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The Generalized Minimum Branch Vertices Problem: Properties and Polyhedral Analysis

Author

Listed:
  • Francesco Carrabs

    (University of Salerno)

  • Raffaele Cerulli

    (University of Salerno)

  • Ciriaco D’Ambrosio

    (University of Salerno)

  • Federica Laureana

    (University of Salerno)

Abstract

This article introduces the Generalized Minimum Branch Vertices problem. Given an undirected graph, where the set of vertices is partitioned into clusters, the Generalized Minimum Branch Vertices problem consists of finding a tree spanning exactly one vertex for each cluster and having the minimum number of branch vertices, namely vertices with degree greater than two. When each cluster is a singleton, the problem reduces to the well-known Minimum Branch Vertices problem, which is NP-hard. We show some properties that any feasible solution to the problem has to satisfy. Some of these properties can be used to determine useless vertices or edges, which can be removed to reduce the size of the instances. We propose an integer linear programming formulation for the problem, we derive the dimension of the polytope, we study the trivial inequalities and introduce two new classes of valid inequalities, that are proved to be facet-defining.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:188:y:2021:i:2:d:10.1007_s10957-020-01783-x
    DOI: 10.1007/s10957-020-01783-x
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    References listed on IDEAS

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    1. 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.
    2. 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.
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    4. Demange, Marc & Ekim, Tınaz & Ries, Bernard & Tanasescu, Cerasela, 2015. "On some applications of the selective graph coloring problem," European Journal of Operational Research, Elsevier, vol. 240(2), pages 307-314.
    5. Matteo Fischetti & Juan José Salazar González & Paolo Toth, 1997. "A Branch-and-Cut Algorithm for the Symmetric Generalized Traveling Salesman Problem," Operations Research, INFORMS, vol. 45(3), pages 378-394, June.
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