<s>
In	O
computational	O
complexity	O
theory	O
,	O
a	O
polynomial-time	B-Algorithm
reduction	I-Algorithm
is	O
a	O
method	O
for	O
solving	O
one	O
problem	O
using	O
another	O
.	O
</s>
<s>
One	O
shows	O
that	O
if	O
a	O
hypothetical	O
subroutine	O
solving	O
the	O
second	O
problem	O
exists	O
,	O
then	O
the	O
first	O
problem	O
can	O
be	O
solved	O
by	O
transforming	O
or	O
reducing	B-Algorithm
it	O
to	O
inputs	O
for	O
the	O
second	O
problem	O
and	O
calling	O
the	O
subroutine	O
one	O
or	O
more	O
times	O
.	O
</s>
<s>
If	O
both	O
the	O
time	O
required	O
to	O
transform	O
the	O
first	O
problem	O
to	O
the	O
second	O
,	O
and	O
the	O
number	O
of	O
times	O
the	O
subroutine	O
is	O
called	O
is	O
polynomial	O
,	O
then	O
the	O
first	O
problem	O
is	O
polynomial-time	O
reducible	B-Algorithm
to	O
the	O
second	O
.	O
</s>
<s>
A	O
polynomial-time	B-Algorithm
reduction	I-Algorithm
proves	O
that	O
the	O
first	O
problem	O
is	O
no	O
more	O
difficult	O
than	O
the	O
second	O
one	O
,	O
because	O
whenever	O
an	O
efficient	O
algorithm	O
exists	O
for	O
the	O
second	O
problem	O
,	O
one	O
exists	O
for	O
the	O
first	O
problem	O
as	O
well	O
.	O
</s>
<s>
Polynomial-time	B-Algorithm
reductions	I-Algorithm
are	O
frequently	O
used	O
in	O
complexity	O
theory	O
for	O
defining	O
both	O
complexity	O
classes	O
and	O
complete	O
problems	O
for	O
those	O
classes	O
.	O
</s>
<s>
The	O
three	O
most	O
common	O
types	O
of	O
polynomial-time	B-Algorithm
reduction	I-Algorithm
,	O
from	O
the	O
most	O
to	O
the	O
least	O
restrictive	O
,	O
are	O
polynomial-time	B-Algorithm
many-one	I-Algorithm
reductions	I-Algorithm
,	O
truth-table	B-Algorithm
reductions	I-Algorithm
,	O
and	O
Turing	B-Algorithm
reductions	I-Algorithm
.	O
</s>
<s>
The	O
most	O
frequently	O
used	O
of	O
these	O
are	O
the	O
many-one	B-Algorithm
reductions	I-Algorithm
,	O
and	O
in	O
some	O
cases	O
the	O
phrase	O
"	O
polynomial-time	B-Algorithm
reduction	I-Algorithm
"	O
may	O
be	O
used	O
to	O
mean	O
a	O
polynomial-time	B-Algorithm
many-one	I-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
The	O
most	O
general	O
reductions	O
are	O
the	O
Turing	B-Algorithm
reductions	I-Algorithm
and	O
the	O
most	O
restrictive	O
are	O
the	O
many-one	B-Algorithm
reductions	I-Algorithm
with	O
truth-table	B-Algorithm
reductions	I-Algorithm
occupying	O
the	O
space	O
in	O
between	O
.	O
</s>
<s>
A	O
polynomial-time	B-Algorithm
many-one	I-Algorithm
reduction	I-Algorithm
from	O
a	O
problem	O
A	O
to	O
a	O
problem	O
B	O
(	O
both	O
of	O
which	O
are	O
usually	O
required	O
to	O
be	O
decision	O
problems	O
)	O
is	O
a	O
polynomial-time	O
algorithm	O
for	O
transforming	O
inputs	O
to	O
problem	O
A	O
into	O
inputs	O
to	O
problem	O
B	O
,	O
such	O
that	O
the	O
transformed	O
problem	O
has	O
the	O
same	O
output	O
as	O
the	O
original	O
problem	O
.	O
</s>
<s>
Polynomial-time	B-Algorithm
many-one	I-Algorithm
reductions	I-Algorithm
may	O
also	O
be	O
known	O
as	O
polynomial	O
transformations	O
or	O
Karp	B-Algorithm
reductions	I-Algorithm
,	O
named	O
after	O
Richard	O
Karp	O
.	O
</s>
<s>
A	O
polynomial-time	O
truth-table	B-Algorithm
reduction	I-Algorithm
from	O
a	O
problem	O
A	O
to	O
a	O
problem	O
B	O
(	O
both	O
decision	O
problems	O
)	O
is	O
a	O
polynomial	O
time	O
algorithm	O
for	O
transforming	O
inputs	O
to	O
problem	O
A	O
into	O
a	O
fixed	O
number	O
of	O
inputs	O
to	O
problem	O
B	O
,	O
such	O
that	O
the	O
output	O
for	O
the	O
original	O
problem	O
can	O
be	O
expressed	O
as	O
a	O
function	O
of	O
the	O
outputs	O
for	O
B	O
.	O
</s>
<s>
A	O
polynomial-time	B-Algorithm
Turing	I-Algorithm
reduction	I-Algorithm
from	O
a	O
problem	O
A	O
to	O
a	O
problem	O
B	O
is	O
an	O
algorithm	O
that	O
solves	O
problem	O
A	O
using	O
a	O
polynomial	O
number	O
of	O
calls	O
to	O
a	O
subroutine	O
for	O
problem	O
B	O
,	O
and	O
polynomial	O
time	O
outside	O
of	O
those	O
subroutine	O
calls	O
.	O
</s>
<s>
Polynomial-time	B-Algorithm
Turing	I-Algorithm
reductions	I-Algorithm
are	O
also	O
known	O
as	O
Cook	B-Algorithm
reductions	I-Algorithm
,	O
named	O
after	O
Stephen	O
Cook	O
.	O
</s>
<s>
Many-one	B-Algorithm
reductions	I-Algorithm
can	O
be	O
regarded	O
as	O
restricted	O
variants	O
of	O
Turing	B-Algorithm
reductions	I-Algorithm
where	O
the	O
number	O
of	O
calls	O
made	O
to	O
the	O
subroutine	O
for	O
problem	O
B	O
is	O
exactly	O
one	O
and	O
the	O
value	O
returned	O
by	O
the	O
reduction	O
is	O
the	O
same	O
value	O
as	O
the	O
one	O
returned	O
by	O
the	O
subroutine	O
.	O
</s>
<s>
A	O
complete	O
problem	O
for	O
a	O
given	O
complexity	O
class	O
C	O
and	O
reduction	O
≤	O
is	O
a	O
problem	O
P	O
that	O
belongs	O
to	O
C	O
,	O
such	O
that	O
every	O
problem	O
A	O
in	O
C	O
has	O
a	O
reduction	O
A	O
≤	O
P	O
.	O
For	O
instance	O
,	O
a	O
problem	O
is	O
NP-complete	O
if	O
it	O
belongs	O
to	O
NP	O
and	O
all	O
problems	O
in	O
NP	O
have	O
polynomial-time	B-Algorithm
many-one	I-Algorithm
reductions	I-Algorithm
to	O
it	O
.	O
</s>
<s>
A	O
problem	O
that	O
belongs	O
to	O
NP	O
can	O
be	O
proven	O
to	O
be	O
NP-complete	O
by	O
finding	O
a	O
single	O
polynomial-time	B-Algorithm
many-one	I-Algorithm
reduction	I-Algorithm
to	O
it	O
from	O
a	O
known	O
NP-complete	O
problem	O
.	O
</s>
<s>
Polynomial-time	B-Algorithm
many-one	I-Algorithm
reductions	I-Algorithm
have	O
been	O
used	O
to	O
define	O
complete	O
problems	O
for	O
other	O
complexity	O
classes	O
,	O
including	O
the	O
PSPACE-complete	O
languages	O
and	O
EXPTIME-complete	O
languages	O
.	O
</s>
<s>
Every	O
decision	O
problem	O
in	O
P	O
(	O
the	O
class	O
of	O
polynomial-time	O
decision	O
problems	O
)	O
may	O
be	O
reduced	O
to	O
every	O
other	O
nontrivial	O
decision	O
problem	O
(	O
where	O
nontrivial	O
means	O
that	O
not	O
every	O
input	O
has	O
the	O
same	O
output	O
)	O
,	O
by	O
a	O
polynomial-time	B-Algorithm
many-one	I-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
for	O
complexity	O
classes	O
within	O
P	O
such	O
as	O
L	O
,	O
NL	O
,	O
NC	O
,	O
and	O
P	O
itself	O
,	O
polynomial-time	B-Algorithm
reductions	I-Algorithm
cannot	O
be	O
used	O
to	O
define	O
complete	O
languages	O
:	O
if	O
they	O
were	O
used	O
in	O
this	O
way	O
,	O
every	O
nontrivial	O
problem	O
in	O
P	O
would	O
be	O
complete	O
.	O
</s>
<s>
Instead	O
,	O
weaker	O
reductions	O
such	O
as	O
log-space	B-Algorithm
reductions	I-Algorithm
or	O
NC	O
reductions	O
are	O
used	O
for	O
defining	O
classes	O
of	O
complete	O
problems	O
for	O
these	O
classes	O
,	O
such	O
as	O
the	O
P-complete	O
problems	O
.	O
</s>
<s>
is	O
the	O
set	O
of	O
problems	O
having	O
a	O
polynomial-time	B-Algorithm
many-one	I-Algorithm
reduction	I-Algorithm
to	O
the	O
existential	O
theory	O
of	O
the	O
reals	O
;	O
it	O
has	O
several	O
other	O
complete	O
problems	O
such	O
as	O
determining	O
the	O
rectilinear	O
crossing	O
number	O
of	O
an	O
undirected	O
graph	O
.	O
</s>
