Improved Lower Bound for Estimating the Number of Defective Items
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Let X be a set of items of size n that contains some defective items, denoted by I, where I ⊆ X. In group testing, a test refers to a subset of items Q ⊂ X. The outcome of a test is 1 if Q contains at least one defective item, i.e., Q∩ I ≠∅, and 0 otherwise. We give a novel approach to obtaining lower bounds in non-adaptive randomized group testing. The technique produced lower bounds that are within a factor of 1/loglogk⋯log n of the existing upper bounds for any constant k. Employing this new method, we can prove the following result. For any fixed constants k, any non-adaptive randomized algorithm that, for any set of defective items I, with probability at least 2/3, returns an estimate of the number of defective items |I| to within a constant factor requires at least Ω(log n/loglogk⋯log n) tests. Our result almost matches the upper bound of O(log n) and solves the open problem posed by Damaschke and Sheikh Muhammad [COCOA 2010 and Discrete Math., Alg. and Appl., 2010]. Additionally, it improves upon the lower bound of Ω(log n/loglog n) previously established by Bshouty [ISAAC 2019].
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