Tight bounds for the distribution-free testing of monotone conjunctions

Abstract

We improve both upper and lower bounds for the distributionfree 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 V over {0,1}", we present an Ō(n1/3/ϵ5)-query algorithm that tests whether f is a monotone conjunction versus E-far from any monotone conjunction with respect to D. This improves the previous best upper bound of Ō(n1/2/ϵ) by Dolev and Ron [DR11], when 1/ϵ is small compared to n. For some constant ϵo > 0, we also prove a lower bound of Ω(nM1/3) for the query complexity, improving the previous best lower bound of Ω(nM1/5) by Glasner and Servedio [GS09]. Our upper and lower bounds are tight, up to a polylogarithmic factor, when the distance parameter e 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.