Advantage of Overlapping Clusters for Minimizing Conductance

Graph clustering is an important problem with applications to bioinformatics, community discovery in social networks, distributed computing, etc. 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 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 really 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 conductances. This

uncrossing

technique is of independent interest and may find further applications in the future.