{
  "abstract": "We study deterministic approximation algorithms for the permanent of a nonnegative matrix through the Bethe permanent, an approximation computable in polynomial time. The tight analysis of Anari and Rezaei gives a universal comparison between the permanent and the Bethe permanent within a factor $(\\sqrt 2)^n$. The simple example of the unweighted $4$-cycle $C_4$ (or a union of disjoint $C_4$'s) shows that this bound is tight. We show that such $4$-cycle obstructions can be identified and exploited algorithmically. Given a Bethe optimizer, our algorithm identifies nearly isolated weighted $2\\times2$ blocks and peels off a vertex-disjoint family of them. If the total weighted correction is large, we can improve the Bethe approximation; if it is small, we show that the Bethe permanent is within a factor of $(\\sqrt2 - \\varepsilon)^n$ of the truth. Combining these facts, we obtain a deterministic polynomial time $(\\sqrt2-\\varepsilon)^n$-approximation algorithm for the permanent of an arbitrary nonnegative $n\\times n$ matrix, where $\\varepsilon>0$ is some absolute constant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.31061",
  "author_total": 2,
  "authors": [
    {
      "name": "Ijay Narang"
    },
    {
      "name": "Will Perkins"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/75t8xf3w2715z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.31061",
  "fulltext_url": null,
  "id": "75t8xf3w2715z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.31061",
  "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",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7204916796",
  "pdf_url": "https://arxiv.org/pdf/2608.31061",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/75t8xf3w2715z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.31061",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.31061"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Structural Corrections to the Bethe Approximation of the Permanent",
  "title_text": "Structural Corrections to the Bethe Approximation of the Permanent",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/75t8xf3w2715z",
  "venue": "arXiv",
  "versions": []
}