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.