New Algorithms and Lower Bounds for Monotonicity Testing.
X. Chen and R. Servedio and L.-Y. Tan.
In 54th Annual Symposium on Foundations of Computer Science (FOCS), 2014, pp. 286--295.

Abstract: We consider the problem of testing whether an unknown Boolean function f:\{-1,1\}^n \to \{-1,1\} is monotone versus epsilon-far from every monotone function. The two main results of this paper are a new lower bound and a new algorithm for this well-studied problem.

Lower bound: We prove an \Omega(n^{1/5}) lower bound on the query complexity of any non-adaptive two-sided error algorithm for testing whether an unknown Boolean function f is monotone versus constant-far from monotone. This gives an exponential improvement on the previous lower bound of \Omega(log n) due to Fischer et al.. We show that the same lower bound holds for monotonicity testing of Boolean-valued functions over hypergrid domains \{1,...,m\}^n for all m \geq 2.

Upper bound: We present an \tilde{O}(n^{5/6})\poly(1/\epsilon)-query algorithm that tests whether an unknown Boolean function f is monotone versus \epsilon-far from monotone. Our algorithm, which is non-adaptive and makes one-sided error, is a modified version of the algorithm of Chakrabarty and Seshadhri [2], which makes \tilde{O}(n^{7/8})\poly(1/\epsilon) queries.

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