On a Hypergraph Approach to Multistage Group Testing Problems

TL;DR

This paper introduces a hypergraph-based method for multistage group testing, providing explicit constructions that optimize the number of tests and stages for detecting defective elements.

Contribution

It presents a novel hypergraph approach for multistage group testing, with explicit constructions for cases s=2 and s>2 that optimize test count and stages.

Findings

For s=2, uses 2 log2 t tests over 4 stages.

For s>2, uses (2s-1) log2 t tests over 2s-1 stages.

Provides explicit, efficient testing strategies.

Abstract

Group testing is a well known search problem that consists in detecting up to $s$ defective elements of the set $[t]=\{1,\ldots,t\}$ by carrying out tests on properly chosen subsets of $[t]$. In classical group testing the goal is to find all defective elements by using the minimal possible number of tests. In this paper we consider multistage group testing. We propose a general idea how to use a hypergraph approach to searching defects. For the case $s=2$, we design an explicit construction, which makes use of $2\log_2t(1+o(1))$ tests in the worst case and consists of $4$ stages. For the general case $s>2$, we provide an explicit construction, which uses $(2s-1)\log_2t(1+o(1))$ tests and consists of $2s-1$ rounds.