## Abstract

A function f : F_{2}^{n} {0, 1} is odd-cycle-free if there are no x_{1},⋯,x_{k} ε F_{2}^{n} with k an odd integer such that f(x_{1}) = ⋯ = f(x_{k}) = 1 and x_{1} + ⋯ + X_{k} = 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 language | English |
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Title of host publication | Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 |

Publisher | Association for Computing Machinery |

Pages | 1140-1149 |

Number of pages | 10 |

ISBN (Print) | 9781611972108 |

DOIs | |

State | Published - 2012 |

Externally published | Yes |

Event | 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 - Kyoto, Japan Duration: 17 Jan 2012 → 19 Jan 2012 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Conference

Conference | 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 |
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Country/Territory | Japan |

City | Kyoto |

Period | 17/01/12 → 19/01/12 |

## Keywords

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