<s>
The	O
vertex	B-Algorithm
k-center	I-Algorithm
problem	I-Algorithm
is	O
a	O
classical	O
NP-hard	O
problem	O
in	O
computer	B-General_Concept
science	I-General_Concept
.	O
</s>
<s>
It	O
has	O
application	O
in	O
facility	O
location	O
and	O
clustering	B-Algorithm
.	O
</s>
<s>
Basically	O
,	O
the	O
vertex	B-Algorithm
k-center	I-Algorithm
problem	I-Algorithm
models	O
the	O
following	O
real	O
problem	O
:	O
given	O
a	O
city	O
with	O
facilities	O
,	O
find	O
the	O
best	O
facilities	O
where	O
to	O
build	O
fire	O
stations	O
.	O
</s>
<s>
The	O
vertex	B-Algorithm
k-center	I-Algorithm
problem	I-Algorithm
is	O
a	O
classical	O
NP-Hard	O
problem	O
in	O
computer	B-General_Concept
science	I-General_Concept
.	O
</s>
<s>
Formally	O
,	O
the	O
vertex	B-Algorithm
k-center	I-Algorithm
problem	I-Algorithm
consists	O
in	O
:	O
given	O
a	O
complete	O
undirected	O
graph	O
in	O
a	O
metric	O
space	O
,	O
and	O
a	O
positive	O
integer	O
,	O
find	O
a	O
subset	O
such	O
that	O
and	O
the	O
objective	O
function	O
is	O
minimized	O
.	O
</s>
<s>
If	O
,	O
the	O
vertex	B-Algorithm
k-center	I-Algorithm
problem	I-Algorithm
can	O
not	O
be	O
(	O
optimally	O
)	O
solved	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
However	O
,	O
there	O
are	O
some	O
polynomial	O
time	O
approximation	B-Algorithm
algorithms	I-Algorithm
that	O
get	O
near-optimal	O
solutions	O
.	O
</s>
<s>
Specifically	O
,	O
2-approximated	B-Algorithm
solutions	I-Algorithm
.	O
</s>
<s>
In	O
the	O
context	O
of	O
a	O
minimization	O
problem	O
,	O
such	O
as	O
the	O
vertex	B-Algorithm
k-center	I-Algorithm
problem	I-Algorithm
,	O
a	O
2-approximated	O
solution	O
is	O
any	O
solution	O
such	O
that	O
,	O
where	O
is	O
the	O
size	O
of	O
an	O
optimal	O
solution	O
.	O
</s>
<s>
An	O
algorithm	O
that	O
guarantees	O
to	O
generate	O
2-approximated	B-Algorithm
solutions	I-Algorithm
is	O
known	O
as	O
a	O
2-approximation	O
algorithm	O
.	O
</s>
<s>
The	O
main	O
2-approximated	O
algorithms	O
for	O
the	O
vertex	B-Algorithm
k-center	I-Algorithm
problem	I-Algorithm
reported	O
in	O
the	O
literature	O
are	O
the	O
Sh	O
algorithm	O
,	O
the	O
HS	O
algorithm	O
,	O
and	O
the	O
Gon	O
algorithm	O
.	O
</s>
<s>
Because	O
of	O
this	O
,	O
many	O
heuristics	B-Algorithm
and	O
metaheuristics	B-Algorithm
have	O
been	O
developed	O
through	O
the	O
time	O
.	O
</s>
<s>
in	O
2017	O
,	O
the	O
CDS	O
algorithm	O
is	O
a	O
3-approximation	O
algorithm	O
that	O
takes	O
ideas	O
from	O
the	O
Gon	O
algorithm	O
(	O
farthest	O
point	O
heuristic	B-Algorithm
)	O
,	O
the	O
HS	O
algorithm	O
(	O
parametric	O
pruning	O
)	O
,	O
and	O
the	O
relationship	O
between	O
the	O
vertex	B-Algorithm
k-center	I-Algorithm
problem	I-Algorithm
and	O
the	O
Dominating	O
Set	O
problem	O
.	O
</s>
<s>
However	O
,	O
by	O
performing	O
a	O
binary	O
search	O
over	O
the	O
ordered	O
set	O
of	O
edge	O
costs	O
,	O
a	O
more	O
efficiente	O
heuristic	B-Algorithm
named	O
CDSh	O
is	O
proposed	O
.	O
</s>
<s>
Despite	O
the	O
suboptimal	O
performance	O
of	O
the	O
CDS	O
algorithm	O
,	O
and	O
the	O
heuristic	B-Algorithm
performance	O
of	O
CDSh	O
,	O
both	O
present	O
a	O
much	O
better	O
performance	O
than	O
the	O
Sh	O
,	O
HS	O
,	O
and	O
Gon	O
algorithms	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
the	O
k-Center	O
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	O
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>
<s>
Some	O
of	O
the	O
most	O
widely	O
used	O
benchmark	O
datasets	O
for	O
the	O
vertex	B-Algorithm
k-center	I-Algorithm
problem	I-Algorithm
are	O
the	O
pmed	O
instances	O
from	O
OR-Lib.	O
,	O
and	O
some	O
instances	O
from	O
TSP-Lib	O
.	O
</s>
<s>
The	O
greedy	O
pure	O
algorithm	O
(	O
or	O
Gr	O
)	O
follows	O
the	O
core	O
idea	O
of	O
greedy	B-Algorithm
algorithms	I-Algorithm
:	O
to	O
take	O
optimal	O
local	O
decisions	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
the	O
vertex	B-Algorithm
k-center	I-Algorithm
problem	I-Algorithm
,	O
the	O
optimal	O
local	O
decision	O
consists	O
in	O
selecting	O
each	O
center	O
in	O
such	O
a	O
way	O
that	O
the	O
size	O
of	O
the	O
solution	O
(	O
covering	O
radius	O
)	O
is	O
minimum	O
at	O
each	O
iteration	O
.	O
</s>
<s>
This	O
algorithm	O
takes	O
advantage	O
of	O
the	O
reduction	O
from	O
the	O
vertex	B-Algorithm
k-center	I-Algorithm
problem	I-Algorithm
to	O
the	O
minimum	O
dominating	O
set	O
problem	O
.	O
</s>
<s>
This	O
heuristic	B-Algorithm
follows	O
the	O
lazy	O
principle	O
,	O
which	O
takes	O
every	O
decision	O
as	O
slow	O
as	O
possible	O
(	O
opossed	O
to	O
the	O
greedy	O
strategy	O
)	O
.	O
</s>
