  Increasing the connectivity of a graph is a pivotal challenge in robust
network design. The weighted connectivity augmentation problem is a common
version of the problem that takes link costs into consideration. The problem is
then to find a minimum cost subset of a given set of weighted links that
increases the connectivity of a graph by one when the links are added to the
edge set of the input instance. In this work, we give a first implementation of
recently discovered better-than-2 approximations. Furthermore, we propose three
new heuristic and one exact approach. These include a greedy algorithm
considering link costs and the number of unique cuts covered, an approach based
on minimum spanning trees and a local search algorithm that may improve a given
solution by swapping links of paths. Our exact approach uses an ILP formulation
with efficient cut enumeration as well as a fast initialization routine. We
then perform an extensive experimental evaluation which shows that our
algorithms are faster and yield the best solutions compared to the current
state-of-the-art as well as the recently discovered better-than-2 approximation
algorithms. Our novel local search algorithm can improve solution quality even
further.
