Approximating Node-Weighted k-MST on Planar Graphs

We study the problem of finding a minimum weight connected subgraph spanning at least k vertices on planar, node-weighted graphs. We give a (4 + ε)-approximation algorithm for this problem. In the process, we use the recent LMP primal-dual 3-approximation for the node-weighted prize-collecting Steiner tree problem [4] and the Lagrangian relaxation [6]. In particular, we improve the procedure of picking additional vertices given by Sadeghian [17] by taking a constant number of recursive steps and utilizing the limited guessing procedure of Arora and Karakostats [1]. We argue that our approach can be interpreted as a generalization of a result by Könemann et al. [12]. Together with a result by Mestre [14] this implies that our bound is essentially best possible among algorithms that utilize an LMP algorithm for the Lagrangian relaxation as a black box.