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Exact Algorithms for General CNF SAT

1998; Hirsch 2003; Schuler

  • Reference work entry
Encyclopedia of Algorithms

Keywords and Synonyms

SAT ; Boolean satisfiability              

Problem Definition

The satisfiability problem (SAT) for Boolean formulas in conjunctive normal form (CNF) is one of the first NP‐complete problems [2,13]. Since its NP‐completeness currently leaves no hope for polynomial-time algorithms, the progress goes by decreasing the exponent. There are several versions of this parametrized problem that differ in the parameter used for the estimation of the running time.

Problem 1 (SAT)

Input: Formula F in CNF containing n variables, m clauses, and l literals in total.

Output: “Yes” if F has a satisfying assignment, i. e., a substitution of Boolean values for the variables that makes F true. “No” otherwise.

The bounds on the running time of SAT algorithms can be thus given in the form \( |F|^{O(1)} \cdot \alpha^n, |F|^{O(1)} \cdot \beta^m \), or \( |F|^{O(1)} \cdot \gamma^l \), where |F| is the length of a reasonable bit representation of F(i. e., the formal input to the algorithm). In fact,...

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Notes

  1. 1.

    They and other researchers also noticed that α could be made smaller for a special case of the problem where the length of each clause is bounded by a constant; the reader is referred to another entry (Local search algorithms for k-SAT) of the Encyclopedia for relevant references and algorithms.

  2. 2.

    Also called DPLL due to the papers of Davis and Putnam [7] and Davis, Logemann, and Loveland [6].

Recommended Reading

  1. Calabro, C., Impagliazzo, R., Paturi, R.: A duality between clause width and clause density for SAT. In: Proceedings of the 21st Annual IEEE Conference on Computational Complexity (CCC 2006), pp. 252–260. IEEE Computer Society (2006)

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  2. Cook, S.A.: The Complexity of Theorem Proving Procedures. Proceedings of the Third Annual ACM Symposium on Theory of Computing, May 1971, pp. 151–158. ACM (2006)

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  3. Dantsin, E., Goerdt, A., Hirsch, E.A., Kannan, R., Kleinberg, J., Papadimitriou, C., Raghavan, P., Schöning, U.: A deterministic (2–2/(k + 1))n algorithm for k-SAT based on local search. Theor. Comput. Sci. 289(1), 69–83 (2002)

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  4. Dantsin, E., Hirsch, E.A.: Worst-Case Upper Bounds. In: Biere, A., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability. IOS Press (2008) To appear

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  5. Dantsin, E., Hirsch, E.A., Wolpert, A.: Clause shortening combined with pruning yields a new upper bound for deterministic SAT algorithms. In: Proceedings of CIAC-2006. Lecture Notes in Computer Science, vol. 3998, pp. 60–68. Springer, Berlin (2006)

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

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

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  8. Hirsch, E.A.: New worst-case upper bounds for SAT. J. Autom. Reason. 24(4), 397–420 (2000)

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  9. Kojevnikov, A., Kulikov, A.: A New Approach to Proving Upper Bounds for MAX-2-SAT. Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2006), pp. 11–17. ACM, SIAM (2006)

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  10. Kulikov, A.: Automated Generation of Simplification Rules for SAT and MAXSAT. Proceedings of the Eighth International Conference on Theory and Applications of Satisfiability Testing (SAT 2005). Lecture Notes in Computer Science, vol. 3569, pp. 430–436. Springer, Berlin (2005)

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  11. Kullmann, O.: New methods for 3-{SAT} decision and worst-case analysis. Theor. Comput. Sci. 223(1–2):1–72 (1999)

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  12. Kullmann, O., Luckhardt, H.: Algorithms for SAT/TAUT decision based on various measures, preprint, 71 pages, http://cs-svr1.swan.ac.uk/csoliver/papers.html (1998)

  13. Levin, L.A.: Universal Search Problems. Проблемы передачи информации 9(3), 265–266, (1973). In Russian. English translation in: Trakhtenbrot, B.A.: A Survey of Russian Approaches to Perebor (Brute-force Search) Algorithms. Annals of the History of Computing 6(4), 384–400 (1984)

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  14. Pudlák, P.: Satisfiability – algorithms and logic. In: Proceedings of the 23rd International Symposium on Mathematical Foundations of Computer Science, MFCS'98. Lecture Notes in Computer Science, vol. 1450, pp. 129–141. Springer, Berlin (1998)

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  15. Schuler, R.: An algorithm for the satisfiability problem of formulas in conjunctive normal form. J. Algorithms 54(1), 40–44 (2005)

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  16. Wahlström, M.: An algorithm for the SAT problem for formulae of linear length. In: Proceedings of the 13th Annual European Symposium on Algorithms, ESA 2005. Lecture Notes in Computer Science, vol. 3669, pp. 107–118. Springer, Berlin (2005)

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  17. Wang, J.: Generating and solving 3-SAT, MSc Thesis. Rochester Institute of Technology, Rochester (2002)

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Author information

Authors and Affiliations

  1. Laboratory of Mathematical Logic, Steklov Institute of Mathematics at St. Petersburg, St. Petersburg, Russia

    Edward A. Hirsch

Authors
  1. Edward A. Hirsch
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer ScienceMcCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, 60208, USA

    Ming-Yang Kao Professor of Computer Science

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Hirsch, E. (2008). Exact Algorithms for General CNF SAT. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_133

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