<s>
Matroid-constrained	B-Algorithm
number	I-Algorithm
partitioning	I-Algorithm
is	O
a	O
variant	O
of	O
the	O
multiway	B-Algorithm
number	I-Algorithm
partitioning	I-Algorithm
problem	O
,	O
in	O
which	O
the	O
subsets	O
in	O
the	O
partition	O
should	O
be	O
independent	O
sets	O
of	O
a	O
matroid	O
.	O
</s>
<s>
When	O
the	O
items	O
represent	O
jobs	O
and	O
the	O
weights	O
represent	O
their	O
length	O
,	O
this	O
objective	O
is	O
simply	O
the	O
makespan	B-Algorithm
of	O
the	O
schedule	O
.	O
</s>
<s>
Therefore	O
,	O
minimizing	O
this	O
objective	O
is	O
equivalent	O
to	O
minimizing	O
the	O
makespan	B-Algorithm
under	O
matroid	O
constraints	O
.	O
</s>
<s>
The	O
special	O
case	O
in	O
which	O
the	O
matroids	O
are	O
free	O
matroids	O
(	O
no	O
constraints	O
)	O
and	O
the	O
m	O
weight-functions	O
are	O
identical	O
corresponds	O
to	O
identical-machines	B-Algorithm
scheduling	I-Algorithm
,	O
also	O
known	O
as	O
multiway	B-Algorithm
number	I-Algorithm
partitioning	I-Algorithm
.	O
</s>
<s>
The	O
case	O
of	O
free	O
matroids	O
and	O
different	O
weight-functions	O
corresponds	O
to	O
unrelated-machines	B-Algorithm
scheduling	I-Algorithm
.	O
</s>
<s>
These	O
problems	O
are	O
described	O
in	O
the	O
page	O
on	O
balanced	B-Algorithm
number	I-Algorithm
partitioning	I-Algorithm
.	O
</s>
<s>
Minimizing	O
this	O
objective	O
can	O
be	O
reduced	O
to	O
the	O
weighted	O
matroid	B-Algorithm
intersection	I-Algorithm
problem	O
-	O
finding	O
a	O
maximum-weight	O
subset	O
that	O
is	O
simultaneously	O
independent	O
in	O
two	O
given	O
matroids	O
.	O
</s>
<s>
Minimizing	O
this	O
objective	O
with	O
graphic	O
matroids	O
can	O
be	O
used	O
to	O
solve	O
the	O
minimum	B-Algorithm
bottleneck	I-Algorithm
spanning	I-Algorithm
tree	I-Algorithm
problem	O
.	O
</s>
<s>
They	O
showed	O
that	O
minimizing	O
(	O
sum	O
,	O
max	O
)	O
can	O
be	O
done	O
in	O
polynomial	O
time	O
by	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
for	O
a	O
subclass	O
of	O
matroids	O
,	O
which	O
includes	O
partition	O
matroids	O
.	O
</s>
<s>
The	O
other	O
two	O
problems	O
were	O
analyzed	O
in	O
previous	O
works	O
:	O
minimizing	O
(	O
max	O
,	O
sum	O
)	O
is	O
known	O
to	O
be	O
strongly	O
NP-hard	O
(	O
3-partition	B-Algorithm
is	O
a	O
special	O
case	O
)	O
,	O
and	O
minimizing	O
(	O
sum	O
,	O
sum	O
)	O
can	O
be	O
reduced	O
to	O
weighted	O
matroid	B-Algorithm
intersection	I-Algorithm
,	O
which	O
is	O
polynomial	O
.	O
</s>
