Skip to main content
SpringerLink
Log in

Topology-Constrained Network Design

  • Chapter
  • First Online:
Network Design with Applications to Transportation and Logistics

Abstract

This chapter studies models and techniques for long-term planning of networks for which clients demands are not known in advance. It this case, the objective is to build a network at minimum cost, considering only the fixed cost associated with opening a link. Capacity and routing costs are therefore ignored. Nevertheless, the network is subject to topological constraints to ensure its connectivity and survivability. We cover the design of connected networks (and in particular the minimum spanning tree problem), followed by the design of networks requiring a higher level of survivability in terms of number of available node-disjoints paths, to allow re-routing in case of failures. To avoid delays in the networks, we also consider models where the length of paths is bounded, by introducing hop constraints or covering of the links by cycles of bounded lengths.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Balakrishnan, A., & Altinkemer, K. (1992). Using a hop-constrained model to generate alternative communication network design. ORSA Journal on Computing, 4, 192–205.

    Article  Google Scholar 

  • Barahona, F. (1992). Separating from the dominant of the spanning tree polytope. Operations Research Letters, 12, 201–203.

    Article  Google Scholar 

  • Bendali, F., Diarrassouba, I., Mahjoub, A., & Mailfert, J. (2010). The edge-disjoint 3-hop-constrained paths polytope. Discrete Optimization, 7(4), 222–233.

    Article  Google Scholar 

  • Bley, A. (2003). On the complexity of vertex-disjoint length-restricted path problems. Computational Complexity, 12(3–4), 131–149.

    Google Scholar 

  • Bley, A., & Neto, J. (2010). Approximability of 3- and 4-hop bounded disjoint paths problems. In Proceedings of IPCO 2010, Lausanne. Lecture notes in computer science (vol. 6080, pp 205–218). Berlin: Springer.

    Google Scholar 

  • Botton, Q., Fortz, B., & Gouveia, L. (2015). On the hop-constrained survivable network design problem with reliable edges. Computers & Operations Research, 64, 159–167.

    Article  Google Scholar 

  • Botton, Q., Fortz, B., & Gouveia, L. (2018). The 2 edge-disjoint 3-paths polyhedron. Annals of Telecommunications, 73(1), 29–36.

    Article  Google Scholar 

  • Botton, Q., Fortz, B., Gouveia, L., & Poss, M. (2013). Benders decomposition for the hop-constrained survivable network design problem. INFORMS Journal on Computing, 25(1), 13–26.

    Article  Google Scholar 

  • Carroll, P., Fortz, B., Labbé, M., & McGarraghy, S. (2013). A branch-and-cut algorithm for the ring spur assignment problem. Networks, 61(2), 89–103.

    Article  Google Scholar 

  • Cornuéjols, G., Fonlupt, F., & Naddef, D. (1985). The traveling salesman problem on a graph and some related integer polyhedra. Mathematical Programming, 33, 1–27.

    Article  Google Scholar 

  • Cunningham, W. (1985). Optimal attack and reinforcement of a network. Journal of ACM, 32, 549–561.

    Article  Google Scholar 

  • Dahl, G. (1999). Notes on polyhedra associated with hop-constrained walk polytopes. Operations Research Letters, 25, 97–100.

    Article  Google Scholar 

  • Dahl, G., & Johannessen, B. (2004). The 2-path network problem. Networks, 43, 190–199.

    Article  Google Scholar 

  • Dahl, G., Foldnes, N., & Gouveia, L. (2004). A note on hop-constrained walk polytopes. Operations Research Letters, 32(4), 345–349.

    Article  Google Scholar 

  • De Boeck, J., & Fortz, B. (2018). Extended formulation for hop constrained distribution network configuration problems. European Journal of Operational Research, 265(2), 488–502.

    Article  Google Scholar 

  • Edmonds, J. (1971). Matroids and the greedy algorithm. Mathematical Programming, 1(1), 127–136.

    Article  Google Scholar 

  • Eswaran, K., & Tarjan, R. (1976). Augmentation problems. SIAM Journal on Computing, 5, 653–665.

    Article  Google Scholar 

  • Fischetti, M., Leitner, M., Ljubić, I., Luipersbeck, M., Monaci, M., Resch, M., et al. (2017). Thinning out steiner trees: a node-based model for uniform edge costs. Mathematical Programming Computation, 9(2), 203–229.

    Article  Google Scholar 

  • Fortz, B. (2000). Design of survivable networks with bounded rings, network theory and applications (vol 2). Dordrecht: Kluwer Academic Publishers.

    Book  Google Scholar 

  • Fortz, B., & Labbé, M. (2002). Polyhedral results for two-connected networks with bounded rings. Mathematical Programming, 93(1), 27–54.

    Article  Google Scholar 

  • Fortz, B., & Labbé, M. (2004). Two-connected networks with rings of bounded cardinality. Computational Optimization and Applications, 27(2), 123–148.

    Article  Google Scholar 

  • Fortz, B., Labbé, M., & Maffioli, F. (2000). Solving the two-connected network with bounded meshes problem. Operations Research, 48(6), 866–877.

    Article  Google Scholar 

  • Fortz, B., Mahjoub, A., Mc Cormick, S., & Pesneau, P. (2006). Two-edge connected subgraphs with bounded rings: polyhedral results and branch-and-cut. Mathematical Programming, 105, 85–111.

    Article  Google Scholar 

  • Goldschmidt, O., Laugier, A., & Olinick, E. V. (2003). SONET/SDH ring assignment with capacity constraints. Discrete Applied Mathematics, 129, 99–128.

    Article  Google Scholar 

  • Gouveia, L. (1998). Using variable redefinition for computing lower bounds for minimum spanning and steiner trees with hop constraints. INFORMS Journal on Computing, 10, 180–188.

    Article  Google Scholar 

  • Gouveia, L., Joyce-Moniz, M., & Leitner, M. (2018). Branch-and-cut methods for the network design problem with vulnerability constraints. Computers & Operations Research, 91, 190–208.

    Article  Google Scholar 

  • Gouveia, L., & Leitner, M. (2017). Design of survivable networks with vulnerability constraints. European Journal of Operational Research, 258(1), 89–103.

    Article  Google Scholar 

  • Gouveia, L., Leitner, M., & Ruthmair, M. (2019). Layered graph approaches for combinatorial optimization problems. Computers & Operations Research, 102, 22–38.

    Article  Google Scholar 

  • Gouveia, L., & Magnanti, T. L. (2003). Network flow models for designing diameter-constrained minimum-spanning and Steiner trees. Networks, 41(3), 159–173.

    Article  Google Scholar 

  • Gouveia, L., Patrício, P., & Sousa, A. (2006). Compact models for hop-constrained node survivable network design. In Telecommunications planning: Innovations in pricing. Network design and management (pp. 167–180). New York: Springer.

    Google Scholar 

  • Gouveia, L., Patrício, P., & Sousa, A. (2008). Hop-contrained node survivable network design: An application to MPLS over WDM. Networks and Spatial Economics, 8(1), 3–21.

    Article  Google Scholar 

  • Gouveia, L. E., Patrício, P., de Sousa, A., & Valadas, R. (2003). MPLS over WDM network design with packet level QoS constraints based on ILP Models. In Proceedings of IEEE INFOCOM (pp. 576–586).

    Google Scholar 

  • Grötschel, M., & Monma, C. (1990). Integer polyhedra arising from certain design problems with connectivity constraints. SIAM Journal on Discrete Mathematics, 3, 502–523.

    Article  Google Scholar 

  • Grötschel, M., Monma, C., & Stoer, M. (1995a). Design of survivable networks. Handbooks in OR/MS, vol 7 on Network models (chap 10, pp. 617–672). Amsterdam: North-Holland.

    Google Scholar 

  • Grötschel, M., Monma, C., & Stoer, M. (1995b). Polyhedral and computational investigations for designing communication networks with high survivability requirements. Operations Research, 43(6), 1012–1024.

    Article  Google Scholar 

  • Huygens, D., Labbé, M., Mahjoub, A. R., & Pesneau, P. (2007). The two-edge connected hop-constrained network design problem: Valid inequalities and branch-and-cut. Networks, 49(1), 116–133.

    Article  Google Scholar 

  • Huygens, D., & Mahjoub, A. R. (2007). Integer programming formulations for the two 4-hop-constrained paths problem. Networks, 49(2), 135–144.

    Article  Google Scholar 

  • Huygens, D., Mahjoub, A, & Pesneau, P. (2004). Two edge-disjoint hop-constrained paths and polyhedra. SIAM Journal on Discrete Mathematics, 18(2), 287–312.

    Article  Google Scholar 

  • Itaí, A., Perl, Y., & Shiloach, Y. (1982). The complexity of finding maximum disjoint paths with length constraints. Networks, 2, 277–286.

    Article  Google Scholar 

  • Kerivin, H., & Mahjoub, A. R. (2005). Design of survivable networks: A survey. Networks, 46(1), 1–21.

    Article  Google Scholar 

  • Kruskal, J. (1956). On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematical Society, 7, 48–50.

    Google Scholar 

  • Lawler, E. (1976). Combinatorial optimization: Networks and matroids. New-York: Holt, Rinehart and Wilson.

    Google Scholar 

  • Magnanti, T. L., & Raghavan, S. (2005). Strong formulations for network design problems with connectivity requirements. Networks, 45(2), 61–79.

    Article  Google Scholar 

  • Magnanti, T. L., & Wolsey, L. A. (1995). Optimal trees. Handbooks in Operations Research and Management Science, 7, 503–615.

    Article  Google Scholar 

  • Mahjoub, A. (1994). Two-edge connected spanning subgraphs and polyhedra. Mathematical Programming, 64, 199–208.

    Article  Google Scholar 

  • Martin, R. K. (1991). Using separation algorithms to generate mixed integer model reformulations. Operations Research Letters, 10(3), 119–128.

    Article  Google Scholar 

  • Menger, K. (1927). Zur allgemeinen kurventheorie. Fundamenta Mathematicae, 10, 96–115.

    Article  Google Scholar 

  • Monma, C., & Shallcross, D. (1989). Methods for designing communications networks with certain two-connected survivability constraints. Operations Research, 37(4), 531–541.

    Article  Google Scholar 

  • Pirkul, H., & Soni, S. (2003). New formulations and solution procedures for the hop constrained network design problem. European Journal of Operational Research, 148, 126–140.

    Article  Google Scholar 

  • Stoer, M. (1992). Design of survivable networks. Lecture Notes in Mathematics (vol. 1531). Berlin: Springer.

    Book  Google Scholar 

  • Winter, P. (1987). Steiner problems in networks: A survey. Networks, 17, 129–167.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science Department, Université libre de Bruxelles, Brussels, Belgium

    Bernard Fortz

  2. INOCS, INRIA Lille Nord-Europe, Villeneuve-d’Ascq, France

    Bernard Fortz

Authors
  1. Bernard Fortz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bernard Fortz .

Editor information

Editors and Affiliations

  1. CIRRELT and AOTI, Université du Québec à Montréal, Montreal, QC, Canada

    Teodor Gabriel Crainic

  2. CIRRELT and MAGI, Polytechnique Montréal, Montreal, QC, Canada

    Michel Gendreau

  3. CIRRELT and Département d'informatique et de recherche opérationnelle, Université de Montréal, Montreal, QC, Canada

    Bernard Gendron

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s)

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Fortz, B. (2021). Topology-Constrained Network Design. In: Crainic, T.G., Gendreau, M., Gendron, B. (eds) Network Design with Applications to Transportation and Logistics. Springer, Cham. https://doi.org/10.1007/978-3-030-64018-7_7

Download citation

Publish with us

Policies and ethics

Search

Navigation