Abstract
We address the 1-line Steiner tree problem with minimum number of Steiner points. Given a line l, a point set P of n terminals in \({\mathbb {R}}^2\) and a positive constant K, we are asked to find a Steiner tree \(T_{l}\) to interconnect the line l and the n terminals such that the Euclidean length of each edge in \(T_{l}\) is no more than the given positive constant K except those connecting two points on the line l, the objective is to minimize total number of the Steiner points in \(T_{l}\), i.e. \(\min _{T_{l}}\{|S_{out}|+|S_{on}|\}\), where \(|S_{out}|\) and \(|S_{on}|\) are the number of Steiner points located outside of the line l and on this line l, respectively. We design a 4-approximation algorithm with time complexity of \(O(n^3)\) for the 1-line Steiner tree problem with minimum number of Steiner points.
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Acknowledgements
The authors would like to thank the anonymous referees for careful reading of the paper and for constructive comments, which are greatly appreciated. This work was supported by the China Scholarship Council (Grant No. 202107030013), Yunnan Provincial Department of Education Science Research Fund (Grant No. 2022Y050), and Graduate Research and Innovation of Yunnan University (Grant No. 2020Z66).
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Liu, S. Approximation algorithm for solving the 1-line Steiner tree problem with minimum number of Steiner points. Optim Lett (2023). https://doi.org/10.1007/s11590-023-02058-w
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DOI: https://doi.org/10.1007/s11590-023-02058-w