A polynomial lower bound for testing monotonicity

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

Authors

• Aleksandrs Belovs
• Eric Blais

Keywords

• Adaptivity of query algorithms
• Property Testing
• Talagrand's Random DNF