  We consider connectivity-augmentation problems in a setting where each
potential new edge has a nonnegative cost associated with it, and the task is
to achieve a certain connectivity target with at most p new edges of minimum
total cost. The main result is that the minimum cost augmentation of
edge-connectivity from k-1 to k with at most p new edges is fixed-parameter
tractable parameterized by p and admits a polynomial kernel. We also prove the
fixed-parameter tractability of increasing edge-connectivity from 0 to 2, and
increasing node-connectivity from 1 to 2.
