IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/vyid10.1007_s10878-020-00549-5.html
   My bibliography  Save this article

Max–min dispersion on a line

Author

Listed:
  • Tetsuya Araki

    (Gunma University)

  • Shin-ichi Nakano

    (Gunma University)

Abstract

Given a set P of n locations on which facilities can be placed and an integer k, we want to place k facilities on some locations so that a designated objective function is maximized. The problem is called the k-dispersion problem. For instance it is desirable to locate fire departments far away each other. In this paper we give a simple $$O((2k^2)^k n)$$O((2k2)kn) time algorithm to solve the max–min version of the k-dispersion problem if P is a set of points on a line. If k is a constant then this is an O(n) time algorithm. This is the first O(n) time algorithm to solve the max–min k-dispersion problem for the set of “unsorted” points on a line. If P is a set of sorted points on a line, and the input is given as an array in which the coordinates of the points are stored in the sorted order, then by slightly modifying the algorithm above one can solve the dispersion problem in $$O(\log n)$$O(logn) time. This is the first sublinear time algorithm to solve the max–min k-dispersion problem for the set of sorted points on a line.

Suggested Citation

  • Tetsuya Araki & Shin-ichi Nakano, 0. "Max–min dispersion on a line," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-7.
    • Handle: RePEc:spr:jcomop:v::y::i::d:10.1007_s10878-020-00549-5
    DOI: 10.1007/s10878-020-00549-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-020-00549-5
    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/s10878-020-00549-5?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. Erkut, Erhan, 1990. "The discrete p-dispersion problem," European Journal of Operational Research, Elsevier, vol. 46(1), pages 48-60, May.
    2. S. S. Ravi & D. J. Rosenkrantz & G. K. Tayi, 1994. "Heuristic and Special Case Algorithms for Dispersion Problems," Operations Research, INFORMS, vol. 42(2), pages 299-310, April.
    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. Rennen, G., 2008. "Subset Selection from Large Datasets for Kriging Modeling," Discussion Paper 2008-26, Tilburg University, Center for Economic Research.
    2. Erkut, E. & ReVelle, C. & Ulkusal, Y., 1996. "Integer-friendly formulations for the r-separation problem," European Journal of Operational Research, Elsevier, vol. 92(2), pages 342-351, July.
    3. Spiers, Sandy & Bui, Hoa T. & Loxton, Ryan, 2023. "An exact cutting plane method for the Euclidean max-sum diversity problem," European Journal of Operational Research, Elsevier, vol. 311(2), pages 444-454.
    4. Prokopyev, Oleg A. & Kong, Nan & Martinez-Torres, Dayna L., 2009. "The equitable dispersion problem," European Journal of Operational Research, Elsevier, vol. 197(1), pages 59-67, August.
    5. Anna Martínez-Gavara & Vicente Campos & Manuel Laguna & Rafael Martí, 2017. "Heuristic solution approaches for the maximum minsum dispersion problem," Journal of Global Optimization, Springer, vol. 67(3), pages 671-686, March.
    6. Nicolas Dupin & Frank Nielsen & El-Ghazali Talbi, 2021. "Unified Polynomial Dynamic Programming Algorithms for P-Center Variants in a 2D Pareto Front," Mathematics, MDPI, vol. 9(4), pages 1-30, February.
    7. Lei, Ting L. & Church, Richard L., 2015. "On the unified dispersion problem: Efficient formulations and exact algorithms," European Journal of Operational Research, Elsevier, vol. 241(3), pages 622-630.
    8. Rennen, G., 2008. "Subset Selection from Large Datasets for Kriging Modeling," Other publications TiSEM 9dfe6396-1933-45c0-b4e3-5, Tilburg University, School of Economics and Management.
    9. Sayyady, Fatemeh & Fathi, Yahya, 2016. "An integer programming approach for solving the p-dispersion problem," European Journal of Operational Research, Elsevier, vol. 253(1), pages 216-225.
    10. Tetsuya Araki & Shin-ichi Nakano, 2022. "Max–min dispersion on a line," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1824-1830, October.
    11. Marilène Cherkesly & Claudio Contardo, 2021. "The conditional p-dispersion problem," Journal of Global Optimization, Springer, vol. 81(1), pages 23-83, September.
    12. Aringhieri, Roberto & Cordone, Roberto & Grosso, Andrea, 2015. "Construction and improvement algorithms for dispersion problems," European Journal of Operational Research, Elsevier, vol. 242(1), pages 21-33.
    13. Maria Liazi & Ioannis Milis & Fanny Pascual & Vassilis Zissimopoulos, 2007. "The densest k-subgraph problem on clique graphs," Journal of Combinatorial Optimization, Springer, vol. 14(4), pages 465-474, November.
    14. Sergey Kovalev & Isabelle Chalamon & Fabio J. Petani, 2023. "Maximizing single attribute diversity in group selection," Annals of Operations Research, Springer, vol. 320(1), pages 535-540, January.
    15. Felix Prause & Kai Hoppmann-Baum & Boris Defourny & Thorsten Koch, 2021. "The maximum diversity assortment selection problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(3), pages 521-554, June.
    16. Hunting, Marcel & Faigle, Ulrich & Kern, Walter, 2001. "A Lagrangian relaxation approach to the edge-weighted clique problem," European Journal of Operational Research, Elsevier, vol. 131(1), pages 119-131, May.
    17. Parreño, Francisco & Álvarez-Valdés, Ramón & Martí, Rafael, 2021. "Measuring diversity. A review and an empirical analysis," European Journal of Operational Research, Elsevier, vol. 289(2), pages 515-532.
    18. Avella, P. & Benati, S. & Canovas Martinez, L. & Dalby, K. & Di Girolamo, D. & Dimitrijevic, B. & Ghiani, G. & Giannikos, I. & Guttmann, N. & Hultberg, T. H. & Fliege, J. & Marin, A. & Munoz Marquez, , 1998. "Some personal views on the current state and the future of locational analysis," European Journal of Operational Research, Elsevier, vol. 104(2), pages 269-287, January.
    19. W. David Pisinger & Anders Bo Rasmussen & Rune Sandvik, 2007. "Solution of Large Quadratic Knapsack Problems Through Aggressive Reduction," INFORMS Journal on Computing, INFORMS, vol. 19(2), pages 280-290, May.
    20. Jia Wang & Shiyan Hu, 2013. "A linear time approximation scheme for computing geometric maximum k-star," Journal of Global Optimization, Springer, vol. 55(4), pages 849-855, April.

    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::y::i::d:10.1007_s10878-020-00549-5. 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.