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Semidefinite programming lower bounds and branch-and-bound algorithms for the quadratic minimum spanning tree problem

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  • Guimarães, Dilson Almeida
  • da Cunha, Alexandre Salles
  • Pereira, Dilson Lucas

Abstract

In this paper, we investigate Semidefinite Programming (SDP) lower bounds for the Quadratic Minimum Spanning Tree Problem (QMSTP). Two SDP lower bounding approaches are introduced here. Both apply Lagrangian Relaxation to an SDP relaxation for the problem. The first one explicitly dualizes the semidefiniteness constraint, attaching to it a positive semidefinite matrix of Lagrangian multipliers. The second relies on a semi-infinite reformulation for the cone of positive semidefinite matrices and dualizes a dynamically updated finite set of inequalities that approximate the cone. These lower bounding procedures are the core ingredient of two QMSTP Branch-and-bound algorithms. Our computational experiments indicate that the SDP bounds computed here are very strong, being able to close at least 70% of the gaps of the most competitive formulation in the literature. As a result, their accompanying Branch-and-bound algorithms are competitive with the best previously available QMSTP exact algorithm in the literature. In fact, one of these new Branch-and-bound algorithms stands out as the new best exact solution approach for the problem.

Suggested Citation

  • Guimarães, Dilson Almeida & da Cunha, Alexandre Salles & Pereira, Dilson Lucas, 2020. "Semidefinite programming lower bounds and branch-and-bound algorithms for the quadratic minimum spanning tree problem," European Journal of Operational Research, Elsevier, vol. 280(1), pages 46-58.
    • Handle: RePEc:eee:ejores:v:280:y:2020:i:1:p:46-58
    DOI: 10.1016/j.ejor.2019.07.038
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    Cited by:

    1. Nihal Berktaş & Hande Yaman, 2021. "A Branch-and-Bound Algorithm for Team Formation on Social Networks," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 1162-1176, July.
    2. Pereira, Dilson Lucas & Salles da Cunha, Alexandre, 2020. "Dynamic intersection of multiple implicit Dantzig–Wolfe decompositions applied to the adjacent only quadratic minimum spanning tree problem," European Journal of Operational Research, Elsevier, vol. 284(2), pages 413-426.

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