A class of asymptotically optimal group screening strategies with limited item participation

Abstract

In many fault-detection problems, we want to identify defective items from a set of n items using the minimum number of tests. Group testing is a scenario in which each test is on a subset of items and determines whether the subset contains at least one defective item. We study a variant of the classical group testing problem with an unknown number of defective items, in which each item participates in no more than a fixed number of tests. Besides being interesting on its own right, investigating the above problem provides an approach to tackle another variant of the group testing problem with an unknown number of defective items, in which the number of positive responses is limited. For the latter problem, existing works all assume that the number of defective items d is known in advance. However, in practice d is usually unknown.For both the above two group testing problems with unknown number of defective items, based on previous work in De Bonis (2016), we give a conditional lower bound on the number of tests required in the worst case. Our main contribution is proposing a class of recursive testing strategies Af for integers f≥1, such that for strategy Af each item participates in at most f tests. For constant f, strategy Af is asymptotically optimal for both the above two problems, as long as the lower bound condition n≥22fd holds.