The Prize-Collecting k-Steiner Tree Problem

Lu Han, Changjun Wang, Dachuan Xu, Dongmei Zhang

Abstract: In this paper, we study the prize-collecting k-Steiner tree problem (PC k-ST), which is an interesting generalization of both the k-Steiner tree problem (k-ST) and the prize-collecting Steiner tree problem (PCST). In the PC k-ST, we are given an undirected connected graph \(G =(V, E)\), a subset \(R \subseteq V\) called terminals, a root vertex \(r \in V\) and an integer k. Every edge has a non-negative edge cost and every vertex has a non-negative penalty cost. We wish to find an r-rooted tree F that spans at least k vertices in R so as to minimize the total edge costs of F as well as the penalty costs of the vertices not in F. As our main contribution, we propose two approximation algorithms for the PC k-ST with ratios of 5.9672 and 5. The first algorithm is based on an observation of the solutions for the k-ST and the PCST, and the second one is based on the technique of primal-dual.