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The diameters of network-flow polytopes satisfy the Hirsch conjecture

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Abstract

We solve a problem in the combinatorics of polyhedra motivated by the network simplex method. We show that the Hirsch conjecture holds for the diameter of the graphs of all network-flow polytopes, in particular the diameter of a network-flow polytope for a network with n nodes and m arcs is never more than \(m+n-1\). A key step to prove this is to show the same result for classical transportation polytopes.

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Authors and Affiliations

  1. University of Colorado, Denver, CO, USA

    S. Borgwardt

  2. University of California, Davis, CA, USA

    J. A. De Loera

  3. Fraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWM, Kaiserslautern, Germany

    E. Finhold

Authors
  1. S. Borgwardt
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  2. J. A. De Loera
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  3. E. Finhold
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Corresponding author

Correspondence to S. Borgwardt.

Additional information

S. Borgwardt gratefully acknowledges support from the Alexander-von-Humboldt Foundation. J. A. De Loera is grateful for the support received through NSF Grant DMS-1522158. J. A. De Loera and E. Finhold gratefully acknowledge the support from the Hausdorff Research Institute for Mathematics (HIM) in Bonn.

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Borgwardt, S., De Loera, J.A. & Finhold, E. The diameters of network-flow polytopes satisfy the Hirsch conjecture. Math. Program. 171, 283–309 (2018). https://doi.org/10.1007/s10107-017-1176-x

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  • DOI: https://doi.org/10.1007/s10107-017-1176-x

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