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Inverse Generalized Maximum Flow Problems

Author

Listed:
  • Javad Tayyebi

    (Department of Industrial Engineering, Birjand University of Technology, Birjand 9719866981, Iran
    The authors contributed equally to this work.)

  • Adrian Deaconu

    (Department of Mathematics and Computer Science, Transilvania University, Brasov 500091, Romania
    The authors contributed equally to this work.)

Abstract

A natural extension of maximum flow problems is called the generalized maximum flow problem taking into account the gain and loss factors for arcs. This paper investigates an inverse problem corresponding to this problem. It is to increase arc capacities as less cost as possible in a way that a prescribed flow becomes a maximum flow with respect to the modified capacities. The problem is referred to as the generalized maximum flow problem (IGMF). At first, we present a fast method that determines whether the problem is feasible or not. Then, we develop an algorithm to solve the problem under the max-type distances in O ( m n · log n ) time. Furthermore, we prove that the problem is strongly NP-hard under sum-type distances and propose a heuristic algorithm to find a near-optimum solution to these NP-hard problems. The computational experiments show the accuracy and the efficiency of the algorithm.

Suggested Citation

  • Javad Tayyebi & Adrian Deaconu, 2019. "Inverse Generalized Maximum Flow Problems," Mathematics, MDPI, vol. 7(10), pages 1-15, September.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:10:p:899-:d:270721
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    References listed on IDEAS

    as
    1. Longcheng Liu & Jianzhong Zhang, 2006. "Inverse maximum flow problems under the weighted Hamming distance," Journal of Combinatorial Optimization, Springer, vol. 12(4), pages 395-408, December.
    2. László A. Végh, 2017. "A Strongly Polynomial Algorithm for Generalized Flow Maximization," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 179-211, January.
    3. Clemens Heuberger, 2004. "Inverse Combinatorial Optimization: A Survey on Problems, Methods, and Results," Journal of Combinatorial Optimization, Springer, vol. 8(3), pages 329-361, September.
    Full references (including those not matched with items on IDEAS)

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