IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v20y2010i1d10.1007_s10878-008-9187-4.html
   My bibliography  Save this article

Median problems with positive and negative weights on cycles and cacti

Author

Listed:
  • Rainer E. Burkard

    (Graz University of Technology)

  • Johannes Hatzl

    (Graz University of Technology)

Abstract

This paper deals with facility location problems on graphs with positive and negative vertex weights. We consider two different objective functions: In the first one (MWD) vertices with positive weight are assigned to the closest facility, whereas vertices with negative weight are assigned to the farthest facility. In the second one (WMD) all the vertices are assigned to the nearest facility. For the MWD model it is shown that there exists a finite set of points in the graph which contains the locations of facilities in an optimal solution. Furthermore, algorithms for both models for the 2-median problem on a cycle are developed. The algorithm for the MWD model runs in linear time, whereas the algorithm for the WMD model has a time complexity of $\mathcal{O}(n^{2})$ .

Suggested Citation

  • Rainer E. Burkard & Johannes Hatzl, 2010. "Median problems with positive and negative weights on cycles and cacti," Journal of Combinatorial Optimization, Springer, vol. 20(1), pages 27-46, July.
    • Handle: RePEc:spr:jcomop:v:20:y:2010:i:1:d:10.1007_s10878-008-9187-4
    DOI: 10.1007/s10878-008-9187-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-008-9187-4
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10878-008-9187-4?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Rainer Burkard & Jafar Fathali, 2007. "A polynomial method for the pos/neg weighted 3-median problem on a tree," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(2), pages 229-238, April.
    2. Richard L. Church & Robert S. Garfinkel, 1978. "Locating an Obnoxious Facility on a Network," Transportation Science, INFORMS, vol. 12(2), pages 107-118, May.
    3. S. L. Hakimi, 1964. "Optimum Locations of Switching Centers and the Absolute Centers and Medians of a Graph," Operations Research, INFORMS, vol. 12(3), pages 450-459, June.
    4. A. J. Goldman, 1971. "Optimal Center Location in Simple Networks," Transportation Science, INFORMS, vol. 5(2), pages 212-221, May.
    5. G. Y. Handler, 1973. "Minimax Location of a Facility in an Undirected Tree Graph," Transportation Science, INFORMS, vol. 7(3), pages 287-293, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Wei Ding & Ke Qiu, 2018. "A quadratic time exact algorithm for continuous connected 2-facility location problem in trees," Journal of Combinatorial Optimization, Springer, vol. 36(4), pages 1262-1298, November.
    2. Mulder, H.M. & Pelsmajer, M.J. & Reid, K.B., 2006. "Generalized centrality in trees," Econometric Institute Research Papers EI 2006-16, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    3. Igor Averbakh & Oded Berman, 2000. "Minmax Regret Median Location on a Network Under Uncertainty," INFORMS Journal on Computing, INFORMS, vol. 12(2), pages 104-110, May.
    4. ReVelle, C.S. & Eiselt, H.A. & Daskin, M.S., 2008. "A bibliography for some fundamental problem categories in discrete location science," European Journal of Operational Research, Elsevier, vol. 184(3), pages 817-848, February.
    5. Mehdi Zaferanieh & Jafar Fathali, 2012. "Finding a core of a tree with pos/neg weight," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 76(2), pages 147-160, October.
    6. Esmaeil Afrashteh & Behrooz Alizadeh & Fahimeh Baroughi, 2020. "Optimal approaches for upgrading selective obnoxious p-median location problems on tree networks," Annals of Operations Research, Springer, vol. 289(2), pages 153-172, June.
    7. Balakrishnan, K. & Changat, M. & Mulder, H.M. & Subhamathi, A.R., 2011. "Consensus Strategies for Signed Profiles on Graphs," Econometric Institute Research Papers EI2011-34, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    8. Zvi Drezner & G. O. Wesolowsky, 1991. "Facility location when demand is time dependent," Naval Research Logistics (NRL), John Wiley & Sons, vol. 38(5), pages 763-777, October.
    9. Rainer Burkard & Jafar Fathali, 2007. "A polynomial method for the pos/neg weighted 3-median problem on a tree," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(2), pages 229-238, April.
    10. Chunsong Bai & Jun Du, 2024. "The Constrained 2-Maxian Problem on Cycles," Mathematics, MDPI, vol. 12(6), pages 1-9, March.
    11. Oded Berman & Dmitry Krass & Mozart B. C. Menezes, 2007. "Facility Reliability Issues in Network p -Median Problems: Strategic Centralization and Co-Location Effects," Operations Research, INFORMS, vol. 55(2), pages 332-350, April.
    12. Hakimi, S.Louis, 1983. "Network location theory and contingency planning," Energy, Elsevier, vol. 8(8), pages 697-702.
    13. Xinqiang Qian & Xiucui Guan & Junhua Jia & Qiao Zhang & Panos M. Pardalos, 2023. "Vertex quickest 1-center location problem on trees and its inverse problem under weighted $$l_\infty $$ l ∞ norm," Journal of Global Optimization, Springer, vol. 85(2), pages 461-485, February.
    14. Ting Zeng & James Ward, 2005. "The Stochastic Location-Assignment Problem on a Tree," Annals of Operations Research, Springer, vol. 136(1), pages 81-97, April.
    15. ReVelle, C. S. & Eiselt, H. A., 2005. "Location analysis: A synthesis and survey," European Journal of Operational Research, Elsevier, vol. 165(1), pages 1-19, August.
    16. H K Smith & G Laporte & P R Harper, 2009. "Locational analysis: highlights of growth to maturity," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 60(1), pages 140-148, May.
    17. Vladimir Marianov & Daniel Serra, 2009. "Median problems in networks," Economics Working Papers 1151, Department of Economics and Business, Universitat Pompeu Fabra.
    18. Van Huy Pham & Nguyen Chi Tam, 2019. "A combinatorial algorithm for the ordered 1-median problem on cactus graphs," OPSEARCH, Springer;Operational Research Society of India, vol. 56(3), pages 780-789, September.
    19. Drezner, Zvi & Eiselt, H.A., 2024. "Competitive location models: A review," European Journal of Operational Research, Elsevier, vol. 316(1), pages 5-18.
    20. Carrizosa, Emilio & Conde, Eduardo, 2002. "A fractional model for locating semi-desirable facilities on networks," European Journal of Operational Research, Elsevier, vol. 136(1), pages 67-80, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jcomop:v:20:y:2010:i:1:d:10.1007_s10878-008-9187-4. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.