Optimal pooling designs with error detection
Abstract
Consider 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 known {\em a priori}. Here, practical considerations lead us to impose the additional requirement of {\em 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 maximumsize collections of subsets of a finite set satisfying a union condition which generalizes that considered by Erdös \etal \cite{erd}. 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 1994
 arXiv:
 arXiv:math/9410210
 Bibcode:
 1994math.....10210B
 Keywords:

 Mathematics  Combinatorics;
 05B40
 EPrint:
 10 pages