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

Author

Listed:
  • David Gamarnik

    (Operations Research Center and Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

  • Devavrat Shah

    (Laboratory for Information and Decision Systems (LIDS) and Operations Research Center, Department of EECS, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

  • Yehua Wei

    (Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

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.

Suggested Citation

  • David Gamarnik & Devavrat Shah & Yehua Wei, 2012. "Belief Propagation for Min-Cost Network Flow: Convergence and Correctness," Operations Research, INFORMS, vol. 60(2), pages 410-428, April.
  • Handle: RePEc:inm:oropre:v:60:y:2012:i:2:p:410-428
    DOI: 10.1287/opre.1110.1025
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    References listed on IDEAS

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    1. James B. Orlin, 1993. "A Faster Strongly Polynomial Minimum Cost Flow Algorithm," Operations Research, INFORMS, vol. 41(2), pages 338-350, April.
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    Cited by:

    1. Guowei Dai & Fengwei Li & Yuefang Sun & Dachuan Xu & Xiaoyan Zhang, 2019. "Convergence and correctness of belief propagation for the Chinese postman problem," Journal of Global Optimization, Springer, vol. 75(3), pages 813-831, November.
    2. Shuvomoy Das Gupta & Lacra Pavel, 2019. "On seeking efficient Pareto optimal points in multi-player minimum cost flow problems with application to transportation systems," Journal of Global Optimization, Springer, vol. 74(3), pages 523-548, July.

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