# Local Search Yields Approximation Schemes for $k$-Means and $k$-Median in Euclidean and Minor-Free Metrics

Abstract : We give the first polynomial-time approximation schemes (PTASs) for the following problems: (1) uniform facility location in edge-weighted planar graphs; (2) $k$-median and $k$-means in edge-weighted planar graphs; and (3) $k$-means in Euclidean space of bounded dimension. Our first and second results extend to minor-closed families of graphs. All our results extend to cost functions that are the $p$th power of the shortest-path distance. The algorithm is local search, where the local neighborhood of a solution $S$ consists of all solutions obtained from $S$ by removing and adding $1/\varepsilon^{\Theta(1)}$ centers.
Document type :
Journal articles

https://hal.archives-ouvertes.fr/hal-02169573
Submitted on : Monday, July 1, 2019 - 12:26:19 PM
Last modification on : Friday, July 5, 2019 - 3:26:03 PM

### Citation

Vincent Cohen-Addad, Philip Klein, Claire Mathieu. Local Search Yields Approximation Schemes for $k$-Means and $k$-Median in Euclidean and Minor-Free Metrics. SIAM Journal on Computing, Society for Industrial and Applied Mathematics, 2019, 48 (2), pp.644-667. ⟨10.1137/17M112717X⟩. ⟨hal-02169573⟩

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