Approximation Algorithms for Graph Augmentation

The problem of increasing both edge and vertex connectivity of a graph at an optimal cost is studied. Since the general problem is NP-hard, we focus on efficient approximation schemes that come within a constant factor from the optima]. Previous algorithms either do not take edge costs into consideration, or they run slower than our algorithm. Our algorithm takes as input an undirected graph G0 = (V, E0) on n vertices, that is not necessarily connected, and a set Feasible of m weighted edges on V, and outputs a subset Aug of edges which when added to G0 make it two-connected. The weight of Aug, when G0 is initially connected, is no more than twice the weight of the least weight subset of edges of Feasible that increases the connectivity to two. The running time of our algorithm is O(m + n log n). As a consequence of our results, we can find an approximation to the least-weight two-connected spanning subgraph of a two-connected weighted graph.