A polynomial lower bound for testing monotonicity

Abstract

We show that every algorithm for testing n-variate Boolean functions for monotonicityhas query complexity Ω(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 Ω(logn). 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.