<s>
The	O
travelling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
(	O
also	O
called	O
the	O
travelling	B-Algorithm
salesperson	I-Algorithm
problem	I-Algorithm
or	O
TSP	O
)	O
asks	O
the	O
following	O
question	O
:	O
"	O
Given	O
a	O
list	O
of	O
cities	O
and	O
the	O
distances	O
between	O
each	O
pair	O
of	O
cities	O
,	O
what	O
is	O
the	O
shortest	O
possible	O
route	O
that	O
visits	O
each	O
city	O
exactly	O
once	O
and	O
returns	O
to	O
the	O
origin	O
city	O
?	O
"	O
</s>
<s>
It	O
is	O
an	O
NP-hard	O
problem	O
in	O
combinatorial	O
optimization	O
,	O
important	O
in	O
theoretical	O
computer	B-General_Concept
science	I-General_Concept
and	O
operations	O
research	O
.	O
</s>
<s>
The	O
travelling	B-Algorithm
purchaser	I-Algorithm
problem	I-Algorithm
and	O
the	O
vehicle	B-Algorithm
routing	I-Algorithm
problem	I-Algorithm
are	O
both	O
generalizations	O
of	O
TSP	O
.	O
</s>
<s>
Thus	O
,	O
it	O
is	O
possible	O
that	O
the	O
worst-case	B-General_Concept
running	O
time	O
for	O
any	O
algorithm	O
for	O
the	O
TSP	O
increases	O
superpolynomially	O
(	O
but	O
no	O
more	O
than	O
exponentially	O
)	O
with	O
the	O
number	O
of	O
cities	O
.	O
</s>
<s>
Even	O
though	O
the	O
problem	O
is	O
computationally	O
difficult	O
,	O
many	O
heuristics	B-Algorithm
and	O
exact	B-Algorithm
algorithms	I-Algorithm
are	O
known	O
,	O
so	O
that	O
some	O
instances	O
with	O
tens	O
of	O
thousands	O
of	O
cities	O
can	O
be	O
solved	O
completely	O
and	O
even	O
problems	O
with	O
millions	O
of	O
cities	O
can	O
be	O
approximated	O
within	O
a	O
small	O
fraction	O
of	O
1%	O
.	O
</s>
<s>
The	O
origins	O
of	O
the	O
travelling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
are	O
unclear	O
.	O
</s>
<s>
A	O
handbook	O
for	O
travelling	O
salesmen	O
from	O
1832	O
mentions	O
the	O
problem	O
and	O
includes	O
example	O
tours	O
through	O
Germany	O
and	O
Switzerland	B-Protocol
,	O
but	O
contains	O
no	O
mathematical	O
treatment	O
.	O
</s>
<s>
The	O
general	O
form	O
of	O
the	O
TSP	O
appears	O
to	O
have	O
been	O
first	O
studied	O
by	O
mathematicians	O
during	O
the	O
1930s	O
in	O
Vienna	O
and	O
at	O
Harvard	O
,	O
notably	O
by	O
Karl	O
Menger	O
,	O
who	O
defines	O
the	O
problem	O
,	O
considers	O
the	O
obvious	O
brute-force	O
algorithm	O
,	O
and	O
observes	O
the	O
non-optimality	O
of	O
the	O
nearest	O
neighbour	O
heuristic	B-Algorithm
:	O
</s>
<s>
The	O
earliest	O
publication	O
using	O
the	O
phrase	O
"	O
travelling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
"	O
was	O
the	O
1949	O
RAND	O
Corporation	O
report	O
by	O
Julia	O
Robinson	O
,	O
"	O
On	O
the	O
Hamiltonian	O
game	O
(	O
a	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
)	O
.	O
"	O
</s>
<s>
Notable	O
contributions	O
were	O
made	O
by	O
George	O
Dantzig	O
,	O
Delbert	O
Ray	O
Fulkerson	O
and	O
Selmer	O
M	O
.	O
Johnson	O
from	O
the	O
RAND	O
Corporation	O
,	O
who	O
expressed	O
the	O
problem	O
as	O
an	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
and	O
developed	O
the	O
cutting	B-Algorithm
plane	I-Algorithm
method	I-Algorithm
for	O
its	O
solution	O
.	O
</s>
<s>
While	O
this	O
paper	O
did	O
not	O
give	O
an	O
algorithmic	O
approach	O
to	O
TSP	B-Algorithm
problems	I-Algorithm
,	O
the	O
ideas	O
that	O
lay	O
within	O
it	O
were	O
indispensable	O
to	O
later	O
creating	O
exact	O
solution	O
methods	O
for	O
the	O
TSP	O
,	O
though	O
it	O
would	O
take	O
15	O
years	O
to	O
find	O
an	O
algorithmic	O
approach	O
in	O
creating	O
these	O
cuts	O
.	O
</s>
<s>
As	O
well	O
as	O
cutting	B-Algorithm
plane	I-Algorithm
methods	I-Algorithm
,	O
Dantzig	O
,	O
Fulkerson	O
and	O
Johnson	O
used	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
algorithms	I-Algorithm
perhaps	O
for	O
the	O
first	O
time	O
.	O
</s>
<s>
In	O
the	O
following	O
decades	O
,	O
the	O
problem	O
was	O
studied	O
by	O
many	O
researchers	O
from	O
mathematics	O
,	O
computer	B-General_Concept
science	I-General_Concept
,	O
chemistry	O
,	O
physics	O
,	O
and	O
other	O
sciences	O
.	O
</s>
<s>
In	O
the	O
1960s	O
,	O
however	O
,	O
a	O
new	O
approach	O
was	O
created	O
,	O
that	O
instead	O
of	O
seeking	O
optimal	O
solutions	O
would	O
produce	O
a	O
solution	O
whose	O
length	O
is	O
provably	O
bounded	O
by	O
a	O
multiple	O
of	O
the	O
optimal	O
length	O
,	O
and	O
in	O
doing	O
so	O
would	O
create	O
lower	O
bounds	O
for	O
the	O
problem	O
;	O
these	O
lower	O
bounds	O
would	O
then	O
be	O
used	O
with	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
approaches	O
.	O
</s>
<s>
In	O
1976	O
,	O
Christofides	O
and	O
Serdyukov	O
independently	O
of	O
each	O
other	O
made	O
a	O
big	O
advance	O
in	O
this	O
direction	O
:	O
the	O
Christofides-Serdyukov	B-Algorithm
algorithm	I-Algorithm
yields	O
a	O
solution	O
that	O
,	O
in	O
the	O
worst	B-General_Concept
case	I-General_Concept
,	O
is	O
at	O
most	O
1.5	O
times	O
longer	O
than	O
the	O
optimal	O
solution	O
.	O
</s>
<s>
However	O
,	O
this	O
hope	O
for	O
improvement	O
did	O
not	O
immediately	O
materialize	O
,	O
and	O
Christofides-Serdyukov	O
remained	O
the	O
method	O
with	O
the	O
best	O
worst-case	B-General_Concept
scenario	O
until	O
2011	O
,	O
when	O
a	O
(	O
very	O
)	O
slightly	O
improved	O
approximation	B-Algorithm
algorithm	I-Algorithm
was	O
developed	O
for	O
the	O
subset	O
of	O
"	O
graphical	O
"	O
TSPs	O
.	O
</s>
<s>
Great	O
progress	O
was	O
made	O
in	O
the	O
late	O
1970s	O
and	O
1980	O
,	O
when	O
Grötschel	O
,	O
Padberg	O
,	O
Rinaldi	O
and	O
others	O
managed	O
to	O
exactly	O
solve	O
instances	O
with	O
up	O
to	O
2,392	O
cities	O
,	O
using	O
cutting	B-Algorithm
planes	I-Algorithm
and	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
.	O
</s>
<s>
Another	O
related	O
problem	O
is	O
the	O
bottleneck	B-Algorithm
travelling	I-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
(	O
bottleneck	B-Algorithm
TSP	I-Algorithm
)	O
:	O
Find	O
a	O
Hamiltonian	O
cycle	O
in	O
a	O
weighted	O
graph	O
with	O
the	O
minimal	O
weight	O
of	O
the	O
weightiest	O
edge	O
.	O
</s>
<s>
The	O
generalized	B-Algorithm
travelling	I-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
,	O
also	O
known	O
as	O
the	O
"	O
travelling	O
politician	O
problem	O
"	O
,	O
deals	O
with	O
"	O
states	O
"	O
that	O
have	O
(	O
one	O
or	O
more	O
)	O
"	O
cities	O
"	O
and	O
the	O
salesman	O
has	O
to	O
visit	O
exactly	O
one	O
"	O
city	O
"	O
from	O
each	O
"	O
state	O
"	O
.	O
</s>
<s>
Noon	O
and	O
Bean	O
demonstrated	O
that	O
the	O
generalized	B-Algorithm
travelling	I-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
can	O
be	O
transformed	O
into	O
a	O
standard	O
TSP	O
with	O
the	O
same	O
number	O
of	O
cities	O
,	O
but	O
a	O
modified	O
distance	O
matrix	O
.	O
</s>
<s>
The	O
travelling	B-Algorithm
purchaser	I-Algorithm
problem	I-Algorithm
deals	O
with	O
a	O
purchaser	O
who	O
is	O
charged	O
with	O
purchasing	O
a	O
set	O
of	O
products	O
.	O
</s>
<s>
The	O
TSP	O
can	O
be	O
formulated	O
as	O
an	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
.	O
</s>
<s>
It	O
is	O
because	O
these	O
are	O
0/1	O
variables	O
that	O
the	O
formulations	O
become	O
integer	B-Algorithm
programs	I-Algorithm
;	O
all	O
other	O
constraints	O
are	O
purely	O
linear	O
.	O
</s>
<s>
The	O
MTZ	O
formulation	O
of	O
TSP	O
is	O
thus	O
the	O
following	O
integer	B-Algorithm
linear	I-Algorithm
programming	I-Algorithm
problem	O
:	O
</s>
<s>
Then	O
TSP	O
can	O
be	O
written	O
as	O
the	O
following	O
integer	B-Algorithm
linear	I-Algorithm
programming	I-Algorithm
problem	O
:	O
</s>
<s>
Because	O
this	O
leads	O
to	O
an	O
exponential	O
number	O
of	O
possible	O
constraints	O
,	O
in	O
practice	O
it	O
is	O
solved	O
with	O
row	B-Algorithm
generation	I-Algorithm
.	O
</s>
<s>
Devising	O
exact	B-Algorithm
algorithms	I-Algorithm
,	O
which	O
work	O
reasonably	O
fast	O
only	O
for	O
small	O
problem	O
sizes	O
.	O
</s>
<s>
Devising	O
"	O
suboptimal	O
"	O
or	O
heuristic	B-Algorithm
algorithms	I-Algorithm
,	O
i.e.	O
,	O
algorithms	O
that	O
deliver	O
approximated	O
solutions	O
in	O
a	O
reasonable	O
time	O
.	O
</s>
<s>
Finding	O
special	O
cases	O
for	O
the	O
problem	O
(	O
"	O
subproblems	O
"	O
)	O
for	O
which	O
either	O
better	O
or	O
exact	O
heuristics	B-Algorithm
are	O
possible	O
.	O
</s>
<s>
The	O
most	O
direct	O
solution	O
would	O
be	O
to	O
try	O
all	O
permutations	B-Algorithm
(	O
ordered	O
combinations	O
)	O
and	O
see	O
which	O
one	O
is	O
cheapest	O
(	O
using	O
brute-force	B-Algorithm
search	I-Algorithm
)	O
.	O
</s>
<s>
One	O
of	O
the	O
earliest	O
applications	O
of	O
dynamic	B-Algorithm
programming	I-Algorithm
is	O
the	O
Held	O
–	O
Karp	O
algorithm	O
that	O
solves	O
the	O
problem	O
in	O
time	O
.	O
</s>
<s>
This	O
bound	O
has	O
also	O
been	O
reached	O
by	O
Exclusion-Inclusion	O
in	O
an	O
attempt	O
preceding	O
the	O
dynamic	B-Algorithm
programming	I-Algorithm
approach	O
.	O
</s>
<s>
For	O
example	O
,	O
it	O
has	O
not	O
been	O
determined	O
whether	O
a	O
classical	O
exact	B-Algorithm
algorithm	I-Algorithm
for	O
TSP	O
that	O
runs	O
in	O
time	O
exists	O
.	O
</s>
<s>
The	O
currently	O
best	O
quantum	O
exact	B-Algorithm
algorithm	I-Algorithm
for	O
TSP	O
due	O
to	O
Ambainis	O
et	O
al	O
.	O
</s>
<s>
Various	O
branch-and-bound	B-Algorithm
algorithms	I-Algorithm
,	O
which	O
can	O
be	O
used	O
to	O
process	O
TSPs	O
containing	O
40	O
–	O
60	O
cities	O
.	O
</s>
<s>
Progressive	O
improvement	O
algorithms	O
which	O
use	O
techniques	O
reminiscent	O
of	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
Implementations	O
of	O
branch-and-bound	B-Algorithm
and	O
problem-specific	O
cut	O
generation	O
(	O
branch-and-cut	B-Algorithm
)	O
;	O
this	O
is	O
the	O
method	O
of	O
choice	O
for	O
solving	O
large	O
instances	O
.	O
</s>
<s>
An	O
exact	O
solution	O
for	O
15,112	O
German	O
towns	O
from	O
TSPLIB	O
was	O
found	O
in	O
2001	O
using	O
the	O
cutting-plane	B-Algorithm
method	I-Algorithm
proposed	O
by	O
George	O
Dantzig	O
,	O
Ray	O
Fulkerson	O
,	O
and	O
Selmer	O
M	O
.	O
Johnson	O
in	O
1954	O
,	O
based	O
on	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
The	O
total	O
computation	O
time	O
was	O
equivalent	O
to	O
22.6years	O
on	O
a	O
single	O
500MHz	O
Alpha	B-Device
processor	I-Device
.	O
</s>
<s>
In	O
May	O
2004	O
,	O
the	O
travelling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
of	O
visiting	O
all	O
24,978	O
towns	O
in	O
Sweden	O
was	O
solved	O
:	O
a	O
tour	O
of	O
length	O
approximately	O
72,500	O
kilometres	O
was	O
found	O
and	O
it	O
was	O
proven	O
that	O
no	O
shorter	O
tour	O
exists	O
.	O
</s>
<s>
In	O
March	O
2005	O
,	O
the	O
travelling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
of	O
visiting	O
all	O
33,810	O
points	O
in	O
a	O
circuit	O
board	O
was	O
solved	O
using	O
Concorde	B-Algorithm
TSP	I-Algorithm
Solver	I-Algorithm
:	O
a	O
tour	O
of	O
length	O
66,048,945	O
units	O
was	O
found	O
and	O
it	O
was	O
proven	O
that	O
no	O
shorter	O
tour	O
exists	O
.	O
</s>
<s>
In	O
April	O
2006	O
an	O
instance	O
with	O
85,900	O
points	O
was	O
solved	O
using	O
Concorde	B-Algorithm
TSP	I-Algorithm
Solver	I-Algorithm
,	O
taking	O
over	O
136	O
CPU-years	O
,	O
see	O
.	O
</s>
<s>
Various	O
heuristics	B-Algorithm
and	O
approximation	B-Algorithm
algorithms	I-Algorithm
,	O
which	O
quickly	O
yield	O
good	O
solutions	O
,	O
have	O
been	O
devised	O
.	O
</s>
<s>
These	O
include	O
the	O
Multi-fragment	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Several	O
categories	O
of	O
heuristics	B-Algorithm
are	O
recognized	O
.	O
</s>
<s>
The	O
nearest	O
neighbour	O
(	O
NN	O
)	O
algorithm	O
(	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
)	O
lets	O
the	O
salesman	O
choose	O
the	O
nearest	O
unvisited	O
city	O
as	O
his	O
next	O
move	O
.	O
</s>
<s>
The	O
bitonic	B-Algorithm
tour	I-Algorithm
of	O
a	O
set	O
of	O
points	O
is	O
the	O
minimum-perimeter	O
monotone	B-Algorithm
polygon	I-Algorithm
that	O
has	O
the	O
points	O
as	O
its	O
vertices	O
;	O
it	O
can	O
be	O
computed	O
efficiently	O
by	O
dynamic	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
Another	O
constructive	B-Algorithm
heuristic	I-Algorithm
,	O
Match	O
Twice	O
and	O
Stitch	O
(	O
MTS	O
)	O
,	O
performs	O
two	O
sequential	O
matchings	O
,	O
where	O
the	O
second	O
matching	O
is	O
executed	O
after	O
deleting	O
all	O
the	O
edges	O
of	O
the	O
first	O
matching	O
,	O
to	O
yield	O
a	O
set	O
of	O
cycles	O
.	O
</s>
<s>
The	B-Algorithm
algorithm	I-Algorithm
of	I-Algorithm
Christofides	I-Algorithm
and	I-Algorithm
Serdyukov	I-Algorithm
follows	O
a	O
similar	O
outline	O
but	O
combines	O
the	O
minimum	O
spanning	O
tree	O
with	O
a	O
solution	O
of	O
another	O
problem	O
,	O
minimum-weight	O
perfect	O
matching	O
.	O
</s>
<s>
It	O
was	O
one	O
of	O
the	O
first	O
approximation	B-Algorithm
algorithms	I-Algorithm
,	O
and	O
was	O
in	O
part	O
responsible	O
for	O
drawing	O
attention	O
to	O
approximation	B-Algorithm
algorithms	I-Algorithm
as	O
a	O
practical	O
approach	O
to	O
intractable	O
problems	O
.	O
</s>
<s>
As	O
a	O
matter	O
of	O
fact	O
,	O
the	O
term	O
"	O
algorithm	O
"	O
was	O
not	O
commonly	O
extended	O
to	O
approximation	B-Algorithm
algorithms	I-Algorithm
until	O
later	O
;	O
the	O
Christofides	B-Algorithm
algorithm	I-Algorithm
was	O
initially	O
referred	O
to	O
as	O
the	O
Christofides	B-Algorithm
heuristic	I-Algorithm
.	O
</s>
<s>
Adapting	O
the	O
above	O
method	O
gives	O
the	B-Algorithm
algorithm	I-Algorithm
of	I-Algorithm
Christofides	I-Algorithm
and	I-Algorithm
Serdyukov	I-Algorithm
.	O
</s>
<s>
The	O
pairwise	O
exchange	O
or	O
2-opt	B-Algorithm
technique	O
involves	O
iteratively	O
removing	O
two	O
edges	O
and	O
replacing	O
these	O
with	O
two	O
different	O
edges	O
that	O
reconnect	O
the	O
fragments	O
created	O
by	O
edge	O
removal	O
into	O
a	O
new	O
and	O
shorter	O
tour	O
.	O
</s>
<s>
Similarly	O
,	O
the	O
3-opt	B-Algorithm
technique	O
removes	O
3	O
edges	O
and	O
reconnects	O
them	O
to	O
form	O
a	O
shorter	O
tour	O
.	O
</s>
<s>
The	O
label	O
Lin	B-Algorithm
–	I-Algorithm
Kernighan	I-Algorithm
is	O
an	O
often	O
heard	O
misnomer	O
for	O
2-opt	B-Algorithm
.	O
</s>
<s>
Lin	B-Algorithm
–	I-Algorithm
Kernighan	I-Algorithm
is	O
actually	O
the	O
more	O
general	O
k-opt	O
method	O
.	O
</s>
<s>
For	O
Euclidean	O
instances	O
,	O
2-opt	B-Algorithm
heuristics	B-Algorithm
give	O
on	O
average	O
solutions	O
that	O
are	O
about	O
5%	O
better	O
than	O
Christofides	O
 '	O
algorithm	O
.	O
</s>
<s>
If	O
we	O
start	O
with	O
an	O
initial	O
solution	O
made	O
with	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
,	O
the	O
average	O
number	O
of	O
moves	O
greatly	O
decreases	O
again	O
and	O
is	O
.	O
</s>
<s>
However	O
whilst	O
in	O
order	O
this	O
is	O
a	O
small	O
increase	O
in	O
size	O
,	O
the	O
initial	O
number	O
of	O
moves	O
for	O
small	O
problems	O
is	O
10	O
times	O
as	O
big	O
for	O
a	O
random	O
start	O
compared	O
to	O
one	O
made	O
from	O
a	O
greedy	B-Algorithm
heuristic	I-Algorithm
.	O
</s>
<s>
This	O
is	O
because	O
such	O
2-opt	B-Algorithm
heuristics	B-Algorithm
exploit	O
'	O
bad	O
 '	O
parts	O
of	O
a	O
solution	O
such	O
as	O
crossings	O
.	O
</s>
<s>
These	O
types	O
of	O
heuristics	B-Algorithm
are	O
often	O
used	O
within	O
Vehicle	B-Algorithm
routing	I-Algorithm
problem	I-Algorithm
heuristics	B-Algorithm
to	O
reoptimize	O
route	O
solutions	O
.	O
</s>
<s>
The	O
Lin	B-Algorithm
–	I-Algorithm
Kernighan	I-Algorithm
heuristic	I-Algorithm
is	O
a	O
special	O
case	O
of	O
the	O
V-opt	O
or	O
variable-opt	O
technique	O
.	O
</s>
<s>
The	O
most	O
popular	O
of	O
the	O
k-opt	O
methods	O
are	O
3-opt	B-Algorithm
,	O
as	O
introduced	O
by	O
Shen	O
Lin	O
of	O
Bell	O
Labs	O
in	O
1965	O
.	O
</s>
<s>
A	O
special	O
case	O
of	O
3-opt	B-Algorithm
is	O
where	O
the	O
edges	O
are	O
not	O
disjoint	O
(	O
two	O
of	O
the	O
edges	O
are	O
adjacent	O
to	O
one	O
another	O
)	O
.	O
</s>
<s>
In	O
practice	O
,	O
it	O
is	O
often	O
possible	O
to	O
achieve	O
substantial	O
improvement	O
over	O
2-opt	B-Algorithm
without	O
the	O
combinatorial	O
cost	O
of	O
the	O
general	O
3-opt	B-Algorithm
by	O
restricting	O
the	O
3-changes	O
to	O
this	O
special	O
subset	O
where	O
two	O
of	O
the	O
removed	O
edges	O
are	O
adjacent	O
.	O
</s>
<s>
This	O
so-called	O
two-and-a-half-opt	O
typically	O
falls	O
roughly	O
midway	O
between	O
2-opt	B-Algorithm
and	O
3-opt	B-Algorithm
,	O
both	O
in	O
terms	O
of	O
the	O
quality	O
of	O
tours	O
achieved	O
and	O
the	O
time	O
required	O
to	O
achieve	O
those	O
tours	O
.	O
</s>
<s>
The	O
best-known	O
method	O
in	O
this	O
family	O
is	O
the	O
Lin	B-Algorithm
–	I-Algorithm
Kernighan	I-Algorithm
method	O
(	O
mentioned	O
above	O
as	O
a	O
misnomer	O
for	O
2-opt	B-Algorithm
)	O
.	O
</s>
<s>
Shen	O
Lin	O
and	O
Brian	O
Kernighan	O
first	O
published	O
their	O
method	O
in	O
1972	O
,	O
and	O
it	O
was	O
the	O
most	O
reliable	O
heuristic	B-Algorithm
for	O
solving	O
travelling	B-Algorithm
salesman	I-Algorithm
problems	I-Algorithm
for	O
nearly	O
two	O
decades	O
.	O
</s>
<s>
These	O
methods	O
(	O
sometimes	O
called	O
Lin	B-Algorithm
–	I-Algorithm
Kernighan	I-Algorithm
–	O
Johnson	O
)	O
build	O
on	O
the	O
Lin	B-Algorithm
–	I-Algorithm
Kernighan	I-Algorithm
method	O
,	O
adding	O
ideas	O
from	O
tabu	B-Algorithm
search	I-Algorithm
and	O
evolutionary	O
computing	O
.	O
</s>
<s>
The	O
basic	O
Lin	B-Algorithm
–	I-Algorithm
Kernighan	I-Algorithm
technique	O
gives	O
results	O
that	O
are	O
guaranteed	O
to	O
be	O
at	O
least	O
3-opt	B-Algorithm
.	O
</s>
<s>
The	O
Lin	B-Algorithm
–	I-Algorithm
Kernighan	I-Algorithm
–	O
Johnson	O
methods	O
compute	O
a	O
Lin	B-Algorithm
–	I-Algorithm
Kernighan	I-Algorithm
tour	O
,	O
and	O
then	O
perturb	O
the	O
tour	O
by	O
what	O
has	O
been	O
described	O
as	O
a	O
mutation	O
that	O
removes	O
at	O
least	O
four	O
edges	O
and	O
reconnects	O
the	O
tour	O
in	O
a	O
different	O
way	O
,	O
then	O
V-opting	O
the	O
new	O
tour	O
.	O
</s>
<s>
The	O
mutation	O
is	O
often	O
enough	O
to	O
move	O
the	O
tour	O
from	O
the	O
local	O
minimum	O
identified	O
by	O
Lin	B-Algorithm
–	I-Algorithm
Kernighan	I-Algorithm
.	O
</s>
<s>
V-opt	O
methods	O
are	O
widely	O
considered	O
the	O
most	O
powerful	O
heuristics	B-Algorithm
for	O
the	O
problem	O
,	O
and	O
are	O
able	O
to	O
address	O
special	O
cases	O
,	O
such	O
as	O
the	O
Hamilton	O
Cycle	O
Problem	O
and	O
other	O
non-metric	O
TSPs	O
that	O
other	O
heuristics	B-Algorithm
fail	O
on	O
.	O
</s>
<s>
For	O
many	O
years	O
Lin	B-Algorithm
–	I-Algorithm
Kernighan	I-Algorithm
–	O
Johnson	O
had	O
identified	O
optimal	O
solutions	O
for	O
all	O
TSPs	O
where	O
an	O
optimal	O
solution	O
was	O
known	O
and	O
had	O
identified	O
the	O
best-known	O
solutions	O
for	O
all	O
other	O
TSPs	O
on	O
which	O
the	O
method	O
had	O
been	O
tried	O
.	O
</s>
<s>
Optimized	O
Markov	O
chain	O
algorithms	O
which	O
use	O
local	O
searching	O
heuristic	B-Algorithm
sub-algorithms	O
can	O
find	O
a	O
route	O
extremely	O
close	O
to	O
the	O
optimal	O
route	O
for	O
700	O
to	O
800	O
cities	O
.	O
</s>
<s>
TSP	O
is	O
a	O
touchstone	O
for	O
many	O
general	O
heuristics	B-Algorithm
devised	O
for	O
combinatorial	O
optimization	O
such	O
as	O
genetic	B-Algorithm
algorithms	I-Algorithm
,	O
simulated	B-Algorithm
annealing	I-Algorithm
,	O
tabu	B-Algorithm
search	I-Algorithm
,	O
ant	B-Algorithm
colony	I-Algorithm
optimization	I-Algorithm
,	O
river	O
formation	O
dynamics	O
(	O
see	O
swarm	B-Architecture
intelligence	I-Architecture
)	O
and	O
the	O
cross	B-Algorithm
entropy	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Artificial	B-Application
intelligence	I-Application
researcher	O
Marco	O
Dorigo	O
described	O
in	O
1993	O
a	O
method	O
of	O
heuristically	O
generating	O
"	O
good	O
solutions	O
"	O
to	O
the	O
TSP	O
using	O
a	O
simulation	B-Algorithm
of	I-Algorithm
an	I-Algorithm
ant	I-Algorithm
colony	I-Algorithm
called	O
ACS	O
(	O
ant	O
colony	O
system	O
)	O
.	O
</s>
<s>
Each	O
ant	O
probabilistically	O
chooses	O
the	O
next	O
city	O
to	O
visit	O
based	O
on	O
a	O
heuristic	B-Algorithm
combining	O
the	O
distance	O
to	O
the	O
city	O
and	O
the	O
amount	O
of	O
virtual	O
pheromone	O
deposited	O
on	O
the	O
edge	O
to	O
the	O
city	O
.	O
</s>
<s>
For	O
points	O
in	O
the	O
Euclidean	O
plane	O
,	O
the	O
optimal	O
solution	O
to	O
the	O
travelling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
forms	O
a	O
simple	O
polygon	O
through	O
all	O
of	O
the	O
points	O
,	O
a	O
polygonalization	O
of	O
the	O
points	O
.	O
</s>
<s>
However	O
,	O
even	O
when	O
the	O
input	O
points	O
have	O
integer	O
coordinates	O
,	O
their	O
distances	O
generally	O
take	O
the	O
form	O
of	O
square	O
roots	O
,	O
and	O
the	O
length	O
of	O
a	O
tour	O
is	O
a	O
sum	O
of	O
radicals	O
,	O
making	O
it	O
difficult	O
to	O
perform	O
the	O
symbolic	B-Algorithm
computation	I-Algorithm
needed	O
to	O
perform	O
exact	O
comparisons	O
of	O
the	O
lengths	O
of	O
different	O
tours	O
.	O
</s>
<s>
time	O
;	O
this	O
is	O
called	O
a	O
polynomial-time	B-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
(	O
PTAS	O
)	O
.	O
</s>
<s>
In	O
practice	O
,	O
simpler	O
heuristics	B-Algorithm
with	O
weaker	O
guarantees	O
continue	O
to	O
be	O
used	O
.	O
</s>
<s>
The	O
case	O
where	O
the	O
distance	O
from	O
A	O
to	O
B	O
is	O
not	O
equal	O
to	O
the	O
distance	O
from	O
B	O
to	O
A	B-Application
is	I-Application
called	O
asymmetric	O
TSP	O
.	O
</s>
<s>
This	O
problem	O
is	O
known	O
as	O
the	O
analyst	O
's	O
travelling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
bottleneck	B-Algorithm
travelling	I-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
is	O
also	O
NP-hard	O
.	O
</s>
<s>
If	O
the	O
distance	O
measure	O
is	O
a	O
metric	O
(	O
and	O
thus	O
symmetric	O
)	O
,	O
the	O
problem	O
becomes	O
APX-complete	B-Algorithm
and	O
the	B-Algorithm
algorithm	I-Algorithm
of	I-Algorithm
Christofides	I-Algorithm
and	I-Algorithm
Serdyukov	I-Algorithm
approximates	O
it	O
within	O
1.5	O
.	O
</s>
<s>
If	O
the	O
distances	O
are	O
restricted	O
to	O
1	O
and	O
2	O
(	O
but	O
still	O
are	O
a	O
metric	O
)	O
the	O
approximation	B-Algorithm
ratio	I-Algorithm
becomes	O
8/7	O
.	O
</s>
<s>
The	O
apparent	O
ease	O
with	O
which	O
humans	O
accurately	O
generate	O
near-optimal	O
solutions	O
to	O
the	O
problem	O
has	O
led	O
researchers	O
to	O
hypothesize	O
that	O
humans	O
use	O
one	O
or	O
more	O
heuristics	B-Algorithm
,	O
with	O
the	O
two	O
most	O
popular	O
theories	O
arguably	O
being	O
the	O
convex-hull	O
hypothesis	O
and	O
the	O
crossing-avoidance	O
heuristic	B-Algorithm
.	O
</s>
<s>
A	O
2011	O
study	O
in	O
animal	O
cognition	O
titled	O
"	O
Let	O
the	O
Pigeon	O
Drive	O
the	O
Bus	O
,	O
"	O
named	O
after	O
the	O
children	O
's	O
book	O
Do	O
n't	O
Let	O
the	O
Pigeon	O
Drive	O
the	O
Bus	O
!,	O
examined	O
spatial	O
cognition	O
in	O
pigeons	O
by	O
studying	O
their	O
flight	O
patterns	O
between	O
multiple	O
feeders	O
in	O
a	O
laboratory	O
in	O
relation	O
to	O
the	O
travelling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
When	O
presented	O
with	O
a	O
spatial	O
configuration	O
of	O
food	O
sources	O
,	O
the	O
amoeboid	B-Operating_System
Physarum	O
polycephalum	O
adapts	O
its	O
morphology	O
to	O
create	O
an	O
efficient	O
path	O
between	O
the	O
food	O
sources	O
which	O
can	O
also	O
be	O
viewed	O
as	O
an	O
approximate	O
solution	O
to	O
TSP	O
.	O
</s>
<s>
Travelling	O
Salesman	O
,	O
by	O
director	O
Timothy	O
Lanzone	O
,	O
is	O
the	O
story	O
of	O
four	O
mathematicians	O
hired	O
by	O
the	O
U.S.	O
government	O
to	O
solve	O
the	O
most	O
elusive	O
problem	O
in	O
computer-science	B-General_Concept
history	O
:	O
P	O
vs.	O
NP	O
.	O
</s>
