<s>
The	O
Klee	B-Algorithm
–	I-Algorithm
Minty	I-Algorithm
cube	I-Algorithm
or	O
Klee	O
–	O
Minty	O
polytope	O
(	O
named	O
after	O
Victor	O
Klee	O
and	O
George	O
J	O
.	O
Minty	O
)	O
is	O
a	O
unit	O
hypercube	B-Operating_System
of	O
variable	O
dimension	O
whose	O
corners	O
have	O
been	O
perturbed	O
.	O
</s>
<s>
Klee	O
and	O
Minty	O
demonstrated	O
that	O
George	O
Dantzig	O
's	O
simplex	B-Algorithm
algorithm	I-Algorithm
has	O
poor	O
worst-case	B-General_Concept
performance	O
when	O
initialized	O
at	O
one	O
corner	O
of	O
their	O
"	O
squashed	O
cube	O
"	O
.	O
</s>
<s>
On	O
the	O
three-dimensional	O
version	O
,	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
and	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
visit	O
all	O
8	O
corners	O
in	O
the	O
worst	B-General_Concept
case	I-General_Concept
.	O
</s>
<s>
In	O
particular	O
,	O
many	O
optimization	O
algorithms	O
for	O
linear	B-Algorithm
optimization	I-Algorithm
exhibit	O
poor	O
performance	O
when	O
applied	O
to	O
the	O
Klee	B-Algorithm
–	I-Algorithm
Minty	I-Algorithm
cube	I-Algorithm
.	O
</s>
<s>
In	O
1973	O
Klee	O
and	O
Minty	O
showed	O
that	O
Dantzig	O
's	O
simplex	B-Algorithm
algorithm	I-Algorithm
was	O
not	O
a	O
polynomial-time	O
algorithm	O
when	O
applied	O
to	O
their	O
cube	O
.	O
</s>
<s>
Later	O
,	O
modifications	O
of	O
the	O
Klee	B-Algorithm
–	I-Algorithm
Minty	I-Algorithm
cube	I-Algorithm
have	O
shown	O
poor	O
behavior	O
both	O
for	O
other	O
basis-exchange	O
pivoting	O
algorithms	O
and	O
also	O
for	O
interior-point	O
algorithms	O
.	O
</s>
<s>
The	O
Klee	B-Algorithm
–	I-Algorithm
Minty	I-Algorithm
cube	I-Algorithm
was	O
originally	O
specified	O
with	O
a	O
parameterized	O
system	O
of	O
linear	O
inequalities	O
,	O
with	O
the	O
dimension	O
as	O
the	O
parameter	O
.	O
</s>
<s>
This	O
has	O
D	O
variables	O
,	O
D	O
constraints	O
other	O
than	O
the	O
D	O
non-negativity	O
constraints	O
,	O
and	O
2D	O
vertices	O
,	O
just	O
as	O
a	O
D-dimensional	O
hypercube	B-Operating_System
does	O
.	O
</s>
<s>
The	O
Klee	B-Algorithm
–	I-Algorithm
Minty	I-Algorithm
cube	I-Algorithm
has	O
been	O
used	O
to	O
analyze	O
the	O
performance	O
of	O
many	O
algorithms	O
,	O
both	O
in	O
the	O
worst	B-General_Concept
case	I-General_Concept
and	O
on	O
average	O
.	O
</s>
<s>
For	O
example	O
,	O
Gaussian	B-Algorithm
elimination	I-Algorithm
requires	O
on	O
the	O
order	O
of	O
D3	O
operations	O
,	O
and	O
so	O
it	O
is	O
said	O
to	O
have	O
polynomial	O
time-complexity	O
,	O
because	O
its	O
complexity	O
is	O
bounded	O
by	O
a	O
cubic	O
polynomial	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
generalization	O
of	O
Gaussian	B-Algorithm
elimination	I-Algorithm
called	O
Buchberger	O
's	O
algorithm	O
has	O
for	O
its	O
complexity	O
an	O
exponential	O
function	O
of	O
the	O
problem	O
data	O
(	O
the	O
degree	O
of	O
the	O
polynomials	O
and	O
the	O
number	O
of	O
variables	O
of	O
the	O
multivariate	O
polynomials	O
)	O
.	O
</s>
<s>
In	O
mathematical	O
optimization	O
,	O
the	O
Klee	B-Algorithm
–	I-Algorithm
Minty	I-Algorithm
cube	I-Algorithm
is	O
an	O
example	O
that	O
shows	O
the	O
worst-case	B-General_Concept
computational	B-General_Concept
complexity	O
of	O
many	O
algorithms	O
of	O
linear	B-Algorithm
optimization	I-Algorithm
.	O
</s>
<s>
Klee	O
and	O
Minty	O
showed	O
that	O
Dantzig	O
's	O
simplex	B-Algorithm
algorithm	I-Algorithm
visits	O
all	O
corners	O
of	O
a	O
(	O
perturbed	O
)	O
cube	O
in	O
dimensionD	O
in	O
the	O
worst	B-General_Concept
case	I-General_Concept
.	O
</s>
<s>
Modifications	O
of	O
the	O
Klee	O
–	O
Minty	O
construction	O
showed	O
similar	O
exponential	O
time	O
complexity	O
for	O
other	O
pivoting	O
rules	O
of	O
simplex	O
type	O
,	O
which	O
maintain	O
primal	O
feasibility	O
,	O
such	O
as	O
Bland	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
Another	O
modification	O
showed	O
that	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
,	O
which	O
does	O
not	O
maintain	O
primal	O
feasibility	O
,	O
also	O
visits	O
all	O
the	O
corners	O
of	O
a	O
modified	O
Klee	B-Algorithm
–	I-Algorithm
Minty	I-Algorithm
cube	I-Algorithm
.	O
</s>
<s>
Like	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
,	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
visits	O
all8	O
corners	O
of	O
the	O
three-dimensional	O
cube	O
in	O
the	O
worst	B-General_Concept
case	I-General_Concept
.	O
</s>
<s>
Further	O
modifications	O
of	O
the	O
Klee	B-Algorithm
–	I-Algorithm
Minty	I-Algorithm
cube	I-Algorithm
have	O
shown	O
poor	O
performance	O
of	O
central-path	O
–	O
following	O
algorithms	O
for	O
linear	B-Algorithm
optimization	I-Algorithm
,	O
in	O
that	O
the	O
central	O
path	O
comes	O
arbitrarily	O
close	O
to	O
each	O
of	O
the	O
corners	O
of	O
a	O
cube	O
.	O
</s>
<s>
This	O
"	O
vertex-stalking	O
"	O
performance	O
is	O
surprising	O
,	O
because	O
such	O
path-following	O
algorithms	O
have	O
polynomial-time	O
complexity	O
for	O
linear	B-Algorithm
optimization	I-Algorithm
.	O
</s>
<s>
The	O
Klee	B-Algorithm
–	I-Algorithm
Minty	I-Algorithm
cube	I-Algorithm
has	O
also	O
inspired	O
research	O
on	O
average-case	O
complexity	O
.	O
</s>
<s>
When	O
eligible	O
pivots	O
are	O
made	O
randomly	O
(	O
and	O
not	O
by	O
the	O
rule	O
of	O
steepest	O
descent	O
)	O
,	O
Dantzig	O
's	O
simplex	B-Algorithm
algorithm	I-Algorithm
needs	O
on	O
average	O
quadratically	O
many	O
steps	O
(	O
on	O
the	O
order	O
of	O
O(D2 )	O
.	O
</s>
<s>
Standard	O
variants	O
of	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
takes	O
on	O
averageD	O
steps	O
for	O
a	O
cube	O
.	O
</s>
<s>
When	O
it	O
is	O
initialized	O
at	O
a	O
random	O
corner	O
of	O
the	O
cube	O
,	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
visits	O
onlyD	O
additional	O
corners	O
,	O
however	O
,	O
according	O
to	O
a1994	O
paper	O
by	O
Fukuda	O
and	O
Namiki	O
.	O
</s>
<s>
Both	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
and	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
visit	O
exactly3	O
additional	O
corners	O
of	O
the	O
three-dimensional	O
cube	O
onaverage	O
.	O
</s>
