The Node-Weighted Steiner Problem in Graphs of Restricted Node Weights

  • Spyros Angelopoulos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4059)


In this paper we study a variant of the Node-Weighted Steiner Tree problem in which the weights (costs) of vertices are restricted, in the sense that the ratio of the maximum node weight to the minimum node weight is bounded by a quantity α. This problem has applications in multicast routing where the cost of participating routers must be taken into consideration and the network is relatively homogenous in terms of the cost of the routers.

We consider both online and offline versions of the problem. For the offline version we show an upper bound of O( min {logα, logk}) on the approximation ratio of deterministic algorithms (where k is the number of terminals). We also prove that the bound is tight unless P=NP. For the online version we show a tight bound of Θ( max { min {α, k}, logk }), which applies to both deterministic and randomized algorithms. We also show how to apply (and extend to node-weighted graphs) recent work of Alon et al. so as to obtain a randomized online algorithm with competitive ratio O(logm logk), where m is the number of the edges in the graph, independently of the value of α. All our bounds also hold for the Generalized Node-Weighted Steiner Problem, in which only connectivity between pairs of vertices must be guaranteed.


Competitive Ratio Online Algorithm Steiner Tree Problem Node Weight Fractional Solution 
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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  1. 1.
    Agrawal, A., Klein, P.N., Ravi, R.: When trees collide: An approximation algorithm for the generalized steiner tree problem on networks. SIAM Journal on Computing 24, 440–456 (1995)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, J.: The online set cover problem. In: Proceedings of the 35th Annual ACM Symposium on the Theory of Computation, pp. 100–105 (2003)Google Scholar
  3. 3.
    Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, J.: A general approach to online network optimization problems. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 570–579 (2005)Google Scholar
  4. 4.
    Berman, P., Coulston, C.: Online algorithms for Steiner tree problems. In: Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, pp. 344–353 (1997)Google Scholar
  5. 5.
    Bern, M., Plassmann, P.: The Steiner problem with edge lengths 1 and 2. Information Processing Letters 32, 171–176 (1989)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Charikar, M., Chekuri, C., Cheung, T., Dai, Z., Goel, A., Guha, S., Li, M.: Approximation algorithms for directed steiner problems. Journal of Algorithms 1(33), 73–91 (1999)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Faloutsos, M.: The Greedy the Naive and the Optimal Multicast Routing–From Theory to Internet Protocols. PhD thesis, University of Toronto (1998)Google Scholar
  8. 8.
    Faloutsos, M., Pankaj, R., Sevcik, K.C.: The effect of asymmetry on the on-line multicast routing problem. Int. J. Found. Comput. Sci. 13(6), 889–910 (2002)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Feige, U.: A threshold of ln n for approximating set cover. Journal of the ACM 45(4), 634–652 (1998)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM Journal on Computing 6(24) (1995)Google Scholar
  11. 11.
    Guha, S., Khuller, S.: Improved methods for approximating node weighted steiner trees and connected dominating sets. Information and Computation (150), 228–248 (1999)Google Scholar
  12. 12.
    Imase, M., Waxman, B.: The dynamic Steiner tree problem. SIAM Journal on Discrte Mathematics 4(3), 369–384 (1991)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Karp, R.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103 (1972)Google Scholar
  14. 14.
    Klein, P., Ravi, R.: A nearly best-possible approximation algorithm for node-weighted steiner trees. Journal of Algorithms (19), 104–115 (1995)Google Scholar
  15. 15.
    Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pp. 475–484 (1997)Google Scholar
  16. 16.
    Robins, G., Zelikovsky, A.: Improved Steiner tree approximation in graphs. In: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 770–779 (2000)Google Scholar
  17. 17.
    Segev, A.: The node-weighted steiner tree problem. Networks (17), 1–17 (1987)Google Scholar
  18. 18.
    Thimm, M.: On the approximability of the Steiner tree problem. Theoretical Computer Science 295(1), 387–402 (2003)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Trevisan, L.: Non-approximability results for optimization problems on bounded degree instances. In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pp. 453–461 (2001)Google Scholar
  20. 20.
    Vazirani, V.: Approximation Algorithms. Springer, Heidelberg (2001)Google Scholar
  21. 21.
    Yao, A.C.-C.: Probabilistic computations:towards a unified measure of complexity. In: Proceedings of the 17th Annual IEEE Symposium on Foundations of Computer Science, pp. 222–227 (1997)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Spyros Angelopoulos
    • 1
  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

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