An exact (O(n^2)) solution to the Travelling Salesman Problem

This paper proposes an exact (O(n^2)) polynomial-time method of solving the Travelling Salesman Problem (TSP) and building the shortest possible route iteratively. First, building the route for a small number of cities closest to the cities’ centroid or their location’s geometric center and then expanding the route further to more distanced cities until all the cities are joined to the route. This method introduces a measure called Excess Path (EP) that is used to determine how and where a new city must be joined to already existing shortest possible route so the new expanded route may also be considered the shortest. This paper also compares the number of computations needed using this method to the number of computations needed using Brute-force search method and shows this method’s big advantage over the Brute-force search.