Max-Product for Maximum Weight Matching - Revisited

We focus on belief propagation for the assignment problem, also known as the maximum weight bipartite matching problem. We provide a constructive proof that the well-known upper bound on the number of iterations (Bayati, Shah, Sharma 2008) is tight up to a factor of four. Furthermore, we investigate the behavior of belief propagation when convergence is not required. We show that the number of iterations required for a sharp approximation consumes a large portion of the convergence time. Finally, we propose an "approximate belief propagation" algorithm for the assignment problem.