Equivalence of LP Relaxation and Max-Product for Weighted Matching in General Graphs

TL;DR

This paper establishes a precise connection between the convergence of max-product belief propagation and the tightness of LP relaxation for weighted matching in general graphs, providing theoretical guarantees and insights.

Contribution

It proves that max-product converges to the correct maximum weight matching if and only if the LP relaxation is tight, extending previous bipartite graph results to general graphs.

Findings

Max-product converges if LP relaxation is tight.

Max-product does not converge if LP relaxation is loose.

Provides a data-dependent characterization of max-product performance.

Abstract

Max-product belief propagation is a local, iterative algorithm to find the mode/MAP estimate of a probability distribution. While it has been successfully employed in a wide variety of applications, there are relatively few theoretical guarantees of convergence and correctness for general loopy graphs that may have many short cycles. Of these, even fewer provide exact ``necessary and sufficient'' characterizations. In this paper we investigate the problem of using max-product to find the maximum weight matching in an arbitrary graph with edge weights. This is done by first constructing a probability distribution whose mode corresponds to the optimal matching, and then running max-product. Weighted matching can also be posed as an integer program, for which there is an LP relaxation. This relaxation is not always tight. In this paper we show that \begin{enumerate} \item If the LP relaxation is tight, then max-product always converges, and that too to the correct answer. \item If the LP relaxation is loose, then max-product does not converge. \end{enumerate} This provides an exact, data-dependent characterization of max-product performance, and a precise connection to LP relaxation, which is a well-studied optimization technique. Also, since LP relaxation is known to be tight for bipartite graphs, our results generalize other recent results on using max-product to find weighted matchings in bipartite graphs.