Advertisement

Certifying Unsatisfiability of Random 2k-SAT Formulas Using Approximation Techniques

  • Amin Coja-Oghlan
  • Andreas Goerdt
  • André Lanka
  • Frank Schädlich
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2751)

Abstract

Let k be an even integer. We investigate the applicability of approximation techniques to the problem of deciding whether a random k-SAT formula is satisfiable. Let n be the number of propositional variables under consideration. First we show that if the number m of clauses satisfies mCn k/2 for a certain constant C, then unsatisfiability can be certified efficiently using (known) approximation algorithms for MAX CUT or MIN BISECTION. In addition, we present an algorithm based on the Lovász ϑ function that within polynomial expected time decides whether the input formula is satisfiable, provided mCn k/2. These results improve previous work by Goerdt and Krivelevich [14]. Finally, we present an algorithm that approximates random MAX 2-SAT within expected polynomial time.

Keywords

Random Graph Label Edge Satisfying Assignment Independence Number Negligible Discrepancy 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N., Spencer, J.: The Probabilistic Method. John Wiley and Sons, Chichester (1992)zbMATHGoogle Scholar
  2. 2.
    Ben-Sasson, E., Bilu, Y.: A Gap in Average Proof Complexity. In: ECCC 2003 (2002)Google Scholar
  3. 3.
    Chung, F.R.K.: Spectral Graph Theory. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  4. 4.
    Coja-Oghlan, A.: The Lovasz number of random graphs. Hamburger Beiträge zur Mathematik 169Google Scholar
  5. 5.
    Coja-Oghlan, A., Taraz, A.: Colouring random graphs in expected polynomial time. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 487–498. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Feige, U.: Relations between average case complexity and approximation complexity. In: Proc. 34th STOC, pp. 310–332 (2002)Google Scholar
  7. 7.
    Feige, U., Goemans, M.X.: Approximating the value of two prover proof systems, with applications to MAX 2SAT and MAX DICUT. In: Proc. 3rd Israel Symp. on Theory of Computing and Systems, pp. 182–189 (1995)Google Scholar
  8. 8.
    Feige, U., Krauthgamer, R.: A polylogarithmic approximation of the minimum bisection. In: Proc. 41st FOCS, pp. 105–115 (2000)Google Scholar
  9. 9.
    Feige, U., Ofek, E.: Spectral techniques applied to sparse random graphs, report MCS03-01, Weizmann Institute of Science (2003)Google Scholar
  10. 10.
    Friedgut, E.: Necessary and Sufficient Conditions for Sharp Thresholds of Graph Properties and the k-SAT problem. J. Amer. Math. Soc. 12, 1017–1054 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Friedman, J., Goerdt, A.: Recognizing more unsatisfiable random 3-SAT instances efficiently. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, p. 310. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  12. 12.
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145Google Scholar
  13. 13.
    Goerdt, A., Jurdzinski, T.: Some results on random unsatisfiable k-sat instances and approximation algorithms applied to random structures. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 280–291. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  14. 14.
    Goerdt, A., Krivelevich, M.: Efficient recognition of random unsatisfiable k-SAT instances by spectral methods. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 294–304. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  15. 15.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric algorithms and combinatorial optimization. Springer, Heidelberg (1988)zbMATHGoogle Scholar
  16. 16.
    Hofri, M.: Probabilistic Analysis of Algorithms. Springer, Heidelberg (1987)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Amin Coja-Oghlan
    • 1
  • Andreas Goerdt
    • 2
  • André Lanka
    • 2
  • Frank Schädlich
    • 2
  1. 1.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Fakultät für InformatikTechnische Universität ChemnitzChemnitzGermany

Personalised recommendations