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The Pseudoflow Algorithm: A New Algorithm for the Maximum-Flow Problem

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  • Dorit S. Hochbaum

    (Department of Industrial Engineering and Operations Research and Walter A. Haas School of Business, University of California, Berkeley, California 94720)

Abstract

We introduce the pseudoflow algorithm for the maximum-flow problem that employs only pseudoflows and does not generate flows explicitly. The algorithm solves directly a problem equivalent to the minimum-cut problem---the maximum blocking-cut problem. Once the maximum blocking-cut solution is available, the additional complexity required to find the respective maximum-flow is O ( m log n ). A variant of the algorithm is a new parametric maximum-flow algorithm generating all breakpoints in the same complexity required to solve the constant capacities maximum-flow problem. The pseudoflow algorithm has also a simplex variant, pseudoflow-simplex, that can be implemented to solve the maximum-flow problem. One feature of the pseudoflow algorithm is that it can initialize with any pseudoflow. This feature allows it to reach an optimal solution quickly when the initial pseudoflow is “close” to an optimal solution. The complexities of the pseudoflow algorithm, the pseudoflow-simplex, and the parametric variants of pseudoflow and pseudoflow-simplex algorithms are all O ( mn log n ) on a graph with n nodes and m arcs. Therefore, the pseudoflow-simplex algorithm is the fastest simplex algorithm known for the parametric maximum-flow problem. The pseudoflow algorithm is also shown to solve the maximum-flow problem on s , t -tree networks in linear time, where s, t -tree networks are formed by joining a forest of capacitated arcs, with nodes s and t adjacent to any subset of the nodes.

Suggested Citation

  • Dorit S. Hochbaum, 2008. "The Pseudoflow Algorithm: A New Algorithm for the Maximum-Flow Problem," Operations Research, INFORMS, vol. 56(4), pages 992-1009, August.
    • Handle: RePEc:inm:oropre:v:56:y:2008:i:4:p:992-1009
    DOI: 10.1287/opre.1080.0524
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    References listed on IDEAS

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    4. Jean-Claude Picard, 1976. "Maximal Closure of a Graph and Applications to Combinatorial Problems," Management Science, INFORMS, vol. 22(11), pages 1268-1272, July.
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    Cited by:

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    12. Xiangyong Li & Y. P. Aneja, 2020. "A new branch-and-cut approach for the generalized regenerator location problem," Annals of Operations Research, Springer, vol. 295(1), pages 229-255, December.
    13. Whittle, D. & Brazil, M. & Grossman, P.A. & Rubinstein, J.H. & Thomas, D.A., 2018. "Combined optimisation of an open-pit mine outline and the transition depth to underground mining," European Journal of Operational Research, Elsevier, vol. 268(2), pages 624-634.
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