<s>
The	O
k-Center	B-Algorithm
Clustering	O
problem	O
can	O
also	O
be	O
defined	O
on	O
a	O
complete	O
undirected	O
graph	O
G	O
=(	O
V	O
,	O
E	O
)	O
as	O
follows	O
:	O
</s>
<s>
The	O
k-center	B-Algorithm
problem	O
is	O
equivalent	O
to	O
finding	O
the	O
smallest	O
index	O
i	O
such	O
that	O
Gi	O
has	O
a	O
dominating	O
set	O
of	O
size	O
at	O
most	O
k	O
.	O
</s>
<s>
Although	O
Dominating	O
Set	O
is	O
NP-complete	O
,	O
the	O
k-center	B-Algorithm
problem	O
remains	O
NP-hard	O
.	O
</s>
<s>
This	O
is	O
clear	O
,	O
since	O
the	O
optimality	O
of	O
a	O
given	O
feasible	O
solution	O
for	O
the	O
k-center	B-Algorithm
problem	O
can	O
be	O
determined	O
through	O
the	O
Dominating	O
Set	O
reduction	O
only	O
if	O
we	O
know	O
in	O
first	O
place	O
the	O
size	O
of	O
the	O
optimal	O
solution	O
(	O
i.e.	O
</s>
<s>
Although	O
a	O
Turing	B-Algorithm
reduction	I-Algorithm
can	O
get	O
around	O
this	O
issue	O
by	O
trying	O
all	O
values	O
of	O
k	O
.	O
</s>
<s>
A	O
simple	O
greedy	B-Algorithm
approximation	B-Algorithm
algorithm	I-Algorithm
that	O
achieves	O
an	O
approximation	O
factor	O
of	O
2	O
builds	O
using	O
a	O
farthest-first	B-Algorithm
traversal	I-Algorithm
in	O
k	O
iterations	O
.	O
</s>
<s>
The	O
solution	O
obtained	O
using	O
the	O
simple	O
greedy	B-Algorithm
algorithm	I-Algorithm
is	O
a	O
2-approximation	O
to	O
the	O
optimal	O
solution	O
.	O
</s>
<s>
Given	O
a	O
set	O
of	O
n	O
points	O
,	O
belonging	O
to	O
a	O
metric	O
space	O
(	O
,	O
d	O
)	O
,	O
the	O
greedy	B-Algorithm
K-center	B-Algorithm
algorithm	O
computes	O
a	O
set	O
K	O
of	O
k	O
centers	O
,	O
such	O
that	O
K	O
is	O
a	O
2-approximation	O
to	O
the	O
optimal	O
k-center	B-Algorithm
clustering	O
of	O
V	O
.	O
</s>
<s>
Assume	O
,	O
without	O
loss	O
of	O
generality	O
,	O
that	O
was	O
added	O
later	O
to	O
the	O
center	O
set	O
by	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
,	O
say	O
in	O
ith	O
iteration	O
.	O
</s>
<s>
But	O
since	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
always	O
chooses	O
the	O
point	O
furthest	O
away	O
from	O
the	O
current	O
set	O
of	O
centers	O
,	O
we	O
have	O
that	O
and	O
,	O
</s>
<s>
Another	O
algorithm	O
with	O
the	O
same	O
approximation	O
factor	O
takes	O
advantage	O
of	O
the	O
fact	O
that	O
the	O
k-Center	B-Algorithm
problem	O
is	O
equivalent	O
to	O
finding	O
the	O
smallest	O
index	O
i	O
such	O
that	O
Gi	O
has	O
a	O
dominating	O
set	O
of	O
size	O
at	O
most	O
k	O
and	O
computes	O
a	O
maximal	O
independent	O
set	O
of	O
Gi	O
,	O
looking	O
for	O
the	O
smallest	O
index	O
i	O
that	O
has	O
a	O
maximal	O
independent	O
set	O
with	O
a	O
size	O
of	O
at	O
least	O
k	O
.	O
</s>
<s>
It	O
is	O
not	O
possible	O
to	O
find	O
an	O
approximation	B-Algorithm
algorithm	I-Algorithm
with	O
an	O
approximation	O
factor	O
of	O
2	O
for	O
any	O
>0	O
,	O
unless	O
P	O
=	O
NP	O
.	O
</s>
<s>
Furthermore	O
,	O
the	O
distances	O
of	O
all	O
edges	O
in	O
G	O
must	O
satisfy	O
the	O
triangle	O
inequality	O
if	O
the	O
k-center	B-Algorithm
problem	O
is	O
to	O
be	O
approximated	O
within	O
any	O
constant	O
factor	O
,	O
unless	O
P	O
=	O
NP	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
the	O
k-Center	B-Algorithm
problem	O
is	O
W[2]-hard	B-General_Concept
to	O
approximate	O
within	O
a	O
factor	O
of	O
2	O
for	O
any	O
>0	O
,	O
when	O
using	O
k	O
as	O
the	O
parameter	O
.	O
</s>
<s>
When	O
considering	O
the	O
combined	O
parameter	O
given	O
by	O
k	O
and	O
the	O
doubling	O
dimension	O
,	O
k-Center	B-Algorithm
is	O
still	O
W[1]-hard	O
but	O
it	O
is	O
possible	O
to	O
obtain	O
a	O
parameterized	B-General_Concept
approximation	O
scheme	O
.	O
</s>
