Optimal group testing

Abstract

In the group testing problem the aim is to identify a small set of k ⁓ n θ infected individuals out of a population size n, 0 < θ < 1. We avail ourselves of a test procedure capable of testing groups of individuals, with the test returning a positive result if and only if 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 minf for non-adaptive group testing, where all tests are conducted in parallel. Thus with more than minf tests the infected individuals can be identified in polynomial time with high probability, 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.

Dynamic group testing to control and monitor disease progression in a population

“…Bernoulli [28]- [30] and near-constant tests-per-item [31], [32] have been proved to be order-optimal in a sparse regime where k = Θ(N α ) and α ∈ (0, 1). In fact, in the same regime, [26] has provided the precise constants for optimal non-adaptive group testing. Conversely, classic individual testing has been proved to be optimal in the linear ( k = Θ(N )) [33] and the mildy sublinear regime ( k = ω( N log N )) [25].…”

“…• For the probabilistic model (ii), any non-adaptive algorithm with a success probability bounded away from zero as N → ∞ must have T = Ω min{ k log N, N } [25, Theorem 1], [26]. This means that either any non-adaptive group testing with a number of tests O( k log N ) is order optimal, or individual testing is order optimal 2 .…”

HYPER: Group testing via hypergraph factorization applied to COVID-19

“…In fact, this asymptotic efficiency matches that of three-stage Dorfman testing 21;64 . While more efficient designs for diminishing prevalence are available in the literature, they rely either on multiple stages 10;21 or on taking q much larger [48][49][50][51][52][53][54][55][56] ; each of which is outside our constraints. See Supplementary Methods for more discussion of some of these methods.…”

“…In particular, under the same statistical model as in our paper, Mezard and Toninelli 51 construct certain tests where each sample is placed into q = ln(1/p)/ ln 2 pools, and show that these attain asymptotic efficiency p ln(1/p). Coja-Oghlan et al 54 constructs a 2-stage algorithm with asymptotically optimal efficiency, requiring q = m ln(2)/(np). Gebhard et al 55 discusses similar proposals for the noisy case.…”

“…Numerous works provide statistical [21][22][23][24][25] , combinatorial [26][27][28][29][30][31] , as well as information and coding theoretic [32][33][34][35][36][37][38][39][40][41][42][43][44][45] perspectives, as well as software 46;47 to aid implementation, to name just a few. Additionally, there has been a lot of work on analyzing and optimizing group testing for various constraints and evaluation criteria [48][49][50][51][52][53][54][55][56][57][58][59] , often in the low prevalence regime. Broadly speaking, the approaches fall into three categories: i) single-stage (or nonadaptive) approaches that identify positive individuals after only one round of pooled tests by using pools with carefully designed overlaps; ii) two-stage [58][59][60][61] approaches that declare putative positives after one round of pooled tests.…”