Small-Error Cascaded Group Testing

Group testing concerns itself with the accurate recovery of a set of "defective" items from a larger population via a series of tests. While most works in this area have considered the classical group testing model, where tests are binary and indicate the presence of at least one defective item in the test, we study the cascaded group testing model. In cascaded group testing, tests admit an ordering, and test outcomes indicate the first defective item in the test under this ordering. Under this model, we establish various achievability bounds for several different recovery criteria using both non-adaptive and adaptive test designs when assuming both unconstrained and constrained test sizes. In the constrained test size setting, we also provide a lower bound showing our achievability result is optimal up to logarithmic factors.