Consider the group testing problem with an input set of size $n$ and the number of defective items being $d=1$, that allows at most $y$ positive responses. Let $A_{n,y}$ denote the minimum expected number of tests required by a randomized testing strategy for this problem. The testing strategy is required to be Las Vagas, that is, guaranteed to give the correct classification of the items. Based on Yao’s minimax principle, $A_{n,y}$ is equal to the minimum average number of tests for a deterministic group testing strategy allowing at most $y$ positive responses, where the average is taken over all the $n$ possible input sets of size $n$ with $d=1$.

The main contribution of our paper is to show that $A_{n,y}$ is $\frac{y}{y+1}{(ny!)}^{\frac{1}{y}}+O\left(y\right)$, for $1\le y<\lceil {log}_{2}n\rceil$. This solves an open question left in Damaschke (2016).

考虑一组输入大小的组测试问题 $ñ$ 并且有缺陷的物品数量是 $d=\mathrm{1个}$，最多允许 $ÿ$积极回应。让${\mathrm{一种}}_{ñ，ÿ}$表示此问题的随机测试策略所需的最低预期测试次数。测试策略必须为Las Vagas，也就是说，必须保证对项目进行正确的分类。根据姚的极小极大原则，${\mathrm{一种}}_{ñ，ÿ}$ 等于确定性组测试策略允许的最大平均测试次数的最小值 $ÿ$ 积极的回应，其中平均值是所有 $ñ$ 可能的输入集大小 $ñ$ 与 $d=\mathrm{1个}$。

我们论文的主要贡献是表明 ${\mathrm{一种}}_{ñ，ÿ}$ 是 $\frac{ÿ}{ÿ+\mathrm{1个}}{（ñÿ！）}^{\frac{\mathrm{1个}}{ÿ}}+Ø（ÿ）$，对于 $\mathrm{1个}\le ÿ<\lceil {日志}_2ñ\rceil$。这解决了Damaschke（2016）中留下的一个开放性问题。