In group testing, the task is to identify defective items by testing groups of them together using as few tests as possible. We consider the setting where each item is defective with a constant probability $\alpha$, independent of all other items. In the (over-)idealized noiseless setting, tests are positive exactly if any of the tested items are defective. We study a more realistic model in which observed test results are subject to noise, i.e., tests can display false positive or false negative results with constant positive probabilities. We determine precise constants $c$ such that $cn\log n$ tests are required to recover the infection status of every individual for both adaptive and non-adaptive group testing: in the former, the selection of groups to test can depend on previously observed test results, whereas it cannot in the latter. Additionally, for both settings, we provide efficient algorithms that identify all defective items with the optimal amount of tests with high probability. Thus, we completely solve the problem of binary noisy group testing in the studied setting.

翻译：在群组检测中，任务是通过对物品进行分组测试，以尽可能少的测试次数识别出缺陷物品。我们考虑每个物品以恒定概率 $\alpha$ 独立于其他物品存在缺陷的场景。在（过度）理想化的无噪声设置中，当测试组中存在任何缺陷物品时，测试结果恰好为阳性。我们研究了一个更现实的模型，其中观测到的测试结果受到噪声影响，即测试可能以恒定的正概率出现假阳性或假阴性结果。我们确定了精确常数 $c$，使得在自适应和非自适应群组检测中均需要 $cn\log n$ 次测试才能恢复每个个体的感染状态：在前者中，测试组的选择可以依赖于先前观测到的测试结果，而后者则不能。此外，针对这两种设置，我们提供了高效算法，能以高概率使用最优测试次数识别所有缺陷物品。因此，我们在所研究的场景中完全解决了二进制噪声群组检测问题。