<s>
Parallel	B-Algorithm
task	I-Algorithm
scheduling	I-Algorithm
(	O
also	O
called	O
parallel	O
job	O
scheduling	O
or	O
parallel	O
processing	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>
In	O
a	O
general	O
job	O
scheduling	O
problem	O
,	O
we	O
are	O
given	O
n	O
jobs	O
J1	O
,	O
J2	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
machines	O
while	O
trying	O
to	O
minimize	O
the	O
makespan	B-Algorithm
-	O
the	O
total	O
length	O
of	O
the	O
schedule	O
(	O
that	O
is	O
,	O
when	O
all	O
the	O
jobs	O
have	O
finished	O
processing	O
)	O
.	O
</s>
<s>
and	O
Drozdowski	O
denote	O
this	O
problem	O
by	O
in	O
the	O
three-field	B-Algorithm
notation	I-Algorithm
introduced	O
by	O
Graham	O
et	O
al	O
.	O
</s>
<s>
Note	O
that	O
the	O
problem	O
of	O
parallel-machines	B-Algorithm
scheduling	I-Algorithm
is	O
a	O
special	O
case	O
of	O
parallel-task	O
scheduling	O
where	O
for	O
all	O
j	O
,	O
that	O
is	O
,	O
each	O
job	O
should	O
run	O
on	O
a	O
single	O
machine	O
.	O
</s>
<s>
The	O
objective	O
of	O
the	O
problem	O
denoted	O
by	O
is	O
to	O
find	O
a	O
schedule	O
with	O
minimum	O
length	O
,	O
also	O
called	O
the	O
makespan	B-Algorithm
of	O
the	O
schedule	O
.	O
</s>
<s>
This	O
special	O
case	O
,	O
denoted	O
by	O
,	O
is	O
a	O
variant	O
of	O
the	O
partition	B-Algorithm
problem	I-Algorithm
,	O
which	O
is	O
known	O
to	O
be	O
NP-hard	O
.	O
</s>
<s>
When	O
the	O
number	O
of	O
machines	O
m	O
is	O
at	O
most	O
3	O
,	O
that	O
is	O
:	O
for	O
the	O
variants	O
and	O
,	O
there	O
exists	O
a	O
pseudo-polynomial	B-Algorithm
time	I-Algorithm
algorithm	O
,	O
which	O
solves	O
the	O
problem	O
exactly	O
.	O
</s>
<s>
This	O
holds	O
even	O
for	O
the	O
special	O
case	O
in	O
which	O
the	O
processing	O
time	O
of	O
all	O
jobs	O
is	O
,	O
since	O
this	O
special	O
case	O
is	O
equivalent	O
to	O
the	O
bin	O
packing	O
problem	O
:	O
each	O
time-step	O
corresponds	O
to	O
a	O
bin	O
,	O
m	O
is	O
the	O
bin	O
size	O
,	O
each	O
job	O
corresponds	O
to	O
an	O
item	O
of	O
size	O
qj	O
,	O
and	O
minimizing	O
the	O
makespan	B-Algorithm
corresponds	O
to	O
minimizing	O
the	O
number	O
of	O
bins	O
.	O
</s>
<s>
Preemption	B-Operating_System
:	O
In	O
this	O
variant	O
,	O
denoted	O
by	O
,	O
it	O
is	O
possible	O
to	O
interrupt	O
jobs	O
that	O
are	O
already	O
running	O
,	O
and	O
schedule	O
other	O
jobs	O
that	O
become	O
available	O
at	O
that	O
time	O
.	O
</s>
<s>
Feldmann	O
,	O
Sgall	O
and	O
Teng	O
observed	O
that	O
the	O
length	O
of	O
a	O
non-preemptive	O
schedule	O
produced	O
by	O
the	O
list	O
scheduling	O
algorithm	O
is	O
actually	O
at	O
most	O
times	O
the	O
optimum	O
preemptive	O
makespan	B-Algorithm
.	O
</s>
<s>
Since	O
,	O
in	O
general	O
,	O
the	O
number	O
of	O
machines	O
appears	O
only	O
in	O
logarithmic	O
in	O
the	O
size	O
of	O
the	O
instance	O
,	O
this	O
algorithm	O
is	O
a	O
pseudo-polynomial	B-Algorithm
time	I-Algorithm
approximation	O
scheme	O
as	O
well	O
.	O
</s>
<s>
Given	O
an	O
instance	O
of	O
the	O
parallel	B-Algorithm
task	I-Algorithm
scheduling	I-Algorithm
problem	I-Algorithm
,	O
the	O
optimal	O
makespan	B-Algorithm
can	O
differ	O
depending	O
on	O
the	O
constraint	O
to	O
the	O
contiguity	O
of	O
the	O
machines	O
.	O
</s>
<s>
If	O
the	O
jobs	O
can	O
be	O
scheduled	O
on	O
non-contiguous	O
machines	O
,	O
the	O
optimal	O
makespan	B-Algorithm
can	O
be	O
smaller	O
than	O
in	O
the	O
case	O
that	O
they	O
have	O
to	O
be	O
scheduled	O
on	O
contiguous	O
ones	O
.	O
</s>
<s>
These	O
are	O
the	O
problems	O
of	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>
