Algorithmic strategies for a fast exploration of the TSP 4 -OPT neighborhood

We describe an effective algorithm for exploring the 4 -OPT neighborhood for the Traveling Salesman Problem. 4 -OPT moves change a tour into another by replacing four of its edges. The best move can be found by a Θ (n4) algorithm by complete enumeration, but a Θ (n3) dynamic programming algorithm exists in the literature. Furthermore a Θ (n2) algorithm also exists for a particular subset of symmetric 4 -OPT moves. In this work we describe a new procedure which behaves, on average, slightly worse than a quadratic algorithm over all moves (estimated at O(n2.5)) and like a quadratic algorithm on the symmetric moves. Computational results are reported which show the effectiveness of our strategy compared to other algorithms for finding the best 4 -OPT move, and discuss the strength of the 4 -OPT neighborhood compared to 2- and 3 -OPT.