IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v56y2013i2p405-438.html
   My bibliography  Save this article

Lower and upper bounds for the spanning tree with minimum branch vertices

Author

Listed:
  • Francesco Carrabs
  • Raffaele Cerulli
  • Manlio Gaudioso
  • Monica Gentili

Abstract

We study a variant of the spanning tree problem where we require that, for a given connected graph, the spanning tree to be found has the minimum number of branch vertices (that is vertices of the tree whose degree is greater than two). We provide four different formulations of the problem and compare different relaxations of them, namely Lagrangian relaxation, continuous relaxation, mixed integer-continuous relaxation. We approach the solution of the Lagrangian dual both by means of a standard subgradient method and an ad-hoc finite ascent algorithm based on updating one multiplier at the time. We provide numerical result comparison of all the considered relaxations on a wide set of benchmark instances. A useful follow-up of tackling the Lagrangian dual is the possibility of getting a feasible solution for the original problem with no extra costs. We evaluate the quality of the resulting upper bound by comparison either with the optimal solution, whenever available, or with the feasible solution provided by some existing heuristic algorithms. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Francesco Carrabs & Raffaele Cerulli & Manlio Gaudioso & Monica Gentili, 2013. "Lower and upper bounds for the spanning tree with minimum branch vertices," Computational Optimization and Applications, Springer, vol. 56(2), pages 405-438, October.
  • Handle: RePEc:spr:coopap:v:56:y:2013:i:2:p:405-438
    DOI: 10.1007/s10589-013-9556-5
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10589-013-9556-5
    Download Restriction: Access to full text is restricted to subscribers.

    File URL: https://libkey.io/10.1007/s10589-013-9556-5?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. R. Cerulli & M. Gentili & A. Iossa, 2009. "Bounded-degree spanning tree problems: models and new algorithms," Computational Optimization and Applications, Springer, vol. 42(3), pages 353-370, April.
    2. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2009. "On solving the Lagrangian dual of integer programs via an incremental approach," Computational Optimization and Applications, Springer, vol. 44(1), pages 117-138, October.
    3. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2006. "An Incremental Method for Solving Convex Finite Min-Max Problems," Mathematics of Operations Research, INFORMS, vol. 31(1), pages 173-187, February.
    4. Akgün, Ibrahim & Tansel, Barbaros Ç., 2011. "New formulations of the Hop-Constrained Minimum Spanning Tree problem via Miller-Tucker-Zemlin constraints," European Journal of Operational Research, Elsevier, vol. 212(2), pages 263-276, July.
    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. Melo, Rafael A. & Queiroz, Michell F. & Ribeiro, Celso C., 2021. "Compact formulations and an iterated local search-based matheuristic for the minimum weighted feedback vertex set problem," European Journal of Operational Research, Elsevier, vol. 289(1), pages 75-92.
    2. Cerrone, C. & Cerulli, R. & Raiconi, A., 2014. "Relations, models and a memetic approach for three degree-dependent spanning tree problems," European Journal of Operational Research, Elsevier, vol. 232(3), pages 442-453.
    3. Mercedes Landete & Alfredo Marín & José Luis Sainz-Pardo, 2017. "Decomposition methods based on articulation vertices for degree-dependent spanning tree problems," Computational Optimization and Applications, Springer, vol. 68(3), pages 749-773, December.
    4. Rafael A. Melo & Phillippe Samer & Sebastián Urrutia, 2016. "An effective decomposition approach and heuristics to generate spanning trees with a small number of branch vertices," Computational Optimization and Applications, Springer, vol. 65(3), pages 821-844, December.
    5. Jorge Moreno & Yuri Frota & Simone Martins, 2018. "An exact and heuristic approach for the d-minimum branch vertices problem," Computational Optimization and Applications, Springer, vol. 71(3), pages 829-855, December.
    6. Antonino Chiarello & Manlio Gaudioso & Marcello Sammarra, 2018. "Truck synchronization at single door cross-docking terminals," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 40(2), pages 395-447, March.
    7. Marín, Alfredo, 2015. "Exact and heuristic solutions for the Minimum Number of Branch Vertices Spanning Tree Problem," European Journal of Operational Research, Elsevier, vol. 245(3), pages 680-689.
    8. Francesco Carrabs & Raffaele Cerulli & Ciriaco D’Ambrosio & Federica Laureana, 2021. "The Generalized Minimum Branch Vertices Problem: Properties and Polyhedral Analysis," Journal of Optimization Theory and Applications, Springer, vol. 188(2), pages 356-377, February.

    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. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2020. "Essentials of numerical nonsmooth optimization," 4OR, Springer, vol. 18(1), pages 1-47, March.
    2. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2022. "Essentials of numerical nonsmooth optimization," Annals of Operations Research, Springer, vol. 314(1), pages 213-253, July.
    3. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2018. "Minimizing Piecewise-Concave Functions Over Polyhedra," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 580-597, May.
    4. Gaudioso, Manlio & Monaco, Maria Flavia & Sammarra, Marcello, 2021. "A Lagrangian heuristics for the truck scheduling problem in multi-door, multi-product Cross-Docking with constant processing time," Omega, Elsevier, vol. 101(C).
    5. G. Rius-Sorolla & J. Maheut & Jairo R. Coronado-Hernandez & J. P. Garcia-Sabater, 2020. "Lagrangian relaxation of the generic materials and operations planning model," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 28(1), pages 105-123, March.
    6. Cerrone, C. & Cerulli, R. & Raiconi, A., 2014. "Relations, models and a memetic approach for three degree-dependent spanning tree problems," European Journal of Operational Research, Elsevier, vol. 232(3), pages 442-453.
    7. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2015. "Optimal Replenishment Order Placement in a Finite Time Horizon," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 1078-1089, March.
    8. Rafael A. Melo & Phillippe Samer & Sebastián Urrutia, 2016. "An effective decomposition approach and heuristics to generate spanning trees with a small number of branch vertices," Computational Optimization and Applications, Springer, vol. 65(3), pages 821-844, December.
    9. Jorge Moreno & Yuri Frota & Simone Martins, 2018. "An exact and heuristic approach for the d-minimum branch vertices problem," Computational Optimization and Applications, Springer, vol. 71(3), pages 829-855, December.
    10. Martina Kuchlbauer & Frauke Liers & Michael Stingl, 2022. "Adaptive Bundle Methods for Nonlinear Robust Optimization," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 2106-2124, July.
    11. Li-Ping Pang & Jian Lv & Jin-He Wang, 2016. "Constrained incremental bundle method with partial inexact oracle for nonsmooth convex semi-infinite programming problems," Computational Optimization and Applications, Springer, vol. 64(2), pages 433-465, June.
    12. Yang Zhan & Chuangyin Dang, 2018. "A smooth path-following algorithm for market equilibrium under a class of piecewise-smooth concave utilities," Computational Optimization and Applications, Springer, vol. 71(2), pages 381-402, November.
    13. Marín, Alfredo, 2015. "Exact and heuristic solutions for the Minimum Number of Branch Vertices Spanning Tree Problem," European Journal of Operational Research, Elsevier, vol. 245(3), pages 680-689.
    14. M. Gaudioso & L. Moccia & M. F. Monaco, 2010. "Repulsive Assignment Problem," Journal of Optimization Theory and Applications, Springer, vol. 144(2), pages 255-273, February.
    15. Antonino Chiarello & Manlio Gaudioso & Marcello Sammarra, 2018. "Truck synchronization at single door cross-docking terminals," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 40(2), pages 395-447, March.
    16. Iago A. Carvalho & Marco A. Ribeiro, 2020. "An exact approach for the Minimum-Cost Bounded-Error Calibration Tree problem," Annals of Operations Research, Springer, vol. 287(1), pages 109-126, April.
    17. Astorino, Annabella & Gaudioso, Manlio & Miglionico, Giovanna, 2018. "Lagrangian relaxation for the directional sensor coverage problem with continuous orientation," Omega, Elsevier, vol. 75(C), pages 77-86.
    18. Mercedes Landete & Alfredo Marín & José Luis Sainz-Pardo, 2017. "Decomposition methods based on articulation vertices for degree-dependent spanning tree problems," Computational Optimization and Applications, Springer, vol. 68(3), pages 749-773, December.
    19. Massinissa Merabet & Miklos Molnar & Sylvain Durand, 2018. "ILP formulation of the degree-constrained minimum spanning hierarchy problem," Journal of Combinatorial Optimization, Springer, vol. 36(3), pages 789-811, October.

    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:coopap:v:56:y:2013:i:2:p:405-438. 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.