【摘要】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.