Traveling Salesman problems

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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