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research
Degree-3 Treewidth Sparsifiers
Authors
Chandra Chekuri
Julia Chuzhoy
Publication date
1 January 2014
Publisher
Doi
Cite
View
on
arXiv
Abstract
We study treewidth sparsifiers. Informally, given a graph
G
G
G
of treewidth
k
k
k
, a treewidth sparsifier
H
H
H
is a minor of
G
G
G
, whose treewidth is close to
k
k
k
,
β£
V
(
H
)
β£
|V(H)|
β£
V
(
H
)
β£
is small, and the maximum vertex degree in
H
H
H
is bounded. Treewidth sparsifiers of degree
3
3
3
are of particular interest, as routing on node-disjoint paths, and computing minors seems easier in sub-cubic graphs than in general graphs. In this paper we describe an algorithm that, given a graph
G
G
G
of treewidth
k
k
k
, computes a topological minor
H
H
H
of
G
G
G
such that (i) the treewidth of
H
H
H
is
Ξ©
(
k
/
polylog
(
k
)
)
\Omega(k/\text{polylog}(k))
Ξ©
(
k
/
polylog
(
k
))
; (ii)
β£
V
(
H
)
β£
=
O
(
k
4
)
|V(H)| = O(k^4)
β£
V
(
H
)
β£
=
O
(
k
4
)
; and (iii) the maximum vertex degree in
H
H
H
is
3
3
3
. The running time of the algorithm is polynomial in
β£
V
(
G
)
β£
|V(G)|
β£
V
(
G
)
β£
and
k
k
k
. Our result is in contrast to the known fact that unless
N
P
β
c
o
N
P
/
p
o
l
y
NP \subseteq coNP/{\sf poly}
NP
β
co
NP
/
poly
, treewidth does not admit polynomial-size kernels. One of our key technical tools, which is of independent interest, is a construction of a small minor that preserves node-disjoint routability between two pairs of vertex subsets. This is closely related to the open question of computing small good-quality vertex-cut sparsifiers that are also minors of the original graph.Comment: Extended abstract to appear in Proceedings of ACM-SIAM SODA 201
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