Belief Propagation for Min-Cost Network Flow: Convergence and Correctness
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Shah_Belief propagation.pdf
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389.29 KB
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Author(s)
Gamarnik, David
Shah, Devavrat
Wei, Yehua
Date Issued
March 2012
Journal
Operations Research
Publisher
Institute for Operations Research and the Management Sciences (INFORMS)
Citation
Gamarnik, D., D. Shah, and Y. Wei. “Belief Propagation for Min-Cost Network Flow: Convergence and Correctness.” Operations Research 60.2 (2012): 410–428.
Version
Author's final manuscript
Abstract
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 cornerstone in the development of the theory of polynomial-time algorithms for optimization problems and widely used in the practice of operations research. As the main result of this paper, we prove that BP converges to the optimal solution in pseudopolynomial 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.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
Massachusetts Institute of Technology. Operations Research Center
Sloan School of Management
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1287/opre.1110.1025
