A well-known bottleneck of Min-Sum-of-Square Clustering (MSSC, the celebrated $k$-means problem) is to tackle the presence of outliers. In this paper, we propose a Partial clustering variant termed PMSSC which considers a fixed number of outliers to remove. We solve PMSSC by Integer Programming formulations and complexity results extending the ones from MSSC are studied. PMSSC is NP-hard in Euclidean space when the dimension or the number of clusters is greater than $2$. Finally, one-dimensional cases are studied: Unweighted PMSSC is polynomial in that case and solved with a dynamic programming algorithm, extending the optimality property of MSSC with interval clustering. This result holds also for unweighted $k$-medoids with outliers. A weaker optimality property holds for weighted PMSSC, but NP-hardness or not remains an open question in dimension one.

翻译：众所周知，最小平方和聚类（MSSC，即著名的$k$-means问题）的一个主要瓶颈在于处理离群点。本文提出一种名为PMSSC的部分聚类变体，该变体考虑移除固定数量的离群点。我们通过整数规划公式求解PMSSC，并研究了从MSSC扩展而来的复杂度结果。当维度或聚类数大于$2$时，PMSSC在欧几里得空间中属于NP难问题。最后，我们研究了单维情形：无权重PMSSC在该情形下为多项式时间可解，并通过动态规划算法求解，该算法扩展了基于区间聚类的MSSC最优性性质。该结论也适用于带离群点的无权重$k$-中心点问题。对于权重PMSSC，存在较弱的最优性性质，但其在单维情形下是否为NP难问题仍为开放性问题。