Convergence and correctness of belief propagation for weighted min-max flow

Guowei Dai, Longkun Guo, Gregory Z. Gutin, Xiaoyan Zhang, Zan-Bo Zhang

Abstract: In this paper, we investigate the performance of message-passing algorithms for the weighted min–max flow (WMMF) problem which was introduced by Ichimori et al. (1980). WMMF was well studied in combinational optimization, as it provides important applications in time transportation problem and the storage management problem. We develop a message-passing algorithm called min–max belief propagation (BP) for determining the optimal solution of WMMF. As the main result of this paper, we prove that for a digraph of size n, BP converges to the optimal solution within O(n3) time after O(n) iterations if the optimal solution of the underlying min–max flow problem instance is unique. To the best of our knowledge, the fastest polynomial time algorithm for WMMF runs in essentially O(n6) time among the known algorithms, where n is the number of vertices. On the other hand, it is one of a very few instances where BP is proved correct with fully-polynomial running time.