Abstract
Given an undirected weighted graph G with n nodes, the k-Undirected Steiner Tree problem is to find a minimum cost tree spanning a specified set of k nodes. If this problem and its directed version have several applications in multicast routing in packet switching networks, the modeling is not adapted anymore in networks based upon the circuit switching principle in which not all nodes are able to duplicate packets. In such networks, the number of branching nodes (with outdegree > 1) in the multicast tree must be limited.
We introduce the (k,p) −Steiner Tree with Limited Number of Branching nodes problems where the goal is to find an optimal Steiner tree with at most p branching nodes. We study, when p is fixed, its complexity depending on two criteria: the graph topology and the parameter k. In particular, we propose a polynomial algorithm when the input graph is acyclic and an other algorithm when k is fixed in an input graph of bounded treewidth. Moreover, in directed graphs where p ≤ k − 2, or in planar graphs, we provide an n ε-inapproximability proof, for any ε < 1.
The original version of this chapter was revised: The copyright line was incorrect. This has been corrected. The Erratum to this chapter is available at DOI: 10.1007/978-3-319-03578-9_29
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References
Cheng, X., Du, D.Z.: Steiner trees in industry, vol. 11. Kluwer (2001)
Voß, S.: Steiner tree problems in telecommunications, pp. 459–492 (January 2006)
Rugeli, J., Novak, R.: Steiner tree algorithms for multicast protocols (1995)
Reinhard, V., Tomasik, J., Barth, D., Weisser, M.-A.: Bandwidth Optimization for Multicast Transmissions in Virtual Circuit Networks. In: Fratta, L., Schulzrinne, H., Takahashi, Y., Spaniol, O. (eds.) NETWORKING 2009. LNCS, vol. 5550, pp. 859–870. Springer, Heidelberg (2009)
Reinhard, V., Cohen, J., Tomasik, J., Barth, D., Weisser, M.A.: Optimal configuration of an optical network providing predefined multicast transmissions. Comput. Netw. 56(8), 2097–2106 (2012)
Gargano, L., Hell, P., Stacho, L., Vaccaro, U.: Spanning trees with bounded number of branch vertices. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 355–365. Springer, Heidelberg (2002)
Salazar-González, J.J.: The Steiner cycle polytope. EJOR 147(3), 671–679 (2003)
Steinová, M.: Approximability of the Minimum Steiner Cycle Problem. Computing and Informatics 29(6+), 1349–1357 (2010)
Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphism problem. Theoretical Computer Science 10(111), 111–121 (1980)
Robertson, N., Seymour, P.: The disjoint paths problem. Journal of Combinatorial Theory, Series B, 65–110 (1995)
Schrijver, A.: Finding k disjoint paths in a directed planar graph. SIAM Journal on Computing, 1–10 (1994)
Scheffler, P.: A Practical Linear Time Algorithm for Disjoint Paths in Graphs with Bounded Tree Width. Technical Report 396/1994, Fachbereich Mathematik (1994)
Kou, L., Markowsky, G., Berman, L.: A fast algorithm for steiner trees. Acta informatica 15(2), 141–145 (1981)
Zelikovsky, A.: An 11/6-approximation algorithm for the network steiner problem. Algorithmica 9(5), 463–470 (1993)
Hougardy, S., Prömel, H.: A 1.598 approximation algorithm for the Steiner problem in graphs. In: Proc. SODA, pp. 448–453 (1999)
Du, D., Lu, B., Ngo, H., Pardalos, P.: Steiner tree problems. Encyclopedia of Optimization 5, 227–290 (2000)
Hsu, T.S., Tsai, K., Wang, D., Lee, D.: Steiner problems on directed acyclic graphs. Computing and Combinatorics, 21–30 (1996)
Charikar, M., et al.: Approximation algorithms for directed steiner problems. In: Proc. SODA, pp. 192–200 (1998)
Ming-IHsieh, E., Tsai, M.: Fasterdsp: A faster approximation algorithm for directed steiner tree problem. JISE 22, 1409–1425 (2006)
Rothvoß, T.: Directed steiner tree and the lasserre hierarchy. CoRR abs/1111.5473 (2011)
Feige, U.: A threshold of ln n for approximating set cover. J. of the ACM 45(4), 634–652 (1998)
Halperin, E., Krauthgamer, R.: Polylogarithmic inapproximability. In: Proc. STOC, pp. 585–594. ACM (2003)
Garg, N., Konjevod, G., Ravi, R.: A polylogarithmic approximation algorithm for the group steiner tree problem. In: Proc. SODA, pp. 253–259 (1998)
Ding, B., Yu, J.X., Wang, S., Qin, L., Zhang, X., Lin, X.: Finding top-k min-cost connected trees in databases. In: Chirkova, R., Dogac, A., Özsu, M.T., Sellis, T.K. (eds.) ICDE, pp. 836–845. IEEE (2007)
Cheriyan, J., Laekhanukit, B., Naves, G., Vetta, A.: Approximating rooted steiner networks. In: Proc. SODA, pp. 1499–1511 (2012)
Watel, D., Weisser, M.A., Bentz, C.: Inapproximability proof of DSTLB and USTLB in planar graphs, http://hal-supelec.archives-ouvertes.fr/hal-00793424
Downey, R.G., Fellows, M.R.: Parameterized complexity, vol. 3. Springer (1999)
Goldberg, A.V., Tarjan, R.E.: Finding minimum-cost circulations by canceling negative cycles. Journal of the ACM 36(4), 873–886 (1989)
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Watel, D., Weisser, MA., Bentz, C., Barth, D. (2013). Steiner Problems with Limited Number of Branching Nodes. In: Moscibroda, T., Rescigno, A.A. (eds) Structural Information and Communication Complexity. SIROCCO 2013. Lecture Notes in Computer Science, vol 8179. Springer, Cham. https://doi.org/10.1007/978-3-319-03578-9_26
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DOI: https://doi.org/10.1007/978-3-319-03578-9_26
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