I am trying the understand the following statement from the book of Grotschel, Lovasz and Schrijver:

They define an optimization problem whose solution is a nonnegative real vector. If we add a constraint that all elements of the vector must be integers, then the problem is equivalent to the minimum spanning tree problem, and therefore is solvable in polynomial time. However, if the constraint says that all elements of the vector must be half-integers (that is, either an integer or an integer plus $1/2$), the problem becomes NP-complete, as it includes the symmetric travelling salesman problem.

I do not see the connection: why does the problem with half-integer constraints contain the TSP?

I am trying the understand the following statement from the book of Grotschel, Lovasz and Schrijver:

Here, $\delta(W)$ is the set of edges incident to a set of vertices $W$.

They define an optimization problem whose solution is a nonnegative real vector. If we add a constraint that all elements of the vector must be integers, then the problem is equivalent to the minimum spanning tree problem, and therefore is solvable in polynomial time. However, if the constraint says that all elements of the vector must be half-integers (that is, either an integer or an integer plus $1/2$), the problem becomes NP-complete, as it includes the symmetric travelling salesman problem.

I do not see the connection: why does the problem with half-integer constraints contain the TSP?