A Tight Lower Bound of $\mathrm{\Omega }(\log n)$ for the Estimation of the Number of Defective Items

Let $X$ be a set of items of size $n$ , which may contain some defective items denoted by $I$, where $I\subseteq X$. In group testing, a {\it test} refers to a subset of items $Q\subset X$. The test outcome is $1$ (positive) if $Q$ contains at least one defective item, i.e., $Q\cap I\mathrm{\ne }\mathrm{\varnothing }$, and $0$ (negative) otherwise. We give a novel approach to obtaining tight lower bounds in non-adaptive randomized group testing. Employing this new method, we can prove the following result. Any non-adaptive randomized algorithm that, for any set of defective items $I$, with probability at least $2\mathrm{/}3$, returns an estimate of the number of defective items $\mathrm{\mid }I\mathrm{\mid }$ to within a constant factor requires at least $\mathrm{\Omega }(\log n)$ tests. Our result matches the upper bound of $O(\log n)$ and solves the open problem posed by Damaschke and Sheikh Muhammad.