Tolerant Junta Testing and the Connection to Submodular Optimization and Function Isomorphism

Eric Blais, Clement L. Canonne, Talya Eden, Amit Levi, Dana Ron

Research output: Contribution to journalArticlepeer-review


A 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 €-close to some k-junta and reject any function that is €-far from every k-junta for some € = O(€ ) and k = O(k). Our first result is an algorithm that solves this problemwith query complexity polynomial in k and 1/€. This result is obtained via a new polynomial-Time 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 k = 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 p (0, 1), accepts any function that is €p 16-close to some k-junta and rejects any function that is €-far from every k-junta. The query complexity of the algorithm is O( k log k €p(1-p)k ). Finally, we show how to apply the second result to the problem of tolerant isomorphism testing between two unknown Boolean functions f and. We give an algorithm for this problem whose query complexity only depends on the (unknown) smallest k such that either f or is close to being a k-junta.

Original languageEnglish
Article number24
JournalACM Transactions on Computation Theory
Issue number4
StatePublished - Sep 2019
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2019 Association for Computing Machinery. All rights reserved.


  • Boolean functions
  • Property testing
  • function isomorphism
  • juntas
  • submodular optimization
  • tolerant testing

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics


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