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. 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.