A tight lower bound on non-adaptive group testing estimation

Efficiently counting or detecting defective items is a crucial task in various fields ranging from biological testing to quality control to streaming algorithms. The group testing estimation problem concerns estimating the number of defective elements d in a collection of n total within a given factor. We primarily consider the classical query model, in which a query reveals whether the selected group of elements contains a defective one. We show that any non-adaptive randomized algorithm that estimates the value of d within a constant factor requires Ω(logn) queries. This confirms that a known O(logn) upper bound by Bshouty (2019) is tight and resolves a conjecture by Damaschke and Sheikh Muhammad (2010). Additionally, we prove similar matching upper and lower bounds in the threshold query model.