{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R550",
  "content_hash": null,
  "slug": "nngm10-claim-certified-interval",
  "type": "claim",
  "title": "The certified ratio lies between 18/11 and 11/5",
  "summary": "An explicit graph gives 18/11, while a finite integer refinement of the classical nearest-neighbor analysis gives the universal upper bound 11/5.",
  "relevance": "For Worst nearest-neighbor tour on a ten-vertex graph metric, record nngm10-claim-certified-interval (“The certified ratio lies between 18/11 and 11/5”) records a bound, answer, status fact, or structural consequence. The record states: An explicit graph gives 18/11, while a finite integer refinement of the classical nearest-neighbor analysis gives the universal upper bound 11/5.",
  "relevance_source": "recorded",
  "body": "Write \\(R_{10}\\) for the requested maximum. The current certified interval is\n\\[\n\\boxed{\\frac{18}{11}\\leq R_{10}\\leq\\frac{11}{5}}.\n\\]\nThe lower endpoint comes from the graph with edges\n\\[\n02,03,17,18,19,23,25,26,27,45,48,56,78,89.\n\\]\nIts deterministic nearest-neighbor order is\n\\[\n0,2,3,5,4,8,1,7,6,9,\n\\]\nwith leg lengths \\(1,1,2,1,1,1,1,2,4,4\\), hence total length 18. Exhaustion of all \\(9!/2=181{,}440\\) unoriented tours gives optimum 11.\n\nFor the upper bound, let \\(O\\) be the optimal tour length and associate to every vertex the length of its outgoing edge in the nearest-neighbor tour. Sort these ten integer lengths as \\(l_1\\geq\\cdots\\geq l_{10}\\). The proof of Lemma 1 in Rosenkrantz, Stearns, and Lewis gives\n\\[\nO\\geq2\\sum_{i=k+1}^{\\min(2k,10)}l_i\\qquad(1\\leq k\\leq10),\n\\]\nand each \\(l_i\\leq O/2\\). These facts hold for arbitrary tie resolution, so they cover the specified label rule.\n\nEvery distinct pair in a connected unweighted graph has positive integral distance. Thus \\(O\\geq10\\). Doubling a nine-edge spanning tree and shortcutting its Euler tour gives \\(O\\leq18\\). The exact artifact enumerates every nonincreasing positive integer sequence with \\(l_1\\leq\\lfloor O/2\\rfloor\\) for each integer \\(O\\) between 10 and 18. The largest permitted sums are respectively\n\\[\n22,22,26,26,30,30,34,34,39.\n\\]\nThe largest quotient is \\(22/10=11/5\\). This integer relaxation covers every graph in the stated family, although some sequences admitted by it may have no graph realization. Closing the interval requires a graph-level exclusion certificate.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "shortest-path metrics of all connected simple graphs on the labeled vertex set {0,...,9}, with nearest neighbor starting at 0 and resolving distance ties by smaller label",
    "bounds": {
      "vertices": {
        "min": 10,
        "max": 10
      },
      "optimal_tour_length": {
        "min": 10,
        "max": 18
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1137/0206041",
      "locator": "Daniel J. Rosenkrantz, Richard E. Stearns, and Philip M. Lewis II, An Analysis of Several Heuristics for the Traveling Salesman Problem, SIAM Journal on Computing 6 (1977), 563-581, Theorem 1 and Lemma 1; exact finite certificates in this dataset"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1137/0206041",
    "locator": "Daniel J. Rosenkrantz, Richard E. Stearns, and Philip M. Lewis II, An Analysis of Several Heuristics for the Traveling Salesman Problem, SIAM Journal on Computing 6 (1977), 563-581, Theorem 1 and Lemma 1; exact finite certificates in this dataset"
  },
  "relations": [
    {
      "slug": "R548",
      "title": "Exact verifier for the 18/11 graph",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R547",
      "title": "Exhaustive integer certificate for the 11/5 upper bound",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R549",
      "title": "A graph-level exclusion certificate remains open",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "nearest-neighbor-graph-metric-ten",
      "title": "nearest neighbor graph metric ten",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}