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Journal of Automated Reasoning

, Volume 24, Issue 4, pp 397–420 | Cite as

New Worst-Case Upper Bounds for SAT

  • Edward A. Hirsch
Article

Abstract

In 1980 Monien and Speckenmeyer proved that satisfiability of a propositional formula consisting of K clauses (of arbitrary length) can be checked in time of the order 2K / 3. Recently Kullmann and Luckhardt proved the worst-case upper bound 2L / 9, where L is the length of the input formula. The algorithms leading to these bounds are based on the splitting method, which goes back to the Davis–Putnam procedure. Transformation rules (pure literal elimination, unit propagation, etc.) constitute a substantial part of this method. In this paper we present a new transformation rule and two algorithms using this rule. We prove that these algorithms have the worst-case upper bounds 20. 30897 K and 20. 10299 L, respectively.

SAT worst-case upper bounds 

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References

  1. 1.
    Bansal, N. and Raman, V.: Upper bounds for MaxSat: Further improved, in Proceedings of ISAAC'99. To appear.Google Scholar
  2. 2.
    Dantsin, E.: Tautology proof systems based on the splitting method, Leningrad Division of Steklov Institute of Mathematics (LOMI), Ph.D. Dissertation, Leningrad, 1982 (in Russian).Google Scholar
  3. 3.
    Dantsin, E. and Fuentes, L. O. and Kreinovich, V.: Less than 2n satisfiability algorithm extended to the case when almost all clauses are short, Computer Science Department, University of Texas at El Paso, UTEP-CS-91-5, 1991.Google Scholar
  4. 4.
    Dantsin, E., Gavrilovich, M., Hirsch, E. A. and Konev, B.: Approximation algorithms for Max Sat: A better performance ratio at the cost of a longer running time, PDMI Preprint 14/1998, available from ftp://ftp.pdmi.ras.ru/pub/publicat/preprint/1998/14-98.psGoogle Scholar
  5. 5.
    Dantsin, E. Ya. and Kreinovich, V. Ya.: Exponential upper bounds for the satisfiability problem, in Proc. of the IX USSR Conf. on Math. Logic, Leningrad, 1988 (in Russian).Google Scholar
  6. 6.
    Davis, M., Logemann, G. and Loveland, D.: A machine program for theorem-proving, Comm. ACM 5 (1962), 394–397.Google Scholar
  7. 7.
    Davis, M. and Putnam, H.: A computing procedure for quantification theory, J. ACM 7 (1960), 201–215.Google Scholar
  8. 8.
    Gu, J., Purdom, P. W., Franco, J. and Wah, B. W.: Algorithms for Satisfiability (SAT) Problem: A Survey, DiscreteMath. and Theoret. Comput. Sci.: Satisfiability (SAT) Problem, Amer.Math. Soc., 1997.Google Scholar
  9. 9.
    Hirsch, E. A.: Separating the signs in the satisfiability problem, Zap. Nauchn. Sem. POMI, 241 (1997), 30–71 (in Russian). English translation of this collection is to appear in the J. Math. Sci., 1999.Google Scholar
  10. 10.
    Hirsch, E. A.: Two new upper bounds for SAT, in Proc. of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, 1998, pp. 521–530.Google Scholar
  11. 11.
    Hirsch, E. A.: Local search algorithms for SAT: Worst-case analysis, in Proc. of the 6th Scandinavian Workshop on Algorithm Theory, Lecture Notes in Comput. Sci. 1432, 1998, pp. 246–254.Google Scholar
  12. 12.
    Hirsch, E. A.: Hard formulas for SAT local search algorithms, PDMI Preprint 19/1998, available from ftp://ftp.pdmi.ras.ru/pub/publicat/preprint/19-98.ps.Google Scholar
  13. 13.
    Hirsch, E. A.: SAT local search algorithms: Worst-case study, J. Automated Reasoning special issue “SAT-2000” 24(1–2) (2000).Google Scholar
  14. 14.
    Hirsch, E. A.: A new algorithm for MAX-2-SAT, Technical Report TR99-036, Electronic Colloquium on Computational Complexity, 1999.Google Scholar
  15. 15.
    Kullmann, O.: Worst-case analysis, 3-SAT decision and lower bounds: Approaches for improved SAT algorithms, in DIMACS Proc. SAT Workshop 1996, Amer. Math. Soc., 1996, pp. 261–313.Google Scholar
  16. 16.
    Kullmann, O.: New methods for 3-SAT decision and worst-case analysis, Theoret. Comput. Sci. 223(1-2) (1999), 1–72.Google Scholar
  17. 17.
    Kullmann, O.: Investigations on autark assignments, Submitted to Discrete Appl. Math., 1998, 19 pages.Google Scholar
  18. 18.
    Kullmann, O. and Luckhardt, H.: Deciding propositional tautologies: Algorithms and their complexity, Preprint, 1997, 82 pages; the ps file can be obtained at http://mi.informatik.unifrankfurt. de/people/kullmann/kullmann.html. A journal version, Algorithms for SAT/TAUT decision based on various measures, is to appear in Inform. and Comput., 1998.Google Scholar
  19. 19.
    Luckhardt, H.: Obere Komplexitätsschranken für TAUT-Entscheidungen, in Proc. Frege Conference 1984, Schwerine, Akademie-Verlag, Berlin, pp. 331–337.Google Scholar
  20. 20.
    Mahajan, M. and Raman, V.: Parametrizing above guaranteed values: MaxSat and MaxCut, Technical Report TR97-033, Electronic Colloquium on Computational Complexity, 1997. To appear in J. Algorithms.Google Scholar
  21. 21.
    Niedermeier, R. and Rossmanith, P.: New upper bounds for Max Sat, Technical Report KAMDIMATIA Series 98-401, Charles University, Praha, Faculty of Mathematics and Physics, July 1998. Extended abstract appeared in Proceedings of the 26th International Colloquium on Automata, Languages, and Programming, Lecture Notes in Comput. Sci. 1644, 1999, pp. 575–584.Google Scholar
  22. 22.
    Monien, B. and Speckenmeyer, E.: 3-satisfiability is testable in O(1.62r ) steps, Bericht Nr. 3/1979, Reihe Theoretische Informatik, Universität-Gesamthochschule-Paderborn.Google Scholar
  23. 23.
    Monien, B. and Speckenmeyer, E.: Upper bounds for covering problems, Bericht Nr. 7/1980, Reihe Theoretische Informatik, Universität-Gesamthochschule-Paderborn.Google Scholar
  24. 24.
    Monien, B. and Speckenmeyer, E.: Solving satisfiability in less than 2n steps, Discrete Appl. Math. 10 (1985), 287–295.Google Scholar
  25. 25.
    Paturi, R., Pudlak, P. and Zane, F.: Satisfiability coding lemma, in Proceedings of the 38th Annual Symposium on Foundations of Computer Science, 1997, pp. 566–574.Google Scholar
  26. 26.
    Paturi, R., Pudlak, P., Saks, M. E. and Zane, F.: An improved exponential-time algorithm for k-SAT, in Proceedings of the 39th Annual Symposium on Foundations of Computer Science, 1998, pp. 628–637.Google Scholar
  27. 27.
    Schiermeyer, I.: Solving 3-satisfiability in less than 1:579n steps, in Lecture Notes in Comput. Sci. 702, 1993, pp. 379–394.Google Scholar
  28. 28.
    Schiermeyer, I.: Pure literal look ahead: An O.(1:497n) 3-satisfiability algorithm, Workshop on the Satisfiability Problem, Technical Report, Siena, 29 April-3 May, 1996; University Köln, Report No. 96-230.Google Scholar
  29. 29.
    Schöning, U.: A probabilistic algorithm for k-SAT and constraint satisfaction problems, in Proceedings of the 40th Annual Symposium on Foundations of Computer Science, 1999. To appear.Google Scholar
  30. 30.
    Urquhart, A.: The complexity of propositional proofs, Bull. Symbolic Logic 1(4) (1995), 425–467.Google Scholar
  31. 31.
    Van Gelder, A.: A satisfiability tester for non-clausal propositional calculus, Inform. and Comput. 79 (1988), 1–21.Google Scholar

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© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Edward A. Hirsch

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