2010/02/10 by Cheraghchi, Mahdi
#Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.1002.2244
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(dg+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(du+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(dg+3 (log d) log n). Using state-of-the-art constructions of lossless condensers, however, we obtain explicit testing schemes with O(dg+3 (log d) qpoly(log n)) and O(dg+3+β poly(log n)) measurements, for arbitrary constant β> 0.