Probabilistic Group Testing with a Linear Number of Tests

In probabilistic nonadaptive group testing (PGT), we aim to characterize the number of pooled tests necessary to identify a random $k$-sparse vector of defectives with high probability. Recent work has shown that $n$ tests are necessary when $k =\omega(n/\log n)$. It is also known that $O(k \log n)$ tests are necessary and sufficient in other regimes. This leaves open the important sparsity regime where the probability of a defective item is $\sim 1/\log n$ (or $k = \Theta(n/\log n)$) where the number of tests required is linear in $n$. In this work we aim to exactly characterize the number of tests in this sparsity regime. In particular, we seek to determine the number of defectives $\lambda(\alpha)n / \log n$ that can be identified if the number of tests is $\alpha n$. In the process, we give upper and lower bounds on the exact point at which individual testing becomes suboptimal, and the use of a carefully constructed pooled test design is beneficial.