Abstract
The bounded diameter minimum spanning tree problem is an NP-hard combinatorial optimization problem arising, for example, in network design when quality of service is of concern. We solve a strong integer linear programming formulation based on so-called jump inequalities by a Branch&Cut algorithm. As the separation subproblem of identifying currently violated jump inequalities is difficult, we approach it heuristically by two alternative construction heuristics, local search, and optionally tabu search. We also introduce a new type of cuts, the center connection cuts, to strengthen the formulation in the more difficult to solve odd diameter case. In addition, primal heuristics are used to compute initial solutions and to locally improve incumbent solutions identified during Branch&Cut. The overall algorithm performs excellently, and we were able to obtain proven optimal solutions for some test instances that were too large to be solved so far.
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
K. Bala, K. Petropoulos, and T.E. Stern. Multicasting in a linear lightwave network. In Proc. of the 12th IEEE Conference on Computer Communications, pages 1350–1358. IEEE Press, 1993.
A. Bookstein and S. T. Klein. Compression of correlated bit-vectors. Information Systems, 16(4):387–400, 1991.
B.V. Cherkassky and A.V. Goldberg. On implementing the push-relabel method for the maximum flow problem. Algorithmica, 19(4):390–410, 1997. Code available at http://www.avglab.com/andrew/CATS/maxflow_solvers.htm.
G. Dahl, T. Flatberg, N. Foldnes, and L. Gouveia. Hop-constrained spanning trees: The jump formulation and a relax-and-cut method. Technical report, University of Oslo, Centre of Mathematics for Applications (CMA), 2005.
G. Dahl, L. Gouveia, and C. Requejo. On formulations and methods for the hop-constrained minimum spanning tree problem. In Handbook of Optimization in Telecommunications, chapter 19, pages 493–515. Springer Science + Business Media, 2006.
A.C. dos Santos, A. Lucena, and C.C. Ribeiro. Solving diameter constrained minimum spanning tree problems in dense graphs. In Proceedings of the International Workshop on Experimental Algorithms, volume 3059 of LNCS, pages 458–467. Springer Verlag, Berlin, 2004.
M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York, 1979.
L. Gouveia and T.L. Magnanti. Network flow models for designing diameter-constrained minimum spanning and Steiner trees. Networks, 41(3):159–173, 2003.
L. Gouveia, T.L. Magnanti, and C. Requejo. A 2-path approach for odd-diameter-constrained minimum spanning and Steiner trees. Networks, 44(4):254–265, 2004.
L. Gouveia, L. Simonetti, and E. Uchoa. Modelling the hop-constrained minimum spanning tree problem over a layered graph. In Proceedings of the International Network Optimization Conference, pages 1–6, Spa, Belgium, 2007.
M. Gruber and G.R. Raidl. A new 0–1 ILP approach for the bounded diameter minimum spanning tree problem. In L. Gouveia and C. Mourão, editors, Proceedings of the International Network Optimization Conference, volume 1, pages 178–185, Lisbon, Portugal, 2005.
M. Gruber and G.R. Raidl. Variable neighborhood search for the bounded diameter minimum spanning tree problem. In P. Hansen, N. Mladenović, J.A. Moreno Pérez, editors, Proceedings of the 18th Mini Euro Conference on Variable Neighborhood Search, Tenerife, Spain, 2005.
M. Gruber, J. van Hemert, and G.R. Raidl. Neighborhood searches for the bounded diameter minimum spanning tree problem embedded in a VNS, EA, and ACO. In Proceedings of the Genetic and Evolutionary Computation Conference 2006, volume 2, pages 1187–1194, 2006.
B.A. Julstrom. Greedy heuristics for the bounded-diameter minimum spanning tree problem. Technical report, St. Cloud State University, 2004.
T.L. Magnanti and L.A. Wolsey. Handbooks in Operations Research and Management Science: Network Models, chapter 9. North-Holland, 1995.
T.F. Noronha, A.C. Santos, and C.C. Ribeiro. Constraint programming for the diameter constrained minimum spanning tree problem. Electronic Notes in Discrete Mathematics, 30:93–98, 2008.
R.C. Prim. Shortest connection networks and some generalizations. Bell System Technical Journal, 36:1389–1401, 1957.
G.R. Raidl and B.A. Julstrom. Greedy heuristics and an evolutionary algorithm for the bounded-diameter minimum spanning tree problem. In G. Lamont, H. Haddad, G.A. Papadopoulos, and B. Panda, editors, Proceedings of the ACM Symposium on Applied Computing, pages 747–752. ACM Press, 2003.
K. Raymond. A tree-based algorithm for distributed mutual exclusion. ACM Transactions on Computer Systems, 7(1):61–77, 1989.
A. Singh and A.K. Gupta. Improved heuristics for the bounded-diameter minimum spanning tree problem. Soft Computing, 11(10):911–921, 2007.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Gruber, M., Raidl, G.R. (2009). (Meta-)Heuristic Separation of Jump Cuts in a Branch&Cut Approach for the Bounded Diameter Minimum Spanning Tree Problem. In: Maniezzo, V., Stützle, T., Voß, S. (eds) Matheuristics. Annals of Information Systems, vol 10. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1306-7_8
Download citation
DOI: https://doi.org/10.1007/978-1-4419-1306-7_8
Published:
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-1305-0
Online ISBN: 978-1-4419-1306-7
eBook Packages: Business and EconomicsBusiness and Management (R0)