Exponential neighbourhood local search for the traveling salesman problem

Scope and Purpose { While exact algorithms can only be used for solving small or quite moderate instances of the traveling salesman problem (TSP), local search remains the main practical tool for nding near optimal solutions for large scale instances. Exponential neighbourhood local search (ENLS) is a relatively new direction in local search for the TSP. In ENLS, one nds the best among very large, exponential, number of tours. Computational experiments reported by several researches demonstrate a very high potential of ENLS. In the present paper, we analyze theoretical properties of some exponential neighbourhoods. Abstract { We analyze an approach to the TSP, introduced by Punnen (1996), which is a generalization of approaches by Sarvanov and Doroshko (1981) and Gutin (1984). We show that Punnen's approach allows one to nd the best among (exp(p n=2)(n=2)!=n 1=4) tours in the TSP with n cities (n is even) in O(n 3) time. We describe an O(n 1+)-time algorithm (for any 2 (0; 2]) that constructs the best among 2 (nlog n) tours. This algorithm provides low complexity solutions to a problem by Burkard, Deineko and Woeginger (1996) and may be quite useful for large scale instances of the TSP. We also show that for every positive integer r there exists an O(r 5 n)-time algorithm that nds the best among (r n) tours. This improves a result of Balas and Simonetti (1996) who showed that the best among (r n) tours can be obtained in time O(r 2 2 r n).