Efficiently counting or detecting defective items is a crucial task in various fields ranging from biological testing to quality control to streaming algorithms. The \emph{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 $\Omega(\log n)$ queries. This confirms that a known $O(\log n)$ 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.

翻译：高效计数或检测缺陷物品是生物学测试、质量控制、流算法等多个领域中的关键任务。\emph{群测试估计问题}涉及在给定因子内估计$n$个总元素集合中缺陷元素数量$d$。我们主要考虑经典查询模型，在该模型中，一次查询即可揭示所选元素组是否包含缺陷元素。我们证明，任何用于在常数因子内估计$d$值的非自适应随机算法都需要$\Omega(\log n)$次查询。这确认了Bshouty（2019）已知的$O(\log n)$上界是紧的，并解决了Damaschke和Sheikh Muhammad（2010）的一个猜想。此外，我们还在阈值查询模型中证明了类似的匹配上界和下界。