Belief Propagation for Min-cost Network Flow: Convergence and Correctness

TL;DR

This paper analyzes the Belief Propagation algorithm's convergence and correctness for the minimum-cost network flow problem, proving it converges to the optimal solution under certain conditions and introducing a fully polynomial-time approximation scheme.

Contribution

It provides the first proof of BP's polynomial-time convergence for a classical optimization problem and introduces a new approximation scheme without requiring solution uniqueness.

Findings

BP converges to the optimal solution in pseudo-polynomial time under certain conditions.

A simple modification yields a fully polynomial-time randomized approximation scheme.

This is the first proof of BP's fully-polynomial running time for an optimization problem.

Abstract

Message passing type algorithms such as the so-called Belief Propagation algorithm have recently gained a lot of attention in the statistics, signal processing and machine learning communities as attractive algorithms for solving a variety of optimization and inference problems. As a decentralized, easy to implement and empirically successful algorithm, BP deserves attention from the theoretical standpoint, and here not much is known at the present stage. In order to fill this gap we consider the performance of the BP algorithm in the context of the capacitated minimum-cost network flow problem - the classical problem in the operations research field. We prove that BP converges to the optimal solution in the pseudo-polynomial time, provided that the optimal solution of the underlying problem is unique and the problem input is integral. Moreover, we present a simple modification of the BP algorithm which gives a fully polynomial-time randomized approximation scheme (FPRAS) for the same problem, which no longer requires the uniqueness of the optimal solution. This is the first instance where BP is proved to have fully-polynomial running time. Our results thus provide a theoretical justification for the viability of BP as an attractive method to solve an important class of optimization problems.