Optimal non-adaptive probabilistic group testing in general sparsity regimes

Abstract In this paper, we consider the problem of noiseless non-adaptive probabilistic group testing, in which goal is high-probability recovery defective set. We show that case $n$ items among $k$ are defective, smallest possible number tests equals $\min \{ C_{k,n} k \log n, n\}$ up to lower-order asymptotic terms, where $C_{k,n}$ a uniformly bounded constant (varying depending on scaling with respect $n$) simple explicit expression. The algorithmic upper bound follows from minor adaptation an existing analysis Definite Defectives algorithm, and algorithm-independent lower builds works for regimes $k \le n^{1-\varOmega (1)}$ = \varTheta (n)$. sufficiently sparse (including o\big ( \frac{n}{\log n} \big )$), our main result generalizes Coja-Oghlan et al. (2020) by avoiding assumption (1)}$, whereas dense \omega shows individual testing asymptotically optimal any non-zero target success probability, thus strengthening Aldridge (2019, IEEE Trans. Inf. Theory, 65, 2058–2061) terms both error probability assumed $k$.