# Information-theoretic and algorithmic thresholds for group testing

## TL;DR

This paper determines the minimum number of tests needed for successful group testing using a randomized design, establishing sharp thresholds and analyzing algorithms to solve the problem efficiently.

## Contribution

It precisely characterizes the information-theoretic threshold for group testing and analyzes the performance of inference algorithms, settling prior conjectures.

## Key findings

- Identified sharp thresholds for the number of tests needed.
- Analyzed the performance of two efficient inference algorithms.
- Settled conjectures from previous studies.

## Abstract

In the group testing problem we aim to identify a small number of infected individuals within a large population. We avail ourselves to a procedure that can test a group of multiple individuals, with the test result coming out positive iff at least one individual in the group is infected. With all tests conducted in parallel, what is the least number of tests required to identify the status of all individuals? In a recent test design [Aldridge et al.\ 2016] the individuals are assigned to test groups randomly, with every individual joining an equal number of groups. We pinpoint the sharp threshold for the number of tests required in this randomised design so that it is information-theoretically possible to infer the infection status of every individual. Moreover, we analyse two efficient inference algorithms. These results settle conjectures from [Aldridge et al.\ 2014, Johnson et al.\ 2019].