Linear-Size Universal Discretization of Geometric Center-Based Problems in Fixed Dimensions

TL;DR

This paper demonstrates that for fixed-dimensional spaces, a small set of candidate centers can approximate solutions to various geometric optimization problems within any desired accuracy, enabling efficient discretization.

Contribution

It introduces a universal, linear-size discretization method for geometric center-based problems in fixed dimensions, applicable to various metrics and objective functions.

Findings

Existence of O(n) candidate centers for fixed dimensions

Approximate solutions can be found efficiently in almost linear time

Applicable to fixed doubling dimension metric spaces

Abstract

Many geometric optimization problems can be reduced to finding points in space (centers) minimizing an objective function which continuously depends on the distances from the centers to given input points. Examples are $k$-Means, Geometric $k$-Median/Center, Continuous Facility Location, $m$-Variance, etc. We prove that, for any fixed $\varepsilon>0$, every set of $n$ input points in fixed-dimensional space with the metric induced by any vector norm admits a set of $O(n)$ candidate centers which can be computed in almost linear time and which contains a $(1+\varepsilon)$-approximation of each point of space with respect to the distances to all the input points. It gives a universal approximation-preserving reduction of geometric center-based problems with arbitrary continuity-type objective functions to their discrete versions where the centers are selected from a fairly small set of candidates. The existence of such a linear-size set of candidates is also shown for any metric space of fixed doubling dimension.