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Golovach","submitted_at":"2019-02-22T17:05:18Z","abstract_excerpt":"We consider the $k$-Clustering problem, which is for a given multiset of $n$ vectors $X\\subset \\mathbb{Z}^d$ and a nonnegative number $D$, to decide whether $X$ can be partitioned into $k$ clusters $C_1, \\dots, C_k$ such that the cost\n  \\[\\sum_{i=1}^k \\min_{c_i\\in \\mathbb{R}^d}\\sum_{x \\in C_i} \\|x-c_i\\|_p^p \\leq D,\\] where $\\|\\cdot\\|_p$ is the Minkowski ($L_p$) norm of order $p$. For $p=1$, $k$-Clustering is the well-known $k$-Median. For $p=2$, the case of the Euclidean distance, $k$-Clustering is $k$-Means. 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