On Detecting Some Defective Items in Group Testing

Group testing is an approach aimed at identifying up to d defective items among a total of n elements. This is accomplished by examining subsets to determine if at least one defective item is present. We focus on the problem of identifying a subset of ℓ< d defective items. We develop upper and lower bounds on the number of tests required to detect ℓ defective items in both the adaptive and non-adaptive settings while considering scenarios where no prior knowledge of d is available, and situations where some non-trivial estimate of d is at hand. When no prior knowledge on d is available, we prove a lower bound of Ω(ℓlog2nlogℓ+loglogn) tests in the randomized non-adaptive settings and an upper bound of O(ℓlog ²n) for the same settings. Furthermore, we demonstrate that any non-adaptive deterministic algorithm must ask Θ(n) tests, signifying a fundamental limitation in this scenario. For adaptive algorithms, we establish tight bounds in different scenarios. In the deterministic case, we prove a tight bound of Θ(ℓlog (n/ ℓ) ). Moreover, in the randomized settings, we derive a tight bound of Θ(ℓlog (n/ d) ). When d, or at least some non-trivial estimate of d, is known, we prove a tight bound of Θ(dlog (n/ d) ) for the deterministic non-adaptive settings, and Θ(ℓlog (n/ d) ) for the randomized non-adaptive settings. In the adaptive case, we present an upper bound of O(ℓlog (n/ ℓ) ) for the deterministic settings, and a lower bound of Ω(ℓlog (n/ d) + log n). Additionally, we establish a tight bound of Θ(ℓlog (n/ d) ) for the randomized adaptive settings.