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Packing trees in communication networks

Author

Listed:
  • Mohamed Saad

    (University of Sharjah)

  • Tamás Terlaky

    (McMaster University)

  • Anthony Vannelli

    (University of Guelph)

  • Hu Zhang

    (Canadian Imperial Bank of Commerce)

Abstract

Given an undirected edge-capacitated graph and given (possibly) different subsets of vertices, we consider the problem of selecting a maximum (weighted) set of Steiner trees, each tree spanning a subset of vertices, without violating the capacity constraints. This problem is motivated by applications in multicast communication networks. We give an integer linear programming (ILP) formulation for the problem, and observe that its linear programming (LP) relaxation is a fractional packing problem with exponentially many variables and a block (sub-)problem that cannot be solved in polynomial time. To this end, we take an r-approximate block solver (a weak block solver) to develop a (1−ε)/r-approximation algorithm for the LP relaxation. The algorithm has a polynomial coordination complexity for any ε∈(0,1). To the best of our knowledge, this is the first approximation result for fractional packing problems with only weak block solvers (with arbitrarily large approximation ratio) and a coordination complexity that is polynomial in the input size. This leads also to an approximation algorithm for the underlying tree packing problem. Finally, we extend our results to an important multicast routing and wavelength assignment problem in optical networks, where each Steiner tree is to be assigned one of a limited set of given wavelengths, so that trees crossing the same fiber are assigned different wavelengths.

Suggested Citation

  • Mohamed Saad & Tamás Terlaky & Anthony Vannelli & Hu Zhang, 2008. "Packing trees in communication networks," Journal of Combinatorial Optimization, Springer, vol. 16(4), pages 402-423, November.
    • Handle: RePEc:spr:jcomop:v:16:y:2008:i:4:d:10.1007_s10878-008-9150-4
    DOI: 10.1007/s10878-008-9150-4
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    References listed on IDEAS

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    1. Guohui Lin, 2005. "An Improved Approximation Algorithm for Multicast k-Tree Routing," Journal of Combinatorial Optimization, Springer, vol. 9(4), pages 349-356, June.
    2. Serge A. Plotkin & David B. Shmoys & Éva Tardos, 1995. "Fast Approximation Algorithms for Fractional Packing and Covering Problems," Mathematics of Operations Research, INFORMS, vol. 20(2), pages 257-301, May.
    3. Michael D. Grigoriadis & Leonid G. Khachiyan, 1996. "Coordination Complexity of Parallel Price-Directive Decomposition," Mathematics of Operations Research, INFORMS, vol. 21(2), pages 321-340, May.
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