{
  "abstract": "For an $n\\times n$ nonnegative matrix $A$, the Bethe permanent, which is computable in deterministic polynomial time, satisfies the tight universal comparison \\[\\operatorname{Bethe}(A) \\leq \\operatorname{per}(A) \\leq 2^{n/2}\\operatorname{Bethe}(A).\\] The lower bound, due to Gurvits, is attained on forests. The upper bound, due to Anari and Rezaei, is attained by the adjacency matrix of a disjoint union of $4$-cycles. Confirming a conjecture of Anari, we provide an optimal girth-dependent refinement of the above comparison. More precisely, we show that if the bipartite support graph of $A$ has girth at least an even integer $g \\geq 4$, then \\[\\operatorname{Bethe}(A) \\leq \\operatorname{per}(A) \\leq 2^{2n/g}\\operatorname{Bethe}(A).\\] The upper bound is attained by the adjacency matrix of a disjoint union of $g$-cycles.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02017",
  "author_total": 2,
  "authors": [
    {
      "name": "Dingding Dong"
    },
    {
      "name": "Vishesh Jain"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4wry36hd38atr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02017",
  "fulltext_url": null,
  "id": "4wry36hd38atr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02017",
  "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.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7207798145",
  "pdf_url": "https://arxiv.org/pdf/2609.02017",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4wry36hd38atr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02017",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02017"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Optimal girth-dependent bounds for the Bethe approximation of the permanent",
  "title_text": "Optimal girth-dependent bounds for the Bethe approximation of the permanent",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/4wry36hd38atr",
  "venue": "arXiv",
  "versions": []
}