<s>
In	O
computer	B-General_Concept
science	I-General_Concept
and	O
operations	O
research	O
,	O
approximation	B-Algorithm
algorithms	I-Algorithm
are	O
efficient	O
algorithms	O
that	O
find	O
approximate	B-Algorithm
solutions	I-Algorithm
to	I-Algorithm
optimization	I-Algorithm
problems	I-Algorithm
(	O
in	O
particular	O
NP-hard	O
problems	O
)	O
with	O
provable	O
guarantees	O
on	O
the	O
distance	O
of	O
the	O
returned	O
solution	O
to	O
the	O
optimal	O
one	O
.	O
</s>
<s>
Approximation	B-Algorithm
algorithms	I-Algorithm
naturally	O
arise	O
in	O
the	O
field	O
of	O
theoretical	O
computer	B-General_Concept
science	I-General_Concept
as	O
a	O
consequence	O
of	O
the	O
widely	O
believed	O
P	O
≠	O
NP	O
conjecture	O
.	O
</s>
<s>
The	O
field	O
of	O
approximation	B-Algorithm
algorithms	I-Algorithm
,	O
therefore	O
,	O
tries	O
to	O
understand	O
how	O
closely	O
it	O
is	O
possible	O
to	O
approximate	O
optimal	O
solutions	O
to	O
such	O
problems	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
In	O
an	O
overwhelming	O
majority	O
of	O
the	O
cases	O
,	O
the	O
guarantee	O
of	O
such	O
algorithms	O
is	O
a	O
multiplicative	O
one	O
expressed	O
as	O
an	O
approximation	B-Algorithm
ratio	I-Algorithm
or	O
approximation	O
factor	O
i.e.	O
,	O
the	O
optimal	O
solution	O
is	O
always	O
guaranteed	O
to	O
be	O
within	O
a	O
(	O
predetermined	O
)	O
multiplicative	O
factor	O
of	O
the	O
returned	O
solution	O
.	O
</s>
<s>
However	O
,	O
there	O
are	O
also	O
many	O
approximation	B-Algorithm
algorithms	I-Algorithm
that	O
provide	O
an	O
additive	O
guarantee	O
on	O
the	O
quality	O
of	O
the	O
returned	O
solution	O
.	O
</s>
<s>
A	O
notable	O
example	O
of	O
an	O
approximation	B-Algorithm
algorithm	I-Algorithm
that	O
provides	O
both	O
is	O
the	O
classic	O
approximation	B-Algorithm
algorithm	I-Algorithm
of	O
Lenstra	O
,	O
Shmoys	O
and	O
Tardos	O
for	O
scheduling	O
on	O
unrelated	O
parallel	O
machines	O
.	O
</s>
<s>
The	O
design	O
and	O
analysis	O
of	O
approximation	B-Algorithm
algorithms	I-Algorithm
crucially	O
involves	O
a	O
mathematical	O
proof	O
certifying	O
the	O
quality	O
of	O
the	O
returned	O
solutions	O
in	O
the	O
worst	O
case	O
.	O
</s>
<s>
This	O
distinguishes	O
them	O
from	O
heuristics	B-Algorithm
such	O
as	O
annealing	B-Algorithm
or	O
genetic	B-Algorithm
algorithms	I-Algorithm
,	O
which	O
find	O
reasonably	O
good	O
solutions	O
on	O
some	O
inputs	O
,	O
but	O
provide	O
no	O
clear	O
indication	O
at	O
the	O
outset	O
on	O
when	O
they	O
may	O
succeed	O
or	O
fail	O
.	O
</s>
<s>
There	O
is	O
widespread	O
interest	O
in	O
theoretical	O
computer	B-General_Concept
science	I-General_Concept
to	O
better	O
understand	O
the	O
limits	O
to	O
which	O
we	O
can	O
approximate	O
certain	O
famous	O
optimization	O
problems	O
.	O
</s>
<s>
For	O
example	O
,	O
one	O
of	O
the	O
long-standing	O
open	O
questions	O
in	O
computer	B-General_Concept
science	I-General_Concept
is	O
to	O
determine	O
whether	O
there	O
is	O
an	O
algorithm	O
that	O
outperforms	O
the	O
2-approximation	O
for	O
the	O
Steiner	O
Forest	O
problem	O
by	O
Agrawal	O
et	O
al	O
.	O
</s>
<s>
The	O
desire	O
to	O
understand	O
hard	O
optimization	O
problems	O
from	O
the	O
perspective	O
of	O
approximability	B-Algorithm
is	O
motivated	O
by	O
the	O
discovery	O
of	O
surprising	O
mathematical	O
connections	O
and	O
broadly	O
applicable	O
techniques	O
to	O
design	O
algorithms	O
for	O
hard	O
optimization	O
problems	O
.	O
</s>
<s>
A	O
simple	O
example	O
of	O
an	O
approximation	B-Algorithm
algorithm	I-Algorithm
is	O
one	O
for	O
the	O
minimum	O
vertex	O
cover	O
problem	O
,	O
where	O
the	O
goal	O
is	O
to	O
choose	O
the	O
smallest	O
set	O
of	O
vertices	O
such	O
that	O
every	O
edge	O
in	O
the	O
input	O
graph	O
contains	O
at	O
least	O
one	O
chosen	O
vertex	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
this	O
is	O
a	O
constant-factor	B-Algorithm
approximation	I-Algorithm
algorithm	I-Algorithm
with	O
an	O
approximation	O
factor	O
of	O
2	O
.	O
</s>
<s>
NP-hard	O
problems	O
vary	O
greatly	O
in	O
their	O
approximability	B-Algorithm
;	O
some	O
,	O
such	O
as	O
the	O
knapsack	B-Algorithm
problem	I-Algorithm
,	O
can	O
be	O
approximated	O
within	O
a	O
multiplicative	O
factor	O
,	O
for	O
any	O
fixed	O
,	O
and	O
therefore	O
produce	O
solutions	O
arbitrarily	O
close	O
to	O
the	O
optimum	O
(	O
such	O
a	O
family	O
of	O
approximation	B-Algorithm
algorithms	I-Algorithm
is	O
called	O
a	O
polynomial-time	B-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
or	O
PTAS	B-Algorithm
)	O
.	O
</s>
<s>
Therefore	O
,	O
an	O
important	O
benefit	O
of	O
studying	O
approximation	B-Algorithm
algorithms	I-Algorithm
is	O
a	O
fine-grained	O
classification	O
of	O
the	O
difficulty	O
of	O
various	O
NP-hard	O
problems	O
beyond	O
the	O
one	O
afforded	O
by	O
the	O
theory	O
of	O
NP-completeness	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
although	O
NP-complete	O
problems	O
may	O
be	O
equivalent	O
(	O
under	O
polynomial-time	O
reductions	B-Algorithm
)	O
to	O
each	O
other	O
from	O
the	O
perspective	O
of	O
exact	O
solutions	O
,	O
the	O
corresponding	O
optimization	O
problems	O
behave	O
very	O
differently	O
from	O
the	O
perspective	O
of	O
approximate	O
solutions	O
.	O
</s>
<s>
By	O
now	O
there	O
are	O
several	O
established	O
techniques	O
to	O
design	O
approximation	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
While	O
approximation	B-Algorithm
algorithms	I-Algorithm
always	O
provide	O
an	O
a	O
priori	O
worst	O
case	O
guarantee	O
(	O
be	O
it	O
additive	O
or	O
multiplicative	O
)	O
,	O
in	O
some	O
cases	O
they	O
also	O
provide	O
an	O
a	O
posteriori	O
guarantee	O
that	O
is	O
often	O
much	O
better	O
.	O
</s>
<s>
For	O
example	O
,	O
there	O
is	O
a	O
different	O
approximation	B-Algorithm
algorithm	I-Algorithm
for	O
minimum	O
vertex	O
cover	O
that	O
solves	O
a	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
to	O
find	O
a	O
vertex	O
cover	O
that	O
is	O
at	O
most	O
twice	O
the	O
value	O
of	O
the	O
relaxation	O
.	O
</s>
<s>
While	O
this	O
is	O
similar	O
to	O
the	O
a	O
priori	O
guarantee	O
of	O
the	O
previous	O
approximation	B-Algorithm
algorithm	I-Algorithm
,	O
the	O
guarantee	O
of	O
the	O
latter	O
can	O
be	O
much	O
better	O
(	O
indeed	O
when	O
the	O
value	O
of	O
the	O
LP	B-Algorithm
relaxation	I-Algorithm
is	O
far	O
from	O
the	O
size	O
of	O
the	O
optimal	O
vertex	O
cover	O
)	O
.	O
</s>
<s>
Approximation	B-Algorithm
algorithms	I-Algorithm
as	O
a	O
research	O
area	O
is	O
closely	O
related	O
to	O
and	O
informed	O
by	O
inapproximability	O
theory	O
where	O
the	O
non-existence	O
of	O
efficient	O
algorithms	O
with	O
certain	O
approximation	B-Algorithm
ratios	I-Algorithm
is	O
proved	O
(	O
conditioned	O
on	O
widely	O
believed	O
hypotheses	O
such	O
as	O
the	O
P	O
≠	O
NP	O
conjecture	O
)	O
by	O
means	O
of	O
reductions	B-Algorithm
.	O
</s>
<s>
In	O
the	O
case	O
of	O
the	O
metric	O
traveling	O
salesman	O
problem	O
,	O
the	O
best	O
known	O
inapproximability	O
result	O
rules	O
out	O
algorithms	O
with	O
an	O
approximation	B-Algorithm
ratio	I-Algorithm
less	O
than	O
123/122	O
≈	O
1.008196	O
unless	O
P	O
=	O
NP	O
,	O
Karpinski	O
,	O
Lampis	O
,	O
Schmied	O
.	O
</s>
<s>
Coupled	O
with	O
the	O
knowledge	O
of	O
the	O
existence	O
of	O
Christofides	O
 '	O
1.5	O
approximation	B-Algorithm
algorithm	I-Algorithm
,	O
this	O
tells	O
us	O
that	O
the	O
threshold	O
of	O
approximability	B-Algorithm
for	O
metric	O
traveling	O
salesman	O
(	O
if	O
it	O
exists	O
)	O
is	O
somewhere	O
between	O
123/122	O
and	O
1.5	O
.	O
</s>
<s>
The	O
PCP	O
theorem	O
,	O
for	O
example	O
,	O
shows	O
that	O
Johnson	O
's	O
1974	O
approximation	B-Algorithm
algorithms	I-Algorithm
for	O
Max	B-Application
SAT	I-Application
,	O
set	B-Algorithm
cover	I-Algorithm
,	O
independent	O
set	O
and	O
coloring	O
all	O
achieve	O
the	O
optimal	O
approximation	B-Algorithm
ratio	I-Algorithm
,	O
assuming	O
P	O
≠	O
NP	O
.	O
</s>
<s>
Not	O
all	O
approximation	B-Algorithm
algorithms	I-Algorithm
are	O
suitable	O
for	O
direct	O
practical	O
applications	O
.	O
</s>
<s>
Some	O
involve	O
solving	O
non-trivial	O
linear	O
programming/semidefinite	O
relaxations	O
(	O
which	O
may	O
themselves	O
invoke	O
the	O
ellipsoid	B-Algorithm
algorithm	I-Algorithm
)	O
,	O
complex	O
data	O
structures	O
,	O
or	O
sophisticated	O
algorithmic	O
techniques	O
,	O
leading	O
to	O
difficult	O
implementation	O
issues	O
or	O
improved	O
running	O
time	O
performance	O
(	O
over	O
exact	B-Algorithm
algorithms	I-Algorithm
)	O
only	O
on	O
impractically	O
large	O
inputs	O
.	O
</s>
<s>
Implementation	O
and	O
running	O
time	O
issues	O
aside	O
,	O
the	O
guarantees	O
provided	O
by	O
approximation	B-Algorithm
algorithms	I-Algorithm
may	O
themselves	O
not	O
be	O
strong	O
enough	O
to	O
justify	O
their	O
consideration	O
in	O
practice	O
.	O
</s>
<s>
One	O
such	O
example	O
is	O
the	O
initial	O
PTAS	B-Algorithm
for	O
Euclidean	O
TSP	O
by	O
Sanjeev	O
Arora	O
(	O
and	O
independently	O
by	O
Joseph	O
Mitchell	O
)	O
which	O
had	O
a	O
prohibitive	O
running	O
time	O
of	O
for	O
a	O
approximation	O
.	O
</s>
<s>
For	O
some	O
approximation	B-Algorithm
algorithms	I-Algorithm
it	O
is	O
possible	O
to	O
prove	O
certain	O
properties	O
about	O
the	O
approximation	O
of	O
the	O
optimum	O
result	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
ρ-approximation	B-Algorithm
algorithm	I-Algorithm
A	O
is	O
defined	O
to	O
be	O
an	O
algorithm	O
for	O
which	O
it	O
has	O
been	O
proven	O
that	O
the	O
value/cost	O
,	O
f(x )	O
,	O
of	O
the	O
approximate	O
solution	O
A(x )	O
to	O
an	O
instance	O
x	O
will	O
not	O
be	O
more	O
(	O
or	O
less	O
,	O
depending	O
on	O
the	O
situation	O
)	O
than	O
a	O
factor	O
ρ	O
times	O
the	O
value	O
,	O
OPT	O
,	O
of	O
an	O
optimum	O
solution	O
.	O
</s>
<s>
The	O
factor	O
ρ	O
is	O
called	O
the	O
relative	B-Algorithm
performance	I-Algorithm
guarantee	I-Algorithm
.	O
</s>
<s>
If	O
an	O
algorithm	O
A	O
guarantees	O
to	O
return	O
solutions	O
with	O
a	O
performance	O
guarantee	O
of	O
at	O
most	O
r(n )	O
,	O
then	O
A	O
is	O
said	O
to	O
be	O
an	O
r(n )	O
-approximation	O
algorithm	O
and	O
has	O
an	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
r(n )	O
.	O
</s>
<s>
Likewise	O
,	O
a	O
problem	O
with	O
an	O
r(n )	O
-approximation	O
algorithm	O
is	O
said	O
to	O
be	O
r(n )	O
-approximable	O
or	O
have	O
an	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
r(n )	O
.	O
</s>
<s>
For	O
minimization	O
problems	O
,	O
the	O
two	O
different	O
guarantees	O
provide	O
the	O
same	O
result	O
and	O
that	O
for	O
maximization	O
problems	O
,	O
a	O
relative	B-Algorithm
performance	I-Algorithm
guarantee	I-Algorithm
of	O
ρ	O
is	O
equivalent	O
to	O
a	O
performance	O
guarantee	O
of	O
.	O
</s>
<s>
The	O
absolute	B-Algorithm
performance	I-Algorithm
guarantee	I-Algorithm
of	O
some	O
approximation	B-Algorithm
algorithm	I-Algorithm
A	O
,	O
where	O
x	O
refers	O
to	O
an	O
instance	O
of	O
a	O
problem	O
,	O
and	O
where	O
is	O
the	O
performance	O
guarantee	O
of	O
A	O
on	O
x	O
(	O
i.e.	O
</s>
<s>
That	O
is	O
to	O
say	O
that	O
is	O
the	O
largest	O
bound	O
on	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
,	O
r	O
,	O
that	O
one	O
sees	O
over	O
all	O
possible	O
instances	O
of	O
the	O
problem	O
.	O
</s>
<s>
In	O
the	O
literature	O
,	O
an	O
approximation	B-Algorithm
ratio	I-Algorithm
for	O
a	O
maximization	O
(	O
minimization	O
)	O
problem	O
of	O
c	O
-	O
ϵ	O
(	O
min	O
:	O
c	O
+	O
ϵ	O
)	O
means	O
that	O
the	O
algorithm	O
has	O
an	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
c	O
∓	O
ϵ	O
for	O
arbitrary	O
ϵ	O
>	O
0	O
but	O
that	O
the	O
ratio	O
has	O
not	O
(	O
or	O
cannot	O
)	O
be	O
shown	O
for	O
ϵ	O
=	O
0	O
.	O
</s>
<s>
An	O
example	O
of	O
this	O
is	O
the	O
optimal	O
inapproximability	O
—	O
inexistence	O
of	O
approximation	O
—	O
ratio	O
of	O
7	O
/	O
8	O
+	O
ϵ	O
for	O
satisfiable	O
MAX-3SAT	B-Application
instances	O
due	O
to	O
Johan	O
Håstad	O
.	O
</s>
<s>
As	O
mentioned	O
previously	O
,	O
when	O
c	O
=	O
1	O
,	O
the	O
problem	O
is	O
said	O
to	O
have	O
a	O
polynomial-time	B-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
.	O
</s>
<s>
An	O
ϵ-term	O
may	O
appear	O
when	O
an	O
approximation	B-Algorithm
algorithm	I-Algorithm
introduces	O
a	O
multiplicative	O
error	O
and	O
a	O
constant	O
error	O
while	O
the	O
minimum	O
optimum	O
of	O
instances	O
of	O
size	O
n	O
goes	O
to	O
infinity	O
as	O
n	O
does	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
is	O
c	O
∓	O
k	O
/	O
OPT	O
=	O
c	O
∓	O
o(1 )	O
for	O
some	O
constants	O
c	O
and	O
k	O
.	O
Given	O
arbitrary	O
ϵ	O
>	O
0	O
,	O
one	O
can	O
choose	O
a	O
large	O
enough	O
N	O
such	O
that	O
the	O
term	O
k	O
/	O
OPT	O
< ϵ for every n ≥ N. For every fixed ϵ, instances of size n < N can be solved by brute force, thereby showing an approximation ratio — existence of approximation algorithms with a guarantee — of c ∓ ϵ for every ϵ >	O
0	O
.	O
</s>
