A filtering algorithm for global sequencing constraints

Abstract

Sequencing constraints have proved very useful in many real-life problems such as rostering or car sequencing problems. They are used to express constraints such as: every sequence of 7 days of work must contain at least 2 days off. More precisely, a global sequencing constraint (gsc) C is specified in terms of an ordered set of variables X (C) = {x ₁,..., x _p} which take their values in D(C) = {v ₁,..., v _d}, some integers q, min and max and a given subset V of D(C). On one hand, a gsc constrains the number of variables in X(C) instantiated to a value v _i ε D(C) be in an interval [1 _i, u _i]. On the other hand, a gsc constrains for each sequence S _i of q consecutive variables of X(C), that at least min and at most max variables of Si are instantiated to a value of V. In this paper, we propose an automatic reformulation of a gsc in terms of global cardinality constraints. This is equivalent to defining a powerful filtering algorithm for a gsc which deals with a part of the globality of the constraint. We illustrate the power of our approach on a set of difficult car sequencing problems.