The power of local search for clustering

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 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{εOp1q centers. ̊Research funded by the French ANR Blanc project ANR-12-BS02-005 (RDAM) :Research funded by NSF Grant CCF-10-12254 with additional support from the Radcliffe Institute of Advanced Study, Harvard University ;Research funded by the French ANR Blanc project ANR-12-BS02-005 (RDAM) ar X iv :1 60 3. 09 53 5v 2 [ cs .D S] 7 A pr 2 01 6