{
  "abstract": "Lee, Skipper, and Speakman (2018) asked for formulae or asymptotic expressions for the volumes of the strong and weak relaxations $\\mathcal{P}_{m,n}\\cap \\mathcal{D}_{m,n}$ and $\\mathcal{Q}_{m,n}\\cap \\mathcal{D}_{m,n}$ of the uncapacitated facility-location polytope with $m$ facilities and $n$ customers (their Problem~3, ``Texas Hot''). We give exact formulae for all $m$ and $n$, and we give asymptotics that hold uniformly as $m+n\\to\\infty$, whatever the relative growth of $m$ and $n$. The volume ratio satisfies $Vol(\\mathcal{Q}_{m,n}\\cap \\mathcal{D}_{m,n})/Vol(\\mathcal{P}_{m,n}\\cap \\mathcal{D}_{m,n})\\sim \\prod_{j=2}^{n}\\mleft(1+\\tfrac{1}{(m-1)j}\\mright)^{m}$. From this, we read off the behavior in every growth regime of $m$, and we identify a sharp transition at $m\\asymp\\ln n$. Below it, in a uniformly random point of the strong relaxation, each facility variable is close to $1$ with high probability, and the ratio grows superlinearly in $n$. Above it, in a uniformly random point of either relaxation, each facility variable is asymptotically uniform on $[0,1]$, and the ratio grows linearly in $n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25981",
  "author_total": 1,
  "authors": [
    {
      "name": "Jon Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/815aatwme3h67.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25981",
  "fulltext_url": null,
  "id": "815aatwme3h67",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25981",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7214068566",
  "pdf_url": "https://arxiv.org/pdf/2609.25981",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/815aatwme3h67.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25981",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25981"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Volumes of the strong and weak relaxations of the uncapacitated facility-location polytope",
  "title_text": "Volumes of the strong and weak relaxations of the uncapacitated facility-location polytope",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/815aatwme3h67",
  "venue": "arXiv",
  "versions": []
}