Solving Necklace Constraint Problems

Abstract

Some constraint problems have a combinatorial structure where the constraints allow the sequence of variables to be rotated (necklaces), if not also the domain values to be permuted (unlabelled necklaces), without getting an essentially different solution. We bring together the fields of combinatorial enumeration, where efficient algorithms have been designed for (special cases of) some of these combinatorial objects, and constraint programming, where the requisite symmetry breaking has at best been done statically so far. We design the first search procedure and identify the first symmetry-breaking constraints for the general case of unlabelled necklaces. Further, we compare dynamic and static symmetry breaking on real-life scheduling problems featuring (unlabelled) necklaces.