Generalized Arc Consistency for Global Cardinality Constraint

Abstract

A global cardinality constraint (gcc) is specified in terms of a set of variables X = fx1 ; :::; xpg which take their values in a subset of V = fv1 ; :::; vdg. It constrains the number of times a value v i 2 V is assigned toavariable in X to be in an interval (l i ;c i ). Cardinality constraints have proved very useful in many real-life problems, suchas scheduling, timetabling, or resource allocation. A gcc is more general than a constraint of difference, which requires each interval to be #0; 1#. In this paper, we present an efficient way of implementing generalized arc consistency for a gcc. The algorithm we propose is based on a new theorem of flow theory. Its space complexity is O(#Xj#jVj) and its time complexity is O(jXj 2 #jVj). We also show how this algorithm can efficiently be combined with other filtering techniques.