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|).

翻译：近期论文开创了对群体检测泛化问题的研究，其中潜在污染集合是给定超图F=(V,E)中的成员。这种泛化在污染可能受某些社会和地理聚类影响的场景中具有应用价值。本文聚焦于少阶段群体检测算法，即具有一定适应性的算法：测试按阶段进行，且同一阶段的所有测试需在该阶段开始时确定。具体而言，本文首次提出一种两阶段算法，对于超边规模不超过d的一般超图，该算法使用o(dlog|E|)次测试；同时提出一种三阶段算法，其测试次数比已知最佳三阶段算法提升了d^{1/6}因子。这些算法是针对任意正整数s≤d设计的s阶段算法的特例。该算法的设计借鉴了一种新型非自适应算法（单阶段算法），即所有测试必须事先确定的算法。此外，我们推导了非自适应群体检测的下界。当E足够大时，该下界非常接近文献中已知最佳非自适应群体检测算法的测试次数上界，并且是首个优于信息论下界Ω(log|E|)的下界。