Advertisement

Easily Refutable Subformulas of Large Random 3CNF Formulas

  • Uriel Feige
  • Eran Ofek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3142)

Abstract

A simple nonconstructive argument shows that most 3CNF formulas with cn clauses (where c is a large enough constant) are not satisfiable. It is an open question whether there is an efficient refutation algorithm that for most formulas with cn clauses proves that they are not satisfiable. We present a polynomial time algorithm that for most 3CNF formulas with cn 3/2 clauses (where c is a large enough constant) finds a subformula with O(c 2 n) clauses and then proves that this subformula is not satisfiable (and hence that the original formula is not satisfiable). Previously, it was only known how to efficiently certify the unsatisfiability of random 3CNF formulas with at least poly(log(n)) · n 3/2 clauses. Our algorithm is simple enough to run in practice. We present some preliminary experimental results.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ben-Sasson, E., Wigderson, A.: Short proofs are narrow-resolution made simple. Journal of the ACM (JACM) 48(2), 149–169 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Chvatal, V., Szemeredi, E.: Many hard examples for resolution. Journal of the ACM 35(4), 759–768 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Coja-Oghlan, A., Goerdt, A., Lanka, A., Schadlich, F.: Certifying unsatisfiability of random 2k-sat formulas using approximation techniques. In: Proc. of the 14th International Symposium on Fundamentals of Computation Theory (2003)Google Scholar
  4. 4.
    Feige, U.: Relations between average case complexity and approximation complexity. In: Proc. of the 34th Annual ACM Symposium on Theory of Computing, pp. 534–543 (2002)Google Scholar
  5. 5.
    Feige, U., Ofek, E.: Spectral techniques applied to sparse random graphs. Technical report, Weizmann Institute of Science (2003)Google Scholar
  6. 6.
    Friedgut, E., Bourgain, J.: Sharp thresholds of graph properties, and the k-sat problem. JAMS: Journal of the American Mathematical Society 12(4), 1017–1054 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Friedman, J., Goerdt, A., Krivelevich, M.: Recognizing more unsatisfiable random 3-sat instances efficiently. Technical report (2003)Google Scholar
  8. 8.
    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
  9. 9.
    Goerdt, A., Lanka, A.: Recognizing more random 3-sat instances efficiently (2003) (manuscript)Google Scholar
  10. 10.
    Hajiaghayi, M., Sorkin, G.B.: The satisfiability threshold for random 3-SAT is at least 3.52 (2003), http://arxiv.org/abs/math.CO/0310193
  11. 11.
    Janson, S., Stamatiou, Y.C., Vamvakari, M.: Bounding the unsatisfiability threshold of random 3-sat. Random Structures and Algorithms 17(2), 103–116 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kaporis, A.C., Kirousis, L.M., Lalas, E.G.: Selecting complementary pairs of literals. In: Proc. LICS 2003 Workshop on Typical Case Complexity and Phase Transitions (June 2003)Google Scholar
  13. 13.
    Zwick, U.: Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems. In: Proc. of the 31st Annual ACM Symposium on Theory of Computing, pp. 679–687 (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Uriel Feige
    • 1
  • Eran Ofek
    • 1
  1. 1.Department of Computer Science and Applied MathematicsWeizmann Institute of ScienceRehovotIsrael

Personalised recommendations