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

2026-03-24

AI Summary

This work addresses the practical challenge in $k$-of-$n$ sequential testing where failure probabilities are often known only to lie within given intervals rather than as exact values. To tackle this uncertainty, the paper introduces distributionally robust optimization into the $k$-of-$n$ testing framework, focusing on minimizing the worst-case expected cost under non-adaptive test ordering. By modeling interval-based uncertainty and designing efficient approximation algorithms, the authors obtain a 2-approximation algorithm for the unit-cost setting. For general costs under $\varepsilon$-bounded instances, they achieve an $O(1/\sqrt{\varepsilon})$-approximation guarantee and further provide a quasipolynomial-time approximation scheme for the inner maximization problem.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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.

Problem

• distributionally robust
• k-of-n testing
• sequential testing
• uncertainty intervals
• stochastic optimization

Innovation

• distributionally robust optimization
• k-of-n testing
• approximation algorithms
• uncertainty intervals
• non-adaptive strategies