Performance Bounds for Group Testing With Doubly-Regular Designs

TL;DR

This paper analyzes the performance of a doubly-regular group testing design combined with the DD decoding algorithm across various regimes, providing new bounds and matching known optimality results in certain settings.

Contribution

It introduces a detailed analysis of doubly-regular designs with DD decoding, establishing new bounds and matching asymptotic optimality in some regimes.

Findings

Matches existing optimality results in the sub-linear regime.

Provides new approximate recovery bounds in the linear regime.

Improves bounds for size-constrained testing with growing test sizes.

Abstract

In the group testing problem, the goal is to identify a subset of defective items within a larger set of items based on tests whose outcomes indicate whether any defective item is present. This problem is relevant in areas such as medical testing, DNA sequencing, and communications. In this paper, we study a doubly-regular design in which the number of tests-per-item and the number of items-per-test are fixed. We analyze the performance of this test design alongside the Definite Defectives (DD) decoding algorithm in several settings, namely, (i) the sub-linear regime $k=o(n)$ with exact recovery, (ii) the linear regime $k=\Theta(n)$ with approximate recovery, and (iii) the size-constrained setting, where the number of items per test is constrained. Under setting (i), we show that our design together with the DD algorithm, matches an existing achievability result for the DD algorithm with the near-constant tests-per-item design, which is known to be asymptotically optimal in broad scaling regimes. Under setting (ii), we provide novel approximate recovery bounds that complement a hardness result regarding exact recovery. Lastly, under setting (iii), we improve on the best known upper and lower bounds in scaling regimes where the maximum test size grows with the total number of items.