{
  "abstract": "We establish counting and sampling analogues of the Lovász Local Lemma: we give efficient algorithms for approximately counting and exactly sampling satisfying assignments of general constraint satisfaction problems (CSPs) in the local lemma regime $$4 \\mathrm{e} p (D+1)^2\\leq 1, $$ where $p$ is the maximum constraint violation probability and $D$ is the maximum dependency degree. This condition is tight up to constant factors under $\\mathbf{NP}\\neq\\mathbf{RP}$, matching known hardness bounds for counting and sampling in natural subclasses of CSPs. Our key ingredient is a novel $2$-tree expansion for constraint marginal probabilities that exhibits exponential decay of correlations throughout this regime. This expansion yields deterministic polynomial-time approximate counting for fixed local parameters, randomized approximate counting with quadratic cost, and exact sampling in expected near-linear time when the local parameters are fixed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.08616",
  "author_total": 5,
  "authors": [
    {
      "name": "Hongyang Liu"
    },
    {
      "name": "Chunyang Wang"
    },
    {
      "name": "Yitong Yin"
    },
    {
      "name": "Yiyao Zhang"
    },
    {
      "name": "Can Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dg16f2j96108z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.08616",
  "fulltext_url": null,
  "id": "dg16f2j96108z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.08616",
  "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.DM"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "1705|Constraint Satisfaction and Optimization"
    ]
  ],
  "openalex": "W7202175940",
  "pdf_url": "https://arxiv.org/pdf/2608.08616",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dg16f2j96108z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2608.08616",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.08616"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A Counting and Sampling Lovász Local Lemma",
  "title_text": "A Counting and Sampling Lovász Local Lemma",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/dg16f2j96108z",
  "venue": "arXiv",
  "versions": []
}