{
  "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"
  ]
}