Global Constraints and Filtering Algorithms

Abstract

Constraint programming (CP) is mainly based on filtering algorithms; their association with global constraints is one of the main strengths of CP. This chapter is an overview of these two techniques. Some of the most frequently used global constraints are presented. In addition, the filtering algorithms establishing arc consistency for two useful constraints, the alldiff and the global cardinality constraints, are fully detailed. Filtering algorithms are also considered from a theoretical point of view: three different ways to design filtering algorithms are described and the quality of the filtering algorithms studied so far is discussed. A categorization is then proposed. Over-constrained problems are also mentioned and global soft constraints are introduced.

A global constraint for total weighted completion time for unary resources

“…This implies that, unless P = NP, there is no efficient algorithm that could find and remove all inconsistent domain values. The common approach in such cases is to consider a polynomially solvable relaxation of the property expressed by the constraint, and exploit the relaxation to prove the inconsistency of some of the domain values [30].…”

“…It initializes by determining the earliest start time t of A i , building a schedule σ for S i = t , computing its cost c =Č S i = t , and defining the variable c min that will be used to tighten the lower bound of C. Note that the direct approach to build σ (line 21) is a standard preemptive scheduling algorithm, therefore it is not presented here in detail. Then, the algorithms iteratively recomputes schedule σ by calling RecomputeSchedule (lines [24][25][26][27][28][29][30][31][32][33][34]. Note that RecomputeSchedule updates the values of σ , t, and c. At this point, variables t prev and t store the values of t j and t j+1 , respectively, while c prev and c contain the corresponding lower bound costs.…”

“…The algorithm is implemented in two procedures: the main procedure Propagate() calls procedure RecomputeSchedule(σ, t, c) iteratively, whose parameters are a schedule σ , an integer t that represents the start time of the activity investigated, and a real number c, which equals the cost of σ . Procedure Propagate performs propagation using the recomputation approach for each activity A i separately (lines [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. It initializes by determining the earliest start time t of A i , building a schedule σ for S i = t , computing its cost c =Č S i = t , and defining the variable c min that will be used to tighten the lower bound of C. Note that the direct approach to build σ (line 21) is a standard preemptive scheduling algorithm, therefore it is not presented here in detail.…”

Integrating Operations Research in Constraint Programming

“…An exhaustive list of global constraints implemented in commercial CP solvers can be found in Regin (2004).…”

Constraint programming for stochastic inventory systems under shortage cost

“…"x or y is true", where x and y are boolean decision variables), linear constraints, and global constraints (Regin 2003).…”