Subtour elimination constraints (${\sum }_{i,j\in S,i\ne j}{x}_{ij}\le |S|-1$)

Binary arc variables with one outgoing and one incoming arc at each vertex describe a cycle cover of a directed graph. For every nonempty proper vertex subset $S$, the displayed inequality forbids a directed cycle confined to $S$. Together these inequalities make the cycle cover one Hamiltonian cycle. They can be generated as violated cuts or used in branch and bound, rather than all being stored initially.