  The goal in the min-\# curve simplification problem is to reduce the number
of the vertices of a polygonal curve without changing its shape significantly.
We study curve-restricted min-\# simplification of polygonal curves, in which
the vertices of the simplified curve can be placed on any point of the input
curve, provided that they respect the order along that curve. For local
directed Hausdorff distance from the input to the simplified curve in
$\mathbb{R}^2$, we present an approximation algorithm that computes a curve
whose number of links is at most twice the minimum possible.
