Computational Geometry pp 126-137

Part of the Lecture Notes in Computer Science book series (LNCS, volume 7579)

# Locating a Service Facility and a Rapid Transit Line

• José Miguel Díaz-Báñez
• Matias Korman
• Pablo Pérez-Lantero
Chapter

## Abstract

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 L1 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 [8]. They gave both a characterization of the optimal solutions and an algorithm running in O(n3logn) 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(n3) time.

### Keywords

Geometric optimization Facility location Time distance

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### References

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## Authors and Affiliations

• José Miguel Díaz-Báñez
• 1
• Matias Korman
• 2
• Pablo Pérez-Lantero
• 3