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：近期研究开始探讨群组检测的一种推广形式，其中可能被污染的集合是给定超图 F=(V,E) 中的成员。该推广适用于污染可能受某些社会和地理聚类影响的情境。本文聚焦于少阶段群组检测算法，即轻度自适应算法，其中检测按阶段进行，且同一阶段的所有检测必须在阶段开始时确定。特别地，本文提出了首个适用于超边规模不超过 d 的一般超图的两阶段算法，其检测次数为 o(dlog|E|)；同时提出一个三阶段算法，相比已知最优三阶段算法的检测次数改进了 d^{1/6} 倍。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。该算法的设计依赖于一种新的非自适应算法（单阶段算法），即所有检测必须预先确定的算法。此外，我们推导了非自适应群组检测的下界。当 E 足够大时，该下界与文献中已知最优非自适应群组检测算法的检测次数上界非常接近，并且是首个改进信息论下界 Omega(log |E|) 的下界。