Group Testing in Arbitrary Hypergraphs and Related Combinatorial Structures

Abstract

We consider 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. We study non-adaptive algorithms, two-stage algorithms, and three-stage algorithms. Non-adaptive group testing algorithms are algorithms in which all tests are decided beforehand and therefore can be performed in parallel, whereas two-stage and three-stage group testing algorithms consist of two stages and three stages, respectively, with each stage being a completely non-adaptive algorithm. In classical group testing, the potentially infected sets are all subsets of up to a certain number of elements of the given input set. For classical group testing, it is known that there exists a correspondence between non-adaptive algorithms and superimposed codes, and between two-stage group testing and disjunctive list-decoding codes and selectors. Bounds on the number of tests for those algorithms are derived from the bounds on the dimensions of the corresponding combinatorial structures. Obviously, the upper bounds for the classical case apply also to our group testing model. In the present paper, we aim at improving on those upper bounds by leveraging on the characteristics of the particular hypergraph at hand. In order to cope with our version of the problem, we introduce new combinatorial structures that generalize the notions of classical selectors and superimposed codes.