Journal of Heuristics, 5, 47–51 (1999) c  1999 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. A Note on Characterizing the k-OPT Neighborhood Via Group Theory BRUCE W. COLLETTI bcolletti@compuserve.com Graduate Program in Operations Research and Industrial Engineering, Department of Mechanical Engineering, ETC 5.110, The University of Texas at Austin, Austin TX 78712 J. WESLEY BARNES wbarnes@mail.utexas.edu Cullen Trust for Higher Education Endowed Professor in Engineering; Graduate Program in Operations Research and Industrial Engineering, Department of Mechanical Engineering, ETC 5.128D, The University of Texas at Austin, Austin TX 78712 STEFTCHO DOKOV stefcho@blacksea.me.utexas.edu Graduate Program in Operations Research and Industrial Engineering, Department of Mechanical Engineering, ETC 5.110, The University of Texas at Austin, Austin TX 78712 Abstract. Group theory can be used to model and synthesize the neighborhood of Traveling Salesman tours reachable through k-OPT exchanges. A primary concept is that a dihedral group action partitions the sets of cut arcs so that k-OPT exchanges of orbital elements are conjugate. Also presented is a method to produce all k-OPT exchanges for a given set of cut arcs. Keywords: tabu search, group theory, k-OPT , Traveling Salesman In this paper, we consider group theory perspective associated with a metaheuristic applied to permutation selection problems in combinatorial optimization. Specifically, we show how group theory can succinctly represent the move neighborhood reachable by k-OPT move strategies (Lin and Kernighan, 1973) for an n-city Asymmetric Traveling Salesman Problem (n-ATSP). When solving an n-ATSP using a metaheuristic method such as tabu search or genetic algorithms, one can use many move strategies that transition from one solution tour to another, e.g., swap two cities, reverse subpaths, or change a city’s position. In the k-OPT strategy, k arcs, some or all of which may be adjacent, are cut from the current tour. k possibly different arcs are then used to connect the endpoints of the disconnected subpaths to form a new tour. In this paper, some or all of the cut arcs may be present in the new tour. Given k and n, this paper shows how group theory may be used to find the k-OPT patterns of a dense n-ATSP and the k-OPT exchanges for a specific set of cut arcs. Since the connection topology alone determines permissible tours, other parameters such as distances between city pairs are not considered here. 1. Group Theory Perspective The solution space of the dense n-ATSP are the n! permutations of the letters {1 ... n}. However, we can take advantage of useful algebraic structure by viewing this space as