A tight lower bound on non-adaptive group testing estimation

TL;DR

This paper establishes a tight lower bound of  queries for non-adaptive algorithms estimating defective items in group testing, confirming the optimality of existing methods and resolving longstanding conjectures.

Contribution

The paper proves a tight  lower bound for non-adaptive group testing estimation, matching known upper bounds and settling open questions.

Findings

Any non-adaptive randomized algorithm needs  queries to estimate defectives within a constant factor

The  bound is tight, confirming the optimality of previous algorithms

Matching bounds are established in the threshold query model

Abstract

Efficiently counting or detecting defective items is a crucial task in various fields ranging from biological testing to quality control to streaming algorithms. The \emph{group testing estimation problem} concerns estimating the number of defective elements $d$ in a collection of $n$ total within a given factor. We primarily consider the classical query model, in which a query reveals whether the selected group of elements contains a defective one. We show that any non-adaptive randomized algorithm that estimates the value of $d$ within a constant factor requires $\Omega(\log n)$ queries. This confirms that a known $O(\log n)$ upper bound by Bshouty (2019) is tight and resolves a conjecture by Damaschke and Sheikh Muhammad (2010). Additionally, we prove similar matching upper and lower bounds in the threshold query model.