<s>
The	O
configuration	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
(	O
configuration-LP	O
)	O
is	O
a	O
particular	O
linear	B-Algorithm
programming	I-Algorithm
used	O
for	O
solving	O
combinatorial	O
optimization	O
problems	O
.	O
</s>
<s>
Later	O
,	O
it	O
has	O
been	O
applied	O
to	O
bin	O
packing	O
and	O
job	B-Algorithm
scheduling	I-Algorithm
.	O
</s>
<s>
Then	O
,	O
the	O
configuration	B-Algorithm
LP	I-Algorithm
of	O
bin-packing	O
is	O
:	O
</s>
<s>
The	O
configuration	B-Algorithm
LP	I-Algorithm
is	O
an	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
,	O
so	O
in	O
general	O
it	O
is	O
NP-hard	O
.	O
</s>
<s>
The	O
fractional	O
configuration	B-Algorithm
LP	I-Algorithm
of	O
bin-packing	O
It	O
is	O
the	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
of	O
the	O
above	O
ILP	O
.	O
</s>
<s>
The	O
relaxation	O
was	O
first	O
presented	O
by	O
Gilmore	O
and	O
Gomory	O
,	O
and	O
it	O
is	O
often	O
called	O
the	O
Gilmore-Gomory	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
.	O
</s>
<s>
A	O
linear	B-Algorithm
program	I-Algorithm
with	O
no	O
integrality	O
constraints	O
can	O
be	O
solved	O
in	O
time	O
polynomial	O
in	O
the	O
number	O
of	O
variables	O
and	O
constraints	O
.	O
</s>
<s>
The	O
problem	O
is	O
that	O
the	O
number	O
of	O
variables	O
in	O
the	O
fractional	O
configuration	B-Algorithm
LP	I-Algorithm
is	O
equal	O
to	O
the	O
number	O
of	O
possible	O
configurations	O
,	O
which	O
might	O
be	O
huge	O
.	O
</s>
<s>
First	O
,	O
they	O
construct	O
the	O
dual	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
of	O
the	O
fractional	O
LP:.It	O
has	O
S	O
variables	O
y1	O
,...,	O
yS	O
,	O
and	O
C	O
constraints	O
:	O
for	O
each	O
configuration	O
c	O
,	O
there	O
is	O
a	O
constraint	O
,	O
where	O
is	O
the	O
column	O
of	O
A	O
representing	O
the	O
configuration	O
c	O
.	O
3It	O
has	O
the	O
following	O
economic	O
interpretation	O
.	O
</s>
<s>
Second	O
,	O
they	O
apply	O
a	O
variant	O
of	O
the	O
ellipsoid	B-Algorithm
method	I-Algorithm
,	O
which	O
does	O
not	O
need	O
to	O
list	O
all	O
the	O
constraints	O
-	O
it	O
just	O
needs	O
a	O
separation	O
oracle	O
.	O
</s>
<s>
The	O
separation	O
oracle	O
for	O
the	O
dual	O
LP	O
can	O
be	O
implemented	O
by	O
solving	O
the	O
knapsack	B-Algorithm
problem	I-Algorithm
with	O
sizes	O
s	O
and	O
values	O
y	O
:	O
if	O
the	O
optimal	O
solution	O
of	O
the	O
knapsack	B-Algorithm
problem	I-Algorithm
has	O
a	O
total	O
value	O
at	O
most	O
1	O
,	O
then	O
y	O
is	O
feasible	O
;	O
if	O
it	O
is	O
larger	O
than	O
1	O
,	O
than	O
y	O
is	O
not	O
feasible	O
,	O
and	O
the	O
optimal	O
solution	O
of	O
the	O
knapsack	B-Algorithm
problem	I-Algorithm
identifies	O
a	O
configuration	O
for	O
which	O
the	O
constraint	O
is	O
violated	O
.	O
</s>
<s>
Third	O
,	O
they	O
show	O
that	O
,	O
with	O
an	O
approximate	O
solution	O
to	O
the	O
knapsack	B-Algorithm
problem	I-Algorithm
,	O
one	O
can	O
get	O
an	O
approximate	O
solution	O
to	O
the	O
dual	O
LP	O
,	O
and	O
from	O
this	O
,	O
an	O
approximate	O
solution	O
to	O
the	O
primal	O
LP	O
;	O
see	O
Karmarkar-Karp	O
bin	O
packing	O
algorithms	O
.	O
</s>
<s>
A	O
natural	O
configuration	B-Algorithm
LP	I-Algorithm
for	O
this	O
problem	O
could	O
be:where	O
A	O
represents	O
all	O
configurations	O
of	O
items	O
with	O
sum	O
at	O
least	O
B	O
(	O
one	O
can	O
take	O
only	O
the	O
inclusion-minimal	O
configurations	O
)	O
.	O
</s>
<s>
In	O
the	O
problem	O
of	O
unrelated-machines	B-Algorithm
scheduling	I-Algorithm
,	O
there	O
are	O
some	O
m	O
different	O
machines	O
that	O
should	O
process	O
some	O
n	O
different	O
jobs	O
.	O
</s>
<s>
The	O
integrality	O
gap	O
of	O
the	O
configuration-LP	O
for	O
unrelated-machines	B-Algorithm
scheduling	I-Algorithm
is	O
2	O
.	O
</s>
