New Approximation Algorithms for Minimum Enclosing Convex Shapes

TL;DR

This paper introduces improved approximation algorithms for the Minimum Enclosing Ball and Convex Polytope problems in high-dimensional spaces, leveraging convex duality and non-smooth optimization techniques to outperform previous coreset-based methods.

Contribution

It presents novel approximation algorithms with better complexity bounds for MEB and MECP problems using convex duality and optimization frameworks, surpassing prior geometric approaches.

Findings

Achieves an $O(nd ilde{Q}/\sqrt{\epsilon})$ approximation for MEB.

Provides an $O(mnd ilde{Q}/\epsilon)$ approximation for MECP.

Outperforms existing coreset-based algorithms in efficiency.

Abstract

Given $n$ points in a $d$ dimensional Euclidean space, the Minimum Enclosing Ball (MEB) problem is to find the ball with the smallest radius which contains all $n$ points. We give a $O(nd\Qcal/\sqrt{\epsilon})$ approximation algorithm for producing an enclosing ball whose radius is at most $\epsilon$ away from the optimum (where $\Qcal$ is an upper bound on the norm of the points). This improves existing results using \emph{coresets}, which yield a $O(nd/\epsilon)$ greedy algorithm. Finding the Minimum Enclosing Convex Polytope (MECP) is a related problem wherein a convex polytope of a fixed shape is given and the aim is to find the smallest magnification of the polytope which encloses the given points. For this problem we present a $O(mnd\Qcal/\epsilon)$ approximation algorithm, where $m$ is the number of faces of the polytope. Our algorithms borrow heavily from convex duality and recently developed techniques in non-smooth optimization, and are in contrast with existing methods which rely on geometric arguments. In particular, we specialize the excessive gap framework of \citet{Nesterov05a} to obtain our results.