BeliefPropagation for Weighted bMatchings 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 (BayatiShahSharma 2005 and HuangJebara 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2007
 arXiv:
 arXiv:0709.1190
 Bibcode:
 2007arXiv0709.1190B
 Keywords:

 Computer Science  Information Theory;
 Computer Science  Artificial Intelligence
 EPrint:
 28 pages, 2 figures. Submitted to SIAM journal on Discrete Mathematics on March 19, 2009