{
  "abstract": "Montanari's incremental approximate message passing (IAMP) algorithm [FOCS 2019; SIAM J. Comput., 2025] and the zero-temperature formulation of El Alaoui, Montanari and Sellke (AMS) [Ann. Probab., 2021] find a $(1-\\varepsilon)$ approximation to the maximum Sherrington-Kirkpatrick (SK) energy in $C(\\varepsilon)N^2$ operations for fixed accuracy, with probability tending to one as $N\\to\\infty$, under their respective full-support and access-model assumptions. However, it was unknown whether this approximation could be achieved with work polynomial in both $N$ and $1/\\varepsilon$, together with quantitative guarantees at finite dimension. Under the same assumptions as AMS, we introduce grouped IAMP with exact Gaussian query outputs (GEG-IAMP), which is the first SK approximation algorithm with total work polynomial in $N$ and $1/\\varepsilon$ and a finite-size guarantee above a dimension threshold polynomial in $1/\\varepsilon$ and $\\log(1/α)$. More precisely, for every $α\\in(0,1/2)$, it returns a $(1-\\varepsilon)$ approximation with probability at least $1-α$ whenever $N\\ge \\mathsf{K}\\varepsilon^{-2}\\log^2(e/\\varepsilon)\\log(\\mathsf{K}/α)$, using $O(N^2/\\varepsilon)$ work and $O(N^2)$ storage. GEG-IAMP replaces the Onsager correction by Reeves's coupling construction [arXiv:2508.10782] and groups Euler-Maruyama steps into $O(1/\\varepsilon)$ matrix-vector products. We also prove a relative guarantee for AMS IAMP with $O(N^2\\varepsilon^{-3}\\log^2(e/\\varepsilon))$ work but a sufficient dimension exponential in inverse accuracy. By returning a fixed configuration below its dimension threshold, GEG-IAMP also provides an asymptotic fully polynomial randomized approximation scheme (AFPRAS), with an additive error independent of dimension.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05446",
  "author_total": 4,
  "authors": [
    {
      "name": "Dylan Herman"
    },
    {
      "name": "Shouvanik Chakrabarti"
    },
    {
      "name": "Junhyung Lyle Kim"
    },
    {
      "name": "Guneykan Ozgul"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c81yk7xmwvf6a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05446",
  "fulltext_url": null,
  "id": "c81yk7xmwvf6a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05446",
  "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",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "quant-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05446",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c81yk7xmwvf6a.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:20:56Z",
      "native_id": "2611.05446",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05446"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "A Fully Polynomial Approximation Algorithm for the Sherrington-Kirkpatrick Model",
  "title_text": "A Fully Polynomial Approximation Algorithm for the Sherrington-Kirkpatrick Model",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:29Z",
  "url": "https://research.draconitic.com/paper/c81yk7xmwvf6a",
  "venue": "arXiv",
  "versions": []
}