arXiv math/06061432006-06math.COmath.PR

图的列表着色计数的相关衰减和确定性FPTAS

Correlation decay and deterministic FPTAS for counting list-colorings of a graph

David Gamarnik, Dmitriy Katz

备注：: 15 pages

摘要：

我们提出了一个确定性的算法，近似计数的数量列表着色的图。在图是无三角形的，每个列表的大小至少为$αΔ$的条件下，得到了如下结果，其中$α$是大于$α^{**}=2.8432...$的任意常数，$Δ$是图的最大度.对于每个列表的大小是一个大常数的情况下，我们显示了存在一个\emph{deterministic} FPTAS计算列表着色的总数。相同的确定性算法具有复杂性$2^{O(\log^2 n)}$，没有对列表大小的任何假设，其中$n$是实例大小。我们进一步扩展我们的方法，离散马尔可夫随机场（MRF）模型。在一定的假设下，有关的字母表的大小，度的图形和相互作用的潜力，我们再次构建一个确定性的FPTAS计算的MRF的分区函数。我们的结果不是基于现有的最强大的计数技术-快速混合马尔可夫链方法。相反，我们建立在统计物理学的概念，特别是相关现象的衰减及其对无限图中吉布斯测度唯一性的影响。这种方法在最近的两篇论文\cite{BandyopadhyayGamarnikCounting}和\cite{weitzCounting}中提出。这种方法的原理是，相关性衰减特性可以相对于某些\emph{computation tree}来建立，这与相对于给定节点的图论邻域的传统相关性衰减特性相反。这允许在对数深度处截断计算，以便在多项式时间内获得多项式精度。

英文摘要：

We propose a deterministic algorithm for approximately counting the number of list colorings of a graph. Under the assumption that the graph is triangle free, the size of every list is at least $αΔ$, where $α$ is an arbitrary constant bigger than $α^{**}=2.8432...$, and $Δ$ is the maximum degree of the graph, we obtain the following results. For the case when the size of the each list is a large constant, we show the existence of a \emph{deterministic} FPTAS for computing the total number of list colorings. The same deterministic algorithm has complexity $2^{O(\log^2 n)}$, without any assumptions on the sizes of the lists, where $n$ is the instance size. We further extend our method to a discrete Markov random field (MRF) model. Under certain assumptions relating the size of the alphabet, the degree of the graph and the interacting potentials we again construct a deterministic FPTAS for computing the partition function of a MRF. Our results are not based on the most powerful existing counting technique -- rapidly mixing Markov chain method. Rather we build upon concepts from statistical physics, in particular, the decay of correlation phenomena and its implication for the uniqueness of Gibbs measures in infinite graphs. This approach was proposed in two recent papers \cite{BandyopadhyayGamarnikCounting} and \cite{weitzCounting}. The principle insight of this approach is that the correlation decay property can be established with respect to certain \emph{computation tree}, as opposed to the conventional correlation decay property with respect to graph theoretic neighborhoods of a given node. This allows truncation of computation at a logarithmic depth in order to obtain polynomial accuracy in polynomial time.

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