Noisy Nonadaptive Group Testing with Binary Splitting: New Test Design and Improvement on Price-Scarlett-Tan's Scheme

TL;DR

This paper extends binary splitting group testing methods to noisy scenarios with false positives and negatives, achieving near-optimal test counts and decoding complexity in probabilistic models with noise.

Contribution

It introduces new NAPGT schemes that handle noise in test results, generalizing previous noiseless methods with asymptotic optimality and improved decoding complexity.

Findings

Achieves asymptotically optimal tests and decoding complexity under noise.

Provides algorithms for both false positive and false negative noise models.

Improves upon previous work by Price, Scarlett, and Tan in noisy settings.

Abstract

In Group Testing, the objective is to identify $K$ defective items out of $N$, $K\ll N$, by testing pools of items together and using the least amount of tests possible. Recently, a fast decoding method based on binary splitting (Price and Scarlett, 2020) has been proposed that simultaneously achieve optimal number of tests and decoding complexity for Non-Adaptive Probabilistic Group Testing (NAPGT). However, the method works only when the test results are noiseless. In this paper, we further study the binary splitting method and propose (1) A NAPGT scheme that generalizes the original binary splitting method from the noiseless case into tests with $\rho$ proportion of false positives (the $\rho$-False Positive Channel), where $\rho$ is a constant, with asymptotically-optimal number of tests and decoding complexity, i.e. $\mathcal{O}(K\log N)$, and (2) A NAPGT scheme in the presence of both false positives and false negatives in test outcomes, improving and generalizing the work of Price, Scarlett and Tan~\cite{price2023fast} in two ways: First, under $\rho$-proportion of test results flipped ($\rho$-Binary Symmetric Channel) and within the general sublinear regime $K=\Theta(N^\alpha)$ where $0<\alpha<1$, our algorithm has a decoding complexity of $\mathcal{O}(\epsilon^{-2}K^{1+\epsilon})$ where $\epsilon>0$ is a constant parameter. Second, when the false negative flipping probability $\rho'$ satisfies $\rho'=\mathcal{O}(K^{-\epsilon})$ and the false positive flipping probability $\rho$ is a constant, we can simultaneously achieve $\mathcal{O}(\epsilon^{-1}K\log N)$ for both the number of tests and the decoding complexity. It remains open to achieve these optimals under the general BSC.