The Cyclic SEQUENCE Constraint

In rostering problems, we need to find a schedule that satisfies various constraints. Ergonomic constraints, that place restrictions on a number of consecutive days, are common (Bourdais, Galinier, & Pesant 2003) . To model such rules, several global constraints have been introduced. For example, the STRETCH constraint (Hellsten, Pesant, & Beek 2004) is useful for limiting the length of stretches of variables (e.g., no more than 3 consecutive night shifts for an individual employee). The SEQUENCE constraint (Beldiceanu & Contejean 1994) is used to restrict the number of occurrences of specific values in a given period (e.g., each employee has to have 2 days-off in any 7 consecutive days). The REGULAR constraint (Pesant 2004) ensures that an assignment of a given sequence of variables belongs to a regular language. This can express, for example, that an employee has to have at least two consecutive days on any shift. Many public services, like hospitals, police departments, some factories, work 7 days per week. Such organizations require a schedule than can be repeated with a given period. To express rules for such cyclic schedules the Cyclic REGULAR constraint (Quimper & Walsh 2006) and the Cyclic STRETCH constraint (Hellsten, Pesant, & Beek 2004) have been introduced. Propagating the Cyclic REGULAR constraint is NP-hard, whilst there is a polynomial filtering algorithm for the Cyclic STRETCH constraint. Recently, the Cyclic SEQUENCE has been introduced but no polynomial filtering algorithms have been proposed (Brand et al. 2007). In this paper we present a polynomial filtering algorithm for the Cyclic SEQUENCE. We also consider a special case of SEQUENCE, the ATMOSTSEQ constraint, and prove that the Cyclic ATMOSTSEQ constraint can be decomposed into ATMOST constraints without hindering propagation.