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

TL;DR

This paper presents the first polynomial-time approximation schemes for k-means and k-median problems in Euclidean and minor-free metrics, using a local search approach with neighborhood size depending on epsilon.

Contribution

It introduces novel PTAS algorithms for k-means and k-median in Euclidean and planar graphs, extending to minor-closed families and p-th power cost functions.

Findings

First PTAS for k-median and k-means in planar graphs

Extension of algorithms to minor-closed graph families

Applicability to p-th power shortest-path cost functions

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; (3) $k$-means in Euclidean spaces 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 $1/\epsilon^{O(1)}$ centers.