Belief propagation for min-cost network flow: convergence & correctness

Distributed, iterative algorithms operating with minimal data structure while performing little computation per iteration are popularly known as message-passing in the recent literature. Belief Propagation (BP), a prototypical message-passing algorithm, has gained a lot of attention across disciplines including communications, statistics, signal processing and machine learning as an attractive scalable, general purpose heuristic for a wide class of optimization and statistical inference problems. Despite its empirical success, the theoretical understanding of BP is far from complete. With the goal of advancing the state-of-art of our understanding of BP, we study the performance of BP in the context of the capacitated minimum-cost network flow problem – a corner stone in the development of theory of polynomial time algorithms for optimization problems as well as widely used in practice of operations research. As the main result of this paper, we prove that BP converges to the optimal solution in the pseudo-polynomial-time, provided that the optimal solution of the underlying network flow problem instance is unique and the problem parameters are integral. We further provide a simple modification of the BP to obtain a fully polynomial-time randomized approximation scheme (FPRAS) without requiring 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.