A Polynomial Lower Bound for Testing Monotonicity

TL;DR

This paper establishes a near-quadratic lower bound on the query complexity for adaptive algorithms testing monotonicity of Boolean functions, and demonstrates an exponential gap between adaptive and non-adaptive testing for linear threshold functions.

Contribution

It provides the first polynomial lower bound for adaptive monotonicity testing and shows an exponential separation between adaptive and non-adaptive algorithms for certain function classes.

Findings

Any adaptive algorithm requires at least rac{1}{4} ilde{ olinebreak} n^{1/4} queries.

Non-adaptive algorithms have a lower bound of rac{1}{2} olinebreak ext{queries for linear threshold functions.

An adaptive algorithm can test monotonicity of regular LTFs with only O( olinebreak ext{log} n) queries.

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.