Approximation algorithms for node-weighted prize-collecting Steiner tree problems on planar graphs

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 ($2.88 + \epsilon$)-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 for the version without penalties. We also give a primal-dual 4-approximation algorithm for the more general forest version using techniques introduced by Hajiaghay and Jain

A New Parallel Algorithm for Planarity Testing

Determining whether a graph is planar is both theoretically and practically interesting. Although several sequential algorithms have been introduced which accomplish planarity testing in O(V ) time for graphs with V vertices, very few of these have been parallelized. In a recent comparison of sequential planarity testing algorithms, the newest algorithms were found to be fastest; however, these are the ones which have not been parallelized. The goal of this thesis is to introduce a method for parallelizing one of the newest planarity testing algorithms