Optimal group testing

In the group testing problem the aim is to identify a small set of $k\sim n^\theta$infectedindividualsoutofapopulationsize$n$,$0<\theta<1$. We avail ourselves of a test procedure capable of testing groups of individuals, with the test returning a positive result iff at least one individual in the group is infected. The aim is to devise a test design with as few tests as possible so that the set of infected individuals can be identified correctly with high probability. We establish an explicit sharp information-theoretic/algorithmic phase transition $\text{minf}$ for non-adaptive group testing, where all tests are conducted in parallel. Thus, with more than $\text{minf}$ tests the infected individuals can be identified in polynomial time \whp, while learning the set of infected individuals is information-theoretically impossible with fewer tests. In addition, we develop an optimal adaptive scheme where the tests are conducted in two stages.