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

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 some number of items (up to a certain specified threshold) to be misclassified. In particular, we considerone-sidedapproximate 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, albeit with exponential runtime. 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.