Approximation algorithms for connected facility location problems

We study Connected Facility Location problems. We are given a connected graph G=(V,E) with nonnegative edge cost ce for each edge e∈E, a set of clients D⊆V such that each client j∈D has positive demand dj and a set of facilities F⊆V each has nonnegative opening cost fi and capacity to serve all client demands. The objective is to open a subset of facilities, say \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat{F}$\end{document} , to assign each client j∈D to exactly one open facility i(j) and to connect all open facilities by a Steiner tree T such that the cost \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sum_{i\in \hat{F}}f_{i}+\sum_{j\in D}d_{j}c_{i(j)j}+M\sum_{e\in T}c_{e}$\end{document} is minimized for a given input parameter M≥1. We propose a LP-rounding based 8.29 approximation algorithm which improves the previous bound 8.55 (Swamy and Kumar in Algorithmica, 40:245–269, 2004). We also consider the problem when opening cost of all facilities are equal. In this case we give a 7.0 approximation algorithm.