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

TL;DR

The paper introduces Hybrid $k$-Clustering, a problem that interpolates between $k$-Center and $k$-Median by selecting $k$ balls of radius $r$ and minimizing the sum of distances outside these balls. It presents a randomized $(1+\varepsilon,1+\varepsilon)$-bicriteria approximation with radius $(1+\varepsilon)r$ and objective cost within $(1+\varepsilon)$ of ${\sf OPT}_r$, running in time $2^{(kd/\varepsilon)^{O(1)}}n^{O(1)}$, supported by preprocessing and a recursive sampling strategy that blends $k$-Center and $k$-Median techniques. The approach combines grid-based discretization, component-wise analysis, and a sampling-driven recursion inspired by Kumar, Sabharwal, and Sen to handle mixed cluster types, achieving strong theoretical guarantees under Euclidean assumptions. The work highlights theoretical optimality relative to known lower bounds and points to future directions in dimensionality reduction, coresets, and metric-space generalizations to broaden applicability and efficiency.

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 ε>0, produces a hybrid k-clustering with balls of radius (1+ε)r. This algorithm achieves a cost at most 1+εof the optimum, and it operates in time 2^{(kd/ε)^{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.

Figures (5)

• Figure 1: Two disks of radius $2$ cover all except four points that are colored red. The total sum of distances from these points to the yellow disks is $2(1+ \sqrt{8}-2)$.
• Figure 2: Left: $k$-Center clustering, a special case of Hybrid $k$-Clustering with $r = r^\star$. All points are covered by $k$ balls of radius $r^\star$ and ${\sf OPT}_{r^\star} = 0$. Right: $k$-Median clustering, a special case of Hybrid $k$-Clustering with $r = 0$, and every point contributes its distance to the closest center (some are shown as brown arrows). Middle: A general instance of Hybrid $k$-Clustering lies somewhere in between the two cases, where points outside radius-$r$ balls contribute the distance to the boundary (shown in blue).
• Figure 3: Example of two different types of clusters. In each figure, we show the cluster center in red, a ball of radius $r$ around the center in green, and a larger ball of radius $\mathcal{O}(r/\epsilon)$ in cyan with a dashed outline. Left: A $1$-center-like cluster. Note that a large chunk of points lies within the radius $\mathcal{O}(r/\varepsilon)$ ball around the center. Right: A $1$-median-like cluster. Note that most of the points lie outside the $\mathcal{O}(r/\epsilon)$ radius ball around ${\color{red} c}$, and for any such point, e.g., $p$ that is outside the $\mathcal{O}(r/\varepsilon)$ radius ball, $\mathsf{dist}_r({\color{blue} p}, {\color{red} c}) \approx \mathsf{dist}({\color{blue} p}, {\color{red} c})$.
• Figure 4: Illustration for Case 1. Centers in $F'$ are shown as red squares and unseen centers of $F \setminus F'$ are shown as purple crosses. $c$ is the closest center to $F'$ and $\mathsf{dist}({\color{Plum}c},{\color{red} c'}) \le 16r$. Then, a nearby center $\tilde{c}'$ can be found using a $\delta r$ grid.
• Figure 5: Illustration for Case 2. Centers of ${\color{red} F'}$ are shown as red squares and unseen centers of ${\color{Plum}F \setminus F'}$ are shown as purple crosses. Balls of radius $q^\star$ around $F'$ are shown in dashed orange. $P'$ are the points lying outside these balls. Among the points of $P'$, $D$ is the set of points belonging to clusters around $F'$, and shown as green-orange filled dots. Finally, the cluster around ${\color{Plum} c}$ is the largest unseen cluster (marked in dashed blue shape), $L$. We analyze different cases depending on the relative sizes of $L$ and $D$.

Theorems & Definitions (14)

• Proposition 1
• Theorem 2
• Lemma 4
• Lemma 5
• Lemma 6
• Claim 7
• Claim 9
• Claim 10
• Claim 11
• Claim 12