Affiliation:
1. National Institutes of Health, Bethesda, MD
2. University of Texas at Dallas, Richardson, TX
Abstract
Given an undirected graph
G
= (
V,E
) with
nonnegative costs on its edges, a root node
r
V
, a
set of demands
D
V
with demand
v
D
wishing to route
w(v)
units of flow (weight) to
r
,
and a positive number
k
, the
Capacitated Minimum Steiner
Tree
(CMStT) problem asks for a minimum Steiner tree, rooted at
r
, spanning the vertices in
D
*
{
r
}, in which the sum of the vertex
weights in every subtree connected to
r
is at most
k
.
When
D
=
V
, this problem is known as the
Capacitated Minimum Spanning Tree
(CMST) problem. Both CMsT
and CMST problems are NP-hard. In this article, we present
approximation algorithms for these problems and several of their
variants in network design. Our main results are the following:
---We present a (³ Á
ST
+
2)-approximation algorithm for the CMStT problem, where ³ is
the
inverse Steiner ratio
, and Á
ST
is the best achievable approximation ratio for the Steiner tree
problem. Our ratio improves the current best ratio of
2Á
ST
+ 2 for this problem.
---In particular, we obtain (³ + 2)-approximation ratio for
the CMST problem, which is an improvement over the current best
ratio of 4 for this problem. For points in Euclidean and
rectilinear planes, our result translates into ratios of 3.1548 and
3.5, respectively.
---For instances in the plane, under the
L
p
norm, with the vertices in
D
having uniform weights, we present a nontrivial
(7/5Á
ST
+ 3/2)-approximation algorithm for
the CMStT problem. This translates into a ratio of 2.9 for the CMST
problem with uniform vertex weights in the
L
p
metric plane. Our ratio of 2.9 solves
the long-standing open problem of obtaining any ratio better than 3
for this case.
---For the CMST problem, we show how to obtain a 2-approximation
for graphs in metric spaces with unit vertex weights and
k
=
3,4.
---For the
budgeted
CMST problem, in which the weights of
the subtrees connected to
r
could be up to ±
k
instead of
k
(± e 1), we obtain a ratio of ³
+ 2/±.
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
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