Cost-Based Arc Consistency for Global Cardinality Constraints

  • Authors:

  • Jean-Charles Régin

  • Affiliations:

  • ILOG, Les Taissounières HB2, 1661, route des Dolines, Sophia Antipolis, 06560 Valbonne, France regin@ilog.fr

  • Venue:
  • Constraints
  • Year:
  • 2002

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Abstract

A global cardinality constraint (gcc) is specified in terms of a set of variables iX={ix1,…,ixp} which take their values in a subset of iV={iv1,…,ivd}. It constrains the number of times each value ivi∈iV is assigned to a variable in iX to be in an interval [ili,iui]. A gcc with costs (costgcc) is a generalization of a gcc in which a cost is associated with each value of each variable. Then, each solution of the underlying gcc is associated with a global cost equal to the sum of the costs associated with the assigned values of the solution. A costgcc constrains the global cost to be less than a given value. Cardinality constraints with costs have proved very useful in many real-life problems, such as traveling salesman problems, scheduling, rostering, or resource allocation. For instance, they are useful for expressing preferences or for defining constraints such as a constraint on the sum of all different variables. In this paper, we present an efficient way of implementing arc consistency for a costgcc. We also study the incremental behavior of the proposed algorithm.