<s>
In	O
combinatorial	O
optimization	O
,	O
the	O
matroid	B-Algorithm
parity	I-Algorithm
problem	I-Algorithm
is	O
a	O
problem	O
of	O
finding	O
the	O
largest	O
independent	O
set	O
of	O
paired	O
elements	O
in	O
a	O
matroid	O
.	O
</s>
<s>
The	O
problem	O
was	O
formulated	O
by	O
as	O
a	O
common	O
generalization	O
of	O
graph	B-General_Concept
matching	I-General_Concept
and	O
matroid	B-Algorithm
intersection	I-Algorithm
.	O
</s>
<s>
However	O
,	O
it	O
is	O
NP-hard	O
for	O
certain	O
compactly-represented	O
matroids	O
,	O
and	O
requires	O
more	O
than	O
a	O
polynomial	O
number	O
of	O
steps	O
in	O
the	O
matroid	B-Application
oracle	I-Application
model	O
.	O
</s>
<s>
Applications	O
of	O
matroid	O
parity	O
algorithms	O
include	O
finding	O
large	O
planar	O
subgraphs	O
and	O
finding	O
graph	B-Algorithm
embeddings	I-Algorithm
of	O
maximum	O
genus	O
.	O
</s>
<s>
Examples	O
of	O
matroids	O
include	O
the	O
linear	O
matroids	O
(	O
in	O
which	O
the	O
elements	O
are	O
vectors	O
in	O
a	O
vector	O
space	O
,	O
with	O
linear	O
independence	O
)	O
,	O
the	O
graphic	O
matroids	O
(	O
in	O
which	O
the	O
elements	O
are	O
edges	O
in	O
an	O
undirected	O
graph	O
,	O
independent	O
when	O
they	O
contain	O
no	O
cycle	O
)	O
,	O
and	O
the	O
partition	O
matroids	O
(	O
in	O
which	O
the	O
elements	O
belong	O
to	O
a	O
family	O
of	O
disjoint	B-Algorithm
sets	I-Algorithm
,	O
and	O
are	O
independent	O
when	O
they	O
contain	O
at	O
most	O
one	O
element	O
in	O
each	O
set	O
)	O
.	O
</s>
<s>
In	O
the	O
matroid	B-Algorithm
parity	I-Algorithm
problem	I-Algorithm
,	O
the	O
input	O
consists	O
of	O
a	O
matroid	O
together	O
with	O
a	O
pairing	O
on	O
its	O
elements	O
,	O
so	O
that	O
each	O
element	O
belongs	O
to	O
one	O
pair	O
.	O
</s>
<s>
The	O
matroid	B-Algorithm
parity	I-Algorithm
problem	I-Algorithm
for	O
linear	O
matroids	O
can	O
be	O
solved	O
by	O
a	O
randomized	B-General_Concept
algorithm	I-General_Concept
in	O
time	O
,	O
where	O
is	O
the	O
number	O
of	O
elements	O
of	O
the	O
matroid	O
,	O
is	O
its	O
rank	O
(	O
the	O
size	O
of	O
the	O
largest	O
independent	O
set	O
)	O
,	O
and	O
is	O
the	O
exponent	O
in	O
the	O
time	O
bounds	O
for	O
fast	O
matrix	B-Architecture
multiplication	O
.	O
</s>
<s>
In	O
particular	O
,	O
using	O
a	O
matrix	B-Architecture
multiplication	O
algorithm	O
of	O
Le	O
Gall	O
,	O
it	O
can	O
be	O
solved	O
in	O
time	O
.	O
</s>
<s>
Without	O
using	O
fast	O
matrix	B-Architecture
multiplication	O
,	O
the	O
linear	O
matroid	B-Algorithm
parity	I-Algorithm
problem	I-Algorithm
can	O
be	O
solved	O
in	O
time	O
.	O
</s>
<s>
These	O
algorithms	O
are	O
based	O
on	O
a	O
linear	B-Language
algebra	I-Language
formulation	O
of	O
the	O
problem	O
by	O
.	O
</s>
<s>
Suppose	O
that	O
an	O
input	O
to	O
the	O
problem	O
consists	O
of	O
pairs	O
of	O
-dimensional	O
vectors	O
(	O
arranged	O
as	O
column	O
vectors	O
in	O
a	O
matrix	B-Architecture
of	O
size	O
)	O
.	O
</s>
<s>
The	O
Schwartz	O
–	O
Zippel	O
lemma	O
can	O
be	O
used	O
to	O
test	O
whether	O
this	O
matrix	B-Architecture
has	O
full	O
rank	O
or	O
not	O
(	O
that	O
is	O
,	O
whether	O
the	O
solution	O
has	O
size	O
or	O
not	O
)	O
,	O
by	O
assigning	O
random	O
values	O
to	O
the	O
variables	O
and	O
testing	O
whether	O
the	O
resulting	O
matrix	B-Architecture
has	O
determinant	O
zero	O
.	O
</s>
<s>
By	O
applying	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
that	O
removes	O
pairs	O
one	O
at	O
a	O
time	O
by	O
setting	O
their	O
indeterminates	O
to	O
zero	O
as	O
long	O
as	O
the	O
matrix	B-Architecture
remains	O
of	O
full	O
rank	O
(	O
maintaining	O
the	O
inverse	O
matrix	B-Architecture
using	O
the	O
Sherman	O
–	O
Morrison	O
formula	O
to	O
check	O
the	O
rank	O
after	O
each	O
removal	O
)	O
,	O
one	O
can	O
find	O
a	O
solution	O
whenever	O
this	O
test	O
shows	O
that	O
it	O
exists	O
.	O
</s>
<s>
Additional	O
methods	O
extend	O
this	O
algorithm	O
to	O
the	O
case	O
that	O
the	O
optimal	O
solution	O
to	O
the	O
matroid	B-Algorithm
parity	I-Algorithm
problem	I-Algorithm
has	O
fewer	O
than	O
pairs	O
.	O
</s>
<s>
For	O
simple	O
graphs	O
,	O
is	O
,	O
but	O
for	O
multigraphs	B-Language
,	O
it	O
may	O
be	O
larger	O
,	O
so	O
it	O
is	O
also	O
of	O
interest	O
to	O
have	O
algorithms	O
with	O
smaller	O
or	O
no	O
dependence	O
on	O
and	O
worse	O
dependence	O
on	O
.	O
</s>
<s>
In	O
these	O
cases	O
,	O
it	O
is	O
also	O
possible	O
to	O
solve	O
the	O
graphic	O
matroid	B-Algorithm
parity	I-Algorithm
problem	I-Algorithm
in	O
randomized	O
expected	O
time	O
,	O
or	O
in	O
time	O
when	O
each	O
pair	O
of	O
edges	O
forms	O
a	O
path	O
.	O
</s>
<s>
Although	O
the	O
matroid	B-Algorithm
parity	I-Algorithm
problem	I-Algorithm
is	O
NP-hard	O
for	O
arbitrary	O
matroids	O
,	O
it	O
can	O
still	O
be	O
approximated	O
efficiently	O
.	O
</s>
<s>
Simple	O
local	B-Algorithm
search	I-Algorithm
algorithms	I-Algorithm
provide	O
a	O
polynomial-time	B-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
for	O
this	O
problem	O
,	O
and	O
find	O
solutions	O
whose	O
size	O
,	O
as	O
a	O
fraction	O
of	O
the	O
optimal	O
solution	O
size	O
,	O
is	O
arbitrarily	O
close	O
to	O
one	O
.	O
</s>
<s>
To	O
achieve	O
an	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
,	O
it	O
suffices	O
to	O
choose	O
to	O
be	O
approximately	O
.	O
</s>
<s>
Many	O
other	O
optimization	O
problems	O
can	O
be	O
formulated	O
as	O
linear	O
matroid	B-Algorithm
parity	I-Algorithm
problems	I-Algorithm
,	O
and	O
solved	O
in	O
polynomial	O
time	O
using	O
this	O
formulation	O
.	O
</s>
<s>
Then	O
there	O
is	O
a	O
solution	O
to	O
the	O
matroid	B-Algorithm
parity	I-Algorithm
problem	I-Algorithm
for	O
this	O
matroid	O
,	O
of	O
size	O
,	O
if	O
and	O
only	O
if	O
has	O
a	O
clique	O
of	O
size	O
.	O
</s>
<s>
Since	O
finding	O
cliques	O
of	O
a	O
given	O
size	O
is	O
NP-complete	O
,	O
it	O
follows	O
that	O
determining	O
whether	O
this	O
type	O
of	O
matrix	B-Architecture
parity	O
problem	O
has	O
a	O
solution	O
of	O
size	O
is	O
also	O
NP-complete	O
.	O
</s>
<s>
By	O
applying	O
it	O
to	O
a	O
randomly-permuted	O
graph	O
that	O
contains	O
exactly	O
one	O
clique	O
of	O
size	O
,	O
one	O
can	O
show	O
that	O
any	O
deterministic	O
or	O
randomized	B-General_Concept
algorithm	I-General_Concept
for	O
matroid	O
parity	O
that	O
accesses	O
its	O
matroid	O
only	O
by	O
independence	O
tests	O
needs	O
to	O
make	O
an	O
exponential	O
number	O
of	O
tests	O
.	O
</s>
