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

Vincent Cohen-Addad; Philip Klein; Claire Mathieu

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

LIP6

2 Brown University - Department of Computer Science

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

Domain :

Computer Science [cs] / Data Structures and Algorithms [cs.DS]

Complete list of metadatas

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

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

Links full text

Identifiers

Collections

Citation

Export

Metrics

22

Contact

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

Links full text

Identifiers

Collections

Citation

Export

Share

Metrics

22

Contact