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 log 2 n + o ( log 2 n ) .