On Detecting Some Defective Items in Group Testing

IF 0.8 4区数学 Q4 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Combinatorial Optimization Pub Date : 2026-06-06 DOI:10.1007/s10878-026-01394-8

Nader H. Bshouty, Catherine A. Haddad-Zaknoon

引用次数: 0

Abstract

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell < d$$\end{document}

defective items. We develop upper and lower bounds on the number of tests required to detect ℓ

defective items in both the adaptive and non-adaptive settings while considering scenarios where no prior knowledge of d is available, and situations where some non-trivial estimate of d is at hand. When d is unknown, we prove a lower bound of Ω(ℓlog2nlogℓ+loglogn)

tests in the randomized non-adaptive settings and an upper bound of O(ℓlog2n)

for the same settings. Furthermore, we demonstrate that any non-adaptive deterministic algorithm must make Θ(n)

tests, signifying a fundamental limitation in this scenario. For adaptive algorithms, we establish tight bounds in different scenarios. In the deterministic case, we prove a tight bound of Θ(ℓlog(n/ℓ))

. Moreover, in the randomized settings, we derive a tight bound of Θ(ℓlog(n/d))

. When d, or at least some non-trivial estimate of d, is known, we prove a tight bound of Θ(dlog(n/d))

for the deterministic non-adaptive settings, and Θ(ℓlog(n/d))

for the randomized non-adaptive settings. In the adaptive case, we present an upper bound of O(ℓlog(n/ℓ))

for the deterministic settings, and a lower bound of Ω(ℓlog(n/d)+logn)

. Additionally, we establish a tight bound of Θ(ℓlog(n/d))

for the randomized adaptive settings.

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浅谈成组试验中缺陷项的检测

Group testing is an approach aimed at identifying up to d defective items among a total of n elements. This is accomplished by examining subsets to determine if at least one defective item is present. We focus on the problem of identifying a subset of ℓ<d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell < d$$\end{document} defective items. We develop upper and lower bounds on the number of tests required to detect ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document} defective items in both the adaptive and non-adaptive settings while considering scenarios where no prior knowledge of d is available, and situations where some non-trivial estimate of d is at hand. When d is unknown, we prove a lower bound of Ω(ℓlog2nlogℓ+loglogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varOmega (\frac{\ell \log ^2n}{\log \ell +\log \log n})$$\end{document} tests in the randomized non-adaptive settings and an upper bound of O(ℓlog2n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\ell \log ^2 n)$$\end{document} for the same settings. Furthermore, we demonstrate that any non-adaptive deterministic algorithm must make Θ(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varTheta (n)$$\end{document} tests, signifying a fundamental limitation in this scenario. For adaptive algorithms, we establish tight bounds in different scenarios. In the deterministic case, we prove a tight bound of Θ(ℓlog(n/ℓ))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varTheta (\ell \log {(n/\ell )})$$\end{document}. Moreover, in the randomized settings, we derive a tight bound of Θ(ℓlog(n/d))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varTheta (\ell \log {(n/d)})$$\end{document}. When d, or at least some non-trivial estimate of d, is known, we prove a tight bound of Θ(dlog(n/d))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varTheta (d\log (n/d))$$\end{document} for the deterministic non-adaptive settings, and Θ(ℓlog(n/d))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varTheta (\ell \log (n/d))$$\end{document} for the randomized non-adaptive settings. In the adaptive case, we present an upper bound of O(ℓlog(n/ℓ))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\ell \log (n/\ell ))$$\end{document} for the deterministic settings, and a lower bound of Ω(ℓlog(n/d)+logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega (\ell \log (n/d)+\log n)$$\end{document}. Additionally, we establish a tight bound of Θ(ℓlog(n/d))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varTheta (\ell \log (n/d))$$\end{document} for the randomized adaptive settings.

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来源期刊

Journal of Combinatorial Optimization 数学-计算机：跨学科应用

CiteScore

2.00

自引率

10.00%

发文量

审稿时长

6 months

期刊介绍： The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering. The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.