Efficient Group Testing Algorithms with a Constrained Number of Positive Responses

Group testing is a well known search problem that consists in detecting the defective members of a set of objects O by performing tests on properly chosen subsets (pools) of the given set O. In classical group testing the goal is to find all defectives by using as few tests as possible. We consider a variant of classical group testing in which one is concerned not only with minimizing the total number of tests but aims also at reducing the number of tests involving defective elements. The rationale behind this search model is that in many practical applications the devices used for the tests are subject to deterioration due to exposure to or interaction with the defective elements. As an example, consider the leak testing procedures aimed at guaranteeing safety of sealed radioactive sources. Personnel involved in these procedures are at risk of being exposed to radiation whenever a leak in the tested sources is present. The number of positive tests admitted by a leak testing procedure should depend on the dose of radiation which is judged to be of no danger for the health. Obviously, the total number of tests should also be taken into account in order to reduce the costs and the work load of the safety personnel. In this paper we consider both adaptive and nonadaptive group testing and for both scenarios we provide almost matching upper and lower bounds on the number of “yes” responses that must be admitted by any strategy performing at most a certain number t of tests. The lower bound for the non adaptive case follows from the upper bound on the optimal size of a variant of d-cover free families introduced in this paper, which we believe may be of interest also in other contexts.