Original Traveling Salesman Problem Research Articles

Homotopic Convex Transformation: A New Landscape Smoothing Method for the Traveling Salesman Problem.

This article proposes a novel landscape smoothing method for the symmetric traveling salesman problem (TSP). We first define the homotopic convex (HC) transformation of a TSP as a convex combination of a well-constructed simple TSP and the original TSP. The simple TSP, called the convex-hull TSP, is constructed by transforming a known local or global optimum. We observe that controlled by the coefficient of the convex combination, with local or global optimum: 1) the landscape of the HC transformed TSP is smoothed in terms that its number of local optima is reduced compared to the original TSP and 2) the fitness distance correlation of the HC transformed TSP is increased. Furthermore, we observe that the smoothing effect of the HC transformation depends highly on the quality of the used optimum. A high-quality optimum leads to a better smoothing effect than a low-quality optimum. We then propose an iterative algorithmic framework in which the proposed HC transformation is combined within a heuristic TSP solver. It works as an escaping scheme from local optima aiming to improve the global searchability of the combined heuristic. Case studies using the 3-Opt and the Lin-Kernighan local search as the heuristic solver show that the resultant algorithms significantly outperform their counterparts and two other smoothing-based TSP heuristic solvers on most of the test instances with up to 20 000 cities.

Hierarchical Solution of the Traveling Salesman Problem with Random Dyadic Tilings

We propose a hierarchical heuristic approach for solving the Traveling Salesman Problem (TSP) in the unit square. The points are partitioned with a random dyadic tiling and clusters are formed by the points located in the same tile. Each cluster is represented by its geometrical barycenter and a “coarse” TSP solution is calculated for these barycenters. Midpoints are placed at the middle of each edge in the coarse solution. Near-optimal (or optimal) minimum tours are computed for each cluster. The tours are concatenated using the midpoints yielding a solution for the original TSP. The method is tested on random TSPs (independent, identically distributed points in the unit square) up to 10,000 points as well as on a popular benchmark problem (att532 — coordinates of 532 American cities). Our solutions are 8–13% longer than the optimal ones. We also present an optimization algorithm for the partitioning to improve our solutions. This algorithm further reduces the solution errors (by several percent using 1000 iteration steps). The numerical experiments demonstrate the viability of the approach.

Traveling Salesman Problem with Clustering

In the original traveling salesman problem, the traveling salesman has the task to find the shortest closed tour through a proposed set of nodes, touching each node exactly once and returning to the initial node at the end. For the sake of the tour length to be minimized, nodes close to each other might not be visited one after the other but separated in the tour. However, for some practical applications, it is useful to group nodes to clusters, such that all nodes of a cluster are visited contiguously. Here we present an approach which leads to an automatic clustering with a clustering parameter governing the sizes of the clusters.

Triangle inequality and symmetry in connection with the assignment and the traveling salesman problem

Triangle inequality and symmetry play an important role in the treatment of the traveling salesman problem (TSP). Examplarily it is shown how to transform the distance matrix in order to use a given heuristic algorithm; furthermore related performance bounds will be given and studied. The corresponding transformation originates in the dual assignment problem. Finally results concerning the exact case are mentioned as reducing the dimension of the original TSP and as well as the restricted use of non-optimal edges. The paper represents a survey of the above mentioned topic.

Zu verschärfungen der christofibes-schranke fü den wert einer optimalen tour des enndrelseproblems

A necessary and sufficient condition for a distance matrix which can be reduced to a matrix with fulfilled triangle inequality is presented. If the reduced matrix fulfills the triangle inequality then the dimension of the original traveling salesman problem can be at least halved. By using a method presented in this paper it is possible to prove efficiently whether the optimal reduced matrix fulfills the triangle inequality. The number of operations is O(n 3).