Optimal Pooling Designs with Error Detection

AbstractConsider a collection of objects, some of which may be “bad,” and a test which determines whether or not a given subcollection contains no bad objects. The nonadaptive pooling (or group testing) problem involves identifying the bad objects using the least number of tests applied in parallel. The “hypergeometric” case occurs when an upper bound on the number of bad objects is knowna priori. Here, practical considerations lead us to impose the additional requirement ofa posterioriconfirmation that the bound is satisfied. A generalization of the problem in which occasional errors in the test outcomes can occur is also considered. Optimal solutions to the general problem are shown to be equivalent to maximum-size collections of subsets of a finite set satisfying a union condition which generalizes that considered by Erdős and co-workers. Lower bounds on the number of tests required are derived when the number of bad objects is believed to be either 1 or 2. Steiner systems are shown to be optimal solutions in some cases