2007/09/30 by Mohsen Bayati, Christian Borgs, Jennifer Chayes +1 · 76 citations
Computer Science · Mathematics · #Algorithm #Asynchronous communication #Belief propagation #Combinatorics #Computer science #Convergence (economics) #Cooperative Communication and Network Coding #Correctness #Discrete mathematics #Error Correcting Code Techniques #Linear programming #Matching (statistics) #Mathematical proof #Mathematics #Optimization and Search Problems #Relaxation (psychology) #cs.AI #cs.IT #math.IT
paper · pdf · doi:10.1137/090753115
published in SIAM Journal on Discrete Mathematics 25(2), 989-1011 (Society for Industrial and Applied Mathematics) · 28 pages, 2 figures. Submitted to SIAM journal on Discrete Mathematics on March 19, 2009; accepted for publication (in revised form) August 30, 2010; published electronically July 1, 2011
openalex publication_date 2011/01/01 · arxiv created 2011/08/04 · arxiv updated 2015/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider the general problem of finding the minimum weight -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.