Hard-capacitated k-means (HCKM) is one of the fundamental problems remaining open in combinatorial optimization and engineering. In HCKM, one is required to partition a given n-point set into k disjoint clusters with known capacity so as to minimize the sum of within-cluster variances. It is known to be at least APX-hard, and most of the work on it has been done from a meta heuristic or bi-criteria approximation perspective. To the best our knowledge, no constant approximation algorithm or existence proof of such an algorithm is known. As our main contribution, we propose an FPT(k) approximation algorithm with constant performance guarantee for HCKM in this paper.

中文翻译：

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硬电容k均值的恒定FPT近似算法

硬电容k均值（HCKM）是组合优化和工程中仍然存在的基本问题之一。在HCKM中，需要将给定的n点集划分为具有已知容量的k个不相交的簇，以使簇内方差之和最小。众所周知，它至少对APX很困难，并且它的大部分工作都是从元启发式或双准则近似的角度完成的。据我们所知，尚无常数近似算法或这种算法的存在性证明。作为我们的主要贡献，本文提出了一种具有恒定性能保证HCKM的FPT（k）近似算法。