Approximation algorithm for minimum-dilation paths

Construct an approximation algorithm for computing, on a given finite metric or Euclidean point set, a spanning path with minimum geometric dilation.

Background

The paper states that the minimum-dilation problem for paths is unresolved. Since paths have path-width one, an approximation algorithm for minimum-dilation paths would also address the corresponding special case of spanners with bounded path-width. The paper provides no such algorithm and explicitly identifies this as an open problem.