arXiv 0909.52242009-09cs.DM

利用子图序列确定图着色数

Deterministic counting of graph colourings using sequences of subgraphs

Charilaos Efthymiou

发表信息：: Combinator. Probab. Comp. 29 (2020) 555-586
DOI：: 10.1017/S0963548320000255

摘要：

本文提出了一个稀疏随机图$G(n,d/n)$的$k$ -着色的确定性近似计数算法.特别是，我们的算法计算在多项式时间的$(1\pm n^{-Ω(1)})$近似的$k$ -着色的$G(n,d/n)$的$k\geq (2+ε) d$的概率高的图形实例的数量的对数。我们的算法与A. Bandyopadhyay等人在SODA '06中，以及A. Montanari等人在SODA '06中的方法，即它使用{ \em空间相关衰减}来计算{\em Gibbs分布}的{\em确定性地}边缘。我们开发了一个计划，其准确性取决于{ \em非重建}的着色的$G(n,d/n)$，而不是{ \em的唯一性}，在以前的作品中所需要的。这使得我们的模式即使对于$k<d$也足够准确的可能性。建立相关衰减的设置如下：给定$G(n,d/n)$，我们通过删除$Λ$的顶点之间的边来改变图的某些特定区域$Λ$中的图结构。然后，我们表明，这种变化对吉布斯分布的边缘的影响，减少，因为我们远离$Λ$。我们的方法是新颖的，并提出了一个新的背景下的确定性计数算法的研究。

英文摘要：

In this paper we propose a deterministic algorithm for approximately counting the $k$-colourings of sparse random graphs $G(n,d/n)$. In particular, our algorithm computes in polynomial time a $(1\pm n^{-Ω(1)})$approximation of the logarithm of the number of $k$-colourings of $G(n,d/n)$ for $k\geq (2+ε) d$ with high probability over the graph instances. Our algorithm is related to the algorithms of A. Bandyopadhyay et al. in SODA '06, and A. Montanari et al. in SODA '06, i.e. it uses {\em spatial correlation decay} to compute {\em deterministically} marginals of {\em Gibbs distribution}. We develop a scheme whose accuracy depends on {\em non-reconstruction} of the colourings of $G(n,d/n)$, rather than {\em uniqueness} that are required in previous works. This leaves open the possibility for our schema to be sufficiently accurate even for $k<d$. The set up for establishing correlation decay is as follows: Given $G(n,d/n)$, we alter the graph structure in some specific region $Λ$ of the graph by deleting edges between vertices of $Λ$. Then we show that the effect of this change on the marginals of Gibbs distribution, diminishes as we move away from $Λ$. Our approach is novel and suggests a new context for the study of deterministic counting algorithms.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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