Approximate the Lower-Bounded Connected Facility Location Problem

Abstract

This paper studies the lower-bounded connected facility location (LB ConFL) problem, which extends the well-known connected facility location (ConFL) and lower-bounded facility location (LBFL) problems. In the LB ConFL, we are given a graph \(G = (V, E)\), where V and E are all the vertices and edges, respectively. A facility set \(\mathcal {F} \subseteq V\), a client set \(\mathcal {D} \subseteq V\), a parameter \(M \ge 1\), and an integer lower bound L are also given. Each facility has an opening cost \(f_i\), and each edge \(e \in E\) has a connection cost \(c_e\). Denote by \(c_{uv}\) the shortest path with respect to the connection costs from vertex u to v. Opening a facility i incurs its opening cost. Assigning a client j to some facility i incurs a connection cost \(c_{ij}\). Connecting a facility subset \(S \subseteq \mathcal {F}\) by a Steiner tree T incurs a cost of \(M\sum _{e \in T} c_e\) called Steiner cost. The goal is to open some facilities \(S \subseteq \mathcal {F}\), assign each client j to some opened facility in S and connect all the opened facilities S by a Steiner tree, such that the number of clients connected to any opened facility is at least L, and the total incurred cost (i.e., the total opening, connection, and Steiner cost) is minimized.