In the group testing problem we aim to identify a small number of infected individuals within a large population. We avail ourselves to a procedure that can test a group of multiple individuals, with the test result coming out positive iff at least one individual in the group is infected. With all tests conducted in parallel, what is the least number of tests required to identify the status of all individuals? In a recent test design [Aldridge et al. 2016] the individuals are assigned to test groups randomly with replacement, with every individual joining an almost equal number of groups. We pinpoint the sharp threshold for the number of tests required in this randomised design so that it is information-theoretically possible to infer the infection status of every individual. Moreover, we analyse two efficient inference algorithms. These results settle conjectures from [Aldridge et al. 2014, Johnson et al. 2019].

中文翻译：

---

组测试的信息论和算法阈值

在群体测试问题中，我们的目标是在大量人口中识别少数受感染的个体。我们利用一种可以测试一组多人的程序，如果该组中至少有一个人被感染，则测试结果为阳性。在并行进行所有测试的情况下，确定所有个人状态所需的最少测试次数是多少？在最近的测试设计中 [Aldridge 等人。2016] 个人被随机分配到测试组中，每个人加入的组数几乎相等。我们为这个随机设计所需的测试数量确定了一个尖锐的阈值，以便从信息理论上推断每个人的感染状态是可能的。此外，我们分析了两种有效的推理算法。这些结果解决了 [Aldridge et al. 2014 年，约翰逊等人。2019]。