DYNAMIC-PROGRAMMING AND THE GRAPHICAL TRAVELING SALESMAN PROBLEM

In this paper, we study cases of polynomial resolution of the Traveling Salesman problem. First, we define a generalization of the Traveling Salesman problem where the cost of an edge depends on the number of times this edge is visited in a tour; here, an edge may be visited 0, 1, or 2 times. By this generalization, we can find polynomial algorithms to solve the Traveling Salesman problem on certain classes of graphs that are built from basic graphs by operations called r-sum, with r fixed. The underlying ideas of these algorithms are similar to dynamic programming techniques. Our classes contain most of the known instances for which the graphical and classical salesman problems are polynomial.