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：近期论文开创了群体检测推广研究，其中潜在污染集合为给定超图F=(V,E)的成员。此类推广适用于污染受社会与地理聚类条件影响的场景。本文聚焦于少阶段群体检测算法，即各阶段测试需在阶段开始时决定的轻度自适应算法。具体而言，首次提出针对超边规模至多为d的一般超图采用o(dlog|E|)次测试的两阶段算法，以及将已知最优三阶段算法测试次数改进d^{1/6}因子的三阶段算法。这些算法是为任意正整数s≤d设计的s阶段算法的特例，其设计依赖于新型非自适应算法（单阶段算法），即所有测试需预先确定。此外，我们推导了非自适应群体检测的下界。当E足够大时，该下界与文献中已知最优非自适应群体检测算法的测试次数上界非常接近，且是首个超越信息论下界Ω(log |E|)的下界。