Given a path of 2d points and a maximum distance, find the minimum number of line segments needed to "connect" all points
Try polyline simplification.
For heuristics or other simplifications you might want to treat segments as resizeable, but for now objective is not that clear.
Here is demo of RDP.
Try it, merge your answer into question and then explain objective.

There is tradeoff between polyline complexity reduction and number of remaining points. Number of removed points depends on data and $\epsilon$.

If you want to find minimum, you have $N-1$ segments on $N$ points, and exact reduction depends on colinearity of points. Otherwise it is only approximation.