Noisy Linear Group Testing: Exact Thresholds and Efficient Algorithms

TL;DR

This paper analyzes noisy group testing models, establishing exact thresholds for the number of tests needed and providing efficient algorithms for both adaptive and non-adaptive strategies to identify defective items.

Contribution

It determines precise constants for the number of tests required in noisy group testing and offers optimal algorithms for defect detection in both adaptive and non-adaptive settings.

Findings

Exact test thresholds established for noisy group testing.

Efficient algorithms achieve optimal testing with high probability.

Complete solution to binary noisy group testing problem.

Abstract

In group testing, the task is to identify defective items by testing groups of them together using as few tests as possible. We consider the setting where each item is defective with a constant probability $\alpha$, independent of all other items. In the (over-)idealized noiseless setting, tests are positive exactly if any of the tested items are defective. We study a more realistic model in which observed test results are subject to noise, i.e., tests can display false positive or false negative results with constant positive probabilities. We determine precise constants $c$ such that $cn\log n$ tests are required to recover the infection status of every individual for both adaptive and non-adaptive group testing: in the former, the selection of groups to test can depend on previously observed test results, whereas it cannot in the latter. Additionally, for both settings, we provide efficient algorithms that identify all defective items with the optimal amount of tests with high probability. Thus, we completely solve the problem of binary noisy group testing in the studied setting.