Max-Product Belief Propagation for Linear Programming: Applications to Combinatorial Optimization

TL;DR

This paper establishes generic criteria under which max-product belief propagation converges to the optimal solution of linear programming formulations in various combinatorial optimization problems, broadening its applicability.

Contribution

It provides a general convergence criterion for BP solving LPs, applicable to many classical combinatorial optimization problems, enhancing understanding of its theoretical foundations.

Findings

BP converges to LP optima under new generic criteria

Applicable to maximum weight perfect matching, shortest path, TSP, and more

Broadens the scope of BP in combinatorial optimization

Abstract

The max-product {belief propagation} (BP) is a popular message-passing heuristic for approximating a maximum-a-posteriori (MAP) assignment in a joint distribution represented by a graphical model (GM). In the past years, it has been shown that BP can solve a few classes of linear programming (LP) formulations to combinatorial optimization problems including maximum weight matching, shortest path and network flow, i.e., BP can be used as a message-passing solver for certain combinatorial optimizations. However, those LPs and corresponding BP analysis are very sensitive to underlying problem setups, and it has been not clear what extent these results can be generalized to. In this paper, we obtain a generic criteria that BP converges to the optimal solution of given LP, and show that it is satisfied in LP formulations associated to many classical combinatorial optimization problems including maximum weight perfect matching, shortest path, traveling salesman, cycle packing, vertex/edge cover and network flow.