{
  "abstract": "We study how many split cuts, which include the Gomory mixed-integer cuts, are needed to reach the integer hull, and how large split closures can be. In every fixed dimension, a rational pure-integer polytope admits a polynomial-time constructible split proof of its integer hull with polynomial total encoding length; consequently, its full split rank is polynomial in the input length. This establishes the split-rank conjecture of Basu, Conforti, Di Summa, and Jiang for bounded pure-integer polyhedra in fixed dimension. Central to our approach is a preservation result: in arbitrary pure-integer dimension, an inequality whose violating points map, under an integer-affine map, into the interior of a planar lattice-free polyhedron has a polynomial split proof retaining every point already satisfying it. Whereas Chvátal--Gomory closures have polynomial-size inequality descriptions in fixed dimension, split closures need not: tetrahedra of input length $Θ(\\log N)$ can have split closures with at least $2N-1$ facets, yet extension complexity $Θ(\\log N)$. For the two-row system $x=f+Rs$, $s\\ge0$, $x\\in\\mathbb Z^2$ of input length $L$, the split closure has an exact extended formulation with $O(L)$ inequalities and auxiliary variables; this bound is tight. Separation over the split closure is polynomial-time in fixed total dimension. Finally, for mixed-integer feasible rational polytopes with fixed integer dimension $p$ and any continuous dimension, we maximize a primitive integral objective on the integer variables with $O_p(G+1)$ split cuts, each removing a current optimal LP vertex, until an optimal vertex is mixed-integer feasible; here $G$ is the integrality gap of the LP relaxation, rounded down.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02169",
  "author_total": 2,
  "authors": [
    {
      "name": "Avinash Bhardwaj"
    },
    {
      "name": "Vishnu Narayanan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0svffsza32eqs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02169",
  "fulltext_url": null,
  "id": "0svffsza32eqs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02169",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02169",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0svffsza32eqs.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02169",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02169"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Split Cuts in Fixed Integer Dimensions",
  "title_text": "Split Cuts in Fixed Integer Dimensions",
  "type": "preprint",
  "updated_at": "2026-11-05T15:21:00Z",
  "url": "https://research.draconitic.com/paper/0svffsza32eqs",
  "venue": "arXiv",
  "versions": []
}