Conjectures on optimal nested generalized group testing algorithm

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 p 1 = … = p N = p, Yao and Hwang (1990) 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 p ∈ [1 − 1∕ √ 2, (3 − √ 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: (a) when all p i ,i = 1,… ,N belong to the R-range, GPTA is the optimal procedure among nested procedures applied to p i of nondecreasing order; and (b) if all p i ,i = 1,… ,N belong to the R-range, GPTA is the optimal nested procedure, that is, minimizes 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.

Nested Group Testing Procedures for Screening

“…All units have to be classified as either non-defective or defective by group testing. Since its introduction, GGTP has seen considerable theoretical investigation (Lee and Sobel (1972); Nebenzahl and Sobel (1973); Katona (1973); Hwang (1976a); Yao and Hwang (1988a,b); Kurtz and Sidi (1988); Kealy et al (2014); Malinovsky (2019bMalinovsky ( , 2020; Malinovsky et al (2020)).…”

Application-oriented mathematical algorithms for group testing

“…Since then there have been hundreds of papers published about similar questions. However, for example, a recent paper by Malinovsky summarizes the open problems and how few we still know about the optimum even in this idealized model [2]. Also, the classical mathematical definition for the number of rounds is very different: it assumes that all samples arrive at the same time, we can make an unlimited number of tests per round, and we want to minimize the number of rounds for the entire algorithm.…”