Advertisement

A Practical Greedy Approximation for the Directed Steiner Tree Problem

  • Dimitri Watel
  • Marc-Antoine Weisser
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8881)

Abstract

The Directed Steiner Tree (DST) NP-hard problem asks, considering a directed weighted graph with \(n\) nodes and \(m\) arcs, a node \(r\) called root and a set of \(k\) nodes \(X\) called terminals, for a minimum cost directed tree rooted at \(r\) spanning \(X\). The best known polynomial approximation ratio for DST is a \(O(k^\varepsilon )\)-approximation greedy algorithm. However, a much faster \(k\)-approximation, returning the shortest paths from \(r\) to \(X\), is generally used in practice. We give in this paper a new \(O(\sqrt{k})\)-approximation greedy algorithm called Greedy\(_\mathrm{FLAC }\) \(^\triangleright \), derived from a new fast \(k\)-approximation algorithm called Greedy\(_\mathrm{FLAC }\) running in time at most \(O(n m k^2)\).

We provide computational results to show that, Greedy\(_\mathrm{FLAC }\) rivals the running time of the fast \(k\)-approximation and returns solution with smaller cost in practice.

Keywords

Directed steiner tree Approximation algorithm Greedy algorithm 

References

  1. 1.
    Karp, R.M.: Reducibility Among Combinatorial Problems. Springer, New York (1972)Google Scholar
  2. 2.
    Kou, L., Markowsky, G., Berman, L.: A fast algorithm for steiner trees. Acta Inf. 15(2), 141–145 (1981)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Zelikovsky, A.Z.: An 11/6-approximation algorithm for the network steiner problem. Algorithmica 9(5), 463–470 (1993)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Byrka, J., Grandoni, F., Rothvoss, T., Sanità, L.: Steiner tree approximation via iterative randomized rounding. J. ACM (JACM) 60(1), 6:1–6:33 (2013)CrossRefGoogle Scholar
  5. 5.
    Cheng, X., Du, D.Z.: Steiner Trees in Industry, vol. 11. Springer, New York (2001)Google Scholar
  6. 6.
    Voß, S.: Steiner tree problems in telecommunications. In: Resende, M.G.C., Pardalos, P.M. (eds.) Handbook of Optimization in Telecommunications, pp. 459–492. Springer, New York (2006)CrossRefGoogle Scholar
  7. 7.
    Novak, R., Rugelj, J., Kandus, G.: A note on distributed multicast routing in point-to-point networks. Comput. Oper. Res. 28(12), 1149–1164 (2001)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Feige, U.: A threshold of ln n for approximating set cover. J. ACM (JACM) 45(4), 634–652 (1998)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Halperin, E., Krauthgamer, R.: Polylogarithmic inapproximability. In: Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pp. 585–594 (2003)Google Scholar
  10. 10.
    Charikar, M., Chekuri, C., Cheung, T.Y., Dai, Z., Goel, A., Guha, S., Li, M.: Approximation algorithms for directed steiner problems. J. Algorithms 33(1), 73–91 (1999)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Helvig, C.S., Robins, G., Zelikovsky, A.: An improved approximation scheme for the group steiner problem. Networks 37(1), 8–20 (2001)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Johnson, D.S.: Approximation algorithms for combinatorial problems. In: Proceedings of the Fifth Annual ACM Symposium on Theory of Computing, pp. 38–49 (1973)Google Scholar
  13. 13.
    Chvatal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4(3), 233–235 (1979)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Olsson, P.M., Kvarnstrom, J., Doherty, P., Burdakov, O., Holmberg, K.: Generating uav communication networks for monitoring and surveillance. In: 2010 11th International Conference on Control Automation Robotics & Vision (ICARCV), pp. 1070–1077. IEEE (2010)Google Scholar
  15. 15.
    Gundecha, P., Feng, Z., Liu, H.: Seeking provenance of information using social media. In: Proceedings of the 22nd ACM International Conference on Information & Knowledge Management, pp. 1691–1696. ACM (2013)Google Scholar
  16. 16.
    Lappas, T., Terzi, E., Gunopulos, D., Mannila, H.: Finding effectors in social networks. In: Proceedings of the 16th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1059–1068. ACM (2010)Google Scholar
  17. 17.
    Koch, T., Martin, A., Voß, S.: SteinLib: an updated library on Steiner tree problems in graphs. In: Cheng, X.Z., Du, D.-Z. (eds.) Steiner Trees in Industry, pp. 285–325. Springer, New York (2001)CrossRefGoogle Scholar
  18. 18.
    Chimani, M., Woste, M.: Contraction-based steiner tree approximations in practice. In: Asano, T., Nakano, S., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 40–49. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  19. 19.
    Stanojevic, M., Vujosevic, M.: An exact algorithm for steiner tree problem on graphs. Int. J. Comput. Commun. Control 1(1), 41–46 (2006)Google Scholar
  20. 20.
    Uchoa, E., Werneck, R.F.F.: Fast local search for steiner trees in graphs. In: ALENEX, vol. 10, pp. 1–10. SIAM (2010)Google Scholar
  21. 21.
    Drummond, L., Santos, M., Uchoa, E.: A distributed dual ascent algorithm for steiner problems in multicast routing. Networks 53(2), 170–183 (2009)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Hsieh, M.I., Wu, E.H.K., Tsai, M.F.: Fasterdsp: a faster approximation algorithm for directed steiner tree problem. J. Inf. Sci. Eng. 22, 1409–1425 (2006)MathSciNetGoogle Scholar
  23. 23.
    de Aragão, M.P., Uchoa, E., Werneck, R.F.: Dual heuristics on the exact solution of large steiner problems. Electron. Notes Discrete Math. 7, 150–153 (2001)CrossRefGoogle Scholar
  24. 24.
    Wong, R.T.: A dual ascent approach for steiner tree problems on a directed graph. Math. Program. 28(3), 271–287 (1984)CrossRefMATHGoogle Scholar
  25. 25.
    Melkonian, V.: New primal-dual algorithms for steiner tree problems. Comput. Oper. Res. 34(7), 2147–2167 (2007)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Zelikovsky, A.: A series of approximation algorithms for the acyclic directed steiner tree problem. Algorithmica 18(1), 99–110 (1997)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Computer Science DepartmentSUPELEC System SciencesGif Sur YvetteFrance
  2. 2.University of VersaillesVersaillesFrance

Personalised recommendations