Node-weighted Steiner tree approximation 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 aln n for any 0<a<1 unless NP⊆DTIME(n ^O(log n)), where n is the number of nodes in s. In this paper, we are the first to show that even though for unit disk graphs, the problem is still NP-hard and it has a polynomial time constant approximation. We present 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 a polynomial time (9.875+ε)-approximation algorithm for minimum weight connected dominating set in unit disk graphs, and also there is a polynomial time (4.875+ε)-approximation algorithm for minimum weight connected vertex cover in unit disk graphs.