Pruning 2-Connected Graphs

Given an edge-weighted undirected graph $G$ with a specified set of terminals, let the \emph{density} of any subgraph be the ratio of its weight/cost to the number of terminals it contains. If $G$ is 2-connected, does it contain smaller 2-connected subgraphs of density comparable to that of $G$? We answer this question in the affirmative by giving an algorithm to \emph{prune} $G$ and find such subgraphs of any desired size, at the cost of only a logarithmic increase in density (plus a small additive factor). We apply the pruning techniques to give algorithms for two NP-Hard problems on finding large 2-vertex-connected subgraphs of low cost; no previous approximation algorithm was known for either problem. In the \kv problem, we are given an undirected graph $G$ with edge costs and an integer $k$; the goal is to find a minimum-cost 2-vertex-connected subgraph of $G$ containing at least $k$ vertices. In the \bv\ problem, we are given the graph $G$ with edge costs, and a budget $B$; the goal is to find a 2-vertex-connected subgraph $H$ of $G$ with total edge cost at most $B$ that maximizes the number of vertices in $H$. We describe an $O(\log n \log k)$ approximation for the \kv problem, and a bicriteria approximation for the \bv\ problem that gives an $O(\frac{1}{\eps}\log^2 n)$ approximation, while violating the budget by a factor of at most $3+\eps$.