Simple and Improved Parameterized Algorithms for Multiterminal Cuts
Given a graph G =( V , E ) with n vertices and m edges, and a subset T of k vertices called terminals , the Edge (respectively, Vertex ) Multiterminal Cut problem is to find a set of at most l edges (non-terminal vertices), whose removal from G separates each terminal from all the others. These two...
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| Published in: | Theory of computing systems Vol. 46; no. 4; pp. 723 - 736 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
Springer-Verlag
01.05.2010
Springer Nature B.V |
| Subjects: | |
| ISSN: | 1432-4350, 1433-0490 |
| Online Access: | Get full text |
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| Summary: | Given a graph
G
=(
V
,
E
) with
n
vertices and
m
edges, and a subset
T
of
k
vertices called
terminals
, the
Edge
(respectively,
Vertex
)
Multiterminal Cut
problem is to find a set of at most
l
edges (non-terminal vertices), whose removal from
G
separates each terminal from all the others. These two problems are NP-hard for
k
≥3 but well-known to be polynomial-time solvable for
k
=2 by the flow technique. In this paper, based on a notion
farthest minimum isolating cut
, we design several simple and improved algorithms for Multiterminal Cut. We show that Edge Multiterminal Cut can be solved in
O
(2
l
kT
(
n
,
m
)) time and Vertex Multiterminal Cut can be solved in
O
(
k
l
T
(
n
,
m
)) time, where
T
(
n
,
m
)=
O
(min (
n
2/3
,
m
1/2
)
m
) is the running time of finding a minimum (
s
,
t
) cut in an unweighted graph. Furthermore, the running time bounds of our algorithms can be further reduced for small values of
k
: Edge 3-Terminal Cut can be solved in
O
(1.415
l
T
(
n
,
m
)) time, and Vertex {3,4,5,6}-Terminal Cuts can be solved in
O
(2.059
l
T
(
n
,
m
)),
O
(2.772
l
T
(
n
,
m
)),
O
(3.349
l
T
(
n
,
m
)) and
O
(3.857
l
T
(
n
,
m
)) time respectively. Our results on Multiterminal Cut can also be used to obtain faster algorithms for
Multicut
:
-time algorithm for Edge Multicut and
O
((2
k
)
k
+
l
/2
T
(
n
,
m
))-time algorithm for Vertex Multicut. |
|---|---|
| Bibliography: | SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 |
| ISSN: | 1432-4350 1433-0490 |
| DOI: | 10.1007/s00224-009-9215-5 |