Sherali–Adams relaxations of graph isomorphism polytopes

Abstract

We investigate the Sherali–Adams lift & project hierarchy applied to a graph isomorphism polytope whose integer points encode the isomorphisms between two graphs. In particular, the Sherali–Adams relaxations characterize a new vertex classification algorithm for graph isomorphism, which we call the generalized vertex classification algorithm. This algorithm generalizes the classic vertex classification algorithm and generalizes the work of Tinhofer on polyhedral methods for graph automorphism testing. We establish that the Sherali–Adams lift & project hierarchy when applied to a graph isomorphism polytope of a graph with n vertices needs Ω(n) iterations in the worst case before converging to the convex hull of integer points. We also show that this generalized vertex classification algorithm is also strongly related to the well-known Weisfeiler–Lehman algorithm, which we show can also be characterized in terms of the Sherali–Adams relaxations of a semi-algebraic set whose integer points encode graph isomorphisms.