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International Symposium on Combinatorial Optimization

ISCO 2016: Combinatorial Optimization pp 285–296Cite as

Two-Level Polytopes with a Prescribed Facet

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9849))

Abstract

A (convex) polytope is said to be 2-level if for every facet-defining direction of hyperplanes, its vertices can be covered with two hyperplanes of that direction. These polytopes are motivated by questions, e.g., in combinatorial optimization and communication complexity. We study 2-level polytopes with one prescribed facet. Based on new general findings about the structure of 2-level polytopes, we give a complete characterization of the 2-level polytopes with some facet isomorphic to a sequentially Hanner polytope, and improve the enumeration algorithm of Bohn et al. (ESA 2015). We obtain, for the first time, the complete list of d-dimensional 2-level polytopes up to affine equivalence for dimension \(d = 7\). As it turns out, geometric constructions that we call suspensions play a prominent role in both our theoretical and experimental results. This yields exciting new research questions on 2-level polytopes, which we state in the paper.

We acknowledge support from ERC grant FOREFRONT (grant agreement no. 615640) funded by the European Research Council under the EU’s 7th Framework Programme (FP7/2007-2013).

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Notes

  1. 1.

    While in general two polytopes can be combinatorially equivalent without being affinely equivalent, for 2-level polytopes these two notions coincide [2]. We simply say that two 2-level polytopes are isomorphic whenever they are combinatorially (or affinely) equivalent.

  2. 2.

    The computed polytopes in polymake format and more experimental results are available online http://homepages.ulb.ac.be/~vfisikop/data/2-level.html.

  3. 3.

    Hydra balanced cluster: http://cc.ulb.ac.be/hpc/hydra.php.

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Authors and Affiliations

  1. Département de Mathématique, Université libre de Bruxelles, ULB CP 216, Boulevard du Triomphe, 1050, Brussels, Belgium

    Samuel Fiorini, Vissarion Fisikopoulos & Marco Macchia

Authors
  1. Samuel Fiorini
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  2. Vissarion Fisikopoulos
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  3. Marco Macchia
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Corresponding author

Correspondence to Marco Macchia .

Editor information

Editors and Affiliations

  1. University of Salerno , Fisciano, Italy

    Raffaele Cerulli

  2. Kyoto University , Kyoto, Japan

    Satoru Fujishige

  3. LAMSADE, Université Paris-Dauphine , Paris , France

    A. Ridha Mahjoub

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Fiorini, S., Fisikopoulos, V., Macchia, M. (2016). Two-Level Polytopes with a Prescribed Facet. In: Cerulli, R., Fujishige, S., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2016. Lecture Notes in Computer Science(), vol 9849. Springer, Cham. https://doi.org/10.1007/978-3-319-45587-7_25

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  • DOI: https://doi.org/10.1007/978-3-319-45587-7_25

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45586-0

  • Online ISBN: 978-3-319-45587-7

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