An Information-Theoretic Analysis of Threshold Group Testing

2026-06-09

AI Summary

This work investigates the information-theoretic limits of noiseless non-adaptive threshold group testing (TGT), where the goal is to exactly recover a sparse binary vector from pooled measurements whose outcomes are determined by whether the number of defective items meets or exceeds a prescribed threshold. Employing constant-column test designs and information-theoretic analysis, the study establishes—for the first time—the sharp phase transition point for TGT, with a particular focus on the case of threshold two. The theoretical results reveal that under low prevalence, the information-theoretic phase transition constant \(c_{\inf}^{\text{TGT}}\) coincides with that of classical group testing (CGT); however, in high-prevalence or linear-defect regimes, an appropriately chosen threshold can substantially reduce the required number of tests, demonstrating TGT’s potential to outperform CGT under specific conditions.

Abstract

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).

Problem

• Threshold Group Testing
• Sparse recovery
• Information-theoretic limits
• Non-adaptive testing
• Group testing

Innovation

• Threshold Group Testing
• information-theoretic phase transition
• non-adaptive group testing
• constant-column design
• sparse recovery