Conference paper

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(d^{g+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(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 come up with explicit testing schemes with O(d^{g+3} (\log d) quasipoly(log n)) and O(d^{g+3+beta} poly(log n)) measurements, for arbitrary constant beta > 0.


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