We formulate and study a statistical version of Katona's two-round search problem of finding at least one excellent element in a set. A population of $n$ elements is considered, where each element is independently excellent with probability $λ/n$, $λ> 0$. A subset test is noiseless: it returns positive exactly when the queried subset contains at least one excellent element. The goal is to minimize the expected number of tests subject to finding one excellent element with probability at least $1-α$, where $0<α<1$, under the restriction that testing is performed in two rounds. Unlike classical group testing, the objective is not to recover the full set of excellent elements, but only to identify one of them. We first show that success is fundamentally limited by the possibility that no excellent element exists. In the sparse Poisson regime, this imposes the necessary feasibility condition $α\ge e^{-λ}$. When the target success probability is feasible, we prove that the optimal expected number of tests grows logarithmically with the population size. The upper bound is obtained by combining an initial existence test with a second-round separating design; the lower bound follows from an information-counting argument. Numerical illustrations show the feasibility boundary and the resulting logarithmic scaling.

翻译：我们提出并研究Katona两轮搜索问题的一种统计版本，目标是在集合中至少找到一个优秀元素。考虑一个包含$n$个元素的总体，每个元素独立地以概率$λ/n$（$λ>0$）成为优秀元素。子集测试是无噪声的：当且仅当被查询的子集中至少包含一个优秀元素时，测试结果返回阳性。目标是在限制为两轮测试的条件下，以至少$1-α$的概率（其中$0<α<1$）找到一个优秀元素，并使期望测试次数最小化。与经典群体测试不同，目标不是恢复所有优秀元素的完整集合，而仅需识别其中一个。我们首先证明，成功本质上受限于可能不存在优秀元素的情况。在稀疏泊松机制下，这施加了必要的可行性条件$α\ge e^{-λ}$。当目标成功概率可行时，我们证明最优期望测试次数随总体规模呈对数增长。上界通过将初始存在性测试与第二轮分离设计相结合得到；下界则基于信息计数论证。数值示例展示了可行性边界以及由此产生的对数缩放规律。