We consider a version of the classical group testing problem motivated by PCR testing for COVID-19. In the so-called tropical group testing model, the outcome of a test is the lowest cycle threshold (Ct) level of the individuals pooled within it, rather than a simple binary indicator variable. We introduce the tropical counterparts of three classical non-adaptive algorithms (COMP, DD and SCOMP), and analyse their behaviour through both simulations and bounds on error probabilities. By comparing the results of the tropical and classical algorithms, we gain insight into the extra information provided by learning the outcomes (Ct levels) of the tests. We show that in a limiting regime the tropical COMP algorithm requires as many tests as its classical counterpart, but that for sufficiently dense problems tropical DD can recover more information with fewer tests, and can be viewed as essentially optimal in certain regimes.

翻译：我们考虑了一种由COVID-19的PCR检测所启发的经典群组检测问题的变体。在所谓的热带群组检测模型中，一次检测的结果是样本池中个体最低的循环阈值（Ct值），而非简单的二元指示变量。我们引入了三种经典非自适应算法（COMP、DD和SCOMP）的热带对应版本，并通过模拟和错误概率的界限分析了其行为。通过比较热带算法与经典算法的结果，我们深入了解了通过获得检测结果（Ct值）所提供的额外信息。我们表明，在极限状态下，热带COMP算法需要与其经典对应算法相同数量的检测次数，但对于足够密集的问题，热带DD算法可以用更少的检测次数恢复更多信息，并在某些状态下可被视为本质上最优。