Connor Riddlesden

Remco van der Hofstad, Noela Müller, Connor Riddlesden

We study the Threshold Group Testing (TGT) problem in the noiseless and non-adaptive setting, where the objective is to exactly recover a sparse binary vector from pooled tests, using as few tests as possible. In TGT, each test applied to a subset of items returns a positive outcome if the number of 1's (defective items) in that subset meets or exceeds a specified threshold, and has a negative outcome otherwise. We investigate how the complexity of TGT compares to that of Classical Group Testing (CGT), corresponding to the special case of the threshold equal to one, and analyse the impact of increasing the threshold on the required number of tests. Our main contribution is the derivation of a sharp information-theoretic phase transition at $c_{\mathrm{inf}}^{\mathrm{TGT}}k\log(n/k)$ (non-adaptive) tests for TGT within the constant-column test design. The threshold constant $c_{\mathrm{inf}}^{\mathrm{TGT}}$ is expressed as a function of the prevalence of defectives and the threshold value. Our upper bound is derived under an analytic assumption, and we verify that this assumption is satisfied for a threshold value of 2. The value of $c_{\mathrm{inf}}^{\mathrm{TGT}}$ reveals that TGT on the constant-column design has the same information-theoretic behaviour as CGT in the low-prevalence regime. Yet, strikingly, at higher prevalences, the threshold leads to a significant reduction in the number of tests. On the other hand, we provide evidence that when the asymptotic proportion of defective items is positive, TGT actually becomes strictly harder than CGT (excluding trivial reductions).

Amin Coja-Oghlan, Remco van der Hofstad, Lena Krieg et al.

We study the Threshold Group Testing (TGT) problem without a gap in the noiseless, non-adaptive setting, where the goal is to exactly recover a sparse binary vector from pooled test outcomes using as few tests as possible. In TGT, a test applied to a subset of items returns a positive outcome if the number of defective items in the subset reaches a prescribed threshold, and a negative outcome otherwise. Under the assumption of an analytic condition, TGT has been shown to undergo a sharp information-theoretic phase transition for exact recovery on the class of constant-column test designs. In this paper, we develop an efficient inference algorithm that achieves exact recovery with high probability using the minimum number of non-adaptive tests that are needed for the constant-column design, thereby matching the information-theoretic threshold of a natural benchmark test design. Our approach is based on a spatially coupled test design and admits a significantly simpler analysis than existing algorithms for related group testing problems. In particular, unlike previous methods for binary group testing, our algorithm does not rely on the analysis of intricate weighted sums. This leads to a more straightforward proof technique, while still allowing near-optimal performance guarantees.