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arXiv · 2607.14654

Spectral Dual Fitting for $k$-Means

Aditya Anand·Moses Charikar·Vincent Cohen-Addad·Ruiquan Gao·Fabrizio Grandoni·Euiwoong Lee·Amatya Sharma·Ernest van Wijland
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Abstract

We give a new dual fitting algorithm which gives improved approximation ratios of $3+\ln 2 + \epsilon\ (\approx 3.694)$ and $4.9+\epsilon$ for $k$-Means in (high-dimensional) Euclidean and general metrics respectively, improving upon the previously known ratios of $4+\epsilon$ [Charikar, Cohen-Addad, Gao, Grandoni, Lee, and van Wijland STOC'26] and $5+\epsilon$ [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.

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Aditya Anand, Moses Charikar, Vincent Cohen-Addad, Ruiquan Gao, Fabrizio Grandoni, Euiwoong Lee, Amatya Sharma, Ernest van Wijland. 2026-07-16. Spectral Dual Fitting for $k$-Means. https://arxiv.org/abs/2607.14654

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