Large Exponential Neighbourhoods for the Traveling Salesman Problem

An exponential neighbourhood for the traveling salesman problem (TSP) is a set of tours, which grows exponentially in the input size. An exponential neighbourhood is polynomial time searchable if we can nd the best among the exponential number of tours in polynomial time. Deineko and Woeginger asked if there exists polynomial time searchable neighbourhoods of size at least bnc!, for some > 1 2 , where n is the number of vertices in the TSP. In this paper we prove that such neighbourhoods exist for all < 1. In fact we give a neighbourhood of size at least c n! and for some constant c (depending on k), which can be searched in O(n 3) time. Using a slight variation of the above algorithm we can search neighbourhoods of size bnc! in time O(n 1+2), for any 0 < < 1. Deineko and Woeginger proved (indirectly) that if P 6 = NP then no algorithm for searching a neighbourhood of size bnc! can run faster then O(n 1+).