Testing odd-cycle-freeness in Boolean functions

Arnab Bhattacharyya*, Elena Grigorescu, Prasad Raghavendra, Asaf Shapira

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


A function f : F2n {0, 1} is odd-cycle-free if there are no x1,⋯,xk ε F2n with k an odd integer such that f(x1) = ⋯ = f(xk) = 1 and x1 + ⋯ + Xk = 0. We show that one can distinguish odd-cycle-free functions from those ε-far from being odd-cycle-free by making poly(1/ε) queries to an evaluation oracle. We give two proofs of this result, each shedding light on a different connection between testability of properties of Boolean functions and of dense graphs. The first problem we study is directly reducing testing linear-invariant properties of Boolean functions to testing associated graph properties. We show a black-box reduction from testing odd-cycle-freeness to testing bipartiteness of graphs. Such reductions have been shown previously (Král-Serra-Vena, Israel J. Math 2011; Shapira, STOC 2009) for monotone linear-invariant properties defined by forbidding solutions to a finite number of equations. But for odd-cycle-freeness whose description involves an infinite number of forbidden equations, a reduction to graph property testing was not previously known. If one could show such a reduction more generally for any linear-invariant property closed under restrictions to subspaces, then it would likely lead to a characterization of the one-sided testable linear-invariant properties, an open problem raised by Sudan. The second issue we study is whether there is an efficient canonical tester for linear-invariant properties of Boolean functions. A canonical tester for linear-invariant properties operates by picking a random linear subspace and then checking if the restriction of the input function to the subspace satisfies a fixed property or not. The question is whether for every linear-invariant property, there is a canonical tester for which there is only a polynomial blowup from the optimal query complexity. We answer the question affirmatively for odd-cycle-freeness. The general question still remains open.

Original languageEnglish
Title of host publicationProceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012
PublisherAssociation for Computing Machinery
Number of pages10
ISBN (Print)9781611972108
StatePublished - 2012
Externally publishedYes
Event23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 - Kyoto, Japan
Duration: 17 Jan 201219 Jan 2012

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012


  • Boolean functions
  • Cayley graphs
  • Fourier analysis
  • Property testing


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