Sharp Estimates for Optimal Multistage Group Partition Testing

TL;DR

This paper provides exact solutions, bounds, and algorithms for multistage group partition testing problems, improving understanding of optimal testing strategies in various scenarios.

Contribution

It introduces exact solutions for specific cases, develops a dynamic programming approach for general cases, and offers sharp bounds and algorithms for unknown defectives.

Findings

Exact solutions for (n, 1, s) and (n, d, 2) problems.

Sharp upper and lower bounds for testing strategies.

Algorithms with optimal competitive ratios for unknown defectives.

Abstract

In multistage group testing, the tests within the same stage are considered nonadaptive, while those conducted across different stages are adaptive. Specifically, when the pools within the same stage are disjoint, meaning that the entire set is divided into several disjoint subgroups, it is referred to as a multistage group partition testing problem, denoted as the (n, d, s) problem, where n, d, and s represent the total number of items, defectives, and stages respectively. This paper presents exact solutions for the (n, 1, s) and (n, d, 2) problems for the first time. Additionally, a general dynamic programming approach is developed for the (n, d, s) problem. Significantly we give the sharp upper and lower bounds estimates. If the defective number in unknown but bounded, we can provide an algorithm with an optimal competitive ratio in the asymptotic sense. While assuming the prior distribution of the defective items, we also establish a well performing upper and lower bound estimate to the expectation of optimal strategy