Conjectures on Optimal Nested Generalized Group Testing Algorithm

TL;DR

This paper explores the optimality of nested group testing algorithms for identifying defective items in a population, proposing conjectures about the generalized pairwise testing algorithm's optimality within certain probability ranges.

Contribution

It introduces conjectures regarding the optimality of the GPTA for the generalized group testing problem when item defect probabilities are within a specific range.

Findings

Conjecture 1: GPTA is optimal among nested procedures for probabilities in the R-range.

Conjecture 2: GPTA minimizes expected tests among all nested procedures in the R-range.

Empirical verification of the first conjecture up to a certain population size.

Abstract

Consider a finite population of $N$ items, where item $i$ has a probability $p_i$ to be defective. The goal is to identify all items by means of group testing. This is the generalized group testing problem (hereafter GGTP). In the case of $\displaystyle p_1=\cdots=p_{N}=p$ \cite{YH1990} proved that the pairwise testing algorithm is the optimal nested algorithm, with respect to the expected number of tests, for all $N$ if and only if $\displaystyle p \in [1-1/\sqrt{2},\,(3-\sqrt{5})/2]$ (R-range hereafter) (an optimal at the boundary values). In this note, we present a result that helps to define the generalized pairwise testing algorithm (hereafter GPTA) for the GGTP. We present two conjectures: (1) when all $p_i, i=1,\ldots,N$ belong to the R-range, GPTA is the optimal procedure among nested procedures applied to $p_i$ of nondecreasing order; (2) if all $p_i, i=1,\ldots,N$ belong to the R-range, GPTA the optimal nested procedure, i.e., minimises the expected total number of tests with respect to all possible testing orders in the class of nested procedures. Although these conjectures are logically reasonable, we were only able to empirically verify the first one up to a particular level of $N$. We also provide a short survey of GGTP.