eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-01-08
20:1
20:7
10.4230/LIPIcs.ITCS.2019.20
article
Adaptive Boolean Monotonicity Testing in Total Influence Time
Chakrabarty, Deeparnab
1
Seshadhri, C.
2
Dartmouth College, Hanover, NH 03755, USA
University of California, Santa Cruz, CA 95064, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol124-itcs2019/LIPIcs.ITCS.2019.20/LIPIcs.ITCS.2019.20.pdf
Property Testing
Monotonicity Testing
Influence of Boolean Functions