<s>
In	O
combinatorial	O
optimization	O
,	O
the	O
stacker	B-Algorithm
crane	I-Algorithm
problem	I-Algorithm
is	O
an	O
optimization	O
problem	O
closely	O
related	O
to	O
the	O
traveling	B-Algorithm
salesperson	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
stacker	B-Algorithm
crane	I-Algorithm
problem	I-Algorithm
can	O
be	O
viewed	O
as	O
a	O
generalization	O
of	O
the	O
traveling	B-Algorithm
salesperson	I-Algorithm
problem	I-Algorithm
in	O
metric	O
spaces	O
:	O
any	O
instance	O
of	O
the	O
traveling	B-Algorithm
salesperson	I-Algorithm
problem	I-Algorithm
can	O
be	O
transformed	O
into	O
an	O
instance	O
of	O
the	O
stacker	B-Algorithm
crane	I-Algorithm
problem	I-Algorithm
,	O
having	O
a	O
pair	O
for	O
each	O
point	O
in	O
the	O
travelling	O
salesman	O
instance	O
.	O
</s>
<s>
In	O
the	O
other	O
direction	O
,	O
the	O
stacker	B-Algorithm
crane	I-Algorithm
problem	I-Algorithm
can	O
be	O
viewed	O
as	O
a	O
special	O
case	O
of	O
the	O
asymmetric	O
traveling	B-Algorithm
salesperson	I-Algorithm
problem	I-Algorithm
,	O
where	O
the	O
points	O
of	O
the	O
asymmetric	O
traveling	B-Algorithm
salesperson	I-Algorithm
problem	I-Algorithm
are	O
the	O
pairs	O
of	O
a	O
stacker	O
crane	O
instance	O
and	O
the	O
distance	O
from	O
one	O
pair	O
to	O
another	O
is	O
taken	O
as	O
the	O
distance	O
from	O
the	O
delivery	O
point	O
of	O
the	O
first	O
pair	O
,	O
through	O
its	O
pickup	O
point	O
,	O
to	O
the	O
delivery	O
point	O
of	O
the	O
second	O
pair	O
.	O
</s>
<s>
Because	O
it	O
generalizes	O
the	O
traveling	B-Algorithm
salesperson	I-Algorithm
problem	I-Algorithm
,	O
it	O
inherits	O
the	O
same	O
computational	O
complexity	O
:	O
it	O
is	O
NP-hard	O
,	O
and	O
at	O
least	O
as	O
hard	O
to	O
approximate	O
.	O
</s>
<s>
An	O
approximation	B-Algorithm
algorithm	I-Algorithm
based	O
on	O
the	O
Christofides	B-Algorithm
algorithm	I-Algorithm
for	O
the	O
traveling	B-Algorithm
salesperson	I-Algorithm
problem	I-Algorithm
can	O
approximate	O
the	O
solution	O
of	O
the	O
stacker	B-Algorithm
crane	I-Algorithm
problem	I-Algorithm
to	O
within	O
an	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
9/5	O
.	O
</s>
<s>
The	O
problem	O
of	O
designing	O
the	O
back	O
side	O
of	O
an	O
embroidery	O
pattern	O
to	O
minimize	O
the	O
total	O
amount	O
of	O
thread	O
used	O
is	O
closely	O
related	O
to	O
the	O
stacker	B-Algorithm
crane	I-Algorithm
problem	I-Algorithm
,	O
but	O
it	O
allows	O
each	O
of	O
its	O
pairs	O
of	O
points	O
(	O
the	O
ends	O
of	O
the	O
visible	O
stitches	O
on	O
the	O
front	O
side	O
of	O
the	O
pattern	O
)	O
to	O
be	O
traversed	O
in	O
either	O
direction	O
,	O
rather	O
than	O
requiring	O
the	O
traversal	O
to	O
go	O
through	O
all	O
pairs	O
in	O
a	O
consistent	O
direction	O
.	O
</s>
<s>
It	O
is	O
NP-hard	O
by	O
the	O
same	O
transformation	O
from	O
the	O
traveling	B-Algorithm
salesperson	I-Algorithm
problem	I-Algorithm
,	O
and	O
can	O
be	O
approximated	O
to	O
within	O
an	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
2	O
.	O
</s>
<s>
Another	O
variation	O
of	O
the	O
stacker	B-Algorithm
crane	I-Algorithm
problem	I-Algorithm
,	O
called	O
the	O
dial-a-ride	O
problem	O
,	O
asks	O
for	O
the	O
minimum	O
route	O
for	O
a	O
vehicle	O
to	O
perform	O
a	O
collection	O
of	O
pickups	O
and	O
deliveries	O
while	O
allowing	O
it	O
to	O
hold	O
some	O
number	O
of	O
loads	O
at	O
any	O
point	O
along	O
its	O
route	O
.	O
</s>
