Approximating minimum size 1,2-connected networks

The problem of finding the minimum size 2-connected subgraph is a classical problem in network design. It is known to be NP-hard even on cubic planar graphs and Max-SNP hard. We study the generalization of this problem, where requirements of 1 or 2 edge or vertex disjoint paths are specified between every pair of vertices, and the aim is to find a minimum subgraph satisfying these requirements. For both problems we give $3/2$-approximation algorithms. This improves on the straightforward $2$-approximation algorithms for these problems, and generalizes earlier results for 2-connectivity. We also give analyses of the classical local optimization heuristics for these two network design problems