发表机构
School of Computer Science, Shanghai Jiao Tong University; John Hopcroft Center for Computer Science, Shanghai Jiao Tong University（上海交通大学计算机学院; 上海交通大学约翰·霍普克罗夫特计算机科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一个确定性近似计数框架，通过递归构造有理多胞形和线性分数规划，在多项式连通常数图上实现强空间混合与FPTAS，并将正常着色的计数常数从3改进到2。
AI 中文摘要
我们开发了一个用于超越有界度图的多自旋系统确定性近似计数的框架。该算法递归地构造包含真实边际向量的有理多胞形，并使用线性分数规划获得边际比率的认证界。对于具有多项式连通常数$D$的图上的正相互作用，我们建立了强空间混合和完全多项式时间近似方案（\textbf{FPTAS}），只要$Dc<1$，其中$c$界定相互作用的Birkhoff收缩系数。我们进一步将该框架扩展到稀疏Erdős-Rényi随机图的正常着色，使用基于许可块的递归。对于每个固定的$\eta\in(0,1)$、足够大的固定$d$以及固定的整数$q\ge(2+\eta)d$，我们以高概率在$G$上获得对$G\sim\mathcal G(n,d/n)$的正常$q$-着色的\textbf{FPTAS}。这将Yin和Zhang（APPROX/RANDOM, 2016）早期计数保证中的领先常数$3$改进为$2$，并在渐近意义上匹配Yin（ICALP, 2014）建立的空间混合机制。
英文摘要
We develop a framework for deterministic approximate counting of multi-spin systems beyond bounded-degree graphs. The algorithm recursively constructs rational polytopes containing the true marginal vectors and uses linear-fractional programming to obtain certified bounds on marginal ratios. For positive interactions on graphs of polynomial connective constant $D$, we establish strong spatial mixing and a fully polynomial-time approximation scheme (\textbf{FPTAS}) whenever $Dc<1$, where $c$ bounds the Birkhoff contraction coefficients of the interactions. We further extend the framework to proper colorings of sparse Erdős-Rényi random graphs using recursion on permissive blocks. For every fixed $η\in(0,1)$, sufficiently large fixed $d$, and fixed integer $q\ge(2+η)d$, we obtain an \textbf{FPTAS} for counting proper $q$-colorings of $G\sim\mathcal G(n,d/n)$ with high probability over $G$. This improves the leading constant $3$ in the earlier counting guarantee of Yin and Zhang (APPROX/RANDOM, 2016) to $2$, and asymptotically matches the spatial mixing regime established by Yin (ICALP, 2014).
CommentsAdd appropriate references for Lagrange inversion
