Near-Optimal Noisy Group Testing via Separate Decoding of Items

TL;DR

This paper analyzes a simple, separate decoding algorithm for noisy group testing, providing near-optimal performance guarantees and bounds on the number of tests needed for accurate defect identification in low-sparsity regimes.

Contribution

It offers new information-theoretic performance guarantees for separate decoding in noisy group testing, extending results to general noise models and low-sparsity scenarios.

Findings

Achieves test efficiency within a factor of ~0.7 of the information-theoretic optimum at low sparsity.

Extends guarantees to all sublinear sparsity levels with a small fraction of errors.

Provides a converse bound showing limitations of separate decoding beyond certain thresholds.

Abstract

The group testing problem consists of determining a small set of defective items from a larger set of items based on a number of tests, and is relevant in applications such as medical testing, communication protocols, pattern matching, and more. In this paper, we revisit an efficient algorithm for noisy group testing in which each item is decoded separately (Malyutov and Mateev, 1980), and develop novel performance guarantees via an information-theoretic framework for general noise models. For the special cases of no noise and symmetric noise, we find that the asymptotic number of tests required for vanishing error probability is within a factor $\log 2 \approx 0.7$ of the information-theoretic optimum at low sparsity levels, and that with a small fraction of allowed incorrectly decoded items, this guarantee extends to all sublinear sparsity levels. In addition, we provide a converse bound showing that if one tries to move slightly beyond our low-sparsity achievability threshold using separate decoding of items and i.i.d. randomized testing, the average number of items decoded incorrectly approaches that of a trivial decoder.