<s>
In	O
computational	O
complexity	O
theory	O
,	O
a	O
numeric	O
algorithm	O
runs	O
in	O
pseudo-polynomial	B-Algorithm
time	I-Algorithm
if	O
its	O
running	O
time	O
is	O
a	O
polynomial	O
in	O
the	O
numeric	O
value	O
of	O
the	O
input	O
(	O
the	O
largest	O
integer	O
present	O
in	O
the	O
input	O
)	O
—	O
but	O
not	O
necessarily	O
in	O
the	O
length	O
of	O
the	O
input	O
(	O
the	O
number	O
of	O
bits	O
required	O
to	O
represent	O
it	O
)	O
,	O
which	O
is	O
the	O
case	O
for	O
polynomial	O
time	O
algorithms	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
numeric	O
value	O
of	O
the	O
input	O
is	O
exponential	O
in	O
the	O
input	O
length	O
,	O
which	O
is	O
why	O
a	O
pseudo-polynomial	B-Algorithm
time	I-Algorithm
algorithm	O
does	O
not	O
necessarily	O
run	O
in	O
polynomial	O
time	O
with	O
respect	O
to	O
the	O
input	O
length	O
.	O
</s>
<s>
An	O
NP-complete	O
problem	O
with	O
known	O
pseudo-polynomial	B-Algorithm
time	I-Algorithm
algorithms	O
is	O
called	O
weakly	O
NP-complete	O
.	O
</s>
<s>
An	O
NP-complete	O
problem	O
is	O
called	O
strongly	O
NP-complete	O
if	O
it	O
is	O
proven	O
that	O
it	O
cannot	O
be	O
solved	O
by	O
a	O
pseudo-polynomial	B-Algorithm
time	I-Algorithm
algorithm	O
unless	O
.	O
</s>
<s>
It	O
is	O
,	O
however	O
,	O
pseudo-polynomial	B-Algorithm
time	I-Algorithm
.	O
</s>
<s>
In	O
the	O
knapsack	B-Algorithm
problem	I-Algorithm
,	O
we	O
are	O
given	O
items	O
with	O
weight	O
and	O
value	O
,	O
along	O
with	O
a	O
maximum	O
weight	O
capacity	O
of	O
a	O
knapsack	O
.	O
</s>
<s>
However	O
,	O
an	O
time	O
algorithm	O
is	O
possible	O
using	O
dynamic	B-Algorithm
programming	I-Algorithm
;	O
since	O
the	O
number	O
only	O
needs	O
bits	O
to	O
describe	O
,	O
this	O
algorithm	O
runs	O
in	O
pseudo-polynomial	B-Algorithm
time	I-Algorithm
.	O
</s>
<s>
Although	O
the	O
notion	O
of	O
pseudo-polynomial	B-Algorithm
time	I-Algorithm
is	O
used	O
almost	O
exclusively	O
for	O
numeric	O
problems	O
,	O
the	O
concept	O
can	O
be	O
generalized	O
:	O
</s>
<s>
m(n )	O
is	O
no	O
greater	O
than	O
a	O
polynomial	O
function	O
of	O
the	O
problem	B-General_Concept
size	I-General_Concept
n	O
and	O
an	O
additional	O
property	O
of	O
the	O
input	O
,	O
k(n )	O
.	O
</s>
<s>
The	O
distinction	O
between	O
the	O
value	O
of	O
a	O
number	O
and	O
its	O
length	O
is	O
one	O
of	O
encoding	O
:	O
if	O
numeric	O
inputs	O
are	O
always	O
encoded	O
in	O
unary	O
,	O
then	O
pseudo-polynomial	B-Algorithm
would	O
coincide	O
with	O
polynomial	O
.	O
</s>
