Improved Lower Bound for Estimating the Number of Defective Items

TL;DR

This paper introduces a new method for deriving lower bounds in non-adaptive randomized group testing, nearly matching known upper bounds and resolving an open problem in the field.

Contribution

The paper presents a novel approach to establish lower bounds in non-adaptive randomized group testing, improving previous bounds and nearly matching the best known upper bounds.

Findings

Lower bounds within a factor of $1/{ m loglog ext{...}}$ of upper bounds.

Proves that estimating the number of defectives within a constant factor requires $ ilde{ ext{O}}( ext{log} n)$ tests.

Almost matches the upper bound of $O( ext{log} n)$, solving an open problem.

Abstract

Let $X$ be a set of items of size $n$ that contains some defective items, denoted by $I$, where $I \subseteq X$. In group testing, a {\it test} refers to a subset of items $Q \subset X$. The outcome of a test is $1$ if $Q$ contains at least one defective item, i.e., $Q\cap I \neq \emptyset$, and $0$ otherwise. We give a novel approach to obtaining lower bounds in non-adaptive randomized group testing. The technique produced lower bounds that are within a factor of $1/{\log\log\stackrel{k}{\cdots}\log n}$ of the existing upper bounds for any constant~$k$. Employing this new method, we can prove the following result. For any fixed constants $k$, any non-adaptive randomized algorithm that, for any set of defective items $I$, with probability at least $2/3$, returns an estimate of the number of defective items $|I|$ to within a constant factor requires at least $$\Omega\left(\frac{\log n}{\log\log\stackrel{k}{\cdots}\log n}\right)$$ tests. Our result almost matches the upper bound of $O(\log n)$ and solves the open problem posed by Damaschke and Sheikh Muhammad [COCOA 2010 and Discrete Math., Alg. and Appl., 2010]. Additionally, it improves upon the lower bound of $\Omega(\log n/\log\log n)$ previously established by Bshouty [ISAAC 2019].