Consider a finite population of $N$ items, where item $i$ has a probability $p_i$ to be defective. The goal is to identify all items by means of group testing. This is the generalized group testing problem (hereafter GGTP). In the case of $\displaystyle p_1=\cdots=p_{N}=p$ \cite{YH1990} proved that the pairwise testing algorithm is the optimal nested algorithm, with respect to the expected number of tests, for all $N$ if and only if $\displaystyle p \in [1-1/\sqrt{2},\,(3-\sqrt{5})/2]$ (R-range hereafter) (an optimal at the boundary values). In this note, we present a result that helps to define the generalized pairwise testing algorithm (hereafter GPTA) for the GGTP. We present two conjectures: (1) when all $p_i, i=1,\ldots,N$ belong to the R-range, GPTA is the optimal procedure among nested procedures applied to $p_i$ of nondecreasing order; (2) if all $p_i, i=1,\ldots,N$ belong to the R-range, GPTA the optimal nested procedure, i.e., minimises the expected total number of tests with respect to all possible testing orders in the class of nested procedures. Although these conjectures are logically reasonable, we were only able to empirically verify the first one up to a particular level of $N$. We also provide a short survey of GGTP.

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关于最优嵌套广义群测试算法的猜想

考虑$N$ 项的有限总体，其中项$i$ 有缺陷的概率$p_i$。目标是通过小组测试来识别所有项目。这就是广义组测试问题（以下简称 GGTP）。在 $\displaystyle p_1=\cdots=p_{N}=p$ \cite{YH1990} 的情况下，证明了成对测试算法是最优嵌套算法，相对于预期的测试次数，对于所有 $N$当且仅当 $\displaystyle p \in [1-1/\sqrt{2},\,(3-\sqrt{5})/2]$（以下为 R 范围）（边界值处的最优值） . 在本说明中，我们提供了一个有助于定义 GGTP 的广义成对测试算法（以下简称 GPTA）的结果。我们提出两个猜想：（1）当所有 $p_i, i=1,\ldots,N$ 都属于 R 范围时，GPTA 是应用于非递减阶 $p_i$ 的嵌套过程中的最优过程；(2) 如果所有 $p_i, i=1,\ldots,N$ 都属于 R-range，GPTA 是最优嵌套程序，即，对于所有可能的测试顺序，最小化预期的测试总数嵌套过程。尽管这些猜想在逻辑上是合理的，但我们只能凭经验验证第一个达到特定水平的 $N$。我们还提供了 GGTP 的简短调查。