{
  "abstract": "In this paper, we study polyhedral aspects on commutative and noncommutative Edmonds' problems for computing the rank of linear symbolic matrix $A = \\sum_{k=1}^m A_k x_k$. We regard them as linear optimization over integral polytopes ${\\cal P}(A)$ and ${\\cal Q}(A)$, respectively, which are obtained by the convex hulls of exponent vectors of subdeterminants of~$A$ and its blow-ups $A^{\\{d\\}} = \\sum_{k=1}^m A_k \\otimes X_k$ $(d=1,2,\\ldots)$. By extending previously known results on nc-rank, we establish a hierarchy of integral polytopes ${\\cal P}(A) \\subseteq {\\cal P}^{\\leq 2}(A) \\subseteq {\\cal P}^{\\leq 3}(A) \\subseteq \\cdots = {\\cal Q}(A)$ and show that the integrality gap of ${\\cal Q}(A)$ relative to ${\\cal P}(A)$ is at least $1/2$. Further, we show that if each $A_k$ is rank-2 skew-symmetric, then the above hierarchy terminates at the second level and the integrality gap is improved to $2/3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19703",
  "author_total": 1,
  "authors": [
    {
      "name": "Hiroshi Hirai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/31pwc5jym4vet.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19703",
  "fulltext_url": null,
  "id": "31pwc5jym4vet",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19703",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7213591629",
  "pdf_url": "https://arxiv.org/pdf/2609.19703",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/31pwc5jym4vet.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19703",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19703"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On integral polytopes related to Edmonds' problem",
  "title_text": "On integral polytopes related to Edmonds' problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/31pwc5jym4vet",
  "venue": "arXiv",
  "versions": []
}