Chattopadhyay, Arkadev ;
Torán, Jacobo ;
Wagner, Fabian
Graph Isomorphism is not AC^0 reducible to Group Isomorphism
Abstract
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 fanin 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 fanin 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.
BibTeX  Entry
@InProceedings{chattopadhyay_et_al:LIPIcs:2010:2874,
author = {Arkadev Chattopadhyay and Jacobo Tor{\'a}n and Fabian Wagner},
title = {{Graph Isomorphism is not AC^0 reducible to Group Isomorphism}},
booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)},
pages = {317326},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897231},
ISSN = {18688969},
year = {2010},
volume = {8},
editor = {Kamal Lodaya and Meena Mahajan},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2010/2874},
URN = {urn:nbn:de:0030drops28748},
doi = {http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2010.317},
annote = {Keywords: Complexity, Algorithms, Group Isomorphism Problem, Circuit Com plexity}
}
2010
Keywords: 

Complexity, Algorithms, Group Isomorphism Problem, Circuit Com plexity 
Seminar: 

IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

Related Scholarly Article: 


Issue date: 

2010 
Date of publication: 

2010 