<s>
The	O
multi-commodity	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
is	O
a	O
network	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
with	O
multiple	O
commodities	O
(	O
flow	O
demands	O
)	O
between	O
different	O
source	O
and	O
sink	O
nodes	O
.	O
</s>
<s>
Given	O
a	O
flow	B-Algorithm
network	I-Algorithm
,	O
where	O
edge	O
has	O
capacity	O
.	O
</s>
<s>
(	O
2	O
)	O
Flow	B-Algorithm
conservation	I-Algorithm
on	O
transit	O
nodes	O
:	O
The	O
amount	O
of	O
a	O
flow	O
entering	O
an	O
intermediate	O
node	O
is	O
the	O
same	O
that	O
exits	O
the	O
node	O
.	O
</s>
<s>
(	O
3	O
)	O
Flow	B-Algorithm
conservation	I-Algorithm
at	O
the	O
source	O
:	O
A	O
flow	O
must	O
exit	O
its	O
source	O
node	O
completely	O
.	O
</s>
<s>
(	O
4	O
)	O
Flow	B-Algorithm
conservation	I-Algorithm
at	O
the	O
destination	O
:	O
A	O
flow	O
must	O
enter	O
its	O
sink	O
node	O
completely	O
.	O
</s>
<s>
In	O
the	O
minimum	B-Algorithm
cost	I-Algorithm
multi-commodity	I-Algorithm
flow	I-Algorithm
problem	I-Algorithm
,	O
there	O
is	O
a	O
cost	O
for	O
sending	O
a	O
flow	O
on	O
.	O
</s>
<s>
The	O
minimum	O
cost	O
variant	O
of	O
the	O
multi-commodity	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
is	O
a	O
generalization	O
of	O
the	O
minimum	B-Algorithm
cost	I-Algorithm
flow	I-Algorithm
problem	I-Algorithm
(	O
in	O
which	O
there	O
is	O
merely	O
one	O
source	O
and	O
one	O
sink	O
.	O
</s>
<s>
Variants	O
of	O
the	O
circulation	B-Algorithm
problem	I-Algorithm
are	O
generalizations	O
of	O
all	O
flow	B-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
That	O
is	O
,	O
any	O
flow	B-Algorithm
problem	I-Algorithm
can	O
be	O
viewed	O
as	O
a	O
particular	O
circulation	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Routing	B-Protocol
and	I-Protocol
wavelength	I-Protocol
assignment	I-Protocol
(	O
RWA	O
)	O
in	O
optical	B-Protocol
burst	I-Protocol
switching	I-Protocol
of	O
Optical	B-Protocol
Network	I-Protocol
would	O
be	O
approached	O
via	O
multi-commodity	B-Algorithm
flow	I-Algorithm
formulas	O
.	O
</s>
<s>
Register	O
allocation	O
can	O
be	O
modeled	O
as	O
an	O
integer	O
minimum	B-Algorithm
cost	I-Algorithm
multi-commodity	I-Algorithm
flow	I-Algorithm
problem	I-Algorithm
:	O
Values	O
produced	O
by	O
instructions	O
are	O
source	O
nodes	O
,	O
values	O
consumed	O
by	O
instructions	O
are	O
sink	O
nodes	O
and	O
registers	O
as	O
well	O
as	O
stack	O
slots	O
are	O
edges	O
.	O
</s>
<s>
If	O
fractional	O
flows	O
are	O
allowed	O
,	O
the	O
problem	O
can	O
be	O
solved	O
in	O
polynomial	O
time	O
through	O
linear	B-Algorithm
programming	I-Algorithm
,	O
or	O
through	O
(	O
typically	O
much	O
faster	O
)	O
fully	B-Algorithm
polynomial	I-Algorithm
time	I-Algorithm
approximation	I-Algorithm
schemes	I-Algorithm
.	O
</s>
