<s>
The	O
multidimensional	B-Algorithm
assignment	I-Algorithm
problem	I-Algorithm
(	O
MAP	O
)	O
is	O
a	O
fundamental	O
combinatorial	O
optimization	O
problem	O
which	O
was	O
introduced	O
by	O
William	O
Pierskalla	O
.	O
</s>
<s>
This	O
problem	O
can	O
be	O
seen	O
as	O
a	O
generalization	O
of	O
the	O
linear	B-Algorithm
assignment	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
An	O
instance	O
of	O
the	O
problem	O
has	O
a	O
number	O
of	O
agents	O
(	O
i.e.	O
,	O
cardinality	B-Application
parameter	O
)	O
and	O
a	O
number	O
of	O
job	O
characteristics	O
(	O
i.e.	O
,	O
dimensionality	O
parameter	O
)	O
such	O
as	O
task	O
,	O
machine	O
,	O
time	O
interval	O
,	O
etc	O
.	O
</s>
<s>
The	O
multidimensional	B-Algorithm
assignment	I-Algorithm
problem	I-Algorithm
consists	O
of	O
finding	O
,	O
in	O
a	O
weighted	B-Algorithm
multipartite	O
graph	O
,	O
a	O
matching	O
of	O
a	O
given	O
size	O
,	O
in	O
which	O
the	O
sum	O
of	O
weights	O
of	O
the	O
edges	O
is	O
minimum	O
.	O
</s>
<s>
Using	O
cost-functions	O
,	O
the	O
dimensional	O
assignment	B-Algorithm
problem	I-Algorithm
(	O
or	O
MAP	O
)	O
can	O
be	O
stated	O
as	O
follows	O
:	O
</s>
<s>
Given	O
sets	O
,	O
and	O
,	O
of	O
equal	O
size	O
,	O
together	O
with	O
a	O
cost	O
array	O
or	O
multidimensional	O
weight	B-Algorithm
function	I-Algorithm
:	O
,	O
find	O
permutations	B-Algorithm
:	O
A	O
such	O
that	O
the	O
total	O
cost	O
function	O
:	O
</s>
<s>
The	O
multidimensional	B-Algorithm
assignment	I-Algorithm
problem	I-Algorithm
(	O
MAP	O
)	O
has	O
two	O
key	O
parameters	O
that	O
determine	O
the	O
size	O
of	O
a	O
problem	O
instance	O
:	O
</s>
<s>
The	O
cardinality	B-Application
parameter	O
,	O
where	O
denotes	O
the	O
number	O
of	O
elements	O
in	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
there	O
is	O
no	O
known	O
algorithm	O
for	O
solving	O
this	O
problem	O
in	O
polynomial	O
time	O
,	O
and	O
so	O
a	O
long	O
computational	O
time	O
may	O
be	O
needed	O
for	O
solving	O
problem	O
instances	O
of	O
even	O
moderate	O
size	O
(	O
based	O
on	O
dimensionality	O
and	O
cardinality	B-Application
parameters	O
)	O
.	O
</s>
