<s>
Birkhoff	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
(	O
also	O
called	O
Birkhoff-von-Neumann	O
algorithm	O
)	O
is	O
an	O
algorithm	O
for	O
decomposing	O
a	O
bistochastic	B-Algorithm
matrix	I-Algorithm
into	O
a	O
convex	O
combination	O
of	O
permutation	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
One	O
such	O
application	O
is	O
for	O
the	O
problem	O
of	O
fair	O
random	O
assignment	O
:	O
given	O
a	O
randomized	O
allocation	O
of	O
items	O
,	O
Birkhoff	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
can	O
decompose	O
it	O
into	O
a	O
lottery	O
on	O
deterministic	O
allocations	O
.	O
</s>
<s>
A	O
bistochastic	B-Algorithm
matrix	I-Algorithm
(	O
also	O
called	O
:	O
doubly-stochastic	O
)	O
is	O
a	O
matrix	O
in	O
which	O
all	O
elements	O
are	O
greater	O
than	O
or	O
equal	O
to	O
0	O
and	O
the	O
sum	O
of	O
the	O
elements	O
in	O
each	O
row	O
and	O
column	O
equals	O
1	O
.	O
</s>
<s>
A	O
permutation	B-Algorithm
matrix	I-Algorithm
is	O
a	O
special	O
case	O
of	O
a	O
bistochastic	B-Algorithm
matrix	I-Algorithm
,	O
in	O
which	O
each	O
element	O
is	O
either	O
0	O
or	O
1	O
(	O
so	O
there	O
is	O
exactly	O
one	O
"	O
1	O
"	O
in	O
each	O
row	O
and	O
each	O
column	O
)	O
.	O
</s>
<s>
A	O
Birkhoff	O
decomposition	O
(	O
also	O
called	O
:	O
Birkhoff-von-Neumann	O
decomposition	O
)	O
of	O
a	O
bistochastic	B-Algorithm
matrix	I-Algorithm
is	O
a	O
presentation	O
of	O
it	O
as	O
a	O
sum	O
of	O
permutation	B-Algorithm
matrices	I-Algorithm
with	O
non-negative	O
weights	O
.	O
</s>
<s>
Birkhoff	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
receives	O
as	O
input	O
a	O
bistochastic	B-Algorithm
matrix	I-Algorithm
and	O
returns	O
as	O
output	O
a	O
Birkhoff	O
decomposition	O
.	O
</s>
<s>
A	O
theorem	O
by	O
Dénes	O
Kőnig	O
says	O
that	O
:	O
Every	O
bistochastic	B-Algorithm
matrix	I-Algorithm
has	O
a	O
permutation-set	O
in	O
which	O
all	O
entries	O
are	O
positive.The	O
positivity	O
graph	O
of	O
an	O
n-by-n	O
matrix	O
X	O
is	O
a	O
bipartite	O
graph	O
with	O
2n	O
vertices	O
,	O
in	O
which	O
the	O
vertices	O
on	O
one	O
side	O
are	O
n	O
rows	O
and	O
the	O
vertices	O
on	O
the	O
other	O
side	O
are	O
the	O
n	O
columns	O
,	O
and	O
there	O
is	O
an	O
edge	O
between	O
a	O
row	O
and	O
a	O
column	O
iff	O
the	O
entry	O
at	O
that	O
row	O
and	O
column	O
is	O
positive	O
.	O
</s>
<s>
Kőnig	O
's	O
theorem	O
is	O
equivalent	O
to	O
the	O
following:The	O
positivity	O
graph	O
of	O
any	O
bistochastic	B-Algorithm
matrix	I-Algorithm
admits	O
a	O
perfect	O
matching.A	O
matrix	O
is	O
called	O
scaled-bistochastic	O
if	O
all	O
elements	O
are	O
weakly-positive	O
,	O
and	O
the	O
sum	O
of	O
each	O
row	O
and	O
column	O
equals	O
c	O
,	O
where	O
c	O
is	O
some	O
positive	O
constant	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
it	O
is	O
c	O
times	O
a	O
bistochastic	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Birkhoff	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
:	O
it	O
greedily	O
finds	O
perfect	O
matchings	O
and	O
removes	O
them	O
from	O
the	O
fractional	O
matching	O
.	O
</s>
<s>
Let	O
P[i]	O
be	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
with	O
1	O
in	O
the	O
positive	O
permutation	O
set	O
.	O
</s>
<s>
In	O
1960	O
,	O
Joshnson	O
,	O
Dulmage	O
and	O
Mendelsohn	O
showed	O
that	O
Birkhoff	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
actually	O
ends	O
after	O
at	O
most	O
n2	O
−	O
2n	O
+	O
2	O
steps	O
,	O
which	O
is	O
tight	O
in	O
general	O
(	O
that	O
is	O
,	O
in	O
some	O
cases	O
n2	O
−	O
2n	O
+	O
2	O
permutation	B-Algorithm
matrices	I-Algorithm
may	O
be	O
required	O
)	O
.	O
</s>
<s>
The	O
probabilistic-serial	B-Algorithm
procedure	I-Algorithm
can	O
compute	O
the	O
probabilities	O
such	O
that	O
each	O
agent	O
,	O
looking	O
at	O
the	O
matrix	O
of	O
probabilities	O
,	O
prefers	O
his	O
row	O
of	O
probabilities	O
over	O
the	O
rows	O
of	O
all	O
other	O
people	O
(	O
this	O
property	O
is	O
called	O
envy-freeness	O
)	O
.	O
</s>
<s>
Here	O
,	O
Birkhoff	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
useful	O
.	O
</s>
<s>
Birkhoff	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
can	O
decompose	O
it	O
into	O
a	O
convex	O
combination	O
of	O
permutation	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Each	O
permutation	B-Algorithm
matrix	I-Algorithm
represents	O
a	O
deterministic	O
assignment	O
,	O
in	O
which	O
every	O
agent	O
receives	O
exactly	O
one	O
object	O
.	O
</s>
<s>
Budish	O
,	O
Che	O
,	O
Kojima	O
and	O
Milgrom	O
generalize	O
Birkhoff	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
to	O
non-square	O
matrices	O
,	O
with	O
some	O
constraints	O
on	O
the	O
feasible	O
assignments	O
.	O
</s>
<s>
Vazirani	O
generalizes	O
Birkhoff	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
to	O
non-bipartite	O
graphs	O
.	O
</s>
