Maximizing Modularity is hard

TLDR

Algorithms exist to partition networks into communities maximizing modularity, yet none have been proven to compute optimal partitions. This study proves that deciding whether a network can achieve a given modularity score is NP‑complete in the strong sense. Consequently, any polynomial‑time algorithm for modularity maximization must be heuristic and will produce suboptimal partitions on many instances.

Abstract

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.