A toolbox for provably optimal multistage strict group testing strategies

Paper in proceeding, 2013

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 up to 10, and t(n,2,2) for some larger n.