On the Complexity of Breaking Symmetry

TL;DR

This paper proves that alternative total orderings for symmetry breaking, such as Gray code or Snake-Lex, do not reduce the inherent computational complexity, making symmetry breaking generally intractable.

Contribution

It establishes the intractability of symmetry breaking with various total orderings beyond lexicographical ordering.

Findings

Breaking symmetry with alternative orderings is intractable.

Lex-leader method is computationally hard for large symmetry groups.

Other total orderings do not simplify symmetry breaking complexity.

Abstract

We can break symmetry by eliminating solutions within a symmetry class that are not least in the lexicographical ordering. This is often referred to as the lex-leader method. Unfortunately, as symmetry groups can be large, the lexleader method is not tractable in general. We prove that using other total orderings besides the usual lexicographical ordering will not reduce the computational complexity of breaking symmetry in general. It follows that breaking symmetry with other orderings like the Gray code ordering or the Snake-Lex ordering is intractable in general.