Approximation Algorithms for Node-Weighted Prize-Collecting Steiner Tree Problems on Planar Graphs

Abstract. We study the prize-collecting version of the Node-weighted Steiner Tree problem (NWPCST) restricted to planar graphs. We give a new primal-dual Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm for planar NWPCST. We then show a (2.88+ǫ)-approximation which establishes a new best approximation guarantee for planar NWPCST. This is done by combining our LMP algorithm with a threshold rounding technique and utilizing the 2.4-approximation of Berman and Yaroslavtsev [3] for the version without penalties. We also give a primaldual 4-approximation algorithm for the more general forest version using techniques introduced by Hajiaghay and Jain [18].