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

## 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.

## Key words

satisfiability DPLL algorithms## Preview

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## References

- 1.Achlioptas, D., Beame, P. and Molloy, M.: A sharp threshold in proof complexity,
*J. Comput. Syst. Sci.*(2003).Google Scholar - 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.Google Scholar - 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.Google Scholar - 4.Alekhnovich, M. and Ben-Sasson, E.: Analysis of the random walk algorithm on random 3-CNFs, Manuscript, 2002.Google Scholar
- 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 8.Ben-Sasson, E. and Wigderson, A.: Short proofs are narrow – resolution made simple,
*J. ACM***48**(2) (2001), 149–169.zbMATHCrossRefMathSciNetGoogle Scholar - 9.Davis, M., Logemann, G. and Loveland, D.: A machine program for theorem-proving,
*Commun. ACM***5**(1962), 394–397.zbMATHCrossRefMathSciNetGoogle Scholar - 10.Davis, M. and Putnam, H.: A computing procedure for quantification theory,
*J. ACM***7**(1960), 201–215.zbMATHCrossRefMathSciNetGoogle 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.zbMATHCrossRefMathSciNetGoogle 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.Google Scholar - 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.Google Scholar - 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