<s>
The	O
Bottleneck	B-Algorithm
traveling	I-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
(	O
bottleneck	B-Algorithm
TSP	I-Algorithm
)	O
is	O
a	O
problem	O
in	O
discrete	O
or	O
combinatorial	O
optimization	O
.	O
</s>
<s>
NP-completeness	O
follows	O
immediately	O
by	O
a	O
reduction	B-Algorithm
from	O
the	O
problem	O
of	O
finding	O
a	O
Hamiltonian	O
cycle	O
.	O
</s>
<s>
Another	O
reduction	B-Algorithm
,	O
from	O
the	O
bottleneck	B-Algorithm
TSP	I-Algorithm
to	O
the	O
usual	O
TSP	O
(	O
where	O
the	O
goal	O
is	O
to	O
minimize	O
the	O
sum	O
of	O
edge	O
lengths	O
)	O
,	O
allows	O
any	O
algorithm	O
for	O
the	O
usual	O
TSP	O
to	O
also	O
be	O
used	O
to	O
solve	O
the	O
bottleneck	B-Algorithm
TSP	I-Algorithm
.	O
</s>
<s>
If	O
the	O
edge	O
weights	O
of	O
the	O
bottleneck	B-Algorithm
TSP	I-Algorithm
are	O
replaced	O
by	O
any	O
other	O
numbers	O
that	O
have	O
the	O
same	O
relative	O
order	O
,	O
then	O
the	O
bottleneck	O
solution	O
remains	O
unchanged	O
.	O
</s>
<s>
For	O
instance	O
,	O
following	O
this	O
transformation	O
,	O
the	O
Held	B-Algorithm
–	I-Algorithm
Karp	I-Algorithm
algorithm	I-Algorithm
could	O
be	O
used	O
to	O
solve	O
the	O
bottleneck	B-Algorithm
TSP	I-Algorithm
in	O
time	O
.	O
</s>
<s>
Alternatively	O
,	O
the	O
problem	O
can	O
be	O
solved	O
by	O
performing	O
a	O
binary	O
search	O
or	O
sequential	B-Algorithm
search	I-Algorithm
for	O
the	O
smallest	O
such	O
that	O
the	O
subgraph	O
of	O
edges	O
of	O
weight	O
at	O
most	O
has	O
a	O
Hamiltonian	O
cycle	O
.	O
</s>
<s>
In	O
an	O
asymmetric	O
bottleneck	B-Algorithm
TSP	I-Algorithm
,	O
there	O
are	O
cases	O
where	O
the	O
weight	O
from	O
node	O
A	O
to	O
B	O
is	O
different	O
from	O
the	O
weight	O
from	O
B	O
to	O
A	O
(	O
e	O
.	O
g	O
.	O
travel	O
time	O
between	O
two	O
cities	O
with	O
a	O
traffic	O
jam	O
in	O
one	O
direction	O
)	O
.	O
</s>
<s>
The	O
Euclidean	O
bottleneck	B-Algorithm
TSP	I-Algorithm
,	O
or	O
planar	O
bottleneck	B-Algorithm
TSP	I-Algorithm
,	O
is	O
the	O
bottleneck	B-Algorithm
TSP	I-Algorithm
with	O
the	O
distance	O
being	O
the	O
ordinary	O
Euclidean	O
distance	O
.	O
</s>
<s>
The	O
maximum	O
scatter	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
is	O
another	O
variation	O
of	O
the	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
in	O
which	O
the	O
goal	O
is	O
to	O
find	O
a	O
Hamiltonian	O
cycle	O
that	O
maximizes	O
the	O
minimum	O
edge	O
length	O
rather	O
than	O
minimizing	O
the	O
maximum	O
length	O
.	O
</s>
<s>
It	O
can	O
be	O
translated	O
into	O
an	O
instance	O
of	O
the	O
bottleneck	B-Algorithm
TSP	I-Algorithm
problem	O
by	O
negating	O
all	O
edge	O
lengths	O
(	O
or	O
,	O
to	O
keep	O
the	O
results	O
positive	O
,	O
subtracting	O
them	O
all	O
from	O
a	O
large	O
enough	O
constant	O
)	O
.	O
</s>
<s>
If	O
the	O
graph	O
is	O
a	O
metric	O
space	O
then	O
there	O
is	O
an	O
efficient	O
approximation	B-Algorithm
algorithm	I-Algorithm
that	O
finds	O
a	O
Hamiltonian	O
cycle	O
with	O
maximum	O
edge	O
weight	O
being	O
no	O
more	O
than	O
twice	O
the	O
optimum	O
.	O
</s>
<s>
Then	O
provides	O
a	O
valid	O
lower	O
bound	O
on	O
the	O
bottleneck	B-Algorithm
TSP	I-Algorithm
weight	O
,	O
for	O
the	O
bottleneck	B-Algorithm
TSP	I-Algorithm
is	O
itself	O
a	O
2-connected	O
graph	O
and	O
necessarily	O
contains	O
an	O
edge	O
of	O
weight	O
at	O
least	O
.	O
</s>
<s>
This	O
approximation	B-Algorithm
ratio	I-Algorithm
is	O
best	O
possible	O
.	O
</s>
<s>
Without	O
the	O
assumption	O
that	O
the	O
input	O
is	O
a	O
metric	O
space	O
,	O
no	O
finite	O
approximation	B-Algorithm
ratio	I-Algorithm
is	O
possible	O
.	O
</s>
