Abstract
In this paper, we introduce a minimum cost network design problem with a given lower bound requirement for capacity of any cut. First, we consider the subproblem including the lower bound requirement only for each fundamental cut determined by deleting edges from some spanning tree. Then, it is shown that the simplex algorithm finds an optimal solution to standard LP-relaxation of the subproblem in the linear time of the number of edges. The simplex and minimum cut algorithms are used in a Branch-and-Cut type algorithm to pick up a solution to the minimum cost network design problem.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M. Alicherry, R. Bhatia, Y.C. Wan, Designing networks with existing traffic to support fast restoration, in Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. Lecture Notes in Computer Science, vol. 3122 (Springer, Berlin, 2004), pp. 1–22
D. Bienstock, G. Muratore, Strong inequalities for capacitated survivable network design problems. Math. Program. 89, 127–147 (2001)
W.H. Cunningham, Minimum cuts, modular functions, and matroid polyhedra. Networks 15, 205–215 (1985)
W.H. Cunningham, On submodular function minimization. Combinatorica 5(3), 185–192 (1985)
F. Fritzsche, F.B. Holt, More polytope meeting the conjectured Hirsch bound. Discret. Math. 205, 77–84 (1999)
M.X. Goemans, D.P., Williamson, A general approximation technique for constrained forest problems. SIAM J. Comput. 24, 296–317 (1995)
M. Grotschel, C.L. Monma, M. Stoer, Polyhedral and computational investigations for designing communication networks with high survivability requirements. Oper. Res. 43, 1012–1024 (1995)
T.C. Hu, Optimum communication spanning trees. SIAM J. Comput. 3(3) ,188–195 (1974)
H. Kerivin, A.R. Mahjoub, Separation of partition inequalities for (1, 2) survivable network design problem. Oper. Res. Lett. 30(4), 265–268 (2002)
K.V. Marintseva, F.A. Sharifov, G.N. Yun, A problem of airport capacity definition. Aeronautic 5, 1–13 (2013)
G.L. Nemhauser, L.A. Wolsey, M.L. Fisher, An analysis of the approximation for maximizing submodular set functions. Math. Program. 14, 265–294 (1978)
A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency in Algorithms and Combinatorics, vol. 24 (Springer, Berlin, 2003)
F.A. Sharifov, Determination of the minimum cut using the base of an extended polymatroid. Cybern. Syst. Anal. 6, 856–867 (1997). Translated from Cybernetics and Systems Analysis 6, 138–152 (1996) (in Russian)
F.A. Sharifov, Submodular functions in synthesis of networks. Cybern. Syst. Anal. 37(4), 603–609 (2001). Translated from Kibernetika i Systemnyi Analis 4, 166–174 (2001) (in Russian)
F.A. Sharifov, Network Design Problem When Edges of Isomorphic Subgraph Are Deleted, Proceedings (Evry, Paris, 2003), pp. 521–525
F.A. Sharifov, Hulianytskyi L.F., Models and complexity of problems of design and reconstruction of telecommunication and transport systems. Cybern. Syst. Anal. 50(5), 693–700 (2014) (in Russian)
H.D. Sherali, R. Sivanandan, A.G. Hobeika, A linear programming approach for synthesizing origin-destination trip tables from link traffic volumes. Transp. Res. B 28(3), 213–233 (1994)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Sharifov, F., Kutucu, H. (2018). Network Design Problem with Cut Constraints. In: Goldengorin, B. (eds) Optimization Problems in Graph Theory. Springer Optimization and Its Applications, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-319-94830-0_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-94830-0_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-94829-4
Online ISBN: 978-3-319-94830-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)