Loop Calculus and Bootstrap-Belief Propagation for Perfect Matchings on Arbitrary Graphs

This manuscript discusses computation of the Partition Function (PF) and the Minimum Weight Perfect Matching (MWPM) on arbitrary, non-bipartite graphs. We present two novel problem formulations - one for computing the PF of a Perfect Matching (PM) and one for finding MWPMs - that build upon the inter-related Bethe Free Energy, Belief Propagation (BP), Loop Calculus (LC), Integer Linear Programming (ILP) and Linear Programming (LP) frameworks. First, we describe an extension of the LC framework to the PM problem. The resulting formulas, coined (fractional) Bootstrap-BP, express the PF of the original model via the BFE of an alternative PM problem. We then study the zero-temperature version of this Bootstrap-BP formula for approximately solving the MWPM problem. We do so by leveraging the Bootstrap-BP formula to construct a sequence of MWPM problems, where each new problem in the sequence is formed by contracting odd-sized cycles (or blossoms) from the previous problem. This Bootstrap-and-Contract procedure converges reliably and generates an empirically tight upper bound for the MWPM. We conclude by discussing the relationship between our iterative procedure and the famous Blossom Algorithm of Edmonds '65 and demonstrate the performance of the Bootstrap-and-Contract approach on a variety of weighted PM problems.

Belief Propagation for Linear Programming

Belief Propagation (BP) is a popular, distributed heuristic for performing MAP computations in Graphical Models. BP can be interpreted, from a variational perspective, as minimizing the Bethe Free Energy (BFE). BP can also be used to solve a special class of Linear Programming (LP) problems. For this class of problems, MAP inference can be stated as an integer LP with an LP relaxation that coincides with minimization of the BFE at ``zero temperature". We generalize these prior results and establish a tight characterization of the LP problems that can be formulated as an equivalent LP relaxation of MAP inference. Moreover, we suggest an efficient, iterative annealing BP algorithm for solving this broader class of LP problems. We demonstrate the algorithm's performance on a set of weighted matching problems by using it as a cutting plane method to solve a sequence of LPs tightened by adding ``blossom'' inequalities.