## Abstract

A signed majority function is a linear threshold function f:{+1,−1}^{n}→{+1,−1} of the form $f(x)=\operatorname{sign}(\sum_{i=1}^{n} \sigma _{i} x_{i})$ where each σ_{i}∈{+1,−1}. Signed majority functions are a highly symmetrical subclass of the class of all linear threshold functions, which are functions of the form $\operatorname{sign} (\sum_{i=1}^{n} w_{i} x_{i} - \theta)$ for arbitrary real w_{i},θ.

We study the query complexity of testing whether an unknown f:{+1,−1}^{n}→{+1,−1} is a signed majority function versus ϵ-far from every signed majority function. While it is known (SIAM J. Comput. 39(5):2004–2047, 2010) that the broader class of all linear threshold functions is testable with poly(1/ϵ) queries (independent of n), prior to our work the best upper bound for signed majority functions was $O(\sqrt{n}) \cdot \mathrm{poly} (1/\epsilon)$ queries (via a non-adaptive algorithm), and the best lower bound was Ω(logn) queries for non-adaptive algorithms (Proceedings of the 13th International Workshop on Approximation, Randomization and Combinatorial Optimization (RANDOM), pp. 646–657, 2009).

As our main results we exponentially improve both these prior bounds for testing signed majority functions: (Upper bound) We give a poly(logn,1/ϵ)-query adaptive algorithm (which is computationally efficient) for this testing problem;(Lower bound) We show that any non-adaptive algorithm for testing the class of signed majorities to constant accuracy must make n^{Ω(1)} queries. This directly implies a lower bound of Ω(logn) queries for any adaptive algorithm. Our testing algorithm performs a sequence of restrictions together with consistency checks to ensure that each successive restriction is “compatible” with the function prior to restriction. This approach is used to transform the original n-variable testing problem into a testing problem over poly(logn,1/ϵ) variables where a simple direct method can be applied. Analysis of the degree-1 Fourier coefficients plays an important role in our proofs.

Original language | English |
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Pages (from-to) | 400-429 |

Number of pages | 30 |

Journal | Algorithmica |

Volume | 72 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jun 2015 |

## Keywords

- Linear threshold function
- Majority function
- Property testing

## All Science Journal Classification (ASJC) codes

- General Computer Science
- Applied Mathematics
- Computer Science Applications