Adaptive Boolean Monotonicity Testing in Total Influence Time

Authors Deeparnab Chakrabarty, C. Seshadhri

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Author Details

Deeparnab Chakrabarty
  • Dartmouth College, Hanover, NH 03755, USA
C. Seshadhri
  • University of California, Santa Cruz, CA 95064, USA

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Deeparnab Chakrabarty and C. Seshadhri. Adaptive Boolean Monotonicity Testing in Total Influence Time. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 20:1-20:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Property Testing
  • Monotonicity Testing
  • Influence of Boolean Functions


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