<s>
In	O
mathematics	O
,	O
the	O
relaxation	O
of	O
a	O
(	O
mixed	O
)	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
is	O
the	O
problem	O
that	O
arises	O
by	O
removing	O
the	O
integrality	O
constraint	O
of	O
each	O
variable	O
.	O
</s>
<s>
The	O
resulting	O
relaxation	O
is	O
a	O
linear	B-Algorithm
program	I-Algorithm
,	O
hence	O
the	O
name	O
.	O
</s>
<s>
This	O
relaxation	O
technique	O
transforms	O
an	O
NP-hard	O
optimization	O
problem	O
(	O
integer	B-Algorithm
programming	I-Algorithm
)	O
into	O
a	O
related	O
problem	O
that	O
is	O
solvable	O
in	O
polynomial	O
time	O
(	O
linear	B-Algorithm
programming	I-Algorithm
)	O
;	O
the	O
solution	O
to	O
the	O
relaxed	O
linear	B-Algorithm
program	I-Algorithm
can	O
be	O
used	O
to	O
gain	O
information	O
about	O
the	O
solution	O
to	O
the	O
original	O
integer	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
Consider	O
the	O
set	B-Algorithm
cover	I-Algorithm
problem	I-Algorithm
,	O
the	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
of	O
which	O
was	O
first	O
considered	O
by	O
.	O
</s>
<s>
To	O
formulate	O
this	O
as	O
a	O
0	O
–	O
1	O
integer	B-Algorithm
program	I-Algorithm
,	O
form	O
an	O
indicator	O
variable	O
xi	O
for	O
each	O
set	O
Si	O
,	O
that	O
takes	O
the	O
value	O
1	O
when	O
Si	O
belongs	O
to	O
the	O
chosen	O
subfamily	O
and	O
0	O
when	O
it	O
does	O
not	O
.	O
</s>
<s>
The	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
of	O
the	O
set	B-Algorithm
cover	I-Algorithm
problem	I-Algorithm
describes	O
a	O
fractional	O
cover	O
in	O
which	O
the	O
input	O
sets	O
are	O
assigned	O
weights	O
such	O
that	O
the	O
total	O
weight	O
of	O
the	O
sets	O
containing	O
each	O
element	O
is	O
at	O
least	O
one	O
and	O
the	O
total	O
weight	O
of	O
all	O
sets	O
is	O
minimized	O
.	O
</s>
<s>
As	O
a	O
specific	O
example	O
of	O
the	O
set	B-Algorithm
cover	I-Algorithm
problem	I-Algorithm
,	O
consider	O
the	O
instance	O
F	O
=	O
{{	O
a	O
,	O
b}	O
,	O
{	O
b	O
,	O
c}	O
,	O
{	O
a	O
,	O
c}}	O
.	O
</s>
<s>
There	O
are	O
three	O
optimal	O
set	B-Algorithm
covers	I-Algorithm
,	O
each	O
of	O
which	O
includes	O
two	O
of	O
the	O
three	O
given	O
sets	O
.	O
</s>
<s>
Thus	O
,	O
the	O
optimal	O
value	O
of	O
the	O
objective	O
function	O
of	O
the	O
corresponding	O
0	O
–	O
1	O
integer	B-Algorithm
program	I-Algorithm
is	O
2	O
,	O
the	O
number	O
of	O
sets	O
in	O
the	O
optimal	O
covers	O
.	O
</s>
<s>
Thus	O
,	O
in	O
this	O
example	O
,	O
the	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
has	O
a	O
value	O
differing	O
from	O
that	O
of	O
the	O
unrelaxed	O
0	O
–	O
1	O
integer	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
The	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
of	O
an	O
integer	B-Algorithm
program	I-Algorithm
may	O
be	O
solved	O
using	O
any	O
standard	O
linear	B-Algorithm
programming	I-Algorithm
technique	O
.	O
</s>
<s>
If	O
the	O
optimal	O
solution	O
to	O
the	O
linear	B-Algorithm
program	I-Algorithm
happens	O
to	O
have	O
all	O
variables	O
either	O
0	O
or	O
1	O
,	O
it	O
will	O
also	O
be	O
an	O
optimal	O
solution	O
to	O
the	O
original	O
integer	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
problems	O
with	O
totally	B-Algorithm
unimodular	I-Algorithm
matrix	O
specifications	O
.	O
)	O
</s>
<s>
In	O
all	O
cases	O
,	O
though	O
,	O
the	O
solution	O
quality	O
of	O
the	O
linear	B-Algorithm
program	I-Algorithm
is	O
at	O
least	O
as	O
good	O
as	O
that	O
of	O
the	O
integer	B-Algorithm
program	I-Algorithm
,	O
because	O
any	O
integer	B-Algorithm
program	I-Algorithm
solution	O
would	O
also	O
be	O
a	O
valid	O
linear	B-Algorithm
program	I-Algorithm
solution	O
.	O
</s>
<s>
That	O
is	O
,	O
in	O
a	O
maximization	O
problem	O
,	O
the	O
relaxed	O
program	O
has	O
a	O
value	O
greater	O
than	O
or	O
equal	O
to	O
that	O
of	O
the	O
original	O
program	O
,	O
while	O
in	O
a	O
minimization	O
problem	O
such	O
as	O
the	O
set	B-Algorithm
cover	I-Algorithm
problem	I-Algorithm
the	O
relaxed	O
program	O
has	O
a	O
value	O
smaller	O
than	O
or	O
equal	O
to	O
that	O
of	O
the	O
original	O
program	O
.	O
</s>
<s>
Thus	O
,	O
the	O
relaxation	O
provides	O
an	O
optimistic	O
bound	O
on	O
the	O
integer	B-Algorithm
program	I-Algorithm
's	O
solution	O
.	O
</s>
<s>
In	O
the	O
example	O
instance	O
of	O
the	O
set	B-Algorithm
cover	I-Algorithm
problem	I-Algorithm
described	O
above	O
,	O
in	O
which	O
the	O
relaxation	O
has	O
an	O
optimal	O
solution	O
value	O
of	O
3/2	O
,	O
we	O
can	O
deduce	O
that	O
the	O
optimal	O
solution	O
value	O
of	O
the	O
unrelaxed	O
integer	B-Algorithm
program	I-Algorithm
is	O
at	O
least	O
as	O
large	O
.	O
</s>
<s>
Since	O
the	O
set	B-Algorithm
cover	I-Algorithm
problem	I-Algorithm
has	O
solution	O
values	O
that	O
are	O
integers	O
(	O
the	O
numbers	O
of	O
sets	O
chosen	O
in	O
the	O
subfamily	O
)	O
,	O
the	O
optimal	O
solution	O
quality	O
must	O
be	O
at	O
least	O
as	O
large	O
as	O
the	O
next	O
larger	O
integer	O
,	O
2	O
.	O
</s>
<s>
Thus	O
,	O
in	O
this	O
instance	O
,	O
despite	O
having	O
a	O
different	O
value	O
from	O
the	O
unrelaxed	O
problem	O
,	O
the	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
gives	O
us	O
a	O
tight	O
lower	O
bound	O
on	O
the	O
solution	O
quality	O
of	O
the	O
original	O
problem	O
.	O
</s>
<s>
Linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
is	O
a	O
standard	O
technique	O
for	O
designing	O
approximation	B-Algorithm
algorithms	I-Algorithm
for	O
hard	O
optimization	O
problems	O
.	O
</s>
<s>
In	O
this	O
application	O
,	O
an	O
important	O
concept	O
is	O
the	O
integrality	O
gap	O
,	O
the	O
maximum	O
ratio	O
between	O
the	O
solution	O
quality	O
of	O
the	O
integer	B-Algorithm
program	I-Algorithm
and	O
of	O
its	O
relaxation	O
.	O
</s>
<s>
In	O
an	O
instance	O
of	O
a	O
minimization	O
problem	O
,	O
if	O
the	O
real	O
minimum	O
(	O
the	O
minimum	O
of	O
the	O
integer	O
problem	O
)	O
is	O
,	O
and	O
the	O
relaxed	O
minimum	O
(	O
the	O
minimum	O
of	O
the	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
)	O
is	O
,	O
then	O
the	O
integrality	O
gap	O
of	O
that	O
instance	O
is	O
.	O
</s>
<s>
Typically	O
,	O
the	O
integrality	O
gap	O
translates	O
into	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
an	O
approximation	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
This	O
is	O
because	O
an	O
approximation	B-Algorithm
algorithm	I-Algorithm
relies	O
on	O
some	O
rounding	O
strategy	O
that	O
finds	O
,	O
for	O
every	O
relaxed	O
solution	O
of	O
size	O
,	O
an	O
integer	O
solution	O
of	O
size	O
at	O
most	O
(	O
where	O
RR	O
is	O
the	O
rounding	O
ratio	O
)	O
.	O
</s>
<s>
The	O
rounding	O
ratio	O
RR	O
is	O
only	O
an	O
upper	O
bound	O
on	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
,	O
so	O
in	O
theory	O
the	O
actual	O
approximation	B-Algorithm
ratio	I-Algorithm
may	O
be	O
lower	O
than	O
IG	O
,	O
but	O
this	O
may	O
be	O
hard	O
to	O
prove	O
.	O
</s>
<s>
In	O
practice	O
,	O
a	O
large	O
IG	O
usually	O
implies	O
that	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
in	O
the	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
might	O
be	O
bad	O
,	O
and	O
it	O
may	O
be	O
better	O
to	O
look	O
for	O
other	O
approximation	O
schemes	O
for	O
that	O
problem	O
.	O
</s>
<s>
For	O
the	O
set	B-Algorithm
cover	I-Algorithm
problem	I-Algorithm
,	O
Lovász	O
proved	O
that	O
the	O
integrality	O
gap	O
for	O
an	O
instance	O
with	O
n	O
elements	O
is	O
Hn	O
,	O
the	O
nth	O
harmonic	O
number	O
.	O
</s>
<s>
One	O
can	O
turn	O
the	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
for	O
this	O
problem	O
into	O
an	O
approximate	O
solution	O
of	O
the	O
original	O
unrelaxed	O
set	B-Algorithm
cover	I-Algorithm
instance	O
via	O
the	O
technique	O
of	O
randomized	B-Algorithm
rounding	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
this	O
technique	O
leads	O
to	O
a	O
randomized	B-General_Concept
approximation	B-Algorithm
algorithm	I-Algorithm
that	O
finds	O
a	O
set	B-Algorithm
cover	I-Algorithm
within	O
a	O
logarithmic	O
factor	O
of	O
the	O
optimum	O
.	O
</s>
<s>
As	O
showed	O
,	O
both	O
the	O
random	O
part	O
of	O
this	O
algorithm	O
and	O
the	O
need	O
to	O
construct	O
an	O
explicit	O
solution	O
to	O
the	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
may	O
be	O
eliminated	O
using	O
the	O
method	B-Algorithm
of	I-Algorithm
conditional	I-Algorithm
probabilities	I-Algorithm
,	O
leading	O
to	O
a	O
deterministic	O
greedy	B-Algorithm
algorithm	I-Algorithm
for	O
set	B-Algorithm
cover	I-Algorithm
,	O
known	O
already	O
to	O
Lovász	O
,	O
that	O
repeatedly	O
selects	O
the	O
set	O
that	O
covers	O
the	O
largest	O
possible	O
number	O
of	O
remaining	O
uncovered	O
elements	O
.	O
</s>
<s>
This	O
greedy	B-Algorithm
algorithm	I-Algorithm
approximates	O
the	O
set	B-Algorithm
cover	I-Algorithm
to	O
within	O
the	O
same	O
Hn	O
factor	O
that	O
Lovász	O
proved	O
as	O
the	O
integrality	O
gap	O
for	O
set	B-Algorithm
cover	I-Algorithm
.	O
</s>
<s>
There	O
are	O
strong	O
complexity-theoretic	O
reasons	O
for	O
believing	O
that	O
no	O
polynomial	O
time	O
approximation	B-Algorithm
algorithm	I-Algorithm
can	O
achieve	O
a	O
significantly	O
better	O
approximation	B-Algorithm
ratio	I-Algorithm
.	O
</s>
<s>
Similar	O
randomized	B-Algorithm
rounding	I-Algorithm
techniques	O
,	O
and	O
derandomized	O
approximation	B-Algorithm
algorithms	I-Algorithm
,	O
may	O
be	O
used	O
in	O
conjunction	O
with	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
to	O
develop	O
approximation	B-Algorithm
algorithms	I-Algorithm
for	O
many	O
other	O
problems	O
,	O
as	O
described	O
by	O
Raghavan	O
,	O
Tompson	O
,	O
and	O
Young	O
.	O
</s>
<s>
As	O
well	O
as	O
its	O
uses	O
in	O
approximation	O
,	O
linear	B-Algorithm
programming	I-Algorithm
plays	O
an	O
important	O
role	O
in	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
algorithms	I-Algorithm
for	O
computing	O
the	O
true	O
optimum	O
solution	O
to	O
hard	O
optimization	O
problems	O
.	O
</s>
<s>
If	O
some	O
variables	O
in	O
the	O
optimal	O
solution	O
have	O
fractional	O
values	O
,	O
we	O
may	O
start	O
a	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
type	O
process	O
,	O
in	O
which	O
we	O
recursively	O
solve	O
subproblems	O
in	O
which	O
some	O
of	O
the	O
fractional	O
variables	O
have	O
their	O
values	O
fixed	O
to	O
either	O
zero	O
or	O
one	O
.	O
</s>
<s>
In	O
each	O
step	O
of	O
an	O
algorithm	O
of	O
this	O
type	O
,	O
we	O
consider	O
a	O
subproblem	O
of	O
the	O
original	O
0	O
–	O
1	O
integer	B-Algorithm
program	I-Algorithm
in	O
which	O
some	O
of	O
the	O
variables	O
have	O
values	O
assigned	O
to	O
them	O
,	O
either	O
0	O
or	O
1	O
,	O
and	O
the	O
remaining	O
variables	O
are	O
still	O
free	O
to	O
take	O
on	O
either	O
value	O
.	O
</s>
<s>
Compute	O
the	O
optimal	O
solution	O
to	O
the	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
of	O
the	O
current	O
subproblem	O
.	O
</s>
<s>
Two	O
0	O
–	O
1	O
integer	B-Algorithm
programs	I-Algorithm
that	O
are	O
equivalent	O
,	O
in	O
that	O
they	O
have	O
the	O
same	O
objective	O
function	O
and	O
the	O
same	O
set	O
of	O
feasible	O
solutions	O
,	O
may	O
have	O
quite	O
different	O
linear	B-Algorithm
programming	I-Algorithm
relaxations	I-Algorithm
:	O
a	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
can	O
be	O
viewed	O
geometrically	O
,	O
as	O
a	O
convex	O
polytope	O
that	O
includes	O
all	O
feasible	O
solutions	O
and	O
excludes	O
all	O
other	O
0	O
–	O
1	O
vectors	O
,	O
and	O
infinitely	O
many	O
different	O
polytopes	O
have	O
this	O
property	O
.	O
</s>
<s>
Ideally	O
,	O
one	O
would	O
like	O
to	O
use	O
as	O
a	O
relaxation	O
the	O
convex	O
hull	O
of	O
the	O
feasible	O
solutions	O
;	O
linear	B-Algorithm
programming	I-Algorithm
on	O
this	O
polytope	O
would	O
automatically	O
yield	O
the	O
correct	O
solution	O
to	O
the	O
original	O
integer	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
Typical	O
relaxations	O
,	O
such	O
as	O
the	O
relaxation	O
of	O
the	O
set	B-Algorithm
cover	I-Algorithm
problem	I-Algorithm
discussed	O
earlier	O
,	O
form	O
a	O
polytope	O
that	O
strictly	O
contains	O
the	O
convex	O
hull	O
and	O
has	O
vertices	O
other	O
than	O
the	O
0	O
–	O
1	O
vectors	O
that	O
solve	O
the	O
unrelaxed	O
problem	O
.	O
</s>
<s>
The	O
cutting-plane	B-Algorithm
method	I-Algorithm
for	O
solving	O
0	O
–	O
1	O
integer	B-Algorithm
programs	I-Algorithm
,	O
first	O
introduced	O
for	O
the	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
by	O
and	O
generalized	O
to	O
other	O
integer	B-Algorithm
programs	I-Algorithm
by	O
,	O
takes	O
advantage	O
of	O
this	O
multiplicity	O
of	O
possible	O
relaxations	O
by	O
finding	O
a	O
sequence	O
of	O
relaxations	O
that	O
more	O
tightly	O
constrain	O
the	O
solution	O
space	O
until	O
eventually	O
an	O
integer	O
solution	O
is	O
obtained	O
.	O
</s>
<s>
This	O
method	O
starts	O
from	O
any	O
relaxation	O
of	O
the	O
given	O
program	O
,	O
and	O
finds	O
an	O
optimal	O
solution	O
using	O
a	O
linear	B-Algorithm
programming	I-Algorithm
solver	O
.	O
</s>
<s>
Otherwise	O
,	O
an	O
additional	O
linear	O
constraint	O
(	O
a	O
cutting	B-Algorithm
plane	I-Algorithm
or	O
cut	O
)	O
is	O
found	O
that	O
separates	O
the	O
resulting	O
fractional	O
solution	O
from	O
the	O
convex	O
hull	O
of	O
the	O
integer	O
solutions	O
,	O
and	O
the	O
method	O
repeats	O
on	O
this	O
new	O
more	O
tightly	O
constrained	O
problem	O
.	O
</s>
<s>
It	O
is	O
especially	O
desirable	O
to	O
find	O
cutting	B-Algorithm
planes	I-Algorithm
that	O
form	O
facets	O
of	O
the	O
convex	O
hull	O
of	O
the	O
integer	O
solutions	O
,	O
as	O
these	O
planes	O
are	O
the	O
ones	O
that	O
most	O
tightly	O
constrain	O
the	O
solution	O
space	O
;	O
there	O
always	O
exists	O
a	O
cutting	B-Algorithm
plane	I-Algorithm
of	O
this	O
type	O
that	O
separates	O
any	O
fractional	O
solution	O
from	O
the	O
integer	O
solutions	O
.	O
</s>
<s>
The	O
related	O
branch	B-Algorithm
and	I-Algorithm
cut	I-Algorithm
method	O
combines	O
the	O
cutting	B-Algorithm
plane	I-Algorithm
and	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
methods	O
.	O
</s>
<s>
In	O
any	O
subproblem	O
,	O
it	O
runs	O
the	O
cutting	B-Algorithm
plane	I-Algorithm
method	I-Algorithm
until	O
no	O
more	O
cutting	B-Algorithm
planes	I-Algorithm
can	O
be	O
found	O
,	O
and	O
then	O
branches	O
on	O
one	O
of	O
the	O
remaining	O
fractional	O
variables	O
.	O
</s>
