<s>
In	O
graph	O
theory	O
and	O
combinatorial	O
optimization	O
,	O
a	O
closure	O
of	O
a	O
directed	O
graph	O
is	O
a	O
set	O
of	O
vertices	O
C	O
,	O
such	O
that	O
no	O
edges	O
leave	O
C	O
.	O
The	O
closure	B-Algorithm
problem	I-Algorithm
is	O
the	O
task	O
of	O
finding	O
the	O
maximum-weight	O
or	O
minimum-weight	O
closure	O
in	O
a	O
vertex-weighted	O
directed	O
graph	O
.	O
</s>
<s>
It	O
may	O
be	O
solved	O
in	O
polynomial	O
time	O
using	O
a	O
reduction	O
to	O
the	O
maximum	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
the	O
input	O
graph	O
to	O
a	O
closure	B-Algorithm
problem	I-Algorithm
may	O
be	O
replaced	O
by	O
its	O
condensation	O
,	O
in	O
which	O
every	O
strongly	O
connected	O
component	O
is	O
replaced	O
by	O
a	O
single	O
vertex	O
.	O
</s>
<s>
a	O
maximum-weight	O
closure	O
may	O
be	O
obtained	O
from	O
G	O
by	O
solving	O
a	O
maximum	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
on	O
a	O
graph	O
H	O
constructed	O
from	O
G	O
by	O
adding	O
to	O
it	O
two	O
additional	O
vertices	O
s	O
and	O
t	O
.	O
For	O
each	O
vertex	O
v	O
with	O
positive	O
weight	O
in	O
G	O
,	O
the	O
augmented	O
graph	O
H	O
contains	O
an	O
edge	O
from	O
s	O
to	O
v	O
with	O
capacity	O
equal	O
to	O
the	O
weight	O
of	O
v	O
,	O
</s>
<s>
By	O
the	O
max-flow	B-Algorithm
min-cut	I-Algorithm
theorem	I-Algorithm
,	O
a	O
minimum	O
cut	O
,	O
and	O
the	O
optimal	O
closure	O
derived	O
from	O
it	O
,	O
can	O
be	O
found	O
by	O
solving	O
a	O
maximum	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Alternative	O
algorithms	O
for	O
the	O
maximum	O
closure	B-Algorithm
problem	I-Algorithm
that	O
do	O
not	O
compute	O
flows	O
have	O
also	O
been	O
studied	O
.	O
</s>
<s>
Their	O
running	O
time	O
is	O
similar	O
to	O
that	O
of	O
the	O
fastest	O
known	O
flow	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
The	O
weight	O
of	O
each	O
vertex	O
in	O
this	O
network	O
is	O
the	O
total	O
value	O
of	O
its	O
block	O
,	O
and	O
the	O
most	O
profitable	O
plan	O
for	O
mining	O
can	O
be	O
determined	O
by	O
finding	O
a	O
maximum	O
weight	O
closure	O
,	O
and	O
then	O
forming	O
a	O
topological	B-Algorithm
ordering	I-Algorithm
of	O
the	O
blocks	O
in	O
this	O
closure	O
.	O
</s>
<s>
The	O
optimal	O
set	O
of	O
targets	O
to	O
attack	O
,	O
to	O
obtain	O
the	O
most	O
value	O
for	O
the	O
resources	O
expended	O
,	O
can	O
be	O
modeled	O
as	O
a	O
closure	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
problem	O
of	O
designing	O
a	O
network	O
that	O
maximizes	O
the	O
difference	O
between	O
the	O
profits	O
and	O
the	O
costs	O
can	O
be	O
solved	O
as	O
a	O
closure	B-Algorithm
problem	I-Algorithm
,	O
by	O
subdividing	O
each	O
undirected	O
edge	O
into	O
two	O
directed	O
edges	O
,	O
both	O
directed	O
outwards	O
from	O
the	O
subdivision	O
point	O
.	O
</s>
<s>
Together	O
with	O
open	O
pit	O
mining	O
,	O
this	O
was	O
one	O
of	O
the	O
original	O
motivating	O
applications	O
for	O
studying	O
the	O
closure	B-Algorithm
problem	I-Algorithm
;	O
it	O
was	O
originally	O
studied	O
in	O
1970	O
,	O
in	O
two	O
independent	O
papers	O
published	O
in	O
the	O
same	O
issue	O
of	O
the	O
same	O
journal	O
by	O
J	O
.	O
M	O
.	O
W	O
.	O
Rhys	O
and	O
Michel	O
Balinski	O
.	O
</s>
<s>
and	O
describe	O
an	O
application	O
of	O
the	O
closure	B-Algorithm
problem	I-Algorithm
to	O
a	O
version	O
of	O
job	B-Algorithm
shop	I-Algorithm
scheduling	I-Algorithm
in	O
which	O
one	O
is	O
given	O
a	O
collection	O
of	O
tasks	O
to	O
be	O
scheduled	O
to	O
be	O
performed	O
,	O
one	O
at	O
a	O
time	O
.	O
</s>
<s>
The	O
goal	O
is	O
to	O
choose	O
an	O
ordering	O
that	O
is	O
consistent	O
with	O
these	O
constraints	O
(	O
a	O
topological	B-Algorithm
ordering	I-Algorithm
of	O
G	O
)	O
that	O
minimizes	O
the	O
total	O
weighted	O
completion	O
time	O
of	O
the	O
tasks	O
.	O
</s>
<s>
Although	O
S	O
is	O
a	O
closure	O
(	O
for	O
a	O
graph	O
with	O
reversed	O
edges	O
from	O
G	O
)	O
the	O
problem	O
of	O
finding	O
S	O
is	O
not	O
exactly	O
a	O
maximum	O
weight	O
closure	B-Algorithm
problem	I-Algorithm
,	O
because	O
the	O
value	O
of	O
S	O
is	O
a	O
ratio	O
rather	O
than	O
a	O
sum	O
of	O
weights	O
.	O
</s>
<s>
Nevertheless	O
,	O
Lawler	O
shows	O
that	O
S	O
may	O
be	O
found	O
in	O
polynomial	O
time	O
by	O
a	O
binary	O
search	O
algorithm	O
in	O
which	O
each	O
step	O
of	O
the	O
search	O
uses	O
an	O
instance	O
of	O
the	O
closure	B-Algorithm
problem	I-Algorithm
as	O
a	O
subroutine	O
.	O
</s>
