Before revision:  We aim to simplify P by }  another polygonal curve P' with minimum number of vertices satisfying D(P,P' ) \leq \delta. We obtain three main results for this problem: (1) An O(n^4)-time algorithm when D(P,P') is the Fr\'echet distance and vertices in P' are selected from a subsequence of vertices in P. (2) An   \emph{  NP-hardness result for the case that D(P ,P')
After revision:  We aim to simplify P by } another polygonal curve P' with minimum number of vertices such that P' is sufficiently similar to P. Quality guarantees of a simplification are usually given in a \emph{ NP-hardness result for the case that D(P ,P')