<s>
The	O
multiple	B-Algorithm
subset	I-Algorithm
sum	I-Algorithm
problem	I-Algorithm
is	O
an	O
optimization	O
problem	O
in	O
computer	B-General_Concept
science	I-General_Concept
and	O
operations	O
research	O
.	O
</s>
<s>
It	O
is	O
a	O
generalization	O
of	O
the	O
subset	B-Algorithm
sum	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
input	O
to	O
the	O
problem	O
is	O
a	O
multiset	B-Language
of	O
n	O
integers	O
and	O
a	O
positive	O
integer	O
m	O
representing	O
the	O
number	O
of	O
subsets	O
.	O
</s>
<s>
Max-min	O
MSSP	O
(	O
also	O
called	O
bottleneck	O
MSSP	O
or	O
BMSSP	O
)	O
:	O
again	O
each	O
subset	O
has	O
a	O
capacity	O
,	O
but	O
now	O
the	O
goal	O
is	O
to	O
make	O
the	O
smallest	O
subset	B-Algorithm
sum	I-Algorithm
as	O
large	O
as	O
possible	O
.	O
</s>
<s>
When	O
m	O
is	O
variable	O
(	O
a	O
part	O
of	O
the	O
input	O
)	O
,	O
both	O
problems	O
are	O
strongly	O
NP-hard	O
,	O
by	O
reduction	O
from	O
3-partition	B-Algorithm
.	O
</s>
<s>
This	O
can	O
be	O
shown	O
by	O
a	O
reduction	O
from	O
the	O
equal-cardinality	O
partition	B-Algorithm
problem	I-Algorithm
(	O
EPART	O
)	O
:	O
</s>
<s>
No	O
better	O
approximation	O
is	O
possible	O
unless	O
P	O
=	O
NP	O
(	O
by	O
reduction	O
from	O
3-partition	B-Algorithm
)	O
.	O
</s>
<s>
The	O
fair	O
subset	B-Algorithm
sum	I-Algorithm
problem	I-Algorithm
(	O
FSSP	O
)	O
is	O
a	O
generalization	O
of	O
SSP	O
in	O
which	O
,	O
after	O
the	O
subset	O
is	O
selected	O
,	O
its	O
items	O
are	O
allocated	O
among	O
two	O
or	O
more	O
agents	O
.	O
</s>
<s>
The	O
multiple	O
knapsack	B-Algorithm
problem	I-Algorithm
(	O
MKP	O
)	O
is	O
a	O
generalization	O
of	O
both	O
the	O
max-sum	O
MSSP	O
and	O
the	O
knapsack	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
knapsack	B-Algorithm
problem	I-Algorithm
is	O
a	O
special	O
case	O
of	O
MKP	O
in	O
which	O
m	O
=	O
1	O
.	O
</s>
<s>
The	O
subset-sum	B-Algorithm
problem	I-Algorithm
is	O
a	O
special	O
case	O
of	O
MKP	O
in	O
which	O
both	O
the	O
value	O
of	O
each	O
item	O
equals	O
its	O
weight	O
,	O
and	O
m	O
=	O
1	O
.	O
</s>
<s>
The	O
MKP	O
has	O
a	O
Polynomial-time	B-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
.	O
</s>
