Locating a single facility and a high-speed line

In this paper we study a facility location problem in the plane in which a single point (facility) and a rapid transit line (highway) are simultaneously located in order to minimize the total travel time from the clients to the facility, using the $L_1$ or Manhattan metric. The rapid transit line is given by a segment with any length and orientation, and is an alternative transportation line that can be used by the clients to reduce their travel time to the facility. We study the variant of the problem in which clients can enter and exit the highway at any point. We provide an $O(n^3)$-time algorithm that solves this variant, where $n$ is the number of clients. We also present a detailed characterization of the solutions, which depends on the speed given in the highway.