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 ce 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 Σi∈F fi + Σj∈D djci(j)j+M Σe∈T ce. W e 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.