SIAM Journal on Computing, Ahead of Print.
Clustering is a classic topic in optimization with $k$-means being one of the most fundamental such problems. In the absence of any restrictions on the input, the best-known algorithm for $k$-means in Euclidean space with a provable guarantee is a simple local search heuristic yielding an approximation guarantee of $9+\epsilon$, a ratio that is known to be tight with respect to such methods. We overcome this barrier by presenting a new primal-dual approach that allows us to (1) exploit the geometric structure of $k$-means and (2) satisfy the hard constraint that at most $k$ clusters are selected without deteriorating the approximation guarantee. Our main result is a 6.357-approximation algorithm with respect to the standard linear programming (LP) relaxation. Our techniques are quite general, and we also show improved guarantees for $k$-median in Euclidean metrics and for a generalization of $k$-means in which the underlying metric is not required to be Euclidean.

中文翻译：

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Primal-Dual算法为$ k $-均值和欧几里德$ k $-中位数提供更好的保证

《 SIAM计算杂志》，预印本。
聚类是优化中的经典话题，$ k $ -means是最基本的此类问题之一。在没有任何输入限制的情况下，在欧氏空间中具有可证明保证的$ k $ -means最著名的算法是简单的局部搜索试探法，产生的近似保证为$ 9 + \ epsilon $，该比率是已知的对这种方法要严格。通过提出一种新的原始对偶方法，我们克服了这一障碍，该方法允许我们（1）利用$ k $-均值的几何结构，并且（2）满足严格的约束条件，即最多选择$ k $个聚类而不会降低近似值保证。我们的主要结果是相对于标准线性规划（LP）松弛的6.357逼近算法。我们的技术很一般