Solving Over-Constrained Problems with SAT Technology

  • Josep Argelich
  • Felip Manyà
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3569)


We present a new generic problem solving approach for over-constrained problems based on Max-SAT. We first define a clausal form formalism that deals with blocks of clauses instead of individual clauses, and that allows one to declare each block either as hard (i.e., must be satisfied by any solution) or soft (i.e., can be violated by some solution). We then present two Max-SAT solvers that find a truth assignment that satisfies all the hard blocks of clauses and the maximum number of soft blocks of clauses. Our solvers are branch and bound algorithms equipped with original lazy data structures; the first one incorporates static variable selection heuristics while the second one incorporates dynamic variable selection heuristics. Finally, we present an experimental investigation to assess the performance of our approach on a representative sample of instances (random 2-SAT, Max-CSP, and graph coloring).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aloul, F., Ramani, A., Markov, I., Sakallah, K.: PBS: A backtrack search pseudo-Boolean solver. In: Symposium on the Theory and Applications of Satisfiability Testing, SAT 2002 (2002)Google Scholar
  2. 2.
    Alsinet, T., Manyà, F., Planes, J.: Improved branch and bound algorithms for Max-SAT. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 162–171. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Borchers, B., Furman, J.: A two-phase exact algorithm for MAX-SAT and weighted MAX-SAT problems. Journal of Combinatorial Optimization 2, 299–306 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cha, B., Iwama, K., Kambayashi, Y., Miyazaki, S.: Local search algorithms for partial MAXSAT. In: Proceedings of the 14th National Conference on Artificial Intelligence, AAAI 1997, Providence/RI, USA, pp. 263–268. AAAI Press, Menlo Park (1997)Google Scholar
  5. 5.
    Culberson, J.: Graph coloring page: The flat graph generator (1995), See
  6. 6.
    Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. Communications of the ACM 5, 394–397 (1962)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gent, I.P.: Arc consistency in SAT. In: Proceedings of the 15th European Conference on Artificial Intelligence (ECAI), Lyon, France, pp. 121–125 (2002)Google Scholar
  8. 8.
    Jiang, Y., Kautz, H., Selman, B.: Solving problems with hard and soft constraints using a stochastic algorithm for MAX-SAT. In: Proceedings of the 1st International Workshop on Artificial Intelligence and Operations Research (1995)Google Scholar
  9. 9.
    Kasif, S.: On the parallel complexity of discrete relaxation in constraint satisfaction networks. Artificial Intelligence 45, 275–286 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Larrosa, J.: Algorithms and Heuristics for Total and Partial Constraint Satisfaction. PhD thesis, FIB, Universitat Politècnica de Catalunya, Barcelona (1998)Google Scholar
  11. 11.
    Loveland, D.W.: Automated Theorem Proving. A Logical Basis. Fundamental Studies in Computer Science, vol. 6. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  12. 12.
    Meseguer, P., Bouhmala, N., Bouzoubaa, T., Irgens, M., Sánchez, M.: Current approaches for solving over-constrained problems. Constraints 8(1), 9–39 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Moskewicz, M., Madigan, C., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient sat solver. In: 39th Design Automation Conference (2001)Google Scholar
  14. 14.
    Selman, B., Levesque, H., Mitchell, D.: A new method for solving hard satisfiability problems. In: Proceedings of the 10th National Conference on Artificial Intelligence, AAAI 1992, San Jose/CA, USA, pp. 440–446. AAAI Press, Menlo Park (1992)Google Scholar
  15. 15.
    Shen, H., Zhang, H.: Study of lower bound functions for max-2-sat. In: Proceedings of AAAI 2004, pp. 185–190 (2004)Google Scholar
  16. 16.
    Smith, B., Dyer, M.: Locating the phase transition in binary constraint satisfaction problems. Artificial Intelligence 81, 155–181 (1996)CrossRefMathSciNetGoogle Scholar
  17. 17.
    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 (1996)Google Scholar
  18. 18.
    Xing, Z., Zhang, W.: Efficient strategies for (weighted) maximum satisfiability. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 690–705. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Josep Argelich
    • 1
  • Felip Manyà
    • 2
  1. 1.Computer Science DepartmentUniversitat de LleidaLleidaSpain
  2. 2.Artificial Intelligence Research Institute (IIIA-CSIC)BellaterraSpain

Personalised recommendations