The optimum assignments and a new heuristic approach for the traveling salesman problem

Abstract

In this paper the equivalent problem approach to the n × n optimum assignment problem is exploited for providing a heuristic solution to the traveling salesman problem. It is shown that the simplified problem of finding an optimum tour using the n − k arcs of the cycles of an n × n optimum assignment and k other arcs is NP-hard. Next by using the “nearest neighbor rule” for zero cost cycles, an O( n 2) algorithm is presented for obtaining a suboptimal solution to the simplified problem. Using the notion of a path switching operation that always results in a new tour having a lower cost than the original tour, an algorithm for a systematic refinement of the suboptimal tour is given. The algorithm presented in this paper efficiently solves the problem for k = 2,3 for any n. Examples illustrating the algorithm are given, and the time complexities as well as error bounds have been studied. Further work needed in the area is indicated.