Symmetric Cardinality Constraint with Costs

The symmetric cardinality constraint is described in terms of a set of variables X = {x1, . . . , xk}, which take their values as subsets of V = {v1, . . . , vn}. It constraints the cardinality of the set assigned to each variable to be in an interval [lxi , uxi ] and at the same time it restricts the number of occurrences of each value vj ∈ V in the sets assigned to variables in X to be in an other interval [lvj , uvj ]. In this paper we extend the symmetric cardinality constraint with a function which associate with each value of each variable a cost and constraints the global cost of the constraint to the sum of costs associated with assigned values. We also give an algorithm for computing the consistency of a symmetric cardinality constraint with costs and describe filtering methods for this constraint.