Improving WPM2 for (Weighted) Partial MaxSAT

  • Carlos Ansótegui
  • Maria Luisa Bonet
  • Joel Gabàs
  • Jordi Levy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8124)


Weighted Partial MaxSAT (WPMS) is an optimization variant of the Satisfiability (SAT) problem. Several combinatorial optimization problems can be translated into WPMS. In this paper we extend the state-of-the-art WPM2 algorithm by adding several improvements, and implement it on top of an SMT solver. In particular, we show that by focusing search on solving to optimality subformulas of the original WPMS instance we increase the efficiency of WPM2. From the experimental evaluation we conducted on the PMS and WPMS instances at the 2012 MaxSAT Evaluation, we can conclude that the new approach is both the best performing for industrial instances, and for the union of industrial and crafted instances.


Soft Clause Partial MaxSAT MaxSAT Algorithm MaxSAT Problem Linear Integer Arithmetic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Ansótegui, C., Bofill, M., Palahí, M., Suy, J., Villaret, M.: A Proposal for Solving Weighted CSPs with SMT. In: Proceedings of the 10th International Workshop on Constraint Modelling and Reformulation (ModRef 2011), pp. 5–19 (2011)Google Scholar
  2. 2.
    Ansótegui, C., Bonet, M.L., Gabàs, J., Levy, J.: Improving sat-based weighted maxsat solvers. In: Milano, M. (ed.) CP 2012. LNCS, vol. 7514, pp. 86–101. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  3. 3.
    Ansótegui, C., Bonet, M.L., Levy, J.: On solving MaxSAT through SAT. In: Proc. of the 12th Int. Conf. of the Catalan Association for Artificial Intelligence (CCIA 2009), pp. 284–292 (2009)Google Scholar
  4. 4.
    Ansótegui, C., Bonet, M.L., Levy, J.: Solving (weighted) partial MaxSAT through satisfiability testing. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 427–440. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Ansotegui, C., Bonet, M.L., Levy, J.: A new algorithm for weighted partial maxsat. In: Proc. the 24th National Conference on Artificial Intelligence (AAAI 2010) (2010)Google Scholar
  6. 6.
    Ansótegui, C., Bonet, M.L., Levy, J.: Sat-based maxsat algorithms. Artif. Intell. 196, 77–105 (2013)CrossRefzbMATHGoogle Scholar
  7. 7.
    Ansotegui, C., Gabas, J.: Solving maxsat with mip. In: CPAIOR (2013)Google Scholar
  8. 8.
    Bailleux, O., Boufkhad, Y., Roussel, O.: New encodings of pseudo-boolean constraints into CNF. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 181–194. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Barrett, C., Stump, A., Tinelli, C.: The Satisfiability Modulo Theories Library (SMT-LIB) (2010),
  10. 10.
    Berre, D.L.: Sat4j, a satisfiability library for java (2006),
  11. 11.
    Borchers, B., Furman, J.: A two-phase exact algorithm for max-sat and weighted max-sat problems. J. Comb. Optim. 2(4), 299–306 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cimatti, A., Franzén, A., Griggio, A., Sebastiani, R., Stenico, C.: Satisfiability modulo the theory of costs: Foundations and applications. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 99–113. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  13. 13.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press (2009)Google Scholar
  14. 14.
    Davies, J., Bacchus, F.: Solving MAXSAT by solving a sequence of simpler SAT instances. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 225–239. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  15. 15.
    Eén, N., Sörensson, N.: Translating pseudo-boolean constraints into SAT. JSAT 2(1-4), 1–26 (2006)zbMATHGoogle Scholar
  16. 16.
    Fu, Z., Malik, S.: On solving the partial MAX-SAT problem. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 252–265. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Heras, F., Larrosa, J., Oliveras, A.: MiniMaxSat: A new weighted Max-SAT solver. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 41–55. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  18. 18.
    Heras, F., Larrosa, J., Oliveras, A.: Minimaxsat: An efficient weighted max-sat solver. J. Artif. Intell. Res (JAIR) 31, 1–32 (2008)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Heras, F., Morgado, A., Marques-Silva, J.: Core-guided binary search algorithms for maximum satisfiability. In: Proc. the 25th National Conference on Artificial Intelligence (AAAI 2011) (2011)Google Scholar
  20. 20.
    Honjyo, K., Tanjo, T.: Shinmaxsat, a Weighted Partial Max-SAT solver inspired by MiniSat+, Information Science and Technology Center, Kobe UniversityGoogle Scholar
  21. 21.
    Koshimura, M., Zhang, T., Fujita, H., Hasegawa, R.: Qmaxsat: A partial max-sat solver. JSAT 8(1/2), 95–100 (2012)MathSciNetGoogle Scholar
  22. 22.
    Kügel, A.: Improved exact solver for the weighted max-sat problem (to appear)Google Scholar
  23. 23.
    Larrosa, J., Heras, F., de Givry, S.: A logical approach to efficient max-sat solving. Artif. Intell. 172(2-3), 204–233 (2008)CrossRefzbMATHGoogle Scholar
  24. 24.
    Li, C.M., Manyà, F., Mohamedou, N., Planes, J.: Exploiting cycle structures in Max-SAT. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 467–480. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  25. 25.
    Li, C.M., Manyà, F., Planes, J.: New inference rules for Max-SAT. J. Artif. Intell. Res (JAIR) 30, 321–359 (2007)Google Scholar
  26. 26.
    Lin, H., Su, K.: Exploiting inference rules to compute lower bounds for Max-SAT solving. In: IJCAI 2007, pp. 2334–2339 (2007)Google Scholar
  27. 27.
    Lin, H., Su, K., Li, C.M.: Within-problem learning for efficient lower bound computation in Max-SAT solving. In: Proc. the 23rd National Conference on Artificial Intelligence (AAAI 2008), pp. 351–356 (2008)Google Scholar
  28. 28.
    Manquinho, V., Marques-Silva, J., Planes, J.: Algorithms for weighted boolean optimization. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 495–508. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  29. 29.
    Manquinho, V.M., Martins, R., Lynce, I.: Improving unsatisfiability-based algorithms for boolean optimization. In: Strichman, O., Szeider, S. (eds.) SAT 2010. LNCS, vol. 6175, pp. 181–193. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  30. 30.
    Marques-Silva, J., Argelich, J., Graça, A., Lynce, I.: Boolean lexicographic optimization: algorithms & applications. Ann. Math. Artif. Intell. 62(3-4), 317–343 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Martins, R., Manquinho, V.M., Lynce, I.: Exploiting cardinality encodings in parallel maximum satisfiability. In: ICTAI, pp. 313–320 (2011)Google Scholar
  32. 32.
    Martins, R., Manquinho, V., Lynce, I.: Clause sharing in parallel MaxSAT. In: Hamadi, Y., Schoenauer, M. (eds.) LION 2012. LNCS, vol. 7219, pp. 455–460. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  33. 33.
    Morgado, A., Heras, F., Marques-Silva, J.: Improvements to core-guided binary search for MaxSAT. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 284–297. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  34. 34.
    Nieuwenhuis, R., Oliveras, A.: On SAT modulo theories and optimization problems. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 156–169. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  35. 35.
    Sebastiani, R.: Lazy Satisfiability Modulo Theories. Journal on Satisfiability, Boolean Modeling and Computation 3(3-4), 141–224 (2007)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Silva, J.P.M., Sakallah, K.A.: Grasp: A search algorithm for propositional satisfiability. IEEE Trans. Computers 48(5), 506–521 (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Carlos Ansótegui
    • 1
  • Maria Luisa Bonet
    • 2
  • Joel Gabàs
    • 1
  • Jordi Levy
    • 3
  1. 1.DIEIUniv. de LleidaSpain
  2. 2.LSI, UPCSpain
  3. 3.IIIA-CSICSpain

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