The Complexity of Symmetry Breaking Beyond Lex-Leader

TL;DR

This paper investigates the computational complexity of generating symmetry breaking predicates in constraint programming, revealing fundamental barriers and providing bounds for specific symmetry groups.

Contribution

It proves a natural barrier for efficiently computing complete SBPs related to graph non-isomorphism and offers polynomial bounds for certain symmetry groups.

Findings

Efficient certification of graph non-isomorphism is a barrier to computing SBPs.

Complete SBPs are hard to compute for general symmetry groups.

Polynomial bounds are found for automorphism groups of trees and wreath products.

Abstract

Symmetry breaking is a widely popular approach to enhance solvers in constraint programming, such as those for SAT or MIP. Symmetry breaking predicates (SBPs) typically impose an order on variables and single out the lexicographic leader (lex-leader) in each orbit of assignments. Although it is NP-hard to find complete lex-leader SBPs, incomplete lex-leader SBPs are widely used in practice. In this paper, we investigate the complexity of computing complete SBPs, lex-leader or otherwise, for SAT. Our main result proves a natural barrier for efficiently computing SBPs: efficient certification of graph non-isomorphism. Our results explain the difficulty of obtaining short SBPs for important CP problems, such as matrix-models with row-column symmetries and graph generation problems. Our results hold even when SBPs are allowed to introduce additional variables. We show polynomial upper bounds for breaking certain symmetry groups, namely automorphism groups of trees and wreath products of groups with efficient SBPs.