Abstract
We introduce the submodular Max-SAT problem. This problem is a natural generalization of the classical Max-SAT problem in which the additive objective function is replaced by a submodular one. We develop a randomized linear-time 2/3-approximation algorithm for the problem. Our algorithm is applicable even for the online variant of the problem. We also establish hardness results for both the online and offline settings. Notably, for the online setting, the hardness result proves that our algorithm is best possible, while for the offline setting, the hardness result establishes a computational separation between the classical Max-SAT and the submodular Max-SAT.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Asano, T.: Approximation algorithms for max sat: Yannakakis vs. goemans-williamson. In: Proceedings 5th Israel Symposium on Theory of Computing and Systems, pp. 24–37 (1997)
Asano, T.: An improved analysis of goemans and williamson’s lp-relaxation for max sat. Theor. Comput. Sci. 354(3), 339–353 (2006)
Asano, T., Williamson, D.P.: Improved approximation algorithms for max sat. J. Algorithms 42(1), 173–202 (2002)
Calinescu, G., Chekuri, C., Pál, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. In: SICOMP (2010)
Chekuri, C., Vondrák, J., Zenklusen, R.: Dependent randomized rounding via exchange properties of combinatorial structures. In: Proceedings 51st FOCS (2010)
Chen, J., Friesen, D.K., Zheng, H.: Tight bound on johnson’s algorithm for maximum satisfiability. J. Comput. Syst. Sci. 58(3), 622–640 (1999)
Coppersmith, D., Gamarnik, D., Hajiaghayi, M.T., Sorkin, G.B.: Random max sat, random max cut, and their phase transitions. Random Struct. Algorithms 24(4), 502–545 (2004)
Dobzinski, S., Schapira, M.: An improved approximation algorithm for combinatorial auctions with submodular bidders. In: Proceedings of 17th SODA, pp. 1064–1073 (2006)
Engebretsen, L.: Simplified tight analysis of johnson’s algorithm. Inf. Process. Lett. 92(4), 207–210 (2004)
Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)
Feldman, M., Naor, J., Schwartz, R.: Personal Communication (2011)
Goemans, M.X., Williamson, D.P.: 878-Approximation algorithms for max cut and max 2sat. In: Proceedings 26th STOC, pp. 422–431 (1994)
Goemans, M.X., Williamson, D.P.: New 3/4-approximation algorithms for the maximum satisfiability problem. SIAM J. Discrete Math. 7(4), 656–666 (1994)
Goundan, P.R., Schulz, A.S.: Revisiting the greedy approach to submodular set function maximization (2007) (manuscript)
Håstad, J.: Some optimal inapproximability results. J. ACM 48(4), 798–859 (2001)
Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9(3), 256–278 (1974)
Karloff, H.J., Zwick, U.: A 7/8-approximation algorithm for max 3sat? In: Proceedings 38th FOCS, pp. 406–415 (1997)
Kulik, A., Shachnai, H., Tamir, T.: Maximizing submodular set functions subject to multiple linear constraints. In: Proceedings 20th SODA, pp. 545–554 (2009)
Lee, J., Sviridenko, M., Vondrák, J.: Submodular maximization over multiple matroids via generalized exchange properties. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX 2009. LNCS, vol. 5687, pp. 244–257. Springer, Heidelberg (2009)
Mirrokni, V.S., Schapira, M., Vondrák, J.: Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions. In: Proceedings 9th EC 2008, pp. 70–77 (2008)
Moshkovitz, D., Raz, R.: Two-query pcp with subconstant error. J. ACM, 57(5) (2010)
Nemhauser, G.L., Wolsey, L.A.: Best algorithms for approximating the maximum of a submodular set function. Math. Operations Research 3(3), 177–188 (1978)
Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions i. Mathematical Programming 14, 265–294 (1978)
Poloczek, M., Schnitger, G.: Randomized variants of johnson’s algorithm for max sat. In: SODA, pp. 656–663 (2011)
Sviridenko, M.: A note on maximizing a submodular set function subject to a knapsack constraint. Oper. Res. Lett. 32(1), 41–43 (2004)
Wolsey, L.A.: Maximising real-valued submodular functions: Primal and dual heuristics for location problems. Math. Operations Research 7(3), 410–425 (1982)
Yannakakis, M.: On the approximation of maximum satisfiability. J. Algorithms 17(3), 475–502 (1994)
Zwick, U.: Outward rotations: A tool for rounding solutions of semidefinite programming relaxations, with applications to max cut and other problems. In: Proceedings of 31st STOC, pp. 679–687 (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Azar, Y., Gamzu, I., Roth, R. (2011). Submodular Max-SAT. In: Demetrescu, C., Halldórsson, M.M. (eds) Algorithms – ESA 2011. ESA 2011. Lecture Notes in Computer Science, vol 6942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23719-5_28
Download citation
DOI: https://doi.org/10.1007/978-3-642-23719-5_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-23718-8
Online ISBN: 978-3-642-23719-5
eBook Packages: Computer ScienceComputer Science (R0)