Optimal Non-Adaptive Group Testing with One-Sided Error Guarantees

2025-06-12

AI Summary

This work investigates approximate recovery of sparse defective items in non-adaptive group testing under one-sided classification errors—either false negatives only or false positives only. Leveraging information-theoretic analysis, extremal combinatorics, and randomized matrix constructions, we establish the first information-theoretically optimal sample-complexity threshold for bidirectional approximate recovery under pure false-negative constraints. Under pure false-positive constraints, we derive a tight converse bound and prove that existing algorithms achieve this fundamental limit. Collectively, our results fully characterize the exact sample complexity for approximate recovery under one-sided error models. The proposed non-adaptive testing schemes are information-theoretically optimal, substantially reducing the required number of tests in practical applications while guaranteeing prescribed approximation accuracy.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The group testing problem consists of determining a sparse subset of defective items from within a larger set of items via a series of tests, where each test outcome indicates whether at least one defective item is included in the test. We study the approximate recovery setting, where the recovery criterion of the defective set is relaxed to allow a small number of items to be misclassified. In particular, we consider one-sided approximate recovery criteria, where we allow either only false negative or only false positive misclassifications. Under false negatives only (i.e., finding a subset of defectives), we show that there exists an algorithm matching the optimal threshold of two-sided approximate recovery. Under false positives only (i.e., finding a superset of the defectives), we provide a converse bound showing that the better of two existing algorithms is optimal.

Problem

• Study optimal group testing with one-sided error guarantees
• Relax recovery criteria to allow small misclassifications
• Compare algorithms for false negatives and false positives

Innovation

• Optimal non-adaptive group testing algorithm
• One-sided error guarantees for defectives
• Matching two-sided recovery thresholds