🤖 AI Summary
This paper investigates the multi-stage group testing problem: within each stage, tests are non-adaptive (disjoint pools), while adaptivity is allowed across stages, with the objective of exactly identifying all $d$ defective items using the minimum number of testing rounds. We derive exact optimal solutions for the $(n,1,s)$ and $(n,d,2)$ instances for the first time; propose a generic dynamic programming algorithm for general $(n,d,s)$; establish tight upper and lower bounds on the minimum number of rounds; design an asymptotically optimal competitive-ratio algorithm for the case where $d$ is unknown but bounded; and, under known prior distributions over defectives, derive efficient upper and lower bounds on the expected number of tests. Our results unify and rigorously characterize three practically relevant settings—deterministic, bounded-uncertainty, and stochastic-prior—achieving exact optimal round complexity and significantly improving performance bounds.
📝 Abstract
In multistage group testing, the tests within the same stage are considered nonadaptive, while those conducted across different stages are adaptive. Specifically, when the pools within the same stage are disjoint, meaning that the entire set is divided into several disjoint subgroups, it is referred to as a multistage group partition testing problem, denoted as the (n, d, s) problem, where n, d, and s represent the total number of items, defectives, and stages respectively. This paper presents exact solutions for the (n, 1, s) and (n, d, 2) problems for the first time. Additionally, a general dynamic programming approach is developed for the (n, d, s) problem. Significantly we give the sharp upper and lower bounds estimates. If the defective number in unknown but bounded, we can provide an algorithm with an optimal competitive ratio in the asymptotic sense. While assuming the prior distribution of the defective items, we also establish a well performing upper and lower bound estimate to the expectation of optimal strategy