This paper considers correlation clustering on unweighted complete graphs. We give a combinatorial algorithm that returns a single clustering solution that is simultaneously $O(1)$-approximate for all $\ell_p$-norms of the disagreement vector. This proves that minimal sacrifice is needed in order to optimize different norms of the disagreement vector. Our algorithm is the first combinatorial approximation algorithm for the $\ell_2$-norm objective, and more generally the first combinatorial algorithm for the $\ell_p$-norm objective when $2 \leq p < \infty$. It is also faster than all previous algorithms that minimize the $\ell_p$-norm of the disagreement vector, with run-time $O(n^\omega)$, where $O(n^\omega)$ is the time for matrix multiplication on $n \times n$ matrices. When the maximum positive degree in the graph is at most $\Delta$, this can be improved to a run-time of $O(n\Delta^2 \log n)$.

翻译：本文研究无权重完全图上的相关聚类问题。我们提出一种组合算法，返回一个单一聚类解，该解同时对所有不一致向量的 $\ell_p$ 范数具有 $O(1)$ 近似保证。这证明了优化不一致向量的不同范数时只需付出极小代价。我们的算法是首个针对 $\ell_2$ 范数目标的组合近似算法，更一般地，当 $2 \leq p < \infty$ 时，也是首个针对 $\ell_p$ 范数目标的组合算法。此外，该算法运行时间为 $O(n^\omega)$（其中 $O(n^\omega)$ 为 $n \times n$ 矩阵乘法时间），快于所有先前最小化不一致向量 $\ell_p$ 范数的算法。当图中最大正度不超过 $\Delta$ 时，运行时间可改进为 $O(n\Delta^2 \log n)$。