<s>
Intuitively	O
,	O
problem	O
A	O
is	O
reducible	B-Algorithm
to	O
problem	O
B	O
,	O
if	O
an	O
algorithm	O
for	O
solving	O
problem	O
B	O
efficiently	O
(	O
if	O
it	O
existed	O
)	O
could	O
also	O
be	O
used	O
as	O
a	O
subroutine	O
to	O
solve	O
problem	O
A	O
efficiently	O
.	O
</s>
<s>
The	O
existence	O
of	O
a	O
reduction	O
from	O
A	O
to	O
B	O
,	O
can	O
be	O
written	O
in	O
the	O
shorthand	O
notation	O
A	O
≤m	O
B	O
,	O
usually	O
with	O
a	O
subscript	O
on	O
the	O
≤	O
to	O
indicate	O
the	O
type	O
of	O
reduction	O
being	O
used	O
(	O
m	O
:	O
mapping	B-Algorithm
reduction	I-Algorithm
,	O
p	O
:	O
polynomial	B-Algorithm
reduction	I-Algorithm
)	O
.	O
</s>
<s>
This	O
kind	O
of	O
reduction	O
corresponds	O
to	O
Turing	B-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
This	O
corresponds	O
to	O
many-one	B-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
As	O
described	O
in	O
the	O
example	O
above	O
,	O
there	O
are	O
two	O
main	O
types	O
of	O
reductions	O
used	O
in	O
computational	O
complexity	O
,	O
the	O
many-one	B-Algorithm
reduction	I-Algorithm
and	O
the	O
Turing	B-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
Many-one	B-Algorithm
reductions	I-Algorithm
map	O
instances	O
of	O
one	O
problem	O
to	O
instances	O
of	O
another	O
;	O
Turing	B-Algorithm
reductions	I-Algorithm
compute	O
the	O
solution	O
to	O
one	O
problem	O
,	O
assuming	O
the	O
other	O
problem	O
is	O
easy	O
to	O
solve	O
.	O
</s>
<s>
The	O
many-one	B-Algorithm
reduction	I-Algorithm
is	O
a	O
stronger	O
type	O
of	O
Turing	B-Algorithm
reduction	I-Algorithm
,	O
and	O
is	O
more	O
effective	O
at	O
separating	O
problems	O
into	O
distinct	O
complexity	O
classes	O
.	O
</s>
<s>
However	O
,	O
the	O
increased	O
restrictions	O
on	O
many-one	B-Algorithm
reductions	I-Algorithm
make	O
them	O
more	O
difficult	O
to	O
find	O
.	O
</s>
<s>
For	O
example	O
,	O
it	O
's	O
quite	O
possible	O
to	O
reduce	O
a	O
difficult-to-solve	O
NP-complete	O
problem	O
like	O
the	O
boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
to	O
a	O
trivial	O
problem	O
,	O
like	O
determining	O
if	O
a	O
number	O
equals	O
zero	O
,	O
by	O
having	O
the	O
reduction	O
machine	O
solve	O
the	O
problem	O
in	O
exponential	O
time	O
and	O
output	O
zero	O
only	O
if	O
there	O
is	O
a	O
solution	O
.	O
</s>
<s>
When	O
studying	O
the	O
complexity	O
class	O
NP	O
and	O
harder	O
classes	O
such	O
as	O
the	O
polynomial	O
hierarchy	O
,	O
polynomial-time	B-Algorithm
reductions	I-Algorithm
are	O
used	O
.	O
</s>
<s>
When	O
studying	O
classes	O
within	O
P	O
such	O
as	O
NC	O
and	O
NL	O
,	O
log-space	B-Algorithm
reductions	I-Algorithm
are	O
used	O
.	O
</s>
<s>
In	O
case	O
of	O
optimization	O
(	O
maximization	O
or	O
minimization	O
)	O
problems	O
,	O
we	O
often	O
think	O
in	O
terms	O
of	O
approximation-preserving	B-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
This	O
way	O
,	O
if	O
we	O
have	O
an	O
optimization	O
algorithm	O
(	O
or	O
approximation	B-Algorithm
algorithm	I-Algorithm
)	O
that	O
finds	O
near-optimal	O
(	O
or	O
optimal	O
)	O
solutions	O
to	O
instances	O
of	O
problem	O
B	O
,	O
and	O
an	O
efficient	O
approximation-preserving	B-Algorithm
reduction	I-Algorithm
from	O
problem	O
A	O
to	O
problem	O
B	O
,	O
by	O
composition	O
we	O
obtain	O
an	O
optimization	O
algorithm	O
that	O
yields	O
near-optimal	O
solutions	O
to	O
instances	O
of	O
problem	O
A	O
.	O
Approximation-preserving	B-Algorithm
reductions	I-Algorithm
are	O
often	O
used	O
to	O
prove	O
hardness	O
of	O
approximation	O
results	O
:	O
if	O
some	O
optimization	O
problem	O
A	O
is	O
hard	O
to	O
approximate	O
(	O
under	O
some	O
complexity	O
assumption	O
)	O
within	O
a	O
factor	O
better	O
than	O
α	O
for	O
some	O
α	O
,	O
and	O
there	O
is	O
a	O
β-approximation-preserving	O
reduction	O
from	O
problem	O
A	O
to	O
problem	O
B	O
,	O
we	O
can	O
conclude	O
that	O
problem	O
B	O
is	O
hard	O
to	O
approximate	O
within	O
factor	O
α/β	O
.	O
</s>
<s>
The	O
complexity	O
classes	O
P	O
,	O
NP	O
and	O
PSPACE	O
are	O
closed	O
under	O
(	O
many-one	B-Algorithm
,	O
"	O
Karp	O
"	O
)	O
polynomial-time	B-Algorithm
reductions	I-Algorithm
.	O
</s>
<s>
The	O
complexity	O
classes	O
L	O
,	O
NL	O
,	O
P	O
,	O
NP	O
and	O
PSPACE	O
are	O
closed	O
under	O
log-space	B-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
Suppose	O
H(M, w )	O
is	O
the	O
problem	O
of	O
determining	O
whether	O
a	O
given	O
Turing	B-Architecture
machine	I-Architecture
M	O
halts	O
(	O
by	O
accepting	O
or	O
rejecting	O
)	O
on	O
input	O
string	O
w	O
.	O
This	O
language	O
is	O
known	O
to	O
be	O
undecidable	O
.	O
</s>
<s>
Suppose	O
E(M )	O
is	O
the	O
problem	O
of	O
determining	O
whether	O
the	O
language	O
a	O
given	O
Turing	B-Architecture
machine	I-Architecture
M	O
accepts	O
is	O
empty	O
(	O
in	O
other	O
words	O
,	O
whether	O
M	O
accepts	O
any	O
strings	O
at	O
all	O
)	O
.	O
</s>
<s>
Given	O
input	O
M	O
and	O
w	O
(	O
a	O
Turing	B-Architecture
machine	I-Architecture
and	O
some	O
input	O
string	O
)	O
,	O
define	O
S(M, w )	O
with	O
the	O
following	O
behavior	O
:	O
S	O
creates	O
a	O
Turing	B-Architecture
machine	I-Architecture
N	O
that	O
accepts	O
only	O
if	O
the	O
input	O
string	O
to	O
N	O
is	O
w	O
and	O
M	O
halts	O
on	O
input	O
w	O
,	O
and	O
does	O
not	O
halt	O
otherwise	O
.	O
</s>
