eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2010-12-14
317
326
10.4230/LIPIcs.FSTTCS.2010.317
article
Graph Isomorphism is not AC^0 reducible to Group Isomorphism
Chattopadhyay, Arkadev
Torán, Jacobo
Wagner, Fabian
We give a new upper bound for the Group and Quasigroup Isomorphism problems when the input structures are given explicitly by multiplication tables. We show that these problems can be computed by polynomial size nondeterministic circuits of unbounded fan-in with $O(\log\log n)$ depth and $O(\log^2 n)$ nondeterministic bits,
where $n$ is the number of group elements. This improves the existing upper bound from \cite{Wolf 94} for the problems. In the previous upper bound the circuits have bounded fan-in but depth $O(\log^2 n)$ and also $O(\log^2 n)$ nondeterministic bits. We then prove that the kind of circuits from our upper bound cannot compute the Parity function. Since Parity is AC0 reducible to Graph Isomorphism, this implies that Graph Isomorphism is strictly harder than Group or Quasigroup Isomorphism under the ordering defined by AC0 reductions.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol008-fsttcs2010/LIPIcs.FSTTCS.2010.317/LIPIcs.FSTTCS.2010.317.pdf
Complexity
Algorithms
Group Isomorphism Problem
Circuit Com plexity