In this paper, we introduce a variation of the group testing problem capturing the idea that a positive test requires a combination of multiple ``types'' of item. Specifically, we assume that there are multiple disjoint \emph{semi-defective sets}, and a test is positive if and only if it contains at least one item from each of these sets. The goal is to reliably identify all of the semi-defective sets using as few tests as possible, and we refer to this problem as \textit{Concomitant Group Testing} (ConcGT). We derive a variety of algorithms for this task, focusing primarily on the case that there are two semi-defective sets. Our algorithms are distinguished by (i) whether they are deterministic (zero-error) or randomized (small-error), and (ii) whether they are non-adaptive, fully adaptive, or have limited adaptivity (e.g., 2 or 3 stages). Both our deterministic adaptive algorithm and our randomized algorithms (non-adaptive or limited adaptivity) are order-optimal in broad scaling regimes of interest, and improve significantly over baseline results that are based on solving a more general problem as an intermediate step (e.g., hypergraph learning).

翻译：本文提出分组检测问题的一种变体，旨在刻画阳性检测结果需由多种“类型”项目组合引发的现象。具体而言，我们假设存在多个互不相交的\emph{半缺陷集}，当且仅当检测样本包含每个半缺陷集中至少一个项目时，该检测结果为阳性。研究目标是在尽可能减少检测次数的条件下可靠识别所有半缺陷集，我们将此问题称为\textit{伴随式分组检测}（ConcGT）。针对该问题，我们推导了多种算法，重点聚焦于存在两个半缺陷集的情形。这些算法的区别在于：（i）确定性（零错误）或随机性（低错误）；（ii）非自适应、完全自适应或有限自适应（例如2阶段或3阶段）。无论采用确定性自适应算法还是随机算法（非自适应或有限自适应），在相关的大尺度缩放机制下均能达到阶数最优，且显著优于以求解更一般问题（如超图学习）为中间步骤的基线方法。