On the exactness of the cavity method for Weighted b-Matchings on Arbitrary Graphs and its Relation to Linear Programs

TL;DR

This paper proves that the cavity method accurately solves the minimum weight b-matching problem on arbitrary graphs when the LP relaxation is integral, establishing a link between belief propagation convergence and LP optimality.

Contribution

It establishes a rigorous connection between the convergence of belief propagation and the integrality of LP relaxations for b-matching problems on arbitrary graphs.

Findings

Belief propagation converges to the correct solution when LP relaxation has no fractional solutions.

The cavity method's exactness is proven under certain LP conditions.

The results apply to both synchronous and asynchronous belief propagation updates.

Abstract

We consider the general problem of finding the minimum weight b-matching on arbitrary graphs. We prove that, whenever the linear programming relaxation of the problem has no fractional solutions, then the cavity or belief propagation equations converge to the correct solution both for synchronous and asynchronous updating.