Small diameter neighbourhood graphs for the traveling salesman problem: at most four moves from tour to tour

A neighbourhood N(T) of a tour T (in the TSP with n cities) is polynomially searchable if the best among tours in N(T) can be found in time polynomial in n. Using Punnen’s neighbourhoods introduced in 1996, we construct polynomially searchable neighbourhoods of exponential size satisfying the following property: for any pair of tours T1 and T5, there exist tours View the MathML source and T4 such that Ti is in the neighbourhood of Ti-1 for all i=2, 3, 4, 5. In contrast, for pyramidal neighbourhoods considered by Carlier and Villon (1990, RAIRO 24, 245–253), one needs up to View the MathML source intermediate tours to ‘move’ from a tour to another one.