Optimal non-adaptive group testing
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In non-adaptive group testing we aim to identify a small set of k∼ n^θ infected individuals out of a population size n, 0<θ<1. We avail ourselves to a test procedure that can test a group of individuals, with the test rendering a positive result iff at least one individual in the group is infected. All tests are conducted in parallel. The aim is to devise a (possibly randomised) test design with as few tests as possible so that the infected individuals can be identified with high probability. We prove that there occurs a sharp information-theoretic/algorithmic phase transition as the number of tests passes an explicit threshold m_inf. Hence, if more than (1+ϵ)m_inf tests are conducted, then there exist a test design and a polynomial time algorithm that identifies the set of infected individuals with high probability. By contrast, identifying the infected individuals is information-theoretically impossible with fewer than (1-ϵ)m_inf tests. These results resolve problems prominently posed in [Aldridge et al. 2019, Johnson et al. 2018].
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