<s>
In	O
the	O
study	O
of	O
algorithms	O
,	O
an	O
LP-type	B-Algorithm
problem	I-Algorithm
(	O
also	O
called	O
a	O
generalized	O
linear	B-Algorithm
program	I-Algorithm
)	O
is	O
an	O
optimization	O
problem	O
that	O
shares	O
certain	O
properties	O
with	O
low-dimensional	O
linear	B-Algorithm
programs	I-Algorithm
and	O
that	O
may	O
be	O
solved	O
by	O
similar	O
algorithms	O
.	O
</s>
<s>
LP-type	B-Algorithm
problems	I-Algorithm
include	O
many	O
important	O
optimization	O
problems	O
that	O
are	O
not	O
themselves	O
linear	B-Algorithm
programs	I-Algorithm
,	O
such	O
as	O
the	O
problem	O
of	O
finding	O
the	O
smallest	O
circle	O
containing	O
a	O
given	O
set	O
of	O
planar	O
points	O
.	O
</s>
<s>
They	O
may	O
be	O
solved	O
by	O
a	O
combination	O
of	O
randomized	B-General_Concept
algorithms	I-General_Concept
in	O
an	O
amount	O
of	O
time	O
that	O
is	O
linear	O
in	O
the	O
number	O
of	O
elements	O
defining	O
the	O
problem	O
,	O
and	O
subexponential	O
in	O
the	O
dimension	O
of	O
the	O
problem	O
.	O
</s>
<s>
LP-type	B-Algorithm
problems	I-Algorithm
were	O
defined	O
by	O
as	O
problems	O
in	O
which	O
one	O
is	O
given	O
as	O
input	O
a	O
finite	O
set	O
of	O
elements	O
,	O
and	O
a	O
function	O
that	O
maps	O
subsets	O
of	O
to	O
values	O
from	O
a	O
totally	O
ordered	O
set	O
.	O
</s>
<s>
A	O
basis	O
of	O
an	O
LP-type	B-Algorithm
problem	I-Algorithm
is	O
a	O
set	O
with	O
the	O
property	O
that	O
every	O
proper	O
subset	O
of	O
has	O
a	O
smaller	O
value	O
of	O
than	O
itself	O
,	O
and	O
the	O
dimension	O
(	O
or	O
combinatorial	O
dimension	O
)	O
of	O
an	O
LP-type	B-Algorithm
problem	I-Algorithm
is	O
defined	O
to	O
be	O
the	O
maximum	O
cardinality	B-Application
of	O
a	O
basis	O
.	O
</s>
<s>
A	O
linear	B-Algorithm
program	I-Algorithm
may	O
be	O
defined	O
by	O
a	O
system	O
of	O
non-negative	O
real	O
variables	O
,	O
subject	O
to	O
linear	O
inequality	O
constraints	O
,	O
together	O
with	O
a	O
non-negative	O
linear	O
objective	O
function	O
to	O
be	O
minimized	O
.	O
</s>
<s>
This	O
may	O
be	O
placed	O
into	O
the	O
framework	O
of	O
LP-type	B-Algorithm
problems	I-Algorithm
by	O
letting	O
be	O
the	O
set	O
of	O
constraints	O
,	O
and	O
defining	O
(	O
for	O
a	O
subset	O
of	O
the	O
constraints	O
)	O
to	O
be	O
the	O
minimum	O
objective	O
function	O
value	O
of	O
the	O
smaller	O
linear	B-Algorithm
program	I-Algorithm
defined	O
by	O
.	O
</s>
<s>
With	O
suitable	O
general	O
position	O
assumptions	O
(	O
in	O
order	O
to	O
prevent	O
multiple	O
solution	O
points	O
having	O
the	O
same	O
optimal	O
objective	O
function	O
value	O
)	O
,	O
this	O
satisfies	O
the	O
monotonicity	O
and	O
locality	O
requirements	O
of	O
an	O
LP-type	B-Algorithm
problem	I-Algorithm
,	O
and	O
has	O
combinatorial	O
dimension	O
equal	O
to	O
the	O
number	O
of	O
variables	O
.	O
</s>
<s>
Similarly	O
,	O
an	O
integer	B-Algorithm
program	I-Algorithm
(	O
consisting	O
of	O
a	O
collection	O
of	O
linear	O
constraints	O
and	O
a	O
linear	O
objective	O
function	O
,	O
as	O
in	O
a	O
linear	B-Algorithm
program	I-Algorithm
,	O
but	O
with	O
the	O
additional	O
restriction	O
that	O
the	O
variables	O
must	O
take	O
on	O
only	O
integer	O
values	O
)	O
satisfies	O
both	O
the	O
monotonicity	O
and	O
locality	O
properties	O
of	O
an	O
LP-type	B-Algorithm
problem	I-Algorithm
,	O
with	O
the	O
same	O
general	O
position	O
assumptions	O
as	O
for	O
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
Theorems	O
of	O
and	O
show	O
that	O
,	O
for	O
an	O
integer	B-Algorithm
program	I-Algorithm
with	O
variables	O
,	O
the	O
combinatorial	O
dimension	O
is	O
at	O
most	O
.	O
</s>
<s>
More	O
generally	O
,	O
the	O
smallest	O
enclosing	O
ball	O
of	O
points	O
in	O
dimensions	O
forms	O
an	O
LP-type	B-Algorithm
problem	I-Algorithm
of	O
combinatorial	O
dimension	O
.	O
</s>
<s>
The	O
smallest	O
circle	O
problem	O
can	O
be	O
generalized	O
to	O
the	O
smallest	O
ball	O
enclosing	O
a	O
set	O
of	O
balls	O
,	O
to	O
the	O
smallest	O
ball	O
that	O
touches	O
or	O
surrounds	O
each	O
of	O
a	O
set	O
of	O
balls	O
,	O
to	O
the	O
weighted	O
1-center	B-Algorithm
problem	I-Algorithm
,	O
or	O
to	O
similar	O
smaller	O
enclosing	O
ball	O
problems	O
in	O
non-Euclidean	O
spaces	O
such	O
as	O
the	O
space	O
with	O
distances	O
defined	O
by	O
Bregman	B-Algorithm
divergence	I-Algorithm
.	O
</s>
<s>
The	O
related	O
problem	O
of	O
finding	O
the	O
smallest	O
enclosing	O
ellipsoid	O
is	O
also	O
an	O
LP-type	B-Algorithm
problem	I-Algorithm
,	O
but	O
with	O
a	O
larger	O
combinatorial	O
dimension	O
,	O
.	O
</s>
<s>
This	O
may	O
be	O
expressed	O
as	O
an	O
LP-type	B-Algorithm
problem	I-Algorithm
in	O
which	O
where	O
Ki	O
is	O
the	O
first	O
member	O
of	O
A	O
that	O
does	O
not	O
belong	O
to	O
an	O
intersecting	O
prefix	O
of	O
A	O
,	O
and	O
where	O
if	O
there	O
is	O
no	O
such	O
member	O
.	O
</s>
<s>
This	O
may	O
be	O
expressed	O
as	O
an	O
LP-type	B-Algorithm
problem	I-Algorithm
with	O
combinatorial	O
dimension	O
4	O
.	O
</s>
<s>
The	O
problem	O
of	O
finding	O
the	O
closest	O
distance	O
between	O
two	O
convex	O
polytopes	O
,	O
specified	O
by	O
their	O
sets	O
of	O
vertices	O
,	O
may	O
be	O
represented	O
as	O
an	O
LP-type	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Then	O
the	O
pointwise	B-Algorithm
maximum	I-Algorithm
is	O
itself	O
quasiconvex	O
,	O
and	O
the	O
problem	O
of	O
finding	O
the	O
minimum	O
value	O
of	O
is	O
an	O
LP-type	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Many	O
other	O
LP-type	B-Algorithm
problems	I-Algorithm
can	O
also	O
be	O
expressed	O
using	O
quasiconvex	O
functions	O
in	O
this	O
way	O
;	O
for	O
instance	O
,	O
the	O
smallest	O
enclosing	O
circle	O
problem	O
is	O
the	O
problem	O
of	O
minimizing	O
where	O
each	O
of	O
the	O
functions	O
measures	O
the	O
Euclidean	O
distance	O
from	O
one	O
of	O
the	O
given	O
points	O
.	O
</s>
<s>
LP-type	B-Algorithm
problems	I-Algorithm
have	O
also	O
been	O
used	O
to	O
determine	O
the	O
optimal	O
outcomes	O
of	O
certain	O
games	O
in	O
algorithmic	O
game	O
theory	O
,	O
improve	O
vertex	O
placement	O
in	O
finite	B-Application
element	I-Application
method	I-Application
meshes	O
,	O
solve	O
facility	O
location	O
problems	O
,	O
analyze	O
the	O
time	O
complexity	O
of	O
certain	O
exponential-time	O
search	O
algorithms	O
,	O
and	O
reconstruct	O
the	O
three-dimensional	O
positions	O
of	O
objects	O
from	O
their	O
two-dimensional	O
images	O
.	O
</s>
<s>
gave	O
an	O
algorithm	O
for	O
low-dimensional	O
linear	B-Algorithm
programming	I-Algorithm
that	O
may	O
be	O
adapted	O
to	O
the	O
LP-type	B-Algorithm
problem	I-Algorithm
framework	O
.	O
</s>
<s>
defines	O
two	O
algorithms	O
,	O
a	O
recursive	O
algorithm	O
and	O
an	O
iterative	O
algorithm	O
,	O
for	O
linear	B-Algorithm
programming	I-Algorithm
based	O
on	O
random	O
sampling	O
techniques	O
,	O
and	O
suggests	O
a	O
combination	O
of	O
the	O
two	O
that	O
calls	O
the	O
iterative	O
algorithm	O
from	O
the	O
recursive	O
algorithm	O
.	O
</s>
<s>
By	O
using	O
the	O
recursive	O
algorithm	O
to	O
solve	O
a	O
given	O
problem	O
,	O
switching	O
to	O
the	O
iterative	O
algorithm	O
for	O
its	O
recursive	O
calls	O
,	O
and	O
then	O
switching	O
again	O
to	O
Seidel	O
's	O
algorithm	O
for	O
the	O
calls	O
made	O
by	O
the	O
iterative	O
algorithm	O
,	O
it	O
is	O
possible	O
solve	O
a	O
given	O
LP-type	B-Algorithm
problem	I-Algorithm
using	O
violation	O
tests	O
.	O
</s>
<s>
When	O
applied	O
to	O
a	O
linear	B-Algorithm
program	I-Algorithm
,	O
this	O
algorithm	O
can	O
be	O
interpreted	O
as	O
being	O
a	O
dual	O
simplex	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
describe	O
an	O
algorithm	O
that	O
uses	O
an	O
additional	O
property	O
of	O
linear	B-Algorithm
programs	I-Algorithm
that	O
is	O
not	O
always	O
held	O
by	O
other	O
LP-type	B-Algorithm
problems	I-Algorithm
,	O
that	O
all	O
bases	O
have	O
the	O
same	O
cardinality	B-Application
of	O
each	O
other	O
.	O
</s>
<s>
If	O
an	O
LP-type	B-Algorithm
problem	I-Algorithm
does	O
not	O
have	O
this	O
property	O
,	O
it	O
can	O
be	O
made	O
to	O
have	O
it	O
by	O
adding	O
new	O
dummy	O
elements	O
and	O
by	O
modifying	O
the	O
function	O
to	O
return	O
the	O
ordered	O
pair	O
of	O
its	O
old	O
value	O
and	O
of	O
the	O
number	O
,	O
ordered	O
lexicographically	O
.	O
</s>
<s>
He	O
shows	O
that	O
,	O
for	O
all	O
non-degenerate	O
LP-type	B-Algorithm
problems	I-Algorithm
(	O
that	O
is	O
,	O
problems	O
in	O
which	O
all	O
bases	O
have	O
distinct	O
values	O
)	O
this	O
problem	O
may	O
be	O
solved	O
in	O
time	O
,	O
by	O
solving	O
a	O
set	O
of	O
LP-type	B-Algorithm
problems	I-Algorithm
defined	O
by	O
subsets	O
of	O
.	O
</s>
<s>
Some	O
geometric	O
optimization	O
problems	O
may	O
be	O
expressed	O
as	O
LP-type	B-Algorithm
problems	I-Algorithm
in	O
which	O
the	O
number	O
of	O
elements	O
in	O
the	O
LP-type	O
formulation	O
is	O
significantly	O
greater	O
than	O
the	O
number	O
of	O
input	O
data	O
values	O
for	O
the	O
optimization	O
problem	O
.	O
</s>
<s>
The	O
problem	O
of	O
finding	O
a	O
time	O
at	O
which	O
the	O
diameter	O
is	O
minimized	O
can	O
be	O
formulated	O
as	O
minimizing	O
the	O
pointwise	B-Algorithm
maximum	I-Algorithm
of	O
quasiconvex	O
functions	O
,	O
one	O
for	O
each	O
pair	O
of	O
points	O
,	O
measuring	O
the	O
Euclidean	O
distance	O
between	O
the	O
pair	O
as	O
a	O
function	O
of	O
time	O
.	O
</s>
<s>
Thus	O
,	O
it	O
can	O
be	O
solved	O
as	O
an	O
LP-type	B-Algorithm
problem	I-Algorithm
of	O
combinatorial	O
dimension	O
two	O
on	O
a	O
set	O
of	O
elements	O
,	O
but	O
this	O
set	O
is	O
significantly	O
larger	O
than	O
the	O
number	O
of	O
input	O
points	O
.	O
</s>
<s>
describes	O
an	O
algorithm	O
for	O
solving	O
implicitly	O
defined	O
LP-type	B-Algorithm
problems	I-Algorithm
such	O
as	O
this	O
one	O
in	O
which	O
each	O
LP-type	O
element	O
is	O
determined	O
by	O
a	O
-tuple	O
of	O
input	O
values	O
,	O
for	O
some	O
constant	O
.	O
</s>
<s>
In	O
order	O
to	O
apply	O
his	O
approach	O
,	O
there	O
must	O
exist	O
a	O
decision	O
algorithm	O
that	O
can	O
determine	O
,	O
for	O
a	O
given	O
LP-type	O
basis	O
and	O
set	O
of	O
input	O
values	O
,	O
whether	O
is	O
a	O
basis	O
for	O
the	O
LP-type	B-Algorithm
problem	I-Algorithm
determined	O
by	O
.	O
</s>
<s>
If	O
the	O
number	O
of	O
input	O
values	O
is	O
below	O
some	O
threshold	O
value	O
,	O
find	O
the	O
set	O
of	O
LP-type	O
elements	O
that	O
it	O
determines	O
and	O
solve	O
the	O
resulting	O
explicit	O
LP-type	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
If	O
is	O
the	O
objective	O
function	O
for	O
the	O
implicitly	O
defined	O
LP-type	B-Algorithm
problem	I-Algorithm
to	O
be	O
solved	O
,	O
then	O
define	O
a	O
function	O
that	O
maps	O
collections	O
of	O
subsets	O
to	O
the	O
value	O
of	O
on	O
the	O
union	O
of	O
the	O
collection	O
.	O
</s>
<s>
Then	O
the	O
collection	O
of	O
subsets	O
and	O
the	O
objective	O
function	O
itself	O
defines	O
an	O
LP-type	B-Algorithm
problem	I-Algorithm
,	O
of	O
the	O
same	O
dimension	O
as	O
the	O
implicit	O
problem	O
to	O
be	O
solved	O
.	O
</s>
<s>
Solve	O
the	O
(	O
explicit	O
)	O
LP-type	B-Algorithm
problem	I-Algorithm
defined	O
by	O
using	O
Clarkson	O
's	O
algorithm	O
,	O
which	O
performs	O
a	O
linear	O
number	O
of	O
violation	O
tests	O
and	O
a	O
polylogarithmic	O
number	O
of	O
basis	O
evaluations	O
.	O
</s>
<s>
For	O
instance	O
,	O
for	O
the	O
minimum	O
diameter	O
of	O
moving	O
points	O
,	O
the	O
decision	O
algorithm	O
needs	O
only	O
to	O
calculate	O
the	O
diameter	O
of	O
a	O
set	O
of	O
points	O
at	O
a	O
fixed	O
time	O
,	O
a	O
problem	O
that	O
can	O
be	O
solved	O
in	O
time	O
using	O
the	O
rotating	B-Algorithm
calipers	I-Algorithm
technique	O
.	O
</s>
<s>
The	O
discovery	O
of	O
linear	O
time	O
algorithms	O
for	O
linear	B-Algorithm
programming	I-Algorithm
and	O
the	O
observation	O
that	O
the	O
same	O
algorithms	O
could	O
in	O
many	O
cases	O
be	O
used	O
to	O
solve	O
geometric	O
optimization	O
problems	O
that	O
were	O
not	O
linear	B-Algorithm
programs	I-Algorithm
goes	O
back	O
at	O
least	O
to	O
,	O
who	O
gave	O
a	O
linear	O
expected	O
time	O
algorithm	O
for	O
both	O
three-variable	O
linear	B-Algorithm
programs	I-Algorithm
and	O
the	O
smallest	O
circle	O
problem	O
.	O
</s>
<s>
However	O
,	O
Megiddo	O
formulated	O
the	O
generalization	O
of	O
linear	B-Algorithm
programming	I-Algorithm
geometrically	O
rather	O
than	O
combinatorially	O
,	O
as	O
a	O
convex	O
optimization	O
problem	O
rather	O
than	O
as	O
an	O
abstract	O
problem	O
on	O
systems	O
of	O
sets	O
.	O
</s>
<s>
Similarly	O
,	O
and	O
Clarkson	O
(	O
in	O
the	O
1988	O
conference	O
version	O
of	O
)	O
observed	O
that	O
their	O
methods	O
could	O
be	O
applied	O
to	O
convex	O
programs	O
as	O
well	O
as	O
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
The	O
use	O
of	O
randomization	O
to	O
improve	O
the	O
time	O
bounds	O
for	O
low	O
dimensional	O
linear	B-Algorithm
programming	I-Algorithm
and	O
related	O
problems	O
was	O
pioneered	O
by	O
Clarkson	O
and	O
by	O
.	O
</s>
<s>
The	O
definition	O
of	O
LP-type	B-Algorithm
problems	I-Algorithm
in	O
terms	O
of	O
functions	O
satisfying	O
the	O
axioms	O
of	O
locality	O
and	O
monotonicity	O
is	O
from	O
,	O
but	O
other	O
authors	O
in	O
the	O
same	O
timeframe	O
formulated	O
alternative	O
combinatorial	O
generalizations	O
of	O
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
break	O
the	O
ties	O
in	O
an	O
LP-type	B-Algorithm
problem	I-Algorithm
to	O
create	O
a	O
total	O
order	O
,	O
but	O
only	O
at	O
the	O
expense	O
of	O
an	O
increase	O
in	O
the	O
combinatorial	O
dimension	O
.	O
</s>
<s>
Additionally	O
,	O
as	O
in	O
LP-type	B-Algorithm
problems	I-Algorithm
,	O
Gärtner	O
defines	O
certain	O
primitives	O
for	O
performing	O
computations	O
on	O
subsets	O
of	O
elements	O
;	O
however	O
,	O
his	O
formalization	O
does	O
not	O
have	O
an	O
analogue	O
of	O
the	O
combinatorial	O
dimension	O
.	O
</s>
<s>
Another	O
abstract	O
generalization	O
of	O
both	O
linear	B-Algorithm
programs	I-Algorithm
and	O
linear	O
complementarity	O
problems	O
,	O
formulated	O
by	O
and	O
later	O
studied	O
by	O
several	O
other	O
authors	O
,	O
concerns	O
orientations	O
of	O
the	O
edges	O
of	O
a	O
hypercube	B-Operating_System
with	O
the	O
property	O
that	O
every	O
face	O
of	O
the	O
hypercube	B-Operating_System
(	O
including	O
the	O
whole	O
hypercube	B-Operating_System
as	O
a	O
face	O
)	O
has	O
a	O
unique	O
sink	O
,	O
a	O
vertex	O
with	O
no	O
outgoing	O
edges	O
.	O
</s>
<s>
An	O
orientation	O
of	O
this	O
type	O
may	O
be	O
formed	O
from	O
an	O
LP-type	B-Algorithm
problem	I-Algorithm
by	O
corresponding	O
the	O
subsets	O
of	O
S	O
with	O
the	O
vertices	O
of	O
a	O
hypercube	B-Operating_System
in	O
such	O
a	O
way	O
that	O
two	O
subsets	O
differ	O
by	O
a	O
single	O
element	O
if	O
and	O
only	O
if	O
the	O
corresponding	O
vertices	O
are	O
adjacent	O
,	O
and	O
by	O
orienting	O
the	O
edge	O
between	O
neighboring	O
sets	O
towards	O
if	O
and	O
towards	O
otherwise	O
.	O
</s>
<s>
The	O
resulting	O
orientation	O
has	O
the	O
additional	O
property	O
that	O
it	O
forms	O
a	O
directed	O
acyclic	O
graph	O
,	O
from	O
which	O
it	O
can	O
be	O
shown	O
that	O
a	O
randomized	B-General_Concept
algorithm	I-General_Concept
can	O
find	O
the	O
unique	O
sink	O
of	O
the	O
whole	O
hypercube	B-Operating_System
(	O
the	O
optimal	O
basis	O
of	O
the	O
LP-type	B-Algorithm
problem	I-Algorithm
)	O
in	O
a	O
number	O
of	O
steps	O
exponential	O
in	O
the	O
square	O
root	O
of	O
.	O
</s>
<s>
The	O
more	O
recently	O
developed	O
framework	O
of	O
violator	O
spaces	O
generalizes	O
LP-type	B-Algorithm
problems	I-Algorithm
,	O
in	O
the	O
sense	O
that	O
every	O
LP-type	B-Algorithm
problem	I-Algorithm
can	O
be	O
modeled	O
by	O
a	O
violator	O
space	O
but	O
not	O
necessarily	O
vice	O
versa	O
.	O
</s>
<s>
Violator	O
spaces	O
are	O
defined	O
similarly	O
to	O
LP-type	B-Algorithm
problems	I-Algorithm
,	O
by	O
a	O
function	O
that	O
maps	O
sets	O
to	O
objective	O
function	O
values	O
,	O
but	O
the	O
values	O
of	O
are	O
not	O
ordered	O
.	O
</s>
<s>
Despite	O
the	O
lack	O
of	O
ordering	O
,	O
every	O
set	O
has	O
a	O
well-defined	O
set	O
of	O
bases	O
(	O
the	O
minimal	O
sets	O
with	O
the	O
same	O
value	O
as	O
the	O
whole	O
set	O
)	O
that	O
can	O
be	O
found	O
by	O
variations	O
of	O
Clarkson	O
's	O
algorithms	O
for	O
LP-type	B-Algorithm
problems	I-Algorithm
.	O
</s>
