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.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
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.
References
Byrka, J., Grandoni, F., Rothvoß, T., Sanità, L.: An improved LP-based approximation for Steiner tree. In: Proc. 42nd STOC, pp. 583–592 (2010)
Berman, P., Ramaiyer, V.: Improved approximations for the Steiner tree problem. In: Proc. 3rd SODA, pp. 325–334 (1992)
Chlebík, M., Chlebíková, J.: The Steiner tree problem on graphs: Inapproximability results. Theory Comput. Syst. 406(3), 207–214 (2008)
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)
Calinescu, G., Zelikovsky, A.: The polymatroid Steiner problems. J. Comb. Opt. 9, 281–294 (2005)
Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)
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)
Halperin, E., Krauthgamer, R.: Polylogarithmic inapproximability. In: Proc. 35th STOC, pp. 585–594 (2003)
Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)
Karpinski, M., Zelikovsky, A.Z.: New approximation algorithms for the Steiner tree problem. J. Comb. Opt. 1, 47–65 (1997)
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)
Konjevod, G.: Directed Steiner trees, linear programs and randomized rounding, 8 pages (2005) (manuscript)
Kortsarz, G., Peleg, D.: Approximating the weight of shallow Steiner trees. Discrete Applied Mathematics 93, 265–285 (1999)
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)
Rothvoß, T.: Directed Steiner tree and the Lasserre hierarchy. ArXiv e-prints (November 2011)
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)
Rajagopalan, S., Vazirani, V.V.: On the bidirected cut relaxation for the metric Steiner tree problem. In: Proc. 10th SODA, pp. 742–751 (1999)
Robins, G., Zelikovsky, A.: Tighter bounds for graph Steiner tree approximation. SIAM J. Discrete Math. 19, 122–134 (2005)
Savage, C.: Depth-first search and the vertex cover problem. Inform. Process. Lett. 14(5), 233–235 (1982)
Zelikovsky, A.: A series of approximation algorithms for the acyclic directed Steiner tree problem. Algorithmica 18, 99–110 (1997)
Zelikovsky, A.: An 11/6-approximation algorithm for the network Steiner problem. Algorithmica 9, 463–470 (1993)
Zosin, L., Khuller, S.: On directed Steiner trees. In: Proc. 13th SODA, pp. 59–63 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
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)