The Lovász Local Lemma and Satisfiability

  • Heidi Gebauer
  • Robin A. Moser
  • Dominik Scheder
  • Emo Welzl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5760)

Abstract

We consider boolean formulas in conjunctive normal form (CNF). If all clauses are large, it needs many clauses to obtain an unsatisfiable formula; moreover, these clauses have to interleave. We review quantitative results for the amount of interleaving required, many of which rely on the Lovász Local Lemma, a probabilistic lemma with many applications in combinatorics.

In positive terms, we are interested in simple combinatorial conditions which guarantee for a CNF formula to be satisfiable. The criteria obtained are nontrivial in the sense that even though they are easy to check, it is by far not obvious how to compute a satisfying assignment efficiently in case the conditions are fulfilled; until recently, it was not known how to do so. It is also remarkable that while deciding satisfiability is trivial for formulas that satisfy the conditions, a slightest relaxation of the conditions leads us into the territory of NP-completeness.

Several open problems remain, some of which we mention in the concluding section.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N., Spencer, J.H.: The Probabilistic Method, 3rd edn. John Wiley & Sons Inc., Chichester (2008)CrossRefMATHGoogle Scholar
  2. 2.
    Erdős, P., Lovász, L.: Problems and results on 3-chromatic hypergraphs and some related questions. In: Hajnal, A., Rado, R., Sós, V.T. (eds.) Infinite and Finite Sets (to Paul Erdős on his 60th birthday), vol. II, pp. 609–627. North-Holland, Amsterdam (1975)Google Scholar
  3. 3.
    Erdős, P.: On a combinatorial problem. Nordisk Mat. Tidskr. 11, 5–10, 40 (1963)MathSciNetMATHGoogle Scholar
  4. 4.
    Erdős, P.: On a combinatorial problem. II. Acta Math. Acad. Sci. Hungar. 15, 445–447 (1964)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Shearer, J.B.: On a problem of Spencer. Combinatorica 5(3), 241–245 (1985)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Beck, J.: An algorithmic approach to the Lovász Local Lemma. I. Random Struct. Algorithms 2(4), 343–365 (1991)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Alon, N.: A parallel algorithmic version of the local lemma. Random Struct. Algorithms 2(4), 367–378 (1991)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Czumaj, A., Scheideler, C.: A new algorithm approach to the general Lovász Local Lemma with applications to scheduling and satisfiability problems. In: Proc. 32nd Ann. ACM Symp. on Theory of Computing, pp. 38–47 (2000)Google Scholar
  9. 9.
    Srinivasan, A.: Improved algorithmic versions of the Lovász Local Lemma. In: Proc. 19th Ann. ACM-SIAM Symp. on Discrete Algorithms, pp. 611–620 (2008)Google Scholar
  10. 10.
    Moser, R.A.: Derandomizing the Lovász Local Lemma more effectively. CoRR abs/0807.2120 (2008)Google Scholar
  11. 11.
    Moser, R.A.: A constructive proof of the Lovász Local Lemma. CoRR abs/0810.4812 (2008); Proc. 41st Ann. ACM Symp. on Theory of Computing (to appear)Google Scholar
  12. 12.
    Moser, R.A., Tardos, G.: A constructive proof of the general Lovász Local Lemma. CoRR abs/0903.0544 (2009)Google Scholar
  13. 13.
    Erdős, P., Spencer, J.: Lopsided Lovász Local Lemma and Latin transversals. Discrete Appl. Math. 30(2-3), 151–154 (1991)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lu, L., Székely, L.: Using Lovász Local Lemma in the space of random injections. Electron. J. Combin. 14(1), 13, Research Paper 63 (2007) (electronic)MATHGoogle Scholar
  15. 15.
    Berman, P., Karpinski, M., Scott, A.D.: Approximation hardness and satisfiability of bounded occurrence instances of SAT. Electronic Colloquium on Computational Complexity (ECCC) 10(022) (2003)Google Scholar
  16. 16.
    Tovey, C.A.: A simplified NP-complete satisfiability problem. Discrete Appl. Math. 8(1), 85–89 (1984)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kratochvíl, J., Savický, P., Tuza, Z.: One more occurrence of variables makes satisfiability jump from trivial to NP-complete. SIAM J. Comput. 22(1), 203–210 (1993)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Savický, P., Sgall, J.: DNF tautologies with a limited number of occurrences of every variable. Theoret. Comput. Sci. 238(1-2), 495–498 (2000)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hoory, S., Szeider, S.: A note on unsatisfiable k-CNF formulas with few occurrences per variable. SIAM J. Discrete Math. 20(2), 523–528 (2006)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gebauer, H.: Disproof of the neighborhood conjecture with implications to SAT. CoRR abs/0904.2541 (2009)Google Scholar
  21. 21.
    Beck, J.: Combinatorial Games: Tic Tac Toe Theory, 1st edn. Encyclopedia of Mathematics and Its Applications, vol. 114. Cambridge University Press, Cambridge (2008)CrossRefMATHGoogle Scholar
  22. 22.
    Stříbrná, J.: Between Combinatorics and Formal Logic, Master’s Thesis. Charles University, Prague (1994)Google Scholar
  23. 23.
    Hoory, S., Szeider, S.: Computing unsatisfiable k-SAT instances with few occurences per variable. Theoret. Comput. Sci. 337(1-3), 347–359 (2005)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Kullmann, O.: New methods for 3-SAT decision and worst-case analysis. Theor. Comput. Sci. 223(1-2), 1–72 (1999)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Aharoni, R., Linial, N.: Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas. J. Comb. Theory, Ser. A 43(2), 196–204 (1986)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Davydov, G., Davydova, I., Kleine Büning, H.: An efficient algorithm for the minimal unsatisfiability problem for a subclass of CNF. Ann. Math. Artificial Intelligence 23(3-4), 229–245 (1998)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Kleine Büning, H.: An upper bound for minimal resolution refutations. In: Gottlob, G., Grandjean, E., Seyr, K. (eds.) CSL 1998. LNCS, vol. 1584, pp. 171–178. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  28. 28.
    Kleine Büning, H.: On subclasses of minimal unsatisfiable formulas. Discrete Appl. Math. 107(1-3), 83–98 (2000)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Szeider, S.: Homomorphisms of conjunctive normal forms. Discrete Appl. Math. 130(2), 351–365 (2003)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Porschen, S., Speckenmeyer, E., Randerath, B.: On linear CNF formulas. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 212–225. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  31. 31.
    Abbott, H.: An application of Ramsey’s Theorem to a problem of Erdős and Hajnal. Canad. Math. Bull. 8, 515–517 (1965)CrossRefMATHGoogle Scholar
  32. 32.
    Kostochka, A., Mubayi, D., Rödl, V., Tetali, P.: On the chromatic number of set systems. Random Struct. Algorithms 19(2), 87–98 (2001)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Scheder, D.: Unsatisfiable linear CNF formulas are large, and difficult to construct explicitely. CoRR abs/0905.1587 (2009)Google Scholar
  34. 34.
    Porschen, S., Speckenmeyer, E., Zhao, X.: Linear CNF formulas and satisfiability. Discrete Appl. Math. 157(5), 1046–1068 (2009)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Scheder, D.: Unsatisfiable linear k-CNFs exist, for every k. CoRR abs/0708.2336 (2007)Google Scholar
  36. 36.
    Ben-Sasson, E., Wigderson, A.: Short proofs are narrow—resolution made simple. J. ACM 48(2), 149–169 (2001)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Heidi Gebauer
    • 1
  • Robin A. Moser
    • 1
  • Dominik Scheder
    • 1
  • Emo Welzl
    • 1
  1. 1.Institute of Theoretical Computer ScienceETH ZürichZürichSwitzerland

Personalised recommendations