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DOI:
URN: urn:nbn:de:0030-drops-17469
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### Pruning 2-Connected Graphs

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### 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
$3+\eps$.

### BibTeX - Entry

@InProceedings{chekuri_et_al:LIPIcs:2008:1746,
author =	{Chandra Chekuri and Nitish Korula},
title =	{{Pruning 2-Connected Graphs}},
booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
pages =	{119--130},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-08-8},
ISSN =	{1868-8969},
year =	{2008},
volume =	{2},
editor =	{Ramesh Hariharan and Madhavan Mukund and V Vinay},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},