<s>
Uniform	B-Algorithm
machine	I-Algorithm
scheduling	I-Algorithm
(	O
also	O
called	O
uniformly-related	O
machine	O
scheduling	O
or	O
related	O
machine	O
scheduling	O
)	O
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>
It	O
is	O
a	O
variant	O
of	O
optimal	B-Algorithm
job	I-Algorithm
scheduling	I-Algorithm
.	O
</s>
<s>
The	O
goal	O
is	O
to	O
minimize	O
the	O
makespan	B-Algorithm
-	O
the	O
total	O
time	O
required	O
to	O
execute	O
the	O
schedule	O
.	O
</s>
<s>
In	O
the	O
specific	O
variant	O
called	O
uniform	B-Algorithm
machine	I-Algorithm
scheduling	I-Algorithm
,	O
some	O
machines	O
are	O
uniformly	O
faster	O
than	O
others	O
.	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
uniform-machine	O
variant	O
is	O
denoted	O
by	O
Q	O
in	O
the	O
first	O
field	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
problem	O
denoted	O
by	O
"	O
Q||	O
"	O
is	O
a	O
uniform	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>
A	O
special	O
case	O
of	O
uniform	B-Algorithm
machine	I-Algorithm
scheduling	I-Algorithm
is	O
identical	B-Algorithm
machine	I-Algorithm
scheduling	I-Algorithm
,	O
in	O
which	O
all	O
machines	O
have	O
the	O
same	O
speed	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>
Exact	O
dynamic	B-Algorithm
programming	I-Algorithm
algorithms	O
for	O
minimizing	O
the	O
weighted-average	O
completion	O
time	O
on	O
uniform	O
machines	O
.	O
</s>
<s>
Polynomial-time	B-Algorithm
approximation	I-Algorithm
schemes	I-Algorithm
,	O
which	O
for	O
any	O
ε>0	O
,	O
attain	O
at	O
most	O
(	O
1+ε	O
)	O
OPT	O
.	O
</s>
<s>
For	O
minimizing	O
the	O
weighted	O
average	O
completion	O
time	O
on	O
two	O
uniform	O
machines	O
,	O
the	O
run-time	O
is	O
=	O
,	O
so	O
it	O
is	O
an	O
FPTAS	B-Algorithm
.	O
</s>
<s>
Minimizing	O
the	O
maximum	O
completion	O
time	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>
A	O
constant-factor	O
approximation	O
is	O
attained	O
by	O
the	O
Longest-processing-time-first	B-Algorithm
algorithm	I-Algorithm
(	O
LPT	O
)	O
.	O
</s>
<s>
Exact	O
dynamic	B-Algorithm
programming	I-Algorithm
algorithms	O
for	O
minimizing	O
the	O
maximum	O
completion	O
time	O
on	O
both	O
uniform	O
and	O
unrelated	O
machines	O
.	O
</s>
<s>
Polynomial-time	B-Algorithm
approximation	I-Algorithm
schemes	I-Algorithm
,	O
which	O
for	O
any	O
ε>0	O
,	O
attain	O
at	O
most	O
(	O
1+ε	O
)	O
OPT	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
run-time	O
is	O
in	O
,	O
so	O
it	O
is	O
an	O
FPTAS	B-Algorithm
.	O
</s>
<s>
Let	O
Ci	O
(	O
for	O
i	O
between	O
1	O
and	O
m	O
)	O
be	O
the	O
makespan	B-Algorithm
of	O
machine	O
i	O
in	O
a	O
given	O
schedule	O
.	O
</s>
<s>
Ambrosio	O
and	O
Auletta	O
proved	O
that	O
the	O
Longest	B-Algorithm
Processing	I-Algorithm
Time	I-Algorithm
algorithm	O
is	O
monotone	O
whenever	O
the	O
machine	O
speeds	O
are	O
powers	O
of	O
some	O
c	O
≥	O
2	O
,	O
but	O
not	O
when	O
c	O
≤	O
1.78	O
.	O
</s>
<s>
In	O
contrast	O
,	O
List	B-Algorithm
scheduling	I-Algorithm
is	O
not	O
monotone	O
for	O
c	O
>	O
2	O
.	O
</s>
<s>
This	O
adds	O
further	O
complication	O
to	O
the	O
multiprocessor	B-Algorithm
scheduling	I-Algorithm
problem	O
.	O
</s>
<s>
For	O
static	O
scheduling	O
algorithms	O
,	O
a	O
typical	O
approach	O
is	O
to	O
rank	O
the	O
tasks	O
according	O
to	O
their	O
precedence	O
relationships	O
and	O
use	O
a	O
list	B-Algorithm
scheduling	I-Algorithm
technique	O
to	O
schedule	O
them	O
onto	O
the	O
processors	O
.	O
</s>
<s>
Some	O
such	O
settings	O
are	O
handled	O
by	O
open	B-Algorithm
shop	I-Algorithm
scheduling	I-Algorithm
,	O
flow	B-Algorithm
shop	I-Algorithm
scheduling	I-Algorithm
and	O
job	B-Algorithm
shop	I-Algorithm
scheduling	I-Algorithm
.	O
</s>
