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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 of the clients to the facility, using the L 1 or Manhattan metric. The rapid transit line is represented by a line segment with fixed length and arbitrary orientation. The highway is an alternative transportation system that can be used by the clients to reduce their travel time to the facility. This problem was introduced by Espejo and Rodríguez-Chía in . They gave both a characterization of the optimal solutions and an algorithm running in O(n 3logn) time, where n represents the number of clients. In this paper we show that the Espejo and Rodríguez-Chía’s algorithm does not always work correctly. At the same time, we provide a proper characterization of the solutions with a simpler proof and give an algorithm solving the problem in O(n 3) time.
CitationDíaz, J. [et al.]. Locating a service facility and a rapid transit line. A: Spanish Meeting on Computational Geometry. "Computational Geometry: XIV Spanish Meeting on Computational Geometry, EGC 2011, dedicated to Ferran Hurtado on the occasion of His 60th birthday, Alcalá de Henares, Spain, June 27-30, 2011: revised selected papers". Alcalá de Henares (Madrid): Springer, 2011, p. 126-137.
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