<s>
The	O
minimum-cost	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
(	O
MCFP	O
)	O
is	O
an	O
optimization	O
and	O
decision	O
problem	O
to	O
find	O
the	O
cheapest	O
possible	O
way	O
of	O
sending	O
a	O
certain	O
amount	O
of	O
flow	O
through	O
a	O
flow	B-Algorithm
network	I-Algorithm
.	O
</s>
<s>
The	O
minimum	B-Algorithm
cost	I-Algorithm
flow	I-Algorithm
problem	I-Algorithm
is	O
one	O
of	O
the	O
most	O
fundamental	O
among	O
all	O
flow	O
and	O
circulation	B-Algorithm
problems	I-Algorithm
because	O
most	O
other	O
such	O
problems	O
can	O
be	O
cast	O
as	O
a	O
minimum	B-Algorithm
cost	I-Algorithm
flow	I-Algorithm
problem	I-Algorithm
and	O
also	O
that	O
it	O
can	O
be	O
solved	O
efficiently	O
using	O
the	O
network	B-Algorithm
simplex	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
A	O
flow	B-Algorithm
network	I-Algorithm
is	O
a	O
directed	O
graph	O
with	O
a	O
source	O
vertex	O
and	O
a	O
sink	O
vertex	O
,	O
where	O
each	O
edge	O
has	O
capacity	O
,	O
flow	O
and	O
cost	O
,	O
with	O
most	O
minimum-cost	O
flow	B-Algorithm
algorithms	I-Algorithm
supporting	O
edges	O
with	O
negative	O
costs	O
.	O
</s>
<s>
A	O
variation	O
of	O
this	O
problem	O
is	O
to	O
find	O
a	O
flow	O
which	O
is	O
maximum	O
,	O
but	O
has	O
the	O
lowest	O
cost	O
among	O
the	O
maximum	B-Algorithm
flow	I-Algorithm
solutions	O
.	O
</s>
<s>
This	O
could	O
be	O
called	O
a	O
minimum-cost	B-Algorithm
maximum-flow	I-Algorithm
problem	I-Algorithm
and	O
is	O
useful	O
for	O
finding	O
minimum	O
cost	O
maximum	O
matchings	O
.	O
</s>
<s>
With	O
some	O
solutions	O
,	O
finding	O
the	O
minimum	O
cost	O
maximum	B-Algorithm
flow	I-Algorithm
instead	O
is	O
straightforward	O
.	O
</s>
<s>
If	O
not	O
,	O
one	O
can	O
find	O
the	O
maximum	B-Algorithm
flow	I-Algorithm
by	O
performing	O
a	O
binary	O
search	O
on	O
.	O
</s>
<s>
A	O
related	O
problem	O
is	O
the	O
minimum	B-Algorithm
cost	I-Algorithm
circulation	I-Algorithm
problem	I-Algorithm
,	O
which	O
can	O
be	O
used	O
for	O
solving	O
minimum	B-Algorithm
cost	I-Algorithm
flow	I-Algorithm
.	O
</s>
<s>
The	O
following	O
problems	O
are	O
special	O
cases	O
of	O
the	O
minimum	B-Algorithm
cost	I-Algorithm
flow	I-Algorithm
problem	I-Algorithm
(	O
we	O
provide	O
brief	O
sketches	O
of	O
each	O
applicable	O
reduction	O
,	O
in	O
turn	O
)	O
:	O
</s>
<s>
Require	O
that	O
a	O
feasible	O
solution	O
to	O
the	O
minimum	B-Algorithm
cost	I-Algorithm
flow	I-Algorithm
problem	I-Algorithm
sends	O
one	O
unit	O
of	O
flow	O
from	O
a	O
designated	O
source	O
to	O
a	O
designated	O
sink	O
.	O
</s>
<s>
Maximum	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
(	O
This	O
reduction	O
is	O
also	O
mentioned	O
in	O
Circulation	B-Algorithm
problem	I-Algorithm
)	O
.	O
</s>
<s>
Assignment	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
minimum	B-Algorithm
cost	I-Algorithm
flow	I-Algorithm
problem	I-Algorithm
can	O
be	O
solved	O
by	O
linear	B-Algorithm
programming	I-Algorithm
,	O
since	O
we	O
optimize	O
a	O
linear	O
function	O
,	O
and	O
all	O
constraints	O
are	O
linear	O
.	O
</s>
<s>
Some	O
of	O
them	O
are	O
generalizations	O
of	O
maximum	B-Algorithm
flow	I-Algorithm
algorithms	I-Algorithm
,	O
others	O
use	O
entirely	O
different	O
approaches	O
.	O
</s>
<s>
Successive	O
shortest	O
path	O
and	O
capacity	O
scaling	O
:	O
dual	O
methods	O
,	O
which	O
can	O
be	O
viewed	O
as	O
the	O
generalization	O
of	O
the	O
Ford	B-Algorithm
–	I-Algorithm
Fulkerson	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Cost	O
scaling	O
:	O
a	O
primal-dual	O
approach	O
,	O
which	O
can	O
be	O
viewed	O
as	O
the	O
generalization	O
of	O
the	O
push-relabel	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Network	B-Algorithm
simplex	I-Algorithm
algorithm	I-Algorithm
:	O
a	O
specialized	O
version	O
of	O
the	O
linear	B-Algorithm
programming	I-Algorithm
simplex	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
idea	O
is	O
to	O
reduce	O
this	O
problem	O
to	O
a	O
network	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
It	O
is	O
proved	O
that	O
there	O
is	O
minimum	O
weight	O
perfect	O
bipartite	O
matching	O
in	O
G	O
if	O
and	O
only	O
if	O
there	O
a	O
minimum	B-Algorithm
cost	I-Algorithm
flow	I-Algorithm
in	O
G′	O
.	O
</s>
