Before revision:  An   \emph{  NP-hardness result for the case that D(P ,P') is the directed Hausdorff distance from P' to P and the vertices of P' can lie anywhere on P while respecting the order of edges along P. (3) For any \epsilon>0, an O^*(n^2\log n \log \log n)-time algorithm that computes P'whose vertices can lie anywhere in the space and whose Fr\'echet distance to P is at most (1+\epsilon)\delta with at most 2m+1 links, where m is the number of links in the optimal simplified curve and O^* hides polynomial factors of 1/\epsilon .
After revision:  An \emph{ NP-hardness for other variants. Through this systematic study we observe, for the first time, some surprising patterns, and suggest directions for future research in this important area .