<s>
In	O
mathematics	O
and	O
computer	O
science	O
,	O
a	O
matroid	B-Application
oracle	I-Application
is	O
a	O
subroutine	O
through	O
which	O
an	O
algorithm	O
may	O
access	O
a	O
matroid	O
,	O
an	O
abstract	O
combinatorial	O
structure	O
that	O
can	O
be	O
used	O
to	O
describe	O
the	O
linear	O
dependencies	O
between	O
vectors	O
in	O
a	O
vector	O
space	O
or	O
the	O
spanning	O
trees	O
of	O
a	O
graph	O
,	O
among	O
other	O
applications	O
.	O
</s>
<s>
The	O
most	O
commonly	O
used	O
oracle	O
of	O
this	O
type	O
is	O
an	O
independence	B-Application
oracle	I-Application
,	O
a	O
subroutine	O
for	O
testing	O
whether	O
a	O
set	O
of	O
matroid	O
elements	O
is	O
independent	O
.	O
</s>
<s>
Several	O
other	O
types	O
of	O
oracle	O
have	O
also	O
been	O
used	O
;	O
some	O
of	O
them	O
have	O
been	O
shown	O
to	O
be	O
weaker	O
than	O
independence	B-Application
oracles	I-Application
,	O
some	O
stronger	O
,	O
and	O
some	O
equivalent	O
in	O
computational	O
power	O
.	O
</s>
<s>
For	O
instance	O
,	O
given	O
an	O
independence	B-Application
oracle	I-Application
for	O
any	O
matroid	O
,	O
it	O
is	O
possible	O
to	O
find	O
the	O
minimum	O
weight	O
basis	O
of	O
the	O
matroid	O
by	O
applying	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
that	O
adds	O
elements	O
to	O
the	O
basis	O
in	O
sorted	O
order	O
by	O
weight	O
,	O
using	O
the	O
independence	B-Application
oracle	I-Application
to	O
test	O
whether	O
each	O
element	O
can	O
be	O
added	O
.	O
</s>
<s>
Instead	O
,	O
different	O
types	O
of	O
matroids	O
may	O
be	O
represented	O
more	O
efficiently	O
from	O
the	O
other	O
structures	O
from	O
which	O
they	O
are	O
defined	O
:	O
uniform	O
matroids	O
from	O
their	O
two	O
numeric	O
parameters	O
,	O
graphic	O
matroids	O
,	O
bicircular	O
matroids	O
,	O
and	O
gammoids	O
from	O
graphs	O
,	O
linear	O
matroids	O
from	O
matrices	B-Architecture
,	O
etc	O
.	O
</s>
<s>
An	O
independence	O
function	O
maps	O
a	O
set	O
of	O
matroid	O
elements	O
to	O
the	O
number	O
if	O
the	O
set	O
is	O
independent	O
or	O
if	O
it	O
is	O
dependent	O
;	O
that	O
is	O
,	O
it	O
is	O
the	O
indicator	O
function	O
of	O
the	O
family	O
of	O
independent	O
sets	O
,	O
essentially	O
the	O
same	O
thing	O
as	O
an	O
independence	B-Application
oracle	I-Application
.	O
</s>
<s>
Matroid	B-Application
oracles	I-Application
have	O
also	O
been	O
part	O
of	O
the	O
earliest	O
algorithmic	O
work	O
on	O
matroids	O
.	O
</s>
<s>
used	O
a	O
subroutine	O
that	O
tests	O
whether	O
a	O
given	O
set	O
is	O
independent	O
(	O
that	O
is	O
,	O
in	O
more	O
modern	O
terminology	O
,	O
an	O
independence	B-Application
oracle	I-Application
)	O
,	O
and	O
observed	O
that	O
the	O
information	O
it	O
provides	O
is	O
sufficient	O
to	O
find	O
the	O
minimum	O
weight	O
basis	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
These	O
two	O
papers	O
by	O
Hausmann	O
and	O
Korte	O
both	O
concerned	O
the	O
problem	O
of	O
finding	O
a	O
maximum	O
cardinality	O
independent	O
set	O
,	O
which	O
is	O
easy	O
for	O
matroids	O
but	O
(	O
as	O
they	O
showed	O
)	O
harder	O
to	O
approximate	O
or	O
compute	O
exactly	O
for	O
more	O
general	O
independence	O
systems	O
represented	O
by	O
an	O
independence	B-Application
oracle	I-Application
.	O
</s>
<s>
This	O
work	O
kicked	O
off	O
a	O
flurry	O
of	O
papers	O
in	O
the	O
late	O
1970s	O
and	O
early	O
1980s	O
showing	O
similar	O
hardness	O
results	O
for	O
problems	O
on	O
matroids	O
and	O
comparing	O
the	O
power	O
of	O
different	O
kinds	O
of	O
matroid	B-Application
oracles	I-Application
.	O
</s>
<s>
Since	O
that	O
time	O
,	O
the	O
independence	B-Application
oracle	I-Application
has	O
become	O
standard	O
for	O
most	O
research	O
on	O
matroid	O
algorithms	O
.	O
</s>
<s>
The	O
following	O
types	O
of	O
matroid	B-Application
oracles	I-Application
have	O
been	O
considered	O
.	O
</s>
<s>
An	O
independence	B-Application
oracle	I-Application
takes	O
as	O
its	O
input	O
a	O
set	O
of	O
matroid	O
elements	O
,	O
and	O
returns	O
as	O
output	O
a	O
Boolean	O
value	O
,	O
true	O
if	O
the	O
given	O
set	O
is	O
independent	O
and	O
false	O
otherwise	O
.	O
</s>
<s>
A	O
circuit	B-Application
oracle	I-Application
takes	O
as	O
its	O
input	O
a	O
set	O
of	O
matroid	O
elements	O
,	O
and	O
returns	O
as	O
output	O
a	O
Boolean	O
value	O
,	O
true	O
if	O
the	O
given	O
set	O
is	O
a	O
circuit	O
and	O
false	O
otherwise	O
.	O
</s>
<s>
The	O
circuit-finding	B-Application
oracle	I-Application
of	O
takes	O
as	O
input	O
an	O
independent	O
set	O
and	O
an	O
additional	O
element	O
,	O
and	O
either	O
determines	O
that	O
their	O
union	O
is	O
independent	O
,	O
or	O
finds	O
a	O
circuit	O
in	O
the	O
union	O
and	O
returns	O
it	O
.	O
</s>
<s>
An	O
oracle	O
is	O
said	O
to	O
be	O
polynomially	O
reducible	O
to	O
another	O
oracle	O
if	O
any	O
call	O
to	O
may	O
be	O
simulated	O
by	O
an	O
algorithm	O
that	O
accesses	O
the	O
matroid	O
using	O
only	O
oracle	O
and	O
takes	O
polynomial	O
time	O
as	O
measured	O
in	O
terms	O
of	O
the	O
number	O
of	O
elements	O
of	O
the	O
matroid	O
;	O
in	O
complexity-theoretic	O
terms	O
,	O
this	O
is	O
a	O
Turing	B-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
independence	B-Application
oracle	I-Application
is	O
polynomially	O
equivalent	O
to	O
the	O
circuit-finding	B-Application
oracle	I-Application
of	O
.	O
</s>
<s>
If	O
a	O
circuit-finding	B-Application
oracle	I-Application
is	O
available	O
,	O
a	O
set	O
may	O
be	O
tested	O
for	O
independence	O
using	O
at	O
most	O
calls	O
to	O
the	O
oracle	O
by	O
starting	O
from	O
an	O
empty	O
set	O
,	O
adding	O
elements	O
of	O
the	O
given	O
set	O
one	O
element	O
at	O
a	O
time	O
,	O
and	O
using	O
the	O
circuit-finding	B-Application
oracle	I-Application
to	O
test	O
whether	O
each	O
addition	O
preserves	O
the	O
independence	O
of	O
the	O
set	O
that	O
has	O
been	O
constructed	O
so	O
far	O
.	O
</s>
<s>
In	O
the	O
other	O
direction	O
,	O
if	O
an	O
independence	B-Application
oracle	I-Application
is	O
available	O
,	O
the	O
circuit	O
in	O
a	O
set	O
may	O
be	O
found	O
using	O
at	O
most	O
calls	O
to	O
the	O
oracle	O
by	O
testing	O
,	O
for	O
each	O
element	O
,	O
whether	O
is	O
independent	O
and	O
returning	O
the	O
elements	O
for	O
which	O
the	O
answer	O
is	O
no	O
.	O
</s>
<s>
The	O
independence	B-Application
oracle	I-Application
is	O
also	O
polynomially	O
equivalent	O
to	O
the	O
rank	O
oracle	O
,	O
the	O
spanning	O
oracle	O
,	O
the	O
first	O
two	O
types	O
of	O
closure	O
oracle	O
,	O
and	O
the	O
port	O
oracle	O
.	O
</s>
<s>
The	O
basis	O
oracle	O
,	O
the	O
circuit	B-Application
oracle	I-Application
,	O
and	O
the	O
oracle	O
that	O
tests	O
whether	O
a	O
given	O
set	O
is	O
closed	O
are	O
all	O
weaker	O
than	O
the	O
independence	B-Application
oracle	I-Application
:	O
they	O
can	O
be	O
simulated	O
in	O
polynomial	O
time	O
by	O
an	O
algorithm	O
that	O
accesses	O
the	O
matroid	O
using	O
an	O
independence	B-Application
oracle	I-Application
,	O
but	O
not	O
vice	O
versa	O
.	O
</s>
<s>
The	O
girth	O
oracle	O
is	O
stronger	O
than	O
the	O
independence	B-Application
oracle	I-Application
,	O
in	O
the	O
same	O
sense	O
.	O
</s>
<s>
As	O
well	O
as	O
polynomial	O
time	O
Turing	B-Algorithm
reductions	I-Algorithm
,	O
other	O
types	O
of	O
reducibility	O
have	O
been	O
considered	O
as	O
well	O
.	O
</s>
<s>
in	O
parallel	B-Operating_System
algorithms	I-Operating_System
,	O
the	O
rank	O
and	O
independence	B-Application
oracles	I-Application
are	O
significantly	O
different	O
in	O
computational	O
power	O
.	O
</s>
<s>
In	O
contrast	O
,	O
finding	O
a	O
minimum	O
basis	O
with	O
an	O
independence	B-Application
oracle	I-Application
is	O
much	O
slower	O
:	O
it	O
can	O
be	O
solved	O
deterministically	O
in	O
time	O
steps	O
,	O
and	O
there	O
is	O
a	O
lower	O
bound	O
of	O
even	O
for	O
randomized	O
parallel	B-Operating_System
algorithms	I-Operating_System
.	O
</s>
<s>
Many	O
problems	O
on	O
matroids	O
are	O
known	O
to	O
be	O
solvable	O
in	O
polynomial	O
time	O
,	O
by	O
algorithms	O
that	O
access	O
the	O
matroid	O
only	O
through	O
an	O
independence	B-Application
oracle	I-Application
or	O
another	O
oracle	O
of	O
equivalent	O
power	O
,	O
without	O
need	O
of	O
any	O
additional	O
assumptions	O
about	O
what	O
kind	O
of	O
matroid	O
has	O
been	O
given	O
to	O
them	O
.	O
</s>
<s>
Finding	O
a	O
minimum	O
or	O
maximum	O
weight	O
basis	O
of	O
a	O
weighted	O
matroid	O
,	O
using	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
latter	O
problem	O
is	O
called	O
matroid	B-Algorithm
intersection	I-Algorithm
,	O
and	O
the	O
solutions	O
to	O
both	O
problems	O
are	O
closely	O
related	O
to	O
each	O
other	O
.	O
</s>
<s>
Approximating	O
the	O
number	O
of	O
bases	O
by	O
a	O
fully	B-Algorithm
polynomial-time	I-Algorithm
randomized	I-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
,	O
for	O
a	O
matroid	O
with	O
elements	O
and	O
rank	O
,	O
with	O
the	O
additional	O
assumption	O
that	O
the	O
number	O
of	O
bases	O
is	O
within	O
a	O
polynomial	O
factor	O
of	O
the	O
number	O
of	O
-element	O
sets	O
.	O
</s>
<s>
For	O
many	O
matroid	O
problems	O
,	O
it	O
is	O
possible	O
to	O
show	O
that	O
an	O
independence	B-Application
oracle	I-Application
does	O
not	O
provide	O
enough	O
power	O
to	O
allow	O
the	O
problem	O
to	O
be	O
solved	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
queries	O
,	O
where	O
denotes	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
,	O
denotes	O
the	O
family	O
of	O
sets	O
whose	O
independence	O
differs	O
from	O
to	O
,	O
and	O
denotes	O
the	O
subgroup	O
of	O
automorphisms	O
that	O
maps	O
to	O
itself	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
the	O
uniform	O
matroid	O
is	O
just	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
with	O
size	O
,	O
and	O
in	O
the	O
problem	O
of	O
testing	O
uniform	O
matroids	O
there	O
was	O
only	O
one	O
set	O
with	O
,	O
smaller	O
by	O
an	O
exponential	O
factor	O
than	O
.	O
</s>
<s>
Problems	O
that	O
have	O
been	O
proven	O
to	O
be	O
impossible	O
for	O
a	O
matroid	B-Application
oracle	I-Application
algorithm	O
to	O
compute	O
in	O
polynomial	O
time	O
include	O
:	O
</s>
