Abstract
Given a graph G = (V,E) with an edge cost and families \(\mathcal{V}_i\subseteq 2^V\), i = 1,2,...,m of disjoint subsets, an edge subset F ⊆ E is called a set connector if, for each \(\mathcal{V}_i\), the graph \((V,F)/\mathcal{V}_i\) obtained from (V,F) by contracting each \(X\in \mathcal{V}_i\) into a single vertex x has a property that every two contracted vertices x and x′ are connected in \((V,F)/\mathcal{V}_i\). In this paper, we introduce a problem of finding a minimum cost set connector, which contains several important network design problems such as the Steiner forest problem, the group Steiner tree problem, and the NA-connectivity augmentation problem as its special cases. We derive an approximate integer decomposition property from a fractional packing theorem of set connectors, and present a strongly polynomial 2α-approximation algorithm for the set connector problem, where \(\alpha=\max_{1 \leq i \leq m}(\sum_{X \in \mathcal{V}_i}|X|)-1\).
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References
Arkin, E.M., Halldórsson, M.M., Hassin, R.: Approximating the tree and tour covers of a graph. Information Processing Letters 47, 275–282 (1993)
Bateman, C.D., Helvig, C.S., Robins, G., Zelikovsky, A.: Provably good routing tree construction with multi-port terminals. In: Proceedings of the 1997 International Symposium on Physical Design, pp. 96–102 (1997)
Charikar, M., Chekuri, C., Goel, A., Guha, S.: Rounding via trees: deterministic approximation algorithms for group Steiner trees and k-median. In: Proceedings of the thirtieth Annual ACM Symposium on Theory of Computing, pp. 114–123 (1998)
Chekuri, C., Even, G., Kortsarzc, G.: A greedy approximation algorithm for the group Steiner problem. Discrete Applied Mathematics 154, 15–34 (2006)
Chekuri, C., Shepherd, F.B.: Approximate integer decompositions for undirected network design problems. Manuscript (2004)
Drake, D.E., Hougardy, S.: On approximation algorithms for the terminal Steiner tree problem. Information Processing Letters 89, 15–18 (2004)
Frank, A.: On a theorem of Mader. Discrete Mathematics 191, 49–57 (1992)
Fujito, T.: On approximability of the independent/connected edge dominating set problems. Information Processing Letters 79, 261–266 (2001)
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)
Garg, N., Konjevod, G., Ravi, R.: A polylogarithmic approximation algorithm for the group Steiner tree problem. Journal of Algorithms 37, 66–84 (2000)
Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM Journal on Computing 24, 296–317 (1995)
Gusfield, D.: Connectivity and edge-disjoint spanning trees. Information Processing Letters 16, 87–89 (1983)
Halperin, E., Kortsarz, G., Krauthgamer, R., Srinivasan, A., Wang, N.: Integrality ratio for group Steiner trees and directed steiner trees. In: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 275–284 (2003)
Halperin, E., Krauthgamer, R.: Polylogarithmic inapproximability. In: Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pp. 585–594 (2003)
Ishii, T., Hagiwara, M.: Augmenting Local Edge-Connectivity between Vertices and Vertex Subsets in Undirected Graphs. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 490–499. Springer, Heidelberg (2003)
Ito, H.: Node-to-area connectivity of graphs. Transactions of the Institute of Electrical Engineers of Japan 11C, 463–469 (1994)
Ito, H., Yokoyama, M.: Edge connectivity between nodes and node-subsets. Networks 31, 157–164 (1998)
Könemann, J., Konjevod, G., Parekh, O., Sinha, A.: Improved approximations for tour and tree covers. Algorithmica 38, 441–449 (2004)
Lin, G., Xue, G.: On the terminal Steiner tree problem. Information Processing Letters 84, 103–107 (2002)
Lu, C.L., Tang, C.Y., Lee, R.C.-T.: The full Steiner tree problem. Theoretical Computer Science 306, 55–67 (2003)
Mader, W.: A reduction method for edge-connectivity in graphs. Annals of Discrete Mathematics 3, 145–164 (1978)
Miwa, H., Ito, H.: Edge augmenting problems for increasing connectivity between vertices and vertex subsets. Technical Report of IPSJ, vol. 99-AL-66, pp. 17–24 (1999)
Nutov, Z.: Approximating connectivity augmentation problems. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 176–185 (2005)
Reich, G., Widmayer, P.: Beyond Steiner’s problem: a VLSI oriented generalization. In: Nagl, M. (ed.) WG 1989. LNCS, vol. 411, pp. 196–210. Springer, Heidelberg (1990)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg (2003)
Slavík, P.: Approximation algorithms for set cover and related problems. PhD thesis, University of New York (1998)
Tardos, E.: A strongly polynomial algorithm to solve combinatorial linear programs. Operations Research 34, 250–256 (1986)
Zosin, L., Khuller, S.: On directed Steiner trees. In: Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms, pp. 59–63 (2002)
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Fukunaga, T., Nagamochi, H. (2007). The Set Connector Problem in Graphs. In: Fischetti, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2007. Lecture Notes in Computer Science, vol 4513. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72792-7_36
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DOI: https://doi.org/10.1007/978-3-540-72792-7_36
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