Reducing Symmetry in Matrix Models

Abstract

Symmetry in a CSP is a permutation of variables, or the values in the domains, or both which preserve the state of the search: either all of them lead to a solution or none does. Hence, elimination of symmetry is essential to avoid exploring equivalent branches in a search tree. An important class of symmetries in constraint programming arises from matrices of decision variables where any two rows can be interchanged, as well as any two columns. Eliminating all such symmetries is not so easy as the effort required may be exponential. We are thus interested in reducing significant amount of row and column symmetries in matrix models with a polynomial effort. In this respect, we have shown that lexicographically ordering both rows and columns of a matrix model reduces much of such symmetries. For an n × n matrix model with row and column symmetry, O(n) lexicographic constraints between adjacent rows and columns are imposed. We have shown that decomposing a lexicographic ordering constraint between a pair of vectors carries a penalty either in the amount or the cost of constraint propagation. We have therefore developed a linear-time global-consistency algorithm which enforces a lexicographic ordering between two vectors. Our experiments confirm the efficiency and value of this new global constraint. As a matrix model has multiple rows and columns, we can treat such a problem as a single global ordering constraint over the whole matrix. Alternatively, we can decompose it into lexicographic ordering constraints between all or adjacent pairs of vectors. Such decompositions hinder constraint propagation in general. However, we identify the special case of a lexicographical ordering on 0/1 variables where it does not.

I am very grateful to Pierre Flener, Alan Frisch, Brahim Hnich, Ian Miguel, Barbara Smith, and Toby Walsh for the development of the work presented here.