<s>
Within	O
computer	B-General_Concept
science	I-General_Concept
and	O
operations	O
research	O
,	O
</s>
<s>
Many	O
such	O
problems	O
do	O
admit	O
fast	O
(	O
polynomial	O
time	O
)	O
approximation	B-Algorithm
algorithms	I-Algorithm
—	O
that	O
is	O
,	O
algorithms	O
that	O
are	O
guaranteed	O
to	O
return	O
an	O
approximately	O
optimal	O
solution	O
given	O
any	O
input	O
.	O
</s>
<s>
is	O
a	O
widely	O
used	O
approach	O
for	O
designing	O
and	O
analyzing	O
such	O
approximation	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
Formulate	O
the	O
problem	O
to	O
be	O
solved	O
as	O
an	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
(	O
ILP	O
)	O
.	O
</s>
<s>
Compute	O
an	O
optimal	O
fractional	O
solution	O
to	O
the	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
(	O
LP	O
)	O
of	O
the	O
ILP	O
.	O
</s>
<s>
(	O
Although	O
the	O
approach	O
is	O
most	O
commonly	O
applied	O
with	O
linear	B-Algorithm
programs	I-Algorithm
,	O
</s>
<s>
Max-Cut	O
approximation	B-Algorithm
algorithm	I-Algorithm
.	O
)	O
</s>
<s>
The	O
challenge	O
in	O
the	O
first	O
step	O
is	O
to	O
choose	O
a	O
suitable	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
.	O
</s>
<s>
Familiarity	O
with	O
linear	B-Algorithm
programming	I-Algorithm
,	O
in	O
particular	O
modelling	O
using	O
linear	B-Algorithm
programs	I-Algorithm
and	O
integer	B-Algorithm
linear	I-Algorithm
programs	I-Algorithm
,	O
is	O
required	O
.	O
</s>
<s>
For	O
many	O
problems	O
,	O
there	O
is	O
a	O
natural	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
that	O
works	O
well	O
,	O
</s>
<s>
such	O
as	O
in	O
the	O
Set	B-Algorithm
Cover	I-Algorithm
example	O
below	O
.	O
</s>
<s>
indeed	O
randomized	B-Algorithm
rounding	I-Algorithm
is	O
often	O
used	O
to	O
prove	O
bounds	O
on	O
integrality	O
gaps	O
.	O
)	O
</s>
<s>
using	O
any	O
standard	O
linear	B-Algorithm
programming	I-Algorithm
algorithm	O
.	O
</s>
<s>
Given	O
any	O
fractional	O
solution	O
of	O
the	O
LP	O
,	O
with	O
positive	O
probability	O
the	O
randomized	B-Algorithm
rounding	I-Algorithm
process	O
produces	O
an	O
integer	O
solution	O
that	O
approximates	O
according	O
to	O
some	O
desired	O
criterion	O
.	O
</s>
<s>
typically	O
using	O
the	O
method	B-Algorithm
of	I-Algorithm
conditional	I-Algorithm
probabilities	I-Algorithm
.	O
</s>
<s>
The	O
randomized	B-Algorithm
rounding	I-Algorithm
step	O
differs	O
from	O
most	O
applications	O
of	O
the	O
probabilistic	O
method	O
in	O
two	O
respects	O
:	O
</s>
<s>
The	O
computational	B-General_Concept
complexity	I-General_Concept
of	O
the	O
rounding	O
step	O
is	O
important	O
.	O
</s>
<s>
The	O
probability	O
distribution	O
underlying	O
the	O
random	O
experiment	O
is	O
a	O
function	O
of	O
the	O
solution	O
of	O
a	O
relaxation	B-Algorithm
of	O
the	O
problem	O
instance	O
.	O
</s>
<s>
This	O
fact	O
is	O
crucial	O
to	O
proving	O
the	O
performance	O
guarantee	O
of	O
the	O
approximation	B-Algorithm
algorithm	I-Algorithm
---	O
that	O
is	O
,	O
that	O
for	O
any	O
problem	O
instance	O
,	O
the	O
algorithm	O
returns	O
a	O
solution	O
that	O
approximates	O
the	O
optimal	O
solution	O
for	O
that	O
specific	O
instance	O
.	O
</s>
<s>
The	O
following	O
example	O
illustrates	O
how	O
randomized	B-Algorithm
rounding	I-Algorithm
can	O
be	O
used	O
to	O
design	O
an	O
approximation	B-Algorithm
algorithm	I-Algorithm
for	O
the	O
Set	B-Algorithm
Cover	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Fix	O
any	O
instance	O
of	O
set	B-Algorithm
cover	I-Algorithm
over	O
a	O
universe	O
.	O
</s>
<s>
For	O
step	O
1	O
,	O
let	O
IP	O
be	O
the	O
standard	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
for	O
set	B-Algorithm
cover	I-Algorithm
for	O
this	O
instance	O
.	O
</s>
<s>
For	O
step	O
2	O
,	O
let	O
LP	O
be	O
the	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
of	O
IP	O
,	O
</s>
<s>
using	O
any	O
standard	O
linear	B-Algorithm
programming	I-Algorithm
algorithm	O
.	O
</s>
<s>
the	O
cost	O
of	O
is	O
a	O
lower	O
bound	O
on	O
the	O
cost	O
of	O
the	O
optimal	O
set	B-Algorithm
cover	I-Algorithm
.	O
</s>
<s>
into	O
a	O
feasible	O
integer	O
solution	O
(	O
corresponding	O
to	O
a	O
true	O
set	B-Algorithm
cover	I-Algorithm
)	O
.	O
</s>
<s>
Then	O
(	O
since	O
the	O
cost	O
of	O
is	O
a	O
lower	O
bound	O
on	O
the	O
cost	O
of	O
the	O
optimal	O
set	B-Algorithm
cover	I-Algorithm
)	O
,	O
</s>
<s>
With	O
positive	O
probability	O
,	O
the	O
rounding	O
scheme	O
returns	O
a	O
set	B-Algorithm
cover	I-Algorithm
of	O
cost	O
at	O
most	O
(	O
and	O
thus	O
of	O
cost	O
times	O
the	O
cost	O
of	O
the	O
optimal	O
set	B-Algorithm
cover	I-Algorithm
)	O
.	O
</s>
<s>
and	O
is	O
a	O
set	B-Algorithm
cover	I-Algorithm
of	O
cost	O
at	O
most	O
.	O
</s>
<s>
In	O
this	O
context	O
our	O
goal	O
is	O
an	O
efficient	O
approximation	B-Algorithm
algorithm	I-Algorithm
,	O
</s>
<s>
That	O
approach	O
weakens	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
.	O
</s>
<s>
match	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
the	O
existence	O
proof	O
above	O
.	O
</s>
<s>
The	O
approach	O
is	O
called	O
the	O
method	B-Algorithm
of	I-Algorithm
conditional	I-Algorithm
probabilities	I-Algorithm
.	O
</s>
<s>
The	O
deterministic	O
algorithm	O
emulates	O
the	O
randomized	B-Algorithm
rounding	I-Algorithm
scheme	O
:	O
</s>
<s>
To	O
apply	O
the	O
method	B-Algorithm
of	I-Algorithm
conditional	I-Algorithm
probabilities	I-Algorithm
,	O
</s>
<s>
Compute	O
a	O
min-cost	O
fractional	O
set	B-Algorithm
cover	I-Algorithm
(	O
an	O
optimal	O
solution	O
to	O
the	O
LP	B-Algorithm
relaxation	I-Algorithm
)	O
.	O
</s>
<s>
The	O
algorithm	O
above	O
returns	O
a	O
set	B-Algorithm
cover	I-Algorithm
of	O
cost	O
at	O
most	O
times	O
the	O
minimum	O
cost	O
of	O
any	O
(	O
fractional	O
)	O
set	B-Algorithm
cover	I-Algorithm
.	O
</s>
<s>
the	O
minimum	O
cost	O
of	O
any	O
(	O
fractional	O
)	O
set	B-Algorithm
cover	I-Algorithm
.	O
</s>
<s>
This	O
is	O
called	O
a	O
pessimistic	B-Algorithm
estimator	I-Algorithm
.	O
</s>
