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A 2n Constraint Formulation for the Capacitated Minimal Spanning Tree Problem

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  • Luis Gouveia

    (Universidade de Lisboa, Lisboa, Portugal)

Abstract

In this paper we present a new formulation for the Capacitated Minimal Spanning Tree ( CMST ) problem. One advantage of the new formulation is that it is more compact (in the number of constraints) than a well-known formulation. Additionally, we show that the linear programming relaxation of both formulations produces optimal solutions with the same cost. We present a brief discussion concerning valid inequalities for the ( CMST ) which are directly derived from the new formulation. We show that some of the new inequalities are not dominated by some sets of facet-inducing inequalities for the ( CMST ). We derive some Lagrangian relaxation-based methods from the new formulation and present computational evidence showing that reasonable improvements on the original linear programming bounds can be obtained if these methods are strengthened by the use of cutting planes. The reported computational results indicate that one of the methods presented in this paper dominates, in most of the cases, the previous best methods reported in the literature. The most significant improvements are obtained in the instances with the root in the corner.

Suggested Citation

  • Luis Gouveia, 1995. "A 2n Constraint Formulation for the Capacitated Minimal Spanning Tree Problem," Operations Research, INFORMS, vol. 43(1), pages 130-141, February.
  • Handle: RePEc:inm:oropre:v:43:y:1995:i:1:p:130-141
    DOI: 10.1287/opre.43.1.130
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    Citations

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    Cited by:

    1. Gouveia, Luis, 1995. "A result on projection for the vehicle routing ptoblem," European Journal of Operational Research, Elsevier, vol. 85(3), pages 610-624, September.
    2. Bernard Gendron & Luis Gouveia, 2017. "Reformulations by Discretization for Piecewise Linear Integer Multicommodity Network Flow Problems," Transportation Science, INFORMS, vol. 51(2), pages 629-649, May.
    3. Amberg, Anita & Domschke, Wolfgang & Vo[ss], Stefan, 2000. "Multiple center capacitated arc routing problems: A tabu search algorithm using capacitated trees," European Journal of Operational Research, Elsevier, vol. 124(2), pages 360-376, July.
    4. Gouveia, Luís & Paias, Ana & Ponte, Mafalda, 2023. "The travelling salesman problem with positional consistency constraints: An application to healthcare services," European Journal of Operational Research, Elsevier, vol. 308(3), pages 960-989.
    5. Rahma Lahyani & Leandro C. Coelho & Jacques Renaud, 2018. "Alternative formulations and improved bounds for the multi-depot fleet size and mix vehicle routing problem," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 40(1), pages 125-157, January.
    6. Kawatra, R. & Bricker, D., 2000. "A multiperiod planning model for the capacitated minimal spanning tree problem," European Journal of Operational Research, Elsevier, vol. 121(2), pages 412-419, March.
    7. Uchoa, Eduardo & Fukasawa, Ricardo & Lysgaard, Jens & Pessoa, Artur & Poggi de Aragão, Marcus & Andrade, Diogo, 2006. "Robust Branch-Cut-and-Price for the Capacitated Minimum Spanning Tree Problem over a Large Extended Formulation," CORAL Working Papers L-2006-08, University of Aarhus, Aarhus School of Business, Department of Business Studies.
    8. Cesar Rego & Frank Mathew & Fred Glover, 2010. "RAMP for the capacitated minimum spanning tree problem," Annals of Operations Research, Springer, vol. 181(1), pages 661-681, December.
    9. Fernandes, Lucinda Matos & Gouveia, Luis, 1998. "Minimal spanning trees with a constraint on the number of leaves," European Journal of Operational Research, Elsevier, vol. 104(1), pages 250-261, January.
    10. Correia, Isabel & Gouveia, Luís & Saldanha-da-Gama, Francisco, 2010. "Discretized formulations for capacitated location problems with modular distribution costs," European Journal of Operational Research, Elsevier, vol. 204(2), pages 237-244, July.
    11. Gen, Mitsuo & Kumar, Anup & Ryul Kim, Jong, 2005. "Recent network design techniques using evolutionary algorithms," International Journal of Production Economics, Elsevier, vol. 98(2), pages 251-261, November.
    12. Ricardo Fukasawa & Qie He & Yongjia Song, 2016. "A Branch-Cut-and-Price Algorithm for the Energy Minimization Vehicle Routing Problem," Transportation Science, INFORMS, vol. 50(1), pages 23-34, February.
    13. Gouveia, Luis & Lopes, Maria Joao, 2000. "Valid inequalities for non-unit demand capacitated spanning tree problems with flow costs," European Journal of Operational Research, Elsevier, vol. 121(2), pages 394-411, March.
    14. Xiaojun Zhu & Shaojie Tang, 2021. "A Branch-and-Bound Algorithm for Building Optimal Data Gathering Tree in Wireless Sensor Networks," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1446-1460, October.
    15. Gouveia, Luís & Lopes, Maria João & de Sousa, Amaro, 2015. "Single PON network design with unconstrained splitting stages," European Journal of Operational Research, Elsevier, vol. 240(2), pages 361-371.
    16. Antonio Frangioni, 2005. "About Lagrangian Methods in Integer Optimization," Annals of Operations Research, Springer, vol. 139(1), pages 163-193, October.
    17. Luís Gouveia & Pedro Moura, 2012. "Enhancing discretized formulations: the knapsack reformulation and the star reformulation," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(1), pages 52-74, April.
    18. Raymond A. Patterson & Erik Rolland, 2002. "Hybrid Fiber Coaxial Network Design," Operations Research, INFORMS, vol. 50(3), pages 538-551, June.
    19. Gouveia, Luis & Leitner, Markus & Ruthmair, Mario, 2017. "Extended formulations and branch-and-cut algorithms for the Black-and-White Traveling Salesman Problem," European Journal of Operational Research, Elsevier, vol. 262(3), pages 908-928.
    20. Erika Buson & Roberto Roberti & Paolo Toth, 2014. "A Reduced-Cost Iterated Local Search Heuristic for the Fixed-Charge Transportation Problem," Operations Research, INFORMS, vol. 62(5), pages 1095-1106, October.
    21. Andrea Bettinelli & Alberto Ceselli & Giovanni Righini, 2010. "A branch-and-price algorithm for the variable size bin packing problem with minimum filling constraint," Annals of Operations Research, Springer, vol. 179(1), pages 221-241, September.

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