Journal of Combinatorial Optimization

, Volume 16, Issue 4, pp 402–423 | Cite as

Packing trees in communication networks

  • Mohamed Saad
  • Tamás Terlaky
  • Anthony Vannelli
  • Hu Zhang
Article

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.

Keywords

Approximation algorithms Mathematical programming Steiner tree packing Communication networks Multicast routing Wavelength assignment 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Mohamed Saad
    • 1
  • Tamás Terlaky
    • 2
  • Anthony Vannelli
    • 3
  • Hu Zhang
    • 4
  1. 1.Department of Electrical and Computer EngineeringUniversity of SharjahSharjahUnited Arab Emirates
  2. 2.School of Computational Engineering and Science, Department of Computing and SoftwareMcMaster UniversityHamiltonCanada
  3. 3.College of Physical and Engineering SciencesUniversity of GuelphGuelphCanada
  4. 4.Canadian Imperial Bank of CommerceTorontoCanada

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