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Primal-Dual Approximation Algorithms for Node-Weighted Steiner Forest on Planar Graphs

  • Carsten Moldenhauer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

Node-Weighted Steiner Forest is the following problem: Given an undirected graph, a set of pairs of terminal vertices, a weight function on the vertices, find a minimum weight set of vertices that includes and connects each pair of terminals. We consider the restriction to planar graphs where the problem remains NP-complete. Demaine et al. [DHK09] showed that the generic primal-dual algorithm of Goemans and Williamson [GW97] is a 6-approximation on planar graphs. We present (1) a different analysis to prove an approximation factor of 3, (2) show that this bound is tight for the generic algorithm, and (3) show how the algorithm can be improved to yield a 9/4-approximation algorithm.

We give a simple proof for the first result using contraction techniques and present an example for the lower bound. Then, we establish a connection to the feedback problems studied by Goemans and Williamson [GW98]. We show how our constructions can be combined with their proof techniques yielding the third result and an alternative, more involved, way of deriving the first result. The third result induces an upper bound on the integrality gap of 9/4. Our analysis implies that improving this bound for Node-Weighted Steiner Forest via the primal-dual algorithm is essentially as difficult as improving the integrality gap for the feedback problems in [GW98].

Keywords

Planar Graph STEINER Tree Steiner Tree Problem Terminal Vertex Black Vertex 
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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References

  1. [BCE+10]
    Bateni, M., Chekuri, C., Ene, A., Hajiaghayi, M., Korula, N., Marx, D.: Prize-collecting Steiner Problems on Planar Graphs. In: 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (2010)Google Scholar
  2. [BGRS10]
    Byrka, J., Grandoni, F., Rothvoß, T., Sanità, L.: An improved LP-based Approximation for Steiner Tree. In: 42nd ACM Symposium on Theory of Computing, pp. 583–592 (2010)Google Scholar
  3. [BHM10]
    Bateni, M., Hajiaghayi, M., Marx, D.: Approximation Schemes for Steiner Forest on Planar Graphs and Graphs of Bounded Treewidth. In: 42nd ACM Symposium on Theory of Computing, pp. 211–220 (2010)Google Scholar
  4. [BKM09]
    Borradaile, G., Klein, P., Mathieu, C.: An O(n logn) Approximation Scheme for Steiner Tree in Planar Graphs. ACM Transactions on Algorithms 5(3), 1–31 (2009)CrossRefzbMATHGoogle Scholar
  5. [CC08]
    Chlebík, M., Chlebíková, J.: The Steiner tree problem on graphs: Inapproximability results. Theoretical Computer Science 406, 207–214 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [DG10]
    Dilkina, B., Gomes, C.P.: Solving Connected Subgraph Problems in Wildlife Conservation. In: Lodi, A., Milano, M., Toth, P. (eds.) CPAIOR 2010. LNCS, vol. 6140, pp. 102–116. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. [DHK09]
    Demaine, E.D., Hajiaghayi, M., Klein, P.N.: Node-Weighted Steiner Tree and Group Steiner Tree in Planar Graphs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 328–340. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. [Fei98]
    Feige, U.: A threshold of ln n for approximating set cover. Journal of the ACM 45, 634–652 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [GJ77]
    Garey, M.R., Johnson, D.S.: The Rectilinear Steiner Tree Problem is NP-Complete. SIAM Journal on Applied Mathematics 32(4), 826–834 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [GMNS99]
    Guha, S., Moss, A., Naor, J.S., Schieber, B.: Efficient recovery from power outage (extended abstract). In: 31st ACM Symposium on Theory of Computing, pp. 574–582 (1999)Google Scholar
  11. [GW97]
    Goemans, M.X., Williamson, D.P.: The Primal-Dual Method for Approximation Algorithms and its Application to Network Design Problems, ch. 4, pp. 144–191 (1997), In Hochbaum [Hoc97]Google Scholar
  12. [GW98]
    Goemans, M.X., Williamson, D.P.: Primal-Dual Approximation Algorithms for Feedback Problems in Planar Graphs. Combinatorica 18, 37–59 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [Hoc97]
    Hochbaum, D.S. (ed.): Approximation Algorithms for NP-hard Problems. PWS Publishing Co., Boston (1997)zbMATHGoogle Scholar
  14. [Kar72]
    Karp, R.M.: Reducibility Among Combinatorial Problems. In: Complexity of Computer Computations, pp. 85–103 (1972)Google Scholar
  15. [KR95]
    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
  16. [MR07]
    Moss, A., Rabani, Y.: Approximation Algorithms for Constrained Node Weighted Steiner Tree Problems. SIAM Journal on Computing 37(2), 460–481 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Carsten Moldenhauer
    • 1
  1. 1.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany

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