近似最优稀疏约束成组测试：改进的界和算法
Near optimal sparsity-constrained group testing: improved bounds and algorithms
摘要：
最近的进展，在无噪声非自适应组测试导致了一个精确的渐近表征的数量的测试所需的高概率恢复的次线性制度$k = n^θ$（与$θ\in (0,1)$），与$n$个人之间的$k$感染。然而，在现实世界的实际约束下，所需的测试数量可能会大幅增加，特别是包括个体可以被置于其中的测试的最大数量$Δ$或给定测试中个体的最大数量$Γ$的界限。虽然以前的作品已经恢复这些设置的保证，显着的差距之间仍然存在的可扩展性和逆边界。在本文中，我们基本上或完全关闭几个最突出的差距。在$Δ$ -可分项目的情况下，我们表明，明确的缺陷（DD）算法加上一个随机的正规设计是渐近最优的密集的缩放制度，并最佳的$\eul$的一个因素内更普遍的，我们建立这两个最好的已知的可分性和逆界加强。在$Γ$大小的测试的情况下，我们提供了一个全面的分析制度$Γ= Θ(1)$，并再次建立一个精确的阈值证明渐近最优的SCOMP（轻微改进DD）配备了一个定制的池计划。最后，对于这两个设置中的每一个，我们提供了基于顺序分裂的近最优自适应算法，并证明了最优自适应算法和非自适应算法的性能之间的差距。
英文摘要：
Recent advances in noiseless non-adaptive group testing have led to a precise asymptotic characterization of the number of tests required for high-probability recovery in the sublinear regime $k = n^θ$ (with $θ\in (0,1)$), with $n$ individuals among which $k$ are infected. However, the required number of tests may increase substantially under real-world practical constraints, notably including bounds on the maximum number $Δ$ of tests an individual can be placed in, or the maximum number $Γ$ of individuals in a given test. While previous works have given recovery guarantees for these settings, significant gaps remain between the achievability and converse bounds. In this paper, we substantially or completely close several of the most prominent gaps. In the case of $Δ$-divisible items, we show that the definite defectives (DD) algorithm coupled with a random regular design is asymptotically optimal in dense scaling regimes, and optimal to within a factor of $\eul$ more generally; we establish this by strengthening both the best known achievability and converse bounds. In the case of $Γ$-sized tests, we provide a comprehensive analysis of the regime $Γ= Θ(1)$, and again establish a precise threshold proving the asymptotic optimality of SCOMP (a slight refinement of DD) equipped with a tailored pooling scheme. Finally, for each of these two settings, we provide near-optimal adaptive algorithms based on sequential splitting, and provably demonstrate gaps between the performance of optimal adaptive and non-adaptive algorithms.
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