On the non-uniform complexity of the Graph Isomorphism problem

Abstract

We study the non-uniform complexity of the Graph Isomorphism (GI) and Graph Automorphism (GA) problems considering the implications of different types of polynomial time reducibilitites from these problems to sparse sets. We show that if GI (or GA) is bounded truth-table or conjunctively reducible to a sparse set, then it is in P; while if we suppose that it is truth-table reducible without restrictions to a sparse set (or, equivalently, that it belongs to P/poly) then the problem is low for MA, the class of sets with publishable proofs. With respect to nondeterministic reductions, contrasting with the fact that GI and GA belong to the class NP¿(co-NP/poly) [Schö 88], we show that if the considered problems are btt strong nondeterministically reducible to a sparse set then they are in co-NP. Some of these results are proved using graph constructions that show new properties of the GI and GA problems.