Skip to main content
SpringerLink
Log in

On the chance-constrained minimum spanning k-core problem

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

A graph is called a k-core if every vertex has at least k neighbors. If the parameter k is sufficiently large relative to the number of vertices, a k-core is guaranteed to possess 2-hop reachability between all pairs of vertices. Furthermore, it is guaranteed to preserve those pairwise distances under arbitrary single-vertex deletion. Hence, the concept of a k-core can be used to produce 2-hop survivable network designs, specifically to design inter-hub networks. Formally, given an edge-weighted graph, the minimum spanningk-core problem seeks a spanning subgraph of the given graph that is a k-core with minimum total edge weight. For any fixed k, this problem is equivalent to a generalized graph matching problem and can be solved in polynomial time. This article focuses on a chance-constrained version of the minimum spanning k-core problem under probabilistic edge failures. We first show that this probabilistic version is NP-hard, and we conduct a polyhedral study to strengthen the formulation. The quality of bounds produced by the strengthened formulation is demonstrated through a computational study.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Andreas, A.K., Smith, J.C.: Mathematical programming algorithms for two-path routing problems with reliability considerations. INFORMS J. Comput. 20(4), 553–564 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andreas, A.K., Smith, J.C., Küçükyavuz, S.: Branch-and-price-and-cut algorithms for solving the reliable h-paths problem. J. Glob. Optim. 42(4), 443–466 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Balakrishnan, A., Altinkemer, K.: Using a hop-constrained model to generate alternative communication network design. ORSA J. Comput. 4(2), 192–205 (1992)

    Article  MATH  Google Scholar 

  4. Balasundaram, B.: Graph theoretic generalizations of clique: Optimization and extensions. Ph.D. thesis, Texas A&M University, College Station, Texas, USA (2007)

  5. Bendali, F., Diarrassouba, I., Mahjoub, A., Mailfert, J.: The \(k\) edge-disjoint 3-hop-constrained paths polytope. Discrete Optim. 7(4), 222–233 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Beraldi, P., Bruni, M.E., Guerriero, F.: Network reliability design via joint probabilistic constraints. IMA J. Manag. Math. 21, 213–226 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Botton, Q., Fortz, B., Gouveia, L., Poss, M.: Benders decomposition for the hop-constrained survivable network design problem. INFORMS J. Comput. 25(1), 13–26 (2013)

    Article  MathSciNet  Google Scholar 

  8. Brueckner, J.K., Spiller, P.T.: Competition and mergers in airline hub-and-spoke networks. Int. J. Ind. Organ. 9, 323–342 (1991)

    Article  Google Scholar 

  9. Carvalho, M., Sorokin, A., Boginski, V., Balasundaram, B.: Topology design for on-demand dual-path routing in wireless networks. Optim. Lett. 7(4), 695–707 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dave, D.B.: Patterns of freight flow and design of a less-than-truckload distribution network. Ph.D. thesis, Georgia Institute of Technology (2004)

  11. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)

    MATH  Google Scholar 

  12. Jaillet, P., Song, G., Yu, G.: Airline network design and hub location problems. Locat. Sci. 4(3), 195–212 (1996)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Kim, Y., Chen, Y.S., Linderman, K.: Supply network disruption and resilience: a network structural perspective. J. Oper. Manag. 33–34, 43–59 (2015)

    Article  Google Scholar 

  15. Koster, A.M., Kutschka, M., Raack, C.: Robust network design: formulations, valid inequalities, and computations. Networks 61(2), 128–149 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Küçükyavuz, S.: On mixing sets arising in chance-constrained programming. Math. Program. 132(1), 31–56 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu, X., Kılınç-Karzan, F., Küçükyavuz, S.: On intersection of two mixing sets with applications to joint chance-constrained programs. Math. Program. (2018). https://doi.org/10.1007/s10107-018-1231-2

  18. Luedtke, J.: An integer programming and decomposition approach to general chance-constrained mathematical programs. In: Eisenbrand, F., Shepherd, F. (eds.) Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, vol. 6080, pp. 271–284. Springer, Berlin (2010)

    Chapter  Google Scholar 

  19. Luedtke, J.: A branch-and-cut decomposition algorithm for solving chance-constrained mathematical programs with finite support. Math. Program. 146(1–2), 219–244 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Luedtke, J., Ahmed, S., Nemhauser, G.: An integer programming approach for linear programs with probabilistic constraints. Math. Program. 12(2), 247–272 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ma, J., Balasundaram, B.: Solving chance-constrained spanning \(k\)-core problem via decomposition and integer programming. In: Proceedings of the 2013 Industrial and Systems Engineering Research Conference (ISERC 2013), pp. 2774–2783. Institute of Industrial Engineers, Norcross, GA (2013)

  22. Ma, J., Pajouh, F.M., Balasundaram, B., Boginski, V.: The minimum spanning \(k\)-core problem with bounded CVaR under probabilistic edge failures. INFORMS J. Comput. 28(2), 295–307 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Mahjoub, A.R., Simonetti, L., Uchoa, E.: Hop-level flow formulation for the survivable network design with hop constraints problem. Networks 61(2), 171–179 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Minoux, M.: Network synthesis and optimum network design problems: models, solution methods, and applications. Networks 19, 313–360 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  26. Monma, C.L., Shallcross, D.F.: Methods for designing communications networks with certain two-connected survivability constraints. Oper. Res. 37(4), 531–541 (1989)

    Article  Google Scholar 

  27. Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1999)

    MATH  Google Scholar 

  28. O’Kelly, M.E., Miller, H.J.: The hub network design problem: a review and synthesis. J. Transp. Geogr. 2(1), 31–40 (1994)

    Article  MathSciNet  Google Scholar 

  29. Pastukhov, G., Veremyev, A., Boginski, V., Pasiliao, E.L.: Optimal design and augmentation of strongly attack-tolerant two-hop clusters in directed networks. J. Comb. Optim. 27(3), 462–486 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rodríguez-Martín, I., Salazar-González, J.J., Yaman, H.: A branch-and-cut algorithm for two-level survivable network design problems. Comput. Oper. Res. 67, 102–112 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Seidman, S.B.: Network structure and minimum degree. Soc. Netw. 5(3), 269–287 (1983)

    Article  MathSciNet  Google Scholar 

  32. Smith, J.C., Lim, C., Sudargho, F.: Survivable network design under optimal and heuristic interdiction scenarios. J. Glob. Optim. 38(2), 181–199 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Song, Y., Luedtke, J.R.: Branch-and-cut approaches for chance-constrained formulations of reliable network design problems. Math. Program. Comput. 5(4), 397–432 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sorokin, A., Boginski, V., Nahapetyan, A., Pardalos, P.M.: Computational risk management techniques for fixed charge network flow problems with uncertain arc failures. J. Comb. Optim. 25(1), 99–122 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Veremyev, A., Boginski, V.: Robustness and strong attack tolerance of low-diameter networks. In: Sorokin, A., Murphey, R., Thai, M.T., Pardalos, P.M. (eds.) Dynamics of Information Systems: Mathematical Foundations, Springer Proceedings in Mathematics & Statistics, vol. 20, pp. 137–156. Springer, New York (2012)

    Google Scholar 

  36. Zheng, Q.P., Shen, S., Shi, Y.: Loss-constrained minimum cost flow under arc failure uncertainty with applications in risk-aware kidney exchange. IIE Trans. 47(9), 961–977 (2015)

    Article  Google Scholar 

Download references

Acknowledgements

The computational experiments reported in this article were conducted at the Oklahoma State University High Performance Computing Center. The authors would also like to thank Foad Mahdavi Pajouh for helpful discussions regarding the complexity and lifting results.

Author information

Authors and Affiliations

  1. Turner Broadcasting System, Inc., Atlanta, GA, 30318, USA

    Juan Ma

  2. School of Industrial Engineering and Management, Oklahoma State University, 322 Engineering North, Stillwater, OK, 74078, USA

    Balabhaskar Balasundaram

Authors
  1. Juan Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Balabhaskar Balasundaram
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Balabhaskar Balasundaram.

Additional information

The authors gratefully acknowledge the support of the Air Force Office of Scientific Research Grant FA9550-12-1-0103, the National Science Foundation Grant CMMI-1404971, and the Air Force Research Laboratory-Mathematical Modeling and Optimization Institute.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ma, J., Balasundaram, B. On the chance-constrained minimum spanning k-core problem. J Glob Optim 74, 783–801 (2019). https://doi.org/10.1007/s10898-018-0714-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-018-0714-2

Keywords

Search

Navigation