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Proof Complexity and the Kneser-Lovász Theorem

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Theory and Applications of Satisfiability Testing – SAT 2014 (SAT 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8561))

Abstract

We investigate the proof complexity of a class of propositional formulas expressing a combinatorial principle known as the Kneser-Lovász Theorem. This is a family of propositional tautologies, indexed by an nonnegative integer parameter k that generalizes the Pigeonhole Principle (obtained for k = 1).

We show, for all fixed k, 2Ω(n) lower bounds on resolution complexity and exponential lower bounds for bounded depth Frege proofs. These results hold even for the more restricted class of formulas encoding Schrijver’s strenghtening of the Kneser-Lovász Theorem. On the other hand for the cases k = 2,3 (for which combinatorial proofs of the Kneser-Lovász Theorem are known) we give polynomial size Frege (k = 2), respectively extended Frege (k = 3) proofs. The paper concludes with a brief announcement of the results (presented in subsequent work) on the complexity of the general case of the Kneser-Lovász theorem.

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Author information

Authors and Affiliations

  1. Dept. of Computer Science, West University of Timişoara, Timişoara, RO-300223, Romania

    Gabriel Istrate & Adrian Crãciun

  2. e-Austria Research Institute, Bd. V. Pârvan 4, cam. 045 B, Timişoara, RO-300223, Romania

    Gabriel Istrate & Adrian Crãciun

Authors
  1. Gabriel Istrate
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  2. Adrian Crãciun
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Editors and Affiliations

  1. Karlsruher Institut für Technologie (KIT), Am Fasanengarten 5, 76131, Karlsruhe, Germany

    Carsten Sinz

  2. TU Wien, Favoritenstraße 9-11, 1040, Wien, Austria

    Uwe Egly

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Istrate, G., Crãciun, A. (2014). Proof Complexity and the Kneser-Lovász Theorem. In: Sinz, C., Egly, U. (eds) Theory and Applications of Satisfiability Testing – SAT 2014. SAT 2014. Lecture Notes in Computer Science, vol 8561. Springer, Cham. https://doi.org/10.1007/978-3-319-09284-3_11

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  • DOI: https://doi.org/10.1007/978-3-319-09284-3_11

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-09283-6

  • Online ISBN: 978-3-319-09284-3

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