New exponential neighbourhood for polynomially solvable TSPs

Many of the well known solvable cases of the symmetric travelling salesman problem (STSP) have been characterized by describing special conditions for the underlying distance matrices. For all these special cases an optimal tour can either be given in advance and without regarding the precise numerical values of the data, or can efficiently be found in the set of so-called pyramidal tours. We introduce a new polynomially solvable case of the STSP where an optimal tour can be found in an exponential neighbourhood which is different from the set of pyramidal tours. Our new case is the first example of "multi-peak" optimization for polynomially solvable STSPs.