<s>
Unrelated-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>
It	O
is	O
a	O
variant	O
of	O
optimal	B-Algorithm
job	I-Algorithm
scheduling	I-Algorithm
.	O
</s>
<s>
We	O
need	O
to	O
schedule	O
n	O
jobs	O
J1	O
,	O
J2	O
,	O
...	O
,	O
Jn	O
on	O
m	O
different	O
machines	O
,	O
such	O
that	O
a	O
certain	O
objective	O
function	O
is	O
optimized	O
(	O
usually	O
,	O
the	O
makespan	B-Algorithm
should	O
be	O
minimized	O
)	O
.	O
</s>
<s>
This	O
is	O
in	O
contrast	O
to	O
two	O
special	O
cases	O
of	O
this	O
problem	O
:	O
uniform-machines	B-Algorithm
scheduling	I-Algorithm
-	O
in	O
which	O
pi	O
,	O
j	O
=	O
pi	O
/	O
sj	O
(	O
where	O
sj	O
is	O
the	O
speed	O
of	O
machine	O
j	O
)	O
,	O
and	O
identical-machines	B-Algorithm
scheduling	I-Algorithm
-	O
in	O
which	O
pi	O
,	O
j	O
=	O
pi	O
(	O
the	O
same	O
run-time	O
on	O
all	O
machines	O
)	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
unrelated-machines	O
variant	O
is	O
denoted	O
by	O
R	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
R||	O
"	O
is	O
an	O
unrelated-machines	B-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
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>
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>
Glass	O
,	O
Potts	O
and	O
Shade	O
compare	O
various	O
local	B-Algorithm
search	I-Algorithm
techniques	O
for	O
minimizing	O
the	O
makespan	B-Algorithm
on	O
unrelated	O
machines	O
.	O
</s>
<s>
Using	O
computerized	O
simulations	O
,	O
they	O
find	O
that	O
tabu	B-Algorithm
search	I-Algorithm
and	O
simulated	B-Algorithm
annealing	I-Algorithm
perform	O
much	O
better	O
than	O
genetic	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
Minimizing	O
the	O
weighted	O
average	O
completion	O
time	O
,	O
R||	O
(	O
where	O
wj	O
is	O
the	O
weight	O
of	O
job	O
j	O
)	O
,	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>
Schulz	O
and	O
Skutella	O
present	O
a	O
(	O
3/2	O
+ε	O
)	O
-approximation	O
algorithm	O
using	O
randomized	B-Algorithm
rounding	I-Algorithm
.	O
</s>
<s>
This	O
problem	O
is	O
equivalent	O
to	O
unrelated-machines	B-Algorithm
scheduling	I-Algorithm
in	O
which	O
the	O
goal	O
is	O
to	O
maximize	O
the	O
minimum	O
completion	O
time	O
.	O
</s>
<s>
Another	O
LP	O
formulation	O
is	O
the	O
configuration	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
.	O
</s>
<s>
Note	O
that	O
the	O
number	O
of	O
configurations	O
is	O
usually	O
exponential	O
in	O
the	O
size	O
of	O
the	O
problem	O
,	O
so	O
the	O
size	O
of	O
the	O
configuration	B-Algorithm
LP	I-Algorithm
is	O
exponential	O
.	O
</s>
<s>
They	O
present	O
a	O
solution	O
using	O
simulated	B-Algorithm
annealing	I-Algorithm
.	O
</s>
<s>
Vallada	O
and	O
Ruiz	O
present	O
a	O
solution	O
using	O
a	O
genetic	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Caragiannis	O
extends	O
the	O
problem	O
in	O
a	O
different	O
way	O
,	O
by	O
assuming	O
that	O
the	O
jobs	O
are	O
owned	O
by	O
selfish	O
agents	O
(	O
see	O
Truthful	B-Algorithm
job	I-Algorithm
scheduling	I-Algorithm
)	O
.	O
</s>
