Optimal adaptive group testing

The group testing problem is concerned with identifying a small number $k \sim n^\theta$ for $\theta \in (0,1)$ of infected individuals in a large population of size $n$. At our disposal is a testing procedure that allows us to test groups of individuals. This paper considers two-stage designs where the test results of the first stage can inform the design of the second stage. We are interested in the minimum number of tests to recover the set of infected individuals w.h.p. Equipped with a novel algorithm for one-stage group testing from [Coja-Oghlan, Gebhard, Hahn-Klimroth \& Loick 2019], we propose a polynomial-time two-stage algorithm that matches the universal information-theoretic lower bound of group testing. This result improves on results from [M\'ezard \& Toninelli 2011] and resolves open problems prominently posed in [Aldridge, Johnson \& Scarlett 2019, Berger \& Levenshtein 2002, Damaschke \& Muhammad 2012].