The capacity of Bernoulli nonadaptive group testing

TL;DR

This paper establishes a tight threshold on the number of tests needed in Bernoulli nonadaptive group testing, showing that optimal Bernoulli testing is fundamentally limited compared to adaptive methods, especially in denser scenarios.

Contribution

It provides a tight converse bound for Bernoulli nonadaptive group testing, confirming the optimality of previous achievability results and highlighting limitations relative to adaptive strategies.

Findings

Bernoulli nonadaptive testing has a fundamental limit on the number of tests needed.

Optimal Bernoulli testing is strictly worse than adaptive strategies in dense cases.

The new bounds unify and extend previous results on group testing performance.

Abstract

We consider nonadaptive group testing with Bernoulli tests, where each item is placed in each test independently with some fixed probability. We give a tight threshold on the maximum number of tests required to find the defective set under optimal Bernoulli testing. Achievability is given by a result of Scarlett and Cevher; here we give a converse bound showing that this result is best possible. Our new converse requires three parts: a typicality bound generalising the trivial counting bound, a converse on the COMP algorithm of Chan et al, and a bound on the SSS algorithm similar to that given by Aldridge, Baldassini, and Johnson. Our result has a number of important corollaries, in particular that, in denser cases, Bernoulli nonadaptive group testing is strictly worse than the best adaptive strategies.