Approximation Algorithms for Quickest Spanning Tree Problems |
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Authors: | Refael Hassin Asaf Levin |
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Affiliation: | (1) Department of Statistics and Operations Research, Tel Aviv University, Tel Aviv 69978, Israel;(2) Faculty of Industrial Engineering and Management, The Technion, Haifa 32000, Israel |
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Abstract: | Let $G=(V,E)$ be an undirected multigraph with a special vertex
${\it root} \in V$, and where each edge $e \in E$ is endowed with a
length $l(e) \geq 0$ and a capacity $c(e) > 0$. For a path $P$
that connects $u$ and $v$, the {\it transmission time} of $P$ is
defined as $t(P)=\mbox{\large$\Sigma$}_{e \in P} l(e) + \max_{e \in P}\!{(1 /
c(e))}$. For a spanning tree $T$, let $P_{u,v}^T$ be the unique $u$--$v$
path in $T$. The {\sc quickest radius spanning tree problem} is to find
a spanning tree $T$ of $G$ such that $\max _{v \in V} t(P^T_{root,v})$ is
minimized. In this paper we present a 2-approximation algorithm for
this problem, and show that unless $P =NP$, there is no approximation
algorithm with a performance guarantee of $2 - \epsilon$ for any
$\epsilon >0$. The {\sc quickest diameter spanning tree problem} is
to find a spanning tree $T$ of $G$ such that $\max_{u,v \in V}
t(P^T_{u,v})$ is minimized. We present a ${3 \over 2}$-approximation
to this problem, and prove that unless $P=NP$
there is no approximation algorithm with a performance guarantee of
${3 \over 2}-\epsilon$ for any $\epsilon >0$. |
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Keywords: | Approximation algorithms Quickest path problem Minimum diameter spanning tree problem |
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