TY - GEN

T1 - Improved Lower Bound for Estimating the Number of Defective Items

AU - Bshouty, Nader H.

N1 - Publisher Copyright:
© 2024, The Author(s), under exclusive license to Springer Nature Switzerland AG.

PY - 2024

Y1 - 2024

N2 - 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 / log log ⋯ klog 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$$\varOmega \left(\frac{\log n}{\log \log {\mathop {\cdots }\limits ^{k}}\log n}\right) $$Ω(lognloglog⋯klogn) tests. Our result almost matches the upper bound of O(log n) and solves the open problem posed by Damaschke and Sheikh Muhammad in [8, 9]. Additionally, it improves upon the lower bound of Ω(log n/ log log n) previously established by Ron and Tsur [21] and independently by Bshouty [2].

AB - 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 / log log ⋯ klog 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$$\varOmega \left(\frac{\log n}{\log \log {\mathop {\cdots }\limits ^{k}}\log n}\right) $$Ω(lognloglog⋯klogn) tests. Our result almost matches the upper bound of O(log n) and solves the open problem posed by Damaschke and Sheikh Muhammad in [8, 9]. Additionally, it improves upon the lower bound of Ω(log n/ log log n) previously established by Ron and Tsur [21] and independently by Bshouty [2].

KW - Estimation

KW - Group Testing

KW - Randomized Algorithm

UR - http://www.scopus.com/inward/record.url?scp=85180539944&partnerID=8YFLogxK

U2 - 10.1007/978-3-031-49611-0_22

DO - 10.1007/978-3-031-49611-0_22

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AN - SCOPUS:85180539944

SN - 9783031496103

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 303

EP - 315

BT - Combinatorial Optimization and Applications - 16th International Conference, COCOA 2023, Proceedings

A2 - Wu, Weili

A2 - Guo, Jianxiong

T2 - 16th Annual International Conference on Combinatorial Optimization and Applications, COCOA 2023

Y2 - 15 December 2023 through 17 December 2023

ER -