Max-Product for Maximum Weight Matching: Convergence, Correctness, and LP Duality

Max-product "belief propagation" (BP) is an iterative, message-passing algorithm for finding the maximum a posteriori (MAP) assignment of a discrete probability distribution specified by a graphical model. Despite the spectacular success of the algorithm in many application areas such as iterative decoding and combinatorial optimization, which involve graphs with many cycles, theoretical results about both the correctness and convergence of the algorithm are known in only a few cases (see section I for references). In this paper, we prove the correctness and convergence of max-product for finding the maximum weight matching (MWM) in bipartite graphs. Even though the underlying graph of the MWM problem has many cycles, somewhat surprisingly we show that the max-product algorithm converges to the correct MWM as long as the MWM is unique. We provide a bound on the number of iterations required and show that for a graph of size n, the computational cost of the algorithm scales as O(n³), which is the same as the computational cost of the best known algorithms for finding the MWM. We also provide an interesting relation between the dynamics of the max-product algorithm and the auction algorithm, which is a well-known distributed algorithm for solving the MWM problem.