# Primal-Dual Algorithms for Connected Facility Location Problems

## Abstract

We consider the *Connected Facility Location* problem. We are given a graph *G* = (*V,E*) with cost *c* _{e} on edge *e*, a set of facilities *F* ⊆ *V*, and a set of demands *D* ⊆ *V*. We are also given a parameter *M* ≥ 1. A solution opens some facilities, say *F*, assigns each demand *j* to an open facility *i(j)*, and connects the open facilities by a Steiner tree *T*. The cost incurred is \(
\sum {_{i \in F} f_i } + \sum {_{j \in \mathcal{D}} d_j c_{i\left( j \right)j} } + M\sum {_{e \in T} c_e }
\)
. We want a solution of minimum cost. A special case is when all opening costs are 0 and facilities may be opened anywhere, i.e.,*F* = *V*. If we *know* a facility *v* that is open, then this problem reduces to the *rent-or-buy* problem. We give the first primal-dual algorithms for these problems and achieve the best known approximation guarantees. We give a 9-approximation algorithm for connected facility location and a 5-approximation for the rent-or-buy problem. Our algorithm integrates the primal-dual approaches for facility location [7] and Steiner trees [1],[2]. We also consider the connected *k*-median problem and give a constant-factor approximation by using our primal-dual algorithm for connected facility location. We generalize our results to an edge capacitated version of these problems.

## Keywords

Facility Location Steiner Tree Facility Location Problem Demand Point Network Design Problem## Preview

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