Two Constant Approximation Algorithms for Node-Weighted Steiner Tree in Unit Disk Graphs
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(nO(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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- 6.Garey, M.R., Johnson, D.S.: Computers and Intractability:A Guide to the Theory of NP-Completeness. Freeman, San Fransico (1978)Google Scholar
- 8.Hougardy, S., Prömel, H.J.: A 1.598 Approximation Algorithm for the Steiner Problem in Graphs. SODA, 448–453 (1998)Google Scholar
- 9.Huang, Y., Gao, X., Zhang, Z., Wu, W.: A Better Constant-Factor Approximation for Weighted Dominating Set in Unit Disk Graph (preprint)Google Scholar
- 11.Kou, L.T., Markowsky, G., Berman, L.: A Fast Algorithm for Steiner Trees, pp. 141–145 (1981)Google Scholar
- 13.Moss, A., Rabani, Y.: Approximation Algorithms for Constrained Node Weighted Steiner Tree Problems. In: STOC (2001)Google Scholar
- 14.Robins, G., Zelikovski, A.: Improved Steiner Tree Approximation in Graphs. In: Proc. of 11th. ACM-SIAM Symposium on Discrete. Algorithms, pp. 770–779 (2000)Google Scholar