New Upper Bounds for MaxSat

  • Rolf Niedermeier
  • Peter Rossmanith
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1644)


Given a boolean formula F in conjunctive normal form and an integer k, is there a truth assignment satisfying at least k clauses? This is the decision version of the Maximum Satisfiability (MaxSax) problem we study in this paper. We improve upper bounds on the worst case running time for MAXSAT. First, Cai and Chen showed that MAXSAT can be solved in time |F|2O(k) when the clause size is bounded by a constant. Imposing no restrictions on clause size, Mahajan and Raman and, independently, Dantsin et al. improved this to O(|F|⌽k), where ⌽ ≈ 1.6181 is the golden ratio. We present an algorithm running in time O(|F|1.3995k). The result extends to finding an optimal assignment and has several applications, in particular, for parameterized complexity and approximation algorithms. Moreover, if F has K clauses, we can find an optimal assignment in O(|F|1.3972K) steps and in O(1.1279|F|) steps, respectively. These are the fastest algorithm in the number of clauses and the length of the formula, respectively.


Transformation Rule Vertex Cover Conjunctive Normal Form Truth Assignment Optimal Assignment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and hardness of approximation problems. In Proc. of 33d FOCS, pages 14–23, 1992.Google Scholar
  2. 2.
    R. Balasubramanian, M.R. Fellows, and V. Raman. An improved fixed parameter algorithm for vertex cover. Information Processing Letters, 65(3):163–168, 1998.CrossRefMathSciNetGoogle Scholar
  3. 3.
    R. Battiti and M. Protasi. Reactive Search, a history-base heuristic for MAX-SAT. ACM Journal of Experimental Algorithmics, 2:Article 2, 1997.Google Scholar
  4. 4.
    R. Battiti and M. Protasi. Approximate algorithms and heuristics for MAX-SAT. In D.-Z. Du and P.M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 1, pages 77–148. Kluwer Academic Publishers, 1998.Google Scholar
  5. 5.
    R. Beigel and D. Eppstein. 3-Coloring in time o(1:3446n): A no MIS algorithm. In Proc. of 36th FOCS, pages 444–452, 1995.Google Scholar
  6. 6.
    L. Cai and J. Chen. On fixed-parameter tractability and approximability of NP optimization problems. J. Comput. Syst. Sci., 54:465–474, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    P. Crescenzi and V. Kann. A compendium of NP optimization problems. Available at, Apr. 1997.
  8. 8.
    E. Dantsin, M. Gavrilovich, E.A. Hirsch, and B. Konev. Approximation algorithms for Max SAT: A better performance ratio at the cost of a longer running time. Technical Report PDMI preprint 14/1998, Steklov Institute of Mathematics at St. Petersburg, 1998.Google Scholar
  9. 9.
    R.G. Downey and M.R. Fellows. Parameterized Complexity. Springer-Verlag, 1999.Google Scholar
  10. 10.
    R.G. Downey, M.R. Fellows, and U. Stege. Parameterized complexity: A framework for systematically confronting computational intractability. In F. Roberts, J. Kratochvíl, and J. Nešetřil, editors, Proc. of 1st DIMATIA Symposium, AMSDIMACS Proceedings Series, 1997. To appear.Google Scholar
  11. 11.
    P. Hansen and B. Jaumard. Algorithms for the maximum satisfiability problem. Computing, 44:279–303, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    E.A. Hirsch. Two new upper bounds for SAT. In Proc. of 9th SODA, pages 521–530, 1998.Google Scholar
  13. 13.
    O. Kullmann and H. Luckhardt. Deciding propositional tautologies: Algorithms and their complexity. 1997. Submitted to Information and Computation.Google Scholar
  14. 14.
    M. Mahajan and V. Raman. Parametrizing above guaranteed values: MaxSat and MaxCut. Technical Report TR97-033, ECCC Trier, 1997. To appear in Journal of Algorithms.Google Scholar
  15. 15.
    B. Monien and E. Speckenmeyer. Upper bounds for covering problems. Technical Report Reihe Theoretische Informatik, Bericht Nr. 7/1980, Universität Gesamthochschule Paderborn, 1980.Google Scholar
  16. 16.
    B. Monien and E. Speckenmeyer. Solving satisfiability in less than 2n steps. Discrete Applied Mathematics, 10:287–295, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    R. Niedermeier. Some prospects for efficient fixed parameter algorithms (invited paper). In B. Rovan, editor, Proceedings of the 25th Conference on Current Trends in Theory and Practice of Informatics (SOFSEM), number 1521 in Lecture Notes in Computer Science, pages 168–185. Springer-Verlag, 1998.Google Scholar
  18. 18.
    R. Niedermeier and P. Rossmanith. New upper bounds for MaxSat. Technical Report KAM-DIMATIA Series 98-401, Faculty of Mathematics and Physics, Charles University, Prague, July 1998.Google Scholar
  19. 19.
    R. Niedermeier and P. Rossmanith. Upper bounds for Vertex Cover further improved. In C. Meinel and S. Tison, editors, Proceedings of the 16th Symposium on Theoretical Aspects of Computer Science, number 1563 in Lecture Notes in Computer Science, pages 561–570._Springer-Verlag, 1999.Google Scholar
  20. 20.
    C.H. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.Google Scholar
  21. 21.
    C.H. Papadimitriou and M. Yannakakis. Optimization, approximation, and complexity classes. J. Comput. Syst. Sci., 43:425–440, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    R. Paturi, P. Pudlák, M. Saks, and F. Zane. An improved exponential-time algorithm for k-SAT. In Proc. of 39th FOCS, pages 628–637, 1998.Google Scholar
  23. 23.
    P. Pudlák. Satisfiability algorithms and logic (invited paper). In Proceedings of the 23d Conference on Mathematical Foundations of Computer Science, number 1450 in Lecture Notes in Computer Science, pages 129–141, Brno, Czech Republic, Aug. 1998. Springer-Verlag.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Rolf Niedermeier
    • 1
  • Peter Rossmanith
    • 2
  1. 1.Wilhelm-Schickard-Institut für InformatikTübingenFederal Republic of Germany
  2. 2.Institut für InformatikTechnische Universität MünchenMünchenFederal Republic of Germany

Personalised recommendations