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A geometric lower bound on the extension complexity of polytopes based on the f-vector

Author

Listed:
  • Dewez, Julien

    (Université catholique de Louvain, LIDAM/CORE, Belgium)

  • Gillis, Nicolas

    (Université de Mons)

  • Glineur, François

    (Université catholique de Louvain, LIDAM/CORE, Belgium)

Abstract

A linear extension of a polytope is any polytope which can be mapped onto via an affine transformation. The extension complexity of a polytope is the minimum number of facets of any linear extension of this polytope. In general, computing the extension complexity of a given polytope is difficult. The extension complexity is also equal to the nonnegative rank of any slack matrix of the polytope. In this paper, we introduce a new geometric lower bound on the extension complexity of a polytope, i.e., which relies only on the knowledge of some of its geometric characteristics. It is based on the monotone behaviour of the -vector of polytopes under affine maps. We present numerical results showing that this bound can improve upon existing geometric lower bounds, and provide a generalization of this lower bound for the nonnegative rank of any matrix.

Suggested Citation

  • Dewez, Julien & Gillis, Nicolas & Glineur, François, 2021. "A geometric lower bound on the extension complexity of polytopes based on the f-vector," LIDAM Reprints CORE 3172, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvrp:3172
    DOI: https://doi.org/10.1016/j.dam.2020.09.028
    Note: In: Discrete Applied Mathematics, vol. 303, p. 22-38 (2021)
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