<s>
In	O
applied	O
mathematics	O
,	O
the	O
maximum	O
generalized	B-Algorithm
assignment	I-Algorithm
problem	I-Algorithm
is	O
a	O
problem	O
in	O
combinatorial	O
optimization	O
.	O
</s>
<s>
This	O
problem	O
is	O
a	O
generalization	B-Algorithm
of	O
the	O
assignment	B-Algorithm
problem	I-Algorithm
in	O
which	O
both	O
tasks	O
and	O
agents	O
have	O
a	O
size	O
.	O
</s>
<s>
In	O
the	O
special	O
case	O
in	O
which	O
all	O
the	O
agents	O
 '	O
budgets	O
and	O
all	O
tasks	O
 '	O
costs	O
are	O
equal	O
to	O
1	O
,	O
this	O
problem	O
reduces	O
to	O
the	O
assignment	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
When	O
the	O
costs	O
and	O
profits	O
of	O
all	O
tasks	O
do	O
not	O
vary	O
between	O
different	O
agents	O
,	O
this	O
problem	O
reduces	O
to	O
the	O
multiple	O
knapsack	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
If	O
there	O
is	O
a	O
single	O
agent	O
,	O
then	O
,	O
this	O
problem	O
reduces	O
to	O
the	O
knapsack	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Mathematically	O
the	O
generalized	B-Algorithm
assignment	I-Algorithm
problem	I-Algorithm
can	O
be	O
formulated	O
as	O
an	O
integer	B-Algorithm
program	I-Algorithm
:	O
</s>
<s>
The	O
generalized	B-Algorithm
assignment	I-Algorithm
problem	I-Algorithm
is	O
NP-hard	O
,	O
However	O
,	O
there	O
are	O
linear-programming	O
relaxations	O
which	O
give	O
a	O
-approximation	O
.	O
</s>
<s>
For	O
the	O
problem	O
variant	O
in	O
which	O
not	O
every	O
item	O
must	O
be	O
assigned	O
to	O
a	O
bin	O
,	O
there	O
is	O
a	O
family	O
of	O
algorithms	O
for	O
solving	O
the	O
GAP	O
by	O
using	O
a	O
combinatorial	O
translation	O
of	O
any	O
algorithm	O
for	O
the	O
knapsack	B-Algorithm
problem	I-Algorithm
into	O
an	O
approximation	O
algorithm	O
for	O
the	O
GAP	O
.	O
</s>
<s>
Using	O
any	O
-approximation	O
algorithm	O
ALG	O
for	O
the	O
knapsack	B-Algorithm
problem	I-Algorithm
,	O
it	O
is	O
possible	O
to	O
construct	O
a	O
(	O
)	O
-approximation	O
for	O
the	O
generalized	B-Algorithm
assignment	I-Algorithm
problem	I-Algorithm
in	O
a	O
greedy	O
manner	O
using	O
a	O
residual	O
profit	O
concept	O
.	O
</s>
