On the Advantage of Overlapping Clusters for Minimizing Conductance

摘要

Graph clustering is an important problem with applications to bioinformat-ics, community discovery in social networks, distributed computing, and more. While most of the research in this area has focused on clustering using disjoint clusters, many real datasets have inherently overlapping clusters. We compare overlapping and non-overlapping clusterings in graphs in the context of minimizing their conductance. It is known that allowing clusters to overlap gives better results in practice. We prove that overlapping clustering may be significantly better than non-overlapping clustering with respect to conductance, even in a theoretical setting. For minimizing the maximum conductance over the clusters, we give examples demonstrating that allowing overlaps can yield significantly better clusterings, namely, one that has much smaller optimum. In addition for the min-max variant, the overlapping version admits a simple approximation algorithm, while our algorithm for the non-overlapping version is complex and yields a worse approximation ratio due to the presence of the additional constraint. Somewhat surprisingly, for the problem of minimizing the sum of conductances, we found out that allowing overlap does not help. We show how to apply a general technique to transform any overlapping clustering into a non-overlapping one with only a modest increase in the sum of con-ductances. This uncrossing technique is of independent interest and may find further applications in the future. We consider this work as a step toward rigorous comparison of overlapping and non-overlapping clusterings and hope that it stimulates further research in this area.