Approximation Preserving Coresets

2026-06-15

AI Summary

This work addresses the computational bottleneck of traditional clustering algorithms in large-scale settings and the gap between overly conservative theoretical guarantees on coreset size and the empirical effectiveness of small coresets. The authors propose “approximate-preserving coresets,” a new paradigm intermediate between strong and weak coresets, which only require approximate preservation of the clustering cost for high-quality (low-cost) solutions rather than for all solutions or solely the optimal one. By carefully characterizing the solution space structure and analyzing the clustering cost function, they establish the existence and lower bounds on the size of such coresets, revealing that even slight perturbations in the approximation factor can destroy their compactness. This study provides both theoretical grounding and a practical pathway for efficient large-scale clustering.

Abstract

Clustering in a big data setting is an intensively studied problem, with coresets emerging as one of the important paradigms in this line of work. Given a cost function $\text{cost}(P,S)$ mapping input points $P$ and a solution $S$ to an objective value, a coreset is a typically weighted sketch $Ω\subseteq P$ such that $\text{cost}(Ω,S)\approx \text{cost}(P,S)$. In practice, coreset sizes much smaller than those suggested by theoretical guarantees are often found to be sufficient. In this paper, we offer an explanation for this phenomenon. Smaller coreset sizes suffice if we only wish to preserve the costs of \emph{good} solutions, i.e., solutions with low cost. We define and devise \emph{approximation-preserving coresets}, which provide a weaker guarantee than strong coresets, which apply to all solutions, while providing stronger guarantees than weak coresets, which apply only to the optimum solution. We complement this result by showing that even a very small distortion in the approximation factor cannot admit coresets of this size.

Problem

• coresets
• clustering
• approximation
• big data
• cost function

Innovation

• approximation-preserving coreset
• clustering
• big data
• coreset theory
• approximation algorithms