Faster parameterized algorithms for minor containment
The H - Minor containment problem asks whether a graph G contains some fixed graph H as a minor, that is, whether H can be obtained by some subgraph of G after contracting edges. The derivation of a polynomial-time algorithm for H - Minor containment is one of the most important and technical parts...
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| Veröffentlicht in: | Theoretical computer science Jg. 412; H. 50; S. 7018 - 7028 |
|---|---|
| Hauptverfasser: | , , , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Oxford
Elsevier B.V
25.11.2011
Elsevier |
| Schlagworte: | |
| ISSN: | 0304-3975, 1879-2294 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | The
H
-
Minor containment problem asks whether a graph
G
contains some fixed graph
H
as a minor, that is, whether
H
can be obtained by some subgraph of
G
after contracting edges. The derivation of a polynomial-time algorithm for
H
-
Minor containment is one of the most important and technical parts of the Graph Minor Theory of Robertson and Seymour and it is a cornerstone for most of the algorithmic applications of this theory.
H
-
Minor containment for graphs of bounded branchwidth is a basic ingredient of this algorithm. The currently fastest solution to this problem, based on the ideas introduced by Robertson and Seymour, was given by Hicks in [I.V. Hicks, Branch decompositions and minor containment, Networks 43 (1) (2004) 1–9], providing an algorithm that in time
O
(
3
k
2
⋅
(
h
+
k
−
1
)
!
⋅
m
)
decides if a graph
G
with
m
edges and branchwidth
k
, contains a fixed graph
H
on
h
vertices as a minor. In this work we improve the dependence on
k
of Hicks’ result by showing that checking if
H
is a minor of
G
can be done in time
O
(
2
(
2
k
+
1
)
⋅
log
k
⋅
h
2
k
⋅
2
2
h
2
⋅
m
)
. We set up an approach based on a combinatorial object called
rooted packing, which captures the properties of the subgraphs of
H
that we seek in our dynamic programming algorithm. This formulation with rooted packings allows us to speed up the algorithm when
G
is embedded in a fixed surface, obtaining the first algorithm for minor containment testing with single-exponential dependence on branchwidth. Namely, it runs in time
2
O
(
k
)
⋅
h
2
k
⋅
2
O
(
h
)
⋅
n
, with
n
=
∣
V
(
G
)
∣
. Finally, we show that slight modifications of our algorithm permit to solve some related problems within the same time bounds, like induced minor or contraction containment. |
|---|---|
| Bibliographie: | ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23 |
| ISSN: | 0304-3975 1879-2294 |
| DOI: | 10.1016/j.tcs.2011.09.015 |