Hardness and Approximation Results for Packing Steiner Trees

  • Joseph Cheriyan
  • Mohammad R. Salavatipour
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3221)


We study approximation algorithms and hardness of approximation for several versions of the problem of packing Steiner trees. For packing edge-disjoint Steiner trees of undirected graphs, we show APX-hardness for 4 terminals. For packing Steiner-node-disjoint Steiner trees of undirected graphs, we show a logarithmic hardness result, and give an approximation guarantee of \(O(\sqrt{n}\log n)\), where n denotes the number of nodes. For the directed setting (packing edge-disjoint Steiner trees of directed graphs), we show a hardness result of \(\Omega(m^{\frac{1}{3}-\epsilon})\) and give an approximation guarantee of \(O(m^{\frac{1}{2}+\epsilon})\), where m denotes the number of edges. The paper has several other results.


Approximation Algorithm Terminal Node Steiner Tree Hardness Result Steiner Tree Problem 
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.
    Arora, S., Lund, C.: Hardness of approximations. In: Hochbaum, D. (ed.) Approximation Algorithms for NP-hard Problems, PWS Publishing (1996)Google Scholar
  2. 2.
    Baveja, A., Srinivasan, A.: Approximation algorithms for disjoint paths and related routing and packing problems. Mathematics of Operations Research 25, 255–280 (1997); Earlier version in FOCS 1997 (1997)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Charikar, M., Chekuri, C., Cheung, T., Dai, Z., Goel, A., Guha, S., Li, M.: Approximation algorithms for directed Steiner problem. J. Algorithms 33(1), 73–91 (1999); Earlier version in SODA 1998 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Charikar, M., Naor, J., Schieber, B.: Resource optimization in QoS multicast routing of real-time multimedia. In: Proc. 19th Annual IEEE INFOCOM (2000)Google Scholar
  5. 5.
    Diestel, R.: Graph Theory. Springer, New York (2000)Google Scholar
  6. 6.
    Feige, U., Halldorsson, M., Kortsarz, G., Srinivasan, A.: Approximating the domatic number. Siam J. Computing 32(1), 172–195 (2002); Earlier version in STOC 2000 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Floréen, P., Kaski, P., Kohonen, J., Orponen, P.: Multicast time maximization in energy constrained wireless networks. In: Proc. 2003 Joint Workshop on Foundations of Mobile Computing (DIALM-POMC 2003), San Diego, CA (2003)Google Scholar
  8. 8.
    Frank, A., Király, T., Kriesell, M.: On decomposing a hypergraph into k connected subhypergraphs. Discrete Applied Mathematics 131(2), 373–383 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Guruswami, V., Khanna, S., Rajaraman, R., Shepherd, B., Yannakakis, M.: Nearoptimal hardness results and approximation algorithms for edge-disjoint paths and related problems. J. Computer and System Sciences 67(3), 473–496 (2003); Earlier version in STOC 1999 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphism problem. Theoretical Computer Science 10(2), 111–121 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Grötschel, M., Martin, A., Weismantel, R.: The Steiner tree packing problem in VLSI design. Mathematical Programming 78, 265–281 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Guha, S., Khuller, S.: Improved methods for approximating node weighted Steiner trees and connected dominating sets. Information and Computation 150, 57–74 (1999); Earlier version in FST&TCS 1998 (1998)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Hazan, E., Safra, S., Schwartz, O.: On the hardness of approximating kdimensional matching, Electronic Colloqium on Computational Complexity, Rep. No. 20 (2003)Google Scholar
  14. 14.
    Jain, K., Mahdian, M., Salavatipour, M.R.: Packing Steiner trees. In: Proc. SODA 2003 (2003)Google Scholar
  15. 15.
    Kann, V.: Maximum bounded 3-dimensional matching is MAX SNP-complete. Information Processing Letters 37, 27–35 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kaski, P.: Packing Steiner trees with identical terminal sets. Information Processing Letters 91(1), 1–5 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kolliopoulos, S.G., Stein, C.: Approximating disjoint-path problems using packing integer programs. Mathematical Programming 99, 63–87 (2004); Earlier version in Bixby, R.E., Boyd, E.A., Ríos-Mercado, R.Z. (eds.): IPCO 1998. LNCS, vol. 1412. Springer, Heidelberg (1998)Google Scholar
  18. 18.
    Kratochvil, J., Tuza, Z.: On the complexity of bicoloring clique hypergraphs of graphs. J. Algorithms 45, 40–54 (2002); Earlier version in SODA 2000 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kriesell, M.: Edge-disjoint trees containing some given vertices in a graph. J. Combinatorial Theory (B) 88, 53–65 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Lau, L.C.: An approximate max-Steiner-tree-packing min-Steiner-cut theorem (2004) (manuscript)Google Scholar
  21. 21.
    Martin, A., Weismantel, R.: Packing paths and Steiner trees: Routing of electronic circuits. CWI Quarterly 6, 185–204 (1993)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Joseph Cheriyan
    • 1
  • Mohammad R. Salavatipour
    • 2
  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  2. 2.Department of Computing ScienceUniversity of AlbertaEdmontonCanada

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