{
  "abstract": "We give deterministic FPTASes for two dense counting problems on which the known deterministic algorithms, based on zero-free interpolation, run in quasipolynomial time. For fixed $0<γ<1/2$ and $0<θ\\leq1$, the first approximates $\\mathrm{haf}(A)$ for a symmetric matrix $A$ when its support graph $G$ has minimum degree at least $(1/2+γ)n$ and its nonzero entries lie in $[θ,1]$. It also approximates permanents under the analogous bipartite condition, including full-support matrices in $[θ,1]$. For fixed $β>0$ and $0<κ\\leq1$, the second approximates the zero-field Ising partition function $Z(J)$ for zero-diagonal real symmetric matrices $J$ satisfying $\\max_{i,j}|J_{ij}|\\leqβ/n$ and $λ_{\\max}(J)\\leq1-κ$. No separate lower-eigenvalue condition is imposed. We further prove $\\log\\mathrm{haf}(A)=h_A(G)-n/2+O_{γ,θ}(1)$ and $Z(J)=2^n\\det(I-J)^{-1/2}(1+O_{β,κ}(1/n))$. Here $h_A(G)$ is the maximum weighted fractional-matching entropy. For unweighted graphs, the first formula improves the Cuckler--Kahn error from $o(n)$ to $O_γ(1)$ on the fixed-margin class and extends it to weights in $[θ,1]$. Both algorithms use a common Gaussian truncation principle. Each problem becomes an integral of a product of a fixed entire function over Gaussian coordinates, with possibly indefinite moment matrix entries of order $1/n$. Cancelling the linear term and exactly resumming the quadratic term leaves a coordinate remainder vanishing to order at least three. Complex dilation handles small supports. For large supports, we bound the recombined tail by a large-deviation rate that beats the entropy of the subsets. The truncation error is at most $(CR/n)^{R/2}+e^{-cn}$. This faster-than-geometric decay permits $R\\log(en/R)=O(\\log n+\\log(1/ε))$ and hence polynomial enumeration.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04079",
  "author_total": 1,
  "authors": [
    {
      "name": "Zihong Yi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cvcdyxzyhjjv5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04079",
  "fulltext_url": null,
  "id": "cvcdyxzyhjjv5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04079",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7208733293",
  "pdf_url": "https://arxiv.org/pdf/2609.04079",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cvcdyxzyhjjv5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.04079",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04079"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Diffuse Gaussian Truncation For Deterministic Approximate Counting",
  "title_text": "Diffuse Gaussian Truncation For Deterministic Approximate Counting",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/cvcdyxzyhjjv5",
  "venue": "arXiv",
  "versions": []
}