<s>
In	O
graph	O
theory	O
,	O
a	O
flow	B-Algorithm
network	I-Algorithm
(	O
also	O
known	O
as	O
a	O
transportation	B-Algorithm
network	I-Algorithm
)	O
is	O
a	O
directed	O
graph	O
where	O
each	O
edge	O
has	O
a	O
capacity	O
and	O
each	O
edge	O
receives	O
a	O
flow	O
.	O
</s>
<s>
If	O
two	O
nodes	O
in	O
are	O
distinguished	O
–	O
one	O
as	O
the	O
source	O
and	O
the	O
other	O
as	O
the	O
sink	O
–	O
then	O
is	O
called	O
a	O
flow	B-Algorithm
network	I-Algorithm
.	O
</s>
<s>
Flow	B-Algorithm
functions	I-Algorithm
model	O
the	O
net	O
flow	O
of	O
units	O
between	O
pairs	O
of	O
nodes	O
,	O
and	O
are	O
useful	O
when	O
asking	O
questions	O
such	O
as	O
what	O
is	O
the	O
maximum	O
number	O
of	O
units	O
that	O
can	O
be	O
transferred	O
from	O
the	O
source	O
node	O
s	O
to	O
the	O
sink	O
node	O
t	O
?	O
</s>
<s>
In	O
flow	B-Algorithm
networks	I-Algorithm
,	O
the	O
source	O
is	O
deficient	O
,	O
and	O
the	O
sink	O
is	O
active	O
.	O
</s>
<s>
Pseudo-flows	O
,	O
feasible	O
flows	O
,	O
and	O
pre-flows	O
are	O
all	O
examples	O
of	O
flow	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
Flow	B-Algorithm
conservation	I-Algorithm
constraint	O
:	O
The	O
total	O
net	O
flow	O
entering	O
a	O
node	O
is	O
zero	O
for	O
all	O
nodes	O
in	O
the	O
network	O
except	O
the	O
source	O
and	O
the	O
sink	O
,	O
that	O
is	O
:	O
for	O
all	O
.	O
</s>
<s>
The	O
value	O
of	O
a	O
feasible	O
flow	O
for	O
a	O
network	O
,	O
is	O
the	O
net	O
flow	O
into	O
the	O
sink	O
of	O
the	O
flow	B-Algorithm
network	I-Algorithm
,	O
that	O
is	O
:	O
.	O
</s>
<s>
This	O
concept	O
is	O
used	O
in	O
Ford	B-Algorithm
–	I-Algorithm
Fulkerson	I-Algorithm
algorithm	I-Algorithm
which	O
computes	O
the	O
maximum	B-Algorithm
flow	I-Algorithm
in	O
a	O
flow	B-Algorithm
network	I-Algorithm
.	O
</s>
<s>
A	O
network	O
is	O
at	O
maximum	B-Algorithm
flow	I-Algorithm
if	O
and	O
only	O
if	O
there	O
is	O
no	O
augmenting	O
path	O
in	O
the	O
residual	O
network	O
.	O
</s>
<s>
The	O
flow	B-Algorithm
network	I-Algorithm
is	O
at	O
maximum	B-Algorithm
flow	I-Algorithm
if	O
and	O
only	O
if	O
it	O
has	O
a	O
bottleneck	O
with	O
a	O
value	O
greater	O
than	O
zero	O
.	O
</s>
<s>
Sometimes	O
,	O
when	O
modeling	O
a	O
network	O
with	O
more	O
than	O
one	O
source	O
,	O
a	O
supersource	B-Algorithm
is	O
introduced	O
to	O
the	O
graph	O
.	O
</s>
<s>
A	O
similar	O
construct	O
for	O
sinks	O
is	O
called	O
a	O
supersink	B-Algorithm
.	O
</s>
<s>
In	O
Figure	O
1	O
you	O
see	O
a	O
flow	B-Algorithm
network	I-Algorithm
with	O
source	O
labeled	O
,	O
sink	O
,	O
and	O
four	O
additional	O
nodes	O
.	O
</s>
<s>
Notice	O
how	O
the	O
network	O
upholds	O
the	O
skew	O
symmetry	O
constraint	O
,	O
capacity	O
constraint	O
,	O
and	O
flow	B-Algorithm
conservation	I-Algorithm
constraint	O
.	O
</s>
<s>
This	O
network	O
is	O
not	O
at	O
maximum	B-Algorithm
flow	I-Algorithm
.	O
</s>
<s>
Flows	O
can	O
pertain	O
to	O
people	O
or	O
material	O
over	O
transportation	B-Algorithm
networks	I-Algorithm
,	O
or	O
to	O
electricity	O
over	O
electrical	O
distribution	O
systems	O
.	O
</s>
<s>
Flow	B-Algorithm
networks	I-Algorithm
also	O
find	O
applications	O
in	O
ecology	O
:	O
flow	B-Algorithm
networks	I-Algorithm
arise	O
naturally	O
when	O
considering	O
the	O
flow	O
of	O
nutrients	O
and	O
energy	O
between	O
different	O
organisms	O
in	O
a	O
food	O
web	O
.	O
</s>
<s>
The	O
mathematical	O
problems	O
associated	O
with	O
such	O
networks	O
are	O
quite	O
different	O
from	O
those	O
that	O
arise	O
in	O
networks	O
of	O
fluid	O
or	O
traffic	B-Operating_System
flow	I-Operating_System
.	O
</s>
<s>
The	O
simplest	O
and	O
most	O
common	O
problem	O
using	O
flow	B-Algorithm
networks	I-Algorithm
is	O
to	O
find	O
what	O
is	O
called	O
the	O
maximum	B-Algorithm
flow	I-Algorithm
,	O
which	O
provides	O
the	O
largest	O
possible	O
total	O
flow	O
from	O
the	O
source	O
to	O
the	O
sink	O
in	O
a	O
given	O
graph	O
.	O
</s>
<s>
There	O
are	O
many	O
other	O
problems	O
which	O
can	O
be	O
solved	O
using	O
max	B-Algorithm
flow	I-Algorithm
algorithms	O
,	O
if	O
they	O
are	O
appropriately	O
modeled	O
as	O
flow	B-Algorithm
networks	I-Algorithm
,	O
such	O
as	O
bipartite	O
matching	O
,	O
the	O
assignment	B-Algorithm
problem	I-Algorithm
and	O
the	O
transportation	O
problem	O
.	O
</s>
<s>
Maximum	B-Algorithm
flow	I-Algorithm
problems	I-Algorithm
can	O
be	O
solved	O
efficiently	O
with	O
the	O
push	B-Algorithm
–	I-Algorithm
relabel	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
max-flow	B-Algorithm
min-cut	I-Algorithm
theorem	I-Algorithm
states	O
that	O
finding	O
a	O
maximal	O
network	B-Algorithm
flow	I-Algorithm
is	O
equivalent	O
to	O
finding	O
a	O
cut	B-Algorithm
of	O
minimum	O
capacity	O
that	O
separates	O
the	O
source	O
and	O
the	O
sink	O
,	O
where	O
a	O
cut	B-Algorithm
is	O
the	O
division	O
of	O
vertices	O
such	O
that	O
the	O
source	O
is	O
in	O
one	O
division	O
and	O
the	O
sink	O
is	O
in	O
another	O
.	O
</s>
<s>
In	O
a	O
multi-commodity	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
,	O
you	O
have	O
multiple	O
sources	O
and	O
sinks	O
,	O
and	O
various	O
"	O
commodities	O
"	O
which	O
are	O
to	O
flow	O
from	O
a	O
given	O
source	O
to	O
a	O
given	O
sink	O
.	O
</s>
<s>
This	O
could	O
be	O
for	O
example	O
various	O
goods	O
that	O
are	O
produced	O
at	O
various	O
factories	O
,	O
and	O
are	O
to	O
be	O
delivered	O
to	O
various	O
given	O
customers	O
through	O
the	O
same	O
transportation	B-Algorithm
network	I-Algorithm
.	O
</s>
<s>
In	O
a	O
minimum	B-Algorithm
cost	I-Algorithm
flow	I-Algorithm
problem	I-Algorithm
,	O
each	O
edge	O
has	O
a	O
given	O
cost	O
,	O
and	O
the	O
cost	O
of	O
sending	O
the	O
flow	O
across	O
the	O
edge	O
is	O
.	O
</s>
<s>
In	O
a	O
circulation	B-Algorithm
problem	I-Algorithm
,	O
you	O
have	O
a	O
lower	O
bound	O
on	O
the	O
edges	O
,	O
in	O
addition	O
to	O
the	O
upper	O
bound	O
.	O
</s>
<s>
Often	O
,	O
flow	B-Algorithm
conservation	I-Algorithm
holds	O
for	O
all	O
nodes	O
in	O
a	O
circulation	B-Algorithm
problem	I-Algorithm
,	O
and	O
there	O
is	O
a	O
connection	O
from	O
the	O
sink	O
back	O
to	O
the	O
source	O
.	O
</s>
