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URN: urn:nbn:de:0030-drops-6218
URL: http://drops.dagstuhl.de/opus/volltexte/2006/621/

Jukna, Stasys

Graphs and Circuits: Some Further Remarks

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Abstract

We consider the power of single level circuits in the context of graph complexity. We first prove that the single level conjecture fails for fanin-$2$ circuits over the basis ${oplus,land,1}$. This shows that the (surpisingly tight) phenomenon, established by Mirwald and Schnorr (1992) for quadratic functions, has no analogon for graphs. We then show that the single level conjecture fails for unbounded fanin circuits over ${lor,land,1}$. This partially answers the question of Pudl'ak, R"odl and Savick'y (1986). We also prove that $Sigma_2 eq Pi_2$ in a restricted version of the hierarhy of communication complexity classes introduced by Babai, Frankl and Simon (1986). Further, we show that even depth-$2$ circuits are surprisingly powerful: every bipartite $n imes n$ graph of maximum degree $Delta$ can be represented by a monotone CNF with $O(Deltalog n)$ clauses. We also discuss a relation between graphs and $ACC$-circuits.

BibTeX - Entry

@InProceedings{jukna:DSP:2006:621,
  author =	{Stasys Jukna},
  title =	{Graphs and Circuits: Some Further Remarks},
  booktitle =	{Complexity of Boolean Functions},
  year =	{2006},
  editor =	{Matthias Krause and Pavel Pudl{\'a}k and R{\"u}diger Reischuk and Dieter van Melkebeek},
  number =	{06111},
  series =	{Dagstuhl Seminar Proceedings},
  ISSN =	{1862-4405},
  publisher =	{Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2006/621},
  annote =	{Keywords: Graph complexity, single level conjecture, Sylvester graphs, communication complexity, ACC-circuits}
}

Keywords: Graph complexity, single level conjecture, Sylvester graphs, communication complexity, ACC-circuits
Seminar: 06111 - Complexity of Boolean Functions
Issue date: 2006
Date of publication: 30.11.2006


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