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Sparsest cut in planar graphs, maximum concurrent flows and their connections with the max-cut problem

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Abstract

We study the sparsest cut problem when the “capacity-demand” graph is planar, and give a combinatorial polynomial algorithm. In this type of graphs there is an edge for each positive capacity and also an edge for each positive demand. We extend this result to graphs with no \(K_5\) minor. We also show how to find a maximum concurrent flow in these two cases. We also prove that the sparsest cut problem is NP-hard if we only impose that the “capacity-demand” graph has no \(K_6\) minor. We use ideas that had been developed for the max-cut problem, and show how to exploit the connections among these problems.

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Acknowledgements

We are grateful to an anonymous referee who helped us to improve the presentation.

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Authors and Affiliations

  1. CNRS, and Université Clermont Auvergne, UCA, Campus des Cézeaux, 1 rue de la Chebarde, 63178, Aubière Cedex, France

    Mourad Baïou

  2. IBM T. J. Watson Research Center, PO Box 218, Route 134, Yorktown Heights, NY, 10589, USA

    Francisco Barahona

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  1. Mourad Baïou
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  2. Francisco Barahona
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Correspondence to Francisco Barahona.

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Baïou, M., Barahona, F. Sparsest cut in planar graphs, maximum concurrent flows and their connections with the max-cut problem. Math. Program. 172, 59–75 (2018). https://doi.org/10.1007/s10107-017-1227-3

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