Ramamohan Paturi
Pavel Pudlák
Abstract
We propose and analyze a simple new randomized algorithm, called ResolveSat, for finding satisfying assignments of Boolean formulas in conjunctive normal form. The algorithm consists of two stages: a preprocessing stage in which resolution is applied to enlarge the set of clauses of the formula, followed by a search stage that uses a simple randomized greedy procedure to look for a satisfying assignment. Currently, this is the fastest known probabilistic algorithm for k-CNF satisfiability for k ≥ 4 (with a running time of O(20.5625n) for 4-CNF). In addition, it is the fastest known probabilistic algorithm for k-CNF, k ≥ 3, that have at most one satisfying assignment (unique k-SAT) (with a running time O(2(2 ln 2 − 1)n + o(n)) = O(20.386 … n) in the case of 3-CNF). The analysis of the algorithm also gives an upper bound on the number of the codewords of a code defined by a k-CNF. This is applied to prove a lower bounds on depth 3 circuits accepting codes with nonconstant distance. In particular we prove a lower bound Ω(21.282…√>i<n>/i<) for an explicitly given Boolean function of n variables. This is the first such lower bound that is asymptotically bigger than 2√>i<n>/i< + o(√>i<n>/i<).
- Alon, N., Spencer, J., and Erdös, P. 1992. The Probabilistic Method. Wiley, New York.Google Scholar
- Dantsin, E., Goerdt, A., Hirsch, E. A., Kannan, R., Kleinberg, J., Papadimitriou, C., Raghavan, P., and Schöning, U. 2002. A deterministic (2 − 2/k + 1)n algorithm for k-SAT based on local search. Theoret. Comput. Sci. 289, 69--83. Google Scholar
- Davis, M., Logemann, G., and Loveland, D. 1962. A machine program for theorem proving, Commun. ACM 5, 394--397. Google Scholar
- Galil, Z. 1975. On the validity and complexity of bounded resolution. In Proceedings of the 7th ACM Symposium on Theory of Computing. ACM, New York, 72--82. Google Scholar
- Harris, T. E. 1960. A lower bound for the critical probability in a certain percolation process. Proc. Camb. Phil. Soc. 56, 13--20.Google Scholar
- Håstad, J. 1986. Almost optimal lower bounds for small depth circuits. In Proceedings of the 18th ACM Symposium on Theory of Computing. ACM, New York, 6--20. Google Scholar
- Håstad, J., Jukna, S., and Pudlák, P. 1993. Top--down lower bounds for depth 3 circuits. In Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, Calif., 124--129.Google Scholar
- Hirsch, E. 2000. New worst-case upper bounds for SAT. J. Automat. Reas. 24, 4, 397--420. Google Scholar
- Hirsch, E., and Kojevnikov, A. 2002. Unit Walk: A new SAT solver that uses local search guided by unit clause elimination. In Proceedings of 5th International Symposium on the Theory and Applications of Satisfiability Testing (SAT 2002). 35--42Google Scholar
- Hofmeister, T., Schöning, U., Schuler, R., and Watanabe, O. 2002. A probabilistic 3-SAT algorithm further improved. In STACS 2002. Lecture Notes in Computer Science, vol. 2285. Springer-Verlag, New York, 192--202. Google Scholar
- Kullmann, O. 1999. New methods for 3-SAT decision and worst-case analysis. Theoret. Comput. Sci. 223, 1-2 (July), 1--72. Google Scholar
- Kullmann, O., and Luckhardt, H. 1998. Algorithms for SAT/TAUT decision based on various measures. Preprint, 81 pages http://www.cs.utoronto.ca/kullmann/.Google Scholar
- MacWilliams, F. J., and Sloane, N. J. 1977. The Theory of Error-Correcting Codes. North-Holland, Amsterdam The Netherland.Google Scholar
- Monien, B., and Speckenmeyer, E. 1985. Solving satisfiability in less than 2n steps. Discr. Appl. Math. 10, 287--295.Google Scholar
- Paturi, R., Pudlák, P., Saks, M., and Zane, F. 1998. An improved exponential-time algorithm for k-SAT. In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, Calif., 628--637. Google Scholar
- Paturi, R., Pudlák, P., and Zane, F. 1997. Satisfiability coding lemma. In Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, Calif., 566--574. (See also Chicago J. Theor. Comput. Sci. (Dec. 1999), http://cjtcs.cs.uchicago.edu/.) Google Scholar
- Razborov, A. A. 1986. Lower bounds on the size of bounded depth networks over a complete basis with logical addition. Matematicheskie Zametki 41, 598--607 (in Russian). (English Translation in Mathematical Notes of the Academy of Sciences of the USSR 41, 333--338.)Google Scholar
- Schiermeyer, I. 1993. Solving 3-satisfiability in less than 1.579n steps. In Selected papers from CSL '92. Lecture Notes in Computer Science, vol. 702. Springer-Verlag, New York, 379--394. Google Scholar
- Schöning, U. 1999. A probabilistic algorithm for k-SAT and constraint satisfaction problems. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, Calif., 410--414. Google Scholar
- Valiant, L. G. 1977. Graph-theoretic arguments in low-level complexity. In Proceedings of the 6th Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, vol. 53, Springer-Verlag, New York, 162--176.Google Scholar
- Zhang, W. 1996. Number of models and satisfiability of sets of clauses. Theoret. Comput. Sci. 155, 277--288. Google Scholar
Index Terms
- An improved exponential-time algorithm for k-SAT
Recommendations
Properties of SLUR formulae
SOFSEM'12: Proceedings of the 38th international conference on Current Trends in Theory and Practice of Computer ScienceSingle look-ahead unit resolution (SLUR) algorithm is a nondeterministic polynomial time algorithm which for a given input formula in a conjunctive normal form (CNF) either outputs its satisfying assignment or gives up. A CNF formula belongs to the SLUR ...
PPSZ for general k-SAT: making Hertli's analysis simpler and 3-SAT faster
CCC '17: Proceedings of the 32nd Computational Complexity ConferenceIn this paper, we achieve three goals. First, we simplify Hertli's 2011 analysis [1] for input formulas with multiple satisfying assignments. Second, we show a "translation result": if you improve PPSZ for k-CNF formulas with a unique satisfying ...
Exponential lower bounds for the PPSZ k-sat algorithm
SODA '13: Proceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithmsIn 1998, Paturi, Pudlák, Saks, and Zane presented PPSZ, an elegant randomized algorithm for k-SAT. Fourteen years on, this algorithm is still the fastest known worst-case algorithm. They proved that its expected running time on k-CNF formulas with n ...
Comments