We study the Threshold Group Testing (TGT) problem in the noiseless and non-adaptive setting, where the objective is to exactly recover a sparse binary vector from pooled tests, using as few tests as possible. In TGT, each test applied to a subset of items returns a positive outcome if the number of 1's (defective items) in that subset meets or exceeds a specified threshold, and has a negative outcome otherwise. We investigate how the complexity of TGT compares to that of Classical Group Testing (CGT), corresponding to the special case of the threshold equal to one, and analyse the impact of increasing the threshold on the required number of tests. Our main contribution is the derivation of a sharp information-theoretic phase transition at $c_{\mathrm{inf}}^{\mathrm{TGT}}k\log(n/k)$ (non-adaptive) tests for TGT within the constant-column test design. The threshold constant $c_{\mathrm{inf}}^{\mathrm{TGT}}$ is expressed as a function of the prevalence of defectives and the threshold value. Our upper bound is derived under an analytic assumption, and we verify that this assumption is satisfied for a threshold value of 2. The value of $c_{\mathrm{inf}}^{\mathrm{TGT}}$ reveals that TGT on the constant-column design has the same information-theoretic behaviour as CGT in the low-prevalence regime. Yet, strikingly, at higher prevalences, the threshold leads to a significant reduction in the number of tests. On the other hand, we provide evidence that when the asymptotic proportion of defective items is positive, TGT actually becomes strictly harder than CGT (excluding trivial reductions).

翻译：我们在无噪声且非自适应设置下研究阈值群测试问题，目标是在尽可能少的测试次数下，通过池化测试精确恢复稀疏二元向量。在TGT中，每个应用于物品子集的测试若该子集中1的数量（缺陷物品）达到或超过指定阈值则返回阳性结果，否则返回阴性结果。我们探究TGT的复杂度与经典群测试（CGT，对应阈值为1的特殊情况）的比较，并分析提高阈值对所需测试次数的影响。主要贡献是在常数列测试设计下，推导出TGT在$c_{\mathrm{inf}}^{\mathrm{TGT}}k\log(n/k)$（非自适应）测试处的尖锐信息论相变。阈值常数$c_{\mathrm{inf}}^{\mathrm{TGT}}$表示为缺陷流行率与阈值的函数。我们的上界在解析假设下导出，并验证该假设在阈值为2时成立。$c_{\mathrm{inf}}^{\mathrm{TGT}}$的值表明，在低流行率下，常数列设计上的TGT与CGT具有相同的信息论特性。然而引人注目的是，在较高流行率下，阈值显著减少了测试次数。另一方面，我们提供证据表明，当缺陷物品的渐近比例为正时，TGT实际上比CGT更严格（排除平凡缩减情况）。