Discuss the Traveling Salesman Problem, its history, importance and complexity. Discuss practical applications (e.g. in Business) which rely on the TSP. Find out what you can about methods that have been used to approach this problem in industry or other applications.

## Expert Answer

**Traveling Salesman Problem:**

Traveling Salesman Problem or TSP implies that for a given a list of cities and the distances between each pair of cities, find the shortest possible route such that we visit each city exactly once and returns to the origin city.

**Example**: The following image is the solution of a travelling salesman problem where the lines represents the shortest possible paths that connects every dot in a loop.

Traveling Salesman Problem has lot of varaiants ,few of which are as under

- a.) As a graph problem:
- b.)The symmetric TSP
- c.)The asymmetric TSP
- d.)The TSP with precedences or time windows
- e.)The online TSP

**History**:Origin of TSP is not very definitive but it is believed that travelling salesman problem was mathematically formulated in the 18th century by W.R. Hamilton ( Irish mathematician) and by the Thomas Kirkman(British mathematician).It has many forms and the general form of the TSP is belived to be studied in 1930 by mathematicians in Vienna and at Harvard.The problem was studied to solve a routing problem of school bus.In later years this problem become famous so much that various institutes started offering prizes to solve it.

**Importance**: Traveling Salesman Problem has importance in various fields like Computers ,Mathematics,solving algorithms and the details are listed in applications of it.

**Complexity** : If n is the number of cities to be visited, the total possibilities of all routes while covering all cities is given as a set of possible solutions of the TSP and is given as (n-1)!/2.

*Practical Applications of TSP*

**Drilling of printed circuit boards:** In drilling of the printed circuit borads we have a condition that we have to drill holes so that we can lay a conducive materail to connect but the catch is that these hole’s diameters are not same so it will be time consuming to change the eqipment every time we visit a new hole with chnaged diameter ,so the solution is to first drill the holes of same diameter then the other and continue like it .Here we can implemt TSP by thinking of hole as cities and distance between them as paths.

Other applications are

- Overhauling gas turbine engines
- Computer wiring
- Xray Crystollography