Travelling salesman tour

A travelling salesman tour is a cyclic visiting order that visits each required vertex exactly once before returning to its start. In a complete graph, it is a Hamiltonian cycle. For an ordering $\pi$ of $n$ vertices and costs $c_{ij}$, its length is ${\sum }_{k=0}^{n-1}{c}_{\pi (k),\pi ((k+1)\mathrm{mod}n)}$.

In a symmetric problem, rotations and reversal represent the same undirected tour, giving $(n-1)!/2$ tours for $n\ge 3$. Thus four labeled locations have three distinct tours under this convention. Asymmetric costs generally distinguish a tour from its reversal. A closed walk allowed to revisit vertices is a different feasible object unless the problem explicitly permits that change.