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Unformatted text preview: The Traveling Salesman Problem Professor Z. Max Shen IEOR151 Preliminary: Eulerian Tour Definition: A Eulerian Tour is a tour that traverses all edges of a graph exactly once The Seven Bridges of Konigsberg In Konigsberg, Germany, a river ran through the city such that in its center was an island, and after passing the island, the river broke into two parts. Seven bridges were built so that the people of the city could get from one part to another. A crude map of the center of Konigsberg might look like this: The people wondered whether or not one could walk around the city in a way that would involve crossing each bridge exactly once. The Seven Bridges of Konigsberg Definition : A vertex is called odd if it has an odd number of arcs leading to it, other wise it is called even. Theorem: If a network has more than two odd vertices, it does not have an Euler tour. Euler also proved the converse: Theorem: If a network has two or less odd vertices, it has at least one Euler tour. TSP Model Let G =( V , E ) be a complete undirected graph with vertices V ,  V = n , and the edges E and let d ij be the length of edge ( i , j ). The objective is to find a tour that visits each vertex exactly once and whose total length is as small as possible. Applications of TSP The TSP naturally arises as a subproblem in many transportation and logistics applications, for example the problem of arranging school bus routes to pick up the children in a school district (Vehicle Routine Problem)....
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This document was uploaded on 02/01/2012.
 Spring '09

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