Constraints for breaking more row and column symmetries

Abstract

Constraint programs containing a matrix of two (or more) dimensions of decision variables often have row and column symmetries: in any assignment to the variables the rows can be swapped and the columns can be swapped without affecting whether or not the assignment is a solution. This introduces an enormous amount of redundancy when searching a space of partial assignments. It has been shown previously that one can remove consistently some of these symmetries by extending such a program with constraints that require the rows and columns to be lexicographically ordered. This paper identifies and studies the properties of a new additional constraint - the first row is less than or equal to all permutations of all other rows - that can be added consistently to break even more symmetries. Two alternative implementations of this stronger symmetry-breaking method are investigated, one of which employs a new algorithm that in time linear in the size of the matrix enforces the constraint that the first row is less than or equal to all permutations of all other rows. It is demonstrated experimentally that our method for breaking more symmetries substantially reduces search effort. © Springer-Verlag 2003.