The Optimality of a Nested Generalized Pairwise Group Testing Procedure

TL;DR

This paper proves the optimality of a nested generalized pairwise group testing procedure for identifying defective units with known probabilities, confirming a conjecture and providing a complete structural characterization and expected test count.

Contribution

It confirms the conjecture that the generalized pairwise testing algorithm is optimal within a specific probability interval and offers a complete structural and analytical characterization.

Findings

Proves the optimality of the generalized pairwise testing algorithm within the specified probability range.

Provides a closed-form expression for the expected number of tests used by the procedure.

Offers new insights into the structure of optimal nested strategies in generalized group testing.

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.