The Query Complexity of Graph Isomorphism: Bypassing Distribution Testing Lower Bounds

We study the query complexity of graph isomorphism in the property testing model for dense graphs. We give an algorithm that makes n(1+o(1)) queries, improving on the previous best bound of (O) over tilde (n(5/4)). Since the problem is known to require Omega(n) queries, our algorithm is optimal up to a subpolynomial factor. While trying to extend a known connection to distribution testing, discovered by Fischer and Matsliah (SICOMP 2008), one encounters a natural obstacle presented by sampling lower bounds such as the Omega(n(2/3))-sample lower bound for distribution closeness testing (Valiant, SICOMP 2011). In the context of graph isomorphism testing, these bounds lead to an n(1+Omega(1)) barrier for Fischer and Matsliah's approach. We circumvent this and other limitations by exploiting a geometric representation of the connectivity of vertices. An approximate representation of similarities between vertices can be learned with a near-linear number of queries and allows relaxed versions of sampling and distribution testing problems to be solved more efficiently.