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A characterization of linearizable instances of the quadratic minimum spanning tree problem

Author

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  • Ante Ćustić

    (Simon Fraser University Surrey)

  • Abraham P. Punnen

    (Simon Fraser University Surrey)

Abstract

We investigate special cases of the quadratic minimum spanning tree problem (QMSTP) on a graph $$G=(V,E)$$ G = ( V , E ) that can be solved as a linear minimum spanning tree problem. We give a characterization of such problems when G is a complete graph, which is the standard case in the QMSTP literature. We extend our characterization to a larger class of graphs that include complete bipartite graphs and cactuses, among others. Our characterization can be verified in $$O(|E|^2)$$ O ( | E | 2 ) time. In the case of complete graphs and when the cost matrix is given in factored form, we show that our characterization can be verified in O(|E|) time. Related open problems are also indicated.

Suggested Citation

  • Ante Ćustić & Abraham P. Punnen, 2018. "A characterization of linearizable instances of the quadratic minimum spanning tree problem," Journal of Combinatorial Optimization, Springer, vol. 35(2), pages 436-453, February.
    • Handle: RePEc:spr:jcomop:v:35:y:2018:i:2:d:10.1007_s10878-017-0184-3
    DOI: 10.1007/s10878-017-0184-3
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    References listed on IDEAS

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    1. Eranda Çela & Vladimir G. Deineko & Gerhard J. Woeginger, 2016. "Linearizable special cases of the QAP," Journal of Combinatorial Optimization, Springer, vol. 31(3), pages 1269-1279, April.
    2. Arjang Assad & Weixuan Xu, 1992. "The quadratic minimum spanning tree problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 39(3), pages 399-417, April.
    3. Manuel Lozano & Fred Glover & Carlos García-Martínez & Francisco Rodríguez & Rafael Martí, 2014. "Tabu search with strategic oscillation for the quadratic minimum spanning tree," IISE Transactions, Taylor & Francis Journals, vol. 46(4), pages 414-428.
    4. Santosh N. Kabadi & Abraham P. Punnen, 2011. "An O ( n 4 ) Algorithm for the QAP Linearization Problem," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 754-761, November.
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    Cited by:

    1. de Meijer, Frank, 2023. "Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization," Other publications TiSEM b1f1088c-95fe-4b8a-9e15-c, Tilburg University, School of Economics and Management.
    2. Hao Hu & Renata Sotirov, 2021. "The linearization problem of a binary quadratic problem and its applications," Annals of Operations Research, Springer, vol. 307(1), pages 229-249, December.
    3. Frank Meijer & Renata Sotirov, 2020. "The quadratic cycle cover problem: special cases and efficient bounds," Journal of Combinatorial Optimization, Springer, vol. 39(4), pages 1096-1128, May.
    4. 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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