An Among constraint holds if the number of variables that belong to a given value domain is between given bounds. This paper focuses on the case where the variable and value domains are intervals. We investigate the conjunction of Among constraints of this type. We prove that checking for satisfiability – and thus, enforcing bound consistency – can be done in polynomial time. The proof is based on a specific decomposition that can be used as such to filter inconsistent bounds from the variable domains. We show that this decomposition is incomparable with the natural conjunction of Among constraints, and that both decompositions do not ensure bound consistency. Still, experiments on randomly generated instances reveal the benefits of this new decomposition in practice. This paper also introduces a generalization of this problem to several dimensions and shows that satisfiability is \(\mathcal{R}\)-complete in the multi-dimensional case


Polynomial Time Variable Domain Domain Versus Global Constraint Cardinality Constraint 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Gilles Chabert
    • 1
  • Sophie Demassey
    • 1
  1. 1.TASC, Mines-Nantes, INRIA, LINA CNRSNantesFrance

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