<s>
The	O
knapsack	B-Algorithm
problem	I-Algorithm
is	O
one	O
of	O
the	O
most	O
studied	O
problems	O
in	O
combinatorial	O
optimization	O
,	O
with	O
many	O
real-life	O
applications	O
.	O
</s>
<s>
The	O
knapsack	B-Algorithm
problem	I-Algorithm
in	O
its	O
most	O
basic	O
form	O
:	O
</s>
<s>
The	O
bounded	O
knapsack	B-Algorithm
problem	I-Algorithm
specifies	O
,	O
for	O
each	O
item	O
j	O
,	O
an	O
upper	O
bound	O
uj	O
(	O
which	O
may	O
be	O
a	O
positive	O
integer	O
,	O
or	O
infinity	O
)	O
on	O
the	O
number	O
of	O
times	O
item	O
j	O
can	O
be	O
selected	O
:	O
</s>
<s>
The	O
unbounded	B-Algorithm
knapsack	I-Algorithm
problem	I-Algorithm
(	O
sometimes	O
called	O
the	O
integer	B-Algorithm
knapsack	I-Algorithm
problem	I-Algorithm
)	O
does	O
not	O
put	O
any	O
upper	O
bounds	O
on	O
the	O
number	O
of	O
times	O
an	O
item	O
may	O
be	O
selected	O
:	O
</s>
<s>
Both	O
the	O
bounded	O
and	O
unbounded	O
variants	O
admit	O
an	O
FPTAS	B-Algorithm
(	O
essentially	O
the	O
same	O
as	O
the	O
one	O
used	O
in	O
the	O
0-1	B-Algorithm
knapsack	I-Algorithm
problem	I-Algorithm
)	O
.	O
</s>
<s>
If	O
the	O
items	O
are	O
subdivided	O
into	O
k	O
classes	O
denoted	O
,	O
and	O
exactly	O
one	O
item	O
must	O
be	O
taken	O
from	O
each	O
class	O
,	O
we	O
get	O
the	O
multiple-choice	O
knapsack	B-Algorithm
problem	I-Algorithm
:	O
</s>
<s>
If	O
for	O
each	O
item	O
the	O
profit	O
and	O
weight	O
are	O
equal	O
,	O
we	O
get	O
the	O
subset	B-Algorithm
sum	I-Algorithm
problem	I-Algorithm
(	O
often	O
the	O
corresponding	O
decision	O
problem	O
is	O
given	O
instead	O
)	O
:	O
</s>
<s>
If	O
we	O
have	O
n	O
items	O
and	O
m	O
knapsacks	O
with	O
capacities	O
,	O
we	O
get	O
the	O
multiple	O
knapsack	B-Algorithm
problem	I-Algorithm
:	O
</s>
<s>
As	O
a	O
special	O
case	O
of	O
the	O
multiple	O
knapsack	B-Algorithm
problem	I-Algorithm
,	O
when	O
the	O
profits	O
are	O
equal	O
to	O
weights	O
and	O
all	O
bins	O
have	O
the	O
same	O
capacity	O
,	O
we	O
can	O
have	O
multiple	O
subset	B-Algorithm
sum	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Quadratic	B-Algorithm
knapsack	I-Algorithm
problem	I-Algorithm
:	O
</s>
<s>
Set-Union	O
Knapsack	B-Algorithm
Problem	I-Algorithm
:	O
</s>
<s>
If	O
there	O
is	O
more	O
than	O
one	O
constraint	O
(	O
for	O
example	O
,	O
both	O
a	O
volume	O
limit	O
and	O
a	O
weight	O
limit	O
,	O
where	O
the	O
volume	O
and	O
weight	O
of	O
each	O
item	O
are	O
not	O
related	O
)	O
,	O
we	O
get	O
the	O
multiple-constrained	O
knapsack	B-Algorithm
problem	I-Algorithm
,	O
multidimensional	O
knapsack	B-Algorithm
problem	I-Algorithm
,	O
or	O
m-dimensional	O
knapsack	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
0-1	O
variant	O
(	O
for	O
any	O
fixed	O
)	O
was	O
shown	O
to	O
be	O
NP-complete	O
around	O
1980	O
and	O
more	O
strongly	O
,	O
has	O
no	O
FPTAS	B-Algorithm
unless	O
P	O
=	O
NP	O
.	O
</s>
<s>
For	O
any	O
fixed	O
,	O
these	O
problems	O
do	O
admit	O
a	O
pseudo-polynomial	B-Algorithm
time	I-Algorithm
algorithm	O
(	O
similar	O
to	O
the	O
one	O
for	O
basic	O
knapsack	O
)	O
and	O
a	O
PTAS	B-Algorithm
.	O
</s>
<s>
If	O
,	O
to	O
the	O
multiple	O
choice	O
knapsack	B-Algorithm
problem	I-Algorithm
,	O
we	O
add	O
the	O
constraint	O
that	O
each	O
subset	O
is	O
of	O
size	O
n	O
and	O
remove	O
the	O
restriction	O
on	O
total	O
weight	O
,	O
we	O
get	O
the	O
assignment	B-Algorithm
problem	I-Algorithm
,	O
which	O
is	O
also	O
the	O
problem	O
of	O
finding	O
a	O
maximal	O
bipartite	O
matching	O
:	O
</s>
<s>
The	O
last	O
three	O
of	O
these	O
are	O
discussed	O
in	O
Kellerer	O
et	O
al	O
's	O
reference	O
work	O
,	O
Knapsack	B-Algorithm
Problems	I-Algorithm
.	O
</s>
