Combinatorial Group Testing and Sparse Recovery Schemes with Near-Optimal Decoding Time

TL;DR

This paper introduces new combinatorial group testing schemes that use an optimal number of measurements and achieve near-linear decoding time, significantly improving efficiency over previous methods for identifying defectives.

Contribution

The authors present novel constructions for group testing that are measurement-optimal and have near-optimal decoding time, avoiding complex existing techniques.

Findings

Achieved near-linear decoding time for group testing schemes

Constructed measurement-optimal schemes with efficient decoding algorithms

Improved upon previous quadratic or linear decoding time methods

Abstract

In the long-studied problem of combinatorial group testing, one is asked to detect a set of $k$ defective items out of a population of size $n$, using $m \ll n$ disjunctive measurements. In the non-adaptive setting, the most widely used combinatorial objects are disjunct and list-disjunct matrices, which define incidence matrices of test schemes. Disjunct matrices allow the identification of the exact set of defectives, whereas list disjunct matrices identify a small superset of the defectives. Apart from the combinatorial guarantees, it is often of key interest to equip measurement designs with efficient decoding algorithms. The most efficient decoders should run in sublinear time in $n$, and ideally near-linear in the number of measurements $m$. In this work, we give several constructions with an optimal number of measurements and near-optimal decoding time for the most fundamental group testing tasks, as well as for central tasks in the compressed sensing and heavy hitters literature. For many of those tasks, the previous measurement-optimal constructions needed time either quadratic in the number of measurements or linear in the universe size. Most of our results are obtained via a clean and novel approach which avoids list-recoverable codes or related complex techniques which were present in almost every state-of-the-art work on efficiently decodable constructions of such objects.