<s>
In	O
combinatorial	O
optimization	O
,	O
the	O
matroid	B-Algorithm
intersection	I-Algorithm
problem	O
is	O
to	O
find	O
a	O
largest	O
common	O
independent	O
set	O
in	O
two	O
matroids	O
over	O
the	O
same	O
ground	O
set	O
.	O
</s>
<s>
If	O
the	O
elements	O
of	O
the	O
matroid	O
are	O
assigned	O
real	O
weights	O
,	O
the	O
weighted	O
matroid	B-Algorithm
intersection	I-Algorithm
problem	O
is	O
to	O
find	O
a	O
common	O
independent	O
set	O
with	O
the	O
maximum	O
possible	O
weight	O
.	O
</s>
<s>
These	O
problems	O
generalize	O
many	O
problems	O
in	O
combinatorial	O
optimization	O
including	O
finding	O
maximum	O
matchings	O
and	O
maximum	B-Algorithm
weight	I-Algorithm
matchings	I-Algorithm
in	O
bipartite	O
graphs	O
and	O
finding	O
arborescences	O
in	O
directed	O
graphs	O
.	O
</s>
<s>
The	O
matroid	B-Algorithm
intersection	I-Algorithm
theorem	O
,	O
due	O
to	O
Jack	O
Edmonds	O
,	O
says	O
that	O
there	O
is	O
always	O
a	O
simple	O
upper	O
bound	O
certificate	O
,	O
consisting	O
of	O
a	O
partitioning	O
of	O
the	O
ground	O
set	O
amongst	O
the	O
two	O
matroids	O
,	O
whose	O
value	O
(	O
sum	O
of	O
respective	O
ranks	O
)	O
equals	O
the	O
size	O
of	O
a	O
maximum	O
common	O
independent	O
set	O
.	O
</s>
<s>
Based	O
on	O
this	O
theorem	O
,	O
the	O
matroid	B-Algorithm
intersection	I-Algorithm
problem	O
for	O
two	O
matroids	O
can	O
be	O
solved	O
in	O
polynomial	O
time	O
using	O
matroid	O
partitioning	O
algorithms	O
.	O
</s>
<s>
Similarly	O
,	O
if	O
each	O
edge	O
has	O
a	O
weight	O
,	O
then	O
the	O
maximum-weight	O
independent	O
set	O
of	O
MU	O
and	O
MV	O
is	O
a	O
Maximum	B-Algorithm
weight	I-Algorithm
matching	I-Algorithm
in	O
G	O
.	O
</s>
<s>
There	O
are	O
several	O
polynomial-time	O
algorithms	O
for	O
weighted	O
matroid	B-Algorithm
intersection	I-Algorithm
,	O
with	O
different	O
run-times	O
.	O
</s>
<s>
The	O
run-times	O
are	O
given	O
in	O
terms	O
of	O
-	O
the	O
number	O
of	O
elements	O
in	O
the	O
common	O
base-set	O
,	O
-	O
the	O
maximum	O
between	O
the	O
ranks	O
of	O
the	O
two	O
matroids	O
,	O
-	O
the	O
number	O
of	O
operations	O
required	O
for	O
a	O
circuit-finding	B-Application
oracle	I-Application
,	O
and	O
-	O
the	O
number	O
of	O
elements	O
in	O
the	O
intersection	O
(	O
in	O
case	O
we	O
want	O
to	O
find	O
an	O
intersection	O
of	O
a	O
specific	O
size	O
)	O
.	O
</s>
<s>
Brezovec	O
,	O
Cornuejos	O
and	O
Glover	O
present	O
two	O
algorithms	O
for	O
weighted	O
matroid	B-Algorithm
intersection	I-Algorithm
.	O
</s>
<s>
Huang	O
,	O
Kakimura	O
and	O
Kakiyama	O
show	O
that	O
the	O
weighted	O
matroid	B-Algorithm
intersection	I-Algorithm
problem	O
can	O
be	O
solved	O
by	O
solving	O
W	O
instances	O
of	O
the	O
unweighted	O
matroid	B-Algorithm
intersection	I-Algorithm
problem	O
,	O
where	O
W	O
is	O
the	O
largest	O
given	O
weight	O
,	O
assuming	O
that	O
all	O
given	O
weights	O
are	O
integral	O
.	O
</s>
<s>
They	O
also	O
present	O
an	O
approximation	O
algorithm	O
that	O
finds	O
an	O
e-approximate	O
solution	O
by	O
solving	O
instances	O
of	O
the	O
unweighted	O
matroid	B-Algorithm
intersection	I-Algorithm
problem	O
,	O
where	O
r	O
is	O
the	O
smaller	O
rank	O
of	O
the	O
two	O
input	O
matroids	O
.	O
</s>
<s>
Ghosh	O
,	O
Gurjar	O
and	O
Raj	O
study	O
the	O
run-time	O
complexity	O
of	O
matroid	B-Algorithm
intersection	I-Algorithm
in	O
the	O
parallel	B-Operating_System
computing	I-Operating_System
model	O
.	O
</s>
<s>
Berczi	O
,	O
Kirali	O
,	O
Yamaguchi	O
and	O
Yokoi	O
present	O
strongly	O
polynomial-time	O
algorithms	O
for	O
weighted	O
matroid	B-Algorithm
intersection	I-Algorithm
using	O
more	O
restricted	O
oracles	O
.	O
</s>
<s>
In	O
a	O
variant	O
of	O
weighted	O
matroid	B-Algorithm
intersection	I-Algorithm
,	O
called	O
"	O
(	O
Pk	O
)	O
"	O
,	O
the	O
goal	O
is	O
to	O
find	O
a	O
common	O
independent	O
set	O
with	O
the	O
maximum	O
possible	O
weight	O
among	O
all	O
such	O
sets	O
with	O
cardinality	O
k	O
,	O
if	O
such	O
a	O
set	O
exists	O
.	O
</s>
<s>
The	O
matroid	B-Algorithm
intersection	I-Algorithm
problem	O
becomes	O
NP-hard	O
when	O
three	O
matroids	O
are	O
involved	O
,	O
instead	O
of	O
only	O
two	O
.	O
</s>
<s>
One	O
proof	O
of	O
this	O
hardness	O
result	O
uses	O
a	O
reduction	B-Algorithm
from	O
the	O
Hamiltonian	O
path	O
problem	O
in	O
directed	O
graphs	O
.	O
</s>
<s>
Another	O
computational	O
problem	O
on	O
matroids	O
,	O
the	O
matroid	B-Algorithm
parity	I-Algorithm
problem	I-Algorithm
,	O
was	O
formulated	O
by	O
Lawler	O
as	O
a	O
common	O
generalization	O
of	O
matroid	B-Algorithm
intersection	I-Algorithm
and	O
non-bipartite	O
graph	O
matching	O
.	O
</s>
<s>
However	O
,	O
although	O
it	O
can	O
be	O
solved	O
in	O
polynomial	O
time	O
for	O
linear	O
matroids	O
,	O
it	O
is	O
NP-hard	O
for	O
other	O
matroids	O
,	O
and	O
requires	O
exponential	O
time	O
in	O
the	O
matroid	B-Application
oracle	I-Application
model	O
.	O
</s>
<s>
The	O
weighted	O
matroid	B-Algorithm
intersection	I-Algorithm
problem	O
is	O
a	O
special	O
case	O
in	O
which	O
the	O
matroid	O
valuations	O
are	O
constant	O
,	O
so	O
we	O
only	O
seek	O
to	O
maximize	O
subject	O
to	O
MX	O
is	O
a	O
base	O
in	O
BX	O
and	O
MY	O
is	O
a	O
base	O
in	O
BY	O
.	O
</s>
