Here's an example that suggests that the stronger claim in the earlier version (Lemma 5.3) is false. I've only given a cursory look at the papers, so please check this carefully to make sure I am understanding correctly, thanks. Consider a cluster $X$ consisting of a rooted star: a root $r$ and $n-1$ nodes $v_1,v_2,\ldots, v_{n-1}$ such that $d(r, v_i) = 1$...


Recently, I came across the following survey: Recent Developments in Approximation Algorithms for Facility Location and Clustering Problems The authors only discuss the major results and mention some interesting open problems at the end.

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