Abstract
Given an edge weighted undirected graph G and a positive integer d, the Min-Degree Constrained Minimum Spanning Tree Problem (MDMST) asks for a minimum cost spanning tree of G, such that each vertex is either a leaf or has degree at least d in the tree. The strongest known MDMST lower bounds, provided by a reformulation by intersection, are very expensive to be evaluated directly, by Linear Programming solvers. Therefore, we propose a Lagrangian Relaxation algorithm for approximating them. The reformulation makes use of a large number of variables and the relaxation involves the dualization of a large number of constraints. Attempting to speed up the computation of the Lagrangian Dual bounds, we implemented a parallel Subgradient Method. We also introduced a Lagrangian heuristic based on a Local Branching algorithm. With the proposed methods, respectively 26 and 14 new best upper and lower bounds are presented.
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References
Akgún, I., Tansel, B.C.: Min-degree constrained minimum spanning tree problem: New formulation via Miller-Tucker-Zemlin constraints. Computers and Operations Research 37(1), 72–82 (2010)
Almeida, A.M., Martins, P., de Souza, M.C.: Min-Degree Constrained Minimum Spanning Tree Problem: Complexity, proprieties and formulations. Centro de Investigação Operacional, Universidade de Lisboa (June 2006)
Almeida, A.M., Martins, P., de Souza, M.C.: md-MST is NP-hard for d ≥ 3. Electronic Notes in Discrete Mathematics 36, 9–15 (2010)
Beasley, J.E.: Lagrangean Relaxation. In: Modern Heuristic Techniques, Collin Reeves. Blackwell Scientific Press, Oxford (1993)
Camerini, P.M., Fratta, L., Maffioli, F.: On improving relaxation methods by modified gradient techniques. Mathematical Programming Study 3, 26–34 (1975)
Fischetti, M., Lodi, A.: Local branching. In: Mathematical Programming, vol. 98, pp. 23–47. Springer, Heidelberg (2003)
Gouveia, L., Telhada, J.: The multi-weighted Steiner Tree problem: A reformulation by intersection. Computers and Operations Research 35, 3599–3611 (2008)
Kruskal, J.B.: On the Shortest Spanning Subtree of a Graph and the Traveling Salesman Problem. American Mathematical Society 7, 48–50 (1956)
Magnanti, T.L., Wolsey, L.: Optimal Trees. In: Handbooks in OR and MS, vol. 7, pp. 503–615. North-Holland (1995)
Martinez, L.C., da Cunha, A.S.: Finding Min-degree Constrained Spanning Trees faster with a Branch-and-cut algorithm. Electronic Notes in Discrete Mathematics 36, 311–318 (2010)
Martinez, L.C., da Cunha, A.S.: The Min-Degree Constrained Minimum Spanning Tree Problem: Formulations and Branch-and-cut algorithm. Discrete Applied Mathematics (2011) (in press), doi:10.1016/j.dam.2011.08.008
Martins, P., de Souza, M.C.: VNS and second order heuristics for the min-degree constrained minimum spanning tree problem. Computers and Operations Research 36(11), 2669–2982 (2009)
Miller, C.E., Tucker, A.W., Zemlin, R.A.: Integer programming formulations and travelling salesman problems. Journal of the Association of Computing Machinery 7, 326–329 (1960)
OpenMp Architecture Review Board, http://www.openmp.org
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Martinez, L.C., da Cunha, A.S. (2012). A Parallel Lagrangian Relaxation Algorithm for the Min-Degree Constrained Minimum Spanning Tree Problem. In: Mahjoub, A.R., Markakis, V., Milis, I., Paschos, V.T. (eds) Combinatorial Optimization. ISCO 2012. Lecture Notes in Computer Science, vol 7422. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32147-4_22
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DOI: https://doi.org/10.1007/978-3-642-32147-4_22
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