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 $$I\subseteq X$$ I ⊆ X , consists of defective items. In the context of group testing, a test is conducted on a subset of items Q, where $$Q \subset X$$ Q ⊂ X . The result of this test is positive, yielding 1, if Q includes at least one defective item, that is if $$Q \cap I \ne \emptyset $$ 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 $$\Omega \left( \dfrac{\log n}{\log \log {\mathop {\cdots }\limits ^{j}}\log n}\right) $$ Ω log n log log ⋯ j log n tests. Our result almost matches the upper bound of $$O(\log n)$$ 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 $$\Omega (\log n/\log \log n)$$ Ω ( 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 $$\alpha (n)$$ α ( n ) , we show: If a constant j exists such that $$\alpha >{\log \log {\mathop {\cdots }\limits ^{j}}\log n}$$ α > log log ⋯ 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 $$\alpha $$ α requires at least $$\Omega \left( \dfrac{\log n}{\log \alpha }\right) .$$ Ω log n log α . In this case, the lower bound is tight.