We investigate the Sherali-Adams lift & project hierarchy applied to a graphisomorphism polytope whose integer points encode the isomorphisms between twographs. In particular, the Sherali-Adams relaxations characterize a new vertexclassification algorithm for graph isomorphism, which we call the generalizedvertex classification algorithm. This algorithm generalizes the classic vertexclassification algorithm and generalizes the work of Tinhofer on polyhedralmethods for graph automorphism testing. We establish that the Sherali-Adamslift & project hierarchy when applied to a graph isomorphism polytope needsOmega(n) iterations in the worst case before converging to the convex hull ofinteger points. We also show that this generalized vertex classificationalgorithm is also strongly related to the well-known Weisfeiler-Lehmanalgorithm, which we show can also be characterized in terms of theSherali-Adams relaxations of a semi-algebraic set whose integer points encodegraph isomorphisms.
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