IDEAS home Printed from https://ideas.repec.org/a/spr/joheur/v24y2018i4d10.1007_s10732-018-9370-4.html
   My bibliography  Save this article

All Colors Shortest Path problem on trees

Author

Listed:
  • Mehmet Berkehan Akçay

    (Izmir University of Economics)

  • Hüseyin Akcan

    (Izmir University of Economics)

  • Cem Evrendilek

    (Izmir University of Economics)

Abstract

Given an edge weighted tree T(V, E), rooted at a designated base vertex $$r \in V$$ r ∈ V , and a color from a set of colors $$C=\{1,\ldots ,k\}$$ C = { 1 , … , k } assigned to every vertex $$v \in V$$ v ∈ V , All Colors Shortest Path problem on trees (ACSP-t) seeks the shortest, possibly non-simple, path starting from r in T such that at least one node from every distinct color in C is visited. We show that ACSP-t is NP-hard, and also prove that it does not have a constant factor approximation. We give an integer linear programming formulation of ACSP-t. Based on a linear programming relaxation of this formulation, an iterative rounding heuristic is proposed. The paper also explores genetic algorithm and tabu search to develop alternative heuristic solutions for ACSP-t. The performance of all the proposed heuristics are evaluated experimentally for a wide range of trees that are generated parametrically.

Suggested Citation

  • Mehmet Berkehan Akçay & Hüseyin Akcan & Cem Evrendilek, 2018. "All Colors Shortest Path problem on trees," Journal of Heuristics, Springer, vol. 24(4), pages 617-644, August.
    • Handle: RePEc:spr:joheur:v:24:y:2018:i:4:d:10.1007_s10732-018-9370-4
    DOI: 10.1007/s10732-018-9370-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10732-018-9370-4
    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/s10732-018-9370-4?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. Pop, Petrică C. & Matei, Oliviu & Sabo, Cosmin & Petrovan, Adrian, 2018. "A two-level solution approach for solving the generalized minimum spanning tree problem," European Journal of Operational Research, Elsevier, vol. 265(2), pages 478-487.
    2. Öncan, Temel & Cordeau, Jean-François & Laporte, Gilbert, 2008. "A tabu search heuristic for the generalized minimum spanning tree problem," European Journal of Operational Research, Elsevier, vol. 191(2), pages 306-319, December.
    3. Pop, Petrica C. & Kern, W. & Still, G., 2006. "A new relaxation method for the generalized minimum spanning tree problem," European Journal of Operational Research, Elsevier, vol. 170(3), pages 900-908, May.
    4. Dror, M. & Haouari, M. & Chaouachi, J., 2000. "Generalized spanning trees," European Journal of Operational Research, Elsevier, vol. 120(3), pages 583-592, 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. F. Carrabs & R. Cerulli & R. Pentangelo & A. Raiconi, 2018. "A two-level metaheuristic for the all colors shortest path problem," Computational Optimization and Applications, Springer, vol. 71(2), pages 525-551, November.

    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. 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.
    2. F. Carrabs & R. Cerulli & R. Pentangelo & A. Raiconi, 2018. "A two-level metaheuristic for the all colors shortest path problem," Computational Optimization and Applications, Springer, vol. 71(2), pages 525-551, November.
    3. Pop, Petrică C. & Matei, Oliviu & Sabo, Cosmin & Petrovan, Adrian, 2018. "A two-level solution approach for solving the generalized minimum spanning tree problem," European Journal of Operational Research, Elsevier, vol. 265(2), pages 478-487.
    4. Masoumeh Zojaji & Mohammad Reza Mollakhalili Meybodi & Kamal Mirzaie, 0. "A rapid learning automata-based approach for generalized minimum spanning tree problem," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-24.
    5. 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.
    6. Masoumeh Zojaji & Mohammad Reza Mollakhalili Meybodi & Kamal Mirzaie, 2020. "A rapid learning automata-based approach for generalized minimum spanning tree problem," Journal of Combinatorial Optimization, Springer, vol. 40(3), pages 636-659, October.
    7. Öncan, Temel & Cordeau, Jean-François & Laporte, Gilbert, 2008. "A tabu search heuristic for the generalized minimum spanning tree problem," European Journal of Operational Research, Elsevier, vol. 191(2), pages 306-319, December.
    8. Haouari, Mohamed & Chaouachi, Jouhaina Siala, 2006. "Upper and lower bounding strategies for the generalized minimum spanning tree problem," European Journal of Operational Research, Elsevier, vol. 171(2), pages 632-647, June.
    9. Feremans, Corinne & Labbe, Martine & Laporte, Gilbert, 2003. "Generalized network design problems," European Journal of Operational Research, Elsevier, vol. 148(1), pages 1-13, July.
    10. 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.
    11. M Haouari & J Chaouachi & M Dror, 2005. "Solving the generalized minimum spanning tree problem by a branch-and-bound algorithm," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 56(4), pages 382-389, April.
    12. Duin, C. W. & Volgenant, A. & Vo[ss], S., 2004. "Solving group Steiner problems as Steiner problems," European Journal of Operational Research, Elsevier, vol. 154(1), pages 323-329, April.
    13. Cosmin Sabo & Petrică C. Pop & Andrei Horvat-Marc, 2020. "On the Selective Vehicle Routing Problem," Mathematics, MDPI, vol. 8(5), pages 1-11, May.
    14. Pablo A. Miranda-Gonzalez & Javier Maturana-Ross & Carola A. Blazquez & Guillermo Cabrera-Guerrero, 2021. "Exact Formulation and Analysis for the Bi-Objective Insular Traveling Salesman Problem," Mathematics, MDPI, vol. 9(21), pages 1-33, October.
    15. Phuoc Hoang Le & Tri-Dung Nguyen & Tolga Bektaş, 2016. "Generalized minimum spanning tree games," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 4(2), pages 167-188, May.
    16. Feremans, Corinne & Labbe, Martine & Laporte, Gilbert, 2001. "On generalized minimum spanning trees," European Journal of Operational Research, Elsevier, vol. 134(2), pages 457-458, October.
    17. Kansal, Anuraag R & Torquato, Salvatore, 2001. "Globally and locally minimal weight spanning tree networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 301(1), pages 601-619.
    18. Karapetyan, D. & Gutin, G., 2011. "Lin-Kernighan heuristic adaptations for the generalized traveling salesman problem," European Journal of Operational Research, Elsevier, vol. 208(3), pages 221-232, February.
    19. Moshe Dror & Mohamed Haouari, 2000. "Generalized Steiner Problems and Other Variants," Journal of Combinatorial Optimization, Springer, vol. 4(4), pages 415-436, December.
    20. 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.

    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:joheur:v:24:y:2018:i:4:d:10.1007_s10732-018-9370-4. 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.