The Optimality of a Nested Generalized Pairwise Group Testing Procedure

2025-06-18

AI Summary

This paper addresses the optimal group testing problem for identifying defective items in a finite population, where items are independent and possess heterogeneous, known defect probabilities; the objective is to minimize the expected number of tests. We propose a nested generalized pairwise testing strategy executed in ascending order of defect probabilities and establish, for the first time, its global optimality among all order-preserving nested strategies over the probability interval $[1-1/sqrt{2},, (3-sqrt{5})/2]$. Through recursive modeling, probabilistic analysis, and optimal decision theory, we fully characterize the strategy’s structural properties and derive an exact closed-form expression for the expected number of tests. Our results not only delineate the theoretical optimality boundary of this strategy but also advance the understanding of optimal nested testing mechanisms under heterogeneous defect probabilities. Furthermore, they provide a computationally tractable and verifiable optimal design paradigm for generalized group testing.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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} ight] ), 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.

Problem

• Identifying defective units in a finite population efficiently
• Proving optimality of nested generalized pairwise testing algorithm
• Characterizing procedure structure and expected test count

Innovation

• Nested generalized pairwise testing algorithm
• Optimal order-preserving nested strategies
• Closed-form expected test count expression