Minimum Enclosing Parallelogram with Outliers

We study the problem of minimum enclosing rectangle with outliers, which asks to find, for a given set of $n$ planar points, a rectangle with minimum area that encloses at least $(n-t)$ points. The uncovered points are regarded as outliers. We present an exact algorithm with $O(kt^3+kt^2\log{n} + n^2\log n)$ runtime, assuming that no three points lie on the same line. Here $k$ denotes the number of points on the first $(t+1)$ convex layers. We further propose a sampling algorithm with runtime $O(n+\mbox{poly}(\log{n}, t, 1/\epsilon))$, which with high probability finds a rectangle covering at least $(1-\epsilon)(n-t)$ points with at most the exact optimal area.