Improved lower bound for estimating the number of defective items

Consider a set of items, X, with a total of n items, among which a subset, denoted as I⊆X, consists of defective items. In the context of group testing, a test is conducted on a subset of items Q, where Q⊂X. The result of this test is positive, yielding 1, if Q includes at least one defective item, that is if Q∩I≠∅. It is negative, yielding 0, if no defective items are present in Q. We introduce a novel method for deriving lower bounds in the context of non-adaptive randomized group testing. For any given constant j, any non-adaptive randomized algorithm that, with probability at least 2/3, estimates the number of defective items |I| within a constant factor requires at least (Formula presented.) tests. Our result almost matches the upper bound of O(logn) and addresses the open problem posed by Damaschke and Sheikh Muhammad in (Combinatorial Optimization and Applications - 4th International Conference, COCOA 2010, pp 117–130, 2010; Discrete Math Alg Appl 2(3):291–312, 2010). Furthermore, it enhances the previously established lower bound of Ω(logn/loglogn) by Ron and Tsur (ACM Trans Comput Theory 8(4): 15:1–15:19, 2016), and independently by Bshouty (30th International Symposium on Algorithms and Computation, ISAAC 2019, LIPIcs, vol 149, pp 2:1–2:9, 2019). For estimation within a non-constant factor α(n), we show: If a constant j exists such that α>loglog⋯jlogn, then any non-adaptive randomized algorithm that, with probability at least 2/3, estimates the number of defective items |I| to within a factor α requires at least (Formula presented.) In this case, the lower bound is tight.