Traveling Salesman Problem Study Guide for ES205 Yu-Chi Ho Jonathan T. Lee Xiaocang Lin Sep. 1, 2000 Version: 3.1 Date: 2003-8-31

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

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

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

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

References: Chvátal, V., The Traveling Salesman Problem, homepage, http://www.cs.rutgers.edu/~chvatal/tsp.html Applegate, D., R. Bixby, V. Chvátal and W. Cook, Traveling Salesman Problem, homepage, http://www.keck.caam.rice.edu/tsp/index.html