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On the Rank of Cutting-Plane Proof Systems

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Integer Programming and Combinatorial Optimization (IPCO 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6080))

Abstract

We introduce a natural abstraction of propositional proof systems that are based on cutting planes. This new class of proof systems includes well-known operators such as Gomory-Chvátal cuts, lift-and-project cuts, Sherali-Adams cuts (for a fixed hierarchy level d), and split cuts. The rank of such a proof system corresponds to the number of rounds needed to show the nonexistence of integral solutions. We exhibit a family of polytopes without integral points contained in the n-dimensional 0/1-cube that has rank Ω(n/logn) for any proof system in our class. In fact, we show that whenever a specific cutting-plane based proof system has (maximal) rank n on a particular family of instances, then any cutting-plane proof system in our class has rank Ω(n/logn) for this family. This shows that the rank complexity of worst-case instances is intrinsic to the problem, and does not depend on specific cutting-plane proof systems, except for log factors. We also construct a new cutting-plane proof system that has worst-case rank O(n/logn) for any polytope without integral points, implying that the universal lower bound is essentially tight.

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Authors and Affiliations

  1. Technische Universität Darmstadt, Germany

    Sebastian Pokutta

  2. Massachusetts Institute of Technology, USA

    Andreas S. Schulz

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  1. Sebastian Pokutta
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  2. Andreas S. Schulz
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Editors and Affiliations

  1. Institute of Mathematics, École Polytechnique Féderale de Lausanne, 1015, Lausanne, Switzerland

    Friedrich Eisenbrand

  2. McGill University, 805 Sherbrooke West, H3A 2K6, Montreal, Quebec, Canada

    F. Bruce Shepherd

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Pokutta, S., Schulz, A.S. (2010). On the Rank of Cutting-Plane Proof Systems. In: Eisenbrand, F., Shepherd, F.B. (eds) Integer Programming and Combinatorial Optimization. IPCO 2010. Lecture Notes in Computer Science, vol 6080. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13036-6_34

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  • DOI: https://doi.org/10.1007/978-3-642-13036-6_34

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13035-9

  • Online ISBN: 978-3-642-13036-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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