Group Testing with General Correlation Using Hypergraphs

TL;DR

This paper introduces a hypergraph-based model for group testing with arbitrary correlations among nodes, proposing a greedy adaptive algorithm that efficiently leverages correlations to reduce testing efforts and extend to noisy and semi-non-adaptive scenarios.

Contribution

It presents a new hypergraph model for correlated group testing and a greedy adaptive algorithm with proven bounds, extending the framework to noisy and semi-non-adaptive settings.

Findings

Algorithm recovers or improves upon previous results

Provides upper bounds based on entropy and infection rate

Extends framework to noisy and semi-non-adaptive testing

Abstract

Group testing, a problem with diverse applications across multiple disciplines, traditionally assumes independence across nodes' states. Recent research, however, focuses on real-world scenarios that often involve correlations among nodes, challenging the simplifying assumptions made in existing models. In this work, we consider a comprehensive model for arbitrary statistical correlation among nodes' states. To capture and leverage these correlations effectively, we model the problem by hypergraphs, inspired by [GLS22], augmented by a probability mass function on the hyper-edges. Using this model, we first design a novel greedy adaptive algorithm capable of conducting informative tests and dynamically updating the distribution. Performance analysis provides upper bounds on the number of tests required, which depend solely on the entropy of the underlying probability distribution and the average number of infections. We demonstrate that the algorithm recovers or improves upon all previously known results for group testing settings with correlation. Additionally, we provide families of graphs where the algorithm is order-wise optimal and give examples where the algorithm or its analysis is not tight. We then generalize the proposed framework of group testing with general correlation in two directions, namely noisy group testing and semi-non-adaptive group testing. In both settings, we provide novel theoretical bounds on the number of tests required.