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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/
Chekuri, Chandra ;
Korula, Nitish
Single-Sink Network Design with Vertex Connectivity Requirements
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},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2008/1747},
URN = {urn:nbn:de:0030-drops-17475},
doi = {10.4230/LIPIcs.FSTTCS.2008.1747},
annote = {Keywords: Network Design, Vertex Connectivity, Buy-at-Bulk, Rent-or-Buy, Approximation}
}
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Keywords: |
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Network Design, Vertex Connectivity, Buy-at-Bulk, Rent-or-Buy, Approximation |
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Collection: |
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IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science |
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Issue Date: |
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2008 |
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Date of publication: |
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05.12.2008 |
