Approximate the Lower-Bounded Connected Facility Location Problem

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 F subset of V, a client set D subset of V, a parameter M >= 1, and an integer lower bound L are also given. Each facility has an opening cost f(i), and each edge e is an element of 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 subset of F by a Steiner tree T incurs a cost of M Sigma(e is an element of T) c(e) called Steiner cost. The goal is to open some facilities S subset of 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. As our main contribution, we propose two approximation algorithms for the LB ConFL with ratios of 696 and 169. The first algorithm is based on an intuitive idea that finding a suitable Steiner tree before considering the lower bound constraints may give a good solution. The second algorithm effectively avoids the shortcoming of the first one and successfully improves the approximation ratio. Moreover, we consider the general LB ConFL (GLB ConFL) problem, in which each facility i has a non-uniform lower bound L-i. We give an approximation algorithm with a ratio related to the parameter M for the GLB ConFL.