Short Proofs of the Kneser-Lovász Coloring Principle

  • James Aisenberg
  • Maria Luisa Bonet
  • Sam Buss
  • Adrian Crãciun
  • Gabriel Istrate
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9135)

Abstract

We prove that the propositional translations of the Kneser-Lovász theorem have polynomial size extended Frege proofs and quasi-polynomial size Frege proofs. We present a new counting-based combinatorial proof of the Kneser-Lovász theorem that avoids the topological arguments of prior proofs for all but finitely many cases for each k. We introduce a miniaturization of the octahedral Tucker lemma, called the truncated Tucker lemma: it is open whether its propositional translations have (quasi-)polynomial size Frege or extended Frege proofs.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aisenberg, J., Bonet, M.L., Buss, S.R.: Quasi-polynomial size Frege proofs of Frankl’s theorem on the trace of finite sets (201?) (to appear in Journal of Symbolic Logic)Google Scholar
  2. 2.
    Bonet, M.L., Buss, S.R., Pitassi, T.: Are there hard examples for Frege systems? In: Clote, P., Remmel, J. (eds.) Feasible Mathematics II, pp. 30–56. Birkhäuser, Boston (1995)CrossRefGoogle Scholar
  3. 3.
    Buss, S.R.: Polynomial size proofs of the propositional pigeonhole principle. Journal of Symbolic Logic 52, 916–927 (1987)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Buss, S.R.: Propositional proof complexity: An introduction. In: Berger, U., Schwichtenberg, H. (eds.) Computational Logic, pp. 127–178. Springer, Berlin (1999)CrossRefGoogle Scholar
  5. 5.
    Buss, S.R.: Towards NP-P via proof complexity and proof search. Annals of Pure and Applied Logic 163(9), 1163–1182 (2012)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Buss, S.R.: Quasipolynomial size proofs of the propositional pigeonhole principle (2014) (submitted for publication)Google Scholar
  7. 7.
    Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. Journal of Symbolic Logic 44, 36–50 (1979)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Istrate, G., Crãciun, A.: Proof complexity and the Kneser-Lovász theorem. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 138–153. Springer, Heidelberg (2014) Google Scholar
  9. 9.
    Jeřábek, E.: Dual weak pigeonhole principle, boolean complexity, and derandomization. Annals of Pure and Applied Logic 124, 1–37 (2004)CrossRefGoogle Scholar
  10. 10.
    Krajíček, J.: Bounded Arithmetic. Propositional Calculus and Complexity Theory. Cambridge University Press, Heidelberg (1995)Google Scholar
  11. 11.
    Lovász, L.: Kneser’s conjecture, chromatic number, and homotopy. Journal of Combinatorial Theory, Series A 25(3), 319–324 (1978)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Matoušek, J.: A combinatorial proof of Kneser’s conjecture. Combinatorica 24(1), 163–170 (2004)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Segerlind, N.: The complexity of propositional proofs. Bulletin of Symbolic Logic 13(4), 417–481 (2007)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Ziegler, G.M.: Generalized Kneser coloring theorems with combinatorial proofs. Inventiones Mathematicae 147(3), 671–691 (2002)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • James Aisenberg
    • 1
  • Maria Luisa Bonet
    • 2
  • Sam Buss
    • 1
  • Adrian Crãciun
    • 3
  • Gabriel Istrate
    • 3
  1. 1.Department of MathematicsUniversity of California, San DiegoLa JollaUSA
  2. 2.Computer Science DepartmentUniversidad Politécnica de CataluñaBarcelonaSpain
  3. 3.West University of Timişoara, and the e-Austria Research InstituteTimişoaraRomania

Personalised recommendations