Breaking Symmetry with Different Orderings

TL;DR

This paper investigates the complexity of breaking symmetry in decision problems using various orderings, proving intractability results for common symmetry types and orderings, while also demonstrating practical benefits of alternative orderings like Gray code.

Contribution

It establishes theoretical intractability results for symmetry breaking with different orderings and shows that alternative orderings can be practically advantageous.

Findings

Lex-Leader method is intractable in general.

Symmetries in matrices are hard to break with Gray code and Snake-Lex orderings.

Alternative orderings can improve practical symmetry breaking.

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.