When cycles collapse: A general approximation technique for constrained two-connectivity problems

We present a general approximation technique for a class of network design problems where we seek a network of minimum cost that satisfies certain communication requirements and is resilient to worst-case single-link failures. Our algorithm runs in $O(n^2 \log n)$ time on a graph ith $n$ nodes and outputs a solution of cost at most thrice the optimum. We extend our technique to obtain approximation algorithms for augmenting a given network so as to satisfy certain communication requirements and achieve resilience to single-link failures. Our technique allows one to find nearly minimum-cost two-connected networks for a variety of connectivity requirements. For example, our result generalizes earlier results on finding a minimum-cost two-connected subgraph of a given edge-weighted graph and an earlier result on finding a minimum-cost subgraph two-connecting a specified subset of the nodes. Using our technique, we can also approximately solve for the first time a two-connected version of the generalized Steiner network problem and a two-connected version of the non-fixed point-to-point connection problem.