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Testing Odd-Cycle-Freeness in Boolean Functions

Published online by Cambridge University Press:  10 August 2012

ARNAB BHATTACHARYYA ,
ELENA GRIGORESCU ,
PRASAD RAGHAVENDRA  and
ASAF SHAPIRA
ARNAB BHATTACHARYYA
Affiliation:
Center for Computational Intractability, Olden Street, Princeton, NJ 08540, USA (e-mail: arnabb@princeton.edu)
ELENA GRIGORESCU
Affiliation:
College of Computing, Georgia Tech, 801 Atlantic Drive, Atlanta, GA 30332, USA (e-mail: elena_g@csail.mit.edu, raghavendra@cc.gatech.edu, asafico@math.gatech.edu
PRASAD RAGHAVENDRA
Affiliation:
College of Computing, Georgia Tech, 801 Atlantic Drive, Atlanta, GA 30332, USA (e-mail: elena_g@csail.mit.edu, raghavendra@cc.gatech.edu, asafico@math.gatech.edu
ASAF SHAPIRA
Affiliation:
College of Computing, Georgia Tech, 801 Atlantic Drive, Atlanta, GA 30332, USA (e-mail: elena_g@csail.mit.edu, raghavendra@cc.gatech.edu, asafico@math.gatech.edu School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
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Abstract

A function f: 2n → {0,1} is odd-cycle-free if there are no x1,. . .,xk ∈ 2n 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 issue we study is directly reducing testing of 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 already been shown (Král’, Serra and Vena, and Shapira) 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 whether the restriction of the input function to the subspace satisfies a fixed property. The question is if, for every linear-invariant property, there is a canonical tester for which there is only a polynomial blow-up from the optimal query complexity. We answer the question affirmatively for odd-cycle-freeness. The general question remains open.

Keywords

Primary 68Q0168Q87
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 6 , November 2012 , pp. 835 - 855
Copyright
Copyright © Cambridge University Press 2012

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