Linear Kernels and Linear-Time Algorithms for Finding Large Cuts
The maximum cut problem in graphs and its generalizations are fundamental combinatorial problems. Several of these cut problems were recently shown to be fixed-parameter tractable and admit polynomial kernels when parameterized above the tight lower bound measured by the size and order of the graph....
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| Veröffentlicht in: | Algorithmica Jg. 80; H. 9; S. 2574 - 2615 |
|---|---|
| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
New York
Springer US
01.09.2018
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 0178-4617, 1432-0541 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | The maximum cut problem in graphs and its generalizations are fundamental combinatorial problems. Several of these cut problems were recently shown to be fixed-parameter tractable and admit polynomial kernels when parameterized above the tight lower bound measured by the size and order of the graph. In this paper we continue this line of research and considerably improve several of those results:
We show that an algorithm by Crowston et al. (Algorithmica 72(3):734–757,
2015
) for
(Signed) Max-Cut Above Edwards−Erd
ő
s Bound
can be implemented so as to run in
linear time
8
k
·
O
(
m
)
; this significantly improves the previous analysis with run time
8
k
·
O
(
n
4
)
.
We give an
asymptotically optimal
kernel for
(Signed) Max-Cut Above Edwards−Erd
ő
s Bound
with
O
(
k
) vertices, improving a kernel with
O
(
k
3
)
vertices by Crowston et al. (Theor Comput Sci 513:53–64,
2013
).
We improve
all
known kernels for parameterizations above strongly
λ
-extendible properties (a generalization of the
Max-Cut
results) by Crowston et al. (Proceedings of FSTTCS 2013, Leibniz international proceedings in informatics, Guwahati,
2013
) from
O
(
k
3
)
vertices to
O
(
k
) vertices.
Therefore,
Max Acyclic Subdigraph
parameterized above Poljak–Turzík bound admits a kernel with
O
(
k
) vertices and can be solved in
2
O
(
k
)
·
n
O
(
1
)
time; this answers an open question by Crowston et al. (Proceedings of FSTTCS 2012, Leibniz international proceedings in informatics, Hyderabad,
2012
).
All presented kernels can be computed in time
O
(
km
). |
|---|---|
| Bibliographie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-017-0388-z |