Minimum Enclosing Parallelogram with Outliers

We study the problem of minimum enclosing parallelogram with outliers, which asks to find, for a given set of $n$ planar points, a parallelogram with minimum area that encloses at least $(n-t)$ points, where the uncovered points are regarded as outliers. We present an exact algorithm with $O(k^{2}t^{4} + n^2\log n)$ runtime and $O(kt^{2})$ space, assuming that no three points lie on the same line. Here $k=k(n,t)$ denotes the number of points on the first $(t+1)$ convex layers. We further propose an sampling algorithm with runtime $O(n+\mbox{poly}(\log{n}, t, 1/\epsilon))$, which with high probability finds a parallelogram covering at least $(1-\epsilon)(n-t)$ and at most $(n-t+1)$ points with at most the exact optimal area.