Algorithmic strategies for finding the best TSP 2-OPT move in average sub-quadratic time

We describe an exact algorithm for finding the best 2-OPT move which, experimentally, was observed to be much faster than the standard quadratic approach. To analyze its average-case complexity, we introduce a family of heuristic procedures and discuss their complexity when applied to a random tour in graphs whose edge costs are either uniform random numbers in [0, 1] or Euclidean distances between random points in the plane. We prove that, for any probability p: (i) there is a heuristic in the family which can find the best move with probability at least p in average-time O(n^3/2) for uniform instances and O(n) for Euclidean instances; (ii) the exact algorithm take lesser time then the above heuristic on all instances on which the heuristic finds the best move. During local search, while the tour becomes less and less random, the speed of our algorithm worsens until it becomes quadratic. We then discuss how to fine tune a successful hybrid approach, made of our algorithm in the beginning followed by the usual quadratic enumeration.