<s>
In	O
mathematics	O
,	O
the	O
vertex	B-Algorithm
enumeration	I-Algorithm
problem	I-Algorithm
for	O
a	O
polytope	O
,	O
a	O
polyhedral	O
cell	O
complex	O
,	O
a	O
hyperplane	O
arrangement	O
,	O
or	O
some	O
other	O
object	O
of	O
discrete	O
geometry	O
,	O
is	O
the	O
problem	O
of	O
determination	O
of	O
the	O
object	O
's	O
vertices	O
given	O
some	O
formal	O
representation	O
of	O
the	O
object	O
.	O
</s>
<s>
The	O
inverse	O
(	O
dual	O
)	O
problem	O
of	O
finding	O
the	O
bounding	O
inequalities	O
given	O
the	O
vertices	O
is	O
called	O
facet	O
enumeration	O
(	O
see	O
convex	B-Algorithm
hull	I-Algorithm
algorithms	I-Algorithm
)	O
.	O
</s>
<s>
The	O
computational	O
complexity	O
of	O
the	O
problem	O
is	O
a	O
subject	O
of	O
research	O
in	O
computer	B-General_Concept
science	I-General_Concept
.	O
</s>
<s>
A	O
1992	O
article	O
by	O
David	O
Avis	O
and	O
Komei	O
Fukuda	O
presents	O
a	O
reverse-search	B-Algorithm
algorithm	I-Algorithm
which	O
finds	O
the	O
v	O
vertices	O
of	O
a	O
polytope	O
defined	O
by	O
a	O
nondegenerate	O
system	O
of	O
n	O
inequalities	O
in	O
d	O
dimensions	O
(	O
or	O
,	O
dually	O
,	O
the	O
v	O
facets	O
of	O
the	O
convex	O
hull	O
of	O
n	O
points	O
in	O
d	O
dimensions	O
,	O
where	O
each	O
facet	O
contains	O
exactly	O
d	O
given	O
points	O
)	O
in	O
time	O
O(ndv )	O
and	O
space	O
O(nd )	O
.	O
</s>
<s>
The	O
Avis	O
–	O
Fukuda	O
algorithm	O
adapted	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
for	O
oriented	O
matroids	O
.	O
</s>
