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Phase Transition for Local Search on Planted SAT

  • Andrei A. Bulatov
  • Evgeny S. Skvortsov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9235)

Abstract

The Local Search algorithm (or Hill Climbing, or Iterative Improvement) is one of the simplest heuristics to solve the Satisfiability and Max-Satisfiability problems. Although it is not the best known Satisfiability algorithm even for the class of problems we study, the Local Search is a part of many satisfiability and max-satisfiability solvers, where it is used to find a good starting point for a more sophisticated heuristics, and to improve a candidate solution. In this paper we give an analysis of Local Search on random planted 3-CNF formulas. We show that a sharp transition of efficiency of Local Search occurs at density \(\varrho = \frac{7}{6} \ln n\). Specifically we show that if there is \(\kappa <\frac{7}{6}\) such that the clause-to-variable ratio is less than \(\kappa \ln n\) (n is the number of variables in a CNF) then Local Search whp does not find a satisfying assignment, and if there is \(\kappa >\frac{7}{6}\) such that the clause-to-variable ratio is greater than \(\kappa \ln n\) then the local search whp finds a satisfying assignment. As a byproduct we also show that for any constant \(\varrho \) there is \(\gamma \) such that Local Search applied to a random (not necessarily planted) 3-CNF with clause-to-variable ratio \(\varrho \) produces an assignment that satisfies at least \(\gamma n\) clauses less than the maximal number of satisfiable clauses.

Notes

Acknowledgment

The fist author was supported by an NSERC Discovery grant.

References

  1. 1.
    Achlioptas, D.: Lower bounds for random 3-SAT via differential equations. Theor. Comput. Sci. 265(1–2), 159–185 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Achlioptas, D., Friedgut, E.: A sharp threshold for k-colorability. Random Struct. Algorithms 14(1), 63–70 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alekhnovich, M., Ben-Sasson, E.: Linear upper bounds for random walk on small density random 3-CNFs. SIAM J. Comput. 36(5), 1248–1263 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alon, N., Spencer, J.: The Probabilistic Method. Wiley, New York (2000)CrossRefzbMATHGoogle Scholar
  5. 5.
    Amiri, E., Skvortsov, E.S.: Pushing random walk beyond golden ratio. In: Diekert, V., Volkov, M.V., Voronkov, A. (eds.) CSR 2007. LNCS, vol. 4649, pp. 44–55. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  6. 6.
    Ben-Sasson, E., Bilu, Y., Gutfreund, D.: Finding a randomly planted assignment in a random 3-CNF (2002). (manuscript)Google Scholar
  7. 7.
    Braunstein, A., Mézard, M., Zecchina, R.: Survey propagation: an algorithm for satisfiability. Random Struct. Algorithms 27(2), 201–226 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bulatov, A.A., Skvortsov, E.S.: Efficiency of local search. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 297–310. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  9. 9.
    Coja-Oghlan, A., Krivelevich, M., Vilenchik, D.: Why almost all k-colorable graphs are easy. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 121–132. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  10. 10.
    Coja-Oghlan, A., Panagiotou, K.: Going after the \(k\)-SAT threshold. In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) STOC, pp. 705–714. ACM (2013)Google Scholar
  11. 11.
    Crawford, J.M., Auton, L.D.: Experimental results on the crossover point in random 3-SAT. Art. Int. 81(1–2), 31–57 (1996)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ding, J., Sly, A., Sun, N.: Proof of the satisfiability conjecture for large \(k\). In: Servedio, R., Rubinfeld, R. (eds.) STOC, pp. 59–68. ACM (2015)Google Scholar
  13. 13.
    Feige, U., Mossel, E., Vilenchik, D.: Complete convergence of message passing algorithms for some satisfiability problems. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds.) APPROX 2006 and RANDOM 2006. LNCS, vol. 4110, pp. 339–350. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  14. 14.
    Flaxman, A.: A spectral technique for random satisfiable 3CNF formulas. In: SODA, pp. 357–363. ACM/SIAM (2003)Google Scholar
  15. 15.
    Friedgut, E.: Sharp thresholds of graph properties, and the \(k\)-SAT problem. J. Amer. Math. Soc. 12, 1017–1054 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hansen, P., Jaumard, B.: Algorithms for the maximum satisfiability problem. Computing 44, 279–303 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Håstad, J.: Some optimal inapproximability results. J. ACM 48(4), 798–859 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Koutsoupias, E., Papadimitriou, C.: On the greedy algorithm for satisfiability. Inf. Process. Lett. 43(1), 53–55 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Krivelevich, M., Vilenchik, D.: Solving random satisfiable 3CNF formulas in expected polynomial time. In: SODA, pp. 454–463. ACM Press (2006)Google Scholar
  20. 20.
    Mézard, M., Mora, T., Zecchina, R.: Clustering of solutions in the random satisfiability problem. CoRR abs/cond-mat/0504070 (2005)Google Scholar
  21. 21.
    Papadimitriou, C.: On selecting a satisfying truth assignment (extended abstract). In: FOCS, pp. 163–169. IEEE Computer Society (1991)Google Scholar
  22. 22.
    Selman, B., Levesque, H., Mitchell, D.: A new method for solving hard satisfiability problems. In: Swartout, W. (ed.) AAAI, pp. 440–446. AAAI Press/The MIT Press (1992)Google Scholar
  23. 23.
    Skvortsov, E.S.: A theoretical analysis of search in GSAT. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 265–275. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  24. 24.
    Vilenchik, D.: It’s all about the support: a new perspective on the satisfiability problem. JSAT 3(3–4), 125–139 (2007)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Simon Fraser UniversityBurnabyCanada
  2. 2.GoogleTorontoCanada

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