The Traveling Salesman Problem

Abstract

The chapter presents a traveling salesman problem, its network properties, convex quadratic formulation, and the solution. In this chapter, it is shown that adding or subtracting a constant to all arcs with special features in a traveling salesman problem (TSP) network model does not change an optimal solution of the TSP. It is also shown that adding or subtracting a constant to all arcs emanating from the same node in a TSP network does not change the TSP optimal solution. In addition, a minimal spanning tree is used to detect sub-tours, and then sub-tour elimination constraints are generated. A convex quadratic program is constructed from the formulated linear integer model of the TSP network. Interior point algorithms are then applied to solve the TSP in polynomial time.