Adaptive Boolean Monotonicity Testing in Total Influence Time
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.
Property Testing
Monotonicity Testing
Influence of Boolean Functions
Theory of computation~Streaming, sublinear and near linear time algorithms
20:1-20:7
Regular Paper
Deeparnab
Chakrabarty
Deeparnab Chakrabarty
Dartmouth College, Hanover, NH 03755, USA
Supported by NSF CCF-1813165.
C.
Seshadhri
C. Seshadhri
University of California, Santa Cruz, CA 95064, USA
Supported by NSF TRIPODS CCF-1740850 and NSF CCF-1813165.
10.4230/LIPIcs.ITCS.2019.20
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Deeparnab Chakrabarty and C. Seshadhri
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