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：我们考虑组测试的一种推广形式，其中潜在污染集是给定超图 ${\cal F}=(V,E)$ 中的成员。该推广适用于污染可能受某种社会及地理聚类影响的情境。我们研究非自适应算法、两阶段算法以及三阶段算法。非自适应组测试算法是指所有测试均事先确定、因而可并行执行的算法；而两阶段组测试算法与三阶段组测试算法则分别由两个阶段和三个阶段组成，每个阶段均为完全非自适应的算法。在经典组测试中，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。已知对于经典组测试，经典叠加码与非自适应算法之间存在对应关系，两阶段算法与选择器之间也存在对应关系。这些算法所需测试次数的界由相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组测试模型。本文旨在利用特定超图的特性，改进这些上界。为处理我们所提出的问题版本，我们引入了新的组合结构，这些结构推广了经典选择器与叠加码的概念。