Improved Constructions for Non-adaptive Threshold Group Testing

TL;DR

This paper advances non-adaptive threshold group testing by providing improved measurement bounds and explicit constructions, enabling efficient identification of defectives in large populations, even in noisy settings.

Contribution

It introduces new upper bounds for measurements needed and develops explicit construction frameworks using lossless condensers for threshold group testing.

Findings

Achieves $O(d^{g+2} (\log d) \log(n/d))$ measurement bound, improving previous results.

Provides explicit measurement schemes with near-optimal bounds using lossless condensers.

Establishes almost matching lower bounds, confirming the efficiency of the proposed methods.

Abstract

The basic goal in combinatorial group testing is to identify a set of up to $d$ defective items within a large population of size $n \gg d$ using a pooling strategy. Namely, the items can be grouped together in pools, and a single measurement would reveal whether there are one or more defectives in the pool. The threshold model is a generalization of this idea where a measurement returns positive if the number of defectives in the pool reaches a fixed threshold $u > 0$, negative if this number is no more than a fixed lower threshold $\ell < u$, and may behave arbitrarily otherwise. We study non-adaptive threshold group testing (in a possibly noisy setting) and show that, for this problem, $O(d^{g+2} (\log d) \log(n/d))$ measurements (where $g := u-\ell-1$ and $u$ is any fixed constant) suffice to identify the defectives, and also present almost matching lower bounds. This significantly improves the previously known (non-constructive) upper bound $O(d^{u+1} \log(n/d))$. Moreover, we obtain a framework for explicit construction of measurement schemes using lossless condensers. The number of measurements resulting from this scheme is ideally bounded by $O(d^{g+3} (\log d) \log n)$. Using state-of-the-art constructions of lossless condensers, however, we obtain explicit testing schemes with $O(d^{g+3} (\log d) qpoly(\log n))$ and $O(d^{g+3+\beta} poly(\log n))$ measurements, for arbitrary constant $\beta > 0$.