Telecommunication Systems

, Volume 6, Issue 1, pp 1–19 | Cite as

Design of capacitated networks with tree configurations

  • Kyungsik Lee
  • Kyungchul Park
  • Sungsoo Park
Article

Abstract

This paper considers the problem of designing a capacitated network with a tree configuration (CTP). For a given set of nodes with their capacities,k types of link facilities with various characteristics, and installation cost for connecting each pair of nodes using each type of link facility, the problem is to find a tree network which satisfies the given traffic requirements between all pairs of nodes and minimizes total installation cost. We formulate (CTP) as an integer programming problem using path variables. To solve the linear programming relaxation which has exponentially many variables, we develop a polynomial-time column generation procedure. Moreover, to tighten the formulation, an efficient preprocessing procedure is devised and some classes of valid inequalities are found. Using the results, we develop a branch- and-cut algorithm with column generation where an efficient branching rule is adopted. Computational results show that the algorithm can solve practically-sized problems to optimality within a reasonable time.

Keywords

Programming Problem Integer Programming Reasonable Time Column Generation Valid Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    D. Adolphson and T.C. Hu, Optimal linear ordering, SAIM J. Appl. Math. 25(1973)403–423.CrossRefGoogle Scholar
  2. [2]
    C. Barnhart, E.L. Johnson, G.L. Nemhauser, M.W.P. Savelsbergh and P.H. Vance, Branch- and-price: Column generation for solving huge integer programs,Mathematical Programming: State of the Art, eds. J.R. Birge and K.G. Murty (1994) pp. 186–207.Google Scholar
  3. [3]
    K.M. Chandy and T. Lo, The capacitated minimum spanning tree, Networks 3(1973)173–182.Google Scholar
  4. [4]
    B. Gavish, Topological design of centralized computer networks — formulation and algorithms, Networks 12(1982)355–377.Google Scholar
  5. [5]
    B. Gavish, Topological design of computer communication networks — the overall design problem, Eur. J. Oper. Res. 58(1992)149–172.CrossRefGoogle Scholar
  6. [6]
    B. Gavish, Augmented Lagrangian based algorithm for centralized network design, IEEE Trans. Commun. COM-33(1985)1247–1257.Google Scholar
  7. [7]
    P.C. Gilmore and R.E. Gomory, A linear programming approach to the cutting-stock problem, Oper. Res. 9(1961)849–859.Google Scholar
  8. [8]
    R.E. Gomory and T.C. Hu, Multi-terminal network flows, J. SIAM (1964)551–570.Google Scholar
  9. [9]
    D. Gusfield, Very simple method for all pairs network flow analysis, SIAM J. Comput. 19(1990)143–155.CrossRefGoogle Scholar
  10. [10]
    T.C. Hu, Optimum communication spanning tree, SIAM J. Comput. 3(1974)188–195.CrossRefGoogle Scholar
  11. [11]
    K.L. Jones, I.J. Lustig, J.M. Farvolden and W.B. Powell, Multicommodity network flowsd: The impact of formulation on decomposition, Math. Progr. 62(1993)95–117.CrossRefGoogle Scholar
  12. [12]
    F. Kaefer and J.S. Park, A method for interconnecting LANs with survivability considerations,Proc. 3rd Int. Conf. on Telecommunication Systems, Nashville, TN (1995) pp. 244–254.Google Scholar
  13. [13]
    A. Kershenbaum and R.R. Boorstyn, Centralized teleprocessing network design, Networks 13(1983)279–293.Google Scholar
  14. [14]
    A. Kersenbaum, P. Kermini and G. Grover, MENTOR: An algorithm for mesh network topological optimization and routing, IEEE Trans. Commun. COM-39(1991)503–513.CrossRefGoogle Scholar
  15. [15]
    E.L. Lawler,Combinatorial Optimization: Networks and Matroids (Holt, Rinehart and Winston, NY, 1976).Google Scholar
  16. [16]
    L. LeBlanc, J.S. Park, V. Sridhar and J. Kalvenes, Topology design and bridge-capacity assignment for interconnecting token ring LANs: A simulated annealing approach, Telecom. Syst., this issue.Google Scholar
  17. [17]
    T.L. Magnanti and P. Mirchandani, Shortest paths, single origin—destination network design, and associated polyhedra, Networks 23(1993)103–121.Google Scholar
  18. [18]
    T.L. Magnanti, P. Mirchandani and R. Vachani, Modeling and solving the two-facility capacitied network loading problem, to appear in Oper. Res.Google Scholar
  19. [19]
    G.L. Nemhauser, The age of optimization: Solving large-scale real-world problems, Oper. Res. 42(1994)5–13.Google Scholar
  20. [20]
    G.L. Nemhauser and S. Park, A polyhedral approach to edge coloring, Oper. Res. Lett. 10(1991) 315–322.CrossRefGoogle Scholar
  21. [21]
    G.L. Nemhauser and L.A. Wolsey,Integer and Combinatorial Optimization (Wiley, New York, 1988).Google Scholar
  22. [22]
    M. Padborg and G. Rinaldi, A branch- and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems, SIAM Rev. 33(1991)60–100.CrossRefGoogle Scholar
  23. [23]
    C.H. Papadimitriou, The complexity of the capacitated tree problem, Networks 8(1978)217–230.Google Scholar
  24. [24]
    K. Park, S. Kang and S. Park, An integer programming approach to the bandwidth packing problem, to appear in Manag. Sci.Google Scholar
  25. [25]
    M. Parker and J. Ryan, A column generation algorithm for bandwidth packing, Telecom. Syst. 2(1994)185–195.CrossRefGoogle Scholar
  26. [26]
    J.F. Shoch and J.A. Hupp, Measured performance of an ethernet local network, Commun. ACM 23(1980)711–721.CrossRefGoogle Scholar
  27. [27]
    V. Sridhar, J.S. Park and B. Gavish, Optimal interconnection of token ring networks using source routing bridges, Working Paper, Department of Management Sciences, University of Iowa, Iowa City, IA (1994).Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1996

Authors and Affiliations

  • Kyungsik Lee
    • 1
  • Kyungchul Park
    • 1
  • Sungsoo Park
    • 1
  1. 1.Department of Industrial EngineeringKorea Advanced Institute of Science and TechnologyTaejonKorea

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