Global Constraints for Lexicographic Orderings

We propose some global constraints for lexicographic orderings on vectors of variables. These constraints are very useful for breaking a certain kind of symmetry arising in matrices of decision variables. We show that decomposing such constraints carries a penalty either in the amount or the cost of constraint propagation. We therefore present a global consistency algorithm which enforces a lexicographic ordering between two vectors of n variables in O(nb) time, where b is the cost of adjusting the bounds of a variable. The algorithm can be modified very slightly to enforce a strict lexicographic ordering. Our experimental results on a number of domains (balanced incomplete block design, social golfer, and sports tournament scheduling) confirm the efficiency and value of these new global constraints.