Finding the best 3-opt move in subcubic time

Given a Traveling Salesman Problem solution, the best 3-OPT move requires us to remove three edges and replace them with three new ones so as to shorten the tour as much as possible. No worst-case algorithm better than the Θ(n3 ) enumeration of all triples is likely to exist for this problem, but algorithms with average case O(n3−ɛ ) are not ruled out. In this paper we describe a strategy for 3-OPT optimization which can find the best move by looking only at a fraction of all possible moves. We extend our approach also to some other types of cubic moves, such as some special 6-OPT and 5-OPT moves. Empirical evidence shows that our algorithm runs in average subcubic time (upper bounded by O(n2.5 )) on a wide class of random graphs as well as Traveling Salesman Problem Library (TSPLIB) instances.