Individual Testing of Independent Items in Optimal Group Testing

Abstract

We consider the group testing problem for a set of independent items I = [I1,… In] where Ii, has probability pi, of being defective and probability qi = 1 – pi of being good. The problem is to classify all items as good or defective with a minimum expected number of group tests where a group test is a test on a subset S of I with two possible outcomes: either S is pure (contains no defective) or S is contaminated (contains at least one defective, with no information provided about which or how many). No polynomial-time algorithm is known for the group testing problem even for the special case pi = p for all i. Hence, any method that reduces the size of the problem is very helpful. In this paper, we give such a method by providing a simple condition to screen items that should be tested (only) individually. This condition leads to a necessary and sufficient condition for the individual testing algorithm to be optimal, generalizing a result of Unger [1] for the special case of identical pi.