Geometric Approximation via Coresets

Summary

The paradigm of coresets has recently emerged as a powerful tool for efficiently approximating various extent measures of a point set P. Using this paradigm, one quickly computes a small subset Q of P, called a coreset, that approximates the original set P and and then solves the problem on Q using a relatively inefficient algorithm. The solution for Q is then translated to an approximate solution to the original point set P. This paper describes the ways in which this paradigm has been successfully applied to various optimization and extent measure problems.

1. Introduction

One of the classical techniques in developing approximation algorithms is the extraction of “small” amount of “most relevant” information from the given data, and performing the computation on this extracted data. Examples of the use of this technique in a geometric context include random sampling [Chazelle 2000; Mulmuley 1993], convex approximation [Dudley 1974; Bronshteyn and Ivanov 1976], surface simplification [Heckbert and Garland 1997], feature extraction and shape descriptors [Dryden and Mardia 1998; Costa and Cesar 2001]. For geometric problems where the input is a set of points, the question reduces to finding a small subset (a coreset) of the points, such that one can perform the desired computation on the coreset.

As a concrete example, consider the problem of computing the diameter of a point set. Here it is clear that, in the worst case, classical sampling techniques like ϵ-approximation and ϵ-net would fail to compute a subset of points that contain a good approximation to the diameter [Vapnik and Chervonenkis 1971; Haussler and Welzl 1987].