On the geometry of symmetry breaking inequalities

Breaking symmetries is a popular way of speeding up the branch-and-bound method for symmetric integer programs. We study symmetry breaking polyhedra, more precisely, fundamental domains. Our long-term goal is to understand the relationship between the complexity of such polyhedra and their symmetry breaking ability. Borrowing ideas from geometric group theory, we provide structural properties that relate the action of the group to the geometry of the facets of fundamental domains. Inspired by these insights, we provide a new generalized construction for fundamental domains, which we call generalized Dirichlet domain (GDD). Our construction is recursive and exploits the coset decomposition of the subgroups that fix given vectors in $\mathbb{R}^n$. We use this construction to analyze a recently introduced set of symmetry breaking inequalities by Salvagnin (2018) and Liberti and Ostrowski (2014), called Schreier-Sims inequalities. In particular, this shows that every permutation group admits a fundamental domain with less than $n$ facets. We also show that this bound is tight. Finally, we prove that the Schreier-Sims inequalities can contain an exponential number of isomorphic binary vectors for a given permutation group $G$, which shows evidence of the lack of symmetry breaking effectiveness of this fundamental domain. Conversely, a suitably constructed GDD for $G$ has linearly many inequalities and contains unique representatives for isomorphic binary vectors.