Abstract
MAX SAT and MAX NAE-SAT are central problems in theoretical computer science. We present an approximation algorithm for MAX NAE-SAT with a conjectured performance guarantee of 0.8279. This improves a previously conjectured performance guarantee of 0.7977 of Zwick [Zwi99]. Using a variant of our MAX NAE-SAT approximation algorithm, combined with other techniques used in [Asa03], we obtain an approximation algorithm for MAX SAT with a conjectured performance guarantee of 0.8434. This improves on an approximation algorithm of Asano [Asa03] with a conjectured performance guarantee of 0.8353. We also obtain a 0.7968-approximation algorithm for MAX SAT which is not based on any conjecture, improving a 0.7877-approximation algorithm of Asano [Asa03].
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Andersson, G., Engebretsen, L.: Better approximation algorithms for Set Splitting and Not-All-Equal Sat. Information Processing Letters 65, 305–311 (1998)
Alon, N., Makarychev, K., Makarychev, Y., Naor, A.: Quadratic forms on graphs. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, Maryland, pp. 486–493 (2005)
Asano, T., Ono, T., Hirata, T.: Approximation algorithms for the maximum satisfiability problem. Nordic Journal of Computing 3, 388–404 (1996)
Asano, T.: Approximation algorithms for MAX SAT: Yannakakis vs. Goemans-Williamson. In: Proceedings of the 3rd Israel Symposium on Theory and Computing Systems, Ramat Gan, Israel, pp. 24–37 (1997)
Asano, T.: An improved analysis of Goemans and Williamson’s \(\textsc{LP}\)-relaxation for \(\textsc{MAX SAT}\). In: Lingas, A., Nilsson, B.J. (eds.) FCT 2003. LNCS, vol. 2751, pp. 2–14. Springer, Heidelberg (2003)
Asano, T., Williamson, D.P.: Improved approximation algorithms for \(\textsc{MAX SAT}\). Journal of Algorithms 42, 173–202 (2002)
Charikar, M., Wirth, A.: Maximizing Quadratic Programs: Extending Grothendieck’s Inequality. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, Rome, Italy, pp. 54–60 (2004)
Feige, U., Goemans, M.X.: Approximating the value of two prover proof systems, with applications to MAX-2SAT and MAX-DICUT. In: Proceedings of the 3rd Israel Symposium on Theory and Computing Systems, Tel Aviv, Israel, pp. 182–189 (1995)
Feige, U., Langberg, M.: The RPR2 rounding technique for semidefinite programs. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 213–224. Springer, Heidelberg (2001)
Goemans, M.X., Williamson, D.P.: New 3/4-approximation algorithms for the maximum satisfiability problem. SIAM Journal on Discrete Mathematics 7, 656–666 (1994)
Goemans, M.X., Williamson, D.P.: Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming. Journal of the ACM 42, 1115–1145 (1995)
Håstad, J.: Some optimal inapproximability results. Journal of the ACM 48(4), 798–859 (2001)
Han, Q., Ye, Y., Zhang, J.: Improved Approximation for Max Set Splitting and Max NAE SAT. Discrete Applied Mathematics 142(1-3), 133–149 (2004)
Halperin, E., Zwick, U.: Approximation algorithms for MAX 4-SAT and rounding procedures for semidefinite programs. Journal of Algorithms 40, 184–211 (2001)
Johnson, D.S.: Approximation algorithms for combinatorical problems. Journal of Computer and System Sciences 9, 256–278 (1974)
Karloff, H., Zwick, U.: A 7/8-approximation algorithm for MAX 3SAT? In: Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, Miami Beach, Florida, pp. 406–415 (1997)
Lewin, M., Livnat, D., Zwick, U.: Improved rounding techniques for the MAX 2-SAT and MAX DI-CUT problems. In: Cook, W.J., Schulz, A.S. (eds.) IPCO 2002. LNCS, vol. 2337, pp. 67–82. Springer, Heidelberg (2002)
Matuura, S., Matsui, T.: 0.863-approximation algorithm for MAX DICUT. In: Goemans, M.X., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) RANDOM 2001 and APPROX 2001. LNCS, vol. 2129, pp. 138–146. Springer, Heidelberg (2001)
Matuura, S., Matsui, T.: 0.935-approximation randomized algorithm for MAX 2SAT and its derandomization. Technical Report METR 2001-03, Department of Mathematical Engineering and Information Physics, the University of Tokyo, Japan (September 2001)
Nesterov, Y.E.: Semidefinite relaxation and nonconvex quadratic optimization. Optimization Methods and Software 9, 141–160 (1998)
Yannakakis, M.: On the approximation of maximum satisfiability. Journal of Algorithms 17, 475–502 (1994)
Ye, Y.: A .699-approximation algorithm for Max-Bisection. Mathematical Programming 90(1, Ser. A), 101–111 (2001)
Zwick, U.: Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems. In: Proceedings of the 31th Annual ACM Symposium on Theory of Computing, Atlanta, Georgia, pp. 679–687 (1999)
Zwick, U.: Computer assisted proof of optimal approximability results. In: Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, California, pp. 496–505 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Avidor, A., Berkovitch, I., Zwick, U. (2006). Improved Approximation Algorithms for MAX NAE-SAT and MAX SAT. In: Erlebach, T., Persinao, G. (eds) Approximation and Online Algorithms. WAOA 2005. Lecture Notes in Computer Science, vol 3879. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11671411_3
Download citation
DOI: https://doi.org/10.1007/11671411_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-32207-8
Online ISBN: 978-3-540-32208-5
eBook Packages: Computer ScienceComputer Science (R0)