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).
A 1.488 approximation algorithm for the uncapacitated facility location problem.Information and Computation, 222:45–58, 2013
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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).