#### Keywords

prize-collecting, Steiner tree, approximation algorithm

#### Abstract

In this paper, we study the prize-collecting k-Steiner tree (PCkST) 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 PCkST 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 PCkST. 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.

#### Recommended Citation

Han, Lu; Wang, Changjun; Xu, Dachuan; and Zhang, Dongmei
(2022)
"Algorithms for the Prize-Collecting k-Steiner Tree Problem,"
*Tsinghua Science and Technology*: Vol. 27:
Iss.
5, Article 2.

DOI: https://doi.org/10.26599/TST.2021.9010053

Available at:
https://dc.tsinghuajournals.com/tsinghua-science-and-technology/vol27/iss5/2