Abstract
Given a graph G = (V,E) with node weight w: V →R + and a subset S ⊆ V, find a minimum total weight tree interconnecting all nodes in S. This is the node-weighted Steiner tree problem which will be studied in this paper. In general, this problem is NP-hard and cannot be approximated by a polynomial time algorithm with performance ratio a ln n for any 0 < a < 1 unless NP ⊆ DTIME(n O(logn)), where n is the number of nodes in s. In this paper, we show that for unit disk graph, the problem is still NP-hard, however it has polynomial time constant approximation. We will present a 4-approximation and a 2.5ρ-approximation where ρ is the best known performance ratio for polynomial time approximation of classical Steiner minimum tree problem in graphs. As a corollary, we obtain that there is polynomial time (9.875+ε)-approximation algorithm for minimum weight connected dominating set in unit disk graphs.
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References
Aneja, Y.P.: An integer linear programming approach to the Steiner problem in graphs. Networks 10, 167–178 (1980)
Beasley, J.E.: An algorithm for the Steiner problem in graphs. Networks 14, 147–159 (1984)
Berman, P., Ramaiyer, V.: Improved approximations for the Steiner tree problem. Journal of Algorithms 17, 381–408 (1994)
Chen, D., Du, D.Z., Hu, X.D., Lin, G.H., Wang, L., Xue, G.: Approximation for Steiner tree with minimum number of Steiner points. Theoretical Computer Science 262, 83–99 (2001)
Feige, U.: A threshold of lnn for approximating set cover. J. ACM 45, 634–652 (1998)
Garey, M.R., Johnson, D.S.: Computers and Intractability:A Guide to the Theory of NP-Completeness. Freeman, San Fransico (1978)
Guha, S., Khuller, S.: Improved Methods for Approximating Node Weighted Steiner Trees and Connected Dominating Sets. Information and Computation 150, 57–74 (1999)
Hougardy, S., Prömel, H.J.: A 1.598 Approximation Algorithm for the Steiner Problem in Graphs. SODA, 448–453 (1998)
Huang, Y., Gao, X., Zhang, Z., Wu, W.: A Better Constant-Factor Approximation for Weighted Dominating Set in Unit Disk Graph (preprint)
Klein, P., Ravi, R.: A nearly best-possible approximation algorithm for node-weighted steiner trees. Journal of Algorithms 19, 104–115 (1995)
Kou, L.T., Markowsky, G., Berman, L.: A Fast Algorithm for Steiner Trees, pp. 141–145 (1981)
Min, M., Du, H., Jia, X., Huang, C.X., Huang, S.C.H., Wu, W.: Improving Construction for Connected Dominating Set with Steiner Tree in Wireless Sensor Networks. Journal of Global Optimizatio 35, 111–119 (2006)
Moss, A., Rabani, Y.: Approximation Algorithms for Constrained Node Weighted Steiner Tree Problems. In: STOC (2001)
Robins, G., Zelikovski, A.: Improved Steiner Tree Approximation in Graphs. In: Proc. of 11th. ACM-SIAM Symposium on Discrete. Algorithms, pp. 770–779 (2000)
Segev, A.: The node-weighted steiner tree problem. Networks 17, 1–17 (1987)
Shore, M.L., Foulds, L.R., Gibbons, R.B.: An algorithm for the Steiner problem in graphs. Networks 12, 323–333 (1982)
Zelikovsky, A.: An 11/6 approximation algorithm for the network Steiner problem. Algorithmica 9, 463–470 (1993)
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Zou, F., Li, X., Kim, D., Wu, W. (2008). Two Constant Approximation Algorithms for Node-Weighted Steiner Tree in Unit Disk Graphs. In: Yang, B., Du, DZ., Wang, C.A. (eds) Combinatorial Optimization and Applications. COCOA 2008. Lecture Notes in Computer Science, vol 5165. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85097-7_26
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DOI: https://doi.org/10.1007/978-3-540-85097-7_26
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