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A two-level solution approach for solving the generalized minimum spanning tree problem

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  • Pop, Petrică C.
  • Matei, Oliviu
  • Sabo, Cosmin
  • Petrovan, Adrian

Abstract

In this paper, we are addressing the generalized minimum spanning tree problem, denoted by GMSTP, which is a variant of the classical minimum spanning tree (MST) problem. The main characteristic of this problem is the fact that the vertices of the graph are partitioned into a given number of clusters and we are looking for a minimum-cost tree spanning a subset of vertices which includes exactly one vertex from each cluster. We describe a two-level solution approach for solving the GMSTP obtained by decomposing the problem into two logical and natural smaller subproblems: an upper-level (global) subproblem and a lower-level (local) subproblem and solving them separately. The goal of the first subproblem is to determine (global) trees spanning the clusters using a genetic algorithm with a diploid representation of the individuals, while the goal of the second subproblem is to determine the best tree (w.r.t. cost minimization), for the above mentioned global trees, spanning exactly one vertex from each cluster. The second subproblem is solved optimally using dynamic programming. Extensive computational results are reported and discussed for an often used set of benchmark instances. The obtained results show an improvement in the quality of the achieved solutions, and demonstrate the efficiency of our approach compared to the existing methods from the literature.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:ejores:v:265:y:2018:i:2:p:478-487
    DOI: 10.1016/j.ejor.2017.08.015
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    References listed on IDEAS

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    1. P. Pop, 2007. "On the prize-collecting generalized minimum spanning tree problem," Annals of Operations Research, Springer, vol. 150(1), pages 193-204, March.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. Bruce Golden & S. Raghavan & Daliborka Stanojević, 2005. "Heuristic Search for the Generalized Minimum Spanning Tree Problem," INFORMS Journal on Computing, INFORMS, vol. 17(3), pages 290-304, August.
    8. Dror, M. & Haouari, M. & Chaouachi, J., 2000. "Generalized spanning trees," European Journal of Operational Research, Elsevier, vol. 120(3), pages 583-592, February.
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    Cited by:

    1. José-Manuel Giménez-Gómez & Josep E Peris & Begoña Subiza, 2020. "An egalitarian approach for sharing the cost of a spanning tree," PLOS ONE, Public Library of Science, vol. 15(7), pages 1-14, July.
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    4. Ze Pan & Xinyun Wu & Caiquan Xiong, 2023. "Dual-Neighborhood Search for Solving the Minimum Dominating Tree Problem," Mathematics, MDPI, vol. 11(19), pages 1-20, October.
    5. Miranda, Pablo A. & Blazquez, Carola A. & Obreque, Carlos & Maturana-Ross, Javier & Gutierrez-Jarpa, Gabriel, 2018. "The bi-objective insular traveling salesman problem with maritime and ground transportation costs," European Journal of Operational Research, Elsevier, vol. 271(3), pages 1014-1036.
    6. 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.
    7. 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.

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