Symmetry handling inequalities (SHIs) are an appealing and popular tool for
handling symmetries in integer programming. Despite their practical
application, little is known about their interaction with optimization
problems. This article focuses on Schreier-Sims (SST) cuts, a recently
introduced family of SHIs, and investigate their impact on the computational
and polyhedral complexity of optimization problems. Given that SST cuts are not
unique, a crucial question is to understand how different constructions of SST
cuts influence the solving process.
First, we observe that SST cuts do not increase the computational complexity
of solving a linear optimization problem over any polytope P. However,
separating the integer hull of P enriched by SST cuts can be NP-hard, even if
P is integral and has a compact formulation. We study this phenomenon more
in-depth for the stable set problem, particularly for subclasses of perfect
graphs. For bipartite graphs, we give a complete characterization of the
integer hull after adding SST cuts based on odd-cycle inequalities. For
trivially perfect graphs, we observe that the separation problem is still
NP-hard after adding a generic set of SST cuts. Our main contribution is to
identify a specific class of SST cuts, called stringent SST cuts, that keeps
the separation problem polynomial and a complete set of inequalities, namely
SST clique cuts, that yield a complete linear description.
We complement these results by giving SST cuts based presolving techniques
and provide a computational study to compare the different approaches. In
particular, our newly identified stringent SST cuts dominate other approaches