Optimal Non-Adaptive Probabilistic Group Testing Requires $Θ(\min\{k \log n, n\})$ Tests

In this paper, we consider the problem of noiseless non-adaptive probabilistic group testing, in which the goal is high-probability recovery of the defective set. We show that any non-adaptive group testing strategy requires $\Omega( \min\{k \log n, n\} )$ tests in the case of $n$ items among which $k$ are defective, thus matching the $O( \min\{k \log n, n\} )$ upper bound obtained by taking the better of random testing and individual testing. This strengthens previous converse results that are only tight/valid in the regimes $k \le n^{1-\Omega(1)}$ and/or $k = \Theta(n)$. When specialized to the regime $k = \omega\big( \frac{n}{ \log n } \big)$ (including the linear regime $k = \Theta(n)$), we additionally prove the stronger statement that individual testing is asymptotically optimal for any non-zero target success probability, thus strengthening an existing result of Aldridge (2019) in terms of both the error probability and the assumed scaling of $k$.