Airports and Railways: Facility Location Meets Network Design

We introduce a new framework of Airport and Railway Problems, which combines capacitated facility location with network design. In this framework we are given a graph with weights on the vertices and on the edges, together with a parameter k. The vertices of the graph represent cities, and weights denote respectively the costs of opening airports in the cities and building railways that connect pairs of cities. The parameter k can be thought of as the capacity of an airport. The goal is to construct a minimum cost network of airports and railways connecting the cities, where each connected component in the network spans at most k vertices, contains an open airport, and the network satisfies some additional requirements specific to the problem in the framework. We consider two problems in this framework. In the ARF problem there are no additional requirements for the network. This problem is related to capacitated facility location. In the ARP problem, we require each component to be a path with airports at both endpoints. ARP is a relaxation of the capacitated vehicle routing problem (CVRP). We consider the problems in the two-dimensional Euclidean setting. We show that both ARF and ARP are NP-hard, even for uniform vertex weights (i. e., when the cost of building an airport is the same for all cities). On the positive side, we provide polynomial time approximation schemes for ARF and ARP when vertex weights are uniform. We also investigate ARF and ARP for k =∞. In this setting we present an exact polynomial time algorithm for ARF with general vertex costs, which also works for general edge costs. In contrast to ARF, ARP remains NP-hard when k =∞, and we present a polynomial time approximation scheme for general vertex weights. We believe that our PTAS for ARP with uniform vertex weights and arbitrary k brings us closer towards a PTAS for Euclidean CVRP, for which the main difficulty is to deal with paths of length at most k. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems