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

TL;DR

This paper investigates optimal non-adaptive group testing algorithms with one-sided error guarantees, providing matching algorithms and bounds for false negative and false positive scenarios.

Contribution

It introduces optimal algorithms and bounds for one-sided approximate recovery in non-adaptive group testing, matching theoretical limits.

Findings

Optimal algorithm for false negatives matches the two-sided recovery threshold.

Converse bound shows existing algorithm is optimal for false positives.

Provides theoretical guarantees for one-sided approximate recovery.

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.