Introducing convex layers to the Traveling Salesman Problem

In this paper, we will propose convex layers to the Traveling Salesman Problem (TSP). Firstly, we will focus on human performance on the TSP. Experimental data shows that untrained humans appear to have the ability to perform well in the TSP. On the other hand, experimental data also supports the hypothesis of convex hull i.e. human relies on convex hull to search for the optimal tour for the TSP. Secondly, from the paper published by Bonabeau, Dorigo and Theraulaz, social insect behavior would be able to help in some of the optimizing problems, especially the TSP. Thus, we propose convex layers to the TSP based on the argument that, by the analogy to the social insect behavior, untrained humans' cognition should be able to help in the TSP. Lastly, we will use Tour Improvement algorithms on convex layers to search for an optimal tour for a 13-cities problem to demonstrate the idea. 1.Introduction Despite the fact that the Traveling Salesman Problem (TSP) is very intuitive and easy to state, it is one of the most widely studied NP-hard combinatorial optimization problem[14]. The following are the statements of the problem. A salesman is required to visit each of n given cities once and only once, starting from any city and returning to the original city of departure. How should he travel in order to minimize the total travel distance?[9] The difficulty becomes obvious when one considers the number of possible tours by the method of brute force searching even for a relatively small number of cities n. For instance, for a problem with 20 cities (n=20) by brute force searching, it would be (20-1)!/2 tours, which is more than 10 18 tours! The TSP is a class of difficult problems whose time complexity is widely believed exponential. Any attempt to construct an algorithm for finding optimal solutions for the TSP in polynomial time (in contrast with exponential time) is also widely believed not possible. Up to date, there are two classes of algorithms in solving the TSP: Exact algorithms and Approximate (or heuristic) algorithms[9]. The main characteristics of Exact algorithms are guaranteed to find the optimal solution in a bounded number of steps but unfortunately also complex with codes and very demanding of computer power[9]. Examples of the most effective Exact algorithms are Cutting-Plane and Facet-Finding algorithms[9]. On the other hand, in contrast, the main characteristics of Approximate algorithms are no …