{
  "id": "KL-FCS-029",
  "version": "1.0.0",
  "domain": "Graph algorithms",
  "kind": "graph",
  "title": "Two equally short paths",
  "problem": "Find the minimum weighted directed path from vertex 0 to vertex 3.",
  "specification": {
    "vertices": 4,
    "edges": [
      [
        0,
        1,
        1
      ],
      [
        0,
        2,
        1
      ],
      [
        1,
        3,
        2
      ],
      [
        2,
        3,
        2
      ]
    ],
    "start": 0,
    "target": 3
  },
  "claim": {
    "minimum_distance": 3
  },
  "witness": {
    "path": [
      0,
      2,
      3
    ]
  },
  "verification_scope": "Complete simple-path enumeration",
  "explanation": "The supplied sequence is a valid path. Every alternative simple path is scored, so equal-cost optima are accepted and longer alternatives are ruled out.",
  "limitations": "Weights are nonnegative. Negative cycles and unreachable targets require distinct result types.",
  "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://networkx.org/documentation/stable/reference/algorithms/generated/networkx.algorithms.flow.minimum_cut.html"
  ],
  "license_status": "not-yet-selected",
  "dataset": {
    "family": "graphs",
    "task": "Find the minimum weighted directed path from vertex 0 to vertex 3.",
    "input_encoding": "Structured JSON; field meanings are stated in the specification.",
    "coverage": "Complete simple-path enumeration",
    "acceptance": [
      "Enumerate every simple source-to-target path in the small graph.",
      "Compute the total weight of each path.",
      "Find the minimum and accept any witness attaining it.",
      "Check the witness edge sequence and objective."
    ],
    "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": "A path is a feasible witness; the minimum-distance claim must also exclude shorter paths.",
    "definitions": [
      {
        "term": "Directed edge",
        "definition": "An ordered pair of vertices with a nonnegative weight."
      },
      {
        "term": "Simple path",
        "definition": "A path visiting no vertex twice."
      },
      {
        "term": "Distance",
        "definition": "The sum of the weights along a path."
      }
    ],
    "reasoning": [
      "Enumerate every simple source-to-target path in the small graph.",
      "Compute the total weight of each path.",
      "Find the minimum and accept any witness attaining it.",
      "Check the witness edge sequence and objective."
    ],
    "worked_example": "The supplied sequence is a valid path. Every alternative simple path is scored, so equal-cost optima are accepted and longer alternatives are ruled out.",
    "complexity": "Simple-path enumeration can be exponential; nonnegative weights ensure a shortest path can be chosen simple.",
    "common_error": "A locally cheapest outgoing edge need not belong to a globally shortest path.",
    "further_work": "Replace enumeration with distance-label certificates and add max-flow/min-cut records."
  },
  "related_ids": [
    "KL-FCS-028",
    "KL-FCS-030"
  ]
}