A Greedy Reconstruction Heuristic for Solving the Minimum Travelling Salesman Tour Problem

Nearest neighbour approach for determination of the minimum travelling salesman tour is a well-documented heuristic for a quick identification of the salesman tour and its associated cost. This heuristic has been used on large sized networks for a quick approximate solution to the travelling salesman problem. For the last several years, an alternative approach to the minimum travelling salesman tour has been developed through the minimum spanning tree (MST). The MST was converted to an index restricted MST (IRMST) and that IRMST was used to find the minimum travelling salesman tour. These two approaches, i.e., the greedy heuristics and the index restricted minimum spanning tree seems to have stronger relations and it is this relationship, which has been explored in this paper. The nearest neighbour approach to find the travelling salesman tour has been modified by incorporating the index balancing theorem. A further modification to this heuristic has been suggested to identify the minimum travelling salesman tour. Many small size problems have been attempted using the heuristic discussed in this paper, and it resulted in an optimal solution but an analytical proof for optimality remains a challenge. This approach further strengthens a natural question about the ‘NP Hard’ category associated with the minimum travelling salesman tour problem.