{
  "id": "configuration-gaps-in-bin-packing",
  "title": "Configuration gaps in bin packing",
  "description": "OpenAI's theorems (2026) that the configuration linear program of bin packing can fall short of the optimum by any additive constant, refuting the modified integer round-up conjecture, and that additive-constant approximation is NP-hard; through competing trees, interval coverage encoded in five-item bins, and a vertex-cover reduction.",
  "units": [
    {
      "id": "bin-packing-and-the-configuration-program",
      "title": "Bin packing and the configuration program",
      "reader": "bin-packing-and-the-configuration-program.html",
      "source": "src/bin-packing-and-the-configuration-program.md",
      "source_sha256": "5eaa43ee5c76c9274058fedc9d142544a122a72ca44a22bed26b9126888beacf",
      "status": "author_self_checked"
    },
    {
      "id": "two-competing-trees",
      "title": "Two competing trees",
      "reader": "two-competing-trees.html",
      "source": "src/two-competing-trees.md",
      "source_sha256": "d2b0529498d672ddaeb9299e062e7854ad4fc682fd6a1747bfc4747fd83ec718",
      "status": "author_self_checked"
    },
    {
      "id": "coordinates-cells-and-items",
      "title": "Coordinates, cells and items",
      "reader": "coordinates-cells-and-items.html",
      "source": "src/coordinates-cells-and-items.md",
      "source_sha256": "694b0a6fd27e0e0a7ebd7858f9b75a64f2a08cc6ba18dd77d85bab626de1f6be",
      "status": "author_self_checked"
    },
    {
      "id": "sizes-that-enforce-the-patterns",
      "title": "Sizes that enforce the patterns",
      "reader": "sizes-that-enforce-the-patterns.html",
      "source": "src/sizes-that-enforce-the-patterns.md",
      "source_sha256": "61ac6136bb151c726688330e4863835b5ca5e56e104af807c46ff87b3505625c",
      "status": "author_self_checked"
    },
    {
      "id": "packings-from-covers-and-a-fractional-packing",
      "title": "Packings from covers and a fractional packing",
      "reader": "packings-from-covers-and-a-fractional-packing.html",
      "source": "src/packings-from-covers-and-a-fractional-packing.md",
      "source_sha256": "aa69bd997eba029a1749bf4a8683f0d52eb07592bc08942f44672ca3bef931e2",
      "status": "author_self_checked"
    },
    {
      "id": "extracting-a-vertex-cover",
      "title": "Extracting a vertex cover",
      "reader": "extracting-a-vertex-cover.html",
      "source": "src/extracting-a-vertex-cover.md",
      "source_sha256": "f54bd8545454853ea7c7f7fd2049ba42bb45167292f35f9ffee482b581f37762",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "none beyond finite sets, rational arithmetic and graphs; the hardness section defines the complexity notions it uses"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}