License: Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license (CC BY-NC-ND 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2011.661
URN: urn:nbn:de:0030-drops-30521
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2011/3052/
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Childs, Andrew M. ; Kothari, Robin

Quantum query complexity of minor-closed graph properties

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Abstract

We study the quantum query complexity of minor-closed graph properties, which include such problems as determining whether an $n$-vertex graph is planar, is a forest, or does not contain a path of a given length. We show that most minor-closed properties -- those that cannot be characterized by a finite set of forbidden subgraphs -- have quantum query complexity Theta(n^(3/2)). To establish this, we prove an adversary lower bound using a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. On the other hand, we show that minor-closed properties (and more generally, sparse graph properties) that can be characterized by finitely many forbidden subgraphs can be solved strictly faster, in o(n^(3/2)) queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.

BibTeX - Entry

@InProceedings{childs_et_al:LIPIcs:2011:3052,
  author =	{Andrew M. Childs and Robin Kothari},
  title =	{{Quantum query complexity of minor-closed graph properties}},
  booktitle =	{28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) },
  pages =	{661--672},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-25-5},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{9},
  editor =	{Thomas Schwentick and Christoph D{\"u}rr},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2011/3052},
  URN =		{urn:nbn:de:0030-drops-30521},
  doi =		{10.4230/LIPIcs.STACS.2011.661},
  annote =	{Keywords: quatum query complexity, quantum algorithms, lower bounds, graph minors, graph properties}
}

Keywords: quatum query complexity, quantum algorithms, lower bounds, graph minors, graph properties
Collection: 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)
Issue Date: 2011
Date of publication: 11.03.2011

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