IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i4p453-d504614.html
   My bibliography  Save this article

Unified Polynomial Dynamic Programming Algorithms for P-Center Variants in a 2D Pareto Front

Author

Listed:
  • Nicolas Dupin

    (Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, 91400 Orsay, France)

  • Frank Nielsen

    (Sony Computer Science Laboratories Inc., Tokyo 141-0022, Japan)

  • El-Ghazali Talbi

    (CNRS UMR 9189-CRIStAL-Centre de Recherche en Informatique Signal et Automatique de Lille, Université Lille, F-59000 Lille, France)

Abstract

With many efficient solutions for a multi-objective optimization problem, this paper aims to cluster the Pareto Front in a given number of clusters K and to detect isolated points. K -center problems and variants are investigated with a unified formulation considering the discrete and continuous versions, partial K -center problems, and their min-sum- K -radii variants. In dimension three (or upper), this induces NP-hard complexities. In the planar case, common optimality property is proven: non-nested optimal solutions exist. This induces a common dynamic programming algorithm running in polynomial time. Specific improvements hold for some variants, such as K -center problems and min-sum K -radii on a line. When applied to N points and allowing to uncover M < N points, K-center and min-sum- K -radii variants are, respectively, solvable in O ( K ( M + 1 ) N log N ) and O ( K ( M + 1 ) N 2 ) time. Such complexity of results allows an efficient straightforward implementation. Parallel implementations can also be designed for a practical speed-up. Their application inside multi-objective heuristics is discussed to archive partial Pareto fronts, with a special interest in partial clustering variants.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:4:p:453-:d:504614
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/4/453/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/4/453/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Sourour Elloumi & Martine Labbé & Yves Pochet, 2004. "A New Formulation and Resolution Method for the p-Center Problem," INFORMS Journal on Computing, INFORMS, vol. 16(1), pages 84-94, February.
    2. Arindam Karmakar & Sandip Das & Subhas C. Nandy & Binay K. Bhattacharya, 2013. "Some variations on constrained minimum enclosing circle problem," Journal of Combinatorial Optimization, Springer, vol. 25(2), pages 176-190, February.
    3. 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.
    4. Callaghan, Becky & Salhi, Said & Nagy, Gábor, 2017. "Speeding up the optimal method of Drezner for the p-centre problem in the plane," European Journal of Operational Research, Elsevier, vol. 257(3), pages 722-734.
    5. Dorit S. Hochbaum & David B. Shmoys, 1985. "A Best Possible Heuristic for the k -Center Problem," Mathematics of Operations Research, INFORMS, vol. 10(2), pages 180-184, May.
    6. Michele Samorani & Yang Wang & Yang Wang & Zhipeng Lv & Fred Glover, 2019. "Clustering-driven evolutionary algorithms: an application of path relinking to the quadratic unconstrained binary optimization problem," Journal of Heuristics, Springer, vol. 25(4), pages 629-642, October.
    7. Zio, E. & Bazzo, R., 2011. "A clustering procedure for reducing the number of representative solutions in the Pareto Front of multiobjective optimization problems," European Journal of Operational Research, Elsevier, vol. 210(3), pages 624-634, May.
    8. Zvi Drezner, 1987. "On the rectangular p‐center problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 34(2), pages 229-234, April.
    9. Erkut, Erhan, 1990. "The discrete p-dispersion problem," European Journal of Operational Research, Elsevier, vol. 46(1), pages 48-60, May.
    10. O. Schütze & C. Hernández & E-G. Talbi & J. Q. Sun & Y. Naranjani & F.-R. Xiong, 2019. "Archivers for the representation of the set of approximate solutions for MOPs," Journal of Heuristics, Springer, vol. 25(1), pages 71-105, February.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Erwin J. Delgado & Xavier Cabezas & Carlos Martin-Barreiro & Víctor Leiva & Fernando Rojas, 2022. "An Equity-Based Optimization Model to Solve the Location Problem for Healthcare Centers Applied to Hospital Beds and COVID-19 Vaccination," Mathematics, MDPI, vol. 10(11), pages 1-24, May.

    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. Yi Xu & Jigen Peng & Wencheng Wang & Binhai Zhu, 2018. "The connected disk covering problem," Journal of Combinatorial Optimization, Springer, vol. 35(2), pages 538-554, February.
    2. Rennen, G., 2008. "Subset Selection from Large Datasets for Kriging Modeling," Discussion Paper 2008-26, Tilburg University, Center for Economic Research.
    3. Tetsuya Araki & Shin-ichi Nakano, 0. "Max–min dispersion on a line," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-7.
    4. Raphael Kramer & Manuel Iori & Thibaut Vidal, 2020. "Mathematical Models and Search Algorithms for the Capacitated p -Center Problem," INFORMS Journal on Computing, INFORMS, vol. 32(2), pages 444-460, April.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. José Alejandro Cornejo Acosta & Jesús García Díaz & Ricardo Menchaca-Méndez & Rolando Menchaca-Méndez, 2020. "Solving the Capacitated Vertex K-Center Problem through the Minimum Capacitated Dominating Set Problem," Mathematics, MDPI, vol. 8(9), pages 1-16, September.
    11. Sayah, David & Irnich, Stefan, 2017. "A new compact formulation for the discrete p-dispersion problem," European Journal of Operational Research, Elsevier, vol. 256(1), pages 62-67.
    12. 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.
    13. 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.
    14. Ríos-Mercado, Roger Z. & Bard, Jonathan F., 2019. "An exact algorithm for designing optimal districts in the collection of waste electric and electronic equipment through an improved reformulation," European Journal of Operational Research, Elsevier, vol. 276(1), pages 259-271.
    15. Alexander Estes & David J. Lovell & Michael O. Ball, 2019. "Unsupervised prototype reduction for data exploration and an application to air traffic management initiatives," EURO Journal on Transportation and Logistics, Springer;EURO - The Association of European Operational Research Societies, vol. 8(5), pages 467-510, December.
    16. 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.
    17. Yi Xu & Jigen Peng & Yinfeng Xu, 2018. "The mixed center location problem," Journal of Combinatorial Optimization, Springer, vol. 36(4), pages 1128-1144, November.
    18. Tetsuya Araki & Shin-ichi Nakano, 2022. "Max–min dispersion on a line," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1824-1830, October.
    19. Alberto Pajares & Xavier Blasco & Juan Manuel Herrero & Miguel A. Martínez, 2021. "A Comparison of Archiving Strategies for Characterization of Nearly Optimal Solutions under Multi-Objective Optimization," Mathematics, MDPI, vol. 9(9), pages 1-28, April.
    20. Marilène Cherkesly & Claudio Contardo, 2021. "The conditional p-dispersion problem," Journal of Global Optimization, Springer, vol. 81(1), pages 23-83, September.

    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:gam:jmathe:v:9:y:2021:i:4:p:453-:d:504614. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.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.