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The Space Complexity of Cutting Planes Refutations

Authors Nicola Galesi, Pavel Pudlák, Neil Thapen

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Nicola Galesi
Pavel Pudlák
Neil Thapen

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Nicola Galesi, Pavel Pudlák, and Neil Thapen. The Space Complexity of Cutting Planes Refutations. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 433-447, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.CCC.2015.433

Abstract

We study the space complexity of the cutting planes proof system, in which the lines in a proof  are integral linear inequalities. We measure the space used by a refutation as the number of linear inequalities that need to be kept on a blackboard while verifying it. We show that any unsatisfiable set of linear inequalities has a cutting planes refutation in space five. This is in contrast to the weaker resolution proof system, for which the  analogous space measure has been well-studied and  many optimal linear lower bounds are known. 

Motivated by this result we consider a natural restriction of cutting planes, in which all coefficients have size bounded by a constant. We show that there is a CNF which requires super-constant space to refute in this system. The system nevertheless already has an exponential speed-up over resolution with respect to size, and  we additionally show that it is stronger than resolution with respect to space, by constructing constant-space cutting planes proofs, with coefficients bounded by two, of the pigeonhole principle.

We also consider  variable instance space for cutting planes, where we count the number of instances of variables on the blackboard, and total space, where we count the total number of symbols.

Subject Classification

Keywords
  • Proof Complexity
  • Cutting Planes
  • Space Complexity

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