<s>
The	O
circulation	B-Algorithm
problem	I-Algorithm
and	O
its	O
variants	O
are	O
a	O
generalisation	O
of	O
network	B-Algorithm
flow	I-Algorithm
problems	O
,	O
with	O
the	O
added	O
constraint	O
of	O
a	O
lower	O
bound	O
on	O
edge	O
flows	O
,	O
and	O
with	O
flow	B-Algorithm
conservation	I-Algorithm
also	O
being	O
required	O
for	O
the	O
source	O
and	O
sink	O
(	O
i.e.	O
</s>
<s>
Given	O
flow	B-Algorithm
network	I-Algorithm
with	O
:	O
</s>
<s>
Finding	O
a	O
flow	O
assignment	O
satisfying	O
the	O
constraints	O
gives	O
a	O
solution	O
to	O
the	O
given	O
circulation	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
In	O
a	O
multi-commodity	O
circulation	B-Algorithm
problem	I-Algorithm
,	O
you	O
also	O
need	O
to	O
keep	O
track	O
of	O
the	O
flow	O
of	O
the	O
individual	O
commodities	O
:	O
</s>
<s>
For	O
the	O
circulation	B-Algorithm
problem	I-Algorithm
,	O
many	O
polynomial	O
algorithms	O
have	O
been	O
developed	O
(	O
e.g.	O
,	O
Edmonds	B-Algorithm
–	I-Algorithm
Karp	I-Algorithm
algorithm	I-Algorithm
,	O
1972	O
;	O
Tarjan	O
1987-1988	O
)	O
.	O
</s>
<s>
For	O
fractional	O
flows	O
,	O
it	O
is	O
solvable	O
in	O
polynomial	O
time	O
,	O
as	O
one	O
can	O
formulate	O
the	O
problem	O
as	O
a	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
Minimum	O
cost	O
multi-commodity	O
circulation	B-Algorithm
problem	I-Algorithm
-	O
Using	O
all	O
constraints	O
given	O
above	O
.	O
</s>
<s>
Multi-commodity	B-Algorithm
flow	I-Algorithm
-	O
If	O
denotes	O
a	O
demand	O
of	O
for	O
commodity	O
from	O
to	O
,	O
create	O
an	O
edge	O
with	O
for	O
all	O
commodities	O
.	O
</s>
<s>
Minimum	B-Algorithm
cost	I-Algorithm
multi-commodity	I-Algorithm
flow	I-Algorithm
problem	I-Algorithm
-	O
As	O
above	O
,	O
but	O
minimize	O
the	O
cost	O
.	O
</s>
<s>
Minimum	B-Algorithm
cost	I-Algorithm
flow	I-Algorithm
problem	I-Algorithm
-	O
As	O
above	O
,	O
with	O
1	O
commodity	O
.	O
</s>
<s>
Maximum	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
-	O
Set	O
all	O
costs	O
to	O
0	O
,	O
and	O
add	O
an	O
edge	O
from	O
the	O
sink	O
to	O
the	O
source	O
with	O
,	O
∞	O
and	O
.	O
</s>
<s>
Minimum	B-Algorithm
cost	I-Algorithm
maximum	I-Algorithm
flow	I-Algorithm
problem	I-Algorithm
-	O
First	O
find	O
the	O
maximum	B-Algorithm
flow	I-Algorithm
amount	O
.	O
</s>
