One-Sided Monge TSP Is NP-Hard

The Travelling Salesman Problem (TSP) is a classical NPhard optimisation problem. There exist, however, special cases of the TSP that can be solved in polynomial time. Many of the well-known TSP special cases have been characterized by imposing special four-point conditions on the underlying distance matrix. Probably the most famous of these special cases is the TSP on a Monge matrix, which is known to be polynomially solvable (as are some other generally NP-hard problems restricted to this class of matrices). By relaxing the four-point conditions corresponding to Monge matrices in different ways, one can define other interesting special cases of the TSP, some of which turn out to be polynomially solvable, and some NP-hard. However, the complexity status of one such relaxation, which we call one-sided Monge TSP (also known as the TSP on a relaxed Supnick matrix), has remained unresolved. In this note, we show that this version of the TSP problem is NP-hard. This completes the full classification of all possible four-point conditions for symmetric TSP.