Abstract
We study three new techniques that will speed up the branch-and-bound algorithm for the MAX-2-SAT problem: The first technique is a group of new lower bound functions for the algorithm and we show that these functions are admissible and consistently better than other known lower bound functions. The other two techniques are based on the strongly connected components of the implication graph of a 2CNF formula: One uses the graph to simplify the formula and the other uses the graph to design a new variable ordering. The experiments show that the simplification can reduce the size of the input substantially no matter what is the clause-to-variable ratio and that the new variable ordering performs much better when the clause-to-variable ratio is less than 2. A direct outcome of this research is a high-performance implementation of an exact algorithm for MAX-2-SAT which outperforms any implementation we know about in the same category. We also show that our implementation is a feasible and effective tool to solve large instances of the Max-Cut problem in graph theory.
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References
T. Alsinet, F. Manyà and J. Planes, Improved branch and bound algorithms for Max-SAT, in: Proc. of 6th International Conference on the Theory and Applications of Satisfiability Testing, SAT2003 (2003) pp. 408–415.
T. Alsinet, F. Manyà and J. Planes, Improved branch and bound algorithms for Max-2-SAT and Weighted Max-2-SAT, in: Catalonian Conference on Artificial Intelligence (2003).
B. Aspvall, M.F. Plass and R.E. Tarjan, A linear-time algorithm for testing the truth of certain quantified Boolean formulas, Information Processing Letters 8(3) (1979) 121–123.
N. Bansal and V. Raman, Upper bounds for MaxSat: Further improved, in: Proc. of the 10th Annual Conference on Algorithms and Computation, ISSAC’99, eds. Aggarwal and Rangan, Lecture Notes in Computer Science, Vol. 1741 (Springer, New York, 1999) pp. 247–258.
B. Borchers and J. Furman, A two-phase exact algorithm for MAX-SAT and weighted MAX-SAT problems, Journal of Combinatorial Optimization 2(4) (1999) 299–306.
R.J. Bayardo and J.D. Pehoushek, Counting models using connected components, in: 17th National Conference on Artificial Intelligence (AAAI) (2000) pp. 157–162.
J. Cheriyan, W.H. Cunningnham, L. Tuncel and Y. Wang, A linear programming and rounding approach to Max 2-Sat, in: Discrete Mathematics and Theoretical Computer Science, Vol. 26 (Science Press, New York, 1996) pp. 395–414.
E. Dantsin, A. Goerdt, E.A. Hirsch, R. Kannan, J. Kleinberg, C. Papadimitriou, P. Raghavan and U. Schöning, A deterministic (2−2/(k+1))n algorithm for k-SAT based on local search, Theoretical Computer Science 289(1) (2002) 69–83.
M. Davis, G. Logemann and D. Loveland, A machine program for theorem-proving, Communications of the Association for Computing Machinery 7 (1962) 394–397.
P. Erdös and A. Rnyi, On the evolution of random graphs, Mat. Kutato Int. Kozl. 5 (1960) 17–61.
J. Gramm, Exact algorithms for Max2Sat and their applications, Diplomarbeit, Universität Tübingen (1999).
S. Givry, J. Larrosa, P. Meseguer and T. Schiex, Solving Max-SAT as weighted CSP, in: Principles and Practice of Constraint Programming – 9th International Conference CP 2003, Kinsale, Ireland (September 2003).
J. Gramm, E.A. Hirsch, R. Niedermeier and P. Rossmanith, Worst-case upper bounds for MAX-2-SAT with application to MAX-CUT, Discrete Applied Mathematics 130(2) (2003) 139–155.
P. Hansen and B. Jaumard, Algorithms for the maximum satisfiability problem, Computing 44 (1990) 279–303.
E.A. Hirsch, A new algorithm for MAX-2-SAT, in: Proc. of the 17th International Symposium on Theoretical Aspects of Computer Science, STACS 2000, Lecture Notes in Computer Science, Vol. 1770 (Springer, New York, 2000) pp. 65–73.
E.A. Hirsch, New worst-case upper bounds for SAT, Journal of Automated Reasoning 24(4) (2000) 397–420.
M. Mahajan and V. Raman, Parameterizing above guaranteed values: Max-Sat and Max-Cut, Journal of Algorithms 31 (1999) 335–354.
R. Niedermeier and P. Rossmanith, New upper bounds for maximum satisfiability, Journal of Algorithms 36 (2000) 63–88.
H. Shen and H. Zhang, An empirical study of Max-2-Sat phase transitions, in: Proc. of LICS’03 Workshop on Typical Case Complexity and Phase Transitions, Ottawa, Canada (June 2003).
H. Shen and H. Zhang, Improving Exact Algorithms for MAX-2-SAT, in: Proc. of 8th International Symposium on Artificial Intelligence and Mathematics (2004).
H. Shen and H. Zhang, Study of Lower Bound Functions for MAX-2-SAT, in: Proc. of the 19th National Conf. on Artificial Intelligence (AAAI), San Jose, CA (July 2004).
R. Wallace and E. Freuder, Comparative studies of constraint satisfaction and Davis–Putnam algorithms for maximum satisfiability problems, in: Cliques, Coloring and Satisfiability, eds.D. Johnson and M. Trick (1996) pp. 587–615.
H. Xu, R.A. Rutenbar and K. Sakallah, Sub-SAT: A formulation for related Boolean satisfiability with applications in routing, in: ISPD’02, San Diego, CA (April 2002).
H. Zhang, H. Shen and F. Manyà, Exact algorithms for MAX-SAT, in: Proc. of International Workshop on First-Order Theorem Proving (FTP 2003).
X. Zhao and W. Zhang, An efficient algorithm for maximum Boolean satisfiability based on unit propagation, linear programming, and dynamic weighting, Preprint, Department of Computer Science, Washington University (2004).
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Preliminary results of this paper appeared in [20,21]. This research was supported in part by NSF under grant CCR-0098093.
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Shen, H., Zhang, H. Improving exact algorithms for MAX-2-SAT. Ann Math Artif Intell 44, 419–436 (2005). https://doi.org/10.1007/s10472-005-7036-z
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DOI: https://doi.org/10.1007/s10472-005-7036-z