Improved lower bound for estimating the number of defective items

Abstract

Consider a set of items, X, with a total of n items, among which a subset, denoted as \(I\subseteq X\), consists of defective items. In the context of group testing, a test is conducted on a subset of items Q, where \(Q \subset X\). The result of this test is positive, yielding 1, if Q includes at least one defective item, that is if \(Q \cap I \ne \emptyset \). 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

$$\Omega \left( \dfrac{\log n}{\log \log {\mathop {\cdots }\limits ^{j}}\log n}\right) $$

tests. Our result almost matches the upper bound of \(O(\log n)\) 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 \(\Omega (\log n/\log \log n)\) 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 \(\alpha (n)\), we show: If a constant j exists such that \(\alpha >{\log \log {\mathop {\cdots }\limits ^{j}}\log n}\), then any non-adaptive randomized algorithm that, with probability at least 2/3, estimates the number of defective items |I| to within a factor \(\alpha \) requires at least

$$\Omega \left( \dfrac{\log n}{\log \alpha }\right) .$$

In this case, the lower bound is tight.