Towards an Optimal CNF Encoding of Boolean Cardinality Constraints

  • Carsten Sinz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3709)


We consider the problem of encoding Boolean cardinality constraints in conjunctive normal form (CNF). Boolean cardinality constraints are formulae expressing that at most (resp. at least) k out of n propositional variables are true. We give two novel encodings that improve upon existing results, one which requires only 7n clauses and 2n auxiliary variables, and another one demanding \(\mathcal{O}(n \cdot k)\) clauses, but with the advantage that inconsistencies can be detected in linear time by unit propagation alone. Moreover, we prove a linear lower bound on the number of required clauses for any such encoding.


Boolean Function Auxiliary Variable Unit Propagation Conjunctive Normal Form Binary Number 
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.
    Küchlin, W., Sinz, C.: Proving consistency assertions for automotive product data management. J. Automated Reasoning 24, 145–163 (2000)zbMATHCrossRefGoogle Scholar
  2. 2.
    Cabon, B., de Givry, S., Lobjois, L., Schiex, T., Warners, J.P.: Radio link frequency assignment. Constraints 4, 79–89 (1999)zbMATHCrossRefGoogle Scholar
  3. 3.
    Bailleux, O., Boufkhad, Y.: Efficient CNF encoding of boolean cardinality constraints. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 108–122. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  4. 4.
    Warners, J.P.: A linear-time transformation of linear inequalities into conjunctive normal form. Inf. Process. Lett. 68, 63–69 (1998)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Tseitin, G.S.: On the complexity of derivation in propositional calculus. In: Slisenko, A.O. (ed.) Studies in Constructive Mathematics and Mathematical Logic, pp. 115–125 (1970)Google Scholar
  6. 6.
    Jackson, P., Sheridan, D.: The optimality of a fast CNF conversion and its use with SAT. Technical Report APES-82-2004, APES Research Group (2004), Available from
  7. 7.
    Muller, D.E., Preparata, F.P.: Bounds to complexities of networks for sorting and for switching. J. ACM 22, 195–201 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Wegener, I.: The Complexity of Boolean Functions. Wiley-Teubner, Chichester (1987)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Carsten Sinz
    • 1
  1. 1.Institute for Formal Models and VerificationJohannes Kepler University LinzLinzAustria

Personalised recommendations