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

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  • 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
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    4. 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.
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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    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.

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