A Two Level Local Search for MAX-SAT Problems with Hard and Soft Constraints

  • John Thornton
  • Stuart Bain
  • Abdul Sattar
  • Duc Nghia Pham
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2557)


Local search techniques have attracted considerable interest in the AI community since the development of GSAT for solving large propositional SAT problems. Newer SAT techniques, such as the Discrete Lagrangian Method (DLM), have further improved on GSAT and can also be applied to general constraint satisfaction and optimisation. However, little work has applied local search to MAX-SAT problems with hard and soft constraints. As many real-world problems are best represented by hard (mandatory) and soft (desirable) constraints, the development of effective local search heuristics for this domain is of significant practical importance.

This paper extends previous work on dynamic constraint weighting by introducing a two-level heuristic that switches search strategy according to whether a current solution contains unsatisfied hard constraints. Using constraint weighting techniques derived from DLM to satisfy hard constraints, we apply a Tabu search to optimise the soft constraint violations. These two heuristics are further combined with a dynamic hard constraint multiplier that changes the relative importance of the hard constraints during the search. We empirically evaluate this new algorithm using a set of randomly generated 3-SAT problems of various sizes and difficulty, and in comparison with various state-of-the-art SAT techniques. The results indicate that our dynamic, two-level heuristic offers significant performance benefits over the standard SAT approaches.


Local Search Tabu Search Soft Constraint Hard Constraint Tabu List 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    U. Bistarelli, S. Montanari and F. Rossi. Semiring-based constraint solving and optimization. Journal of ACM, 44(2):201–236, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A. Borning, B. Freeman-Benson, and M. Wilson. Constraint hierarchies. Lisp and Symbolic Computation, 5(3):223–270, 1992.CrossRefGoogle Scholar
  3. 3.
    B. Cha, K. Iwama, Y. Kambayashi, and S. Miyazaki. Local search algorithms for partial MAX-SAT. In Proceedings of the Fourteenth National Conference on Artificial Intelligence (AAAI-97), pages 332–337, 1997.Google Scholar
  4. 4.
    J. Frank. Learning short term weights for GSAT. In Proceedings of theF ourteenth National Conference on Artificial Intelligence (AAAI-97), pages 384–389, 1997.Google Scholar
  5. 5.
    E. Freuder and R. Wallace. Partial constraint satisfaction. Artificial Intelligence, 58(1):21–70, 1992.CrossRefMathSciNetGoogle Scholar
  6. 6.
    F. Glover. Tabu search: Part 1. ORSA Journal on Computing, 1(3):190–206, 1989.zbMATHGoogle Scholar
  7. 7.
    M. Heinz, L. Fong, L. Chong, S. Ping, J. Walser, and R. Yap. Solving hierarchical constraints over finite domains. In Proceedings of theSixth International Symposium on Artificial Intelligence and Mathematics, 2000.Google Scholar
  8. 8.
    H. Hoos. On the run-time behavior of stochastic local search algorithms for SAT. In Proceedings of the Sixteenth National Conference on Artificial Intelligence (AAAI-99), pages 661–666, 1999.Google Scholar
  9. 9.
    H. Jiang, Y. Kautz and B. Selman. Solving problems with hard and soft constraints using a stochastic algorithm for MAX-SAT. In First International Joint Workshop on Artificial Intelligence and Operations Research, 1995.Google Scholar
  10. 10.
    D. McAllester, B. Selman, and H. Kautz. Evidence for invariance in local search. In Proceedings of the Fourteenth National Conference on Artificial Intelligence (AAAI-97), pages 321–326, 1997.Google Scholar
  11. 11.
    P. Mills and E. Tsang. Guided local search applied to the satisfiability (SAT) problem. In Proceedings of the15th National Conferenceof theA ustralian Society for Operations Research (ASOR’99), pages 872–883, 1999.Google Scholar
  12. 12.
    P. Mills and E. Tsang. Guided local search for solving SAT and weighted MAXSAT problems. Journal of Automated Reasoning, 24:205–223, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    P. Morris. The Breakout method for escaping local minima. In Proceedings of the Eleventh National Conference on Artificial Intelligence (AAAI-93), pages 40–45, 1993.Google Scholar
  14. 14.
    A. Schaerf. Tabu search for large high school timetabling problems. In Proceedings of the Thirteenth National Conference on Artificial Intelligence (AAAI-96), pages 363–368, 1996.Google Scholar
  15. 15.
    D. Schuurmans and F. Southey. Local search characteristics of incomplete SAT procedures. In Proceedings of the Seventeenth National Conference on Artificial Intelligence (AAAI-00), pages 297-302, 2000.Google Scholar
  16. 16.
    Y. Shang and B. Wah. A discrete Lagrangian-based global search method for solving satisfiability problems. J. Global Optimization, 12:61–99, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    J. Thornton, W. Pullan, and J. Terry. Towards fewer parameters for SAT clause weighting algorithms. In Proceedings of the Fifteenth Australian Joint Conference on Artificial Intelligence (AI’2002), To appear, 2002.Google Scholar
  18. 18.
    J. Thornton and A. Sattar. Dynamic constraint weighting for over-constrained problems. In Proceedings of the Fifth Pacific Rim Conference on Artificial Intelligence (PRICAI-98), pages 377–388, 1998.Google Scholar
  19. 19.
    Wu Z. TheThe ory and Applications of DiscreteConstr ained Optimization using LagrangeMultiplie rs. PhD thesis, Department of Computer Science, University of Illinois, 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • John Thornton
    • 1
  • Stuart Bain
    • 1
  • Abdul Sattar
    • 1
  • Duc Nghia Pham
    • 1
  1. 1.School of Information TechnologyGriffith University Gold CoastSouthportAustralia

Personalised recommendations