Computing Instance-Optimal Kernels in Two Dimensions

Let P be a set of n points in ℝ^2 . 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 weak ε -kernel of P if its directional width approximates that of P in every direction. Let _ε(P) (resp. ^_ε(P) ) denote the minimum-size of an ε -kernel (resp. weak ε -kernel) of P . We present an O(n_ε(P)log n) -time algorithm for computing an ε -kernel of P of size _ε(P) , and an O(n^2log n) -time algorithm for computing a weak ε -kernel of P of size ^_ε(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.