Improved approximation algorithms for weighted 2- and 3-vertex connectivity augmentation problems

The problem of finding a minimum augmenting edge-set to make a graph k-vertex connected is considered. This problem is denoted as the minimum k-augmentation problem. For weighted graphs, the minimum k-augmentation problem is NP-complete. Our main result is an approximation algorithm with a performance ratio of 3 for solving the minimum 3-augmentation problem. This improves the best previously known performance guarantee of 11/3. We also have the following marginal result: an approximation algorithm for the minimum 2-augmentation problem that achieves a factor of 2, and thus improves the previously known factor of 2 + (1/n), with n as the number of vertices in the graph. (C) 1997 Academic Press.