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Open Access

Algorithms for the Prize-Collecting k-Steiner Tree Problem

School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing 100124, China
School of Computer Science and Technology, Shandong Jianzhu University, Jinan 250101, China
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Abstract

In this paper, we study the prize-collecting k-Steiner tree (PC kST) problem. We are given a graph G=(V,E) and an integer k. The graph is connected and undirected. A vertex r∈V called root and a subset R⊆V called terminals are also given. A feasible solution for the PC kST is a tree F rooted at r and connecting at least k vertices in R. Excluding a vertex from the tree incurs a penalty cost, and including an edge in the tree incurs an edge cost. We wish to find a feasible solution with minimum total cost. The total cost of a tree F is the sum of the edge costs of the edges in F and the penalty costs of the vertices not in F. We present a simple approximation algorithm with the ratio of 5.9672 for the PC kST. This algorithm uses the approximation algorithms for the prize-collecting Steiner tree (PCST) problem and the k-Steiner tree ( kST) problem as subroutines. Then we propose a primal-dual based approximation algorithm and improve the approximation ratio to 5.

Keywords

approximation algorithmprize-collectingSteiner tree

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Tsinghua Science and Technology
Volume 27 Issue 5,
October 2022
Pages 785-792
DOI: 10.26599/TST.2021.9010053
Cite this article:
Han L, Wang C, Xu D, et al. Algorithms for the Prize-Collecting k-Steiner Tree Problem. Tsinghua Science and Technology, 2022, 27(5): 785-792. https://doi.org/10.26599/TST.2021.9010053

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Received: 10 May 2021
Revised: 17 June 2021
Accepted: 17 July 2021
Published: 17 March 2022
© The author(s) 2022.

The articles published in this open access journal are distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/).
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