Abstract
In this paper we propose an algorithm for the vertex separator problem (VSP) based on Lagrangian Relaxation and on cutting planes derived from valid inequalities presented in [3]. The procedure is used as a preprocessing for the branch-and-cut (B&C) algorithm implemented for VSP in [12], aiming to generate an initial pool of cutting planes and a strong primal bound for the latter. Computational experiments show that the exact method obtained in that way outperforms the pure B&C algorithm recently introduced by de Souza and Balas in [12]. Besides, we show that the Lagrangian phase is a very effective heuristic for the VSP, often producing optimal solutions extremely fast.
Research supported by grants CNPq-307773/2004-3 and FAPESP-03/09925-5.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Balas, E., Christofides, N.: A restricted lagrangean approach to the traveling salesman problem. Mathematical Programming 21, 19–46 (1981)
Balas, E., Ho, C.: Set covering algorithms using cutting planes, heuristics, and subgradient optimization. Mathematical Programming Study 12, 37–60 (1980)
Balas, E., de Souza, C.C.: The vertex separator problem: a polyhedral investigation. Mathematical Programming 103, 583–608 (2005)
Beasley, J.E.: Modern Heuristic Techniques, ch. 6. Blackwell Scientific Press, Oxford (1993)
Calheiros, F., Lucena, A., de Souza, C.: Optimal rectangular partitions. Networks 41, 51–67 (2003)
Guignard, M.: Lagrangean relaxation. In: Lòpez-Cerdá, M.A., García-Jurado, I. (eds.) Top, Madrid, Spain. Sociedad de Estadística e Investigación Operativa, vol. 11, pp. 151–228 (2003)
Kliewer, G., Timajev, L.: Relax-and-cut for capacitated network design. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 47–58. Springer, Heidelberg (2005)
Lucena, A.: Steiner problem in graphs: Lagrangean relaxation and cutting-planes. In: COAL Bulletin. Mathematical Programming Society, vol. 21 (1992)
Lucena, A.: Non delayed relax-and-cut algorithms. Annals of Operations Research 140(1), 375–410 (2005)
Martinhon, C., Lucena, A., Maculan, N.: Stronger k-tree relaxations for the vehicle routing problem. European Journal on Operational Research 158, 56–71 (2004)
Ferreira, C.E., Borndörfer, R., Martin, A.: Decomposing matrices into blocks. SIAM Journal on optimization 9, 236–269 (1998)
de Souza, C.C., Balas, E.: The vertex separator problem: algorithms and computations. Mathematical Programming 103, 609–631 (2005)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cavalcante, V.F., de Souza, C.C. (2007). Lagrangian Relaxation and Cutting Planes for the Vertex Separator Problem. In: Chen, B., Paterson, M., Zhang, G. (eds) Combinatorics, Algorithms, Probabilistic and Experimental Methodologies. ESCAPE 2007. Lecture Notes in Computer Science, vol 4614. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74450-4_42
Download citation
DOI: https://doi.org/10.1007/978-3-540-74450-4_42
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74449-8
Online ISBN: 978-3-540-74450-4
eBook Packages: Computer ScienceComputer Science (R0)