files / problems / hilbert16-degree-8 / compute / q7 / certs / even_components.json

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{
  "what": "Component sizes for q5's pinned even add/drop/swap BFS. Five nonempty fixed-odd components are counted exactly. The all-even component is additional, so the sum is a certified lower bound on the full BFS closure, not a lower bound on real schemes.",
  "split_count": 1254,
  "even_split_count": 1065,
  "fixed_odd_components": 6,
  "exact_nonempty_component_collections": 538188419044,
  "all_even_component_collections_at_least": 249665706104,
  "full_bfs_collections_at_least": 787854125148,
  "lower_bound_reason": "Removing the fixed odd splits injects every even clique from a nonempty-odd component into the all-even component.",
  "full_bfs_complete": false,
  "components": [
    {
      "seeds": [
        "<10v1<2v1<8>>>"
      ],
      "odd_splits": [
        13,
        29,
        69,
        116,
        179,
        245,
        496
      ],
      "compatible_even_splits": 28,
      "collections": 126336,
      "complete": true,
      "memo_states": 144
    },
    {
      "seeds": [
        "<16v3<1>>"
      ],
      "odd_splits": [
        26,
        1121
      ],
      "compatible_even_splits": 340,
      "collections": 249665706104,
      "complete": true,
      "memo_states": 404618
    },
    {
      "seeds": [
        "<17v1<1>v1<2>>"
      ],
      "odd_splits": [
        65,
        116,
        428,
        975
      ],
      "compatible_even_splits": 294,
      "collections": 199741441484,
      "complete": true,
      "memo_states": 347612
    },
    {
      "seeds": [
        "<17v1<2v1<1>>>"
      ],
      "odd_splits": [
        9,
        13,
        65,
        101,
        245,
        496,
        914
      ],
      "compatible_even_splits": 60,
      "collections": 25292736,
      "complete": true,
      "memo_states": 4668
    },
    {
      "seeds": [
        "<18v1<3>>"
      ],
      "odd_splits": [
        93,
        432
      ],
      "compatible_even_splits": 321,
      "collections": 88755852384,
      "complete": true,
      "memo_states": 458813
    },
    {
      "seeds": [
        "bowtie_B8"
      ],
      "odd_splits": [],
      "compatible_even_splits": 1065,
      "collections": null,
      "complete": false,
      "why_incomplete": "The all-even component is larger than the already huge nonempty-odd components; it is not needed for the certified lower bound on the BFS closure size."
    }
  ]
}