Computational Optimization and Applications, 32, 231–257, 2005 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands. Expanding Neighborhood GRASP for the Traveling Salesman Problem YANNIS MARINAKIS marinakis@ergasya.tuc.gr ATHANASIOS MIGDALAS sakis@verenike.ergasya.tuc.gr Decision Support Systems Laboratory, Department of Production Engineering and Management, Technical Uni- versity of Crete, 73100 Chania, Greece PANOS M. PARDALOS pardalos@cao.ise.ufl.edu Department of Industrial and Systems Engineering, University of Florida, USA Received June 17, 2003; Revised May 28, 2004; Accepted October 11, 2004 Abstract. In this paper, we present the application of a modified version of the well known Greedy Randomized Adaptive Search Procedure (GRASP) to the TSP. The proposed GRASP algorithm has two phases: In the first phase the algorithm finds an initial solution of the problem and in the second phase a local search procedure is utilized for the improvement of the initial solution. The local search procedure employs two different local search strategies based on 2-opt and 3-opt methods. The algorithm was tested on numerous benchmark problems from TSPLIB. The results were very satisfactory and for the majority of the instances the results were equal to the best known solution. The algorithm is also compared to the algorithms presented and tested in the DIMACS Implementation Challenge that was organized by David Johnson [18]. Keywords: Traveling Salesman Problem, Greedy Randomized Adaptive Search Procedure, local search, Meta-Heuristics 1. Introduction Consider a salesman who has to visit n cities. The Traveling Salesman Problem (TSP) asks for the shortest tour through all the cities such that no city is visited twice and the salesman returns at the end of the tour back to the starting city. We speak of a symmetric TSP, if for all pairs i , j the distance c ij is equal to the distance c ji . Otherwise, we speak of the asymetric traveling salesman problem. If the cities can be represented as points in the plain such that c ij is the Euclidean distance between point i and point j , then the corresponding TSP is called the Euclidean TSP. Euclidean TSP obeys in particular the triangle inequality c ij ≤ c ik + c kj for all i , j , k . The Traveling Salesman Problem (TSP) is one of the most famous hard combinatorial optimization problems. Since 1950s many algorithms have been proposed, developed and tested for the solution of the problem. The TSP belongs to the class of NP-hard optimization problems [19]. This means that no polynomial time algorithm is known for its solution. Algorithms for solving the TSP may be divided into two classes, exact algorithms and heuristic algorithms.