{
  "name": "Combinatorial optimization",
  "version": "1.0.0",
  "area": {
    "name": "Combinatorial optimization",
    "slug": "combinatorial-optimization",
    "group": "Optimization",
    "kind": "knapsack",
    "summary": "Separate a feasible chosen subset from a certified best objective over all allowed choices.",
    "definitions": [
      {
        "term": "0/1 knapsack",
        "definition": "Each item may be chosen at most once under a total weight limit."
      },
      {
        "term": "Primal witness",
        "definition": "A concrete feasible subset attaining a particular value."
      },
      {
        "term": "Optimality",
        "definition": "No other feasible subset has a larger objective."
      }
    ],
    "methodology": [
      "Enumerate every binary item-selection vector.",
      "Discard selections exceeding capacity.",
      "Compute each remaining total value.",
      "Check that the submitted subset is feasible and attains the maximum."
    ],
    "complexity": "n items produce 2ⁿ subsets. Pseudopolynomial dynamic programming offers a different tradeoff for integral capacity.",
    "common_error": "The highest value-to-weight ratio can fail for indivisible 0/1 items.",
    "next_question": "Add assignment dual certificates, set cover, and branch-and-bound proof logs.",
    "references": [
      "https://developers.google.com/optimization/assignment/linear_assignment"
    ]
  },
  "records": [
    {
      "id": "KL-FCS-043",
      "version": "1.0.0",
      "domain": "Combinatorial optimization",
      "kind": "knapsack",
      "title": "A four-item capacity decision",
      "problem": "Maximize total item value without exceeding the weight capacity.",
      "specification": {
        "items": [
          [
            2,
            3
          ],
          [
            3,
            4
          ],
          [
            4,
            7
          ],
          [
            5,
            8
          ]
        ],
        "capacity": 7,
        "item_encoding": "[weight, value]; index identifies an indivisible item"
      },
      "claim": {
        "maximum_value": 11
      },
      "witness": {
        "selected": [
          1,
          2
        ]
      },
      "verification_scope": "Complete subset enumeration · 16 candidates",
      "explanation": "Each subset is a distinct candidate. The checker independently computes feasibility and objective, accepts any optimal witness, and rejects attractive but overweight selections.",
      "limitations": "Only this finite 0/1 instance is certified; no approximation ratio or measured solver speed is claimed.",
      "verification_status": "mechanically-checked",
      "review_status": "awaiting-independent-review",
      "provenance": {
        "origin": "Original Kenton Labs reference instance, authored with Codex assistance on 2026-10-11.",
        "external_dataset": null,
        "model_run": null
      },
      "references": [
        "https://developers.google.com/optimization/assignment/linear_assignment"
      ],
      "license_status": "not-yet-selected",
      "dataset": {
        "family": "combinatorial-optimization",
        "task": "Maximize total item value without exceeding the weight capacity.",
        "input_encoding": "Structured JSON; field meanings are stated in the specification.",
        "coverage": "Complete subset enumeration · 16 candidates",
        "acceptance": [
          "Enumerate every binary item-selection vector.",
          "Discard selections exceeding capacity.",
          "Compute each remaining total value.",
          "Check that the submitted subset is feasible and attains the maximum."
        ],
        "generation": "Deterministic finite fixture; full enumeration or witness replay as stated.",
        "split_policy": "Reference corpus for exposition and reproduction; no train/test evaluation split is claimed."
      },
      "lesson": {
        "motivation": "Separate a feasible chosen subset from a certified best objective over all allowed choices.",
        "definitions": [
          {
            "term": "0/1 knapsack",
            "definition": "Each item may be chosen at most once under a total weight limit."
          },
          {
            "term": "Primal witness",
            "definition": "A concrete feasible subset attaining a particular value."
          },
          {
            "term": "Optimality",
            "definition": "No other feasible subset has a larger objective."
          }
        ],
        "reasoning": [
          "Enumerate every binary item-selection vector.",
          "Discard selections exceeding capacity.",
          "Compute each remaining total value.",
          "Check that the submitted subset is feasible and attains the maximum."
        ],
        "worked_example": "Each subset is a distinct candidate. The checker independently computes feasibility and objective, accepts any optimal witness, and rejects attractive but overweight selections.",
        "complexity": "n items produce 2ⁿ subsets. Pseudopolynomial dynamic programming offers a different tradeoff for integral capacity.",
        "common_error": "The highest value-to-weight ratio can fail for indivisible 0/1 items.",
        "further_work": "Add assignment dual certificates, set cover, and branch-and-bound proof logs."
      },
      "related_ids": [
        "KL-FCS-044",
        "KL-FCS-045"
      ]
    },
    {
      "id": "KL-FCS-044",
      "version": "1.0.0",
      "domain": "Combinatorial optimization",
      "kind": "knapsack",
      "title": "Value ties in a five-item knapsack",
      "problem": "Maximize total item value without exceeding the weight capacity.",
      "specification": {
        "items": [
          [
            1,
            2
          ],
          [
            2,
            4
          ],
          [
            3,
            4
          ],
          [
            4,
            6
          ],
          [
            5,
            9
          ]
        ],
        "capacity": 8,
        "item_encoding": "[weight, value]; index identifies an indivisible item"
      },
      "claim": {
        "maximum_value": 15
      },
      "witness": {
        "selected": [
          0,
          1,
          4
        ]
      },
      "verification_scope": "Complete subset enumeration · 32 candidates",
      "explanation": "Each subset is a distinct candidate. The checker independently computes feasibility and objective, accepts any optimal witness, and rejects attractive but overweight selections.",
      "limitations": "Only this finite 0/1 instance is certified; no approximation ratio or measured solver speed is claimed.",
      "verification_status": "mechanically-checked",
      "review_status": "awaiting-independent-review",
      "provenance": {
        "origin": "Original Kenton Labs reference instance, authored with Codex assistance on 2026-10-11.",
        "external_dataset": null,
        "model_run": null
      },
      "references": [
        "https://developers.google.com/optimization/assignment/linear_assignment"
      ],
      "license_status": "not-yet-selected",
      "dataset": {
        "family": "combinatorial-optimization",
        "task": "Maximize total item value without exceeding the weight capacity.",
        "input_encoding": "Structured JSON; field meanings are stated in the specification.",
        "coverage": "Complete subset enumeration · 32 candidates",
        "acceptance": [
          "Enumerate every binary item-selection vector.",
          "Discard selections exceeding capacity.",
          "Compute each remaining total value.",
          "Check that the submitted subset is feasible and attains the maximum."
        ],
        "generation": "Deterministic finite fixture; full enumeration or witness replay as stated.",
        "split_policy": "Reference corpus for exposition and reproduction; no train/test evaluation split is claimed."
      },
      "lesson": {
        "motivation": "Separate a feasible chosen subset from a certified best objective over all allowed choices.",
        "definitions": [
          {
            "term": "0/1 knapsack",
            "definition": "Each item may be chosen at most once under a total weight limit."
          },
          {
            "term": "Primal witness",
            "definition": "A concrete feasible subset attaining a particular value."
          },
          {
            "term": "Optimality",
            "definition": "No other feasible subset has a larger objective."
          }
        ],
        "reasoning": [
          "Enumerate every binary item-selection vector.",
          "Discard selections exceeding capacity.",
          "Compute each remaining total value.",
          "Check that the submitted subset is feasible and attains the maximum."
        ],
        "worked_example": "Each subset is a distinct candidate. The checker independently computes feasibility and objective, accepts any optimal witness, and rejects attractive but overweight selections.",
        "complexity": "n items produce 2ⁿ subsets. Pseudopolynomial dynamic programming offers a different tradeoff for integral capacity.",
        "common_error": "The highest value-to-weight ratio can fail for indivisible 0/1 items.",
        "further_work": "Add assignment dual certificates, set cover, and branch-and-bound proof logs."
      },
      "related_ids": [
        "KL-FCS-043",
        "KL-FCS-045"
      ]
    },
    {
      "id": "KL-FCS-045",
      "version": "1.0.0",
      "domain": "Combinatorial optimization",
      "kind": "knapsack",
      "title": "Six items and a larger capacity",
      "problem": "Maximize total item value without exceeding the weight capacity.",
      "specification": {
        "items": [
          [
            2,
            5
          ],
          [
            2,
            4
          ],
          [
            3,
            6
          ],
          [
            4,
            7
          ],
          [
            5,
            11
          ],
          [
            1,
            1
          ]
        ],
        "capacity": 10,
        "item_encoding": "[weight, value]; index identifies an indivisible item"
      },
      "claim": {
        "maximum_value": 22
      },
      "witness": {
        "selected": [
          0,
          2,
          4
        ]
      },
      "verification_scope": "Complete subset enumeration · 64 candidates",
      "explanation": "Each subset is a distinct candidate. The checker independently computes feasibility and objective, accepts any optimal witness, and rejects attractive but overweight selections.",
      "limitations": "Only this finite 0/1 instance is certified; no approximation ratio or measured solver speed is claimed.",
      "verification_status": "mechanically-checked",
      "review_status": "awaiting-independent-review",
      "provenance": {
        "origin": "Original Kenton Labs reference instance, authored with Codex assistance on 2026-10-11.",
        "external_dataset": null,
        "model_run": null
      },
      "references": [
        "https://developers.google.com/optimization/assignment/linear_assignment"
      ],
      "license_status": "not-yet-selected",
      "dataset": {
        "family": "combinatorial-optimization",
        "task": "Maximize total item value without exceeding the weight capacity.",
        "input_encoding": "Structured JSON; field meanings are stated in the specification.",
        "coverage": "Complete subset enumeration · 64 candidates",
        "acceptance": [
          "Enumerate every binary item-selection vector.",
          "Discard selections exceeding capacity.",
          "Compute each remaining total value.",
          "Check that the submitted subset is feasible and attains the maximum."
        ],
        "generation": "Deterministic finite fixture; full enumeration or witness replay as stated.",
        "split_policy": "Reference corpus for exposition and reproduction; no train/test evaluation split is claimed."
      },
      "lesson": {
        "motivation": "Separate a feasible chosen subset from a certified best objective over all allowed choices.",
        "definitions": [
          {
            "term": "0/1 knapsack",
            "definition": "Each item may be chosen at most once under a total weight limit."
          },
          {
            "term": "Primal witness",
            "definition": "A concrete feasible subset attaining a particular value."
          },
          {
            "term": "Optimality",
            "definition": "No other feasible subset has a larger objective."
          }
        ],
        "reasoning": [
          "Enumerate every binary item-selection vector.",
          "Discard selections exceeding capacity.",
          "Compute each remaining total value.",
          "Check that the submitted subset is feasible and attains the maximum."
        ],
        "worked_example": "Each subset is a distinct candidate. The checker independently computes feasibility and objective, accepts any optimal witness, and rejects attractive but overweight selections.",
        "complexity": "n items produce 2ⁿ subsets. Pseudopolynomial dynamic programming offers a different tradeoff for integral capacity.",
        "common_error": "The highest value-to-weight ratio can fail for indivisible 0/1 items.",
        "further_work": "Add assignment dual certificates, set cover, and branch-and-bound proof logs."
      },
      "related_ids": [
        "KL-FCS-043",
        "KL-FCS-044"
      ]
    }
  ],
  "verification": [
    {
      "id": "KL-FCS-043",
      "status": "mechanically-checked",
      "check_units": 16,
      "scope": "Complete subset enumeration · 16 candidates",
      "review_status": "awaiting-independent-review"
    },
    {
      "id": "KL-FCS-044",
      "status": "mechanically-checked",
      "check_units": 32,
      "scope": "Complete subset enumeration · 32 candidates",
      "review_status": "awaiting-independent-review"
    },
    {
      "id": "KL-FCS-045",
      "status": "mechanically-checked",
      "check_units": 64,
      "scope": "Complete subset enumeration · 64 candidates",
      "review_status": "awaiting-independent-review"
    }
  ]
}