Affiliation:
1. University of Bergen, Norway
2. The Institute of Mathematical Sciences, HBNI, Chennai, India
3. The Institute of Mathematical Sciences, HBNI, Chennai, India and University of Bergen, Norway
Abstract
A subfamily
F′
of a set family
F
is said to
q
-
represent
F
if for every
A
∈
F
and
B
of size
q
such that
A
∩
B
= ∅ there exists a set
A′
∈
F′
such that
A′
∩
B
= ∅. Recently, we provided an algorithm that, for a given family
F
of sets of size
p
together with an integer
q
, efficiently computes a
q
-representative family
F′
of
F
of size approximately (p+q p). In this article, we consider the efficient computation of
q
-representative families for
product
families
F
. A family
F
is a product family if there exist families
A
and
B
such that
F
= {
A
, ∪,
B
:
A
∈
A
,
B
∈
B
,
A
, ∩,
B
= ∅}. Our main technical contribution is an algorithm that, given
A
,
B
and
q
, computes a
q
-representative family
F′
of
F
. The running time of our algorithm is
sublinear
in |
F
| for many choices of
A
,
B
, and
q
that occur naturally in several dynamic programming algorithms. We also give an algorithm for the computation of
q
-representative families for product families
F
in the more general setting where
q
-representation also involves independence in a matroid in addition to disjointness. This algorithm considerably outperforms the naive approach where one first computes
F
from
A
and
B
and then computes the
q
-representative family
F′
from
F
.
We give two applications of our new algorithms for computing
q
-representative families for product families. The first is a 3.8408
k
n
O
(1)
deterministic algorithm for the M
ultilinear
M
onomial
D
etection
(
k
-M
l
D) problem. The second is a significant improvement of deterministic dynamic programming algorithms for “connectivity problems” on graphs of bounded treewidth.
Funder
Rigorous Theory of Preprocessing, ERC Advanced Investigator
Parameterized Approximation, ERC Starting
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Reference31 articles.
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4. Andreas Björklund Thore Husfeldt Petteri Kaski and Mikko Koivisto. 2010. Narrow sieves for parameterized paths and packings. CoRR abs/1007.1161 (2010). Andreas Björklund Thore Husfeldt Petteri Kaski and Mikko Koivisto. 2010. Narrow sieves for parameterized paths and packings. CoRR abs/1007.1161 (2010).
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