Approximating Node Connectivity Problems via Set
Covers |
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Authors: | Email author" target="_blank">Guy?KortsarzEmail author Email author" target="_blank">Zeev?NutovEmail author |
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Affiliation: | (1) Rutgers University, Camden, NJ 08102, USA;(2) Open University of Israel, Klauzner 16 Street, Ramat-Aviv 61392, Israel |
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Abstract: | Given a graph (directed or undirected) with costs on the edges,
and an integer $k$, we consider the problem of finding a $k$-node connected
spanning subgraph of minimum cost.
For the general instance of the problem (directed or undirected),
there is a simple $2k$-approximation algorithm.
Better algorithms are known for various ranges of $n,k$.
For undirected graphs with metric costs Khuller and Raghavachari gave a
$( 2+{2(k-1)}/{n})$-approximation algorithm.
We obtain the following results:
(i)
For arbitrary costs, a $k$-approximation algorithm for undirected graphs and
a $(k+1)$-approximation algorithm for directed graphs.
(ii)
For metric costs, a $(2+({k-1})/{n})$-approximation algorithm
for undirected graphs and
a $(2+{k}/{n})$-approximation algorithm for directed graphs.
For undirected graphs and $k=6,7$, we further improve the approximation ratio
from $k$ to $\lceil (k+1)/2 \rceil=4$; previously,
$\lceil (k+1)/2 \rceil$-approximation algorithms were known only for
$k \leq 5$. We also give a fast $3$-approximation algorithm for $k=4$.
The multiroot problem generalizes the min-cost $k$-connected subgraph problem.
In the multiroot problem, requirements $k_u$ for every node $u$ are given, and
the aim is to find a minimum-cost subgraph that contains $\max\{k_u,k_v\}$
internally disjoint paths between every pair of nodes $u,v$.
For the general instance of the problem, the best known algorithm has
approximation ratio $2k$, where $k=\max k_u$.
For metric costs there is a 3-approximation algorithm.
We consider the case of metric costs, and, using our techniques,
improve for $k \leq 7$ the approximation guarantee from $3$ to
$2+{\lfloor (k-1)/2 \rfloor}/{k} < 2.5$. |
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Keywords: | Vertex connected spanning subgraph Approximation algorithms Metric costs |
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