Distributionally Robust $k$-of-$n$ Sequential Testing

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 $O(\frac{1}{\sqrt \epsilon})$-approximation algorithm on $\epsilon$-bounded instances where each uncertainty interval is contained in $[\epsilon, 1-\epsilon]$. 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.