The Optimality of a Nested Generalized Pairwise Group Testing Procedure

Yaakov Malinovsky; Viktor Skorniakov

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arXiv · 2506.15797

The Optimality of a Nested Generalized Pairwise Group Testing Procedure

Yaakov Malinovsky·Viktor Skorniakov

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Abstract

We study the problem of identifying defective units in a finite population of $ n $ units, where each unit $ i $ is independently defective with known probability $ p_i $. This setting is referred to as the \emph{Generalized Group Testing Problem}. A testing procedure is called optimal if it minimizes the expected number of tests. It has been conjectured that, when all probabilities $ p_i $ lie within the interval $ \left[1 - \frac{1}{\sqrt{2}},\, \frac{3 - \sqrt{5}}{2} \right] $, the \emph{generalized pairwise testing {algorithm}}, applied to the $ p_i $ arranged in nondecreasing order, constitutes the optimal nested testing strategy among all such order-preserving nested strategies. In this work, we confirm this conjecture and establish the optimality of the procedure within the specified regime. Additionally, we provide a complete structural characterization of the procedure and derive a closed-form expression for its expected number of tests. These results offer new insights into the theory of optimal nested strategies in generalized group testing.

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Yaakov Malinovsky, Viktor Skorniakov. 2025-06-18. The Optimality of a Nested Generalized Pairwise Group Testing Procedure. https://arxiv.org/abs/2506.15797

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