Abstract
AbstractA directed graph D is semicomplete if for every pair x, y of vertices of D, there is at least one arc between x and y. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph $$D=(V,A)$$
D
=
(
V
,
A
)
and a pair of natural numbers k and $$\ell $$
ℓ
, we are to decide whether there is a subset X of V of size k such that the largest strongly connected component in $$D-X$$
D
-
X
has at most $$\ell $$
ℓ
vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for $$\ell =1.$$
ℓ
=
1
.
We study the parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters: $$k, \ell ,\ell +k$$
k
,
ℓ
,
ℓ
+
k
and $$n-\ell $$
n
-
ℓ
. In particular, we prove that DCOC with parameter k on semicomplete digraphs can be solved in time $$O^*(2^{16k})$$
O
∗
(
2
16
k
)
but not in time $$O^*(2^{o(k)})$$
O
∗
(
2
o
(
k
)
)
unless the Exponential Time Hypothesis (ETH) fails. The upper bound $$O^*(2^{16k})$$
O
∗
(
2
16
k
)
implies the upper bound $$O^*(2^{16(n-\ell )})$$
O
∗
(
2
16
(
n
-
ℓ
)
)
for the parameter $$n-\ell .$$
n
-
ℓ
.
We complement the latter by showing that there is no algorithm of time complexity $$O^*(2^{o({n-\ell })})$$
O
∗
(
2
o
(
n
-
ℓ
)
)
unless ETH fails. Finally, we improve (in dependency on $$\ell $$
ℓ
) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter $$\ell +k$$
ℓ
+
k
on general digraphs from $$O^*(2^{O(k\ell \log (k\ell ))})$$
O
∗
(
2
O
(
k
ℓ
log
(
k
ℓ
)
)
)
to $$O^*(2^{O(k\log (k\ell ))}).$$
O
∗
(
2
O
(
k
log
(
k
ℓ
)
)
)
.
Note that Drange, Dregi and van ’t Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time $$O^*(2^{o(k\log \ell )})$$
O
∗
(
2
o
(
k
log
ℓ
)
)
unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity $$O^*(2^{o(k\log k)}).$$
O
∗
(
2
o
(
k
log
k
)
)
.
Funder
Danmarks Frie Forskningsfond
Leverhulme Trust
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computer Science Applications,General Computer Science
Cited by
1 articles.
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