A Polynomial Lower Bound for Testing Monotonicity

Aleksandrs Belovs; Eric Blais

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arXiv · 1511.05053

A Polynomial Lower Bound for Testing Monotonicity

Aleksandrs Belovs·Eric Blais

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Abstract

We show that every algorithm for testing $n$-variate Boolean functions for monotonicity must have query complexity $\tilde{\Omega}(n^{1/4})$. All previous lower bounds for this problem were designed for non-adaptive algorithms and, as a result, the best previous lower bound for general (possibly adaptive) monotonicity testers was only $\Omega(\log n)$. Combined with the query complexity of the non-adaptive monotonicity tester of Khot, Minzer, and Safra (FOCS 2015), our lower bound shows that adaptivity can result in at most a quadratic reduction in the query complexity for testing monotonicity. By contrast, we show that there is an exponential gap between the query complexity of adaptive and non-adaptive algorithms for testing regular linear threshold functions (LTFs) for monotonicity. Chen, De, Servedio, and Tan (STOC 2015) recently showed that non-adaptive algorithms require almost $\Omega(n^{1/2})$ queries for this task. We introduce a new adaptive monotonicity testing algorithm which has query complexity $O(\log n)$ when the input is a regular LTF.

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BibTeX RIS

Aleksandrs Belovs, Eric Blais. 2015-11-16. A Polynomial Lower Bound for Testing Monotonicity. https://arxiv.org/abs/1511.05053

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