<s>
Identical-machines	B-Algorithm
scheduling	I-Algorithm
is	O
an	O
optimization	O
problem	O
in	O
computer	B-General_Concept
science	I-General_Concept
and	O
operations	O
research	O
.	O
</s>
<s>
We	O
are	O
given	O
n	O
jobs	O
J1	O
,	O
J2	O
,	O
...	O
,	O
Jn	O
of	O
varying	O
processing	O
times	O
,	O
which	O
need	O
to	O
be	O
scheduled	O
on	O
m	O
identical	O
machines	O
,	O
such	O
that	O
a	O
certain	O
objective	O
function	O
is	O
optimized	O
,	O
for	O
example	O
,	O
the	O
makespan	B-Algorithm
is	O
minimized	O
.	O
</s>
<s>
Identical	B-Algorithm
machine	I-Algorithm
scheduling	I-Algorithm
is	O
a	O
special	O
case	O
of	O
uniform	B-Algorithm
machine	I-Algorithm
scheduling	I-Algorithm
,	O
which	O
is	O
itself	O
a	O
special	O
case	O
of	O
optimal	B-Algorithm
job	I-Algorithm
scheduling	I-Algorithm
.	O
</s>
<s>
In	O
the	O
general	O
case	O
,	O
the	O
processing	O
time	O
of	O
each	O
job	O
may	O
be	O
different	O
on	O
different	O
machines	O
;	O
in	O
the	O
case	O
of	O
identical	B-Algorithm
machine	I-Algorithm
scheduling	I-Algorithm
,	O
the	O
processing	O
time	O
of	O
each	O
job	O
is	O
the	O
same	O
on	O
each	O
machine	O
.	O
</s>
<s>
Therefore	O
,	O
identical	B-Algorithm
machine	I-Algorithm
scheduling	I-Algorithm
is	O
equivalent	O
to	O
multiway	B-Algorithm
number	I-Algorithm
partitioning	I-Algorithm
.	O
</s>
<s>
A	O
special	O
case	O
of	O
identical	B-Algorithm
machine	I-Algorithm
scheduling	I-Algorithm
is	O
single-machine	B-Algorithm
scheduling	I-Algorithm
.	O
</s>
<s>
In	O
the	O
standard	O
three-field	B-Algorithm
notation	I-Algorithm
for	I-Algorithm
optimal	I-Algorithm
job	I-Algorithm
scheduling	I-Algorithm
problems	I-Algorithm
,	O
the	O
identical-machines	O
variant	O
is	O
denoted	O
by	O
P	O
in	O
the	O
first	O
field	O
.	O
</s>
<s>
For	O
example	O
,	O
"	O
P||	O
"	O
is	O
an	O
identical	B-Algorithm
machine	I-Algorithm
scheduling	I-Algorithm
problem	O
with	O
no	O
constraints	O
,	O
where	O
the	O
goal	O
is	O
to	O
minimize	O
the	O
maximum	O
completion	O
time	O
.	O
</s>
<s>
Minimizing	O
the	O
weighted	O
average	O
completion	O
time	O
is	O
NP-hard	O
even	O
on	O
identical	O
machines	O
,	O
by	O
reduction	O
from	O
the	O
knapsack	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
It	O
is	O
NP-hard	O
even	O
if	O
the	O
number	O
of	O
machines	O
is	O
fixed	O
and	O
at	O
least	O
2	O
,	O
by	O
reduction	O
from	O
the	O
partition	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Sahni	O
presents	O
an	O
exponential-time	O
algorithm	O
and	O
a	O
polynomial-time	B-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
for	O
solving	O
both	O
these	O
NP-hard	O
problems	O
on	O
identical	O
machines	O
:	O
</s>
<s>
Minimizing	O
the	O
maximum	O
completion	O
time	O
(	O
P||	O
)	O
is	O
NP-hard	O
even	O
for	O
identical	O
machines	O
,	O
by	O
reduction	O
from	O
the	O
partition	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Any	O
list	B-Algorithm
scheduling	I-Algorithm
algorithm	O
(	O
an	O
algorithm	O
that	O
processes	O
the	O
jobs	O
in	O
an	O
arbitrary	O
fixed	O
order	O
,	O
and	O
schedules	O
each	O
job	O
to	O
the	O
first	O
available	O
machine	O
)	O
is	O
a	O
approximation	O
for	O
identical	O
machines	O
.	O
</s>
<s>
The	O
specific	O
list-scheduling	O
algorithm	O
called	O
Longest	B-Algorithm
Processing	I-Algorithm
Time	I-Algorithm
First	I-Algorithm
(	O
LPT	O
)	O
,	O
which	O
sorts	O
the	O
jobs	O
by	O
descending	O
length	O
,	O
is	O
a	O
approximation	O
for	O
identical	O
machines	O
.	O
</s>
<s>
It	O
is	O
also	O
called	O
greedy	B-Algorithm
number	I-Algorithm
partitioning	I-Algorithm
.	O
</s>
<s>
Coffman	O
,	O
Garey	O
and	O
Johnson	O
presented	O
a	O
different	O
algorithm	O
called	O
multifit	B-Algorithm
algorithm	I-Algorithm
,	O
using	O
techniques	O
from	O
bin	O
packing	O
,	O
which	O
has	O
an	O
approximation	O
factor	O
of	O
13/11	O
≈	O
1.182	O
.	O
</s>
<s>
Huang	O
and	O
Lu	O
presented	O
a	O
simple	O
polynomial-time	O
algorithm	O
that	O
attains	O
an	O
11/9	O
≈	O
1.222	O
approximation	O
in	O
time	O
O( m	O
log	O
m	O
+	O
n	O
)	O
,	O
through	O
the	O
more	O
general	O
problem	O
of	O
maximin-share	B-Algorithm
allocation	O
of	O
chores	O
.	O
</s>
<s>
Sahni	O
presented	O
a	O
PTAS	B-Algorithm
that	O
attains	O
(	O
1+ε	O
)	O
OPT	O
in	O
time	O
.	O
</s>
<s>
This	O
is	O
a	O
PTAS	B-Algorithm
.	O
</s>
<s>
The	O
LPT	B-Algorithm
algorithm	I-Algorithm
attains	O
at	O
least	O
of	O
the	O
optimum	O
.	O
</s>
<s>
Woeginger	O
presented	O
a	O
PTAS	B-Algorithm
that	O
attains	O
an	O
approximation	O
factor	O
of	O
in	O
time	O
,	O
where	O
a	O
huge	O
constant	O
that	O
is	O
exponential	O
in	O
the	O
required	O
approximation	O
factor	O
ε	O
.	O
</s>
<s>
The	O
algorithm	O
uses	O
Lenstra	O
's	O
algorithm	O
for	O
integer	B-Algorithm
linear	I-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
They	O
prove	O
that	O
,	O
if	O
f	O
is	O
non-negative	O
,	O
convex	O
,	O
and	O
satisfies	O
a	O
strong	O
continuity	O
assumption	O
that	O
they	O
call	O
"	O
F*	O
"	O
,	O
then	O
both	O
minimization	O
problems	O
have	O
a	O
PTAS	B-Algorithm
.	O
</s>
<s>
Similarly	O
,	O
if	O
f	O
is	O
non-negative	O
,	O
concave	O
,	O
and	O
satisfies	O
F*	O
,	O
then	O
both	O
maximization	O
problems	O
have	O
a	O
PTAS	B-Algorithm
.	O
</s>
<s>
In	O
both	O
cases	O
,	O
the	O
run-time	O
of	O
the	O
PTAS	B-Algorithm
is	O
O(n )	O
,	O
but	O
with	O
constants	O
that	O
are	O
exponential	O
in	O
1/ε	O
.	O
</s>
