{
  "abstract": "In an $n$-dimensional cropped hypercube each of the $2^n$ cropping inequalities chops off a single corner of the $0$--$1$ hypercube by an $\\ell_1$-distance $ρ$. The case $ρ= 1/2$ has been extensively studied in the literature. This paper shows that the Lasserre rank of the $n$-dimensional cropped hypercube where $ρ= 1/2$, $n \\geq 2$, is the smallest integer $0\\leq t \\leq n$ such that $Δ_t < 0$ in the recurrence $Δ_{-1} = 1$, $Δ_{0} = n-1$, $Δ_t = (n-1)Δ_{t-1} - t(n-t+1)Δ_{t-2}$. It follows that the Lasserre rank can be computed in time $O(n^2 \\log^2 n)$. Asymptotically, the rank is $\\frac{n}{2} + c_{1/2}\\sqrt{n} + o(\\sqrt{n})$, where $c_{1/2}$ is the unique zero of a given function. Numerically, $c_{1/2} \\approx 0.3825$. In fact, we prove such results for any fixed $0 < ρ< 1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27748",
  "author_total": 3,
  "authors": [
    {
      "name": "Gérard Cornuéjols"
    },
    {
      "name": "Vrishabh Patil"
    },
    {
      "name": "Jiaye Wei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dtdp9q3q1x5n5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27748",
  "fulltext_url": null,
  "id": "dtdp9q3q1x5n5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27748",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7214209475",
  "pdf_url": "https://arxiv.org/pdf/2609.27748",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dtdp9q3q1x5n5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2609.27748",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27748"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "The Lasserre Rank of the Cropped Hypercube",
  "title_text": "The Lasserre Rank of the Cropped Hypercube",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/dtdp9q3q1x5n5",
  "venue": "arXiv",
  "versions": []
}