On The Complexity and Completeness of Static Constraints for Breaking Row and Column Symmetry

We consider a common type of symmetry where we have a matrix of decision variables with interchangeable rows and columns. A simple and efficient method to deal with such row and column symmetry is to post symmetry breaking constraints like DOUBLELEX and SNAKELEX. We provide a number of positive and negative results on posting such symmetry breaking constraints. On the positive side, we prove that we can compute in polynomial time a unique representative of an equivalence class in a matrix model with row and column symmetry if the number of rows (or of columns) is bounded and in a number of other special cases. On the negative side, we show that whilst DOUBLELEX and SNAKELEX are often effective in practice, they can leave a large number of symmetric solutions in the worst case. In addition, we prove that propagating DOUBLELEX completely is NP-hard. Finally we consider how to break row, column and value symmetry, correcting a result in the literature about the safeness of combining different symmetry breaking constraints. We end with the first experimental study on how much symmetry is left by DOUBLELEX and SNAKELEX on some benchmark problems.Comment: To appear in the Proceedings of the 16th International Conference on Principles and Practice of Constraint Programming (CP 2010

Breaking Symmetry with Different Orderings

Abstract. We can break symmetry by eliminating solutions within each symmetry class. For instance, the Lex-Leader method eliminates all but the smallest solution in the lexicographical ordering. Unfortunately, the Lex-Leader method is intractable in general. We prove that, under modest assumptions, we cannot reduce the worst case complexity of breaking symmetry by using other orderings on solutions. We also prove that a common type of symmetry, where rows and columns in a matrix of decision variables are interchangeable, is intractable to break when we use two promising alternatives to the lexicographical ordering: the Gray code ordering (which uses a different ordering on solutions), and the Snake-Lex ordering (which is a variant of the lexicographical ordering that re-orders the variables). Nevertheless, we show experimentally that using other orderings like the Gray code to break symmetry can be beneficial in practice as they may better align with the objective function and branching heuristic.