License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2016.17
URN: urn:nbn:de:0030-drops-63133
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6313/
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Thaler, Justin

Lower Bounds for the Approximate Degree of Block-Composed Functions

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LIPIcs-ICALP-2016-17.pdf (0.6 MB)

Abstract

We describe a new hardness amplification result for point-wise approximation of Boolean functions by low-degree polynomials.
Specifically, for any function f on N bits, define F(x_1,...,x_M) = OMB(f(x_1),...,f(x_M)) to be the function on M*N bits obtained by block-composing f with a function known as ODD-MAX-BIT. We show that, if f requires large degree to approximate to error 2/3 in a certain one-sided sense (captured by a complexity measure known as positive one-sided approximate degree), then F requires large degree to approximate even to error 1-2^{-M}. This generalizes a result of Beigel (Computational Complexity, 1994), who proved an identical result for the special case f=OR.

Unlike related prior work, our result implies strong approximate degree lower bounds even for many functions F that have low threshold degree. Our proof is constructive: we exhibit a solution to the dual of an appropriate linear program capturing the approximate degree of any function. We describe several applications, including improved separations between the complexity classes P^{NP} and PP in both the query and communication complexity settings. Our separations improve on work of Beigel (1994) and Buhrman, Vereshchagin, and de Wolf (CCC, 2007).

BibTeX - Entry

@InProceedings{thaler:LIPIcs:2016:6313,
  author =	{Justin Thaler},
  title =	{{Lower Bounds for the Approximate Degree of Block-Composed Functions}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{17:1--17:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6313},
  URN =		{urn:nbn:de:0030-drops-63133},
  doi =		{10.4230/LIPIcs.ICALP.2016.17},
  annote =	{Keywords: approximate degree, one-sided approximate degree, polynomial approx- imations, threshold degree, communication complexity}
}

Keywords: approximate degree, one-sided approximate degree, polynomial approx- imations, threshold degree, communication complexity
Collection: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 23.08.2016

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