The standard TSP can be stated mathematically as follows:

2 Traveling Salesman Problem(TSP)

2.1 TSP

The ideal of the TSP is to find a tour of a given number of cities, visiting each city exactly once and returning to the starting city where the length of this tour is minimized. The traveling salesman first gained fame in a book written by German salesman BF Voigt in 1832 on how to be a successful traveling salesman ([12]). He mentions the TSP although not by that name, by suggesting that to cover as many locations as possible without visiting any location twice is the most important aspect of the scheduling of a tour.

The standard TSP can be stated mathematically as follows:

Given a weighted graph G = (V, E) where the weight c

on the edge between the nodes i and j is a non-negative value, find the tour of all nodes that has the minimum total cost.

Currently the only known method guaranteed to optimally solve the TSP of any size, is by enumerating each possible tour and searching for the tour with smallest cost. Each possible tour is a permutation of 1, 2,...,n, where n is the number of cities, so therefore the number of tours is n!. When n gets large, it becomes impossible to find the cost of every tour in polynomial time. The problem is that the search can be very complicated. One may not know where to look for a solution or where to start. There are many methods one can use for finding a suitable solution, but these methods do not necessarily provide the best solution. Some of these methods are hill climbing, tabu search, simulated annealing and the genetic algorithm. The solutions found by these methods are often considered as good solutions, because it is not often possible to prove what the optimum is.

2.2 Different Forms of TSP

Currently there are many forms of TSP different from the standard TSP:

The asymmetric TSP ([14]) is when the cost of traveling from city i to city j is not the same as the cost from city j to city i. This can be solved in the same way as the standard TSP.

The multisalesman problem ([11]) is the same as the standard TSP except that we have more than one salesman. When each city is visited exactly once and each salesman returns to the original city, we need to find the minimum of total cost of those tours.

The bottleneck TSP ([14]) is where we want to minimize the largest edge cost in the tour instead of the total cost. That is, we want to minimize the maximum cost the salesman travels between any two connected cities.

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The maximum scatter (max-min 1-neighbor) TSP ([1]) is closely related to the bottleneck TSP, and is motivated by applications in manufacturing and medical imaging. It is based on the objective of finding a path that is most ”scattered”. Specifically, the goal is to maximize the length of the shortest edge in the path.

The time dependent TSP ([11]) is the same as the standard TSP except we now have time periods. The cost c

is the cost of traveling from city i to city j in time period t.

Various forms of TSP are still increasing. Each of them has many implications. Homaifar states, ” One approach which would certainly find the optimal solution of any TSP is the application of exhaustive enumeration and evaluation.”([9]) Thus we can solve a large class of problems which is subject to the TSP can be solved.

2.3 TSP is an NP-complete problem

There are classes of problems exist algorithms that are efficient to solve them. We evaluate the difficulty of a problem by computational complexity. A decision problem is one that takes the form of a question with a Yes/No answer. For TSP

Given a tour and a positive number L, is there a tour of length ≤ L?

This differs from the optimization problem: find the shortest tour, or the evaluation problem: find the length of the shortest tour. Of course the three types of problem are closely related. We focus on decision problems. We say that a decision problem belongs to P if an algorithm that runs in polynomial time on an ordinary computer can solve it. We say that a decision problem belongs to NP if every YES instance has a certificate whose validity can be checked in polynomial time. ’NP’ stands for nondeterministic polynomial.

A decision problem belongs to NP if it can be solved in polynomial time on a ’nondeterministic computer’.

A nondeterministic computer consists of exponentially many ordinary computers working in parallel, any one of which can answer YES in polynomial time without consulting the others. NP stands for Nondeterministic Polynomial-time. NP was originally defined as the class of problems that can be solved by a nondeterministic algorithm in polynomial time.

Suppose X and Y are two decision problems. We say that X is polynomially reducible to Y (or X is reduced to Y) if there exists a function f , which is computable in polynomial time, such that X can be transformed to Y by f . That means X is not harder to solve than Y. A problem belongs to NP-hard if every problem in NP can be reduced to it. An NP problem is said to be NP-complete if it is also NP-hard. Thus all NP-complete problems can be reduced to one another and are as difficult as all problems in NP.

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P NP

NP-hard

NP-complete

Figure 2.1 The relation of P, NP, NP-hard and NP-complete

No polynomial-time algorithm has yet been discovered for any NP-complete problem; at the same time no NP-complete problem has been shown to have a super polynomial-time (for example exponential time) lower bound. If a polynomial-time algorithm is discovered for even one NP-complete problem, then all NP-complete problems will be solvable in polynomial-time. It is believed (but so far no proof is available) that NP-complete problems do not have polynomial-time algorithms and therefore are intractable.

To prove a problem Y to be NP-complete there are two steps:

1) Prove Y is an NP problem.

2) We use the known NP-complete X and reduce it to the new problem.

We can use the Hamilton cycle problem (HAM) that is already known to be NP-complete. HAM is to find a Hamilton cycle in a graph if one exists. It is clear that TSP is an NP problem. We prove that the Hamilton decision problem HAMD (deciding whether a given graph is Hamiltonian) is reduced to the TSP decision problem TSPD (deciding whether a weighted graph has a cycle with total weight ≤ L).

P roof : Let G be a graph with n nodes. Define the cost function of TSPD on G by:

c

= 1 if the i, j edge is in G; c

= 2 if the i, j edge is not in G.

Let L = n. Any Hamilton cycle in G transforms into a tour in G that cost exactly n. If there is no Hamilton cycle in G, any tour in G must use at least one edge of cost 2, thus the total cost must be at least n + 1.

2.4 Methods of Solving TSP

There are many methods of solving the TSP. As we mention, the TSP is NP-complete, so there is no known algorithm that will solve it in polynomial time. We will probably have to sacrifice optimally in order to get a

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good answer in a short time. Many algorithms have been tried for the TSP.

The greedy algorithm is a method of finding a feasible solution to the TSP ([14]). The algorithm creates a list of all edges in the graph and orders them from the smallest cost to the largest cost. It then chooses the edges with the smallest cost first, providing they do not create a cycle. The greedy algorithm gives feasible solutions however they are not always good.

The nearest neighborhood algorithm is similar to the greedy algorithm in its simple approach ([14]). We arbitrarily choose a starting city and then travel to the city closest to it that does not create a cycle. We continue to do this until all cities are in the tour. This algorithm also does not always give good solutions because often the last edge added to the tour can be quite large.

GA is a powerful algorithm for finding an optimal solution in a large searching space. TSP is a standard problem with the property. So GA is a suitable method for TSP. The first researcher to tackle the TSP with GA was Brady in 1985 ([2]). Many researchers have used GA to tackle the TSP, and get some achievements.

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