Approximation algorithms for solving the line-capacitated minimum Steiner tree problem
In this paper, we address the line-capacitated minimum Steiner tree problem (the Lc-MStT problem, for short), which is a variant of the (Euclidean) capacitated minimum Steiner tree problem and defined as follows. Given a set X = { r 1 , r 2 , … , r n } of n terminals in R 2 , a demand function d : X...
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| Published in: | Journal of global optimization Vol. 84; no. 3; pp. 687 - 714 |
|---|---|
| Main Authors: | , , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
Springer US
01.11.2022
Springer Springer Nature B.V |
| Subjects: | |
| ISSN: | 0925-5001, 1573-2916 |
| Online Access: | Get full text |
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| Summary: | In this paper, we address the line-capacitated minimum Steiner tree problem (the Lc-MStT problem, for short), which is a variant of the (Euclidean) capacitated minimum Steiner tree problem and defined as follows. Given a set
X
=
{
r
1
,
r
2
,
…
,
r
n
}
of
n
terminals in
R
2
, a demand function
d
:
X
→
N
and a positive integer
C
, we are asked to determine the location of a line
l
and a Steiner tree
T
l
to interconnect these
n
terminals in
X
and at least one point located on this line
l
such that the total demand of terminals in each maximal subtree (of
T
l
) connected to the line
l
, where the terminals in such maximal subtree are all located at the same side of this line
l
, does not exceed the bound
C
. The objective is to minimize total weight
∑
e
∈
T
l
w
(
e
)
of such a Steiner tree
T
l
among all line-capacitated Steiner trees mentioned-above, where weight
w
(
e
)
=
0
if two endpoints of that edge
e
∈
T
l
are located on the line
l
and otherwise weight
w
(
e
) is the Euclidean distance between two endpoints of that edge
e
∈
T
l
. In addition, when this line
l
is as an input in
R
2
and
∑
r
∈
X
d
(
r
)
≤
C
holds, we refer to this version as the 1-line-fixed minimum Steiner tree problem (the 1Lf-MStT problem, for short). We obtain three main results. (1) Given a
ρ
st
-approximation algorithm to solve the Euclidean minimum Steiner tree problem and a
ρ
1
L
f
-approximation algorithm to solve the 1Lf-MStT problem, respectively, we design a
(
ρ
st
ρ
1
L
f
+
2
)
-approximation algorithm to solve the Lc-MStT problem. (2) Whenever demand of each terminal
r
∈
X
is less than
C
2
, we provide a
(
ρ
1
L
f
+
2
)
-approximation algorithm to resolve the Lc-MStT problem. (3) Whenever demand of each terminal
r
∈
X
is at least
C
2
, using the Edmonds’ algorithm to solve the minimum weight perfect matching as a subroutine, we present an exact algorithm in polynomial time to resolve the Lc-MStT problem. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0925-5001 1573-2916 |
| DOI: | 10.1007/s10898-022-01163-x |