Maximizing Barber's bipartite modularity is also hard

Modularity introduced by Newman and Girvan (Phys Rev E 69:026113, 2004) is a quality function for community detection in networks. Numerous methods for modularity maximization have been developed so far. In 2007, Barber (Phys Rev E 76:066102, 2007) introduced a variant of modularity called bipartite modularity which is appropriate for bipartite networks. Although maximizing the standard modularity is known to be NP-hard, the computational complexity of maximizing bipartite modularity has yet to be revealed. In this study, we prove that maximizing bipartite modularity is also NP-hard. More specifically, we show the NP-completeness of its decision version by constructing a reduction from a classical partitioning problem.