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On the Continuous Fermat-Weber Problem

Author

Listed:
  • Sándor P. Fekete

    (Institute for Mathematical Optimization, Braunschweig University of Technology, 38106 Braunschweig, Germany)

  • Joseph S. B. Mitchell

    (Department of Applied Mathematics and Statistics, State University of New York, Stony Brook, New York 11794-3600)

  • Karin Beurer

    (SAP AG, 69190 Walldorf, Germany)

Abstract

We give the first exact algorithmic study of facility location problems that deal with finding a median for a continuum of demand points. In particular, we consider versions of the “continuous k -median (Fermat-Weber) problem” where the goal is to select one or more center points that minimize the average distance to a set of points in a demand region . In such problems, the average is computed as an integral over the relevant region, versus the usual discrete sum of distances. The resulting facility location problems are inherently geometric, requiring analysis techniques of computational geometry. We provide polynomial-time algorithms for various versions of the L 1 1-median (Fermat-Weber) problem. We also consider the multiple-center version of the L 1 k -median problem, which we prove is NP-hard for large k .

Suggested Citation

  • Sándor P. Fekete & Joseph S. B. Mitchell & Karin Beurer, 2005. "On the Continuous Fermat-Weber Problem," Operations Research, INFORMS, vol. 53(1), pages 61-76, February.
    • Handle: RePEc:inm:oropre:v:53:y:2005:i:1:p:61-76
    DOI: 10.1287/opre.1040.0137
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    References listed on IDEAS

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    Cited by:

    1. Blanco, Víctor & Gázquez, Ricardo & Ponce, Diego & Puerto, Justo, 2023. "A branch-and-price approach for the continuous multifacility monotone ordered median problem," European Journal of Operational Research, Elsevier, vol. 306(1), pages 105-126.
    2. Jing Yao & Alan T. Murray, 2014. "Serving regional demand in facility location," Papers in Regional Science, Wiley Blackwell, vol. 93(3), pages 643-662, August.
    3. Valentin Hartmann & Dominic Schuhmacher, 2020. "Semi-discrete optimal transport: a solution procedure for the unsquared Euclidean distance case," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(1), pages 133-163, August.
    4. Gérard P. Cachon, 2014. "Retail Store Density and the Cost of Greenhouse Gas Emissions," Management Science, INFORMS, vol. 60(8), pages 1907-1925, August.
    5. Thomas Byrne & Sándor P. Fekete & Jörg Kalcsics & Linda Kleist, 2023. "Competitive location problems: balanced facility location and the One-Round Manhattan Voronoi Game," Annals of Operations Research, Springer, vol. 321(1), pages 79-101, February.
    6. Igor Averbakh & Oded Berman & Jörg Kalcsics & Dmitry Krass, 2015. "Structural Properties of Voronoi Diagrams in Facility Location Problems with Continuous Demand," Operations Research, INFORMS, vol. 63(2), pages 394-411, April.
    7. John Gunnar Carlsson & Fan Jia & Ying Li, 2014. "An Approximation Algorithm for the Continuous k -Medians Problem in a Convex Polygon," INFORMS Journal on Computing, INFORMS, vol. 26(2), pages 280-289, May.
    8. John Gunnar Carlsson & Raghuveer Devulapalli, 2013. "Dividing a Territory Among Several Facilities," INFORMS Journal on Computing, INFORMS, vol. 25(4), pages 730-742, November.
    9. Daoqin Tong & Alan T. Murray, 2009. "Maximising coverage of spatial demand for service," Papers in Regional Science, Wiley Blackwell, vol. 88(1), pages 85-97, March.
    10. Byrne, Thomas & Kalcsics, Jörg, 2022. "Conditional facility location problems with continuous demand and a polygonal barrier," European Journal of Operational Research, Elsevier, vol. 296(1), pages 22-43.
    11. Thomas Byrne & S'andor P. Fekete & Jorg Kalcsics & Linda Kleist, 2020. "Competitive Location Problems: Balanced Facility Location and the One-Round Manhattan Voronoi Game," Papers 2011.13275, arXiv.org, revised Sep 2022.

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