A randomized competitive group testing procedure

In many fault detection problems, we want to identify all defective items from a set of n items using the minimum number of tests. Group testing is for the scenario where each test is on a subset of items, and tells whether the subset contains at least one defective item or not. In practice, the number d of defective items is often unknown in advance. In this paper, we propose a randomized group testing procedure RGT for the scenario where the number d of defectives is unknown in advance, and prove that RGT is competitive. By incorporating numerical results, we obtain improved upper bounds on the expected number of tests performed by RGT, for \(1\le d\le 10^6\). In particular, for \(1\le d\le 10^6\) and the special case where n is a power of 2, we obtain an upper bound of \(d\log \frac{n}{d}+Cd+O(\log d)\) with \(C\approx 2.67\) on the expected number of tests performed by RGT, which is better than the currently best upper bound in Cheng et al. (INFORMS J Comput 26(4):677–689, 2014). We conjecture that the above improved upper bounds based on numerical results from \(1\le d\le 10^6\) actually hold for all \(d\ge 1\).