In recent years, the mathematical limits and algorithmic bounds for probabilistic group testing having become increasingly well-understood, with exact asymptotic thresholds now being known in general scaling regimes for the noiseless setting. In the noisy setting where each test outcome is flipped with constant probability, there have been similar developments, but the overall understanding has lagged significantly behind the noiseless setting. In this paper, we substantially narrow this gap by deriving exact asymptotic thresholds for the noisy setting under two widely-studied random test designs: i.i.d. Bernoulli and near-constant tests-per-item. These thresholds are established by combining components of an existing information-theoretic threshold decoder with a novel analysis of maximum-likelihood decoding (upper bounds), and deriving a novel set of impossibility results by analyzing certain failure events for optimal maximum-likelihood decoding (lower bounds). Our results show that existing algorithmic upper bounds for the noisy setting are strictly suboptimal, and leave open the interesting question of whether our thresholds can be attained using computationally efficient algorithms.

翻译：近年来，概率群组测试的数学极限与算法边界已得到日益清晰的理解，在无噪声场景的一般缩放条件下，其精确渐近阈值现已明确。在每次测试结果以恒定概率翻转的噪声场景中，相关研究虽取得进展，但整体认知显著落后于无噪声场景。本文通过推导两种广泛研究的随机测试设计（独立同分布伯努利设计及近似恒定每项测试次数设计）下噪声场景的精确渐近阈值，显著缩小了这一差距。这些阈值的建立，一方面结合了现有信息论阈值解码器的组件与对最大似然解码（上界）的新颖分析，另一方面通过分析最优最大似然解码（下界）的特定失效事件推导出一系列新的不可能性结论。我们的结果表明，现有噪声场景的算法上界严格次优，并揭示了这些阈值能否通过计算高效算法实现的开放性有趣问题。