Annals of Operations Research

, Volume 246, Issue 1–2, pp 167–179 | Cite as

The 1-Center and 1-Highway problem revisited

  • J. M. Díaz-Báñez
  • M. Korman
  • P. Pérez-Lantero
  • I. Ventura
Article
  • 156 Downloads

Abstract

In this paper we extend the Rectilinear 1-center problem as follows: given a set \(S\) of \(n\) demand points in the plane, simultaneously locate a facility point \(f\) and a rapid transit line (i.e. highway) \(h\) that together minimize the expression \(\max _{p\in S}T_{h}(p,f)\), where \(T_{h}(p,f)\) denotes the travel time between \(p\) and \(f\). A point of \(S\) uses \(h\) to reach \(f\) if \(h\) saves time: every point \(p\in S\) moves outside \(h\) at unit speed under the \(L_1\) metric, and moves along \(h\) at a given speed \(v>1\). We consider two types of highways: (1) a turnpike in which the demand points can enter/exit the highway only at the endpoints; and (2) a freeway problem in which the demand points can enter/exit the highway at any point. We solve the location problem for the turnpike case in \(O(n^2)\) or \(O(n\log n)\) time, depending on whether or not the highway’s length is fixed. In the freeway case, independently of whether the highway’s length is fixed or not, the location problem can be solved in \(O(n\log n)\) time.

Keywords

Geometric optimization Facility location Time metric Rectilinear 1-center problem 

Notes

Acknowledgments

J. M. Díaz-Báñez, P. Pérez-Lantero and I. Ventura were partially supported by the project FEDER MEC MTM2009-08652. J. M. Díaz-Báñez, M. Korman and I. Ventura were partially supported by ESF EUROCORES programme EuroGIGA, CRP ComPoSe: grant EUI-EURC-2011-4306. M. Korman was partially supported by the Secretary for Universities and Research of the Ministry of Economy and Knowledge of the Government of Catalonia and the European Union. P. Pérez-Lantero was partially supported by project Millennium Nucleus Information and Coordination in Networks ICM/FIC RC130003 (Chile).

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • J. M. Díaz-Báñez
    • 1
  • M. Korman
    • 2
    • 3
  • P. Pérez-Lantero
    • 4
  • I. Ventura
    • 1
  1. 1.Departamento de Matemática Aplicada IIUniversidad de SevillaSevilleSpain
  2. 2.National Institute of InformaticsTokyoJapan
  3. 3.JST, ERATO, Kawarabayashi Large Graph ProjectTokyoJapan
  4. 4.Escuela de Ingeniería Civil en InformáticaUniversidad de ValparaísoValparaisoChile

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