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A Strongly Polynomial Cut Canceling Algorithm for the Submodular Flow Problem

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Integer Programming and Combinatorial Optimization (IPCO 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1610))

Abstract

This paper presents a new strongly polynomial cut canceling algorithm for minimum cost submodular flow. The algorithm is a generalization of our similar cut canceling algorithm for ordinary mincost flow. The advantage of cut canceling over cycle canceling is that cut canceling seems to generalize to other problems more readily than cycle canceling. The algorithm scales a relaxed optimality parameter, and creates a second, inner relaxation that is a kind of submodular max flow problem. The outer relaxation uses a novel technique for relaxing the submodular constraints that allows our previous proof techniques to work. The algorithm uses the min cuts from the max flow subproblem as the relaxed most positive cuts it chooses to cancel. We show that this algorithm needs to cancel only O(n 3) cuts per scaling phase, where n is the number of nodes. Furthermore, we also show how to slightly modify this algorithm to get a strongly polynomial running time.

Research supported by a Grant-in-Aid for Scientific Research from Ministry of Education, Science, Sports and Culture of Japan.

Research supported by an NSERC Operating Grant; part of this research was done during visits to RIMS at Kyoto University and SORIE at Cornell University.

Research supported by a Grant-in-Aid for Scientific Research from Ministry of Education, Science, Sports and Culture of Japan.

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Author information

Authors and Affiliations

  1. Graduate School of Engineering Sciences, Osaka University, Osaka, 560, Japan

    Satoru Iwata

  2. Faculty of Commerce and Business Administration, University of British Columbia, BC, V6T 1Z2, Canada

    S. Thomas McCormick

  3. Institute of Policy and Planning Sciences, University of Tsukuba, Ibaraki, 305, Japan

    Maiko Shigeno

Authors
  1. Satoru Iwata
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  2. S. Thomas McCormick
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  3. Maiko Shigeno
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Editor information

Editors and Affiliations

  1. GSIA, Carnegie Mellon University, Schenley Park, Pittsburgh, PA, 15213, USA

    Gérard Cornuéjols

  2. Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, A-8010, Graz, Austria

    Rainer E. Burkard  & Gerhard J. Woeginger  & 

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© 1999 Springer-Verlag Berlin Heidelberg

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Iwata, S., McCormick, S.T., Shigeno, M. (1999). A Strongly Polynomial Cut Canceling Algorithm for the Submodular Flow Problem. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds) Integer Programming and Combinatorial Optimization. IPCO 1999. Lecture Notes in Computer Science, vol 1610. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48777-8_20

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  • DOI: https://doi.org/10.1007/3-540-48777-8_20

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66019-4

  • Online ISBN: 978-3-540-48777-7

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