Maximizing Modularity is hard

TL;DR

This paper proves that maximizing modularity in network community detection is NP-complete, establishing the computational difficulty and implying that efficient algorithms can only produce heuristic, suboptimal solutions.

Contribution

It establishes the NP-completeness of the modularity maximization problem, clarifying its computational complexity status for the first time.

Findings

Modularity maximization is NP-complete in the strong sense.

Efficient algorithms can only produce heuristic, suboptimal solutions.

The result explains the difficulty in finding optimal community partitions.

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.