  We prove that all spaces of finite Assouad-Nagata dimension admit a good
order for Travelling Salesman Problem, and provide sufficient conditions under
which the converse is true. We formulate a conjectural characterisation of
spaces of finite $AN$-dimension, which would yield a gap statement for the
efficiency of orders on metric spaces. Under assumption of doubling, we prove a
stronger gap phenomenon about all orders on a given metric space.
