Directed Steiner Tree with Branching Constraint

  • Dimitri Watel
  • Marc-Antoine Weisser
  • Cédric Bentz
  • Dominique Barth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8591)


Given a directed weighted graph G, a root r and k terminals, the k-Directed Steiner Tree problem is to find a minimum cost tree rooted at r and spanning all terminals. If this problem has 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 some nodes, called non diffusing nodes, are not able to duplicate packets. We define a more general problem, named Directed Steiner Tree with Limited number of Diffusing nodes (DSTLD), able to model the multicast in a network containing at most d diffusing nodes. We show that DSTLD is XP with respect to d, and use this result to build a \(\lceil \frac{k-1}{d} \rceil\)-approximation XP in d for DST. Finally, we prove that, under the assumption that NP \(\not\subseteq\) DTIME[n O(loglogn)], there is no polynomial approximation algorithm for DSTLD with ratio \(1+(\frac{1}{e} - \varepsilon) \cdot \frac{k}{d-1}\) for every constant ε > 0.


Directed Steiner Tree Approximation Diffusing node 


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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. In: Handbook of Optimization 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.
    Novak, R.: A note on distributed multicast routing in point-to-point networks. Computers & Operations Research, 1149–1164 (October 2001)Google Scholar
  5. 5.
    Karp, R.: Reducibility among combinatorial problems. Springer (1972)Google Scholar
  6. 6.
    Kou, L., Markowsky, G., Berman, L.: A fast algorithm for Steiner trees. Acta Informatica, 141–145 (1981)Google Scholar
  7. 7.
    Robins, G., Zelikovsky, A.: Improved Steiner tree approximation in graphs. In: Proc. SODA, pp. 770–779 (2000)Google Scholar
  8. 8.
    Feige, U.: A threshold of ln n for approximating set cover. JACM, 634–652 (1998)Google Scholar
  9. 9.
    Halperin, E., Krauthgamer, R.: Polylogarithmic inapproximability. In: Proc. STOC, pp. 585–594. ACM (2003)Google Scholar
  10. 10.
    Charikar, M., Chekuri, C., Cheung, T., Dai, Z.: Approximation algorithms for directed Steiner problems. In: Proc. SODA, pp. 192–200 (1998)Google Scholar
  11. 11.
    Zelikovsky, A.: A series of approximation algorithms for the acyclic directed Steiner tree problem. Algorithmica, 99–110 (1997)Google Scholar
  12. 12.
    Helvig, C., Robins, G., Zelikovsky, A.: An improved approximation scheme for the group Steiner problem. Networks (2001)Google Scholar
  13. 13.
    Dreyfus, S.E., Wagner, R.A.: The steiner problem in graphs. Networks 1(3), 195–207 (1971)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Ding, B., Yu, J.X., Wang, S., Qin, L.: Finding top-k min-cost connected trees in databases. In: ICDE (2007)Google Scholar
  15. 15.
    Downey, R.G., Fellows, M.R.: Parameterized complexity. Monographs in computer science edn. Springer (1999)Google Scholar
  16. 16.
    Jones, M., Lokshtanov, D., Ramanujan, M.S., Saurabh, S., Suchý, O.: Parameterized complexity of directed steiner tree on sparse graphs. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 671–682. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  17. 17.
    Malli, R., Zhang, X., Qiao, C.: Benefit of Multicasting in All-Optical Networks. In: SPIE Proc. Conf. All-Optical Networking (1998)Google Scholar
  18. 18.
    Lin, H.-c., Wang, S.-W.: Splitter Placement in All-Optical WDM Networks. In: Global Telecommunications Conference (2005)Google Scholar
  19. 19.
    Du, H., Jia, X., Wang, F., Thai, M.Y., Li, Y.: A Note on Optical Network with Nonsplitting Nodes. JCO (2005)Google Scholar
  20. 20.
    Guo, L., Wu, W., Wang, F., Thai, M.: Approximation for Minimum Multicast Route in Optical Network with Nonsplitting Nodes. JCO (2005)Google Scholar
  21. 21.
    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
  22. 22.
    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
  23. 23.
    Watel, D., Weisser, M.-A., Bentz, C., Barth, D.: Steiner Problems with Limited Number of Branching Nodes. In: Moscibroda, T., Rescigno, A.A. (eds.) SIROCCO 2013. LNCS, vol. 8179, pp. 310–321. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  24. 24.
    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
  25. 25.
    Salazar-González, J.J.: The Steiner cycle polytope. European Journal of Operational Research 147(3), 671–679 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Steinová, M.: Approximability of the Minimum Steiner Cycle Problem (2010)Google Scholar
  27. 27.
    Tarjan, R.: Finding optimum branchings. Networks 7(1), 25–35 (1977)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Dimitri Watel
    • 1
    • 3
  • Marc-Antoine Weisser
    • 1
  • Cédric Bentz
    • 2
  • Dominique Barth
    • 3
  1. 1.SUPELEC System Sciences, Computer Science DPT.Gif Sur YvetteFrance
  2. 2.CEDRIC-CNAM 292Paris Cédex 03France
  3. 3.University of VersaillesVersaillesFrance

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