Sherali-Adams Relaxations of Graph Isomorphism Polytopes

TL;DR

This paper explores Sherali-Adams relaxations applied to graph isomorphism polytopes, introducing a new vertex classification algorithm that generalizes existing methods and relates to the Weisfeiler-Lehman algorithm, with complexity bounds.

Contribution

It introduces a generalized vertex classification algorithm based on Sherali-Adams relaxations, connecting it to existing graph isomorphism techniques and analyzing its convergence properties.

Findings

Sherali-Adams relaxations characterize a new vertex classification algorithm.

The hierarchy requires Omega(n) iterations to converge in the worst case.

The generalized algorithm relates closely to the Weisfeiler-Lehman algorithm.

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 needs Omega(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.