Paper
Distributionally Robust $k$-of-$n$ Sequential Testing
Authors
Rayen Tan, Viswanath Nagarajan
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.
Metadata
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Raw Data (Debug)
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.23705v1</id>\n <title>Distributionally Robust $k$-of-$n$ Sequential Testing</title>\n <updated>2026-03-24T20:46:03Z</updated>\n <link href='https://arxiv.org/abs/2603.23705v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.23705v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>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.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='cs.DS'/>\n <published>2026-03-24T20:46:03Z</published>\n <arxiv:comment>28 pages, 3 figures</arxiv:comment>\n <arxiv:primary_category term='cs.DS'/>\n <author>\n <name>Rayen Tan</name>\n </author>\n <author>\n <name>Viswanath Nagarajan</name>\n </author>\n </entry>"
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