The Christofides Algorithm mentioned by educated foo guaranties a solution better than 1.5 times the optimum.

Nota bene: since you have a fixed start and end for an "open" hamiltonian path (i.e. not a circle like in the TSP) you'll need to adjust the algorithm accordingly. ¹

update

¹) after reading two SO discussions (search "minimal hamilton path") you just need to add a dummy point neighboring (only) the desired start- and end-point with distance 0 (!)

Like this any TSP algorithm will chose start and end as neighbors for a hamilton circle with the same accumulated length like a optimized hamilton path, you just need to ignore the two dummy edges at the end.