Arc consistency for global cardinality constraints with costs

A global cardinality constraint (gcc) is specified in terms of a set of variables, X = {x(1),...,x(p),} which take their values in a subset of V = {upsilon(1),...,upsilon(d)} It constrains the number of times each value upsilon(i) is an element of V is assigned to a variable in X to be in an interval [l(i), u(i)]. A gee with costs (costgcc) is a generalization of a gee in which a cost is associated with each value of each variable. Then, each solution of the underlying gee 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 are consistency for a costgcc. We also study the incremental behavior of the proposed algorithm.