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(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.
KeywordsApproximation Ratio Minimum Span Tree Steiner Tree Steiner Point Steiner Tree Problem
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