Belief Propagation for Weighted b-Matchings on Arbitrary Graphs and its Relation to Linear Programs with Integer Solutions

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.