任意图上完美匹配的循环演算和Bootstrap-置信传播
Loop Calculus and Bootstrap-Belief Propagation for Perfect Matchings on Arbitrary Graphs
摘要：
本文讨论了任意非二部图的配分函数和最小权完美匹配的计算。我们提出了两个新的问题配方-一个用于计算PF的完美匹配（PM）和一个用于寻找MWPMs -建立在相互关联的贝特自由能，置信传播（BP），循环演算（LC），线性规划（ILP）和线性规划（LP）框架。首先，我们描述了一个扩展的LC框架PM问题。由此产生的公式，创造（分数）Bootstrap-BP，通过一个替代PM问题的BFE表示原始模型的PF。然后，我们研究了该Bootstrap-BP公式的零温度版本，以近似解决MWPM问题。我们这样做，利用Bootstrap-BP公式来构建一个序列的MWPM问题，其中每个新的问题序列是由收缩奇数大小的周期（或开花）从以前的问题。这个Bootstrap和合同程序可靠地收敛，并产生一个经验严格的上限MWPM。最后，我们讨论了我们的迭代过程和著名的Blossom算法的埃德蒙兹'65之间的关系，并证明了各种加权PM问题的Bootstrap和合同方法的性能。
英文摘要：
This manuscript discusses computation of the Partition Function (PF) and the Minimum Weight Perfect Matching (MWPM) on arbitrary, non-bipartite graphs. We present two novel problem formulations - one for computing the PF of a Perfect Matching (PM) and one for finding MWPMs - that build upon the inter-related Bethe Free Energy, Belief Propagation (BP), Loop Calculus (LC), Integer Linear Programming (ILP) and Linear Programming (LP) frameworks. First, we describe an extension of the LC framework to the PM problem. The resulting formulas, coined (fractional) Bootstrap-BP, express the PF of the original model via the BFE of an alternative PM problem. We then study the zero-temperature version of this Bootstrap-BP formula for approximately solving the MWPM problem. We do so by leveraging the Bootstrap-BP formula to construct a sequence of MWPM problems, where each new problem in the sequence is formed by contracting odd-sized cycles (or blossoms) from the previous problem. This Bootstrap-and-Contract procedure converges reliably and generates an empirically tight upper bound for the MWPM. We conclude by discussing the relationship between our iterative procedure and the famous Blossom Algorithm of Edmonds '65 and demonstrate the performance of the Bootstrap-and-Contract approach on a variety of weighted PM problems.
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