Optimal pooling designs with error detection

Consider a collection of objects, some of which may be ''bad,'' and a lest 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 known a priori. Here, practical considerations lead us to impose the additional requirement of a posteriori confirmation 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 Erdos 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. (C) 1996 Academic Press, Inc.