We consider a generalization of group testing where the potentially contaminated sets are the members of a given hypergraph ${\cal F}=(V,E)$. 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 group testing algorithms and three-stage group testing algorithms are algorithms that 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 classical superimposed codes and non-adaptive algorithms, and between two stage algorithms 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.

翻译：我们考虑群组测试的一种推广，其中潜在污染集是给定超图${\cal F}=(V,E)$中的成员。这一推广可应用于污染受某种社会与地理聚类条件影响的场景。我们研究非自适应算法、两阶段算法和三阶段算法。非自适应群组测试算法中所有测试事先确定并可并行执行，而两阶段与三阶段群组测试算法分别由两个及三个完全非自适应算法的阶段组成。经典群组测试中，潜在感染集是给定输入集中不超过特定元素数量的所有子集。已知经典群组测试中，非自适应算法对应经典叠加码，两阶段算法对应选择器。这些算法所需测试次数的界由对应组合结构的维度界推导得出。显然，经典情形下的上界同样适用于我们的群组测试模型。本文旨在通过利用特定超图的特性改进这些上界。为应对问题的变体，我们引入推广了经典选择器与叠加码概念的新型组合结构。