<s>
Karmarkar	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
an	O
algorithm	O
introduced	O
by	O
Narendra	O
Karmarkar	O
in	O
1984	O
for	O
solving	O
linear	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
The	O
ellipsoid	B-Algorithm
method	I-Algorithm
is	O
also	O
polynomial	O
time	O
but	O
proved	O
to	O
be	O
inefficient	O
in	O
practice	O
.	O
</s>
<s>
Denoting	O
as	O
the	O
number	O
of	O
variables	O
and	O
as	O
the	O
number	O
of	O
bits	O
of	O
input	O
to	O
the	O
algorithm	O
,	O
Karmarkar	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
requires	O
operations	O
on	O
-digit	O
numbers	O
,	O
as	O
compared	O
to	O
such	O
operations	O
for	O
the	O
ellipsoid	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
using	O
FFT-based	B-Algorithm
multiplication	I-Algorithm
(	O
see	O
Big	O
O	O
notation	O
)	O
.	O
</s>
<s>
Karmarkar	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
falls	O
within	O
the	O
class	O
of	O
interior-point	B-Algorithm
methods	I-Algorithm
:	O
the	O
current	O
guess	O
for	O
the	O
solution	O
does	O
not	O
follow	O
the	O
boundary	O
of	O
the	O
feasible	O
set	O
as	O
in	O
the	O
simplex	B-Algorithm
method	I-Algorithm
,	O
but	O
moves	O
through	O
the	O
interior	O
of	O
the	O
feasible	O
region	O
,	O
improving	O
the	O
approximation	O
of	O
the	O
optimal	O
solution	O
by	O
a	O
definite	O
fraction	O
with	O
every	O
iteration	O
and	O
converging	O
to	O
an	O
optimal	O
solution	O
with	O
rational	O
data	O
.	O
</s>
<s>
Consider	O
a	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
in	O
matrix	O
form	O
:	O
</s>
<s>
Karmarkar	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
determines	O
the	O
next	O
feasible	O
direction	O
toward	O
optimality	O
and	O
scales	O
back	O
by	O
a	O
factor	O
.	O
</s>
<s>
Since	O
the	O
actual	O
algorithm	O
is	O
rather	O
complicated	O
,	O
researchers	O
looked	O
for	O
a	O
more	O
intuitive	O
version	O
of	O
it	O
,	O
and	O
in	O
1985	O
developed	O
affine	B-Algorithm
scaling	I-Algorithm
,	O
a	O
version	O
of	O
Karmarkar	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
that	O
uses	O
affine	B-Algorithm
transformations	I-Algorithm
where	O
Karmarkar	O
used	O
projective	O
ones	O
,	O
only	O
to	O
realize	O
four	O
years	O
later	O
that	O
they	O
had	O
rediscovered	O
an	O
algorithm	O
published	O
by	O
Soviet	O
mathematician	O
I	O
.	O
I	O
.	O
Dikin	O
in	O
1967	O
.	O
</s>
<s>
This	O
left	O
many	O
mathematicians	O
uneasy	O
,	O
such	O
as	O
Ronald	O
Rivest	O
(	O
himself	O
one	O
of	O
the	O
holders	O
of	O
the	O
patent	O
on	O
the	O
RSA	B-Architecture
algorithm	I-Architecture
)	O
,	O
who	O
expressed	O
the	O
opinion	O
that	O
research	O
proceeded	O
on	O
the	O
basis	O
that	O
algorithms	O
should	O
be	O
free	O
.	O
</s>
<s>
Mathematicians	O
who	O
specialized	O
in	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
including	O
Philip	O
Gill	O
and	O
others	O
,	O
claimed	O
that	O
Karmarkar	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
equivalent	O
to	O
a	O
projected	O
Newton	O
barrier	B-Algorithm
method	I-Algorithm
with	O
a	O
logarithmic	O
barrier	O
function	O
,	O
if	O
the	O
parameters	O
are	O
chosen	O
suitably	O
.	O
</s>
<s>
Legal	O
scholar	O
Andrew	O
Chin	O
opines	O
that	O
Gill	O
's	O
argument	O
was	O
flawed	O
,	O
insofar	O
as	O
the	O
method	O
they	O
describe	O
does	O
not	O
constitute	O
an	O
"	O
algorithm	O
"	O
,	O
since	O
it	O
requires	O
choices	O
of	O
parameters	O
that	O
do	O
n't	O
follow	O
from	O
the	O
internal	O
logic	O
of	O
the	O
method	O
,	O
but	O
rely	O
on	O
external	O
guidance	O
,	O
essentially	O
from	O
Karmarkar	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
AT&T	O
designed	O
a	O
vector	B-Operating_System
multi-processor	B-Operating_System
computer	O
system	O
specifically	O
to	O
run	O
Karmarkar	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
calling	O
the	O
resulting	O
combination	O
of	O
hardware	O
and	O
software	O
KORBX	O
,	O
and	O
marketed	O
this	O
system	O
at	O
a	O
price	O
of	O
US	O
$8.9	O
million	O
.	O
</s>
<s>
Opponents	O
of	O
software	O
patents	O
have	O
further	O
argued	O
that	O
the	O
patents	O
ruined	O
the	O
positive	O
interaction	O
cycles	O
that	O
previously	O
characterized	O
the	O
relationship	O
between	O
researchers	O
in	O
linear	B-Algorithm
programming	I-Algorithm
and	O
industry	O
,	O
and	O
specifically	O
it	O
isolated	O
Karmarkar	O
himself	O
from	O
the	O
network	O
of	O
mathematical	O
researchers	O
in	O
his	O
field	O
.	O
</s>
