Abstract
We show that the graph isomorphism problem is hard under DLOGTIME uniform AC{$^0$} many-one reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class {Mod}$_k$L and for the class DET of problems NC{$^1$} reducible to the determinant. These are the strongest known hardness results for the graph isomorphism problem and imply a randomized logarithmic space reduction from the perfect matching problem to graph isomorphism. We also investigate hardness results for the graph automorphism problem.
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