Computing Instance-Optimal Kernels in Two Dimensions

Abstract

Let \(P\) be a set of n points in \(\mathbb {R}^2\). For a parameter \(\varepsilon \in (0,1)\), a subset \(C\subseteq P\) is an \(\varepsilon \)-kernel of \(P\) if the projection of the convex hull of \(C\) approximates that of \(P\) within \((1-\varepsilon )\)-factor in every direction. The set \(C\) is a weak \(\varepsilon \)-kernel of \(P\) if its directional width approximates that of \(P\) in every direction. Let \(\textsf{k}_{\varepsilon }(P)\) (resp. \(\textsf{k}^{\textsf{w}}_{\varepsilon }(P)\)) denote the minimum-size of an \(\varepsilon \)-kernel (resp. weak \(\varepsilon \)-kernel) of \(P\). We present an \(O(n\textsf{k}_{\varepsilon }(P)\log n)\)-time algorithm for computing an \(\varepsilon \)-kernel of \(P\) of size \(\textsf{k}_{\varepsilon }(P)\), and an \(O(n^2\log n)\)-time algorithm for computing a weak \(\varepsilon \)-kernel of \(P\) of size \(\textsf{k}^{\textsf{w}}_{\varepsilon }(P)\). We also present a fast algorithm for the Hausdorff variant of this problem. In addition, we introduce the notion of \(\varepsilon \)-core, a convex polygon lying inside , prove that it is a good approximation of the optimal \(\varepsilon \)-kernel, present an efficient algorithm for computing it, and use it to compute an \(\varepsilon \)-kernel of small size.