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Abstract

We present primal-dual algorithms which give a 2.4 approximation for a class of node-weighted network design problems in planar graphs, introduced by Demaine, Hajiaghayi and Klein (ICALP’09). This class includes Node-Weighted Steiner Forest problem studied recently by Moldenhauer (ICALP’11) and other node-weighted problems in planar graphs that can be expressed using (0,1)-proper functions introduced by Goemans and Williamson. We show that these problems can be equivalently formulated as feedback vertex set problems and analyze approximation factors guaranteed by different violation oracles within the primal-dual framework developed by Goemans and Williamson.

Keywords

Planar Graph Proper Function Approximation Factor Full Version 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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References

  1. 1.
    Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. Discrete Math. 12(3), 289–297 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bar-Yehuda, R., Bendel, K., Freund, A., Rawitz, D.: Local ratio: A unified framework for approxmation algrithms in memoriam: Shimon even 1935-2004. ACM Comput. Surv. 36(4), 422–463 (2004)CrossRefGoogle Scholar
  3. 3.
    Bar-Yehuda, R., Geiger, D., Naor, J.S., Roth, R.M.: Approximation algorithms for the vertex feedback set problem with applications to constraint satisfaction and bayesian inference. In: SODA 1994, pp. 344–354. SIAM, Philadelphia (1994), http://dl.acm.org/citation.cfm?id=314464.314514 Google Scholar
  4. 4.
    Becker, A., Geiger, D.: Optimization of pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artif. Intell. 83(1), 167–188 (1996)MathSciNetCrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    Demaine, E.D., Hajiaghayi, M.: Bidimensionality: new connections between fpt algorithms and ptass. In: SODA 2005, pp. 590–601. SIAM, Philadelphia (2005), http://dl.acm.org/citation.cfm?id=1070432.1070514 Google Scholar
  7. 7.
    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
  8. 8.
    Even, G., (Seffi) Naor, J., Schieber, B., Sudan, M.: Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica 20, 151–174 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Garg, N., Vazirani, V.V., Yannakakis, M.: Approximate max-flow min-(multi)cut theorems and their applications. SIAM J. Comput. 25, 235–251 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM J. Comput. 24(2), 296–317 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Goemans, M.X., Williamson, D.P.: Primal-dual approximation algorithms for feedback problems in planar graphs. Combinatorica 18, 37–59 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Guha, S., Moss, A., Naor, J., Schieber, B.: Efficient recovery from power outage (extended abstract). In: STOC 1999, pp. 574–582 (1999)Google Scholar
  13. 13.
    Kahng, A.B., Vaya, S., Zelikovsky, A.: New graph bipartizations for double-exposure, bright field alternating phase-shift mask layout. In: ASP-DAC 2001, pp. 133–138. ACM, New York (2001)CrossRefGoogle Scholar
  14. 14.
    Klein, P.: Optimization Algorithms for Planar Graphs, http://www.planarity.org/
  15. 15.
    Klein, P.N., Ravi, R.: A nearly best-possible approximation algorithm for node-weighted steiner trees. J. Algorithms 19(1), 104–115 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Li, X., Xu, X.-H., Zou, F., Du, H., Wan, P., Wang, Y., Wu, W.: A PTAS for Node-Weighted Steiner Tree in Unit Disk Graphs. In: Du, D.-Z., Hu, X., Pardalos, P.M. (eds.) COCOA 2009. LNCS, vol. 5573, pp. 36–48. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  17. 17.
    Moldenhauer, C.: Primal-Dual Approximation Algorithms for Node-Weighted Steiner Forest on Planar Graphs. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 748–759. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. 18.
    Moss, A., Rabani, Y.: Approximation algorithms for constrained node weighted steiner tree problems. SIAM J. Comput. 37(2), 460–481 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Remy, J., Steger, A.: Approximation Schemes for Node-Weighted Geometric Steiner Tree Problems. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX and RANDOM 2005. LNCS, vol. 3624, pp. 221–232. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  20. 20.
    Yannakakis, M.: Node-and edge-deletion np-complete problems. In: STOC 1978, pp. 253–264. ACM, New York (1978)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Piotr Berman
    • 1
  • Grigory Yaroslavtsev
    • 1
  1. 1.Pennsylvania State UniversityUSA

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