Hybrid k-Clustering: Blending k-Median and k-Center

TL;DR

This paper introduces a new hybrid clustering model combining k-center and k-median clustering, along with an approximation algorithm that balances coverage and cost in high-dimensional spaces.

Contribution

It presents the first bicriteria approximation algorithm for Hybrid k-Clustering, achieving near-optimal cost with a controlled increase in radius.

Findings

Achieves a (1+ε)-approximate cost for hybrid clustering.

Operates efficiently with a runtime polynomial in input size.

Provides the best possible approximation considering existing lower bounds.

Abstract

We propose a novel clustering model encompassing two well-known clustering models: k-center clustering and k-median clustering. In the Hybrid k-Clusetring problem, given a set P of points in R^d, an integer k, and a non-negative real r, our objective is to position k closed balls of radius r to minimize the sum of distances from points not covered by the balls to their closest balls. Equivalently, we seek an optimal L_1-fitting of a union of k balls of radius r to a set of points in the Euclidean space. When r=0, this corresponds to k-median; when the minimum sum is zero, indicating complete coverage of all points, it is k-center. Our primary result is a bicriteria approximation algorithm that, for a given \epsilon>0, produces a hybrid k-clustering with balls of radius (1+\epsilon)r. This algorithm achieves a cost at most 1+\epsilon of the optimum, and it operates in time 2^{(kd/\epsilon)^{O(1)}} n^{O(1)}. Notably, considering the established lower bounds on k-center and k-median, our bicriteria approximation stands as the best possible result for Hybrid k-Clusetring.