<s>
Guillotine	B-Algorithm
cutting	I-Algorithm
is	O
the	O
process	O
of	O
producing	O
small	O
rectangular	O
items	O
of	O
fixed	O
dimensions	O
from	O
a	O
given	O
large	O
rectangular	O
sheet	O
,	O
using	O
only	O
guillotine-cuts	O
.	O
</s>
<s>
Guillotine	B-Algorithm
cutting	I-Algorithm
is	O
particularly	O
common	O
in	O
the	O
glass	O
industry	O
.	O
</s>
<s>
There	O
are	O
various	O
optimization	O
problems	O
related	O
to	O
guillotine	B-Algorithm
cutting	I-Algorithm
,	O
such	O
as	O
:	O
maximize	O
the	O
total	O
area	O
of	O
the	O
produced	O
pieces	O
,	O
or	O
their	O
total	O
value	O
;	O
minimize	O
the	O
amount	O
of	O
waste	O
(	O
unused	O
parts	O
)	O
of	O
the	O
large	O
sheet	O
,	O
or	O
the	O
total	O
number	O
of	O
sheets	O
.	O
</s>
<s>
A	O
related	O
but	O
different	O
problem	O
is	O
guillotine	B-Algorithm
partition	I-Algorithm
.	O
</s>
<s>
The	O
following	O
terms	O
and	O
notations	O
are	O
often	O
used	O
in	O
the	O
literature	O
on	O
guillotine	B-Algorithm
cutting	I-Algorithm
.	O
</s>
<s>
The	O
goal	O
is	O
to	O
decide	O
whether	O
this	O
pattern	O
can	O
be	O
implemented	O
using	O
only	O
guillotine	B-Algorithm
cuts	I-Algorithm
,	O
and	O
if	O
so	O
,	O
find	O
a	O
sequence	O
of	O
such	O
cuts	O
.	O
</s>
<s>
All	O
conditions	O
together	O
imply	O
that	O
,	O
if	O
any	O
set	O
of	O
adjacent	O
rectangles	O
contains	O
more	O
than	O
one	O
element	O
,	O
then	O
they	O
can	O
be	O
separated	O
by	O
some	O
guillotine	B-Algorithm
cut	I-Algorithm
.	O
</s>
<s>
At	O
each	O
iteration	O
,	O
divide	O
a	O
given	O
pattern	O
,	O
containing	O
at	O
least	O
two	O
rectangles	O
,	O
into	O
two	O
disjoint	O
sub-patterns	O
using	O
a	O
guillotine	B-Algorithm
cut	I-Algorithm
,	O
and	O
recurse	O
on	O
each	O
sub-pattern	O
.	O
</s>
<s>
Stop	O
when	O
either	O
all	O
subpatterns	O
contain	O
one	O
rectangle	O
(	O
in	O
which	O
case	O
the	O
answer	O
is	O
"	O
yes	O
"	O
)	O
or	O
no	O
more	O
guillotine	B-Algorithm
cuts	I-Algorithm
are	O
possible	O
(	O
in	O
which	O
case	O
the	O
answer	O
is	O
"	O
no	O
"	O
)	O
.	O
</s>
<s>
Finding	O
a	O
guillotine	B-Algorithm
cut	I-Algorithm
for	O
a	O
given	O
pattern	O
is	O
done	O
as	O
follows	O
:	O
</s>
<s>
If	O
,	O
after	O
merging	O
,	O
there	O
are	O
at	O
least	O
two	O
disjoint	O
horizontal	O
intervals	O
,	O
then	O
a	O
vertical	O
guillotine	B-Algorithm
cut	I-Algorithm
is	O
possible	O
;	O
if	O
there	O
are	O
at	O
least	O
two	O
disjoint	O
vertical	O
intervals	O
,	O
then	O
a	O
horizontal	O
cut	O
is	O
possible	O
;	O
otherwise	O
,	O
no	O
cut	O
is	O
possible	O
.	O
</s>
<s>
When	O
the	O
algorithm	O
returns	O
"	O
yes	O
"	O
,	O
it	O
also	O
produces	O
a	O
sequence	O
of	O
guillotine	B-Algorithm
cuts	I-Algorithm
;	O
when	O
it	O
returns	O
"	O
no	O
"	O
,	O
it	O
also	O
produces	O
specific	O
subsets	O
of	O
rectangles	O
that	O
cannot	O
be	O
separated	O
by	O
guillotine	B-Algorithm
cuts	I-Algorithm
.	O
</s>
<s>
These	O
are	O
variants	O
of	O
the	O
two-dimensional	O
cutting	O
stock	O
,	O
bin	O
packing	O
and	O
rectangle	O
packing	O
problems	O
,	O
where	O
the	O
cuts	O
are	O
constrained	O
to	O
be	O
guillotine	B-Algorithm
cuts	I-Algorithm
.	O
</s>
<s>
In	O
the	O
basic	O
(	O
unweighted	O
)	O
guillotine-cutting	O
problem	O
,	O
the	O
required	O
output	O
is	O
a	O
sequence	O
of	O
guillotine	B-Algorithm
cuts	I-Algorithm
producing	O
pieces	O
of	O
the	O
target	O
dimensions	O
,	O
such	O
that	O
the	O
total	O
area	O
of	O
the	O
produced	O
pieces	O
is	O
maximized	O
(	O
equivalently	O
,	O
the	O
waste	O
from	O
the	O
raw	O
rectangle	O
is	O
minimized	O
)	O
.	O
</s>
<s>
This	O
case	O
is	O
associated	O
with	O
a	O
decision	O
problem	O
,	O
where	O
the	O
goal	O
is	O
to	O
decide	O
whether	O
it	O
is	O
possible	O
to	O
produce	O
a	O
single	O
element	O
of	O
each	O
target	O
dimension	O
using	O
guillotine	B-Algorithm
cuts	I-Algorithm
.	O
</s>
<s>
k-staged	O
guillotine	B-Algorithm
cutting	I-Algorithm
is	O
a	O
restricted	O
variant	O
of	O
guillotine	B-Algorithm
cutting	I-Algorithm
where	O
the	O
cutting	O
is	O
made	O
in	O
at	O
most	O
k	O
stages	O
:	O
in	O
the	O
first	O
stage	O
,	O
only	O
horizontal	O
cuts	O
are	O
made	O
;	O
in	O
the	O
second	O
stage	O
,	O
only	O
vertical	O
cuts	O
are	O
made	O
;	O
and	O
so	O
on	O
.	O
</s>
<s>
1-simple	O
guillotine	B-Algorithm
cutting	I-Algorithm
is	O
a	O
restricted	O
variant	O
of	O
guillotine-cutting	O
in	O
which	O
each	O
cut	O
separates	O
a	O
single	O
rectangle	O
.	O
</s>
<s>
A	O
2-simple	O
guillotine	B-Algorithm
cutting	I-Algorithm
is	O
a	O
1-simple	O
pattern	O
such	O
that	O
each	O
part	O
is	O
itself	O
a	O
1-simple	O
pattern	O
.	O
</s>
<s>
However	O
,	O
when	O
there	O
are	O
two	O
or	O
more	O
types	O
,	O
all	O
optimization	O
problems	O
related	O
to	O
guillotine	B-Algorithm
cutting	I-Algorithm
are	O
NP	O
hard	O
.	O
</s>
<s>
Due	O
to	O
its	O
practical	O
importance	O
,	O
various	O
exact	B-Algorithm
algorithms	I-Algorithm
and	O
approximation	B-Algorithm
algorithms	I-Algorithm
have	O
been	O
devised	O
.	O
</s>
<s>
Gilmore	O
and	O
Gomory	O
presented	O
a	O
dynamic	B-Algorithm
programming	I-Algorithm
recursion	O
for	O
both	O
staged	O
and	O
unstaged	O
guillotine	B-Algorithm
cutting	I-Algorithm
.	O
</s>
<s>
Herz	O
and	O
Christofides	O
and	O
Whitlock	O
presented	O
tree-search	B-Algorithm
procedures	O
for	O
unstaged	O
guillotine	B-Algorithm
cutting	I-Algorithm
.	O
</s>
<s>
Masden	O
and	O
Wang	O
presented	O
heuristic	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
It	O
is	O
a	O
bottom-up	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
algorithm	I-Algorithm
using	O
best-first	B-Algorithm
search	I-Algorithm
.	O
</s>
<s>
Clautiaux	O
,	O
Jouglet	O
and	O
Moukrim	O
propose	O
an	O
exact	B-Algorithm
algorithm	I-Algorithm
for	O
the	O
decision	O
problem	O
.	O
</s>
<s>
They	O
then	O
solve	O
the	O
optimization	O
problem	O
using	O
constraint	B-Application
programming	I-Application
on	O
the	O
space	O
of	O
well-sorted	O
normal	O
guillotine	O
graphs	O
.	O
</s>
<s>
There	O
are	O
two	O
main	O
approaches	O
for	O
exact	O
solutions	O
:	O
dynamic	B-Algorithm
programming	I-Algorithm
and	O
tree-search	B-Algorithm
(	O
branch-and-bound	B-Algorithm
)	O
.	O
</s>
<s>
The	O
tree-search	B-Algorithm
approaches	O
are	O
further	O
categorized	O
as	O
bottom-up	O
(	O
starting	O
with	O
single	O
rectangles	O
and	O
using	O
builds	O
to	O
construct	O
the	O
entire	O
sheet	O
)	O
or	O
top-down	O
.	O
</s>
<s>
"	O
A	O
new	O
guillotine	O
placement	O
heuristic	B-Algorithm
combined	O
with	O
an	O
improved	O
genetic	O
algorithm	O
for	O
the	O
orthogonal	O
cutting-stock	O
problem.	O
"	O
</s>
<s>
"	O
A	O
Controlled	O
Stability	O
Genetic	O
Algorithm	O
With	O
the	O
New	O
BLF2G	O
Guillotine	O
Placement	O
Heuristic	B-Algorithm
for	O
the	O
Orthogonal	O
Cutting-Stock	O
Problem.	O
"	O
</s>
<s>
McHale	O
and	O
Shah	O
wrote	O
a	O
Prolog	B-Language
program	O
implementing	O
an	O
anytime	B-Algorithm
algorithm	I-Algorithm
:	O
it	O
generates	O
approximately-optimal	O
solutions	O
in	O
a	O
given	O
amount	O
of	O
time	O
,	O
and	O
then	O
improves	O
it	O
if	O
the	O
user	O
allows	O
more	O
time	O
.	O
</s>
<s>
Guillotine	O
separation	O
is	O
a	O
related	O
problem	O
in	O
which	O
the	O
input	O
is	O
a	O
collection	O
of	O
n	O
pairwise-disjoint	O
convex	O
objects	O
in	O
the	O
plane	O
,	O
and	O
the	O
goal	O
is	O
to	O
separate	O
them	O
using	O
a	O
sequence	O
of	O
guillotine	B-Algorithm
cuts	I-Algorithm
.	O
</s>
<s>
Regarding	O
the	O
conjecture	O
that	O
it	O
is	O
possible	O
separate	O
axes-parallel	O
rectangle	O
,	O
while	O
they	O
do	O
not	O
settle	O
it	O
,	O
they	O
show	O
that	O
,	O
if	O
it	O
is	O
correct	O
,	O
then	O
it	O
implies	O
an	O
O(1 )	O
approximation	B-Algorithm
algorithm	I-Algorithm
to	O
the	O
problem	O
of	O
maximum	O
disjoint	O
set	O
of	O
axes-parallel	O
rectangles	O
in	O
time	O
.	O
</s>
<s>
Each	O
level	O
can	O
be	O
separated	O
by	O
two	O
guillotine	B-Algorithm
cuts	I-Algorithm
.	O
</s>
