Improved Constructions for Non-adaptive Threshold Group Testing

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(dg+2(log d) 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(du+1log(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(dg+3(log d) log n). Using state-of-the-art constructions of lossless condensers, however, we come up with explicit testing schemes with O(dg+3(log d)quasipoly(log n)) and O(dg+3+βpoly(log n)) measurements, for arbitrary constant β < 0.