Abstract
To solve a problem on a given CNF formula F a splitting algorithm recursively calls for F[v] and F[¬v] for a variable v. Obviously, after the first call an algorithm obtains some information on the structure of the formula that can be used in the second call. We use this idea to design new surprisingly simple algorithms for the MAX-SAT problem. Namely, we show that MAX-SAT for formulas with constant clause density can be solved in time c n, where c < 2 is a constant and n is the number of variables, and within polynomial space (the only known such algorithm by Dantsin and Wolpert uses exponential space). We also prove that MAX-2-SAT can be solved in time 2m/5.88, where m is the number of clauses (this improves the bound 2m/5.769 proved independently by Kneis et al. and by Scott and Sorkin).
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References
Williams, R.: On computing k-CNF formula properties. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 330–340. Springer, Heidelberg (2004)
Robson, J.: Algorithms for maximum independent sets. Journal of Algorithms 7(3), 425–440 (1986)
Marques-Silva, J., Sakallah, K.: Grasp: a search algorithm for propositional satisfiability. IEEE Transactionon Computers 48(5), 506–521 (1999)
Moskewicz, M., Madigan, C., Zhao, Y., Zhang, L., Malik, S.: Chaff: engineering an efficient SAT solver. In: Proceedings of the 38th Design Automation Conference, pp. 530–535 (2001)
Zhang, H.: Sato: An efficient propositional prover. In: McCune, W. (ed.) Automated Deduction - CADE-14. LNCS, vol. 1249, pp. 272–275. Springer, Heidelberg (1997)
Beame, P., Impagliazzo, R., Pitassi, T., Segerlind, N.: Memoization and DPLL: formula caching proof systems. In: Proceedings of 18th IEEE Annual Conference on Computational Complexity, pp. 248–259. IEEE Computer Society Press, Los Alamitos (2003)
Dantsin, E., Wolpert, A.: MAX-SAT for formulas with constant clause density can be solved faster than in O(2n) time. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 266–276. Springer, Heidelberg (2006)
Williams, R.: A new algorithm for optimal 2-constraint satisfaction and its implications. Theoretical Computer Science 348, 357–365 (2005)
Niedermeier, R., Rossmanith, P.: New upper bounds for MaxSat. In: Wiedermann, J., van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 575–585. Springer, Heidelberg (1999)
Hirsch, E.A.: A 2K/4-time algorithm for MAX-2-SAT: Corrected version. ECCC Report TR99-036, Revision 02 (2000)
Gramm, J., Hirsch, E.A., Niedermeier, R., Rossmanith, P.: Worst-case upper bounds for MAX-2-SAT with an application to MAX-CUT. Discrete Applied Mathematics 130(2), 139–155 (2003)
Scott, A., Sorkin, G.: Faster algorithms for MAX CUT and MAX CSP, with polynomial expected time for sparse instances. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 382–395. Springer, Heidelberg (2003)
Kneis, J., Rossmanith, P.: A new satisfiability algorithm with applications to Max-Cut. Technical Report AIB2005-08, Dept. of Computer Science, RWTH Aachen University (2005)
Kojevnikov, A., Kulikov, A.S.: A new approach to proving upper bounds for MAX-2-SAT. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 11–17. ACM Press, New York (2006)
Kneis, J., Moelle, D., Richter, S., Rossmanith, P.: Algorithms based on the treewidth of sparse graphs. In: Proceedings of the 31st International Workshop on Graph-Theoretic Concepts in Computer Science. LNCS, pp. 385–396 (2005)
Scott, A.D., Sorkin, G.B.: Linear-programming design and analysis of fast algorithms for Max 2-SAT and Max 2-CSP. Discrete Optimization 2006 (to appear)
Kullmann, O., Luckhardt, H.: Algorithms for SAT/TAUT decision based on various measures (preprint)
Kulikov, A.S.: Automated generation of simplification rules for SAT and MAXSAT. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 430–436. Springer, Heidelberg (2005)
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Kulikov, A.S., Kutzkov, K. (2007). New Bounds for MAX-SAT by Clause Learning. In: Diekert, V., Volkov, M.V., Voronkov, A. (eds) Computer Science – Theory and Applications. CSR 2007. Lecture Notes in Computer Science, vol 4649. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74510-5_21
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DOI: https://doi.org/10.1007/978-3-540-74510-5_21
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