Skip to main content
SpringerLink
Log in

Exponential Lower Bounds for the Running Time of DPLL Algorithms on Satisfiable Formulas

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

DPLL (for Davis, Putnam, Logemann, and Loveland) algorithms form the largest family of contemporary algorithms for SAT (the propositional satisfiability problem) and are widely used in applications. The recursion trees of DPLL algorithm executions on unsatisfiable formulas are equivalent to treelike resolution proofs. Therefore, lower bounds for treelike resolution (known since the 1960s) apply to them. However, these lower bounds say nothing about the behavior of such algorithms on satisfiable formulas. Proving exponential lower bounds for them in the most general setting is impossible without proving P ≠ NP; therefore, to prove lower bounds, one has to restrict the power of branching heuristics. In this paper, we give exponential lower bounds for two families of DPLL algorithms: generalized myopic algorithms, which read up to n 1−ε of clauses at each step and see the remaining part of the formula without negations, and drunk algorithms, which choose a variable using any complicated rule and then pick its value at random.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Achlioptas, D., Beame, P. and Molloy, M.: A sharp threshold in proof complexity, J. Comput. Syst. Sci. (2003).

  2. Achlioptas, D., Beame, P. and Molloy, M.: Exponential bounds for DPLL below the satisfiability threshold, in Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'04, 2004, pp. 139–140.

  3. Achlioptas, D. and Sorkin, G. B.: Optimal myopic algorithms for random 3-SAT, in Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, FOCS'00, 2000.

  4. Alekhnovich, M. and Ben-Sasson, E.: Analysis of the random walk algorithm on random 3-CNFs, Manuscript, 2002.

  5. Alekhnovich, M., Ben-Sasson, E., Razborov, A. and Wigderson, A.: Pseudorandom generators in propositional complexity, in Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, FOCS'00, Journal version is to appear in SIAM Journal on Computing, 2000.

  6. Alekhnovich, M., Borodin, A., Buresh-Oppenheim, J., Impagliazzo, R., Magen, A. and Pitassi, T.: Toward a model for backtracking and dynamic programming, in Proceedings of the 20th Annual Conference on Computational Complexity, 2005, pp. 308–322.

  7. Alekhnovich, M. and Razborov, A.: Lower bounds for the polynomial calculus: Non-binomial case, in Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science, 2001.

  8. Ben-Sasson, E. and Wigderson, A.: Short proofs are narrow – resolution made simple, J. ACM 48(2) (2001), 149–169.

    Article  MATH  MathSciNet  Google Scholar 

  9. Davis, M., Logemann, G. and Loveland, D.: A machine program for theorem-proving, Commun. ACM 5 (1962), 394–397.

    Article  MATH  MathSciNet  Google Scholar 

  10. Davis, M. and Putnam, H.: A computing procedure for quantification theory, J. ACM 7 (1960), 201–215.

    Article  MATH  MathSciNet  Google Scholar 

  11. Hirsch, E. A.: SAT local search algorithms: Worst-case study, J. Autom. Reason. 24(1/2) (2000), 127–143. Also reprinted in “Highlights of Satisfiability Research in the Year 2000”, Volume 63 in Frontiers in Artificial Intelligence and Applications, IOS.

    Article  MATH  MathSciNet  Google Scholar 

  12. Iwama, K. and Tamaki, S.: Improved upper bounds for 3-SAT, in Proceedings of the Fifteenth Annual ACM–SIAM Symposium on Discrete Algorithms, SODA'04, 2004, pp. 328–328.

  13. Nikolenko, S. I.: Hard satisfiable formulas for DPLL-type algorithms, Zap. Nauc. Semin. POMI 293 (2002), 139–148. English translation is to appear in Journal of Mathematical Sciences: Consultants Bureau, N.Y., March 2005, Vol. 126, No. 3, pp. 1205–1209.

    Google Scholar 

  14. Pudlák, P. and Impagliazzo, R.: A lower bound for DLL algorithms for k-SAT, in Proceedings of the 11th Annual ACM–SIAM Symposium on Discrete Algorithms, SODA'00, 2000.

  15. Simon, L., Le Berre, D. and Hirsch, E. A.: The SAT 2002 Competition, Ann. Math. Artif. Intell. 43 (2005), 307–342.

    Google Scholar 

  16. Tseitin, G. S.: On the complexity of derivation in the propositional calculus, Zap. Nauc. Semin. LOMI 8 (1968), 234–259. English translation of this volume: Consultants Bureau, N.Y., 1970, pp. 115–125.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Advanced Study, Princeton, NJ, USA

    Michael Alekhnovich

  2. St. Petersburg Department of Steklov, Institute of Mathematics, St. Petersburg, 191011, Russia

    Edward A. Hirsch

  3. Faculty of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia

    Dmitry Itsykson

Authors
  1. Michael Alekhnovich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Edward A. Hirsch
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dmitry Itsykson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Alekhnovich.

Additional information

★Extended abstract of this paper appeared in Proceedings of ICALP 2004, LNCS 3142, Springer, 2004, pp. 84–96.

†Supported by CCR grant \(\mathcal{N}\)CCR-0324906.

‡Supported in part by Russian Science Support Foundation, RAS program of fundamental research “Research in principal areas of contemporary mathematics,” and INTAS grant \(\mathcal{N}\) 04-77-7173.

§Supported in part by INTAS grant \(\mathcal{N}\)04-77-7173.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alekhnovich, M., Hirsch, E.A. & Itsykson, D. Exponential Lower Bounds for the Running Time of DPLL Algorithms on Satisfiable Formulas. J Autom Reasoning 35, 51–72 (2005). https://doi.org/10.1007/s10817-005-9006-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10817-005-9006-x

Key words

Search

Navigation