Journal of Combinatorial Optimization

, Volume 31, Issue 3, pp 1061–1089 | Cite as

Cardinality constraints and systems of restricted representatives



Cardinality constraints have received considerable attention from the Constraint Programming community as (so-called) global constraints that appear in the formulation of several real-life problems, while also having an interesting combinatorial structure. After discussing the relation of cardinality constraints with well-known combinatorial problems (e.g., systems of restricted representatives), we study the polytope defined by the convex hull of vectors satisfying two such constraints, in the case where all variables share a common domain. We provide families of facet-defining inequalities that are polytime separable, together with a condition for when these families of inequalities define a convex hull relaxation. Our results also hold for the case of a single such constraint.


Global cardinality constraint Polyhedral combinatorics Constraint programming 



We would like to thank a reviewer, whose valuable and constructive comments have helped simplifying certain proofs and improving the readability of several others. This research has been co-financed by the European Union (European Social Fund-ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: Thales. Investing in knowledge society through the European Social Fund.


  1. Balas E, Bockmayr A, Pisaruk N, Wolsey L (2004) On unions and dominants of polytopes. Math Program 99:223–239MathSciNetCrossRefMATHGoogle Scholar
  2. Bergman D, Hooker JN (2014) Graph coloring inequalities from all-different systems. Constraints 19:404–433MathSciNetCrossRefMATHGoogle Scholar
  3. Bulatov AA, Marx D (2010) Constraint satisfaction problems and global cardinality constraints. Commun ACM 53:99–106CrossRefGoogle Scholar
  4. Ford LR Jr, Fulkerson DR (1958) Network flow and systems of representatives. Can J Math 10:78–84MathSciNetCrossRefMATHGoogle Scholar
  5. Grötschel M, Lovász L, Schrijver A (1981) The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1:169–197MathSciNetCrossRefMATHGoogle Scholar
  6. Hooker JN (2012) Integrated methods for optimization, international series in operations research & management science, vol 170. SpringerGoogle Scholar
  7. Hooker JN, Yan H (2002) A relaxation of the cumulative constraint. Proceedings of CP2002. Lect. Notes Comput. Sci., vol. 2470, pp 80–92Google Scholar
  8. Katriel I, Thiel S (2005) Complete bound consistency for the global cardinality constraint. Constraints 10:191–217MathSciNetCrossRefMATHGoogle Scholar
  9. Kaya LG, Hooker JN (2011) The circuit polytope.
  10. Kutz M, Elbassioni K, Katriel I, Mahajan M (2008) Simultaneous matchings: hardness and approximation. J Comput Syst Sci 74:884–897MathSciNetCrossRefMATHGoogle Scholar
  11. Magos D, Mourtos I (2011) On the facial structure of the AllDifferent system. SIAM J Discrete Math 25:130–158MathSciNetCrossRefMATHGoogle Scholar
  12. Magos D, Mourtos I, Appa G (2012) A polyhedral approach to the alldifferent system. Math Program 132:209–260MathSciNetCrossRefMATHGoogle Scholar
  13. Milano M, Ottosson G, Refalo P, Thorsteinsson E (2002) The role of integer programming techniques in constraint programming’s global constraints. INFORMS J Comput 14:387–402MathSciNetCrossRefMATHGoogle Scholar
  14. Mirsky L (1971) Transversal theory. Math. Sci. Eng., vol 75. Academic Press, LondonGoogle Scholar
  15. Mourtos I (2013) Tight LP-relaxations of overlapping global cardinality constraints. Proceedings of CPAIOR’13. Lect. Notes Comput. Sci., vol 7874, pp 362–368Google Scholar
  16. Quimper CG, Golynski A, López-Ortiz A, van Beek P (2005) An efficient bounds consistency algorithm for the global cardinality constraint. Constraints 10:115–135MathSciNetCrossRefMATHGoogle Scholar
  17. Quimper CG, Walsh T (2006) The all different and global cardinality constraints on set, multiset and tuple variables. In: Hnich B et al (eds) CSCLP 2005. Lect. Notes Artificial Intelligence, vol 3978. Springer pp 1–13Google Scholar
  18. Regin JC (1996) Generalized arc consistency for global cardinality constraint. Proceedings of AAAI-96, Portland, OR, pp 209–215Google Scholar
  19. Regin JC (2002) Cost-based Arc consistency for global cardinality constraints. Constraints 7:387–405MathSciNetCrossRefMATHGoogle Scholar
  20. Samer M, Szeider S (2011) Tractable cases of the extended global cardinality constraint. Constraints 16:1–24MathSciNetCrossRefMATHGoogle Scholar
  21. Schrijver A (2004) Polyhedra and efficiency (algorithms and combinatorics). Springer, BerlinMATHGoogle Scholar
  22. van Beek P, Wilken K (2001) Fast optimal instruction scheduling for single-issue processors with arbitrary latencies. Proceedings of CP-01. Paphos, Cyprus, pp 625–639Google Scholar
  23. Williams HP, Yan H (2001) Representations of the all-different predicate of constraint satisfaction in integer programming. INFORMS J Comput 13:96–103MathSciNetCrossRefMATHGoogle Scholar
  24. Yan H, Hooker JN (1999) Tight representation of logic constraints as cardinality rules. Math Program 85:363–377MathSciNetCrossRefMATHGoogle Scholar
  25. Yunes TH (2002) On the sum constraint: relaxation and applications. In: van Hentenryck P (ed) Proceedings of CP2002. Lect. Notes Comput. Sci., vol 2470, pp 80–92Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Management Science and TechnologyAthens University of Economics and BusinessAthensGreece

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