AI 中文总结
提出用于k均值的谱对偶拟合算法，在欧几里得和一般度量下分别给出更好近似比，打破硬度障碍，且与此前方法不同，紧密考虑对偶支付并引入谱分析框架确定近似因子。
AI 中文摘要
我们给出了一种新的对偶拟合算法，对于（高维）欧几里得度量和一般度量下的k均值，分别给出了改进的近似比3 + ln2 + ε（约为3.694）和4.9 + ε，优于之前已知的4 + ε和5 + ε。特别是，我们关于欧几里得k均值的结果打破了度量k均值1 + 8/e≈3.94的硬度障碍。此前对于k中位数、k均值或设施选址，在可近似性方面，一般度量和欧几里得度量之间不存在这样的差异。与之前的k均值对偶拟合方法不同，我们的新算法在有效进行对偶可行性分析的同时，紧密考虑对偶支付。我们引入了一个使用谱分析来确定算法近似因子的新框架。
英文摘要
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.
