An approximation algorithm for the group prize-collecting Steiner tree problem with submodular penalties

Abstract

In this paper, we consider the group prize-collecting Steiner tree problem with submodular penalties (GPCST-SP problem). In this problem, we are given an undirected connected graph $G=(V,E)$ with a pre-specified root r and a partition $\mathcal {V}=\{V_0,V_1,\ldots ,V_k\}$ of V with $r\in V_0$. Assume $c: E\rightarrow \mathbb {R}_+$ is an edge cost function and $p: 2^{\mathcal {V}}\rightarrow \mathbb {R}_+$ is a submodular penalty function, where $\mathbb {R}_+$ is the set of nonnegative real numbers. For a group $V_i\in \mathcal {V}$, we say it is spanned by a tree if the tree contains at least one vertex of that group. The goal of the GPCST-SP problem is to find an r-rooted tree that minimizes the costs of the edges in the tree plus the penalty cost of the subcollection $\mathcal {S}$ containing these groups not spanned by the tree. Our main result is a 2I-approximation algorithm for the problem, where $I=\max \{ |V_i| \mid i=0,1,2,\ldots ,k\}$.