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 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.Note that Q G,H ∩ {0, 1} n×n = Ψ G,H because the equations X u1v1 X u2v2 = 0 enforce that edges must map onto edges and non-edges (2-vertex independent sets) must map onto non-edges. We define Q G = Q G,G , and Q k G,H (respectively Q k G ) as the kth Sherali-Adams relaxations of Q G,H (respectively Q G ). The following theorem summarizes the relationship between the W-L algorithm and the Sherali-Adams relaxations of Q G,H . Theorem 1.2. Let k > 1. The polytope Q k+1 G = {I n } if and only if the k-dim W-L algorithm determines that G is asymmetric. The polytope Q k+1 G,H = ∅ if and only if the k-dim W-L algorithm determines that G and H are not isomorphic.Again, we will actually prove a more general and stronger result than the above theorem that illustrates that the the polytopes Q k G and Q G,H are geometric analogues of the k-dim W-L automorphism and isomorphism algorithms respectively (see Section 4).

Limitations of Algebraic Approaches to Graph Isomorphism Testing

“…Complementing recent research on linear and semidefinite programming approaches to GI [1,10,15,19,20], we investigate the power of GI-algorithms based on algebraic reasoning techniques like Gröbner basis computation. The idea of all these approaches is to encode isomorphisms between two graphs as solutions to a system of equations and possibly inequalities and then try to solve this system or relaxations of it.…”

Pebble Games and Linear Equations

“…We study a surprising connection between equivalence in finite variable logics and a linear programming approach to the graph isomorphism problem. This connection has recently been uncovered by Atserias and Maneva [1] and, independently, Malkin [14], building on earlier work of Tinhofer [22,23] and Ramana, Scheinerman and Ullman [17] that just concerns the 2-variable case.…”

“…Those of type (15) are actually subsumed by those of type (14) with J = ∅. It is an easy exercise to prove that such systems of linear boolean equations can be solved in polynomial time.…”

“…Furthermore, we prove that a natural combination of equalities from ISO(k) and ISO(k − 1) gives a linear program ISO(k − 1/2) that characterises C k -equivalence exactly. Malkin [14] gives an alternative characterisation of C k -equivalence in terms of the Sherali-Adams relaxations of a different polytope. Building on our work, yet another algebraic characterisation of C k -equivalence has recently been given in [3].…”

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We give a new, simplified and detailed account of the correspondence between levels of the Sherali-Adams relaxation of graph isomorphism and levels of pebble-game equivalence with counting (higher-dimensional Weisfeiler-Lehman colour refinement). The correspondence between basic colour refinement and fractional isomorphism, due to Tinhofer [22,23] and Ramana, Scheinerman and Ullman [17], is re-interpreted as the base level of Sherali-Adams and generalised to higher levels in this sense by Atserias and Maneva [1] and Malkin [14], who prove that the two resulting hierarchies interleave. In carrying this analysis further, we here give (a) a precise characterisation of the level k Sherali-Adams relaxation in terms of a modified counting pebble game; (b) a variant of the Sherali-Adams levels that precisely match the k-pebble counting game; (c) a proof that the interleaving between these two hierarchies is strict.

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