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Solving the weighted MAX-SAT problem using the dynamic convexized method

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Abstract

Satisfiability (SAT) and maximum satisfiability (MAX-SAT) are difficult combinatorial problems that have many important real-world applications. In this paper we investigate the performance of the dynamic convexized method based heuristics on the weighted MAX-SAT problem. We first present an auxiliary function which is constructed based on a penalty function, and minimize the function by a local search method which can escape successfully from previously converged local minimizers by increasing the value of a parameter. Two algorithms of the approach are implemented and compared with the Greedy Randomized Adaptive Search Procedure (GRASP) and the GRASP with Path Relinking (GRASP + PR). Experimental results illustrate efficient and faster convergence of our two algorithms.

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References

  1. Alsinet, T., Many, F., Planes, J.: Improved branch and bound algorithms for Max-SAT. In: Proceedings of the 6th International Conference on the Theory and Applications of Satisfiability Testing. Portofino, Italy (2003)

  2. Alsinet, T., Many, F., Planes, J.: An efficient solver for weighted MAX-SAT. J. Global Optimiz. 41, 61–73 (2008)

    Article  MATH  Google Scholar 

  3. Avidor, A., Berkovitch, I., Zwick, U.: Improved approximation algorithms for MAX NAE-SAT and MAX SAT. In: Proceedings of WAOA, pp. 27–40 (2005)

  4. Battiti, R., Protasi, M.: Reactive search, a history-sensitive heuristic for MAX-SAT. ACM J Exp Algorithms. 2(2) (1997). Artical no 2

  5. Bellare, M., Goldreich, O., Sudan, M.: Free bits, PCPs and non-approximability—towards tight results. Soc. Ind. Appl. Math. 27(3), 804–915 (1998)

    MATH  MathSciNet  Google Scholar 

  6. Borchers, B., Furman, J.: A two-phase exact algorithm for MAX-SAT and weighted MAX-SAT problems. J. Comb. Optim. 2, 299–306 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cohen, D., Cooper, M., Jeavons, P.: A complete characterization of complexity for boolean constraint optimization problems. Lect. Notes Comput. Sci. 3258, 212–226 (2004)

    Article  Google Scholar 

  8. Costello, K.P., Shapira, A., Tetali, P.: On randomizing two derandomized greedy algorithms. J. Combin. 1(3–4), 265–283 (2010)

    Google Scholar 

  9. Feo, T.A., Resende, M.G.C., Smith, S.H.: A greedy randomized adaptive search procedure for maximum independent set. Operat. Res. 42, 860–878 (1994)

    Article  MATH  Google Scholar 

  10. Festa, P., Pardalos, P.M., Pitsoulis, L.S., Resende, M.G.C.: GRASP with Path Relinking for the weighted MAXSAT problem. ACM J. Exp. Algorithm 11, 1–16 (2006)

    MathSciNet  Google Scholar 

  11. Garey, M., Johnson, D.S.: Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman and Company, New York (1979)

    MATH  Google Scholar 

  12. Gu, J.: Local search for satisfiability (SAT) problem. IEEE Trans. Syst. Man Cybern. Part A 23(4), 1108–1129 (1993)

    Google Scholar 

  13. Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9(3), 256–278 (1974)

    Article  MATH  Google Scholar 

  14. Joy, S., Mitchell, J., Borchers, B.: A branch and cut algorithm for MAX-SAT and weighted MAX-SAT. In: Proceedings of the DIMACS Workshop on Satisfiability: theory and Applications. Rutgers University, NJ (1996)

  15. de Klerk, E., Warners, J.P.: Semidefinite programming approaches for MAX-2-SAT and MAX-3-SAT: computational perspectives. Technical report, Delft, The Netherlands (1998)

  16. Mills, P., Tsang, E.: Guided local search for solving SAT and weighted MAX-SAT problems. J. Autom. Reason. 24, 205–223 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Resende, M.G.C., Ribeiro, C.C.: Greedy randomized adaptive search procedures: advances and applications. In: M. Gendreau and J.-Y. Potvin (eds.) Handbook of Metaheuristics, 2nd edn., pp. 281–317. Springer, New York (2010)

  18. Resende, M.G.C.: Algorithm source codes distribution. http://research.att.com/~/data

  19. Resende, M.G.C., Pitsoulis, L.S., Pardalos, P.M.: Approximate solution of weighted MAX-SAT problems using GRASP. In: Du, D.-Z., Gu, J., Pardalos, P. M. (eds.), Satisfiability Problem: Theory and Applications, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pp. 393–405. American Mathematical Society, Providence (1997)

  20. Resende, M.G.C., Pitsoulis, L.S., Pardalos, P.M.: Fortran subroutines for computing approximate solutions of weighted MAX-SAT problems using GRASP. Discret. Appl. Math. 100, 95–113 (2000)

    Article  MATH  Google Scholar 

  21. Resende, M.G.C.: Test problems distribution. http://research.att.com/~/data

  22. Selman, B., Levesque, H., Mitchell, D.: A new method for solving hard satisfiability problems. In: Proceedings of the Tenth National Conference on Artificial Intelligence (AAAI-92), pp. 440–446. San Jose, CA (1992)

  23. Shylo, O.V., Prokopyev, O.A., Shylo, V.P.: Solving weighted MAX-SAT via global equilibrium search. Oper. Res. Lett. 36, 434–438 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. Wallace, R., Freuder, E.: Comparative studies of constraint satisfaction and Davis-Putnam algorithms for maximum satisfiability problems. In: Johnson, D., Trick, M. (eds.) Cliques, Coloring and Satisfiability, vol. 26, pp. 587–615. American Mathematical Society, Providence (1996)

  25. Xing, Z., Zhang, W.: Efficient strategies for (weighted) maximum satisfiability. In: Proceedings of CP- 2004, pp. 690–705. Toronto, Canada (2004)

  26. Spears, W.M.: Simulated annealing for hard satisfiability problems. In: Johnson, D.S., Trick, M.A. (eds.) Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, DIMACS Series in . Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, Providence 26, 533–555 (1996)

  27. Zhu, W.X., Ali, M.M.: Discrete dynamic convexized method for nonlinearly constrained nonlinear integer programming. Comput. Oper. Res. 36, 2723–2728 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  28. Zhu, W.X., Fan, H.: A discrete dynamic convexized method for nonlinear integer programming. J. Comput. Appl. Math. 223(1), 356–373 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. Chen, J., Zhu, W.X.: A dynamic convexized method for VLSI circuit partitioning. Optim. Methods Softw (to appear)

  30. Zhu, W.X., Lin, G., Ali, M.M.: Max-k-Cut by the discrete dynamic convexized method, INFORMS. J. Comput (to appear)

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Authors and Affiliations

  1. Center for Discrete Mathematics and Theoretical Computer Science, Fuzhou University, Fuzhou , 350108, China

    Wenxing Zhu & Yuanhui Yan

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  1. Wenxing Zhu
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  2. Yuanhui Yan
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Correspondence to Wenxing Zhu.

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This work was supported in part by the National Science Foundation of China (NSFC) under Grants 61170308 and 10931003, in part by the National Key Basic Research Special Foundation (NKBRSF) of China under Grant 2011CB808000.

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Zhu, W., Yan, Y. Solving the weighted MAX-SAT problem using the dynamic convexized method. Optim Lett 8, 359–374 (2014). https://doi.org/10.1007/s11590-012-0583-4

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  • DOI: https://doi.org/10.1007/s11590-012-0583-4

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