On the Rank of Cutting-Plane Proof Systems

  • Sebastian Pokutta
  • Andreas S. Schulz
Conference paper

DOI: 10.1007/978-3-642-13036-6_34

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6080)
Cite this paper as:
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

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.

Keywords

Cutting planes proof systems Gomory-Chvátal cuts lift-and-project cuts split cuts 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Sebastian Pokutta
    • 1
  • Andreas S. Schulz
    • 2
  1. 1.Technische Universität DarmstadtGermany
  2. 2.Massachusetts Institute of TechnologyUSA

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