An Improved Algorithm for Fixed-Hub Single Allocation Problems

  • Dong-Dong Ge
  • Zi-Zhuo Wang
  • Lai Wei
  • Jia-Wei Zhang


This paper discusses the fixed-hub single allocation problem (FHSAP). In this problem, a network consists of hub nodes and terminal nodes. Hubs are fixed and fully connected; each terminal node is assigned to a single hub which routes all its traffic. The goal is to minimize the cost of routing the traffic in the network. In this paper, we propose a new linear programming (LP) relaxation for this problem by incorporating a set of validity constraints into the classical formulations by Ernst and Krishnamoorthy (Locat Sci 4:139–154, Ann Op Res 86:141–159). A geometric rounding algorithm is then used to obtain an integral solution from the fractional solution. We show that by incorporating the validity constraints, the strengthened LP often provides much tighter upper bounds than the previous methods with a little more computational effort and the solution obtained often has a much smaller gap with the optimal solution. We also formulate a robust version of the FHSAP and show that it can guard against data uncertainty with little costs.


Hub location Network design Linear programming Worst-case analysis 

Mathematics Subject Classification




The authors would like to thank anonymous referees for their helpful comments.


  1. 1.
    Campbell, J., Ernst, A., Krishnamoorthy, M.: Hub location problems. In: Drezner, Z., Hamcher, H.W. (eds.) Facility Location: Application and Theory. Springer, Berlin (2002)Google Scholar
  2. 2.
    Sohn, J., Park, S.: The single-allocation problem in the interacting three-hub network. Networks 35(1), 17–25 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    O’Kelly, M.: A quadratic integer program for the location of interacting hub facilities. Eur. J. Oper. Res. 33, 393–402 (1987)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Klincewicz, J.: Heuristics for the \(p\)-hub location problem. Eur. J. Oper. Res. 53, 25–37 (1991)CrossRefMATHGoogle Scholar
  5. 5.
    Campbell, J.: Hub location and the \(p\)-hub median problem. Oper. Res. 44, 925–935 (1996)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Skorin-Kapov, D., Skorin-Kapov, J., O’kelly, M.: Tight linear programming relaxations of uncapacitated \(p\)-hub median problems. Eur. J. Oper. Res. 94, 582–593 (1996)CrossRefMATHGoogle Scholar
  7. 7.
    Ernst, A., Krishnamoorthy, M.: Efficient algorithms for the uncapacitated single allocation \(p\)-hub median problem. Locat. Sci. 4(3), 139–154 (1996)CrossRefMATHGoogle Scholar
  8. 8.
    Ernst, A., Krishnamoorthy, M.: Solution algorithms for the capacitated single allocation hub location problem. Ann. Oper. Res. 86, 141–159 (1999)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Campbell, J.: Integer programming formulation of discrete hub location problems. Eur. J. Oper. Res. 72, 387–405 (1994)CrossRefMATHGoogle Scholar
  10. 10.
    O’Kelly, M., Bryan, D., Skorin-Kapov, D., Skorin-Kapov, J.: Hub network design with single and multiple allocation: a computational study. Locat. Sci. 4, 125–138 (1996)CrossRefMATHGoogle Scholar
  11. 11.
    O’Kelly, M., Skorin-Kapov, D., Skorin-Kapov, J.: Lower bounds for the hub location problem. Manag. Sci. 41, 713–721 (1995)CrossRefMATHGoogle Scholar
  12. 12.
    Ge, D., He, S., Ye, Y., Zhang, J.: Geometric rounding: dependent randomized rounding scheme. J. Comb. Optim. 22(4), 699–725 (2011)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kleinberg, J., Tardos, E.: Approximation algorithms for classification problems with pairwise relationships: metric labeling and markov random fields. J. ACM 49(5), 616–639 (2002)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)CrossRefMATHGoogle Scholar

Copyright information

© Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Information Management and EngineeringShanghai University of Finance and EconomicsShanghaiChina
  2. 2.Department of Industrial and Systems EngineeringUniversity of MinnesotaMinneapolisUSA
  3. 3.Stephen M. Ross School of BusinessUniversity of MichiganAnn ArborUSA
  4. 4.Leonard N. Stern School of BusinessNew York UniversityNew YorkUSA

Personalised recommendations