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.

Research partially supported by NSF grant CCR-9912422.

Research supported by NSF grant CCR-9820951 and NSF ITR/IM grant IIS- 0081334.