Improved Constructions for Non-adaptive Threshold Group Testing

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 ≫ 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 passes a fixed threshold u, negative if this number is below a fixed lower threshold ℓ ≤ 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 (logd) log(n/d)) measurements (where g : = u − ℓ) 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 (logd) logn). Using state-of-the-art constructions of lossless condensers, however, we come up with explicit testing schemes with O(d ^g + 3 (logd) quasipoly(logn)) and O(d ^{g + 3 + β} poly(logn)) measurements, for arbitrary constant β> 0.