A PTAS for the Cluster Editing Problem on Planar Graphs

Abstract

The goal of the cluster editing problem is to add or delete a minimum number of edges from a given graph, so that the resulting graph becomes a union of disjoint cliques. The cluster editing problem is closely related to correlation clustering and has applications, e.g. in image segmentation. For general graphs this problem is \({\mathbb {APX}}\)-hard. In this paper we present an efficient polynomial time approximation scheme for the cluster editing problem on graphs embeddable in the plane with a few edge crossings. The running time of the algorithm is \({2^{O\left( \epsilon ^{-1} \log (\epsilon ^{-1})\right) }n}\) for planar graphs and \(2^{O\left( k^2\epsilon ^{-1}\log \left( k^2\epsilon ^{-1}\right) \right) }n\) for planar graphs with at most k crossings.