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Proof Complexity and the Kneser-Lovász Theorem

  • Gabriel Istrate
  • Adrian Crãciun
Part of the Lecture Notes in Computer Science book series (LNCS, 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.

Keywords

Chromatic Number Proof System Propositional Formula Resolution Complexity Variable Substitution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Gabriel Istrate
    • 1
    • 2
  • Adrian Crãciun
    • 1
    • 2
  1. 1.Dept. of Computer ScienceWest University of TimişoaraTimişoaraRomania
  2. 2.e-Austria Research InstituteTimişoaraRomania

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