What TSP variant doesn't return to start point?
You can reduce to a normal TSP variant by adding a dummy city that is distance $0$ away from each of the existing cities. (See also this answer on StackOverflow.)

Edit: it seems that my suggested modification is not quite appropriate: as I understand it, your Hamiltonian path has a fixed starting point but no fixed end point. One way to solve this is to add two dummy cities $v$ and $w$ such that:


$v$ is only connected to the starting point and to $w$,
$w$ is connected to everything.