An iterative rounding procedure achieves a ((3^p + 1)/2 + ε)-approximation for k-clustering under p-th power distance costs, recovering the 2-approximation for k-median and improving k-means bounds to 5+ε (metric) and 4+ε (Euclidean).
Greedy facility location algorithms analyzed using dual fitting with factor-revealing lp.Journal of the ACM (JACM), 50(6):795–824, 2003
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$k$-Clustering via Iterative Randomized Rounding
An iterative rounding procedure achieves a ((3^p + 1)/2 + ε)-approximation for k-clustering under p-th power distance costs, recovering the 2-approximation for k-median and improving k-means bounds to 5+ε (metric) and 4+ε (Euclidean).