TY - GEN

T1 - Tolerant junta testing and the connection to submodular optimization and function isomorphism

AU - Blais, Eric

AU - Canonne, Clément L.

AU - Eden, Talya

AU - Levi, Amit

AU - Ron, Dana

N1 - Publisher Copyright:
© Copyright 2018 by SIAM.

PY - 2018

Y1 - 2018

N2 - The function f :{-1, 1}n → {-1, 1} is a k-junta if it depends on at most k of its variables. We consider the problem of tolerant testing of k-juntas, where the testing algorithm must accept any function that is ϵ-close to some k-junta and reject any function that is ϵ0-far from every k0-junta for some ϵ0 = O( ϵ) and k0 = O(k). Our first result is an algorithm that solves this problem with query complexity polynomial in k and 1/ϵ. This result is obtained via a new polynomialtime approximation algorithm for submodular function minimization (SFM) under large cardinality constraints, which holds even when only given an approximate oracle access to the function. Our second result considers the case where k0 = k. We show how to obtain a smooth tradeoff between the amount of tolerance and the query complexity in this setting. Specifically, we design an algorithm that given ϵ 2 (0, 1/2) accepts any function that isí 8 -close to some k-junta and rejects any function that is ϵ-far from every k-junta. The query complexity of the algorithm is O( k log k/cp(1- p)ϵ). Finally, we show how to apply the second result to the problem of tolerant isomorphism testing between two unknown Boolean functions f and g. We give an algorithm for this problem whose query complexity only depends on the (unknown) smallest k such that either f or g is close to being a k-junta.

AB - The function f :{-1, 1}n → {-1, 1} is a k-junta if it depends on at most k of its variables. We consider the problem of tolerant testing of k-juntas, where the testing algorithm must accept any function that is ϵ-close to some k-junta and reject any function that is ϵ0-far from every k0-junta for some ϵ0 = O( ϵ) and k0 = O(k). Our first result is an algorithm that solves this problem with query complexity polynomial in k and 1/ϵ. This result is obtained via a new polynomialtime approximation algorithm for submodular function minimization (SFM) under large cardinality constraints, which holds even when only given an approximate oracle access to the function. Our second result considers the case where k0 = k. We show how to obtain a smooth tradeoff between the amount of tolerance and the query complexity in this setting. Specifically, we design an algorithm that given ϵ 2 (0, 1/2) accepts any function that isí 8 -close to some k-junta and rejects any function that is ϵ-far from every k-junta. The query complexity of the algorithm is O( k log k/cp(1- p)ϵ). Finally, we show how to apply the second result to the problem of tolerant isomorphism testing between two unknown Boolean functions f and g. We give an algorithm for this problem whose query complexity only depends on the (unknown) smallest k such that either f or g is close to being a k-junta.

UR - http://www.scopus.com/inward/record.url?scp=85045560370&partnerID=8YFLogxK

U2 - 10.1137/1.9781611975031.138

DO - 10.1137/1.9781611975031.138

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AN - SCOPUS:85045560370

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 2113

EP - 2132

BT - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018

A2 - Czumaj, Artur

PB - Association for Computing Machinery

T2 - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018

Y2 - 7 January 2018 through 10 January 2018

ER -