Orbits, schemes and dynamic programming procedures for the TSP 4-OPT neighborhood

We discuss the way to group all 25 possible 4-OPT moves into 7 orbits of equivalent moves. We then describe two implementations, one for a $\mathrm{\Theta }(n^3)$ algorithm by de Berg’s et al. and one of a $\mathrm{\Theta }(n^2)$ algorithm by Glover, for finding the best 4-OPT move via dynamic programming.