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

Vincent Cohen-Addad (1) , Philip Klein , Claire Mathieu (2)

1 RO - Recherche Opérationnelle
2 Brown University - Department of Computer Science

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.

Domaines: Algorithme et structure de données [cs.DS]

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, 2019, 48 (2), pp.644-667. ⟨10.1137/17M112717X⟩. ⟨hal-02169573⟩