Improved bounds for group testing in arbitrary hypergraphs

Recent papers initiated the study of a generalization of group testing where the potentially contaminated sets are the members of a given hypergraph F=(V,E). This generalization finds application in contexts where contaminations can be conditioned by some kinds of social and geographical clusterings. The paper focuses on few-stage group testing algorithms, i.e., slightly adaptive algorithms where tests are performed in stages and all tests performed in the same stage should be decided at the very beginning of the stage. In particular, the paper presents the first two-stage algorithm that uses o(dlog|E|) tests for general hypergraphs with hyperedges of size at most d, and a three-stage algorithm that improves by a d^1/6 factor on the number of tests of the best known three-stage algorithm. These algorithms are special cases of an s-stage algorithm designed for an arbitrary positive integer s<= d. The design of this algorithm resort to a new non-adaptive algorithm (one-stage algorithm), i.e., an algorithm where all tests must be decided beforehand. Further, we derive a lower bound for non-adaptive group testing. For E sufficiently large, the lower bound is very close to the upper bound on the number of tests of the best non-adaptive group testing algorithm known in the literature, and it is the first lower bound that improves on the information theoretic lower bound Omega(log |E|).