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Simple extensions of polytopes

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Abstract

We introduce the simple extension complexity of a polytope \(P\) as the smallest number of facets of any simple (i.e., non-degenerate in the sense of linear programming) polytope which can be projected onto \(P\). We devise a combinatorial method to establish lower bounds on the simple extension complexity and show for several polytopes that they have large simple extension complexities. These examples include both the spanning tree and the perfect matching polytopes of complete graphs, uncapacitated flow polytopes for non-trivially decomposable directed acyclic graphs, hypersimplices, and random 0/1-polytopes with vertex numbers within a certain range. On our way to obtain the result on perfect matching polytopes we generalize a result of Padberg and Rao’s on the adjacency structures of those polytopes. In addition to the material in the extended abstract (Kaibel and Walter in Integer programming and combinatorial optimization. Lecture Notes in Computer Science, vol 8494. Springer, Berlin, 2014) we include omitted proofs, supporting figures, and an analysis of known upper bounding techniques.

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Acknowledgments

We are greatful to the referees whose comments lead to significant improvements in the presentation of the material.

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Authors and Affiliations

  1. Otto-von-Guericke Universität Magdeburg (FMA/IMO), Universitätsplatz 2, 39106, Magdeburg, Germany

    Volker Kaibel & Matthias Walter

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  1. Volker Kaibel
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  2. Matthias Walter
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Correspondence to Matthias Walter.

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Kaibel, V., Walter, M. Simple extensions of polytopes. Math. Program. 154, 381–406 (2015). https://doi.org/10.1007/s10107-015-0885-2

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