When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSTTCS.2008.1747
URN: urn:nbn:de:0030-drops-17475
URL: https://drops.dagstuhl.de/opus/volltexte/2008/1747/
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### Single-Sink Network Design with Vertex Connectivity Requirements

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### Abstract

We study single-sink network design problems in undirected graphs
with vertex connectivity requirements. The input to these problems
is an edge-weighted undirected graph $G=(V,E)$, a sink/root vertex
$r$, a set of terminals $T \subseteq V$, and integer $k$. The goal is
to connect each terminal $t \in T$ to $r$ via $k$ \emph{vertex-disjoint}
paths. In the {\em connectivity} problem, the objective is to find a
min-cost subgraph of $G$ that contains the desired paths. There is a
$2$-approximation for this problem when $k \le 2$ \cite{FleischerJW}
but for $k \ge 3$, the first non-trivial approximation was obtained
in the recent work of Chakraborty, Chuzhoy and Khanna
\cite{ChakCK08}; they describe and analyze an algorithm with an
approximation ratio of $O(k^{O(k^2)}\log^4 n)$ where $n=|V|$.

In this paper, inspired by the results and ideas in \cite{ChakCK08},
we show an $O(k^{O(k)}\log |T|)$-approximation bound for a simple
greedy algorithm. Our analysis is based on the dual of a natural
linear program and is of independent technical interest. We use the
insights from this analysis to obtain an $O(k^{O(k)}\log |T|)$-approximation for the more general single-sink {\em
rent-or-buy} network design problem with vertex connectivity
requirements. We further extend the ideas to obtain a
poly-logarithmic approximation for the single-sink {\em buy-at-bulk}
problem when $k=2$ and the number of cable-types is a fixed
constant; we believe that this should extend to any fixed $k$. We
also show that for the non-uniform buy-at-bulk problem, for each
fixed $k$, a small variant of a simple algorithm suggested by
Charikar and Kargiazova \cite{CharikarK05} for the case of $k=1$
gives an $2^{O(\sqrt{\log |T|})}$ approximation for larger $k$.
These results show that for each of these problems, simple and
natural algorithms that have been developed for $k=1$ have good
performance for small $k > 1$.

### BibTeX - Entry

@InProceedings{chekuri_et_al:LIPIcs:2008:1747,
author =	{Chandra Chekuri and Nitish Korula},
title =	{{Single-Sink Network Design with Vertex Connectivity Requirements}},
booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
pages =	{131--142},
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},