The group testing problem concerns discovering a small number of defective items within a large population by performing tests on pools of items. A test is positive if the pool contains at least one defective, and negative if it contains no defectives. This is a sparse inference problem with a combinatorial flavour, with applications in medical testing, biology, telecommunications, information technology, data science, and more. In this monograph, we survey recent developments in the group testing problem from an information-theoretic perspective. We cover several related developments: efficient algorithms with practical storage and computation requirements, achievability bounds for optimal decoding methods, and algorithm-independent converse bounds. We assess the theoretical guarantees not only in terms of scaling laws, but also in terms of the constant factors, leading to the notion of the {\em rate} of group testing, indicating the amount of information learned per test. For the noiseless setting, we present a series of results leading to optimal rates, which in turn imply optimality and suboptimality results of various algorithms depending on the sparsity regime. We also survey analogous developments in noisy settings. In addition, we survey results concerning a number of variations on the standard group testing problem, including approximate recovery criteria, adaptive algorithms with a limited number of stages, sublinear-time algorithms, and settings with additional prior information, among others.

翻译：群体检测问题关注通过检测物品池来发现大规模群体中的少量缺陷物品：若池中包含至少一个缺陷物品，则检测结果呈阳性；若无缺陷物品，则呈阴性。这是一个具有组合特性的稀疏推断问题，在医学检测、生物学、电信、信息技术、数据科学等领域均有应用。本综述从信息论视角系统梳理了群体检测问题的最新进展，涵盖以下相关研究方向：具有实际存储与计算需求的高效算法、最优解码方法的可达界，以及与算法无关的逆界。我们不仅从标度律角度，更从常数因子角度评估理论保证，进而引出群体检测"速率"概念——该速率衡量每次检测所能习得的信息量。在无噪场景中，我们给出系列成果以导出最优速率，这些成果同时揭示了不同稀疏度机制下各算法的最优性与次优性。此外，我们亦梳理了含噪场景的相似进展，并综述了标准群体检测问题的多种变体，包括近似恢复准则、有限阶段自适应算法、次线性时间算法及具有额外先验信息的场景等。