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Multi-rooted Greedy Approximation of Directed Steiner Trees with Applications

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Graph-Theoretic Concepts in Computer Science (WG 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7551))

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Abstract

We present a greedy algorithm for the directed Steiner tree problem (DST), where any tree rooted at any (uncovered) terminal can be a candidate for greedy choice. It will be shown that the algorithm, running in polynomial time for any constant l, outputs a directed Steiner tree of cost no larger than 2(l − 1)(ln n + 1) times the cost of the minimum l-restricted Steiner tree. We derive from this result that 1) DST for a class of graphs, including quasi-bipartite graphs, in which the length of paths induced by Steiner vertices is bounded by some constant can be approximated within a factor of O(logn), and 2) the tree cover problem on directed graphs can also be approximated within a factor of O(logn).

Supported in part by a Grant in Aid for Scientific Research of the Ministry of Education, Science, Sports and Culture of Japan.

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References

  1. Byrka, J., Grandoni, F., Rothvoß, T., Sanità, L.: An improved LP-based approximation for Steiner tree. In: Proc. 42nd STOC, pp. 583–592 (2010)

    Google Scholar 

  2. Berman, P., Ramaiyer, V.: Improved approximations for the Steiner tree problem. In: Proc. 3rd SODA, pp. 325–334 (1992)

    Google Scholar 

  3. Chlebík, M., Chlebíková, J.: The Steiner tree problem on graphs: Inapproximability results. Theory Comput. Syst. 406(3), 207–214 (2008)

    Article  MATH  Google Scholar 

  4. Charikar, M., Chekuri, C., Cheung, T., Dai, Z., Goel, A., Guha, S., Li, M.: Approximation algorithms for directed Steiner tree problems. J. Algorithms 33, 73–91 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Calinescu, G., Zelikovsky, A.: The polymatroid Steiner problems. J. Comb. Opt. 9, 281–294 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fujito, T.: How to Trim an MST: A 2-Approximation Algorithm for Minimum Cost Tree Cover. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 431–442. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  8. Halperin, E., Krauthgamer, R.: Polylogarithmic inapproximability. In: Proc. 35th STOC, pp. 585–594 (2003)

    Google Scholar 

  9. Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)

    Chapter  Google Scholar 

  10. Karpinski, M., Zelikovsky, A.Z.: New approximation algorithms for the Steiner tree problem. J. Comb. Opt. 1, 47–65 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Könemann, J., Konjevod, G., Parekh, O., Sinha, A.: Improved Approximations for Tour and Tree Covers. In: Jansen, K., Khuller, S. (eds.) APPROX 2000. LNCS, vol. 1913, pp. 184–193. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  12. Konjevod, G.: Directed Steiner trees, linear programs and randomized rounding, 8 pages (2005) (manuscript)

    Google Scholar 

  13. Kortsarz, G., Peleg, D.: Approximating the weight of shallow Steiner trees. Discrete Applied Mathematics 93, 265–285 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nguyen, V.H.: Approximation Algorithm for the Minimum Directed Tree Cover. In: Wu, W., Daescu, O. (eds.) COCOA 2010, Part II. LNCS, vol. 6509, pp. 144–159. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  15. Rothvoß, T.: Directed Steiner tree and the Lasserre hierarchy. ArXiv e-prints (November 2011)

    Google Scholar 

  16. Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proc. 29th STOC, pp. 475–484 (1997)

    Google Scholar 

  17. Rajagopalan, S., Vazirani, V.V.: On the bidirected cut relaxation for the metric Steiner tree problem. In: Proc. 10th SODA, pp. 742–751 (1999)

    Google Scholar 

  18. Robins, G., Zelikovsky, A.: Tighter bounds for graph Steiner tree approximation. SIAM J. Discrete Math. 19, 122–134 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Savage, C.: Depth-first search and the vertex cover problem. Inform. Process. Lett. 14(5), 233–235 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zelikovsky, A.: A series of approximation algorithms for the acyclic directed Steiner tree problem. Algorithmica 18, 99–110 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zelikovsky, A.: An 11/6-approximation algorithm for the network Steiner problem. Algorithmica 9, 463–470 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zosin, L., Khuller, S.: On directed Steiner trees. In: Proc. 13th SODA, pp. 59–63 (2002)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, Toyohashi University of Technology, Tempaku, Toyohashi, 441-8580, Japan

    Tomoya Hibi & Toshihiro Fujito

Authors
  1. Tomoya Hibi
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  2. Toshihiro Fujito
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Caesarea Rothschild Institute (CRI), 31905, Haifa, Israel

    Martin Charles Golumbic

  2. The Academic College of Tel Aviv, Unit of Statistics and Probability Studies, 64044, Yaffo, Israel

    Michal Stern

  3. Shenkar College and Caesarea Rothschild Institute (CRI), University of Haifa, 31905, Haifa, Israel

    Avivit Levy

  4. Caesarea Rothschild Institute (CRI), 31905, Haifa, Israel

    Gila Morgenstern

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Hibi, T., Fujito, T. (2012). Multi-rooted Greedy Approximation of Directed Steiner Trees with Applications. In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2012. Lecture Notes in Computer Science, vol 7551. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34611-8_23

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  • DOI: https://doi.org/10.1007/978-3-642-34611-8_23

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-34610-1

  • Online ISBN: 978-3-642-34611-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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