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.ITCS.2019.20
URN: urn:nbn:de:0030-drops-101133
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/10113/
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Chakrabarty, Deeparnab ; Seshadhri, C.

Adaptive Boolean Monotonicity Testing in Total Influence Time

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LIPIcs-ITCS-2019-20.pdf (0.4 MB)

Abstract

Testing monotonicity of a Boolean function f:{0,1}^n -> {0,1} is an important problem in the field of property testing. It has led to connections with many interesting combinatorial questions on the directed hypercube: routing, random walks, and new isoperimetric theorems. Denoting the proximity parameter by epsilon, the best tester is the non-adaptive O~(epsilon^{-2}sqrt{n}) tester of Khot-Minzer-Safra (FOCS 2015). A series of recent results by Belovs-Blais (STOC 2016) and Chen-Waingarten-Xie (STOC 2017) have led to Omega~(n^{1/3}) lower bounds for adaptive testers. Reducing this gap is a significant question, that touches on the role of adaptivity in monotonicity testing of Boolean functions.
We approach this question from the perspective of parametrized property testing, a concept recently introduced by Pallavoor-Raskhodnikova-Varma (ACM TOCT 2017), where one seeks to understand performance of testers with respect to parameters other than just the size. Our result is an adaptive monotonicity tester with one-sided error whose query complexity is O(epsilon^{-2}I(f)log^5 n), where I(f) is the total influence of the function. Therefore, adaptivity provably helps monotonicity testing for low influence functions.

BibTeX - Entry

@InProceedings{chakrabarty_et_al:LIPIcs:2018:10113,
  author =	{Deeparnab Chakrabarty and C. Seshadhri},
  title =	{{Adaptive Boolean Monotonicity Testing in Total Influence Time}},
  booktitle =	{10th Innovations in Theoretical Computer Science  Conference (ITCS 2019)},
  pages =	{20:1--20:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{124},
  editor =	{Avrim Blum},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/10113},
  URN =		{urn:nbn:de:0030-drops-101133},
  doi =		{10.4230/LIPIcs.ITCS.2019.20},
  annote =	{Keywords: Property Testing, Monotonicity Testing, Influence of Boolean Functions}
}

Keywords: Property Testing, Monotonicity Testing, Influence of Boolean Functions
Collection: 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)
Issue Date: 2018
Date of publication: 08.01.2019

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