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On the extension complexity of combinatorial polytopes

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Abstract

In this paper we extend recent results of Fiorini et al. on the extension complexity of the cut polytope and related polyhedra. We first describe a lifting argument to show exponential extension complexity for a number of NP-complete problems including subset-sum and three dimensional matching. We then obtain a relationship between the extension complexity of the cut polytope of a graph and that of its graph minors. Using this we are able to show exponential extension complexity for the cut polytope of a large number of graphs, including those used in quantum information and suspensions of cubic planar graphs.

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Notes

  1. The set \(E\) is sometimes not specified explicitly when \(E\) is clear from the context or the choice of \(E\) does not make any difference.

  2. Even though an extension can also be a polyhedron and not necessarily a polytope, we will consider only those extensions that are polytopes. It is not difficult to see that for a polytope the extension with smallest size would indeed be a polytope.

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Acknowledgments

Research of the first author is supported by the NSERC and JSPS. The second author was supported by FNRS, Belgium as a postdoctoral researcher during this research.

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Author notes

  1. Hans Raj Tiwary

    Present address: Department of Applied Mathematics (KAM), Institute of Theoretical Computer Science (ITI), Charles University, Malostranské nám. 25, 118 00 , Prague 1, Czech Republic

Authors and Affiliations

  1. GERAD and School of Computer Science, McGill University, 3480 University Street, Montreal, Quebec, H3A 2A7, Canada

    David Avis

  2. Graduate School of Informatics, Kyoto University, Sakyo-ku, Yoshida Yoshida, Kyoto, 606-8501, Japan

    David Avis

  3. Department of Mathematics, Université Libre de Bruxelles, Boulevard du Triomphe, 1050 , Brussels, Belgium

    Hans Raj Tiwary

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  1. David Avis
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  2. Hans Raj Tiwary
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Correspondence to Hans Raj Tiwary.

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Avis, D., Tiwary, H.R. On the extension complexity of combinatorial polytopes. Math. Program. 153, 95–115 (2015). https://doi.org/10.1007/s10107-014-0764-2

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  • DOI: https://doi.org/10.1007/s10107-014-0764-2

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