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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
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    References listed on IDEAS

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    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.
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