Approximation Algorithms for Single and Multi-Commodity Connected Facility Location

In the classical facility location problemwe are given a set of facilities, with associated opening costs, and a set of clients. The goal is to open a subset of facilities, and to connect each client to the closest open facility, so that the total connection and opening cost is minimized. In some applications, however, open facilities need to be connected via an infrastructure. Furthermore, connecting two facilities among them is typically more expensive than connecting a client to a facility (for a given path length). This scenario motivated the study of the connected facility location problem (CFL). Here we are also given a parameter M ≥ 1. A feasible solution consists of a subset of open facilities and a Steiner tree connecting them. The cost of the solution is now the opening cost, plus the connection cost, plusM times the cost of the Steiner tree. In this paper we investigate the approximability of CFL and related problems. More precisely, we achieve the following results: • We present a new, simple 3.19 approximation algorithm for CFL. The previous best approximation factor is 3.92 [Eisenbrand, Grandoni, Rothvoß, Schäfer-’10]. •We show that SROB, i.e. the special case of CFL where all opening costs are 0, is hard to approximate within 1.28. The previous best lower bound for SROB is 1.01, and derives trivially from Steiner tree inapproximability [Chlebík, Chlebíková-’08]. The same inapproximability result extends to other well-studied problems, such as virtual private network and single-sink buy-at-bulk. •We introduce and study a natural multi-commodity generalizationMCFL of CFL. In MCFL we are given source-sink pairs (rather than clients) that we wish to connect. A feasible solution consists of a subset of open facilities, and a forest (rather than a tree) spanning them. Source-sink connection paths can use several trees in the forest, but must enter and leave each tree at open facilities. We present the first constant approximation for MCFL.