A PTAS for the Cluster Editing Problem on Planar Graphs

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 apx apx{\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(? -1 log(? -1 )) n 2o(?-1log?(?-1))n{2^{o\left( \epsilon ^{-1} \log (\epsilon ^{-1})\right) }n} for planar graphs and 2 o(k 2 ? -1 log(k 2 ? -1 )) n 2o(k2?-1log?(k2?-1))n2^{o\left( k^2\epsilon ^{-1}\log \left( k^2\epsilon ^{-1}\right) \right) }n for planar graphs with at most k crossings.

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