The subdivision-constrained minimum spanning tree problem
Motivated by the constrained minimum spanning tree (CST) problem in Hassin and Levin [R. Hassin, A. Levin, An efficient polynomial time approximation scheme for the constrained minimum spanning tree problem using matroid intersection, SIAM Journal on Computing 33 (2) (2004) 261–268], we study a new...
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| Published in: | Theoretical computer science Vol. 410; no. 8; pp. 877 - 885 |
|---|---|
| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Oxford
Elsevier B.V
01.03.2009
Elsevier |
| Subjects: | |
| ISSN: | 0304-3975, 1879-2294 |
| Online Access: | Get full text |
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| Summary: | Motivated by the constrained minimum spanning tree (CST) problem in Hassin and Levin [R. Hassin, A. Levin, An efficient polynomial time approximation scheme for the constrained minimum spanning tree problem using matroid intersection, SIAM Journal on Computing 33 (2) (2004) 261–268], we study a new combinatorial optimization problem in this paper, called the general subdivision-constrained spanning tree problem (GSCST): given a graph
G
=
(
V
,
E
;
w
,
c
)
with two nonnegative integers
w
(
e
)
and
c
(
e
)
for each edge
e
∈
E
, two positive integers
B
and
d
, the GSCST problem is to first find a spanning tree
T
=
(
V
,
E
T
)
of
G
with weight
∑
e
∈
E
T
w
(
e
)
≤
B
and then to insert some new vertices on some suitable edges in
T
such that each edge in the subdivision tree
T
′
of
T
has its weight not beyond
d
. The objective is to minimize the cost
∑
e
∈
E
T
i
n
s
e
r
t
(
e
)
c
(
e
)
of such new vertices inserted on the suitable edges among all spanning trees of
G
subject to the two preceding constraints, where a subdivision tree
T
′
of
T
is constructed by inserting some new vertices on the suitable edges in
T
, the value
i
n
s
e
r
t
(
e
)
=
⌈
w
(
e
)
d
⌉
−
1
is the least number of vertices inserted and
c
(
e
)
is the cost of each vertex inserted on the edge
e
. We obtain the following main results: (1) the GSCST problem and its variant are still
NP-hard, by a reduction from the 0–1 knapsack problem, respectively; (2) the GSCST problem as well as its variant is polynomially equivalent to the CST problem, which implies the existence of a polynomial time approximation scheme to solve the GSCST problem and its variant; (3) we finally design three strongly polynomial time algorithms to solve the special versions of the GSCST problem and its variant, respectively. |
|---|---|
| ISSN: | 0304-3975 1879-2294 |
| DOI: | 10.1016/j.tcs.2008.12.038 |