Pruning 2-Connected Graphs

Abstract

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
  $3+\eps$.