  Several algorithms have been proposed to compute partitions of networks into
communities that score high on a graph clustering index called modularity.
While publications on these algorithms typically contain experimental
evaluations to emphasize the plausibility of results, none of these algorithms
has been shown to actually compute optimal partitions. We here settle the
unknown complexity status of modularity maximization by showing that the
corresponding decision version is NP-complete in the strong sense. As a
consequence, any efficient, i.e. polynomial-time, algorithm is only heuristic
and yields suboptimal partitions on many instances.
