Abstract
Given a connected, undirected and m-partite complete graph \(G = (V_1 \cup V_2 \cup ... \cup V_m; E)\), the Generalized Minimum Spanning Tree Problem (GMSTP) consists in finding a tree with exactly \(m - 1\) edges, connecting the m clusters \(V_1, V_2, ..., V_m\) through the selection of a unique vertex in each cluster. GMSTP finds applications in network design, irrigation agriculture, smart cities, data science, among others. This paper presents a new multigraph mathematical formulation for GMSTP which is compared to existing formulations from the literature. The proposed model proves optimality for well-known GMSTP instances. In addition, this work opens new directions for future research to the development of sophisticated cutting plane and decomposition algorithms for related problems.
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References
Adasme, P., Andrade, R., Letournel, M., Lisser, A.: Stochastic maximum weight forest problem. Networks 65, 289–305 (2015)
Andrade, R.C.: Appointments on the spanning tree polytope. Annals of the ALIO/EURO - Montevideo (2014)
Afsar, H.M., Prins, C., Santos, A.C.: Exact and heuristic algorithms for solving the generalized vehicle routing problem with flexible fleet size. Int. Trans. Oper. Res. 21(1), 153–175 (2014)
Bhattacharya, B., Ćustić, A., Rafiey, A., Rafiey, A., Sokol, V.: Approximation algorithms for generalized MST and TSP in grid clusters. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, D.-Z. (eds.) COCOA 2015. LNCS, vol. 9486, pp. 110–125. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-26626-8_9
Contreras-Bolton, C., Gatica, G., Barra, C.R., Parada, V.: A multi-operator genetic algorithm for the generalized minimum spanning tree problem. Expert Syst. Appl. 50, 1–8 (2016)
Dror, M., Haouari, M.: Generalized Steiner problems and other variants. J. Comb. Optim. 4, 415–436 (2000)
Duin, C.W., Volgenant, A., Voss, S.: Solving group Steiner problems as Steiner problems. Eur. J. Oper. Res. 154, 323–329 (2004)
Feremans, C.: Generalized spanning tree and extensions. Ph.D. thesis, Universite Libre de Bruxelles (2001)
Feremans, C., Labbé, M., Laporte, G.: A comparative analysis of several formulations for the generalized minimum spanning tree problem. Networks 39(1), 29–34 (2002)
Feremans, C., Labbé, M., Laporte, G.: The generalized minimum spanning tree problem: polyhedral analysis and branch-and-cut algorithm. Networks 43(2), 71–86 (2004)
Ferreira, C.S., Ochi, L.S., Parada, V., Uchoa, E.: A GRASP-based approach to the generalized minimum spanning tree problem. Expert Syst. Appl. 39(3), 3526–3536 (2012)
Ghosh, D.: Solving medium to large sized Euclidean generalized minimum spanning tree problems. Technical report NEP-CMP-2003-09-28 Indian Institute of Management, Research and Publication Department, India (2003)
Golden, B., Raghavan, S., Stanojevic, D.: Heuristic search for the generalized minimum spanning tree problem. INFORMS J. Comput. 17(3), 290–304 (2005)
Hu, B., Leitner, M., Raidl, G.R.: Computing generalized minimum spanning trees with variable neighborhood search. In: Proceedings of the 18th Mini-Euro Conference on Variable Neighborhood Search (2005)
Martin, K.: Using separation algorithms to generate mixed integer model reformulations. Oper. Res. Lett. 10(3), 119–128 (1991)
Myung, Y.S., Lee, C.H., Tcha, D.W.: On the generalized minimum spanning tree problem. Networks 26(4), 231–241 (1995)
Öncan, T., Cordeau, J., Laporte, G.: A tabu search heuristic for the generalized minimum spanning tree problem. Eur. J. Oper. Res. 191(2), 306–319 (2008)
Pop, P.C.: The generalized minimum spanning tree problem. Ph.D. thesis, University of Twente, Netherlands (2002)
Pop, P.C., Matei, O., Sabo, C., Petrovan, A.: A two-level solution approach for solving the generalized minimum spanning tree problem. Eur. J. Oper. Res. 170, 1–10 (2017)
Santos, A.C., Duhamel, C., Andrade, R.: Trees and forests. In: Martí, R., Panos, P., Resende, M.G.C. (eds.) Handbook of Heuristics, pp. 1–27. Springer, Boston (2016)
Sousa, E.G., Andrade, R.C., Santos, A.C.: Algoritmo genético para o problema da árvore geradora generalizada de custo mínimo. SBPO 34(4), 437–444 (2017)
Wang, Z., Che, C.H., Lim, A.: Tabu search for generalized minimum spanning tree problem. In: Yang, Q., Webb, G. (eds.) PRICAI 2006. LNCS, vol. 4099, pp. 918–922. Springer, Heidelberg (2006). https://doi.org/10.1007/978-3-540-36668-3_106
Acknowledgements
The authors are grateful to CNPq (grant 449254/2014-3) and FUNCAP (grant PNE-0112-00061.01.00/16) and to the anonymous referees for their helpful comments.
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de Sousa, E.G., de Andrade, R.C., Santos, A.C. (2018). A Multigraph Formulation for the Generalized Minimum Spanning Tree Problem. In: Lee, J., Rinaldi, G., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2018. Lecture Notes in Computer Science(), vol 10856. Springer, Cham. https://doi.org/10.1007/978-3-319-96151-4_12
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