<s>
Branch	B-Algorithm
and	I-Algorithm
cut	I-Algorithm
is	O
a	O
method	O
of	O
combinatorial	O
optimization	O
for	O
solving	O
integer	B-Algorithm
linear	I-Algorithm
programs	I-Algorithm
(	O
ILPs	O
)	O
,	O
that	O
is	O
,	O
linear	B-Algorithm
programming	I-Algorithm
(	O
LP	O
)	O
problems	O
where	O
some	O
or	O
all	O
the	O
unknowns	O
are	O
restricted	O
to	O
integer	O
values	O
.	O
</s>
<s>
Branch	B-Algorithm
and	I-Algorithm
cut	I-Algorithm
involves	O
running	O
a	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
algorithm	I-Algorithm
and	O
using	O
cutting	B-Algorithm
planes	I-Algorithm
to	O
tighten	O
the	O
linear	B-Algorithm
programming	I-Algorithm
relaxations	I-Algorithm
.	O
</s>
<s>
Note	O
that	O
if	O
cuts	O
are	O
only	O
used	O
to	O
tighten	O
the	O
initial	O
LP	B-Algorithm
relaxation	I-Algorithm
,	O
the	O
algorithm	O
is	O
called	O
cut	O
and	O
branch	O
.	O
</s>
<s>
The	O
method	O
solves	O
the	O
linear	B-Algorithm
program	I-Algorithm
without	I-Algorithm
the	I-Algorithm
integer	I-Algorithm
constraint	I-Algorithm
using	O
the	O
regular	O
simplex	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
When	O
an	O
optimal	O
solution	O
is	O
obtained	O
,	O
and	O
this	O
solution	O
has	O
a	O
non-integer	O
value	O
for	O
a	O
variable	O
that	O
is	O
supposed	O
to	O
be	O
integer	O
,	O
a	O
cutting	B-Algorithm
plane	I-Algorithm
algorithm	I-Algorithm
may	O
be	O
used	O
to	O
find	O
further	O
linear	O
constraints	O
which	O
are	O
satisfied	O
by	O
all	O
feasible	O
integer	O
points	O
but	O
violated	O
by	O
the	O
current	O
fractional	O
solution	O
.	O
</s>
<s>
These	O
inequalities	O
may	O
be	O
added	O
to	O
the	O
linear	B-Algorithm
program	I-Algorithm
,	O
such	O
that	O
resolving	O
it	O
will	O
yield	O
a	O
different	O
solution	O
which	O
is	O
hopefully	O
"	O
less	O
fractional	O
"	O
.	O
</s>
<s>
At	O
this	O
point	O
,	O
the	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
part	O
of	O
the	O
algorithm	O
is	O
started	O
.	O
</s>
<s>
The	O
new	O
linear	B-Algorithm
programs	I-Algorithm
are	O
then	O
solved	O
using	O
the	O
simplex	B-Algorithm
method	I-Algorithm
and	O
the	O
process	O
repeats	O
.	O
</s>
<s>
During	O
the	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
process	O
,	O
non-integral	O
solutions	O
to	O
LP	B-Algorithm
relaxations	I-Algorithm
serve	O
as	O
upper	O
bounds	O
and	O
integral	O
solutions	O
serve	O
as	O
lower	O
bounds	O
.	O
</s>
<s>
A	O
node	O
can	O
be	O
pruned	B-Algorithm
if	O
an	O
upper	O
bound	O
is	O
lower	O
than	O
an	O
existing	O
lower	O
bound	O
.	O
</s>
<s>
Further	O
,	O
when	O
solving	O
the	O
LP	B-Algorithm
relaxations	I-Algorithm
,	O
additional	O
cutting	B-Algorithm
planes	I-Algorithm
may	O
be	O
generated	O
,	O
which	O
may	O
be	O
either	O
global	O
cuts	O
,	O
i.e.	O
,	O
valid	O
for	O
all	O
feasible	O
integer	O
solutions	O
,	O
or	O
local	O
cuts	O
,	O
meaning	O
that	O
they	O
are	O
satisfied	O
by	O
all	O
solutions	O
fulfilling	O
the	O
side	O
constraints	O
from	O
the	O
currently	O
considered	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
subtree	O
.	O
</s>
<s>
Solve	O
the	O
LP	B-Algorithm
relaxation	I-Algorithm
of	O
the	O
problem	O
.	O
</s>
<s>
If	O
desired	O
,	O
search	O
for	O
cutting	B-Algorithm
planes	I-Algorithm
that	O
are	O
violated	O
by	O
.	O
</s>
<s>
If	O
any	O
are	O
found	O
,	O
add	O
them	O
to	O
the	O
LP	B-Algorithm
relaxation	I-Algorithm
and	O
return	O
to	O
3.2	O
.	O
</s>
<s>
In	O
C++	B-Language
-like	O
pseudocode	B-Language
,	O
this	O
could	O
be	O
written	O
:	O
</s>
<s>
In	O
the	O
above	O
pseudocode	B-Language
,	O
the	O
functions	O
LP_relax	O
,	O
LP_solve	O
and	O
branch_partition	O
called	O
as	O
subroutines	O
must	O
be	O
provided	O
as	O
applicable	O
to	O
the	O
problem	O
.	O
</s>
<s>
For	O
example	O
,	O
LP_solve	O
could	O
call	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
An	O
important	O
step	O
in	O
the	O
branch	B-Algorithm
and	I-Algorithm
cut	I-Algorithm
algorithm	O
is	O
the	O
branching	O
step	O
.	O
</s>
<s>
The	O
computational	O
cost	O
can	O
be	O
reduced	O
by	O
only	O
considering	O
a	O
subset	O
of	O
the	O
candidate	O
variables	O
and	O
not	O
solving	O
each	O
of	O
the	O
corresponding	O
LP	B-Algorithm
relaxations	I-Algorithm
to	O
completion	O
.	O
</s>
