Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dimension-Free Parameterized Approximation Schemes for Hybrid Clustering

Ameet Gadekar, Tanmay Inamdar

TL;DR

This work delivers a dimension-free, randomized bicriteria FPT-Approximation Scheme for Hybrid $k$-Clustering, removing the exponential dependence on dimension while maintaining $(1+\varepsilon,1+\varepsilon)$ guarantees. The method centers on a uniform witness-sampling framework driven by the bounded algorithmic $\varepsilon$-scatter dimension, augmented with inflated radii to overcome intrinsic barriers, and provides running-time bounds that are FPT in $k$ and $\varepsilon$ for Euclidean and several structured metric spaces. Extensions cover Hybrid $(k,z)$-clustering and Hybrid Norm clustering, with a practical coreset construction for doubling metrics, collectively answering open questions from prior work. These results broaden the applicability of dimension-free clustering approaches to diverse spaces and objective families, with implications for both theory and practice in scalable clustering.

Abstract

Hybrid $k$-Clustering is a model of clustering that generalizes two of the most widely studied clustering objectives: $k$-Center and $k$-Median. In this model, given a set of $n$ points $P$, the goal is to find $k$ centers such that the sum of the $r$-distances of each point to its nearest center is minimized. The $r$-distance between two points $p$ and $q$ is defined as $\max\{d(p, q)-r, 0\}$ -- this represents the distance of $p$ to the boundary of the $r$-radius ball around $q$ if $p$ is outside the ball, and $0$ otherwise. This problem was recently introduced by Fomin et al. [APPROX 2024], who designed a $(1+\varepsilon, 1+\varepsilon)$-bicrtieria approximation that runs in time $2^{(kd/\varepsilon)^{O(1)}} \cdot n^{O(1)}$ for inputs in $\mathbb{R}^d$; such a bicriteria solution uses balls of radius $(1+\varepsilon)r$ instead of $r$, and has a cost at most $1+\varepsilon$ times the cost of an optimal solution using balls of radius $r$. In this paper we significantly improve upon this result by designing an approximation algorithm with the same bicriteria guarantee, but with running time that is FPT only in $k$ and $\varepsilon$ -- crucially, removing the exponential dependence on the dimension $d$. This resolves an open question posed in their paper. Our results extend further in several directions. First, our approximation scheme works in a broader class of metric spaces, including doubling spaces, minor-free, and bounded treewidth metrics. Secondly, our techniques yield a similar bicriteria FPT-approximation schemes for other variants of Hybrid $k$-Clustering, e.g., when the objective features the sum of $z$-th power of the $r$-distances. Finally, we also design a coreset for Hybrid $k$-Clustering in doubling spaces, answering another open question from the work of Fomin et al.

Dimension-Free Parameterized Approximation Schemes for Hybrid Clustering

TL;DR

This work delivers a dimension-free, randomized bicriteria FPT-Approximation Scheme for Hybrid -Clustering, removing the exponential dependence on dimension while maintaining guarantees. The method centers on a uniform witness-sampling framework driven by the bounded algorithmic -scatter dimension, augmented with inflated radii to overcome intrinsic barriers, and provides running-time bounds that are FPT in and for Euclidean and several structured metric spaces. Extensions cover Hybrid -clustering and Hybrid Norm clustering, with a practical coreset construction for doubling metrics, collectively answering open questions from prior work. These results broaden the applicability of dimension-free clustering approaches to diverse spaces and objective families, with implications for both theory and practice in scalable clustering.

Abstract

Hybrid -Clustering is a model of clustering that generalizes two of the most widely studied clustering objectives: -Center and -Median. In this model, given a set of points , the goal is to find centers such that the sum of the -distances of each point to its nearest center is minimized. The -distance between two points and is defined as -- this represents the distance of to the boundary of the -radius ball around if is outside the ball, and otherwise. This problem was recently introduced by Fomin et al. [APPROX 2024], who designed a -bicrtieria approximation that runs in time for inputs in ; such a bicriteria solution uses balls of radius instead of , and has a cost at most times the cost of an optimal solution using balls of radius . In this paper we significantly improve upon this result by designing an approximation algorithm with the same bicriteria guarantee, but with running time that is FPT only in and -- crucially, removing the exponential dependence on the dimension . This resolves an open question posed in their paper. Our results extend further in several directions. First, our approximation scheme works in a broader class of metric spaces, including doubling spaces, minor-free, and bounded treewidth metrics. Secondly, our techniques yield a similar bicriteria FPT-approximation schemes for other variants of Hybrid -Clustering, e.g., when the objective features the sum of -th power of the -distances. Finally, we also design a coreset for Hybrid -Clustering in doubling spaces, answering another open question from the work of Fomin et al.
Paper Structure (13 sections, 14 theorems, 2 equations, 2 algorithms)

This paper contains 13 sections, 14 theorems, 2 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Technical Overview
  3. Background on Algorithmic Scatter Dimension
  4. Bicriteria FPT Approximation Scheme
  5. Algorithm
  6. Analysis
  7. Bounding runtime using Algorithmic Scatter Dimension
  8. Bounding success probability
  9. Case 1: $C_{W_{\sf near}} \ge \frac{\varepsilon}{100} C_P$.
  10. Case 2: $C_{W_{\sf near}} < \frac{\varepsilon}{100} C_P$.
  11. Extensions of the Bicriteria FPT-AS
  12. Coreset
  13. Conclusion

Key Result

Theorem 1

There exists a randomized algorithm, that, given an instance of Hybrid $k$-Clustering in $\mathbb{R}^d$ for any dimension $d$, runs in time $2^{O(*){k \log k \cdot (1/\varepsilon^5) \log^2 (1/\varepsilon)}} \cdot n^{\mathcal{O}(1)}$, and returns a $(1+\varepsilon, 1+\varepsilon)$-bicrtieria approxi

Theorems & Definitions (20)

  • Theorem 1: Bicriteria FPT-AS for Euclidean Spaces
  • Theorem 2: Informal version of \ref{['thm:main']}
  • Theorem 3: Coreset for
  • Definition 4: Ball Intersection Problem
  • Definition 5: Algorithmic $\varepsilon$-Scatter Dimension
  • Lemma 5
  • Lemma 7
  • Lemma 8
  • Theorem 9: Main Theorem
  • Lemma 10: Feasible upper bounds
  • ...and 10 more