Steiner Problems with Limited Number of Branching Nodes

  • Dimitri Watel
  • Marc-Antoine Weisser
  • Cédric Bentz
  • Dominique Barth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8179)


Given an undirected weighted graph G with n nodes, the k-Undirected Steiner Tree problem is to find a minimum cost tree spanning a specified set of k nodes. If this problem and its directed version have several applications in multicast routing in packet switching networks, the modeling is not adapted anymore in networks based upon the circuit switching principle in which not all nodes are able to duplicate packets. In such networks, the number of branching nodes (with outdegree > 1) in the multicast tree must be limited.

We introduce the (k,p) −Steiner Tree with Limited Number of Branching nodes problems where the goal is to find an optimal Steiner tree with at most p branching nodes. We study, when p is fixed, its complexity depending on two criteria: the graph topology and the parameter k. In particular, we propose a polynomial algorithm when the input graph is acyclic and an other algorithm when k is fixed in an input graph of bounded treewidth. Moreover, in directed graphs where p ≤ k − 2, or in planar graphs, we provide an n ε -inapproximability proof, for any ε < 1.


graph algorithm parameterized complexity Steiner tree 


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  1. 1.
    Cheng, X., Du, D.Z.: Steiner trees in industry, vol. 11. Kluwer (2001)Google Scholar
  2. 2.
    Voß, S.: Steiner tree problems in telecommunications, pp. 459–492 (January 2006)Google Scholar
  3. 3.
    Rugeli, J., Novak, R.: Steiner tree algorithms for multicast protocols (1995)Google Scholar
  4. 4.
    Reinhard, V., Tomasik, J., Barth, D., Weisser, M.-A.: Bandwidth Optimization for Multicast Transmissions in Virtual Circuit Networks. In: Fratta, L., Schulzrinne, H., Takahashi, Y., Spaniol, O. (eds.) NETWORKING 2009. LNCS, vol. 5550, pp. 859–870. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Reinhard, V., Cohen, J., Tomasik, J., Barth, D., Weisser, M.A.: Optimal configuration of an optical network providing predefined multicast transmissions. Comput. Netw. 56(8), 2097–2106 (2012)CrossRefGoogle Scholar
  6. 6.
    Gargano, L., Hell, P., Stacho, L., Vaccaro, U.: Spanning trees with bounded number of branch vertices. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 355–365. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Salazar-González, J.J.: The Steiner cycle polytope. EJOR 147(3), 671–679 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Steinová, M.: Approximability of the Minimum Steiner Cycle Problem. Computing and Informatics 29(6+), 1349–1357 (2010)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphism problem. Theoretical Computer Science 10(111), 111–121 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Robertson, N., Seymour, P.: The disjoint paths problem. Journal of Combinatorial Theory, Series B, 65–110 (1995)Google Scholar
  11. 11.
    Schrijver, A.: Finding k disjoint paths in a directed planar graph. SIAM Journal on Computing, 1–10 (1994)Google Scholar
  12. 12.
    Scheffler, P.: A Practical Linear Time Algorithm for Disjoint Paths in Graphs with Bounded Tree Width. Technical Report 396/1994, Fachbereich Mathematik (1994)Google Scholar
  13. 13.
    Kou, L., Markowsky, G., Berman, L.: A fast algorithm for steiner trees. Acta informatica 15(2), 141–145 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Zelikovsky, A.: An 11/6-approximation algorithm for the network steiner problem. Algorithmica 9(5), 463–470 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hougardy, S., Prömel, H.: A 1.598 approximation algorithm for the Steiner problem in graphs. In: Proc. SODA, pp. 448–453 (1999)Google Scholar
  16. 16.
    Du, D., Lu, B., Ngo, H., Pardalos, P.: Steiner tree problems. Encyclopedia of Optimization 5, 227–290 (2000)Google Scholar
  17. 17.
    Hsu, T.S., Tsai, K., Wang, D., Lee, D.: Steiner problems on directed acyclic graphs. Computing and Combinatorics, 21–30 (1996)Google Scholar
  18. 18.
    Charikar, M., et al.: Approximation algorithms for directed steiner problems. In: Proc. SODA, pp. 192–200 (1998)Google Scholar
  19. 19.
    Ming-IHsieh, E., Tsai, M.: Fasterdsp: A faster approximation algorithm for directed steiner tree problem. JISE 22, 1409–1425 (2006)MathSciNetGoogle Scholar
  20. 20.
    Rothvoß, T.: Directed steiner tree and the lasserre hierarchy. CoRR abs/1111.5473 (2011)Google Scholar
  21. 21.
    Feige, U.: A threshold of ln n for approximating set cover. J. of the ACM 45(4), 634–652 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Halperin, E., Krauthgamer, R.: Polylogarithmic inapproximability. In: Proc. STOC, pp. 585–594. ACM (2003)Google Scholar
  23. 23.
    Garg, N., Konjevod, G., Ravi, R.: A polylogarithmic approximation algorithm for the group steiner tree problem. In: Proc. SODA, pp. 253–259 (1998)Google Scholar
  24. 24.
    Ding, B., Yu, J.X., Wang, S., Qin, L., Zhang, X., Lin, X.: Finding top-k min-cost connected trees in databases. In: Chirkova, R., Dogac, A., Özsu, M.T., Sellis, T.K. (eds.) ICDE, pp. 836–845. IEEE (2007)Google Scholar
  25. 25.
    Cheriyan, J., Laekhanukit, B., Naves, G., Vetta, A.: Approximating rooted steiner networks. In: Proc. SODA, pp. 1499–1511 (2012)Google Scholar
  26. 26.
    Watel, D., Weisser, M.A., Bentz, C.: Inapproximability proof of DSTLB and USTLB in planar graphs,
  27. 27.
    Downey, R.G., Fellows, M.R.: Parameterized complexity, vol. 3. Springer (1999)Google Scholar
  28. 28.
    Goldberg, A.V., Tarjan, R.E.: Finding minimum-cost circulations by canceling negative cycles. Journal of the ACM 36(4), 873–886 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Dimitri Watel
    • 1
    • 3
  • Marc-Antoine Weisser
    • 1
  • Cédric Bentz
    • 2
  • Dominique Barth
    • 3
  1. 1.Computer Science Dpt.SUPELEC System SciencesGif sur YvetteFrance
  2. 2.CEDRIC-CNAMParisFrance
  3. 3.University of VersaillesFrance

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