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 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 fractional solution then BP algorithm can be used to solve the LP. These results are notable in several regards: (1) It is one of a very small number of proofs showing correctness of BP without any constraint on the graph structure. (2) Instead of showing that BP leads to a PTAS, we give a finite bound for the number of iterations after which BP has converged to the exact solution. (3) Variants of the proof work for both synchronous and asynchronous BP; to the best of our knowledge, it is the first proof of convergence and correctness of an asynchronous BP algorithm for a combinatorial optimization problem. (4) It works for both ordinary b-matchings and the more difficult case of perfect b-matchings. (5) Together with the recent work of Sanghavi, Malioutov and Wilskly [41] they are the first complete proofs showing that tightness of LP implies correctness of BP.