Loopy annealing belief propagation for vertex cover and matching: convergence, LP relaxation, correctness and Bethe approximation

Abstract—For the minimum cardinality vertex cover and maximum cardinality matching problems, the max-product form of belief propagation (BP) is known to perform poorly on general graphs. In this paper, we present an iterative annealing BP algorithm which is shown to converge and to solve a Linear Programming relaxation of the vertex cover problem on general graphs. As a consequence, our annealing BP finds (asymptotically) a minimum fractional vertex cover on any graph. We also show that it finds (asymptotically) a minimum size vertex cover for any bipartite graph and as a consequence compute the matching number of the graph. Our approach is based on the Bethe variational interpretation of BP. We show that the Bethe free entropy is concave and that BP maximizes it. Using loop calculus, we also give an exact (also intractable for general graphs) expression of the partition function for matchings in term of our BP messages which can be used to improve mean-field approximations.