Approximate k-Steiner Forests Via the Lagrangian Relaxation Technique with Internal Preprocessing

  • Danny Segev
  • Gil Segev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4168)


An instance of the k-Steiner forest problem consists of an undirected graph G = (V,E), the edges of which are associated with non-negative costs, and a collection \({\cal D} = \{ (s_i, t_i) : 1 \leq i \leq d \}\) of distinct pairs of vertices, interchangeably referred to as demands. We say that a forest \({\cal F} \subseteq G\) connects a demand (s i , t i ) when it contains an s i -t i path. Given a requirement parameter \(k \leq |{\cal D}|\), the goal is to find a minimum cost forest that connects at least k demands in \({\cal D}\). This problem has recently been studied by Hajiaghayi and Jain [SODA ’06], whose main contribution in this context was to relate the inapproximability of k-Steiner forest to that of the dense k -subgraph problem. However, Hajiaghayi and Jain did not provide any algorithmic result for the respective settings, and posed this objective as an important direction for future research.

In this paper, we present the first non-trivial approximation algorithm for the k-Steiner forest problem, which is based on a novel extension of the Lagrangian relaxation technique. Specifically, our algorithm constructs a feasible forest whose cost is within a factor of \(O( {\rm min} \{ n^{ 2/3 }, \sqrt{d} \} \cdot \log d )\) of optimal, where n is the number of vertices in the input graph and d is the number of demands.


Steiner Tree Lagrangian Relaxation Input Graph Forest Problem Steiner Forest 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agrawal, A., Klein, P.N., Ravi, R.: When trees collide: An approximation algorithm for the generalized Steiner problem on networks. SIAM Journal on Computing 24(3), 440–456 (1995)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arora, S., Karakostas, G.: A 2 + ε approximation algorithm for the k-MST problem. In: 11th SODA, pp. 754–759 (2000)Google Scholar
  3. 3.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. Journal of the ACM 45(3), 501–555 (1998)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Asahiro, Y., Iwama, K., Tamaki, H., Tokuyama, T.: Greedily finding a dense subgraph. Journal of Algorithms 34(2), 203–221 (2000)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bienstock, D., Goemans, M.X., Simchi-Levi, D., Williamson, D.P.: A note on the prize collecting traveling salesman problem. Mathematical Programming 59, 413–420 (1993)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Blum, A., Ravi, R., Vempala, S.: A constant-factor approximation algorithm for the k-MST problem. Journal of Computer and System Sciences 58(1), 101–108 (1999)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Feige, U., Kortsarz, G., Peleg, D.: The dense k-subgraph problem. Algorithmica 29(3), 410–421 (2001)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Feige, U., Langberg, M.: Approximation algorithms for maximization problems arising in graph partitioning. Journal of Algorithms 41(2), 174–211 (2001)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Garg, N.: A 3-approximation for the minimum tree spanning k vertices. In: 37th FOCS, pp. 302–309 (1996)Google Scholar
  10. 10.
    Garg, N.: Saving an epsilon: A 2-approximation for the k-MST problem in graphs. In: 37th STOC, pp. 396–402 (2005)Google Scholar
  11. 11.
    Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM Journal on Computing 24(2), 296–317 (1995)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hajiaghayi, M., Jain, K.: The prize-collecting generalized Steiner tree problem via a new approach of primal-dual schema. In: 17th SODA, pp. 631–640 (2006)Google Scholar
  13. 13.
    Han, Q., Ye, Y., Zhang, J.: An improved rounding method and semidefinite programming relaxation for graph partition. Mathematical Programming 92(3), 509–535 (2002)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Levin, A., Segev, D.: Partial multicuts in trees. In: Erlebach, T., Persinao, G. (eds.) WAOA 2005. LNCS, vol. 3879, pp. 320–333. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. Journal of Computer and System Sciences 43(3), 425–440 (1991)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Ravi, R., Goemans, M.X.: The constrained minimum spanning tree problem (extended abstract). In: Karlsson, R., Lingas, A. (eds.) SWAT 1996. LNCS, vol. 1097, pp. 66–75. Springer, Heidelberg (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Danny Segev
    • 1
  • Gil Segev
    • 2
  1. 1.School of Mathematical SciencesTel-Aviv UniversityTel-AvivIsrael
  2. 2.Department of Computer Science and Applied MathematicsWeizmann Institute of ScienceRehovotIsrael

Personalised recommendations