Approximating the distance to monotonicity of Boolean functions

We design a nonadaptive algorithm that, given oracle access to a function f:{0,1} n→{0,1} which is α‐far from monotone, makes poly(n,1/α) queries and returns an estimate that, with high probability, is an Õ(n)‐approximation to the distance of f to monotonicity. The analysis of our algorithm relies o...

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Bibliographic Details
Published in:Random structures & algorithms Vol. 60; no. 2; pp. 233 - 260
Main Authors: Pallavoor, Ramesh Krishnan S., Raskhodnikova, Sofya, Waingarten, Erik
Format: Journal Article
Language:English
Published: New York John Wiley & Sons, Inc 01.03.2022
Wiley Subscription Services, Inc
Subjects:
Algorithms
analysis of Boolean functions
Approximation
Boolean algebra
Boolean functions
Lower bounds
Mathematical analysis
property testing
Queries
sublinear algorithms
tolerant and erasure‐resilient testing
ISSN:1042-9832, 1098-2418
Online Access:Get full text
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Summary:We design a nonadaptive algorithm that, given oracle access to a function f:{0,1} n→{0,1} which is α‐far from monotone, makes poly(n,1/α) queries and returns an estimate that, with high probability, is an Õ(n)‐approximation to the distance of f to monotonicity. The analysis of our algorithm relies on an improvement to the directed isoperimetric inequality of Khot, Minzer, and Safra (SIAM J. Comput., 2018). Furthermore, we rule out a poly(n,1/α)‐query nonadaptive algorithm that approximates the distance to monotonicity significantly better by showing that, for all constant κ>0, every nonadaptive n1/2−κ‐approximation algorithm for this problem requires 2nκ queries. This answers a question of Seshadhri (Property Testing Review, 2014) for the case of nonadaptive algorithms. We obtain our lower bound by proving an analogous bound for erasure‐resilient (and tolerant) testers. Our method also yields the same lower bounds for unateness and being a k‐junta.
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ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.21029