Effective Neighborhood Structures for the Generalized Traveling Salesman Problem

We consider the generalized traveling salesman problem in which a graph with nodes partitioned into clusters is given. The goal is to identify a minimum cost round trip visiting exactly one node from each cluster. For solving difficult instances of this problem heuristically, we present a new Variable Neighborhood Search (VNS) approach that utilizes two complementary, large neighborhood structures. One of them is the already known generalized 2-opt neighborhood for which we propose a new incremental evaluation technique to speed up the search significantly. The second structure is based on node exchanges and the application of the chained Lin-Kernighan heuristic. A comparison with other recently published metaheuristics on TSPlib instances with geographical clustering indicates that our VNS, though requiring more time than two genetic algorithms, is able to find substantially better solutions.