Travel
ing Salesman problem
s
I.
History
The origins of the
Traveling
salesman prob
lem are unclear. A handbook for
Traveling
salesmen
from 1832 mentions the problem and includes example tours through Germany and Switzerland, but
con
tains no mathematical treatment.
Mathematical problems related to the
Traveling
salesman problem
were tr
eated in the 1800s by the Irish mathematician
W. R. Hamilton
and by the British mathematician
Thomas Kirkman
. Hamilton’s
Icosian Game
was a recreational puzzle based on finding a
Hamiltonian cycle
.
The general form of the TSP appears to have been
first stu
died by mathematicians during the 1930s in Vienna and at
Harvard, notably by
Karl Menger
, who defines the problem,
considers the obvious brute

force algorithm, and observes the
non

optimality of the nearest
neighbor
heuristic
.
In the 1950s and 1960s, t
he problem became increasingly popular in scientific circles in
Europe and the USA. Notable contributions were made by
George Dantzig
,
Delbert Ray Fulkerson
and S
elmer M. Johnson at the
RAND Corporation
in
Santa Monica
, who expressed the problem as
an
integer linear program
and developed the
cutting planemethod
for its solution. With these new
methods they solved an instance with 49 cities to optimality by construc
ting a tour and proving that
no other tour could be shorter. In the following decades, the problem was studied by many
researchers from
mathematics, computer science,
chemistry, physics
, and other sciences.
Richard M. Karp
showed in 1972 that the
Hamil
tonian cycle
problem was
NP

complete
,
which implies the
NP

hardness
of TSP. This supplied a scientific explanation for the apparent
computational difficulty of finding optimal tours.
Great progress was made in the late 1970s and 1980, when Grötschel, P
adberg, Rinaldi and
other managed to exactly solve instances with up to 2392 cities, using cutting planes and
branch

and

bound
.
In the 1990s, Applegate, Bixby, Chvátal, and Cook developed the program
Concorde
that has
been used in many recent record so
lutions. Gerhard Reinelt published the TSPLIB in 1991, a
collection of benchmark instances of varying difficulty, which has been used by many research
groups for comparing results. In 2005, Cook and others computed an optimal tour through a
33,810

city ins
tance given by a microchip layout problem, currently the largest solved TSPLIB
instance. For many other instances with millions of cities, solutions can be found that are provably
within 1% of optimal tour.
William Rowan Hamilton
II.
Problem
Introduction
2.1
Definition:
The Traveli
ng Salesman Problem
(it is
also referred to as the TSP)
can be expressed in many
different ways. The first one introduced below is the original version and which will be the focus of
our exploration of the TSP. The examples following demonstrate other ways
to present this
problem.
We will show different example
s
about
the
Traveling
Salesman Problem.
Example 1:
A salesman is planning a business trip that takes him to certain cities in which he
has customers and then brings him back home to the city
in which he started. Between some of the
pairs of cities he has to visit, there is direct air service; between others there is not. Can he plan the
trip so that he (a) begins and ends in the same city while visiting every other city only once, and (b)
pays
the lowest price in airfare possible? The key to this is not just finding a solution, but an
optimal solution, the one with the lowest airfare.
Example 2:
A salesman spends his time visiting n cities (or nodes) cyclically. In one tour he
visits each c
ity just once, and finishes up where he started. In what order should he visit them to
minimize the distance travelled?
Example 3:
Automated Teller Machine Problem
A bank has many ATM machines. Each day, a courier goes from machine to machine to ma
ke
collections, gather computer information, and service the machines. In what order should the
machines be visited so that the courier's route is the shortest possible? This problem arises in
practice at many banks. One of the earliest banks to use the TS
P algorithm, in the early days of
ATMs, was the Shawmutt Bank in Boston.
2.2 Solving
the TSP
There are many difference methods to solving the Traveling Salesman Problem
.
We will list some
methods, and explain
the TSP

answering skills.
2.2.1
Computational
complexity
and Complexity of approximation
Computational complexity
:
The problem has been shown to be NP

hard, and the decision
problem version ("given the costs and a number
x
, decide whether there is a round

trip route
cheaper than
x
") is NP

complete
. The
bottleneck
Traveling
salesman problem
is also NP

hard. The
problem remains NP

hard even for the case when the cities are in the plane with Euclidean
distances, as well as in a number of other restrictive cases. Removing the condition of visiting each
city "only once" does not remove the NP

hardness, since it is easily seen that in the planar case
there is an optimal tour that visits each city only once (otherwise, by the triangle inequality, a
shortcut that skips a repeated visit would not increase th
e tour length).
Complexity of approximation
:
In the general case, finding a shortest
Traveling
salesman tour
is NPO

complete. If the distance measure is a
metric and symmetric
, the problem becomes
APX

complete
and Christofides’s algorithm approximates
it within 3/2.
2.2.2
Exact algorithms
An exact solution for 15,112 German towns from TSPLIB was found in 2001 using the
cutting

plane method proposed by
George Dantzig
,
Ray Fulkerson
, and
Selmer Johnson
in 1954,
based on
linear programming
. The compu
tations were performed on a network of 110 processors
located at Rice University and Princeton University (see the Princeton external link). The total
computation time was equivalent to 22.6
years on a single 500
MHz Alpha processor. In May 2004,
the
Trave
ling
salesman problem of visiting all 24,978 towns in Sweden was solved: a tour of length
approximately 72,500 kilometers was found and it was proven that no shorter tour exists.
In March 2005, the
Traveling
salesman problem of visiting all 33,810 point
s in a circuit board
was solved using
Concorde TSP Solver
: a tour of length 66,048,945 units was found and it was
proven that no shorter tour exists. The computation took approximately 15.7 CPU years (Cook et al.
2006). In April 2006 an instance with 85,90
0 points was solved using
Concorde TSP Solver
, taking
over 136 CPU years, see Applegate (2006).
2.2.3
Heuristic and approximation algorithms
Various heuristics and approximation algorithms, which quickly yield good solutions
,
have
been devised. Modern
methods can find solutions for extremely large problems (millions of cities)
within a reasonable time which are with a high probability just 2

3% away from the optimal
solution.
Several categories of heuristics are recognized.
Constructive heuristics
I
terative improvement
Pairwise exchange
, or
Lin
–
Kernighan
heuristics
k

opt heuristic
V

opt heuristic
Randomised improvement
III.
Applications
In
this chapter we will list some applications about TSP.
Application 1
Source:
Dantzig, G., Fulkerson, R., Johnson, S
., 1954.
Solution of a
l
arge

s
cale
t
raveling

s
alesman
p
roblem.
Journal of the
o
perations
r
esearch
s
ociety of America,
2(4),
393

410
Description:
We
do not claim that this note alters the situation very much; what we shall do
is outline a way of
approachin
g the problem that sometimes, at least, enables one to find an optimal path and prove it
so. In particular, it will be
shown that a certain arrangement of 49 cities, one m each of the 48 states
and Washington, D. C, is best, the du used representing road d
istances as
taken from an atlas.
Application
2
Source:
Crowder, H., Padberg, M.W., 1980. Solving
l
arge

s
cale
s
ymmetric
t
ravelling
s
alesman
p
roblems to
o
ptimality.
Management
s
cience
, 26(5), 495

509.
Description:
A number of related problems, such as, for e
xample, the multiple traveling salesman problem with
m
traveling salesman locations at a central depot and with fixed charge for their deployment, have
been shown elsewhere to fit the standard form of the TSP treated in this paper.
Th
e smallest problem
of this computational study has 48 cities and the large one has 318 cities,
i.e. the corresponding zero

one linear programming problems have been 1,128 and
50,403 zero

one
variable. The algorithmic procedure is a cutting

plane approach coupled with branch

and

bound.
Application
3
Source:
Grotschel
,
M.,
1980.
On the symmetric traveling salesman problem: S
olution of a 120

city problem.
Mathematical
p
rogramming
s
tudy
, 12, 61

77
.
Description:
Having
claimed that a good knowledge of the polytope Q
n
T
is of high
vale for the solution of
traveling salesman problem
s, we are going to demonstrate this by solving a real

world
symmetric
120

city problem using the facets foun
d as cutting planes.
Application
4
Source:
Potvin, J.Y.,1992.
The traveling salesman problem
: a
n
eural
n
etwork
p
erspective
.
Description:
It is worth noting that problems with a few hundred
vertices
can now
be
routinely solved to
optimality. Also,
instance
s involving
more than 2,000 vertices have been addressed. For
example,
the
optimal
solution to a
symm
etric
problem
with 2,392 vertices
was
i
dentified after two hours and
forty minutes of computation time
on
a powerful
vector
computer, the IBM
3090/600.
On
the
other
hand, a classical problem with 532 vertices took five and a half hours
on the
same
machine,
indicating that the size of the problem is
not
the only
determining
factor for computation time. We
refer
the
interested reader to for a complete descript
ion of the state of the
art with respect to exact
algorithms.
Application
5
Source:
Renaud
,
J.
,
Boctor
,
F.F.
,
Ouenniche
,
J.
, 2000.
A heuristic for the pickup and delivery traveling
salesman problem
.
Computers and
o
perations
r
esearch
,
27
(
9
)
, 905

916
.
Descri
ption:
As there are no benchmark problems available to evaluate the performance of the proposed
heuristic, we
generated a set of 108 problem instances divided into 3 subsets (called A, B,
and C) of 36 problems each.
These problems were derived from 36 TSP
LIB problems having
up to 441 vertices. All the 108 problems are
Euclidean and the distances between the vertices were
rounded to the nearest integer. These test problems
can be obtained from the authors upon request.
Application
6
Source:
Applegate
,
D.,
Bixby
,
R., Chvátal
,
V., Cook
,
W.,
2003.
Implementing the
Dantzig

Fulkerson

Johnson algorithm for large traveling salesman problems
.
Mathematical
p
rogramming
.
97(1

2), 91

153
.
Description:
In this paper we discuss an implementation of Dantzig et al.’s meth
od
that is suitable for TSP
instances having 1,000,000 or more cities. Our aim is to use the study of the TSP as
a step towards
understanding the applicability and limits of the general cutting

plane method in large

scale
applications
.
There is muc
h of the work on the TSP. But we list three of TSP applications t
o give the reader
a sample
. By the way, computer codes for the TSP have become increasingly more sophisticated
over the years.
In the paper of
“
On
t
he Solution of Traveling Salesman Problems
”
by Applegate D.,
Bixby R., Chvatal V., and Cook W
. list some of
report
ing
the computer codes of running on
computer code.
IV.
Reference
[1]
Applegate
,
D., Bixby
,
R., Chvatal
,
V., and Cook
,
W., 1998.
On the solution of
t
ravelling
s
alesman
p
roblems
.
Documenta
m
athematica
,
3,
645

656
.
[
2
]
Arora
,
S., 1998.
Polynomial time approximation schemes for Euclidean traveling salesman
and other geometric problems.
Journal of the ACM (JACM)
, 45(5), 753
–
782
.
[
3
]
Boyd
,
A
.
B
., 2002. Discrete
m
athematics
t
opics in the
s
econdary
s
chool
c
urriculum.
[
4
]
Dantzig
,
G., Fulkerson
,
R., Johnson
,
S. 1954.
Solution of a
l
arge

s
cale
t
raveling

s
alesman
p
roblem.
Journal of the
o
perations
r
esearch
s
ociety of America
, 2(4), 393

410
.
[
5
]
Gavish
,
B., Srikanth
,
K
, 1986
.
An
o
ptimal
s
olut
ion
m
ethod for
l
arge

s
cale
m
ultiple
t
raveling
s
alesmen
p
roblems.
Operations
r
esearch
, 34(5), 698

717.
[
6
] Johnson
,
D.S., McGeoch
,
L.A., 1995. The
t
raveling
s
alesman
p
roblem:
a
c
ase
s
tudy in
l
ocal
o
ptimization.
[
7
]
Lee
Po

wing,
2000.
Integrated modern

heu
ristic and B/B approach for the classical traveling
salesman problem on a parallel computer.
[
8
]
Lin
,
S., Kernighan
,
B.
W.
,
1973.
An
e
ffective
h
euristic
a
lgorithm for the
t
raveling

s
alesman
p
roblem.
Operations
r
esearch
, 21(2),
498

516
.
[
9
]
Little
,
J.D.
C.,
Murty
,
K.G., Sweeney
,
D.W., Karel
,
C. 1963.
An
a
lgorithm for the
t
raveling
s
alesman
p
roblem.
Operations
r
esearch
, 11(6), 972

989
.
[
10
]
Martin
,
O., Otto
,
S.
W., Felten
,
E.W., 1991.
Large

step Markov
chains for the TSP
incorporating local search heuristics
.
Complex
s
ystem
, 5(3), 299
.
[
1
1
]
Shapiro
,
J. F., 1989.
Convergent
d
uality for the
t
raveling
s
alesman
p
roblem.
Operations
r
esearch
c
enter
, 1

14
.
[
1
2
]
T
raveling salesman problem.
http://en.wikipedia.org/wiki/Traveling_Salesman_Problem#Metric_TSP
[1
3
]
T
raveling salesman problem.
http://www.tsp.gatech.edu/index.html
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