Approximation Algorithms for Network Design and Facility Location with Service Capacities

We present the first constant-factor approximation algorithms for the following problem: Given a metric space (V,c), a set D ⊆ V of terminals/ customers with demands d:D→ℝ + , a facility opening cost f ∈ ℝ + and a capacity u ∈ ℝ + , find a partition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D=D_1\dot{\cup}\cdots\dot{\cup} D_k$\end{document} and Steiner trees Ti for Di (i=1,...,k) with c(E(Ti))+d(Di)≤ u for i=1,...,k such that ∑\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$_{i=1}^{k}$\end{document}c(E(Ti)) + kf is minimum. This problem arises in VLSI design. It generalizes the bin-packing problem and the Steiner tree problem. In contrast to other network design and facility location problems, it has the additional feature of upper bounds on the service cost that each facility can handle. Among other results, we obtain a 4.1-approximation in polynomial time, a 4.5-approximation in cubic time and a 5-approximation as fast as computing a minimum spanning tree on (D,c).