Generalized Feedback Vertex Set Problems on Bounded-Treewidth Graphs: Chordality is the Key to Single-Exponential Parameterized Algorithms
It has long been known that Feedback Vertex Set can be solved in time 2 O ( w log w ) n O ( 1 ) on n -vertex graphs of treewidth w , but it was only recently that this running time was improved to 2 O ( w ) n O ( 1 ) , that is, to single-exponential parameterized by treewidth. We investigate which g...
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| Vydáno v: | Algorithmica Ročník 81; číslo 10; s. 3890 - 3935 |
|---|---|
| Hlavní autoři: | , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.10.2019
Springer Nature B.V |
| Témata: | |
| ISSN: | 0178-4617, 1432-0541 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | It has long been known that
Feedback Vertex Set
can be solved in time
2
O
(
w
log
w
)
n
O
(
1
)
on
n
-vertex graphs of treewidth
w
, but it was only recently that this running time was improved to
2
O
(
w
)
n
O
(
1
)
, that is, to single-exponential parameterized by treewidth. We investigate which generalizations of
Feedback Vertex Set
can be solved in a similar running time. Formally, for a class
P
of graphs, the
Bounded
P
-Block Vertex Deletion
problem asks, given a graph
G
on
n
vertices and positive integers
k
and
d
, whether
G
contains a set
S
of at most
k
vertices such that each block of
G
-
S
has at most
d
vertices and is in
P
. Assuming that
P
is recognizable in polynomial time and satisfies a certain natural hereditary condition, we give a sharp characterization of when single-exponential parameterized algorithms are possible for fixed values of
d
:
if
P
consists only of chordal graphs, then the problem can be solved in time
2
O
(
w
d
2
)
n
O
(
1
)
,
if
P
contains a graph with an induced cycle of length
ℓ
⩾
4
, then the problem is not solvable in time
2
o
(
w
log
w
)
n
O
(
1
)
even for fixed
d
=
ℓ
, unless the ETH fails.
We also study a similar problem, called
Bounded
P
-
Component Vertex Deletion
, where the target graphs have connected components of small size rather than blocks of small size, and we present analogous results. For this problem, we also show that if
d
is part of the input and
P
contains all chordal graphs, then it cannot be solved in time
f
(
w
)
n
o
(
w
)
for some function
f
, unless the ETH fails. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-019-00579-4 |