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

Authors Amit Levi, Erik Waingarten

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Author Details

Amit Levi
  • University of Waterloo, Canada
Erik Waingarten
  • Columbia University, USA

Funding

  • Levi, Amit: Research supported by NSERC Discovery grant and the David R. Cheriton Graduate Scholarship. Part of this work was done while the author was visiting Columbia University.
  • Waingarten, Erik: This work is supported in part by the NSF Graduate Research Fellowship under Grant No. DGE-16-44869, CCF-1703925, CCF-1563155, and CCF-1420349.

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Amit Levi and Erik Waingarten. Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 52:1-52:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ITCS.2019.52

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Probabilistic computation
Keywords
  • Property Testing
  • Juntas
  • Tolerant Testing
  • Boolean functions

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