In this paper, we introduce a variation of the group testing problem where each test is specified by an ordered subset of items, and returns the first defective item in the specified order. We refer to this as \textit{cascaded group testing} and the goal is to identify a small set of $K$ defective items amongst a collection of size $N$, using as few tests as possible. For the adaptive testing regime, we show that a simple scheme is able to find all defective items in at most $K$ tests, which is optimal. For the non-adaptive setting, we first come up with a necessary and sufficient condition for any collection of tests to be feasible for recovering all the defectives. Using this, we are able to show that any feasible non-adaptive strategy requires at least $\Omega(K^2)$ tests. In terms of achievability, it is easy to show that a collection of $O(K^2 \log (N/K))$ randomly constructed tests is feasible. We show via carefully constructed explicit designs that one can do significantly better. We provide two simple schemes for $K = 1, 2$ which only require one and two tests respectively irrespective of the number of items $N$. Note that this is in contrast to standard binary group testing, where at least $\Omega(\log N)$ tests are required. The case of $K \ge 3$ is more challenging and here we come up with an iterative design which requires only $\text{poly}(\log \log N)$ tests.

翻译：本文提出了一种群组检测问题的变体，其中每个检测由项目的有序子集定义，并返回指定顺序中的首个缺陷项目。我们将此称为\textit{级联群组检测}，其目标是在包含$N$个项目的集合中，用尽可能少的检测识别出包含$K$个缺陷项目的小型集合。对于自适应检测机制，我们证明一种简单方案最多仅需$K$次检测即可找出所有缺陷项目，该结果已达到最优。在非自适应检测场景中，我们首先提出了任何检测集合能够成功恢复所有缺陷项目的充分必要条件。基于此条件，我们证明了任何可行的非自适应策略至少需要$\Omega(K^2)$次检测。在可实现性方面，容易证明由$O(K^2 \log (N/K))$个随机构建的检测组成的集合是可行的。通过精心构建的显式设计方案，我们证明了可以取得显著更优的结果：针对$K = 1, 2$的情况，我们提出了两种简单方案，分别仅需1次和2次检测，且与项目总数$N$无关。值得注意的是，这与标准二进制群组检测形成鲜明对比——后者至少需要$\Omega(\log N)$次检测。$K \ge 3$的情况更具挑战性，为此我们设计了一种迭代方案，仅需$\text{poly}(\log \log N)$次检测即可完成识别。