Strict group testing and the set basis problem

Group testing is the problem to identify up to d defectives out of n elements, by testing subsets for the presence of defectives. Let t(n, d, s) be the optimal number of tests needed by an s-stage strategy in the strict group testing model where the searcher must also verify that at most d defectives are present. We start building a combinatorial theory of strict group testing. We compute many exact t(n, d, s) values, thereby extending known results for s = 1 to multistage strategies. These are interesting since asymptotically nearly optimal group testing is possible already in s = 2 stages. Besides other combinatorial tools we generalize d-disjunct matrices to any candidate hypergraphs, and we reveal connections to the set basis problem and communication complexity. As a proof of concept we apply our tools to determine almost all test numbers for n ≤ 10 and some further t(n, 2, 2) values. We also show t(n, 2, 2) ≤ 2.44 log2 n+ o(log2 n).