A New Space-Filling Curve Based Method for the Traveling Salesman Problems

The Traveling Salesman Problem (TSP) is one of the most intensively studied problems in optimization and it is used as a benchmark for many optimization methods. Given a list of n cities and their pairwise distances, the TSP aims to find a shortest possible tour that visits each city exactly once. As known, there are several applications for the TSPs, including mail/product delivery, production sequencing, planning, logistics, and the manufacture of microchips etc. In addition, some distribution problem, vehicle routing problem and scheduling problem etc can be reduced into a TSP. Moreover, the TSP can be applied in the DNA sequencing. The TSP is an NP-hard problem and several heuristic approaches have been proposed for solving it approximately. As known, the space-filling curve (SFC) method is a very special heuristic approach for solving the TSP. The SFC that can transform a point of two-dimensional space in [0,1]x[0,1] into a point of one-dimensional line in [0,1] was firstly proposed by Peano in 1890. In 1988, based upon the square type SFC, Bartholdi and Platzman developed a heuristic for solving the TSPs and applied it for developing the tour of meal delivery. There are many different basic types for the recursively transformation of SFCs. In this paper, we intend to propose a new type of SFC based method for solving the TSP. Numerical results of one hundred random problems and fourteen benchmark problems show that the new SFC based method performs better and faster than the typical square SFC based method when few CPU time is allowed.