Belief-Propagation for Weighted b-Matchings on Arbitrary Graphs and its Relation to Linear Programs with Integer Solutions
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.
Cite
@article{arxiv.0709.1190,
title = {Belief-Propagation for Weighted b-Matchings on Arbitrary Graphs and its Relation to Linear Programs with Integer Solutions},
author = {Mohsen Bayati and Christian Borgs and Jennifer Chayes and Riccardo Zecchina},
journal= {arXiv preprint arXiv:0709.1190},
year = {2015}
}
Comments
28 pages, 2 figures. Submitted to SIAM journal on Discrete Mathematics on March 19, 2009; accepted for publication (in revised form) August 30, 2010; published electronically July 1, 2011