Belief Propagation for Linear Programming

TL;DR

This paper explores the use of Belief Propagation (BP) as an efficient heuristic for solving a broad class of Linear Programming problems by connecting MAP inference with LP relaxations and introducing an annealing BP algorithm.

Contribution

It generalizes previous results linking BP and LP, providing a tight characterization of solvable LPs and proposing an iterative annealing BP algorithm for broader LP classes.

Findings

The annealing BP algorithm effectively solves weighted matching problems.

BP-based approach can serve as a cutting plane method for LPs.

The method demonstrates competitive performance on benchmark problems.

Abstract

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.