Abstract
We consider a generalization of the (Price Collecting) Steiner Tree Problem on a graph with special redundancy requirements for customer nodes. The problem occurs in the design of the last mile of real-world communication networks. We formulate it as an abstract integer linear program and apply Lagrangian Decomposition to obtain relatively tight lower bounds as well as feasible solutions. Furthermore, a Variable Neighborhood Search and a GRASP approach are described, utilizing a new construction heuristic and special neighborhoods. In particular, hybrids of these methods are also studied and turn out to often perform superior. By comparison to previously published exact methods we show that our approaches are applicable to larger problem instances, while providing high quality solutions together with good lower bounds.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Barahona, F., Anbil, R.: The volume algorithm: producing primal solutions with a subgradient method. Mathematical Programming 87(3), 385–399 (2000)
Hansen, P., Mladenovic, N.: An introduction to variable neighborhood search. In: Voss, S., Martello, S., Osman, I.H., Roucairol, C. (eds.) Meta-heuristics, Advances and trends in local search paradigms for optimization, pp. 433–458. Kluwer Academic Publishers, Dordrecht (1999)
Feo, T., Resende, M.: Greedy randomized adaptive search procedures. Journal of Global Optimization 6(2), 109–133 (1995)
Wagner, D., Raidl, G.R., Pferschy, U., Mutzel, P., Bachhiesl, P.: A multi-commodity flow approach for the design of the last mile in real-world fiber optic networks. In: Waldmann, K.H., Stocker, U.M. (eds.) Operations Research Proceedings 2006, pp. 197–202. Springer, Heidelberg (2007)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press (1972)
Winter, P.: Steiner problem in networks: a survey. Networks 17(2), 129–167 (1987)
Segev, A.: The node-weighted Steiner tree problem. Networks 17(1), 1–17 (1987)
Balas, E.: The prize collecting traveling salesman problem. Networks 19, 621–636 (1989)
Kerivin, H., Mahjoub, A.R.: Design of survivable networks: A survey. Networks 46(1), 1–21 (2005)
ILOG: CPLEX 10.0. (2006), http://www.ilog.com
Wagner, D., Pferschy, U., Mutzel, P., Raidl, G.R., Bachhiesl, P.: A directed cut model for the design of the last mile in real-world fiber optic networks. In: Fortz, B. (ed.) Proceedings of the International Network Optimization Conference 2007, Spa, Belgium, pp. 1–6, 103 (2007)
Chimani, M., Kandyba, M., Mutzel, P.: A new ILP formulation for 2-root-connected prize-collecting Steiner networks. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 681–692. Springer, Heidelberg (2007)
Beasley, J.E.: Lagrangean relaxation. In: Reeves, C. (ed.) Modern heuristic techniques in combinatorial problems, pp. 243–303. Blackwell Scientific Publications, Malden (1993)
Bahiense, L., Barahona, F., Porto, O.: Solving Steiner tree problems in graphs with Lagrangian relaxation. Journal of Combinatorial Optimization 7(3), 259–282 (2003)
Haouari, M., Siala, J.C.: A hybrid Lagrangian genetic algorithm for the prize collecting Steiner tree problem. Computers and Operations Research 33(5), 1274–1288 (2006)
Suurballe, J.W., Tarjan, R.E.: A quick method for finding shortest pairs of disjoint paths. Networks 14, 325–335 (1984)
Kar, K., Kodialam, M., Lakshman, T.V.: Routing restorable bandwidth guaranteed connections using maximum 2-route flows. IEEE/ACM Transactions on Networking 11(5), 772–781 (2003)
Martins, S.L., Resende, M.G.C., Ribeiro, C.C., Pardalos, P.M.: A parallel GRASP for the Steiner tree problem in graphs using a hybrid local search strategy. Journal of Global Optimization 17(1-4), 267–283 (2000)
Mehlhorn, K.: A faster approximation algorithm for the Steiner problem in graphs. Information Processing Letters 27(3), 125–128 (1988)
Raidl, G.R.: A unified view on hybrid metaheuristics. In: Almeida, F., et al. (eds.) HM 2006. LNCS, vol. 4030, pp. 1–12. Springer, Heidelberg (2006)
Bachhiesl, P.: The OPT- and the SST-problems for real world access network design – basic definitions and test instances. Working Report 01/2005, Carinthia Tech Institue, Department of Telematics and Network Engineering, Klagenfurt, Austria (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Leitner, M., Raidl, G.R. (2008). Lagrangian Decomposition, Metaheuristics, and Hybrid Approaches for the Design of the Last Mile in Fiber Optic Networks. In: Blesa, M.J., et al. Hybrid Metaheuristics. HM 2008. Lecture Notes in Computer Science, vol 5296. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88439-2_12
Download citation
DOI: https://doi.org/10.1007/978-3-540-88439-2_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-88438-5
Online ISBN: 978-3-540-88439-2
eBook Packages: Computer ScienceComputer Science (R0)