IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v33y2017i3d10.1007_s10878-016-0026-8.html
   My bibliography  Save this article

On the minimum routing cost clustered tree problem

Author

Listed:
  • Chen-Wan Lin

    (National Chung Cheng University)

  • Bang Ye Wu

    (National Chung Cheng University)

Abstract

For an edge-weighted graph $$G=(V,E,w)$$ G = ( V , E , w ) , in which the vertices are partitioned into k clusters $$\mathcal {R}=\{R_1,R_2,\ldots ,R_k\}$$ R = { R 1 , R 2 , … , R k } , a spanning tree T of G is a clustered spanning tree if T can be cut into k subtrees by removing $$k-1$$ k - 1 edges such that each subtree is a spanning tree for one cluster. In this paper, we show the inapproximability of finding a clustered spanning tree with minimum routing cost, where the routing cost is the total distance summed over all pairs of vertices. We present a 2-approximation for the case that the input is a complete weighted graph whose edge weights obey the triangle inequality. We also study a variant in which the objective function is the total distance summed over all pairs of vertices of different clusters. We show that the problem is polynomial-time solvable when the number of clusters k is 2 and NP-hard for $$k=3$$ k = 3 . Finally, we propose a polynomial-time 2-approximation algorithm for the case of three clusters.

Suggested Citation

  • Chen-Wan Lin & Bang Ye Wu, 2017. "On the minimum routing cost clustered tree problem," Journal of Combinatorial Optimization, Springer, vol. 33(3), pages 1106-1121, April.
    • Handle: RePEc:spr:jcomop:v:33:y:2017:i:3:d:10.1007_s10878-016-0026-8
    DOI: 10.1007/s10878-016-0026-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-016-0026-8
    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/s10878-016-0026-8?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. Bang Ye Wu & Chen-Wan Lin, 2015. "On the clustered Steiner tree problem," Journal of Combinatorial Optimization, Springer, vol. 30(2), pages 370-386, August.
    2. Feremans, Corinne & Labbe, Martine & Laporte, Gilbert, 2003. "Generalized network design problems," European Journal of Operational Research, Elsevier, vol. 148(1), pages 1-13, July.
    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. Mattia D’Emidio & Luca Forlizzi & Daniele Frigioni & Stefano Leucci & Guido Proietti, 2019. "Hardness, approximability, and fixed-parameter tractability of the clustered shortest-path tree problem," Journal of Combinatorial Optimization, Springer, vol. 38(1), pages 165-184, July.
    2. Ghosh, Diptesh, 2003. "Solving Medium to Large Sized Euclidean Generalized Minimum Spanning Tree Problems," IIMA Working Papers WP2003-08-02, Indian Institute of Management Ahmedabad, Research and Publication Department.
    3. Pop, Petrică C., 2020. "The generalized minimum spanning tree problem: An overview of formulations, solution procedures and latest advances," European Journal of Operational Research, Elsevier, vol. 283(1), pages 1-15.
    4. J Renaud & F F Boctor & G Laporte, 2004. "Efficient heuristics for Median Cycle Problems," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 55(2), pages 179-186, February.
    5. Garijo, Delia & Márquez, Alberto & Rodríguez, Natalia & Silveira, Rodrigo I., 2019. "Computing optimal shortcuts for networks," European Journal of Operational Research, Elsevier, vol. 279(1), pages 26-37.
    6. Pop, Petrică C. & Cosma, Ovidiu & Sabo, Cosmin & Sitar, Corina Pop, 2024. "A comprehensive survey on the generalized traveling salesman problem," European Journal of Operational Research, Elsevier, vol. 314(3), pages 819-835.
    7. Cosmin Sabo & Petrică C. Pop & Andrei Horvat-Marc, 2020. "On the Selective Vehicle Routing Problem," Mathematics, MDPI, vol. 8(5), pages 1-11, May.
    8. Ghosh, Diptesh, 2003. "A Probabilistic Tabu Search Algorithm for the Generalized Minimum Spanning Tree Problem," IIMA Working Papers WP2003-07-02, Indian Institute of Management Ahmedabad, Research and Publication Department.
    9. Gallo, Mariano & D'Acierno, Luca & Montella, Bruno, 2010. "A meta-heuristic approach for solving the Urban Network Design Problem," European Journal of Operational Research, Elsevier, vol. 201(1), pages 144-157, February.
    10. Markus Leitner, 2016. "Integer programming models and branch-and-cut approaches to generalized {0,1,2}-survivable network design problems," Computational Optimization and Applications, Springer, vol. 65(1), pages 73-92, September.
    11. Yi Tao & Ek Peng Chew & Loo Hay Lee & Yuran Shi, 2017. "A column generation approach for the route planning problem in fourth party logistics," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 68(2), pages 165-181, February.
    12. Akyüz, M. Hakan & Lee, Chung-Yee, 2016. "Service type assignment and container routing with transit time constraints and empty container repositioning for liner shipping service networks," Transportation Research Part B: Methodological, Elsevier, vol. 88(C), pages 46-71.
    13. Feess, E. & Walzl, M., 2004. "An analysis of corporte leniency programs and lessons to learn for EU and US policies," Research Memorandum 037, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    14. Zhou, Yangming & Qu, Chenhui & Wu, Qinghua & Kou, Yawen & Jiang, Zhibin & Zhou, MengChu, 2024. "A bilevel hybrid iterated search approach to soft-clustered capacitated arc routing problems," Transportation Research Part B: Methodological, Elsevier, vol. 184(C).

    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:33:y:2017:i:3:d:10.1007_s10878-016-0026-8. 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.