The traveling salesman problem

Abstract

EditorialThe traveling salesman problem This Special Issue of Discrete Optimization is devoted to the traveling salesman problem (TSP), its generalizations and modifications. The TSP is a very well studied optimization problem and there have been numerous publications on the TSP and its modifications, including two edited books (the latter was published in 2002 with G. Gutin and A.P. Punnen as editors). The topic continues to attract the attention of researchers from various areas including mathematics and computer science. Generally, TSP publications are scattered among various journals in mathematics, computer science, operations research, artificial intelligence and other related areas. The aim of this Special Issue is to bring to the attention of the reader a selected sample of research papers on the topic. We briefly describe below various papers included in this collection.It is well known that an asymmetric TSP on n vertices could be formulated as a symmetric TSP on 2n vertices. However, the paper by E. Balas, R. Carr, M. Fischetti and N. Simonetti included in this issue is the first article to demonstrate that the transformation can be used to obtain new facets for the symmetric TS polytope from asymmetric TS polytope facets. investigate a new class of polynomial length formulations for the asymmetric TSP. They show that a relaxation of their formulation is tighter than the formulation based on the exponential number of Danzig-Fulkerson-Johnson subtour elimination constraints. They also provide the results of computational experiments showing the efficiency of the proposed formulations.V. Mak and N. Boland study a recently defined generalization of the asymmetric TSP in which all arcs are partitioned into two classes (ordinary and replenishment arcs) and every vertex has a positive weight. A tour is feasible if it uses either type of arc before the total weight of vertices exceeds a certain limit, in which case only a replenishment arc can be used. The authors show that two classes of inequalities, under certain conditions, are facet-defining.A. Grigoriev and J. van de Klundert introduce another generalization of the TSP in which every vertex can be visited several times. The authors investigate how an optimal tour and its value change when the number of visits of each vertex is increased by the same factor.M. Turkensteen, D. Gosh, B. Goldengorin and G. Sierksma discuss an iterative technique for patching within the context of branch-and-bound approaches for the asymmetric TSP. Although already relevant per se, the technique could also be beneficial in other areas such as metaheuristics for the asymmetric TSP.Usually neighborhoods used in local search heuristics are of small polynomial size. This limitation is due to the fact that the best tour in the neighborhood has to be found for the worst case. However, as far back as the 1980s some researchers introduced TSP neighborhoods of exponential size in which the best tour can be computed in polynomial time. The study of exponential neighborhoods for various...

Tsp Solving by Hybridization of GA and AS

“…The postman problem is modeled as traversing a given set of streets in a city, rather than visiting a set of specified locations. Moreover, the TSP plays an important role in general post problem, where the houses or streets are far away from each other [3].…”

“…The goal is to find the shortest tour that visits each city exactly once and then returns to the starting city. Many books and papers introduce the procedure of how to find the shortest tour usually called as the optimum solution in TSP search algorithms [3].…”

A Bi-Modal Routing Problem with Cyclical and One-Way Trips: Formulation and Heuristic Solution

“…For a detailed review of multi-objective problems, see Jozefowiez et al [2]. Excellent references for the Traveling Salesman Problem (TSP), the Vehicle Routing Problem (VRP), and arc routing problems are by Gutin [3], Applegate et al [4], Golden et al [5], Dror [6], and Braekers et al [7].…”

Computing Nonsimple Polygons of Minimum Perimeter

“…The well-established benchmark library TSPLIB [22] of TSP instances has become so widely accepted that it is used as a benchmark for a large variety of other optimization problems. See the books [17,14] for an overview of various aspects of the TSP and the books [3,7] for more details on exact optimization. From a geometric point of view, the TSP asks for the shortest polygonal chain through a given set of vertices in the plane; as a consequence of triangle inequality, the result is always a simple polygon of minimum perimeter.…”