<s>
The	O
Birkhoff	B-Algorithm
polytope	I-Algorithm
Bn	O
(	O
also	O
called	O
the	O
assignment	B-Algorithm
polytope	I-Algorithm
,	O
the	O
polytope	O
of	O
doubly	B-Algorithm
stochastic	I-Algorithm
matrices	I-Algorithm
,	O
or	O
the	O
perfect	O
matching	O
polytope	O
of	O
the	O
complete	O
bipartite	O
graph	O
)	O
is	O
the	O
convex	O
polytope	O
in	O
RN	O
(	O
where	O
N	O
=	O
n2	O
)	O
whose	O
points	O
are	O
the	O
doubly	B-Algorithm
stochastic	I-Algorithm
matrices	I-Algorithm
,	O
i.e.	O
,	O
the	O
matrices	B-Architecture
whose	O
entries	O
are	O
non-negative	O
real	O
numbers	O
and	O
whose	O
rows	O
and	O
columns	O
each	O
add	O
up	O
to	O
1	O
.	O
</s>
<s>
The	O
Birkhoff	B-Algorithm
polytope	I-Algorithm
has	O
n	O
!	O
</s>
<s>
This	O
follows	O
from	O
the	O
Birkhoff	B-Algorithm
–	I-Algorithm
von	I-Algorithm
Neumann	I-Algorithm
theorem	I-Algorithm
,	O
which	O
states	O
that	O
the	O
extreme	O
points	O
of	O
the	O
Birkhoff	B-Algorithm
polytope	I-Algorithm
are	O
the	O
permutation	B-Algorithm
matrices	I-Algorithm
,	O
and	O
therefore	O
that	O
any	O
doubly	B-Algorithm
stochastic	I-Algorithm
matrix	I-Algorithm
may	O
be	O
represented	O
as	O
a	O
convex	O
combination	O
of	O
permutation	B-Algorithm
matrices	I-Algorithm
;	O
this	O
was	O
stated	O
in	O
a	O
1946	O
paper	O
by	O
Garrett	O
Birkhoff	O
,	O
but	O
equivalent	O
results	O
in	O
the	O
languages	O
of	O
projective	O
configurations	O
and	O
of	O
regular	O
bipartite	O
graph	O
matchings	O
,	O
respectively	O
,	O
were	O
shown	O
much	O
earlier	O
in	O
1894	O
in	O
Ernst	O
Steinitz	O
's	O
thesis	O
and	O
in	O
1916	O
by	O
Dénes	O
Kőnig	O
.	O
</s>
<s>
Because	O
all	O
of	O
the	O
vertex	O
coordinates	O
are	O
zero	O
or	O
one	O
,	O
the	O
Birkhoff	B-Algorithm
polytope	I-Algorithm
is	O
an	O
integral	O
polytope	O
.	O
</s>
<s>
The	O
edges	O
of	O
the	O
Birkhoff	B-Algorithm
polytope	I-Algorithm
correspond	O
to	O
pairs	O
of	O
permutations	O
differing	O
by	O
a	O
cycle	O
:	O
</s>
<s>
This	O
implies	O
that	O
the	O
graph	O
of	O
Bn	O
is	O
a	O
Cayley	O
graph	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
Sn	O
.	O
</s>
<s>
The	O
Birkhoff	B-Algorithm
polytope	I-Algorithm
lies	O
within	O
an	O
dimensional	O
affine	O
subspace	O
of	O
the	O
n2-dimensional	O
space	O
of	O
all	O
matrices	B-Architecture
:	O
this	O
subspace	O
is	O
determined	O
by	O
the	O
linear	O
equality	O
constraints	O
that	O
the	O
sum	O
of	O
each	O
row	O
and	O
of	O
each	O
column	O
be	O
one	O
.	O
</s>
<s>
The	O
Birkhoff	B-Algorithm
polytope	I-Algorithm
Bn	O
is	O
both	O
vertex-transitive	O
and	O
facet-transitive	O
(	O
i.e.	O
</s>
<s>
An	O
outstanding	O
problem	O
is	O
to	O
find	O
the	O
volume	O
of	O
the	O
Birkhoff	B-Algorithm
polytopes	I-Algorithm
.	O
</s>
<s>
The	O
Ehrhart	O
polynomial	O
associated	O
with	O
the	O
Birkhoff	B-Algorithm
polytope	I-Algorithm
is	O
only	O
known	O
for	O
small	O
values	O
.	O
</s>
<s>
The	O
Birkhoff	B-Algorithm
polytope	I-Algorithm
is	O
a	O
special	O
case	O
of	O
the	O
transportation	O
polytope	O
,	O
a	O
polytope	O
of	O
nonnegative	O
rectangular	O
matrices	B-Architecture
with	O
given	O
row	O
and	O
column	O
sums	O
.	O
</s>
<s>
The	O
integer	O
points	O
in	O
these	O
polytopes	O
are	O
called	O
contingency	B-Application
tables	I-Application
;	O
they	O
play	O
an	O
important	O
role	O
in	O
Bayesian	O
statistics	O
.	O
</s>
<s>
The	O
Birkhoff	B-Algorithm
polytope	I-Algorithm
is	O
a	O
special	O
case	O
of	O
the	O
matching	O
polytope	O
,	O
defined	O
as	O
a	O
convex	O
hull	O
of	O
the	O
perfect	O
matchings	O
in	O
a	O
finite	O
graph	O
.	O
</s>
<s>
The	O
description	O
of	O
facets	O
in	O
this	O
generality	O
was	O
given	O
by	O
Jack	O
Edmonds	O
(	O
1965	O
)	O
,	O
and	O
is	O
related	O
to	O
Edmonds	B-Algorithm
's	I-Algorithm
matching	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
Birkhoff	B-Algorithm
polytope	I-Algorithm
is	O
a	O
special	O
case	O
of	O
the	O
flow	O
polytope	O
of	O
nonnegative	O
flows	O
through	O
a	O
network	O
.	O
</s>
<s>
It	O
is	O
related	O
to	O
the	O
Ford	B-Algorithm
–	I-Algorithm
Fulkerson	I-Algorithm
algorithm	I-Algorithm
that	O
computes	O
the	O
maximum	O
flow	O
in	O
a	O
flow	O
network	O
.	O
</s>
