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 or returns null if there are no defectives. We refer to this as 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 perfect recovery. For the adaptive testing regime, we show that a simple scheme can 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 show that any feasible non-adaptive strategy requires at least $\Omega(K^2)$ tests. In terms of achievability, it is easy to show the existence of a feasible collection of $O(K^2 \log (N/K))$ tests. We show via carefully constructed explicit designs that one can do significantly better for constant $K$. While the cases $K = 1, 2$ are straightforward, the case $K=3$ is already non-trivial and we come up with an iterative design that is asymptotically optimal and requires $\Theta(\log \log N)$ tests. Note that this is in contrast to standard binary group testing, where at least $\Omega(\log N)$ tests are required. For constant $K \ge 3$, our iterative design requires only poly$(\log \log N)$ tests.

翻译：本文提出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；若无缺陷则返回空值。我们将此称为级联群组检测，其目标是在规模为$N$的项目集合中识别出少量$K$个缺陷项目，并以尽可能少的检测次数实现完全恢复。在自适应检测机制下，我们证明一种简单方案最多通过$K$次检测即可找出所有缺陷项目，该结果已达到最优。对于非自适应场景，我们首先推导出任何检测集合能够完全恢复所有缺陷项目的充要条件。基于此条件，我们证明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，容易证明存在由$O(K^2 \log (N/K))$次检测构成的可行集合。通过精心构建的显式设计方案，我们证明在常数$K$的情况下可取得显著更优的结果。当$K = 1, 2$时问题较为简单，而$K=3$的情况已具有非平凡性：我们提出一种渐进最优的迭代设计方案，仅需$\Theta(\log \log N)$次检测。值得注意的是，这与标准二元群组检测形成鲜明对比——后者至少需要$\Omega(\log N)$次检测。对于常数$K \ge 3$，我们的迭代设计方案仅需多对数级$\text{poly}(\log \log N)$次检测。