分布鲁棒的k-of-n顺序检验
Distributionally Robust $k$-of-$n$ Sequential Testing
AI总结：
本文研究了分布鲁棒的k-of-n顺序检验问题，提出非适应性解法，针对单位成本情况获得2近似算法，一般成本情况下在ε有界实例中获得O(1/√ε)近似算法，并考虑了分布鲁棒k-of-n的内最大化问题。
摘要：
k-of-n检验问题涉及依次执行n个独立检验，以确定是否至少k个检验通过。目标是最小化检验的预期成本。这是一个基础且广泛研究的随机优化问题。然而，该模型的关键限制是假设每个检验的成功/失败概率已知精确。本文放松这一假设，研究了k-of-n检验的分布鲁棒模型。在我们的设定中，每个检验关联一个区间，包含其（未知）失败概率。目标是找到一个解，使其在最坏情况下预期成本最小，其中每个检验的概率从其区间中选择。我们专注于非适应性解，即由固定排列决定的解。当所有检验成本为单位时，我们获得分布鲁棒k-of-n检验的2近似算法。对于一般成本，我们获得O(1/√ε)近似算法，在ε有界实例中每个不确定性区间包含在[ε, 1-ε]。我们还考虑了分布鲁棒k-of-n的内最大化问题：该问题涉及为给定解找到最坏情况的概率。对于此问题，除了上述近似比外，我们还获得在所有成本为多项式有界的情况下，准多项式时间近似方案。
英文摘要：
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.
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