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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2019.64
URN: urn:nbn:de:0030-drops-106407
URL: http://drops.dagstuhl.de/opus/volltexte/2019/10640/
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Gavinsky, Dmitry ; Lee, Troy ; Santha, Miklos ; Sanyal, Swagato

A Composition Theorem for Randomized Query Complexity via Max-Conflict Complexity

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Abstract

For any relation f subseteq {0,1}^n x S and any partial Boolean function g:{0,1}^m -> {0,1,*}, we show that R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * sqrt{R_{1/3}(g)}) , where R_epsilon(*) stands for the bounded-error randomized query complexity with error at most epsilon, and f o g^n subseteq ({0,1}^m)^n x S denotes the composition of f with n instances of g. The new composition theorem is optimal, at least, for the general case of relational problems: A relation f_0 and a partial Boolean function g_0 are constructed, such that R_{4/9}(f_0) in Theta(sqrt n), R_{1/3}(g_0)in Theta(n) and R_{1/3}(f_0 o g_0^n) in Theta(n). The theorem is proved via introducing a new complexity measure, max-conflict complexity, denoted by bar{chi}(*). Its investigation shows that bar{chi}(g) in Omega(sqrt{R_{1/3}(g)}) for any partial Boolean function g and R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * bar{chi}(g)) for any relation f, which readily implies the composition statement. It is further shown that bar{chi}(g) is always at least as large as the sabotage complexity of g.

BibTeX - Entry

@InProceedings{gavinsky_et_al:LIPIcs:2019:10640,
  author =	{Dmitry Gavinsky and Troy Lee and Miklos Santha and Swagato Sanyal},
  title =	{{A Composition Theorem for Randomized Query Complexity via Max-Conflict Complexity}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{64:1--64:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10640},
  URN =		{urn:nbn:de:0030-drops-106407},
  doi =		{10.4230/LIPIcs.ICALP.2019.64},
  annote =	{Keywords: query complexity, lower bounds}
}

Keywords: query complexity, lower bounds
Seminar: 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Issue Date: 2019
Date of publication: 08.07.2019


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