<s>
In	O
combinatorics	O
,	O
a	O
greedoid	B-Algorithm
is	O
a	O
type	O
of	O
set	O
system	O
.	O
</s>
<s>
It	O
arises	O
from	O
the	O
notion	O
of	O
the	O
matroid	O
,	O
which	O
was	O
originally	O
introduced	O
by	O
Whitney	O
in	O
1935	O
to	O
study	O
planar	O
graphs	O
and	O
was	O
later	O
used	O
by	O
Edmonds	O
to	O
characterize	O
a	O
class	O
of	O
optimization	O
problems	O
that	O
can	O
be	O
solved	O
by	O
greedy	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
Around	O
1980	O
,	O
Korte	O
and	O
Lovász	O
introduced	O
the	O
greedoid	B-Algorithm
to	O
further	O
generalize	O
this	O
characterization	O
of	O
greedy	B-Algorithm
algorithms	I-Algorithm
;	O
hence	O
the	O
name	O
greedoid	B-Algorithm
.	O
</s>
<s>
Besides	O
mathematical	O
optimization	O
,	O
greedoids	B-Algorithm
have	O
also	O
been	O
connected	O
to	O
graph	O
theory	O
,	O
language	O
theory	O
,	O
order	O
theory	O
,	O
and	O
other	O
areas	O
of	O
mathematics	O
.	O
</s>
<s>
When	O
considering	O
a	O
greedoid	B-Algorithm
,	O
a	O
member	O
of	O
is	O
called	O
a	O
feasible	O
set	O
.	O
</s>
<s>
An	O
accessible	B-Algorithm
set	I-Algorithm
system	I-Algorithm
is	O
a	O
set	O
system	O
in	O
which	O
every	O
nonempty	O
feasible	O
set	O
contains	O
an	O
element	O
such	O
that	O
is	O
feasible	O
.	O
</s>
<s>
This	O
implies	O
that	O
any	O
nonempty	O
,	O
finite	O
,	O
accessible	B-Algorithm
set	I-Algorithm
system	I-Algorithm
necessarily	O
contains	O
the	O
empty	O
set	O
∅	O
.	O
</s>
<s>
A	O
greedoid	B-Algorithm
is	O
an	O
accessible	B-Algorithm
set	I-Algorithm
system	I-Algorithm
that	O
satisfies	O
the	O
exchange	O
property	O
:	O
</s>
<s>
(	O
Note	O
:	O
Some	O
people	O
reserve	O
the	O
term	O
exchange	O
property	O
for	O
a	O
condition	O
on	O
the	O
bases	O
of	O
a	O
greedoid	B-Algorithm
,	O
and	O
prefer	O
to	O
call	O
the	O
above	O
condition	O
the	O
“	O
augmentation	O
property	O
”	O
.	O
)	O
</s>
<s>
A	O
basis	O
of	O
a	O
greedoid	B-Algorithm
is	O
a	O
maximal	O
feasible	O
set	O
,	O
meaning	O
it	O
is	O
a	O
feasible	O
set	O
but	O
not	O
contained	O
in	O
any	O
other	O
one	O
.	O
</s>
<s>
The	O
rank	O
of	O
a	O
greedoid	B-Algorithm
is	O
the	O
size	O
of	O
a	O
basis	O
.	O
</s>
<s>
Just	O
as	O
with	O
matroids	O
,	O
greedoids	B-Algorithm
have	O
a	O
cryptomorphism	O
in	O
terms	O
of	O
rank	O
functions	O
.	O
</s>
<s>
A	O
function	O
is	O
the	O
rank	O
function	O
of	O
a	O
greedoid	B-Algorithm
on	O
the	O
ground	O
set	O
if	O
and	O
only	O
if	O
is	O
subcardinal	O
,	O
monotonic	O
,	O
and	O
locally	O
semimodular	O
,	O
that	O
is	O
,	O
for	O
any	O
and	O
any	O
we	O
have	O
:	O
</s>
<s>
Most	O
classes	O
of	O
greedoids	B-Algorithm
have	O
many	O
equivalent	O
definitions	O
in	O
terms	O
of	O
set	O
system	O
,	O
language	O
,	O
poset	O
,	O
simplicial	O
complex	O
,	O
and	O
so	O
on	O
.	O
</s>
<s>
An	O
interval	O
greedoid	B-Algorithm
is	O
a	O
greedoid	B-Algorithm
that	O
satisfies	O
the	O
Interval	O
Property	O
:	O
</s>
<s>
Equivalently	O
,	O
an	O
interval	O
greedoid	B-Algorithm
is	O
a	O
greedoid	B-Algorithm
such	O
that	O
the	O
union	O
of	O
any	O
two	O
feasible	O
sets	O
is	O
feasible	O
if	O
it	O
is	O
contained	O
in	O
another	O
feasible	O
set	O
.	O
</s>
<s>
An	O
antimatroid	O
is	O
a	O
greedoid	B-Algorithm
that	O
satisfies	O
the	O
Interval	O
Property	O
without	O
Upper	O
Bounds	O
:	O
</s>
<s>
Equivalently	O
,	O
an	O
antimatroid	O
is	O
(	O
i	O
)	O
a	O
greedoid	B-Algorithm
with	O
a	O
unique	O
basis	O
;	O
or	O
(	O
ii	O
)	O
an	O
accessible	B-Algorithm
set	I-Algorithm
system	I-Algorithm
closed	O
under	O
union	O
.	O
</s>
<s>
It	O
is	O
easy	O
to	O
see	O
that	O
an	O
antimatroid	O
is	O
also	O
an	O
interval	O
greedoid	B-Algorithm
.	O
</s>
<s>
A	O
matroid	O
is	O
a	O
non-empty	O
greedoid	B-Algorithm
that	O
satisfies	O
the	O
Interval	O
Property	O
without	O
Lower	O
Bounds	O
:	O
</s>
<s>
It	O
is	O
easy	O
to	O
see	O
that	O
a	O
matroid	O
is	O
also	O
an	O
interval	O
greedoid	B-Algorithm
.	O
</s>
<s>
This	O
is	O
called	O
the	O
vertex	B-Algorithm
search	I-Algorithm
greedoid	I-Algorithm
and	O
is	O
a	O
kind	O
of	O
antimatroid	O
.	O
</s>
<s>
This	O
is	O
called	O
the	O
line	B-Algorithm
search	I-Algorithm
greedoid	I-Algorithm
,	O
or	O
directed	B-Algorithm
branching	I-Algorithm
greedoid	I-Algorithm
.	O
</s>
<s>
It	O
is	O
an	O
interval	O
greedoid	B-Algorithm
,	O
but	O
neither	O
an	O
antimatroid	O
nor	O
a	O
matroid	O
.	O
</s>
<s>
Consider	O
an	O
matrix	B-Architecture
.	O
</s>
<s>
Let	O
the	O
ground	O
set	O
be	O
the	O
indices	O
of	O
the	O
columns	O
from	O
1	O
to	O
and	O
the	O
feasible	O
sets	O
be	O
This	O
is	O
called	O
the	O
Gaussian	B-Algorithm
elimination	I-Algorithm
greedoid	I-Algorithm
because	O
this	O
structure	O
underlies	O
the	O
Gaussian	B-Algorithm
elimination	I-Algorithm
algorithm	O
.	O
</s>
<s>
It	O
is	O
a	O
greedoid	B-Algorithm
,	O
but	O
not	O
an	O
interval	O
greedoid	B-Algorithm
.	O
</s>
<s>
In	O
general	O
,	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
is	O
just	O
an	O
iterative	O
process	O
in	O
which	O
a	O
locally	O
best	O
choice	O
,	O
usually	O
an	O
input	O
of	O
maximum	O
weight	O
,	O
is	O
chosen	O
each	O
round	O
until	O
all	O
available	O
choices	O
have	O
been	O
exhausted	O
.	O
</s>
<s>
In	O
order	O
to	O
describe	O
a	O
greedoid-based	O
condition	O
in	O
which	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
is	O
optimal	O
(	O
i.e.	O
,	O
obtains	O
a	O
basis	O
of	O
maximum	O
value	O
)	O
,	O
we	O
need	O
some	O
more	O
common	O
terminologies	O
in	O
greedoid	B-Algorithm
theory	I-Algorithm
.	O
</s>
<s>
Without	O
loss	O
of	O
generality	O
,	O
we	O
consider	O
a	O
greedoid	B-Algorithm
with	O
finite	O
.	O
</s>
<s>
But	O
the	O
equality	O
does	O
not	O
hold	O
for	O
greedoids	B-Algorithm
in	O
general	O
.	O
</s>
<s>
An	O
objective	O
function	O
is	O
linear	O
over	O
a	O
set	O
S	O
if	O
,	O
for	O
all	O
we	O
have	O
for	O
some	O
weight	B-Algorithm
function	I-Algorithm
Proposition	O
.	O
</s>
<s>
A	O
greedy	B-Algorithm
algorithm	I-Algorithm
is	O
optimal	O
for	O
every	O
R-compatible	O
linear	O
objective	O
function	O
over	O
a	O
greedoid	B-Algorithm
.	O
</s>
<s>
The	O
intuition	O
behind	O
this	O
proposition	O
is	O
that	O
,	O
during	O
the	O
iterative	O
process	O
,	O
each	O
optimal	O
exchange	O
of	O
minimum	O
weight	O
is	O
made	O
possible	O
by	O
the	O
exchange	O
property	O
,	O
and	O
optimal	O
results	O
are	O
obtainable	O
from	O
the	O
feasible	O
sets	O
in	O
the	O
underlying	O
greedoid	B-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
a	O
minimum	O
spanning	O
tree	O
of	O
a	O
weighted	B-Algorithm
graph	O
may	O
be	O
obtained	O
using	O
Kruskal	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
which	O
is	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
for	O
the	O
cycle	O
matroid	O
.	O
</s>
<s>
Prim	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
can	O
be	O
explained	O
by	O
taking	O
the	O
vertex	B-Algorithm
search	I-Algorithm
greedoid	I-Algorithm
instead	O
.	O
</s>
