A Polynomial Lower Bound for Testing Monotonicity

We show that every algorithm for testing n-variate Boolean functions for monotonicity must have query complexity tildeOmega(n1/4). 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(n1/2) 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.