Affiliation:
1. McGill University, Montréal, Canada and Centre National de la Recherche Scientifique, Sophia-Antipolis, France
2. The University of Melbourne, Melbourne, Australia
Abstract
Let
G
be an
n
-vertex
m
-edge graph with weighted vertices. A pair of vertex sets
A
,
B
⊆
V
(
G
) is a 2/3
-separation
of
order
|
A
∩
B
| if
A
∪
B
=
V
(
G
), there is no edge between
A
−
B
and
B
−
A
, and both
A
−
B
and
B
−
A
have weight at most 2/3 the total weight of
G
. Let ℓ ∈ Z
+
be fixed. Alon et al. [1990] presented an algorithm that in
O
(
n
1/2
m
) time, outputs either a
K
ℓ
-minor of
G
, or a separation of
G
of order
O
(
n
1/2
). Whether there is a
O
(
n
+
m
)-time algorithm for this theorem was left as an open problem. In this article, we obtain a
O
(
n
+
m
)-time algorithm at the expense of a
O
(
n
2/3
) separator. Moreover, our algorithm exhibits a trade-off between time complexity and the order of the separator. In particular, for any given ϵ ∈ [0,1/2], our algorithm outputs either a
K
ℓ
-minor of
G
, or a separation of
G
with order
O
(
n
(2−ϵ)/3
in
O
(
n
1 + ϵ
+
m
) time. As an application we give a fast approximation algorithm for finding an independent set in a graph with no
K
ℓ-minor.
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
21 articles.
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