Belief-Propagation for Weighted b-Matchings on Arbitrary Graphs and its Relation to Linear Programs with Integer Solutions

TL;DR

This paper proves that belief propagation converges to the optimal solution for weighted b-matching problems on arbitrary graphs when the LP relaxation is integral, and relates BP solutions to LP relaxations with fractional solutions.

Contribution

It provides the first proof of BP correctness on arbitrary graphs without structural constraints and extends analysis to asynchronous BP algorithms.

Findings

BP converges to correct solutions when LP relaxation is integral.

BP can solve LP relaxations with fractional solutions.

First proof of asynchronous BP convergence for combinatorial optimization.

Abstract

We consider the general problem of finding the minimum weight $\bm$-matching on arbitrary graphs. We prove that, whenever the linear programming (LP) relaxation of the problem has no fractional solutions, then the belief propagation (BP) algorithm converges to the correct solution. We also show that when the LP relaxation has a fractional solution then the BP algorithm can be used to solve the LP relaxation. Our proof is based on the notion of graph covers and extends the analysis of (Bayati-Shah-Sharma 2005 and Huang-Jebara 2007}. These results are notable in the following regards: (1) It is one of a very small number of proofs showing correctness of BP without any constraint on the graph structure. (2) Variants of the proof work for both synchronous and asynchronous BP; it is the first proof of convergence and correctness of an asynchronous BP algorithm for a combinatorial optimization problem.