Group Testing with Selectable Thresholds

2026-07-15

AI Summary

This work addresses the group testing problem of identifying $k$ defective items among $n$ by introducing an adjustable-threshold model, where each test returns positive if the number of defectives in the tested group meets or exceeds a pre-specified threshold. Building on this framework, the authors design efficient test matrices and decoding algorithms, complemented by information-theoretic analysis. They establish tight achievability and converse bounds under fixed thresholds and demonstrate that, when the maximum threshold is unbounded or sufficiently large, recovery succeeds with high probability near the information-theoretic limit. Notably, in the dense regime where $k = \Theta(n^\theta)$ with $\theta \to 1$, the upper and lower bounds on the number of tests coincide, and in the unbounded-threshold setting, the testing rate approaches the theoretical maximum of one.

Abstract

We consider the problem of group testing, in which one seeks to identify a subset of defective items of size $k$ from a larger set of $n$ items based on pooled tests. We introduce a selectable threshold model, in which each test has an associated threshold that can be chosen, such that the test outcome is 1 if and only if the number of defectives in the test is no smaller than that threshold. In settings with a large or unbounded maximum threshold, we establish conditions under which high-probability recovery can be attained with a rate (i.e., the asymptotic ratio of $\log_2{n \choose k}$ to the number of tests) approaching its maximum possible value of 1. Moreover, in the case of a fixed maximum threshold, we establish an achievable number of tests using simple and computationally efficient decoding methods, and a converse that holds under suitable regularity conditions on the test design, with the two coinciding in the dense limit (i.e., $θ$ approaching one in the scaling $k = Θ(n^θ)$).

Problem

• Group Testing
• Selectable Thresholds
• Defective Items
• High-Probability Recovery
• Test Design

Innovation

• group testing
• selectable thresholds
• high-probability recovery
• achievable rate
• efficient decoding