双正则设计成组检验的性能界
Performance Bounds for Group Testing With Doubly-Regular Designs
摘要：
在分组测试问题中，目标是根据测试结果指示是否存在任何缺陷产品的测试，在更大的产品集中识别缺陷产品的子集。这个问题与医学测试、DNA测序和通信等领域有关。本文研究了一个双正则设计，其中每项试验数和每项试验的项目数都是固定的。我们分析了这种测试设计的性能，以及在几种设置中的确定缺陷（DD）解码算法，即（i）精确恢复的子线性制度$k=o(n)$，（ii）近似恢复的线性制度$k=Θ(n)$，以及（iii）大小受限的设置，其中每个测试的项目数量受到限制。在设置（i）下，我们证明了我们的设计与DD算法一起，匹配DD算法的现有可验证性结果与近常数测试每项设计，这是已知的在宽尺度范围内渐近最优。根据设置（ii），我们提供了新的近似恢复界限，补充了关于确切恢复的硬度结果。最后，在设置（iii）下，我们改进了最大测试大小随项目总数增长的缩放制度中最知名的上界和下界。
英文摘要：
In the group testing problem, the goal is to identify a subset of defective items within a larger set of items based on tests whose outcomes indicate whether any defective item is present. This problem is relevant in areas such as medical testing, DNA sequencing, and communications. In this paper, we study a doubly-regular design in which the number of tests-per-item and the number of items-per-test are fixed. We analyze the performance of this test design alongside the Definite Defectives (DD) decoding algorithm in several settings, namely, (i) the sub-linear regime $k=o(n)$ with exact recovery, (ii) the linear regime $k=Θ(n)$ with approximate recovery, and (iii) the size-constrained setting, where the number of items per test is constrained. Under setting (i), we show that our design together with the DD algorithm, matches an existing achievability result for the DD algorithm with the near-constant tests-per-item design, which is known to be asymptotically optimal in broad scaling regimes. Under setting (ii), we provide novel approximate recovery bounds that complement a hardness result regarding exact recovery. Lastly, under setting (iii), we improve on the best known upper and lower bounds in scaling regimes where the maximum test size grows with the total number of items.
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