Belief Propagation and LP Relaxation for Weighted Matching in General Graphs

Loopy belief propagation has been employed in a wide variety of applications with great empirical success, but it comes with few theoretical guarantees. In this paper, we analyze the performance of the max-product form of belief propagation for the weighted matching problem on general graphs. We show that the performance of max-product is exactly characterized by the natural linear programming (LP) relaxation of the problem. In particular, we first show that if the LP relaxation has no fractional optima then max-product always converges to the correct answer. This establishes the extension of the recent result by Bayati, Shah and Sharma, which considered bipartite graphs, to general graphs. Perhaps more interestingly, we also establish a tight converse, namely that the presence of any fractional LP optimum implies that max-product will fail to yield useful estimates on some of the edges. We extend our results to the weighted b-matching and r-edge-cover problems. We also demonstrate how to simplify the max-product message-update equations for weighted matching, making it easily deployable in distributed settings like wireless or sensor networks.