A Toolbox for Provably Optimal Multistage Strict Group Testing Strategies

Group testing is the problem of identifying up to d defectives in a set 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 no more than d defectives are present. We develop combinatorial tools that are powerful enough to compute many exact t(n, d, s) values. This extends the work of Huang and Hwang (2001) for s = 1 to multistage strategies. The latter are interesting since it is known that asymptotically nearly optimal group testing is possible already in s = 2 stages. Besides other tools we generalize d-disjunct matrices to any candidate hypergraphs, which enables us to express optimal test numbers for s = 2 as chromatic numbers of certain conflict graphs. As a proof of concept we determine almost all test numbers for n ≤ 10, and t(n, 2, 2) for some