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

Scope and Purpose { One of the main characteristics of a neighbourhood structure imposed on the solution set of a combinatorial optimization problem is the diameter of the corresponding neighbourhood graph. The diameter reeects the 'closeness' of one solution to another. We study the diameter of the neighbourhood graph of some exponential size neighbourhood structure for the TSP and show that the diameter of the graph is surprisingly small: at most four. This demonstrates a high potential of some exponential size neighbourhoods. Abstract { 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 T 1 and T 5 , there exist tours T 2 ; T 3 and T 4 such that T i is in the neighbourhood of T i?1 for all i = 2; 3; 4; 5: In contrast, for pyramidal neighbourhoods considered by J. Carlier and P. Villon (1990), one needs up to (log n) intermediate tours to 'move' from a tour to another one.