Correlation Clustering and Two-Edge-Connected Augmentation for Planar Graphs

Abstract

We study two problems. In correlation clustering, the input is a weighted graph, where every edge is labelled either $$\langle +\rangle $$ or $$\langle -\rangle $$ according to whether its endpoints are in the same category or in different categories. The goal is to produce a partition of the vertices into categories that tries to respect the labels of the edges. In two-edge-connected augmentation, the input is a weighted graph and a subset R of edges of the graph. The goal is to produce a minimum weight subset S of edges of the graph, such that for every edge in R, its endpoints are two-edge-connected in $$R\cup S$$ . In this paper, we study these problems under the restriction that the input graph must be planar. We give an approximation-preserving reduction from correlation clustering on planar graphs to two-edge-connected augmentation on planar graphs. We give a polynomial-time approximation scheme (PTAS) for the latter problem, yielding a PTAS for the former problem as well. The approximation scheme employs brick decompositions, which have been used in previous approximation schemes for planar graphs, but the way it uses brick decompositions is fundamentally different from previous uses.