Abstract
This paper considers the Optimum Communication Spanning Tree Problem. An integer programming formulation that yields tight LP bounds is proposed. Given that the computational effort required to obtain the LP bounds considerably increases with the size of the instances when using commercial solvers, we propose a Lagrangean relaxation that exploits the structure of the formulation. Since feasible solutions to the Lagrangean function are spanning trees, upper bounds are also obtained. These bounds are later improved with a simple local search. Computational experiments have been run on several benchmark instances from the literature. The results confirm the interest of the proposal since tight lower and upper bounds are obtained, for instances up to 100 nodes, in competitive computational times.
Similar content being viewed by others
References
Ahuja R, Murty V (1987) Exact and heuristic algorithms for the optimum communication spanning tree problem. Transp Sci 21:163–170
Balakrishnan A, Magnanti T, Wong R (1989) A dual-ascent procedure for large-scale uncapacitated network design. Oper Res 37:716–740
Berry L, Murtagh B, McMahon G (1995) Applications of a genetic-based algorithm for optimal design of tree-structured communication networks. In: Proceedings of the regional teletraffic engineering conference of the international teletraffic congress. Telkom South Africa, Pretoria, South Africa, pp 361–370
Contreras I (2009) Network hub location: models, algorithms and related problems. PhD thesis, Department of Statistics and Operations Research, Technical University of Catalonia, Spain
Contreras I, Fernández E, Marín A (2009a) Tight bounds from a path based formulation for the tree of hubs location problem. Comput Oper Res 36:3117–3127
Contreras I, Fernández E, Marín A (2009b) The tree of hubs location problem. Eur J Oper Res. doi:10.1016/j.ejor.2009.05.044
Fischer T (2007) Improved local search for large optimum communication spanning. In: MIC2007: 7th metaheuristics international conference, Montréal
Fischer T, Merz P (2007) A memetic algorithm for the optimum communication spanning tree problem. In: Bartz-Beielstein T, Aguilera M, Blum C, Naujoks B, Roli A, Rudolph G, Sampels M (eds) HM 2007: 4th international workshop on hybrid metaheuristics, vol 4771. Dortmund, Germany. Springer, Berlin, pp 170–184
Fischetti M, Lancia G, Serafini P (2002) Exact algorithms for minimum routing cost trees. Networks 39:161–173
Guignard M (2003) Lagrangean relaxation. TOP 11:151–228
Hu T (1974) Optimum communication spanning trees. SIAM J Comput 3(3):188–195
Hu T (1982) Combinatorial algorithms. Addison-Wesley, Reading
Johnson D, Lenstra J, Kan JR (1987) The complexity of the network design problem. Networks 8:279–285
Palmer C (1994a) An approach to a problem in network design using genetic algorithms. PhD thesis, Polytechnic University, Troy, NY
Palmer C (1994b) Two algorithms for finding optimal communication spanning trees. Tech Rep RC 19394, IBM TJ Watson Research Center NY, USA
Papadimitriou C, Yannakakis M (1988) Optimization, approximation, and complexity classes. In: STOC 88: Proceedings of the twentieth annual. ACM symposium on theory of computing, New York, USA
Peleg D, Reshef E (1998) Deterministic polylog approximation for minimum communication spanning trees. In: Lecture notes in computer science, vol 1443. Springer, Berlin, pp 670–686
Prim R (1957) Shortest connection networks and some generalisations. Bell Syst Tech J 36:1389–1401
Rothlauf F (2006) Representations for genetic and evolutionary algorithms, 2nd edn. Springer, Berlin
Rothlauf F (2007) Design and applications of metaheuristics. Habilitationsschrift, Universität Mannheim
Rothlauf F (2009) On the optimal solutions for the optimal communication spanning tree problem. Oper Res 57(2):413–425
Rothlauf F, Heinzl A (2008). Orientation matters: How to efficiently solve cost problems with problem-specific eas. Tech rep, Johannes Gutenberg-University Mainz, Germany
Sharma P (2006) Algorithms for the optimum communication spanning tree problem. Ann Oper Res 143(1):203–209
Soak S (2006) A new evolutionary approach for the optimal communication spanning tree problem. E89-A 10:2882–2893
Wolsey L, Magnanti T (1995) Optimal trees. In: Networks models. Handbooks operations research and management science, vol 7. North-Holland, Amsterdam, pp 503–615
Wu B (2002) A polynomial time approximation scheme for the two-source minimum routing cost spanning trees. J Algorithms 44:359–378
Wu B, Chao KM, Tang C (2000a) Approximation algorithms for some optimum communication spanning trees. Discrete Appl Math 102:245–266
Wu B, Chao KM, Tang C (2000b) A polynomial time approximation scheme for optimal product-requirement communication spanning trees. J Algorithms 102:245–266
Wu B, Lancia G, Bafna V, Chao KM, Ravi R, Tang C (2000c) A polynomial time approximation scheme for minimum routing cost spanning trees. J Comput 29:761–778
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Contreras, I., Fernández, E. & Marín, A. Lagrangean bounds for the optimum communication spanning tree problem. TOP 18, 140–157 (2010). https://doi.org/10.1007/s11750-009-0112-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11750-009-0112-5