We show that the graph isomorphism problem, is low for PP and for C=P, i.e., it does not provide a PP or C=P computation with any additional power when used as an oracle. Furthermore, we show that graph isomorphism belongs to the class LWPP (see Fenner, Fortnow, Kurtz ). A similar result holds for the (apparently more difficult) problem Group Factorization. The problem of determining whether a given graph has a nontrivial automorphism, Graph Automorphism, is shown to be in SPP, and is therefore low for PP, C=P, and ModkP,k≥2.
Key wordsgraph isomorphism complexity classes lowness counting properties
Subject classifications68Q15 68Q25 05C60 68R10
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