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.2019.52
URN: urn:nbn:de:0030-drops-101452
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/10145/
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Levi, Amit ; Waingarten, Erik

Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs

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Abstract

We introduce a new model for testing graph properties which we call the rejection sampling model. We show that testing bipartiteness of n-nodes graphs using rejection sampling queries requires complexity Omega~(n^2). Via reductions from the rejection sampling model, we give three new lower bounds for tolerant testing of Boolean functions of the form f : {0,1}^n -> {0,1}:
- Tolerant k-junta testing with non-adaptive queries requires Omega~(k^2) queries.
- Tolerant unateness testing requires Omega~(n) queries.
- Tolerant unateness testing with non-adaptive queries requires Omega~(n^{3/2}) queries.
Given the O~(k^{3/2})-query non-adaptive junta tester of Blais [Eric Blais, 2008], we conclude that non-adaptive tolerant junta testing requires more queries than non-tolerant junta testing. In addition, given the O~(n^{3/4})-query unateness tester of Chen, Waingarten, and Xie [Xi Chen et al., 2017] and the O~(n)-query non-adaptive unateness tester of Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova, and Seshadhri [Roksana Baleshzar et al., 2017], we conclude that tolerant unateness testing requires more queries than non-tolerant unateness testing, in both adaptive and non-adaptive settings. These lower bounds provide the first separation between tolerant and non-tolerant testing for a natural property of Boolean functions.

BibTeX - Entry

@InProceedings{levi_et_al:LIPIcs:2018:10145,
  author =	{Amit Levi and Erik Waingarten},
  title =	{{Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs}},
  booktitle =	{10th Innovations in Theoretical Computer Science  Conference (ITCS 2019)},
  pages =	{52:1--52:20},
  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/2018/10145},
  URN =		{urn:nbn:de:0030-drops-101452},
  doi =		{10.4230/LIPIcs.ITCS.2019.52},
  annote =	{Keywords: Property Testing, Juntas, Tolerant Testing, Boolean functions}
}

Keywords: Property Testing, Juntas, Tolerant Testing, 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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