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We consider the general problem of finding the minimum weight $\bm$-matchingon arbitrary graphs. We prove that, whenever the linear programming (LP)relaxation of the problem has no fractional solutions, then the beliefpropagation (BP) algorithm converges to the correct solution. We also show thatwhen the LP relaxation has a fractional solution then the BP algorithm can beused to solve the LP relaxation. Our proof is based on the notion of graphcovers and extends the analysis of (Bayati-Shah-Sharma 2005 and Huang-Jebara2007}.  These results are notable in the following regards: (1) It is one of a verysmall number of proofs showing correctness of BP without any constraint on thegraph structure. (2) Variants of the proof work for both synchronous andasynchronous BP; it is the first proof of convergence and correctness of anasynchronous BP algorithm for a combinatorial optimization problem.

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