Anti-factor is FPT Parameterized by Treewidth and List Size (but Counting is Hard)
In the general AntiFactor problem, a graph G and, for every vertex v of G , a set X v ⊆ N of forbidden degrees is given. The task is to find a set S of edges such that the degree of v in S is not in the set X v . Standard techniques (dynamic programming plus fast convolution) can be used to show tha...
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| Veröffentlicht in: | Algorithmica Jg. 87; H. 1; S. 22 - 88 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
New York
Springer US
01.01.2025
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 0178-4617, 1432-0541 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | In the general
AntiFactor
problem, a graph
G
and, for every vertex
v
of
G
, a set
X
v
⊆
N
of forbidden degrees is given. The task is to find a set
S
of edges such that the degree of
v
in
S
is
not
in the set
X
v
. Standard techniques (dynamic programming plus fast convolution) can be used to show that if
M
is the largest forbidden degree, then the problem can be solved in time
(
M
+
2
)
tw
·
n
O
(
1
)
if a tree decomposition of width
tw
is given. However, significantly faster algorithms are possible if the sets
X
v
are sparse: our main algorithmic result shows that if every vertex has at most
x
forbidden degrees (we call this special case
AntiFactor
x
), then the problem can be solved in time
(
x
+
1
)
O
(
tw
)
·
n
O
(
1
)
. That is,
AntiFactor
x
is fixed-parameter tractable parameterized by treewidth
tw
and the maximum number
x
of excluded degrees. Our algorithm uses the technique of representative sets, which can be generalized to the optimization version, but (as expected) not to the counting version of the problem. In fact, we show that #
AntiFactor
1
is already #
W
[
1
]
-hard parameterized by the width of the given decomposition. Moreover, we show that, unlike for the decision version, the standard dynamic programming algorithm is essentially optimal for the counting version. Formally, for a fixed nonempty set
X
, we denote by
X
-
AntiFactor
the special case where every vertex
v
has the same set
X
v
=
X
of forbidden degrees. We show the following lower bound for every fixed set
X
: if there is an
ϵ
>
0
such that #
X
-
AntiFactor
can be solved in time
(
max
X
+
2
-
ϵ
)
tw
·
n
O
(
1
)
given a tree decomposition of width
tw
, then the counting strong exponential-time hypothesis (#SETH) fails. |
|---|---|
| Bibliographie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-024-01265-w |