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Dense subgraph discovery (DSD) is a key primitive in graph mining that typically deals with extracting cliques and near-cliques. In this paper, we revisit the optimal quasi-clique (OQC) formulation for DSD and establish that it is NP--hard. In addition, we reveal the hitherto unknown property that OQC can be used to explore the entire spectrum of densest subgraphs of all distinct sizes by appropriately varying a single hyperparameter, thereby forging an intimate link with the classic densest-k-subgraph problem (DkS). We corroborate these findings on real-world graphs by applying the simple greedy algorithm for OQC with improved hyperparameter tuning, to quickly generate high-quality approximations of the size-density frontier. Our findings indicate that OQC not only extracts high quality (near)-cliques, but also large and loosely-connected subgraphs that exhibit well defined local community structure. The latter discovery is particularly intriguing, since OQC is not explicitly geared towards community detection.
Konar et al. (Sun,) studied this question.