<s>
The	O
assignment	B-Algorithm
problem	I-Algorithm
is	O
a	O
fundamental	O
combinatorial	O
optimization	O
problem	O
.	O
</s>
<s>
The	O
assignment	B-Algorithm
problem	I-Algorithm
consists	O
of	O
finding	O
,	O
in	O
a	O
weighted	B-Algorithm
bipartite	O
graph	O
,	O
a	O
matching	O
of	O
a	O
given	O
size	O
,	O
in	O
which	O
the	O
sum	O
of	O
weights	O
of	O
the	O
edges	O
is	O
minimum	O
.	O
</s>
<s>
Commonly	O
,	O
when	O
speaking	O
of	O
the	O
assignment	B-Algorithm
problem	I-Algorithm
without	O
any	O
additional	O
qualification	O
,	O
then	O
the	O
linear	O
balanced	O
assignment	B-Algorithm
problem	I-Algorithm
is	O
meant	O
.	O
</s>
<s>
This	O
is	O
a	O
balanced	O
assignment	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
This	O
is	O
an	O
unbalanced	O
assignment	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
This	O
reduces	O
the	O
problem	O
to	O
a	O
balanced	O
assignment	B-Algorithm
problem	I-Algorithm
,	O
which	O
can	O
then	O
be	O
solved	O
in	O
the	O
usual	O
way	O
and	O
still	O
give	O
the	O
best	O
solution	O
to	O
the	O
problem	O
.	O
</s>
<s>
Given	O
two	O
sets	O
,	O
A	O
and	O
T	O
,	O
of	O
equal	O
size	O
,	O
together	O
with	O
a	O
weight	B-Algorithm
function	I-Algorithm
C	O
:	O
A	O
T	O
R	O
.	O
Find	O
a	O
bijection	B-Algorithm
f	O
:	O
A	O
T	O
such	O
that	O
the	O
cost	O
function	O
:	O
</s>
<s>
Usually	O
the	O
weight	B-Algorithm
function	I-Algorithm
is	O
viewed	O
as	O
a	O
square	O
real-valued	O
matrix	B-Architecture
C	O
,	O
so	O
that	O
the	O
cost	O
function	O
is	O
written	O
down	O
as	O
:	O
</s>
<s>
A	O
naive	O
solution	O
for	O
the	O
assignment	B-Algorithm
problem	I-Algorithm
is	O
to	O
check	O
all	O
the	O
assignments	O
and	O
calculate	O
the	O
cost	O
of	O
each	O
one	O
.	O
</s>
<s>
The	O
assignment	B-Algorithm
problem	I-Algorithm
is	O
a	O
special	O
case	O
of	O
the	O
transportation	O
problem	O
,	O
which	O
is	O
a	O
special	O
case	O
of	O
the	O
minimum	B-Algorithm
cost	I-Algorithm
flow	I-Algorithm
problem	I-Algorithm
,	O
which	O
in	O
turn	O
is	O
a	O
special	O
case	O
of	O
a	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
While	O
it	O
is	O
possible	O
to	O
solve	O
any	O
of	O
these	O
problems	O
using	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
,	O
each	O
specialization	O
has	O
a	O
smaller	O
solution	O
space	O
and	O
thus	O
more	O
efficient	O
algorithms	O
designed	O
to	O
take	O
advantage	O
of	O
its	O
special	O
structure	O
.	O
</s>
<s>
In	O
the	O
balanced	O
assignment	B-Algorithm
problem	I-Algorithm
,	O
both	O
parts	O
of	O
the	O
bipartite	O
graph	O
have	O
the	O
same	O
number	O
of	O
vertices	O
,	O
denoted	O
by	O
n	O
.	O
</s>
<s>
One	O
of	O
the	O
first	O
polynomial-time	O
algorithms	O
for	O
balanced	O
assignment	O
was	O
the	O
Hungarian	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Its	O
run-time	O
complexity	O
,	O
when	O
using	O
Fibonacci	B-Application
heaps	I-Application
,	O
is	O
,	O
where	O
m	O
is	O
a	O
number	O
of	O
edges	O
.	O
</s>
<s>
These	O
algorithms	O
are	O
called	O
auction	B-Algorithm
algorithms	I-Algorithm
,	O
push-relabel	O
algorithms	O
,	O
or	O
preflow-push	O
algorithms	O
.	O
</s>
<s>
As	O
shown	O
by	O
Mulmuley	O
,	O
Vazirani	O
and	O
Vazirani	O
,	O
the	O
problem	O
of	O
minimum	O
weight	O
perfect	O
matching	O
is	O
converted	O
to	O
finding	O
minors	O
in	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
a	O
graph	O
.	O
</s>
<s>
In	O
the	O
unbalanced	O
assignment	B-Algorithm
problem	I-Algorithm
,	O
the	O
larger	O
part	O
of	O
the	O
bipartite	O
graph	O
has	O
n	O
vertices	O
and	O
the	O
smaller	O
part	O
has	O
r	O
<	O
n	O
vertices	O
.	O
</s>
<s>
Instead	O
of	O
using	O
reduction	O
,	O
the	O
unbalanced	O
assignment	B-Algorithm
problem	I-Algorithm
can	O
be	O
solved	O
by	O
directly	O
generalizing	O
existing	O
algorithms	O
for	O
balanced	O
assignment	O
.	O
</s>
<s>
The	O
Hungarian	B-Algorithm
algorithm	I-Algorithm
can	O
be	O
generalized	O
to	O
solve	O
the	O
problem	O
in	O
strongly-polynomial	O
time	O
.	O
</s>
<s>
The	O
assignment	B-Algorithm
problem	I-Algorithm
can	O
be	O
solved	O
by	O
presenting	O
it	O
as	O
a	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
This	O
is	O
an	O
integer	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
However	O
,	O
we	O
can	O
solve	O
it	O
without	O
the	O
integrality	O
constraints	O
(	O
i.e.	O
,	O
drop	O
the	O
last	O
constraint	O
)	O
,	O
using	O
standard	O
methods	O
for	O
solving	O
continuous	O
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
This	O
is	O
because	O
the	O
constraint	O
matrix	B-Architecture
of	O
the	O
fractional	O
LP	O
is	O
totally	O
unimodular	O
–	O
it	O
satisfies	O
the	O
four	O
conditions	O
of	O
Hoffman	O
and	O
Gale	O
.	O
</s>
<s>
Other	O
approaches	O
for	O
the	O
assignment	B-Algorithm
problem	I-Algorithm
exist	O
and	O
are	O
reviewed	O
by	O
Duan	O
and	O
Pettie	O
(	O
see	O
Table	O
II	O
)	O
.	O
</s>
<s>
Their	O
work	O
proposes	O
an	O
approximation	B-Algorithm
algorithm	I-Algorithm
for	O
the	O
assignment	B-Algorithm
problem	I-Algorithm
(	O
and	O
the	O
more	O
general	O
maximum	B-Algorithm
weight	I-Algorithm
matching	I-Algorithm
problem	I-Algorithm
)	O
,	O
which	O
runs	O
in	O
linear	O
time	O
for	O
any	O
fixed	O
error	O
bound	O
.	O
</s>
<s>
When	O
phrased	O
as	O
a	O
graph	O
theory	O
problem	O
,	O
the	O
assignment	B-Algorithm
problem	I-Algorithm
can	O
be	O
extended	O
from	O
bipartite	O
graphs	O
to	O
arbitrary	O
graphs	O
.	O
</s>
<s>
The	O
corresponding	O
problem	O
,	O
of	O
finding	O
a	O
matching	O
in	O
a	O
weighted	B-Algorithm
graph	O
where	O
the	O
sum	O
of	O
weights	O
is	O
maximized	O
,	O
is	O
called	O
the	O
maximum	B-Algorithm
weight	I-Algorithm
matching	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Another	O
generalization	O
of	O
the	O
assignment	B-Algorithm
problem	I-Algorithm
is	O
extending	O
the	O
number	O
of	O
sets	O
to	O
be	O
matched	O
from	O
two	O
to	O
many	O
.	O
</s>
<s>
This	O
results	O
in	O
Multidimensional	B-Algorithm
assignment	I-Algorithm
problem	I-Algorithm
(	O
MAP	O
)	O
.	O
</s>
