Abstract
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.
Supported by a Feodor Lynen fellowship of the Alexander von Humboldt-Stiftung, Bonn, and the Center for Discrete Mathematics, Theoretical Computer Science and Applications (DIMATIA), Prague.
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References
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.
R. Balasubramanian, M.R. Fellows, and V. Raman. An improved fixed parameter algorithm for vertex cover. Information Processing Letters, 65(3):163–168, 1998.
R. Battiti and M. Protasi. Reactive Search, a history-base heuristic for MAX-SAT. ACM Journal of Experimental Algorithmics, 2:Article 2, 1997.
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.
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.
L. Cai and J. Chen. On fixed-parameter tractability and approximability of NP optimization problems. J. Comput. Syst. Sci., 54:465–474, 1997.
P. Crescenzi and V. Kann. A compendium of NP optimization problems. Available at http://www.nada.kth.se/theory/problemlist.html, Apr. 1997.
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.
R.G. Downey and M.R. Fellows. Parameterized Complexity. Springer-Verlag, 1999.
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.
P. Hansen and B. Jaumard. Algorithms for the maximum satisfiability problem. Computing, 44:279–303, 1990.
E.A. Hirsch. Two new upper bounds for SAT. In Proc. of 9th SODA, pages 521–530, 1998.
O. Kullmann and H. Luckhardt. Deciding propositional tautologies: Algorithms and their complexity. 1997. Submitted to Information and Computation.
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.
B. Monien and E. Speckenmeyer. Upper bounds for covering problems. Technical Report Reihe Theoretische Informatik, Bericht Nr. 7/1980, Universität Gesamthochschule Paderborn, 1980.
B. Monien and E. Speckenmeyer. Solving satisfiability in less than 2n steps. Discrete Applied Mathematics, 10:287–295, 1985.
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.
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.
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.
C.H. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.
C.H. Papadimitriou and M. Yannakakis. Optimization, approximation, and complexity classes. J. Comput. Syst. Sci., 43:425–440, 1991.
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.
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.
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Niedermeier, R., Rossmanith, P. (1999). New Upper Bounds for MaxSat. In: Wiedermann, J., van Emde Boas, P., Nielsen, M. (eds) Automata, Languages and Programming. Lecture Notes in Computer Science, vol 1644. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48523-6_54
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