from_id": 5, "to_id": 13, "distance": 4 }, { "from_id": 5, "to_id": 14, "distance": 4 }, { "from_id": 5, "to_id": 15, "distance": 9 }, { "from_id": 5, "to_id": 16, "distance": 0 }, { "from_id": 5, "to_id": 17, "distance": 8 }, { "from_id": 5, "to_id": 18, "distance": 9 }, { "from_id": 5, "to_id": 19, "distance": 3 }, { "from_id": 5, "to_id": 20, "distance": 6 }, { "from_id": 5, "to_id": 21, "distance": 0 }, { "from_id": 6, "to_id": 0, "distance": 7 }, { "from_id": 6, "to_id": 1, "distance": 3 }, { "from_id": 6, "to_id": 2, "distance": 6 }, { "from_id": 6, "to_id": 3, "distance": 8 }, { "from_id": 6, "to_id": 4, "distance": 5 }, { "from_id": 6, "to_id": 5, "distance": 4 }, { "from_id": 6, "to_id": 7, "distance": 7 }, { "from_id": 6, "to_id": 8, "distance": 4 }, { "from_id": 6, "to_id": 9, "distance": 9 }, { "from_id": 6, "to_id": 10, "distance": 6 }, { "from_id": 6, "to_id": 11, "distance": 6 }, { "from_id": 6, "to_id": 12, "distance": 8 }, { "from_id": 6, "to_id": 13, "distance": 9 }, { "from_id": 6, "to_id": 14, "distance": 3 }, { "from_id": 6, "to_id": 15, "distance": 0 }, { "from_id": 6, "to_id": 16, "distance": 9 }, { "from_id": 6, "to_id": 17, "distance": 7 }, { "from_id": 6, "to_id": 18, "distance": 0 }, { "from_id": 6, "to_id": 19, "distance": 2 }, { "from_id": 6, "to_id": 20, "distance": 8 }, { "from_id": 6, "to_id": 21, "distance": 3 }, { "from_id": 7, "to_id": 0, "distance": 9 }, { "from_id": 7, "to_id": 1, "distance": 0 }, { "from_id": 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"from_id": 21, "to_id": 10, "distance": 1 }, { "from_id": 21, "to_id": 11, "distance": 2 }, { "from_id": 21, "to_id": 12, "distance": 2 }, { "from_id": 21, "to_id": 13, "distance": 1 }, { "from_id": 21, "to_id": 14, "distance": 7 }, { "from_id": 21, "to_id": 15, "distance": 5 }, { "from_id": 21, "to_id": 16, "distance": 6 }, { "from_id": 21, "to_id": 17, "distance": 9 }, { "from_id": 21, "to_id": 18, "distance": 9 }, { "from_id": 21, "to_id": 19, "distance": 0 }, { "from_id": 21, "to_id": 20, "distance": 7 } ] }, "solution_variant": [ 8, 12 ], "context_index": 49, "input_format": "markdown_table", "input_index_base": 0 }, { "task_name": "MDP", "problem_type": "MDP", "instruction": "Recently the festival team decided to focus on variety, so the task became: pick exactly m bands that together sound as different from each other as possible. There’s a number for every pair of bands that measures their musical distance, and the way to compare lineups is to add up all those pairwise numbers for the chosen acts — the larger that total, the broader the range. The lineup must contain exactly m unique bands (no duplicates), and the concrete details are shown below.\n\n# total_available_bands=20\n# lineup_size=2\n# available_band_ids=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\nband_id_first,band_id_second,pairwise_musical_distance\n1,2,6\n1,3,9\n1,4,2\n1,5,5\n1,6,7\n1,7,9\n1,8,1\n1,9,9\n1,10,5\n1,11,2\n1,12,1\n1,13,2\n1,14,0\n1,15,9\n1,16,7\n1,17,5\n1,18,8\n1,19,3\n1,20,4\n2,1,6\n2,3,7\n2,4,0\n2,5,8\n2,6,7\n2,7,0\n2,8,0\n2,9,9\n2,10,0\n2,11,8\n2,12,5\n2,13,8\n2,14,8\n2,15,8\n2,16,8\n2,17,7\n2,18,4\n2,19,5\n2,20,0\n3,1,9\n3,2,7\n3,4,0\n3,5,8\n3,6,0\n3,7,7\n3,8,9\n3,9,2\n3,10,8\n3,11,4\n3,12,6\n3,13,3\n3,14,0\n3,15,2\n3,16,1\n3,17,4\n3,18,2\n3,19,6\n3,20,6\n4,1,2\n4,2,0\n4,3,0\n4,5,0\n4,6,3\n4,7,4\n4,8,0\n4,9,4\n4,10,2\n4,11,5\n4,12,4\n4,13,1\n4,14,9\n4,15,4\n4,16,8\n4,17,6\n4,18,1\n4,19,2\n4,20,4\n5,1,5\n5,2,8\n5,3,8\n5,4,0\n5,6,2\n5,7,3\n5,8,5\n5,9,9\n5,10,9\n5,11,5\n5,12,3\n5,13,8\n5,14,3\n5,15,4\n5,16,4\n5,17,1\n5,18,1\n5,19,0\n5,20,6\n6,1,7\n6,2,7\n6,3,0\n6,4,3\n6,5,2\n6,7,8\n6,8,9\n6,9,0\n6,10,9\n6,11,5\n6,12,2\n6,13,5\n6,14,5\n6,15,9\n6,16,6\n6,17,5\n6,18,6\n6,19,1\n6,20,0\n7,1,9\n7,2,0\n7,3,7\n7,4,4\n7,5,3\n7,6,8\n7,8,3\n7,9,8\n7,10,9\n7,11,3\n7,12,4\n7,13,1\n7,14,1\n7,15,1\n7,16,3\n7,17,2\n7,18,2\n7,19,9\n7,20,8\n8,1,1\n8,2,0\n8,3,9\n8,4,0\n8,5,5\n8,6,9\n8,7,3\n8,9,8\n8,10,8\n8,11,2\n8,12,0\n8,13,7\n8,14,5\n8,15,8\n8,16,8\n8,17,1\n8,18,0\n8,19,5\n8,20,5\n9,1,9\n9,2,9\n9,3,2\n9,4,4\n9,5,9\n9,6,0\n9,7,8\n9,8,8\n9,10,8\n9,11,4\n9,12,4\n9,13,8\n9,14,1\n9,15,9\n9,16,3\n9,17,4\n9,18,0\n9,19,9\n9,20,4\n10,1,5\n10,2,0\n10,3,8\n10,4,2\n10,5,9\n10,6,9\n10,7,9\n10,8,8\n10,9,8\n10,11,7\n10,12,5\n10,13,6\n10,14,0\n10,15,2\n10,16,1\n10,17,2\n10,18,6\n10,19,5\n10,20,1\n11,1,2\n11,2,8\n11,3,4\n11,4,5\n11,5,5\n11,6,5\n11,7,3\n11,8,2\n11,9,4\n11,10,7\n11,12,8\n11,13,3\n11,14,3\n11,15,2\n11,16,3\n11,17,1\n11,18,5\n11,19,7\n11,20,0\n12,1,1\n12,2,5\n12,3,6\n12,4,4\n12,5,3\n12,6,2\n12,7,4\n12,8,0\n12,9,4\n12,10,5\n12,11,8\n12,13,7\n12,14,1\n12,15,9\n12,16,0\n12,17,6\n12,18,0\n12,19,9\n12,20,1\n13,1,2\n13,2,8\n13,3,3\n13,4,1\n13,5,8\n13,6,5\n13,7,1\n13,8,7\n13,9,8\n13,10,6\n13,11,3\n13,12,7\n13,14,9\n13,15,2\n13,16,4\n13,17,3\n13,18,4\n13,19,8\n13,20,4\n14,1,0\n14,2,8\n14,3,0\n14,4,9\n14,5,3\n14,6,5\n14,7,1\n14,8,5\n14,9,1\n14,10,0\n14,11,3\n14,12,1\n14,13,9\n14,15,2\n14,16,6\n14,17,6\n14,18,6\n14,19,2\n14,20,1\n15,1,9\n15,2,8\n15,3,2\n15,4,4\n15,5,4\n15,6,9\n15,7,1\n15,8,8\n15,9,9\n15,10,2\n15,11,2\n15,12,9\n15,13,2\n15,14,2\n15,16,4\n15,17,5\n15,18,3\n15,19,4\n15,20,4\n16,1,7\n16,2,8\n16,3,1\n16,4,8\n16,5,4\n16,6,6\n16,7,3\n16,8,8\n16,9,3\n16,10,1\n16,11,3\n16,12,0\n16,13,4\n16,14,6\n16,15,4\n16,17,2\n16,18,8\n16,19,8\n16,20,8\n17,1,5\n17,2,7\n17,3,4\n17,4,6\n17,5,1\n17,6,5\n17,7,2\n17,8,1\n17,9,4\n17,10,2\n17,11,1\n17,12,6\n17,13,3\n17,14,6\n17,15,5\n17,16,2\n17,18,8\n17,19,6\n17,20,8\n18,1,8\n18,2,4\n18,3,2\n18,4,1\n18,5,1\n18,6,6\n18,7,2\n18,8,0\n18,9,0\n18,10,6\n18,11,5\n18,12,0\n18,13,4\n18,14,6\n18,15,3\n18,16,8\n18,17,8\n18,19,2\n18,20,3\n19,1,3\n19,2,5\n19,3,6\n19,4,2\n19,5,0\n19,6,1\n19,7,9\n19,8,5\n19,9,9\n19,10,5\n19,11,7\n19,12,9\n19,13,8\n19,14,2\n19,15,4\n19,16,8\n19,17,6\n19,18,2\n19,20,8\n20,1,4\n20,2,0\n20,3,6\n20,4,4\n20,5,6\n20,6,0\n20,7,8\n20,8,5\n20,9,4\n20,10,1\n20,11,0\n20,12,1\n20,13,4\n20,14,1\n20,15,4\n20,16,8\n20,17,8\n20,18,3\n20,19,8\n\nAlso, when you send back the final pick, please use this simple JSON layout so it's easy to read and parse:\n\n{\n \"solution\": [, , ...]\n}\n\n\"solution\" is just the list of the m bands you want on the bill. Each is a placeholder for the exact band identifier from the instance (so replace each placeholder with the real ID when you answer). This is just a sketch of the expected shape, not your final answer — think of it like filling in a little form.\n\nPlease be careful 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\".", "instance": { "distance_matrix": [ [ 0, 6, 9, 2, 5, 7, 9, 1, 9, 5, 2, 1, 2, 0, 9, 7, 5, 8, 3, 4 ], [ 6, 0, 7, 0, 8, 7, 0, 0, 9, 0, 8, 5, 8, 8, 8, 8, 7, 4, 5, 0 ], [ 9, 7, 0, 0, 8, 0, 7, 9, 2, 8, 4, 6, 3, 0, 2, 1, 4, 2, 6, 6 ], [ 2, 0, 0, 0, 0, 3, 4, 0, 4, 2, 5, 4, 1, 9, 4, 8, 6, 1, 2, 4 ], [ 5, 8, 8, 0, 0, 2, 3, 5, 9, 9, 5, 3, 8, 3, 4, 4, 1, 1, 0, 6 ], [ 7, 7, 0, 3, 2, 0, 8, 9, 0, 9, 5, 2, 5, 5, 9, 6, 5, 6, 1, 0 ], [ 9, 0, 7, 4, 3, 8, 0, 3, 8, 9, 3, 4, 1, 1, 1, 3, 2, 2, 9, 8 ], [ 1, 0, 9, 0, 5, 9, 3, 0, 8, 8, 2, 0, 7, 5, 8, 8, 1, 0, 5, 5 ], [ 9, 9, 2, 4, 9, 0, 8, 8, 0, 8, 4, 4, 8, 1, 9, 3, 4, 0, 9, 4 ], [ 5, 0, 8, 2, 9, 9, 9, 8, 8, 0, 7, 5, 6, 0, 2, 1, 2, 6, 5, 1 ], [ 2, 8, 4, 5, 5, 5, 3, 2, 4, 7, 0, 8, 3, 3, 2, 3, 1, 5, 7, 0 ], [ 1, 5, 6, 4, 3, 2, 4, 0, 4, 5, 8, 0, 7, 1, 9, 0, 6, 0, 9, 1 ], [ 2, 8, 3, 1, 8, 5, 1, 7, 8, 6, 3, 7, 0, 9, 2, 4, 3, 4, 8, 4 ], [ 0, 8, 0, 9, 3, 5, 1, 5, 1, 0, 3, 1, 9, 0, 2, 6, 6, 6, 2, 1 ], [ 9, 8, 2, 4, 4, 9, 1, 8, 9, 2, 2, 9, 2, 2, 0, 4, 5, 3, 4, 4 ], [ 7, 8, 1, 8, 4, 6, 3, 8, 3, 1, 3, 0, 4, 6, 4, 0, 2, 8, 8, 8 ], [ 5, 7, 4, 6, 1, 5, 2, 1, 4, 2, 1, 6, 3, 6, 5, 2, 0, 8, 6, 8 ], [ 8, 4, 2, 1, 1, 6, 2, 0, 0, 6, 5, 0, 4, 6, 3, 8, 8, 0, 2, 3 ], [ 3, 5, 6, 2, 0, 1, 9, 5, 9, 5, 7, 9, 8, 2, 4, 8, 6, 2, 0, 8 ], [ 4, 0, 6, 4, 6, 0, 8, 5, 4, 1, 0, 1, 4, 1, 4, 8, 8, 3, 8, 0 ] ], "m": 2, "objective": 9.0 }, "solution": [ 8, 18 ], "obj": 9.0, "instance_variant": { "problem_type": "MDP", "num_nodes": 20, "m": 2, "nodes": [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 ], "distance_pairs": [ { "from_id": 1, "to_id": 2, "distance": 6 }, { "from_id": 1, "to_id": 3, "distance": 9 }, { "from_id": 1, "to_id": 4, "distance": 2 }, { "from_id": 1, "to_id": 5, "distance": 5 }, { "from_id": 1, "to_id": 6, "distance": 7 }, { "from_id": 1, "to_id": 7, "distance": 9 }, { "from_id": 1, "to_id": 8, "distance": 1 }, { "from_id": 1, "to_id": 9, "distance": 9 }, { "from_id": 1, "to_id": 10, "distance": 5 }, { "from_id": 1, "to_id": 11, "distance": 2 }, { "from_id": 1, "to_id": 12, "distance": 1 }, { "from_id": 1, "to_id": 13, "distance": 2 }, { "from_id": 1, "to_id": 14, "distance": 0 }, { "from_id": 1, "to_id": 15, "distance": 9 }, { "from_id": 1, "to_id": 16, "distance": 7 }, { "from_id": 1, "to_id": 17, "distance": 5 }, { "from_id": 1, "to_id": 18, "distance": 8 }, { "from_id": 1, "to_id": 19, "distance": 3 }, { "from_id": 1, "to_id": 20, "distance": 4 }, { "from_id": 2, "to_id": 1, "distance": 6 }, { "from_id": 2, "to_id": 3, "distance": 7 }, { "from_id": 2, "to_id": 4, "distance": 0 }, { "from_id": 2, "to_id": 5, "distance": 8 }, { "from_id": 2, "to_id": 6, "distance": 7 }, { "from_id": 2, "to_id": 7, "distance": 0 }, { "from_id": 2, "to_id": 8, "distance": 0 }, { "from_id": 2, "to_id": 9, "distance": 9 }, { "from_id": 2, "to_id": 10, "distance": 0 }, { "from_id": 2, "to_id": 11, "distance": 8 }, { "from_id": 2, "to_id": 12, "distance": 5 }, { "from_id": 2, "to_id": 13, "distance": 8 }, { "from_id": 2, "to_id": 14, "distance": 8 }, { "from_id": 2, "to_id": 15, "distance": 8 }, { "from_id": 2, "to_id": 16, "distance": 8 }, { "from_id": 2, "to_id": 17, "distance": 7 }, { "from_id": 2, "to_id": 18, "distance": 4 }, { "from_id": 2, "to_id": 19, "distance": 5 }, { "from_id": 2, "to_id": 20, "distance": 0 }, { "from_id": 3, "to_id": 1, "distance": 9 }, { "from_id": 3, "to_id": 2, "distance": 7 }, { "from_id": 3, "to_id": 4, "distance": 0 }, { "from_id": 3, "to_id": 5, "distance": 8 }, { "from_id": 3, "to_id": 6, "distance": 0 }, { "from_id": 3, "to_id": 7, "distance": 7 }, { "from_id": 3, "to_id": 8, "distance": 9 }, { "from_id": 3, "to_id": 9, "distance": 2 }, { "from_id": 3, "to_id": 10, "distance": 8 }, { "from_id": 3, "to_id": 11, "distance": 4 }, { "from_id": 3, "to_id": 12, "distance": 6 }, { "from_id": 3, "to_id": 13, "distance": 3 }, { "from_id": 3, "to_id": 14, "distance": 0 }, { "from_id": 3, "to_id": 15, "distance": 2 }, { "from_id": 3, "to_id": 16, "distance": 1 }, { "from_id": 3, "to_id": 17, "distance": 4 }, { "from_id": 3, "to_id": 18, "distance": 2 }, { "from_id": 3, "to_id": 19, "distance": 6 }, { "from_id": 3, "to_id": 20, "distance": 6 }, { "from_id": 4, "to_id": 1, "distance": 2 }, { "from_id": 4, "to_id": 2, "distance": 0 }, { "from_id": 4, "to_id": 3, "distance": 0 }, { "from_id": 4, "to_id": 5, "distance": 0 }, { "from_id": 4, "to_id": 6, "distance": 3 }, { "from_id": 4, "to_id": 7, "distance": 4 }, { "from_id": 4, "to_id": 8, "distance": 0 }, { "from_id": 4, "to_id": 9, "distance": 4 }, { "from_id": 4, "to_id": 10, "distance": 2 }, { "from_id": 4, "to_id": 11, "distance": 5 }, { "from_id": 4, "to_id": 12, "distance": 4 }, { "from_id": 4, "to_id": 13, "distance": 1 }, { "from_id": 4, "to_id": 14, "distance": 9 }, { "from_id": 4, "to_id": 15, "distance": 4 }, { "from_id": 4, "to_id": 16, "distance": 8 }, { "from_id": 4, "to_id": 17, "distance": 6 }, { "from_id": 4, "to_id": 18, "distance": 1 }, { "from_id": 4, "to_id": 19, "distance": 2 }, { "from_id": 4, "to_id": 20, "distance": 4 }, { "from_id": 5, "to_id": 1, "distance": 5 }, { "from_id": 5, "to_id": 2, "distance": 8 }, { "from_id": 5, "to_id": 3, "distance": 8 }, { "from_id": 5, "to_id": 4, "distance": 0 }, { "from_id": 5, "to_id": 6, "distance": 2 }, { "from_id": 5, "to_id": 7, "distance": 3 }, { "from_id": 5, "to_id": 8, "distance": 5 }, { "from_id": 5, "to_id": 9, "distance": 9 }, { "from_id": 5, "to_id": 10, "distance": 9 }, { "from_id": 5, "to_id": 11, "distance": 5 }, { "from_id": 5, "to_id": 12, "distance": 3 }, { "from_id": 5, "to_id": 13, "distance": 8 }, { "from_id": 5, "to_id": 14, "distance": 3 }, { "from_id": 5, "to_id": 15, "distance": 4 }, { "from_id": 5, "to_id": 16, "distance": 4 }, { "from_id": 5, "to_id": 17, "distance": 1 }, { "from_id": 5, "to_id": 18, "distance": 1 }, { "from_id": 5, "to_id": 19, "distance": 0 }, { "from_id": 5, "to_id": 20, "distance": 6 }, { "from_id": 6, "to_id": 1, "distance": 7 }, { "from_id": 6, "to_id": 2, "distance": 7 }, { "from_id": 6, "to_id": 3, "distance": 0 }, { "from_id": 6, "to_id": 4, "distance": 3 }, { "from_id": 6, "to_id": 5, "distance": 2 }, { "from_id": 6, "to_id": 7, "distance": 8 }, { "from_id": 6, "to_id": 8, "distance": 9 }, { "from_id": 6, "to_id": 9, "distance": 0 }, { "from_id": 6, "to_id": 10, "distance": 9 }, { "from_id": 6, "to_id": 11, "distance": 5 }, { "from_id": 6, "to_id": 12, "distance": 2 }, { "from_id": 6, "to_id": 13, "distance": 5 }, { "from_id": 6, "to_id": 14, "distance": 5 }, { "from_id": 6, "to_id": 15, "distance": 9 }, { "from_id": 6, "to_id": 16, "distance": 6 }, { "from_id": 6, "to_id": 17, "distance": 5 }, { "from_id": 6, "to_id": 18, "distance": 6 }, { "from_id": 6, "to_id": 19, "distance": 1 }, { "from_id": 6, "to_id": 20, "distance": 0 }, { "from_id": 7, "to_id": 1, "distance": 9 }, { "from_id": 7, "to_id": 2, "distance": 0 }, { "from_id": 7, "to_id": 3, "distance": 7 }, { "from_id": 7, "to_id": 4, "distance": 4 }, { "from_id": 7, "to_id": 5, "distance": 3 }, { "from_id": 7, "to_id": 6, "distance": 8 }, { "from_id": 7, "to_id": 8, "distance": 3 }, { "from_id": 7, "to_id": 9, "distance": 8 }, { "from_id": 7, "to_id": 10, "distance": 9 }, { "from_id": 7, "to_id": 11, "distance": 3 }, { "from_id": 7, "to_id": 12, "distance": 4 }, { "from_id": 7, "to_id": 13, "distance": 1 }, { "from_id": 7, "to_id": 14, "distance": 1 }, { "from_id": 7, "to_id": 15, "distance": 1 }, { "from_id": 7, "to_id": 16, "distance": 3 }, { "from_id": 7, "to_id": 17, "distance": 2 }, { "from_id": 7, "to_id": 18, "distance": 2 }, { "from_id": 7, "to_id": 19, "distance": 9 }, { "from_id": 7, "to_id": 20, "distance": 8 }, { "from_id": 8, "to_id": 1, "distance": 1 }, { "from_id": 8, "to_id": 2, "distance": 0 }, { "from_id": 8, "to_id": 3, "distance": 9 }, { "from_id": 8, "to_id": 4, "distance": 0 }, { "from_id": 8, 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