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 \(\Theta (n^4)\) algorithm by complete enumeration, but a \(\Theta (n^3)\) dynamic programming algorithm exists in the literature. Furthermore a \(\Theta (n^2)\) 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 \(O(n^{2.5})\)) 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.