Network Reconstruction – A New Approach to the Traveling Salesman Problem and Complexity

Abstract

The paper presents a network reconstruction technique for the traveling salesman problem (TSP). A minimal spanning tree (MST) of the TSP is constructed and used to detect sub-tours. The TSP network is then reconstructed using dummy nodes as bridges in such a way that sub-tours are eliminated and there is no change in optimal TSP solution. A linear programming problem (LP) is formulated from the reconstructed TSP network and the coefficient matrix of the formulated LP is shown to be unimodular. Thus the formulated LP can be solved in polynomial time by interior point algorithms to obtain an optimal solution of the TSP. With the proposed approach there are no dangers of combinatorial explosion. Interior point algorithms can manage any size of the formulated LP.