Combination of Among and Cardinality Constraints

A cardinality constraint imposes that each value of a set V must be taken a certain number of times by a set of variables X, whereas an among constraint imposes that a certain number of variables of a set X must take a value in the set V. This paper studies several combinations of among constraints and several conjunctions of among constraints and cardinality constraints. Some filtering algorithms are proposed and they are characterized when it is possible. Moreover, a weak form of Singleton arc consistency is considered. At last, it is shown how the global sequencing constraint and the global minimum distance constraint can be easily modeled by some conjunctions of cardinality and among constraints. Some results are also given for the global minimum distance constraint. They show that our study outperforms the existing constraints in ILOG Solver.