10.4230/LIPICS.FSTTCS.2008.1747
Chekuri, Chandra
Chandra
Chekuri
Korula, Nitish
Nitish
Korula
Single-Sink Network Design with Vertex Connectivity Requirements
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
2008
Article
Network Design
Vertex Connectivity
Buy-at-Bulk
Rent-or-Buy
Approximation
Hariharan, Ramesh
Ramesh
Hariharan
Mukund, Madhavan
Madhavan
Mukund
Vinay, V
V
Vinay
2008
2008-12-05
2008-12-05
2008-12-05
en
urn:nbn:de:0030-drops-17475
10.4230/LIPIcs.FSTTCS.2008
978-3-939897-08-8
1868-8969
10.4230/LIPIcs.FSTTCS.2008
LIPIcs, Volume 2, FSTTCS 2008
IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science
2013
2
30
131
142
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Hariharan, Ramesh
Ramesh
Hariharan
Mukund, Madhavan
Madhavan
Mukund
Vinay, V
V
Vinay
1868-8969
Leibniz International Proceedings in Informatics (LIPIcs)
2008
2
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
12 pages
451264 bytes
application/pdf
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license
info:eu-repo/semantics/openAccess
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 $k > 1$.
LIPIcs, Vol. 2, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, pages 131-142