1. Traveling Salesman Problem
Study Guide for ES205
Yu-Chi Ho
Jonathan T. Lee
Xiaocang Lin
Sep. 1, 2000
Version: 3.1
Date:

2. Problem Statement
Given a finite number of cities along with the distance/cost between each pair of them, find the shortest route to travel through all of them and to return to the starting city.
The picture shows a problem instance for 5 cities, along with a particular route through all cities. a b c d e N

3. Example The optimal route for 13,509 cities in the USA with
populations > 500 [Applegate et al. 1998].

4. Applications Circuit board designs Deliveries Others N
You want to minimize the distance between the various components on a circuit board so to maximize speed and minimize energy consumption.
In the case of deliveries, the fuel consumption would be minimized if we find the shortest path. N

5. Difficulty
For N cities, there are (N-1)! number of possibilities — exponential growth. a b c d e
There are N! possible number of combination. But N of them represents the same route, e.g., the route [a b c d e] is the same as [b c d e a]. Thus, there are only N!/N = (N-1)! possible routes. N

6. References:
Chvátal, V., The Traveling Salesman Problem, homepage,
Applegate, D., R. Bixby, V. Chvátal and W. Cook, Traveling Salesman Problem, homepage,