Let $X$ be a set of items of size $n$ that contains 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 outcome of a test is $1$ if $Q$ contains at least one defective item, i.e., $Q\cap I \neq \emptyset$, and $0$ otherwise. We give a novel approach to obtaining lower bounds in non-adaptive randomized group testing. The technique produced lower bounds that are within a factor of $1/{\log\log\stackrel{k}{\cdots}\log n}$ of the existing upper bounds for any constant~$k$. Employing this new method, we can prove the following result. For any fixed constants $k$, any non-adaptive randomized algorithm that, for any set of defective items $I$, with probability at least $2/3$, returns an estimate of the number of defective items $|I|$ to within a constant factor requires at least $$\Omega\left(\frac{\log n}{\log\log\stackrel{k}{\cdots}\log n}\right)$$ tests. Our result almost matches the upper bound of $O(\log n)$ and solves the open problem posed by Damaschke and Sheikh Muhammad [COCOA 2010 and Discrete Math., Alg. and Appl., 2010]. Additionally, it improves upon the lower bound of $\Omega(\log n/\log\log n)$ previously established by Bshouty [ISAAC 2019].

翻译：设 $X$ 为包含 $n$ 个物品的集合，其中包含若干缺陷物品，记为 $I$，且 $I \subseteq X$。在群体测试中，测试指物品子集 $Q \subset X$。若 $Q$ 至少包含一个缺陷物品（即 $Q\cap I \neq \emptyset$），则测试结果为 $1$，否则为 $0$。我们提出了一种新颖的方法，用于在非自适应随机群体测试中获取下界。该技术所得到的下界与现有上界（对任意常数 $k$）的差距仅为因子 $1/{\log\log\stackrel{k}{\cdots}\log n}$。运用这一新方法，我们可证明以下结论：对任意固定常数 $k$，任何非自适应随机算法，若对于任意缺陷物品集合 $I$ 能以至少 $2/3$ 的概率将缺陷物品数量 $|I|$ 的估计值控制在常数因子误差范围内，则至少需要 $$\Omega\left(\frac{\log n}{\log\log\stackrel{k}{\cdots}\log n}\right)$$ 次测试。该结果几乎匹配 $O(\log n)$ 的上界，并解决了 Damaschke 与 Sheikh Muhammad [COCOA 2010 及 Discrete Math., Alg. and Appl., 2010] 提出的公开问题。此外，该结果改进了 Bshouty [ISAAC 2019] 此前建立的 $\Omega(\log n/\log\log n)$ 下界。