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We give the first polynomial-time approximation schemes (PTASs) for thefollowing problems: (1) uniform facility location in edge-weighted planargraphs; (2) $k$-median and $k$-means in edge-weighted planar graphs; (3)$k$-means in Euclidean spaces of bounded dimension. Our first and secondresults extend to minor-closed families of graphs. All our results extend tocost functions that are the $p$-th power of the shortest-path distance. Thealgorithm is local search where the local neighborhood of a solution $S$consists of all solutions obtained from $S$ by removing and adding$1/\epsilon^{O(1)}$ centers.

Local search yields approximation schemes for k-means and k-median in Euclidean and minor-free metrics