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DOI: 10.4230/LIPIcs.FSTTCS.2012.148
URN: urn:nbn:de:0030-drops-38523
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2012/3852/
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Ivanyos, Gábor ; Klauck, Hartmut ; Lee, Troy ; Santha, Miklos ; de Wolf, Ronald

New bounds on the classical and quantum communication complexity of some graph properties

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Abstract

We study the communication complexity of a number of graph properties where the edges of the graph G are distributed between Alice and Bob (i.e., each receives some of the edges as input). Our main results are:
1. An Omega(n) lower bound on the quantum communication complexity of deciding whether an n-vertex graph G is connected, nearly matching the trivial classical upper bound of O(n log n) bits of communication.
2. A deterministic upper bound of O(n^{3/2} log n) bits for deciding if a bipartite graph contains a perfect matching, and a quantum lower bound of Omega(n) for this problem.
3. A Theta(n^2) bound for the randomized communication complexity of deciding if a graph has an Eulerian tour, and a Theta(n^{3/2}) bound for its quantum communication complexity.
4. The first two quantum lower bounds are obtained by exhibiting a reduction from the n-bit Inner Product problem to these graph problems, which solves an open question of Babai, Frankl and Simon [Babai et al 1986]. The third quantum lower bound comes from recent results about the quantum communication complexity of composed functions. We also obtain essentially tight bounds for the quantum communication complexity of a few other problems, such as deciding if $G$ is triangle-free, or if G is bipartite, as well as computing the determinant of a distributed matrix.

BibTeX - Entry

@InProceedings{ivanyos_et_al:LIPIcs:2012:3852,
  author =	{G{\'a}bor Ivanyos and Hartmut Klauck and Troy Lee and Miklos Santha and Ronald de Wolf},
  title =	{{New bounds on the classical and quantum communication complexity of some graph properties}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012) },
  pages =	{148--159},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-47-7},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{18},
  editor =	{Deepak D'Souza and Telikepalli Kavitha and Jaikumar Radhakrishnan},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2012/3852},
  URN =		{urn:nbn:de:0030-drops-38523},
  doi =		{10.4230/LIPIcs.FSTTCS.2012.148},
  annote =	{Keywords: Graph properties, communication complexity, quantum communication}
}

Keywords: Graph properties, communication complexity, quantum communication
Collection: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)
Issue Date: 2012
Date of publication: 14.12.2012

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