Group Testing with Pools of Fixed Size

TL;DR

This paper investigates the minimum number of tests needed in a fixed-size pool group testing problem, extending classical adaptive group testing by restricting subset sizes to a fixed positive integer.

Contribution

It introduces and begins analysis of the function M^{[k]}(d, n), representing optimal testing strategies with fixed pool sizes, a variant not extensively studied before.

Findings

Initial bounds and properties of M^{[k]}(d, n) are established.

The problem's complexity increases with fixed pool size constraints.

Foundations for future algorithms and bounds are laid out.

Abstract

In the classical combinatorial (adaptive) group testing problem, one is given two integers \(d\) and \(n\), where \(0\le d\le n\), and a population of \(n\) items, exactly \(d\) of which are known to be defective. The question is to devise an optimal sequential algorithm that, at each step, tests a subset of the population and determines whether such subset is contaminated (i.e. contains defective items) or otherwise. The problem is solved only when the \(d\) defective items are identified. The minimum number of steps that an optimal sequential algorithm takes in general (i.e. in the worst case) to solve the problem is denoted by \(M(d, n)\). The computation of \(M(d, n)\) appears to be very difficult and a general formula is known only for \(d = 1\). We consider here a variant of the original problem, where the size of the subsets to be tested is restricted to be a fixed positive integer \(k\). The corresponding minimum number of tests by a sequential optimal algorithm is denoted by \(M^{\lbrack k\rbrack}(d, n)\). In this paper we start the investigation of the function \(M^{\lbrack k\rbrack}(d, n)\).