Annals of Telecommunications

, Volume 73, Issue 1–2, pp 5–28 | Cite as

k-node-disjoint hop-constrained survivable networks: polyhedral analysis and branch and cut

  • Ibrahima DiarrassoubaEmail author
  • Meriem Mahjoub
  • A. Ridha Mahjoub
  • Hande Yaman


Given a graph with weights on the edges, a set of origin and destination pairs of nodes, and two integers L ≥ 2 and k ≥ 2, the k-node-disjoint hop-constrained network design problem is to find a minimum weight subgraph of G such that between every origin and destination there exist at least k node-disjoint paths of length at most L. In this paper, we consider this problem from a polyhedral point of view. We propose an integer linear programming formulation for the problem for L ∈{2,3} and arbitrary k, and investigate the associated polytope. We introduce new valid inequalities for the problem for L ∈{2,3,4}, and give necessary and sufficient conditions for these inequalities to be facet defining. We also devise separation algorithms for these inequalities. Using these results, we propose a branch-and-cut algorithm for solving the problem for both L = 3 and L = 4 along with some computational results.


k-node-disjoint hop-constrained paths Survivable network Polytope Valid inequalities Facets Separation Branch-and-cut 



We would like to thank the anonymous referees for their valuable comments that permitted to correct some flaw in the previous version and improve the presentation of the paper.


  1. 1.
    Barahona F, Mahjoub AR (1995) On two-connected subgraph polytopes. Discret Math 147:19–34MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bendali F, Diarrassouba I, Mahjoub AR, Mailfert J (2010) The k edge-disjoint 3-hop-constrained paths polytope. Discret Optim 7:222–233MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bendali F, Diarrassouba I, Didi Biha M, Mahjoub AR, Mailfert J (2010) A branch-and-cut algorithm for the k-edge-connected subgraph problem. Networks 55:13–32MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Botton Q, Fortz B, Gouveia L (2015) On the hop-constrained survivable network design problem with reliable edges. Comput Oper Res 64:159–167MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Botton Q, Fortz B, Gouveia L, Poss M (2013) Benders decomposition for the hop-constrained survivable network design problem. INFORMS J Comput 25:13–26MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chimani M, Kandyba M, Ljubic I, Mutzel P (2010) Orientation-based models for 0, 1, 2-survivable network design: theory and practice. Math Program 124(1-2):413–439MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dahl G (1999) Notes on polyhedra associated with hop-constrained paths. Oper Res Lett 25:97–100MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dahl G, Foldnes N, Gouveia L (2004) A note on hop-constrained walk polytopes. Oper Res Lett 32:345–349MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dahl G, Gouveia L (2004) On the directed hop-constrained shortest path problem. Oper Res Lett 32:15–22MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    IBM, IBM ILOG CPLEX Optimization studio 12.5 documentation (2013). Available at:
  11. 11.
    Diarrassouba I (2009) Survivable network design problems with high connectivity requirements, PhD Thesis, Université Blaise Pascal, FranceGoogle Scholar
  12. 12.
    Diarrassouba I, Kutucu H, Mahjoub AR (2016) Two node-disjoint hop-constrained survivable network design and polyhedra. Networks 67:316–337MathSciNetCrossRefGoogle Scholar
  13. 13.
    Diarrassouba I, Gabrel V, Mahjoub AR, Gouveia L, Pesneau P (2016) Integer programming formulations for the k-edge-connected 3-hop-constrained network design problem. Networks 67:148–169MathSciNetCrossRefGoogle Scholar
  14. 14.
    Didi Biha M, Mahjoub AR (2004) The k-edge connected subgraph problem I: polytopes and critical extreme points. Linear Algebra Appl 381:117–139MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Didi Biha M, Mahjoub AR (1996) K-edge connected polyhedra on series-parallel graphs. Oper Res Lett 19:71–78MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Edmonds J, Karp RM (1972) Theoretical improvements in algorithmic efficiency for network flow problems. J ACM 19:248–264CrossRefzbMATHGoogle Scholar
  17. 17.
    Gomory RE, Hu TC (1961) Multi-terminal network flows. JSoc Ind Appl Math 9:551–570MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gouveia Luis, Leitner Markus (2017) Design of survivable networks with vulnerability constraints. Eur J Oper Res 258(1):89–103MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gouveia L, Patricio P, de Sousa A (2005) Compact models for hop-constrained node survivable network design, an application to MPLS, telecommunications planning: innovations in pricing, network design and management. Springer 33:167–180Google Scholar
  20. 20.
    Grötschel M, Monma CL, Stoer M (1991) Polyhedral approaches to network survivability. Series in Discrete Mathematics & Theoretical Computer Science 5:121–141MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Grötschel M, Monma CL (1990) Integer polyhedra arising from certain network design problems with connectivity constraints. SIAM J Discret Math 3:502–523MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Grötschel M, Monma CL, Stoer M (1992) Computational results with a cutting plane algorithm for designing communication networks with low-connectivity constraints. Oper Res 40:309–330MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Grötschel M, Monma CL, Stoer M (1995) Polyhedral and computational investigations for designing communication networks with high survivability requirements. Oper Res 43:1012–1024MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Huygens D, Mahjoub AR (2007) Integer programming formulations for the two 4-hop-constrained paths problem. Networks 49:135–144MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Huygens D, Labbé M., Mahjoub AR, Pesneau P (2007) The two-edge connected hop-constrained network design problem: valid inequalities and branch-and-cut. Networks 49:116–133MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Huygens D, Mahjoub AR, Pesneau P (2004) Two edge-disjoint hop-constrained paths and polyhedra. SIAM J Disc Math 18:287–312MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kerivin H, Mahjoub AR (2015) Design of survivable networks: a survey. Networks 46:1–21MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kerivin H, Mahjoub AR, Nocq C (2004) (1,2)-survivable networks: facets and branch and cut, The sharpest cut. In: Grötschel M (ed) MPS/SIAM optimization, pp 121–152Google Scholar
  29. 29.
    Mahjoub M, Diarrassouba I, Mahjoub AR, Taktak R (2017) The survivable k-node-connected network design problem: valid inequalities and Branch-and-Cut. Comput Ind Eng 112:690– 705CrossRefzbMATHGoogle Scholar
  30. 30.
    Mahjoub AR (1994) Two-edge connected spanning subgraphs and polyhedra. Math Program 64:199–208MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Mahjoub AR, Nocq C (1999) On the linear relaxation of the 2-node connected subgraph polytope. Discret Appl Math 95(1–3):389–416MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Mahjoub AR, Simonetti L, Uchoa E (2011) Hop-level flow formulation for the hop constrained survivable network design problem. Lect Notes Comput Sci 6701:176–181CrossRefzbMATHGoogle Scholar
  33. 33.
    Menger K (1927) Zur allgemeinen kurventhorie. Fundamanta Mathematicae 10:96–115CrossRefzbMATHGoogle Scholar
  34. 34.

Copyright information

© Institut Mines-Télécom and Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Normandie Univ, UNIHAVRE, LMAHLe HavreFrance
  2. 2.PSL, CNRS UMR 7243 LAMSADEUniversité Paris-DauphineParisFrance
  3. 3.Faculté des Sciences de Tunis, URAPOPUniversité Tunis El ManarTunisTunisia
  4. 4.Department of Industrial EngineeringBilkent UniversityBilkentTurkey

Personalised recommendations