Information-Theoretic and Algorithmic Thresholds for Group Testing

In the group testing problem we aim to identify a small number of infected individuals within a large population. We avail ourselves to a procedure that can test a group of multiple individuals, with the test result coming out positive iff at least one individual in the group is infected. With all tests conducted in parallel, what is the least number of tests required to identify the status of all individuals? In a recent test design [Aldridge et al. 2016] the individuals are assigned to test groups randomly, with every individual joining an equal number of groups. We pinpoint the sharp threshold for the number of tests required in this randomised design so that it is information-theoretically possible to infer the infection status of every individual. Moreover, we analyse two efficient inference algorithms. These results settle conjectures from [Aldridge et al. 2014, Johnson et al. 2019]

On the All-or-Nothing Behavior of Bernoulli Group Testing

In this paper, we study the problem of non-adaptive group testing, in which one seeks to identify which items are defective given a set of suitably-designed tests whose outcomes indicate whether or not at least one defective item was included in the test. The most widespread recovery criterion seeks to exactly recover the entire defective set, and relaxed criteria such as approximate recovery and list decoding have also been considered. In this paper, we study the fundamental limits of group testing under the significantly relaxed {\em weak recovery} criterion, which only seeks to identify a small fraction (e.g., $0.01$) of the defective items. Given the near-optimality of i.i.d.~Bernoulli testing for exact recovery in sufficiently sparse scaling regimes, it is natural to ask whether this design additionally succeeds with much fewer tests under weak recovery. Our main negative result shows that this is not the case, and in fact, under i.i.d.~Bernoulli random testing in the sufficiently sparse regime, an {\em all-or-nothing} phenomenon occurs: When the number of tests is slightly below a threshold, weak recovery is impossible, whereas when the number of tests is slightly above the same threshold, high-probability exact recovery is possible. In establishing this result, we additionally prove similar negative results under Bernoulli designs for the weak detection problem (distinguishing between the group testing model vs.~completely random outcomes) and the problem of identifying a single item that is definitely defective. On the positive side, we show that all three relaxed recovery criteria can be attained using considerably fewer tests under suitably-chosen non-Bernoulli designs.Comment: (v2) Added section on non-i.i.d. test matrices, including optimal approximate recovery threshold. (v3) Final version accepted to IEEE Journal on Selected Areas in Information Theory (JSAIT

A Fast Binary Splitting Approach to Non-Adaptive Group Testing

In this paper, we consider the problem of noiseless non-adaptive group testing under the for-each recovery guarantee, also known as probabilistic group testing. In the case of $n$ items and $k$ defectives, we provide an algorithm attaining high-probability recovery with $O(k \log n)$ scaling in both the number of tests and runtime, improving on the best known $O(k^2 \log k \cdot \log n)$ runtime previously available for any algorithm that only uses $O(k \log n)$ tests. Our algorithm bears resemblance to Hwang's adaptive generalized binary splitting algorithm (Hwang, 1972); we recursively work with groups of items of geometrically vanishing sizes, while maintaining a list of "possibly defective" groups and circumventing the need for adaptivity. While the most basic form of our algorithm requires $\Omega(n)$ storage, we also provide a low-storage variant based on hashing, with similar recovery guarantees.Comment: Accepted to RANDOM 202