On Modularity - NP-Completeness and Beyond

Modularity is a recently introduced quality measure for graph clusterings. It has immediately received considerable attention in several disciplines, and in particular in the complex systems literature, although its properties are not well understood. We here present first results on the computational and analytical properties of modularity. The complexity status of modularity maximization is resolved showing that the corresponding decision version is NP-complete in the strong sense. We also give a formulation as an Integer Linear Program (ILP) to facilitate exact optimization, and provide results on the approximation factor of the commonly used greedy algorithm. Completing our investigation, we characterize clusterings with maximum modularity for several graph families.