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TR23-141 | 19th September 2023 21:54
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#### A Tight Lower Bound of $\Omega(\log n)$ for the Estimation of the Number of Defective Items

**Abstract:**
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., $Q\cap I \neq \emptyset$, 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 $2/3$, returns an estimate of the number of defective items $|I|$ to within a constant factor requires at least

$\Omega({\log n})$ tests.

Our result matches the upper bound of $O(\log n)$ and solves the open problem posed by Damaschke and Sheikh Muhammad.