The prediction of protein complexes from protein-protein interactions (PPIs) is a well-studied problem in bioinformatics. However, the currently available PPI data is not enough to describe all known protein complexes. In this paper, we express the problem of determining the minimum number of (additional) required protein-protein interactions as a graph theoretic problem under the constraint that each complex constitutes a connected component in a PPI network. For this problem, we develop two computational methods: one is based on integer linear programming (ILPMinPPI) and the other one is based on an existing greedy-type approximation algorithm (GreedyMinPPI) originally developed in the context of communication and social networks. Since the former method is only applicable to datasets of small size, we apply the latter method to a combination of the CYC2008 protein complex dataset and each of eight PPI datasets (STRING, MINT, BioGRID, IntAct, DIP, BIND, WI-PHI, iRefIndex). The results show that the minimum number of additional required PPIs ranges from 51 (STRING) to 964 (BIND), and that even the four best PPI databases, STRING (51), BioGRID (67), WI-PHI (93) and iRefIndex (85), do not include enough PPIs to form all CYC2008 protein complexes. We also demonstrate that the proposed problem framework and our solutions can enhance the prediction accuracy of existing PPI prediction methods. ILPMinPPI can be freely downloaded from http://sunflower.kuicr.kyoto-u.ac.jp/~nakajima/.