Traveling salesman problem,Traveling salesman salesman problem , Traveling problem

1. Traveling Salesman problems
I. History
The origins of the Traveling salesman problem are unclear. A handbook for Traveling salesmen
from 1832 mentions the problem and includes example tours through Germany and Switzerland, but
contains no mathematical treatment.
Mathematical problems related to the Traveling salesman problem were treated 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 studied 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, the problem became increasingly popular in scientific circles in Europe
and the USA. Notable contributions were made by George Dantzig, Delbert Ray Fulkerson and
Selmer 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 constructing 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 Hamiltonian 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, Padberg, 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 solutions. Gerhard Reinelt published the TSPLIB in 1991, a
William Rowan Hamilton

2. 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 instance 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.

3. II.Problem Introduction
2.1 Definition:
The Traveling 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 examples 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 city 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 make
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 TSP 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 the tour length).

4. 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 computations 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
Traveling 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 points 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 85,900 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
• Iterative improvement
 Pairwise exchange, or Lin–Kernighan heuristics
 k-opt heuristic
 V-opt heuristic
• Randomised improvement

5. 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 large-scale traveling-salesman problem.
Journal of the operations research society 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
approaching 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 distances as taken from an atlas.
Application 2
Source:
Crowder, H., Padberg, M.W., 1980. Solving large-scale symmetric travelling salesman problems to
optimality. Management science, 26(5), 495-509.
Description:
A number of related problems, such as, for example, 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.
The 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: Solution of a 120-city problem.
Mathematical programming study, 12, 61-77.
Description:
Having claimed that a good knowledge of the polytope Qn
T is of high vale for the solution of
traveling salesman problems, we are going to demonstrate this by solving a real-world symmetric
120-city problem using the facets found as cutting planes.

6. Application 4
Source:
Potvin, J.Y.,1992. The traveling salesman problem: a neural network perspective.
Description:
It is worth noting that problems with a few hundred vertices can now be routinely solved to
optimality. Also, instances involving more than 2,000 vertices have been addressed. For example,
the optimal solution to a symmetric problem with 2,392 vertices was identified 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 description 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 operations research, 27(9), 905-916.
Description:
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 TSPLIB 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 programming. 97(1-2), 91-
153.
Description:
In this paper we discuss an implementation of Dantzig et al.’s method 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 much of the work on the TSP. But we list three of TSP applications to 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 the Solution of Traveling Salesman Problems” by Applegate D.,
Bixby R., Chvatal V., and Cook W. list some of reporting the computer codes of running on

7. computer code.

8. IV. Reference
[1] Applegate, D., Bixby, R., Chvatal, V., and Cook, W., 1998. On the solution of travelling
salesman problems. Documenta mathematica, 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 mathematics topics in the secondary school curriculum.
[4] Dantzig, G., Fulkerson, R., Johnson, S. 1954. Solution of a large-scale traveling-salesman
problem. Journal of the operations research society of America, 2(4), 393-410.
[5] Gavish, B., Srikanth, K, 1986. An optimal solution method for large-scale multiple traveling
salesmen problems. Operations research, 34(5), 698-717.
[6] Johnson, D.S., McGeoch, L.A., 1995. The traveling salesman problem: a case study in local
optimization.
[7] Lee Po-wing, 2000. Integrated modern-heuristic and B/B approach for the classical traveling
salesman problem on a parallel computer.
[8] Lin, S., Kernighan, B.W., 1973. An effective heuristic algorithm for the traveling-salesman
problem. Operations research, 21(2), 498-516.
[9] Little, J.D.C., Murty, K.G., Sweeney, D.W., Karel, C. 1963. An algorithm for the traveling
salesman problem. Operations research, 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 system, 5(3), 299.
[11] Shapiro, J. F., 1989. Convergent duality for the traveling salesman problem. Operations
research center, 1-14.
[12] Traveling salesman problem.
http://en.wikipedia.org/wiki/Traveling_Salesman_Problem#Metric_TSP
[13] Traveling salesman problem. http://www.tsp.gatech.edu/index.html