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A polyhedral approach to the generalized minimum labeling spanning tree problem

Author

Listed:
  • Thiago Gouveia da Silva

    (IFPB, Instituto Federal de Educação, Ciência e Tecnologia da Paraíba
    UFF, Universidade Federal Fluminense
    Université d’Avignon)

  • Serigne Gueye

    (Université d’Avignon)

  • Philippe Michelon

    (Université d’Avignon)

  • Luiz Satoru Ochi

    (UFF, Universidade Federal Fluminense)

  • Lucídio dos Anjos Formiga Cabral

    (UFPB, Universidade Federal da Paraíba)

Abstract

The minimum labeling spanning tree problem (MLSTP) is a combinatorial optimization problem that consists in finding a spanning tree in a simple graph G, in which each edge has one label, by using a minimum number of labels. It is an NP-hard problem that was introduced by Chang and Leu (Inf Process Lett 63(5):277–282, 1997. https://doi.org/10.1016/S0020-0190(97)00127-0 ). Chen et al. (Comparison of heuristics for solving the gmlst problem, in: Raghavan, Golden, Wasil (eds) Telecommunications modeling, policy, and technology, Springer, Boston, pp 191–217, 2008) subsequently proposed a generalization of the MLSTP, called the generalized minimum labeling spanning tree problem (GMLSTP), that allows a situation in which multiple labels can be assigned to an edge. Here, we show how the GMLSTP can be expressed as an MLSTP in a multigraph. Both problems have applications in various areas such as computer network design, multimodal transportation network design, and data compression. We propose a new compact binary integer programming model to solve exactly the GMLSTP and analyze the polytope associated with the formulation. The paper introduces new concepts, gives the polytope dimension, and describes five new facet families. The polyhedral comparison results for the studied polytope show that the new model is theoretically superior to current state-of-the-art formulations.

Suggested Citation

  • Thiago Gouveia da Silva & Serigne Gueye & Philippe Michelon & Luiz Satoru Ochi & Lucídio dos Anjos Formiga Cabral, 2019. "A polyhedral approach to the generalized minimum labeling spanning tree problem," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 7(1), pages 47-77, March.
    • Handle: RePEc:spr:eurjco:v:7:y:2019:i:1:d:10.1007_s13675-018-0099-5
    DOI: 10.1007/s13675-018-0099-5
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    References listed on IDEAS

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    1. Consoli, S. & Darby-Dowman, K. & Mladenovic, N. & Moreno Pérez, J.A., 2009. "Greedy Randomized Adaptive Search and Variable Neighbourhood Search for the minimum labelling spanning tree problem," European Journal of Operational Research, Elsevier, vol. 196(2), pages 440-449, July.
    2. Andreas M. Chwatal & Günther R. Raidl, 2011. "Solving the Minimum Label Spanning Tree Problem by Mathematical Programming Techniques," Advances in Operations Research, Hindawi, vol. 2011, pages 1-38, June.
    3. Sergio Consoli & Kenneth Darby-Dowman & Nenad Mladenović & José Moreno-Pérez, 2009. "Variable neighbourhood search for the minimum labelling Steiner tree problem," Annals of Operations Research, Springer, vol. 172(1), pages 71-96, November.
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