A New Algorithm for MAX-2-SAT
Recently there was a significant progress in proving (exponential-time) worst-case upper bounds for the propositional satisfiability problem (SAT) and related problems. In particular, for MAX-2-SAT Niedermeier and Rossmanith recently presented an algorithm with worstcase upper bound O(K ·2K/2.88...), and the bound O(K ·2K/3.44...) is implicit from the paper by Bansal and Raman (K is the number of clauses). In this paper we improve this bound to p(K)2 K 2/4, where K 2 is the number of 2-clauses, and p is a polynomial. In addition, our algorithm and the proof are much simpler than the previous ones. The key ideas are to use the symmetric flow algorithm of Yannakakis and to count only 2-clauses (and not 1-clauses).
Unable to display preview. Download preview PDF.
- 1.T. Asano, D. P. Williamson, Improved Approximation Algorithms for MAX SAT, To appear in Proc. of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, 2000.Google Scholar
- 2.N. Bansal, V. Raman, Upper Bounds for MaxSat: Further Improved, To appear in Proc. of ISAAC’99.Google Scholar
- 3.S.A. Cook, The complexity of theorem-proving procedure, Proc. 3rd Annual ACM Symposium on the Theory of Computing, 1971, pp. 151–159.Google Scholar
- 4.E. Dantsin, Tautology proof systems based on the splitting method, Leningrad Division of Steklov Institute of Mathematics (LOMI), PhD Dissertation, Leningrad, 1982 (in Russian).Google Scholar
- 5.E. Dantsin and L. O. Fuentes and V. Kreinovich, 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
- 6.E. Dantsin, M. Gavrilovich, E. A. Hirsch, B. Konev, 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.ps
- 7.E. Ya. Dantsin, V. Ya. Kreinovich, Exponential upper bounds for the satisfiability problem, Proc. of the IX USSR conf. on math. logic, Leningrad, 1988 (in Russian).Google Scholar
- 10.U. Feige, M. X. Goemans, Approximating the value of two prover proof systems, with applications to MAX 2SAT and MAX DICUT, Proc. of the Third Israel Symposium on Theory of Computing and Systems, 1995, pp. 182–189.Google Scholar
- 11.J. Håstad, Some optimal inapproximability results, Proc. of the 29th Annual ACM Symposium on Theory of Computing, 1997, pp. 1–10.Google Scholar
- 12.E. A. Hirsch, Two new upper bounds for SAT, Proc. of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, 1998, pp. 521–530.Google Scholar
- 13.E. A. Hirsch, New Worst-Case Upper Bounds for SAT, To appear in the special issue SAT-2000 and in the Journal of Automated Reasoning, Kluwer Academic Publishers, 2000.Google Scholar
- 14.H. Karloff, U. Zwick, A 7/8-approximation algorithm for MAX 3SAT?, In Proc. of the 38th Annual IEEE Symposium on Foundations of Computer Science, pp. 406–415, 1997.Google Scholar
- 16.O. Kullmann, H. Luckhardt, Deciding propositional tautologies: Algorithms and their complexity, Preprint, 1997, 82 pages; The electronic version can be obtained at http://www.cs.utoronto.ca/~kullmann. A journal version, Algorithms for SAT/TAUT decision based on various measures, is to appear in Information and Computation.
- 17.M. Mahajan and V. Raman, Parametrizing above guaranteed values: MaxSat and MaxCut, Technical Report TR97-033, Electronic Colloquium on Computational Complexity, 1997. To appear in Journal of Algorithms.Google Scholar
- 18.B. Monien, E. Speckenmeyer, 3-satisfiability is testable in O(1.62r) steps, Bericht Nr. 3/1979, Reihe Theoretische Informatik, Universität-Gesamthochschule-Paderborn.Google Scholar
- 20.R. Niedermeier and P. Rossmanith. New upper bounds for MaxSat, Technical Report KAM-DIMATIA Series 98–401, Charles University, Praha, Faculty of Mathematics and Physics, July 1998. Extended abstract appeared in Proc. of the 26th International Colloquium on Automata, Languages, and Programming, LNCS 1644, pp. 575–584, 1999. A journal version is to appear in Journal of Algorithms.Google Scholar
- 21.Ch. H. Papadimitriou, Computational Complexity, Addison-Wesley, 1994, 532 p.Google Scholar
- 22.R. Paturi, P. Pudlak, M. E. Saks, F. Zane, An Improved Exponential-time Algorithm for k-SAT, Proc. of the 39th Annual Symposium on Foundations of Computer Science, 1998, pp. 628–637.Google Scholar
- 25.U. Schöning, A probabilistic algorithm for k-SAT and constraint satisfaction problems, Proc. of the 40th Annual Symposium on Foundations of Computer Science, 1999. To appear.Google Scholar