Improved Constructions for Non-adaptive Threshold Group Testing

摘要

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 reaches a fixed threshold u > 0, negative if this number is no more than 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)(log d) log(n/d)) measurements (where g := u - ℓ - 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)(logd) log n). Using state-of-the-art constructions of lossless condensers, however, we obtain 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.