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research
The parameterized complexity of some geometric problems in unbounded dimension
Authors
B. Aronov
D. Bremner
+9Â more
E. Armaldi
K. Varadarajan
N. Megiddo
N. Megiddo
P. Giannopoulos
R. Impagliazzo
R.G. Downey
S. Arora
S. Langerman
Publication date
1 January 2009
Publisher
'Springer Science and Business Media LLC'
Doi
Cite
View
on
arXiv
Abstract
We study the parameterized complexity of the following fundamental geometric problems with respect to the dimension
d
d
d
: i) Given
n
n
n
points in
\Rd
, compute their minimum enclosing cylinder. ii) Given two
n
n
n
-point sets in
\Rd
, decide whether they can be separated by two hyperplanes. iii) Given a system of
n
n
n
linear inequalities with
d
d
d
variables, find a maximum-size feasible subsystem. We show that (the decision versions of) all these problems are W[1]-hard when parameterized by the dimension
d
d
d
. %and hence not solvable in
O
(
f
(
d
)
n
c
)
{O}(f(d)n^c)
O
(
f
(
d
)
n
c
)
time, for any computable function
f
f
f
and constant
c
c
c
%(unless FPT=W[1]). Our reductions also give a
n
Ω
(
d
)
n^{\Omega(d)}
n
Ω
(
d
)
-time lower bound (under the Exponential Time Hypothesis)
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Last time updated on 03/12/2019