Exact and Approximate Digraph Bandwidth
In this paper, we introduce a directed variant of the classical Bandwidth problem and study it from the view-point of moderately exponential time algorithms, both exactly and approximately. Motivated by the definitions of the directed variants of the classical Cutwidth and Pathwidth problems, we def...
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| Vydáno v: | Theory of computing systems Ročník 69; číslo 1; s. 10 |
|---|---|
| Hlavní autoři: | , , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.03.2025
Springer Nature B.V |
| Témata: | |
| ISSN: | 1432-4350, 1433-0490 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | In this paper, we introduce a directed variant of the classical
Bandwidth
problem and study it from the view-point of moderately exponential time algorithms, both exactly and approximately. Motivated by the definitions of the directed variants of the classical
Cutwidth
and
Pathwidth
problems, we define
Digraph Bandwidth
as follows. Given a digraph
D
and an ordering
σ
of its vertices, the
digraph bandwidth
of
σ
with respect to
D
is equal to the maximum value of
σ
(
v
)
-
σ
(
u
)
over all arcs
(
u
,
v
)
of
D
going forward along
σ
(that is, when
σ
(
u
)
<
σ
(
v
)
). The
Digraph Bandwidth
problem takes as input a digraph
D
and asks to output an ordering with the minimum digraph bandwidth. The undirected
Bandwidth
easily reduces to
Digraph Bandwidth
and thus, it immediately implies that
Digraph Bandwidth
is
NP
-hard. While an
O
⋆
(
n
!
)
time algorithm for the problem is trivial, the goal of this paper is to design algorithms for
Digraph Bandwidth
which have running times of the form
2
O
(
n
)
. In particular, we obtain the following results. Here,
n
and
m
denote the number of vertices and arcs of the input digraph
D
, respectively.
Digraph Bandwidth
can be solved in
O
⋆
(
3
n
·
2
m
)
time. This result implies a
2
O
(
n
)
time algorithm on sparse graphs, such as graphs of bounded average degree (planar graphs).
Let
G
be the underlying undirected graph of the input digraph. If the treewidth of
G
is at most
t
, then
Digraph Bandwidth
can be solved in time
O
⋆
(
2
n
+
(
t
+
2
)
log
n
)
. This result implies a
2
n
+
O
(
n
log
n
)
algorithm, for directed planar graphs and, in general, for the class of digraphs whose underlying undirected graph excludes some fixed graph
H
as a minor.
Digraph Bandwidth
can be solved in
min
{
O
⋆
(
4
n
·
b
n
)
,
O
⋆
(
4
n
·
2
b
log
b
log
n
)
}
time, where
b
denotes the optimal digraph bandwidth of
D
. This allow us to deduce a
2
O
(
n
)
algorithm in many cases, for example when
b
≤
n
log
2
n
.
Finally, we give a
(Single) Exponential Time Approximation Scheme
for
Digraph Bandwidth
. In particular, we show that for any fixed real
ϵ
>
0
, we can find an ordering whose digraph bandwidth is at most
(
1
+
ϵ
)
times the optimal digraph bandwidth, in time
O
⋆
(
4
n
·
(
⌈
4
/
ϵ
⌉
)
n
)
. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1432-4350 1433-0490 |
| DOI: | 10.1007/s00224-024-10202-x |