BELIEF PROPAGATION FOR WEIGHTED b-MATCHINGS ON ARBITRARY GRAPHS AND ITS RELATION TO LINEAR PROGRAMS WITH INTEGER SOLUTIONS

作     者：Bayati, Mohsen Borgs, Christian Chayes, Jennifer Zecchina, Riccardo 

作者机构：Stanford Univ Stanford CA 94305 USA Microsoft Res Cambridge MA 02142 USA Politecn Torino I-01029 Turin Italy 

出 版 物：《SIAM JOURNAL ON DISCRETE MATHEMATICS》 (工业与应用数学会离散数学杂志)

年 卷 期：2011年第25卷第2期

页      面：989-1011页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：Microsoft Technical Computing Initiative 

主　　题：belief propagation linear program matching cavity method graph cover 

摘      要：We consider the general problem of finding the minimum weight b-matching on arbitrary graphs. We prove that, whenever the linear programming (LP) relaxation of the problem has no fractional solutions, then the belief propagation (BP) algorithm converges to the correct solution. We also show that when the LP relaxation has a fractional solution then the BP algorithm can be used to solve the LP relaxation. Our proof is based on the notion of graph covers and extends the analyses of [M. Bayati, D. Shah, and M. Sharma, in Proceedings of the IEEE Int. Symp. Information Theory, 2005] and [B. Huang and T. Jebara, in Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, 2007]. The result is notable in the following regards: (1) It is one of a very small number of proofs showing correctness of BP without any constraint on the graph structure;(2) Variants of the proof work for both synchronous and asynchronous BP;it is the first proof of convergence and correctness of an asynchronous BP algorithm for a combinatorial optimization problem.