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A matheuristic for the minimum weight rooted arborescence problem

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Abstract

The combinatorial optimization problem tackled in this work is from the family of minimum weight rooted arborescence problems. The problem is NP-hard and has applications, for example, in computer vision and in multistage production planning. We describe an algorithm which makes use of a mathematical programming solver in order to find near-optimal solutions to the problem both in acyclic directed graphs and in directed graphs possibly containing directed circuits. It is shown that the proposed technique compares favorably to competiting approaches published in the related literature. Moreover, the experimental evaluation demonstrates that, although mathematical programming solvers are very powerful for this problem, with growing graph size and density they become unpractical due to excessive memory requirements.

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Notes

  1. In fact, M-DAG is a \(\mathcal {O}(n^2)\) model, while M-DG is a \(\mathcal {O}(n^3)\) model.

References

  • Applegate, D., Bixby, R., Chvátal, V., Cook, W.: Finding tours in the TSP. Techical Report, Forschungsinstitut für Diskrete Mathematik, University of Bonn, Germany (1999)

  • Blum, C., Puchinger, J., Raidl, G., Roli, A.: Hybrid metaheuristics in combinatorial optimization: a survey. Appl. Soft Comput. 11(6), 4135–4151 (2011)

    Article  MATH  Google Scholar 

  • Boschetti, M.A., Maniezzo, V., Roffilli, M., Bolufé Röhler, A.: Matheuristics: Optimization, simulation and control. In: Blesa, M.J., Blum, C., Di Gaspero, L., Roli, A., Sampels, M., Schaerf, A. (eds.) Proceedings of the HM 2009 Sixth International Workshop on Hybrid Metaheuristics. Lecture Notes in Computer Science, vol. 5818, pp. 171–177. Springer, Berlin (2009)

  • Cook, W., Seymour, P.: Tour merging via branch-decomposition. INFORMS J. Comput. 15(3), 233–248 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Coudert, D., Nepomuceno, N., Rivano, H.: Power-efficient radio configuration in fixed broadband wireless networks. Comput. Commun. 33(8), 898–906 (2010)

    Article  Google Scholar 

  • Dorigo, M., Stützle, T.: Ant Colony Optimization. MIT Press, Cambridge, MA (2004)

    Book  MATH  Google Scholar 

  • Duhamel, C., Gouveia, L., Moura, P., Souza, M.: Models and heuristics for a minimum arborescence problem. Networks 51(1), 34–47 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Klau, G.W., Ljubić, I., Moser, A., Mutzel, P., Neuner, P., Pferschy, U., Raidl, G., Weiskircher, R.: Combining a memetic algorithm with integer programming to solve the prize-collecting Steiner tree problem. In: Proceedings of the GECCO 2004 -Genetic and Evolutionary Computation Conference. Lecture Notes in Computer Science, vol. 3102, pp. 1304–1315. Springer, Berlin (2004)

  • Mateo, S., Blum, C, Türetken, E., Fua, P.: Hybrid algorithms for the minimum-weight rooted arborescence problem. In: Dorigo M, Birattari M, Blum C, Christensen AL, Engelbrecht AP, Groß R, Stützle T (eds) Proceedings of ANTS 2012–8th International Conference on Swarm Intelligence. Lecture Notes in Computer Science, vol. 7461, pp. 61–72. Springer, Berlin (2012)

  • Nepomuceno, N., Pinheiro, P., Coelho, A.L.V.: A Hybrid Optimization Framework for Cutting and Packing Problems. Recent Advances in Evolutionary Computation for Combinatorial Optimization, Studies in Computational Intelligence, vol. 153, pp. 87–99. Springer, Berlin (2008)

  • Pinheiro, P.R., Coelho, A.L.V., de Aguiar, A.B., Bonates, T.O.: On the concept of density control and its application to a hybrid optimization framework: investigation into cutting problems. Comput. Industr. Eng. 61(3), 463–472 (2011)

    Article  Google Scholar 

  • Raidl, G.R.: Decomposition based hybrid metaheuristics. Eur. J. Oper. Res. (2014). In press

  • Talbi, E.G. (ed.): Hybrid Metaheuristics. Studies in Computational Intelligence. Springer, Berlin (2013)

    Google Scholar 

  • Türetken, E., González, G., Blum, C., Fua, P.: Automated reconstruction of dendritic and axonal trees by global optimization with geometric priors. Neuroinformatics 9(2–3), 279–302 (2011)

    Article  Google Scholar 

  • Tutte, W.T.: Graph Theory. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  • Venkata Rao, V., Sridharan, R.: The minimum weight rooted arborescence problem: weights on arcs case. Technical Report, Indian Institute of Management Ahmedabad, Research and Publication Department (1992)

  • Venkata Rao, V., Sridharan, R.: Minimum-weight rooted not-necessarily-spanning arborescence problem. Networks 39(2), 77–87 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This work was supported by Grants TIN2012-37930-02 and TIN2013-41272P of the Spanish Government and by Grant IT-609-13 of the Basque Government. In addition, support is acknowledged from IKERBASQUE (Basque Foundation for Science). Our experiments have been executed in the High Performance Computing environment managed by RDlab (http://rdlab.lsi.upc.edu) and we would like to thank them for their support.

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Authors and Affiliations

  1. University of the Basque Country UPV/EHU, San Sebastian, Spain

    Christian Blum & Borja Calvo

  2. Ikerbasque, Basque Foundation for Science, Bilbao, Spain

    Christian Blum

Authors
  1. Christian Blum
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  2. Borja Calvo
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Correspondence to Christian Blum.

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Blum, C., Calvo, B. A matheuristic for the minimum weight rooted arborescence problem. J Heuristics 21, 479–499 (2015). https://doi.org/10.1007/s10732-015-9286-1

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  • DOI: https://doi.org/10.1007/s10732-015-9286-1

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