Spectral Dual Fitting for $k$-Means

2026-07-16

AI Summary

This work addresses the $k$-means clustering problem by proposing a novel algorithmic framework that integrates dual fitting with spectral analysis. The approach refines the accounting of dual payments while preserving dual feasibility, thereby improving the approximation guarantee. Notably, it is the first to achieve a separation in approximation performance between Euclidean and general metric spaces, breaking the long-standing hardness barrier of $1 + 8/e \approx 3.94$ in Euclidean space. The method yields approximation ratios of approximately $3.694$ in high-dimensional Euclidean space and $4.9$ in general metric spaces, surpassing the previous best-known bounds of $4 + \varepsilon$ and $5 + \varepsilon$, respectively.

Abstract

We give a new dual fitting algorithm which gives improved approximation ratios of $3+\ln 2 + ε (\approx 3.694)$ and $4.9+ε$ for $k$-Means in (high-dimensional) Euclidean and general metrics respectively, improving upon the previously known ratios of $4+ε$ [Charikar, Cohen-Addad, Gao, Grandoni, Lee, and van Wijland STOC'26] and $5+ε$ [Byrka, Guo, Hu, Li, Wan, Wang FOCS'26], resp. In particular, our result for Euclidean $k$-Means breaks the hardness barrier of $1+8/e\approx 3.94$ for Metric $k$-Means. Prior to our work, no such separation between general and Euclidean metrics was known for $k$-Median, $k$-Means, or Facility Location in terms of their approximability. Unlike prior dual fitting approaches for $k$-Means, our new dual fitting algorithm tightly accounts for dual payments while still facilitating an effective dual feasibility analysis. We introduce a new framework that uses spectral analysis for determining the approximation factor of our algorithm.

Problem

• k-Means
• approximation ratio
• Euclidean metric
• general metric
• hardness barrier

Innovation

• spectral dual fitting
• k-Means approximation
• Euclidean metrics
• dual feasibility
• approximation ratio