Computing Instance-Optimal Kernels in Two Dimensions

Let P be a set of n points in R². For a parameter ε∈(0,1), a subset C⊆P is an ε-kernel of P if the projection of the convex hull of C approximates that of P within (1-ε)-factor in every direction. The set C is a weakε-kernel of P if its directional width approximates that of P in every direction. Let k_ε(P) (resp. k_ε^w(P)) denote the minimum-size of an ε-kernel (resp. weak ε-kernel) of P. We present an O(nk_ε(P)logn)-time algorithm for computing an ε-kernel of P of size k_ε(P), and an O(n²logn)-time algorithm for computing a weak ε-kernel of P of size k_ε^w(P). We also present a fast algorithm for the Hausdorff variant of this problem. In addition, we introduce the notion of ε-core, a convex polygon lying inside, prove that it is a good approximation of the optimal ε-kernel, present an efficient algorithm for computing it, and use it to compute an ε-kernel of small size.