Advertisement

Efficient CNF Encoding of Boolean Cardinality Constraints

  • Olivier Bailleux
  • Yacine Boufkhad
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2833)

Abstract

In this paper, we address the encoding into CNF clauses of Boolean cardinality constraints that arise in many practical applications. The proposed encoding is efficient with respect to unit propagation, which is implemented in almost all complete CNF satisfiability solvers. We prove the practical efficiency of this encoding on some problems arising in discrete tomography that involve many cardinality constraints. This encoding is also used together with a trivial variable elimination in order to re-encode parity learning benchmarks so that a simple Davis and Putnam procedure can solve them.

Keywords

Encode Scheme Unit Propagation Boolean Variable Cardinality Constraint Unit Clause 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barcucci, E., Dellungo, A., Nivat, M., Pinzani, R.: Reconstructing convex polyominoes from horizontal and vertical projections. Theoret. Comput. Sci, pp. 321– 347 (1996)Google Scholar
  2. 2.
    Beldiceanu, N., Contjean, E.: Introducing global constraints in CHIP. Mathematical and Computer Modelling 12, 97–123 (1994)CrossRefGoogle Scholar
  3. 3.
    Boufkhad, Y., Dubois, O., Nivat, M.: Reconstructing (h, v)-convex 2-dimensional patterns of objects from approximate horizontal and vertical projections. Theoret. Comput. Sci. 290(3), 1647–1664 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chang, S.: The reconstruction of binary patterns from their projections. Comm. ACM, 21–25 (1971)Google Scholar
  5. 5.
    Cosytec, S.A.: CHIP C++ Library. Reference Manual, Version 5.4 (October 2001)Google Scholar
  6. 6.
  7. 7.
    Davis, M., Logemann, G., Loveland, D.: A machine program for theorem proving. Communications of the ACM 5, 394–397 (1962)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Davis, M., Putnam, H.: A computing procedure for quantification theory. Journal of the ACM 7, 201–215 (1960)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dellungo, A.: Polyominoes defined by two vectors. Theoret. Comput. Sci. 127, 187–198 (1994)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Gardner, R.G., Gritzmann, P., Prangenberg, D.: Ont the computational complexity of reconstructing lattice sets from their x-rays. Discrete Mathematics, 45–71 (1999)Google Scholar
  11. 11.
    Gent, P.: Arc consistency in sat. In: Proceedings of the Fifteenth European Conference on Artificial Intelligence, (ECAI 2002) (2002)Google Scholar
  12. 12.
    Johnson, D., Trick, M. (eds.): DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 26. American Mathematical Society, Providence (1996)Google Scholar
  13. 13.
    Kuba, A.: The reconstruction of two-directionaly connected binary patterns. Comput. Graph. Image Process 27, 249–265 (1984)CrossRefGoogle Scholar
  14. 14.
    Li, C.: Integrating equivalency reasoning into davis-putnam procedure. In: AAAI: 17th National Conference on Artificial Intelligence. AAAI / MIT Press (2000)Google Scholar
  15. 15.
    Moskewicz, M., Madigan, C., Zhao, L., Zhang, L., Malik, S.: Chaff: Engineering an efficient sat solver. In: 39th Design Automation Conference (June 2001)Google Scholar
  16. 16.
    Régin, J.-C.: Generalized arc consistency for global cardinality constraint. In: AAAI 1996, pp. 209–215 (1996)Google Scholar
  17. 17.
    Ryser, H.: Combinatorial Mathematics. The Carus Mathematical Monographs (1963)Google Scholar
  18. 18.
    Selman, B., Kautz, H.A., Mcallester, D.A.: Ten challenges in propositional reasoning and search. In: Proceedings of the Fifteenth International Joint Conference on Artificial Intelligence (IJCAI 1997), pp. 50–54 (1997)Google Scholar
  19. 19.
    Wang, Y.: Characterization of binary patterns and their projections. IEEE Trans. Compt.  C-24, 1032–1035 (1975)zbMATHCrossRefGoogle Scholar
  20. 20.
    Warners, J., van Maaren, H.: A two phase algorithm for solving a class of hard satisfiability problems. Op. Res. Lett. 23(3-5), 81–88 (1999)CrossRefGoogle Scholar
  21. 21.
    Woeginger, G.: The reconstruction of polyominoes from their orthogonal projections, tech. rep., TU Graz (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Olivier Bailleux
    • 1
  • Yacine Boufkhad
    • 2
  1. 1.LERSIAUniversité de BourgogneDijon
  2. 2.LIAFAUniversité Paris 7Paris

Personalised recommendations