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Universal Gröbner basis associated with the maximum flow problem

Japan Journal of Industrial and Applied Mathematics volume 30, pages 39–50 (2013)Cite this article

Abstract

We give a formulation of the maximum flow problem as an integer programming problem in the standard form. We characterize elementary vectors of the kernel lattice of the matrix coefficient in our formulation in terms of the combinatorial property of the graph, such as circuits and s–t paths, and we determine the universal Gröbner basis of the toric ideal associated with the maximum flow problem. Next, we examine the kernel lattice of the reduced incidence matrix of the digraph. Under suitable assumptions for the digraph, we prove that it is generated by all incidence vectors of s–t directed paths, which yield the generator of the toric ideal associated with the reduced incidence matrix of the digraph.

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Authors and Affiliations

  1. Graduate School of Engineering, Science of Environment and Mathematical Modeling, Doshisha University, 1-3 Tatara Miyakodani, Kyotanabe, 610-0394, Japan

    Sennosuke Watanabe

  2. Department of Mathematical Sciences, Faculty of Science and Engineering, Doshisha University, 1-3 Tatara Miyakodani, Kyotanabe, 610-0394, Japan

    Yoshihide Watanabe

  3. Kyotanabe, Japan

    Daisuke Ikegami

Authors
  1. Sennosuke Watanabe
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  2. Yoshihide Watanabe
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  3. Daisuke Ikegami
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Correspondence to Sennosuke Watanabe.

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Watanabe, S., Watanabe, Y. & Ikegami, D. Universal Gröbner basis associated with the maximum flow problem. Japan J. Indust. Appl. Math. 30, 39–50 (2013). https://doi.org/10.1007/s13160-012-0080-2

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  • DOI: https://doi.org/10.1007/s13160-012-0080-2

Keywords

  • Maximum flow problem
  • Circuit
  • Universal Gröbner basis

Mathematics Subject Classification

  • 05C21
  • 05E45
  • 90C35