Random and quasi-random designs in group testing

TL;DR

This paper analyzes the effectiveness of random and quasi-random designs in group testing, deriving bounds and proposing improved design methods with better separability, supported by theoretical and simulation results.

Contribution

It introduces new bounds for the probability of successful item identification and proposes design constructions with enhanced separability over traditional methods.

Findings

Random designs have quantifiable success probabilities.

Proposed designs outperform disjunct matrix-based methods.

Simulation confirms improved separability in binary group testing.

Abstract

For large classes of group testing problems, we derive lower bounds for the probability that all significant items are uniquely identified using specially constructed random designs. These bounds allow us to optimize parameters of the randomization schemes. We also suggest and numerically justify a procedure of constructing designs with better separability properties than pure random designs. We illustrate theoretical considerations with a large simulation-based study. This study indicates, in particular, that in the case of the common binary group testing, the suggested families of designs have better separability than the popular designs constructed from disjunct matrices. We also derive several asymptotic expansions and discuss the situations when the resulting approximations achieve high accuracy.