Max-product belief propagation (BP) is a popular message-passing algorithm for computing a maximum-a-posteriori (MAP) assignment in a joint distribution represented by a graphical model (GM). It has been shown that BP can solve a few classes of Linear Programming (LP) formulations to combinatorial optimization problems including maximum weight matching and shortest path, i.e., BP can be a distributed solver for certain LPs. However, those LPs and corresponding BP analysis are very sensitive to underlying problem setups, and it has been not clear what extent these results can be generalized to. In this paper, we obtain a generic criteria that BP converges to the optimal solution of given LP, and show that it is satisfied in LP formulations associated to many classical combinatorial optimization problems including maximum weight perfect matching, shortest path, traveling salesman, cycle packing and vertex cover. More importantly, our criteria can guide the BP design to compute fractional LP solutions, while most prior results focus on integral ones. Our results provide new tools on BP analysis and new directions on efficient solvers for large-scale LPs.
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