Parameterized Analysis of the Online Priority and Node-Weighted Steiner Tree Problems

  • Spyros AngelopoulosEmail author


In this paper we study the online variant of two well-known Steiner tree problems. In the online setting, the input consists of a sequence of terminals; upon arrival of a terminal, the online algorithm must irrevocably buy a subset of edges and vertices of the graph so as to guarantee the connectivity of the currently revealed part of the input. More precisely, we first study the online node-weighted Steiner tree problem, in which both edges and vertices are weighted, and the objective is to minimize the total cost of edges and vertices in the solution. We then address the online priority Steiner tree problem, in which each edge and each request are associated with a priority value, which corresponds to their bandwidth support and requirement, respectively. Both problems have applications in the domain of multicast network communications and have been studied from the point of view of approximation algorithms. Motivated by the observation that competitive analysis gives very pessimistic and unsatisfactory results when the only relevant parameter is the number of terminals, we introduce an approach based on parameterized analysis of online algorithms. In particular, we base the analysis on additional parameters that help reveal the true complexity of the underlying problem, and allow a much finer classification of online algorithms based on their performance. More specifically, for the online node-weighted Steiner tree problem, we show a tight bound of Θ(max{min{α,k},log k}) on the competitive ratio, where α is the ratio of the maximum node weight to the minimum node weight and k is the number of terminals. For the online priority Steiner tree problem, we show corresponding tight bounds of \({\Theta }(b\log \frac {k}{b})\), when k > b and Θ(k), when kb, where b is the number of priority levels and k is the number of terminals. Our main results apply to both deterministic and randomized algorithms, as well as to generalized versions of the problems (i.e., to Steiner forest variants).


Online algorithms Competitive analysis Steiner tree problems Multicasting Models of communication networks 



  1. 1.
    Agrawal, A., Klein, P.N., Ravi, R.: When trees collide: an approximation algorithm for the generalized steiner tree problem on networks. SIAM J. Comput. 24, 440–456 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, J.: A general approach to online network optimization problems. Trans. Algorithm. 2(4), 640–660 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alon, N., Azar, Y.: On-line Steiner trees in the Euclidean plane. Discret. Comput. Geom. 10, 113–121 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Angelopoulos, S.: Improved bounds for the online steiner tree problem in graphs of bounded edge-asymmetry. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 248–257 (2007)Google Scholar
  5. 5.
    Angelopoulos, S.: A near-tight bound for the online steiner tree problem in graphs of bounded asymmetry. In: Proceedings of the 16th Annual European Symposium on Algorithms (ESA), pp. 76–87 (2008)Google Scholar
  6. 6.
    Angelopoulos, S.: Ion the competitiveness of the online asymmetric and euclidean steiner tree problems. In: Proceedings of the 7th International Workshop on Approximation and Online Algorithms (WAOA), pp. 1–12 (2009)Google Scholar
  7. 7.
    Awerbuch, B., Azar, Y., Bartal, Y.: On-line generalized Steiner problem. Theor. Comput. Sci. 324(2–3), 313–324 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Basu, P., Chau, C., Gibbens, R.J., Guha, S., Irwin, R.E.: Multicasting under multi-domain and hierarchical constraints. In: 11th International Symposium and Workshops on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks, pp. 524–531 (2013)Google Scholar
  9. 9.
    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
  10. 10.
    Bern, M., Plassmann, P.: The Steiner problem with edge lengths 1 and 2. Inf. Process. Lett. 32, 171–176 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Borodin, A., El-Yaniv, R.: Online computation and competitive analysis. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  12. 12.
    Byrka, J., Grandoni, F., Rothvoß, T., Sanitȧ, L.: Steiner tree approximation via iterative randomized rounding. J. ACM 60(1), 6 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Charikar, M., Chekuri, C., Cheung, T., Dai, Z., Goel, A., Guha, S., Li, M.: Approximation algorithms for directed steiner problems. J. Algorithm. 1 (33), 73–91 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Charikar, M., Naor, J., Schieber, B.: Resource optimization in QoS multicast routing of real-time multimedia. IEEE/ACM Trans. Netw. 12(2), 340–348 (2004)CrossRefGoogle Scholar
  15. 15.
    Chuzhoy, J., Gupta, A., Naor, J., Sinha, A.: On the approximability of some network design problems. Transactions on Algorithms 4(2), 23:1–23:17 (2008)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Claffy, K., Polyzos, G., Braun, H.W.: Traffic characteristics of the t1 nsfnet backbone. In: Proceedings of INFOCOM (1993)Google Scholar
  17. 17.
    Faloutsos, M.: The Greedy the Naive and the Optimal Multicast Routing–From Theory to Internet Protocols. PhD thesis, University of Toronto (1998)Google Scholar
  18. 18.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Garg, N., Gupta, A., Leonardi, S., Sankowski, P.: Stochastic analyses for online combinatorial optimization problems. In: Proceedings of the Nineteenth Annual ACM-SIAM, Symposium on Discrete Algorithms, (SODA), pp. 942–951 (2008)Google Scholar
  21. 21.
    Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM Journal on Computing 24(2), 296–317 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gu, A., Gupta, A., Kumar, A.: The power of deferral: maintaining a constant-competitive steiner tree online. In: Proceedings of the 45th Symposium on Theory of Computing Conference (STOC), pp. 525–534 (2013)Google Scholar
  23. 23.
    Gu, A., Gupta, A., Kumar, A.: The power of deferral: maintaining a constant-competitive steiner tree online. SIAM J. Comput. 45(1), 1–28 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Guha, S., Khuller, S.: Improved methods for approximating node weighted steiner trees and connected dominating sets. Information and Computation 150(1), 57–74 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gupta, A., Kumar, A.: Online steiner tree with deletions. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM, Symposium on Discrete Algorithms (SODA), pp. 455–467 (2014)Google Scholar
  26. 26.
    Hajiaghayi, M.T., Liaghat, V., Panigrahi, D.: Online node-weighted steiner forest and extensions via disk paintings (2013)Google Scholar
  27. 27.
    Imase, M., Waxman, B.: The dynamic Steiner tree problem. SIAM J. Discret. Math. 4(3), 369–384 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp 85–103. Springer, Boston (1972)CrossRefGoogle Scholar
  29. 29.
    Klein, P., Ravi, R.: A nearly best-possible approximation algorithm for node-weighted steiner trees. Journal of Algorithms 19(1), 104–115 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lai, K.J., Gomes, C.P., Schwartz, M.K., McKelvey, Kevin S., Calkin, D.E., Montgomery, C.A.: The steiner multigraph problem: Wildlife corridor design for multiple species. In: Proceedings of the Twenty-Fifth AAAI Conference on Artificial Intelligence (AAAI) (2011)Google Scholar
  31. 31.
    Naor, J., Panigrahi, D., Singh, M.: Online node-weighted steiner tree and related problems. In: IEEE, 52nd Annual Symposium on Foundations of Computer Science, FOCS, pp. 210–219 (2011)Google Scholar
  32. 32.
    Oliveira, C.A.S., Pardalos, P.M.: A survey of combinatorial optimization problems in multicast routing. Comput. Oper Res. 32(8), 1953–1981 (2005)CrossRefzbMATHGoogle Scholar
  33. 33.
    Ramanathan, S.: Multicast tree generation in networks with asymmetric links. IEEE/ACM Trans. Netw. 4(4), 558–568 (1996)CrossRefGoogle Scholar
  34. 34.
    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
  35. 35.
    Thimm, M.: On the approximability of the Steiner tree problem. Theor. Comput. Sci. 295(1), 387–402 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    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
  37. 37.
    Vazirani, V.: Approximation algorithms. Springer, Berlin (2001)zbMATHGoogle Scholar
  38. 38.
    Westbrook, J., Yan, D.C.K.: Linear bounds for on-line Steiner problems. Inf. Process. Lett. 55(2), 59–63 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Westbrook, J., Yan, D.C.K.: The performance of greedy algorithms for the on-line Steiner tree and related problems. Math. Syst. Theory 28(5), 451–468 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    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

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Sorbonne Université, CNRS, Laboratoire d’Informatique de Paris 6ParisFrance

Personalised recommendations