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

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 s...

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Bibliographic Details
Published in:Discrete Applied Mathematics Vol. 354; pp. 122 - 130
Main Authors: Dai, Guowei, Guo, Longkun, Gutin, Gregory, Zhang, Xiaoyan, Zhang, Zan-Bo
Format: Journal Article
Language:English
Published: Elsevier B.V 15.09.2024
Subjects:
Belief propagation
Message-passing algorithm
Min–max BP algorithm
Min–max flow
Min–max BP algorithm
Message-passing algorithm
Belief propagation
Min–max flow
ISSN:0166-218X, 1872-6771
Online Access:Get full text
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Summary: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.
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2021.12.025