The $k$-of-$n$ testing problem involves performing $n$ independent tests sequentially, in order to determine whether/not at least $k$ tests pass. The objective is to minimize the expected cost of testing. This is a fundamental and well-studied stochastic optimization problem. However, a key limitation of this model is that the success/failure probability of each test is assumed to be known precisely. In this paper, we relax this assumption and study a distributionally-robust model for $k$-of-$n$ testing. In our setting, each test is associated with an interval that contains its (unknown) failure probability. The goal is to find a solution that minimizes the worst-case expected cost, where each test's probability is chosen from its interval. We focus on non-adaptive solutions, that are specified by a fixed permutation of the tests. When all test costs are unit, we obtain a $2$-approximation algorithm for distributionally-robust $k$-of-$n$ testing. For general costs, we obtain an $O(\frac{1}{\sqrt ε})$-approximation algorithm on $ε$-bounded instances where each uncertainty interval is contained in $[ε, 1-ε]$. We also consider the inner maximization problem for distributionally-robust $k$-of-$n$: this involves finding the worst-case probabilities from the uncertainty intervals for a given solution. For this problem, in addition to the above approximation ratios, we obtain a quasi-polynomial time approximation scheme under the assumption that all costs are polynomially bounded.

翻译：$k$-of-$n$测试问题涉及序贯执行$n$次独立测试，以确定是否至少有$k$次测试通过。其目标是最小化测试的期望成本。这是一个基础且被广泛研究的随机优化问题。然而，该模型的一个关键局限性在于假设每次测试的成功/失败概率是精确已知的。本文放宽了这一假设，研究了$k$-of-$n$测试的分布鲁棒模型。在我们的设定中，每个测试关联一个包含其（未知）失败概率的区间。目标是找到一个解，使得最坏情况下的期望成本最小化，其中每个测试的概率从对应区间中选取。我们关注非自适应解，由测试的固定排列指定。当所有测试成本为单位成本时，我们为分布鲁棒$k$-of-$n$测试提供了一个$2$-近似算法。对于一般成本，我们在每个不确定性区间包含于$[ε, 1-ε]$的$ε$-有界实例上，获得了$O(\frac{1}{\sqrt ε})$-近似算法。我们还考虑了分布鲁棒$k$-of-$n$的内层最大化问题：这涉及从给定解的不确定性区间中找出最坏情况概率。对于此问题，除上述近似比外，我们在所有成本均为多项式有界的假设下，得到了一个拟多项式时间近似方案。