Max-Product for Maximum Weight Matching - Revisited

TL;DR

This paper revisits belief propagation for maximum weight bipartite matching, providing tight bounds on iteration complexity, analyzing its behavior without convergence, and proposing an approximate algorithm for practical use.

Contribution

It offers a constructive proof of iteration bounds, insights into belief propagation's behavior without convergence, and introduces an approximate algorithm for assignment problems.

Findings

The upper bound on iterations is tight up to a factor of four.

Approximate belief propagation can achieve sharp solutions efficiently.

Convergence time is dominated by the iterations needed for a good approximation.

Abstract

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.