Solving Max-SAT as Weighted CSP

  • Simon de Givry
  • Javier Larrosa
  • Pedro Meseguer
  • Thomas Schiex
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2833)


For the last ten years, a significant amount of work in the constraint community has been devoted to the improvement of complete methods for solving soft constraints networks. We wanted to see how recent progress in the weighted CSP (WCSP) field could compete with other approaches in related fields. One of these fields is propositional logic and the well-known Max-SAT problem. In this paper, we show how Max-SAT can be encoded as a weighted constraint network, either directly or using a dual encoding. We then solve Max-SAT instances using state-of-the-art algorithms for weighted Max-CSP, dedicated Max-SAT solvers and the state-of-the-art MIP solver CPLEX. The results show that, despite a limited adaptation to CNF structure, WCSP-solver based methods are competitive with existing methods and can even outperform them, especially on the hardest, most over-constrained problems.


Mixed Integer Linear Program Constraint Satisfaction Problem Conjunctive Normal Form Local Consistency Conjunctive Normal Form Formula 
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.
    Aloul, F., Ramani, A., Markov, I., Sakallah, K.: Pbs: A backtracksearch pseudo-boolean solver and optimizer. In: Symposium on the Theory and Applications of Satisfiability Testing (SAT), Cincinnati (OH), pp. 346–353 (2002)Google Scholar
  2. 2.
    Barth, P.: A davis-putnam based enumeration algorithm for linear pseudo-boolean optimization. Tech. Rep. MPI-I-95-2-003, Max-Planck Institut Für Informatik (1995)Google Scholar
  3. 3.
    Borchers, B., Mitchell, J., Joy, S.: A branch-and-cut algorithm for MAXSAT and weighted MAX-SAT. In: Du, D., Gu, J., Pardalos, P. (eds.) Satisfiability Problem: Theory and Applications. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 35, pp. 519–536. AMS, Providence (1997)Google Scholar
  4. 4.
    Borning, A., Mahert, M., Martindale, A., Wilson, M.: Constraint hierarchies and logic programming. In: Int. conf. on logic programming, pp. 149–164 (1989)Google Scholar
  5. 5.
    Dixon, H., Ginsberg, M.: Inference methods for a pseudo-boolean satisfiability solver. In: Proceedings of the Eighteenth National Conference on Artificial Intelligence (AAAI 2002), pp. 635–640 (2002)Google Scholar
  6. 6.
    Freuder, E., Wallace, R.: Partial constraint satisfaction. Artificial Intelligence 58, 21–70 (1992)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Freuder, E.C.: Partial constraint satisfaction. In: Proc. of the 11th IJCAI, Detroit, MI, pp. 278–283 (1989)Google Scholar
  8. 8.
    Gramm, J., Hirsch, E.A., Niedermeier, R., Rossmanith, P.: New worstcase upper bounds for MAX-2-SAT with application to MAX-CUT. Tech. Rep. TR00-037, Electronic Colloquium on Computational Complexity (2000)Google Scholar
  9. 9.
    Gramm, J., Niedermeier, R.: Faster exact solutions for max2Sat. In: Bongiovanni, G., Petreschi, R., Gambosi, G. (eds.) CIAC 2000. LNCS, vol. 1767, pp. 174–186. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  10. 10.
    Hansen, P., Jaumard, B.: Algorithms for the maximum satisfiability problem. Computing 44, 279–303 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    ILOG. Cplex solver 8.1.0 (2002),
  12. 12.
    Johnson, D.S., Trick, M.A. (eds.): Second DIMACS implementation challenge:cliques, coloring and satisfiability. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 26. AMS, Providence (1996)zbMATHGoogle Scholar
  13. 13.
    Larrosa, J.: On arc and node consistency in weighted CSP. In: Proc. AAAI 2002, Edmondton, CA (2002)Google Scholar
  14. 14.
    Larrosa, J., Dechter, R.: CP 2000. LNCS, vol. 1894, p. 531. Springer, Heidelberg (2000)Google Scholar
  15. 15.
    Larrosa, J., Schiex, T.: In: the quest of the best form of local consistency for weighted CSP. In: Proc. of the 18th IJCAI, Acapulco, Mexico (August 2003), see
  16. 16.
    Moskewicz, M., Madigan, C., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient sat solver. In: 38th Design Automation Conference (DAC 2001), June 2001, pp. 530–535 (2001)Google Scholar
  17. 17.
    Resende, M., Pitsoulis, L., Pardalos, P.: Approximate solution of weighted max-SAT problems using GRASP. In: Du, D., Gu, J., Pardalos, P. (eds.) Satisfiability problem: Theory and Applications, pp. 393–405. AMS, Providence (1997)Google Scholar
  18. 18.
    Rosenfeld, A., Hummel, R., Zucker, S.: Scene labeling by relaxation operations. IEEE Trans. on Systems, Man, and Cybernetics 6(6), 173–184 (1976)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Schiex, T. Arc cohérence pour contraintes molles. In: Actes de JNPC 2000, Marseille (June 2000) Google Scholar
  20. 20.
    Schiex, T.: Arc consistency for soft constraints. In: Dechter, R. (ed.) CP 2000. LNCS, vol. 1894, pp. 411–424. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  21. 21.
    Schiex, T., Fargier, H., Verfaillie, G.: Valued constraint satisfaction problems: hard and easy problems. In: Proc. of the 14th IJCAI, Montréal, Canada, August 1995, pp. 631–637 (1995)Google Scholar
  22. 22.
    Selman, B., Kautz, H., Cohen, B.: Noise strategies for improving local search. In: Proc. of AAAI 1994, Seattle, WA, pp. 337–343 (1994)Google Scholar
  23. 23.
  24. 24.
    van Hentenryck, P., Deville, Y.: The cardinality operator: A new logical connective for constraint logic programming. In: Proc. of the 8th international conference on logic programming, Paris, France (June 1991)Google Scholar
  25. 25.
    Whittemore, J., Kim, J., Sakallah, K.: SATIRE: A new incremental satisfiability engine. In: Proceedings of the 38th conference on Design automation, Las Vegas, NV, pp. 542–545. ACM, New York (2001)Google Scholar
  26. 26.
    Xu, H., Rutenbar, R.A., Sakallah, K.: sub-SAT: A formulation for relaxed boolean satisfiability with applications in routing. In: Proc. Int. Symp. on Physical Design, San Diego, CA (April 2002)Google Scholar
  27. 27.
    Zhang, W.: Phase transitions and backbones of 3-SAT and maximum 3-SAT. In: Walsh, T. (ed.) CP 2001. LNCS, vol. 2239, pp. 153–167. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Simon de Givry
    • 1
  • Javier Larrosa
    • 2
  • Pedro Meseguer
    • 3
  • Thomas Schiex
    • 1
  1. 1.INRAToulouseFrance
  2. 2.Dep. LSIUPCBarcelonaSpain
  3. 3.IIIA-CSICCampus UABBellaterraSpain

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