Tight Bounds for the Distribution-Free Testing of Monotone Conjunctions

We improve both upper and lower bounds for the distribution-free testing of monotone conjunctions. Given oracle access to an unknown Boolean function f : {0, 1}n →{0, 1} and sampling oracle access to an unknown distribution D over {0, 1}n, we present an Õ(n/ǫ)-query algorithm that tests whether f is a monotone conjunction versus ǫ-far from any monotone conjunction with respect to D. This improves the previous best upper bound of Õ(n/ǫ) by Dolev and Ron [DR11] when 1/ǫ is small compared to n. For some constant ǫ0 > 0, we also prove a lower bound of Ω̃(n) for the query complexity, improving the previous best lower bound of Ω̃(n) by Glasner and Servedio [GS09]. Our upper and lower bounds are tight, up to a poly-logarithmic factor, when the distance parameter ǫ is a constant. Furthermore, the same upper and lower bounds can be extended to the distribution-free testing of general conjunctions, and the lower bound can be extended to that of decision lists and linear threshold functions.