Densest k-Subgraph Mining via a Provably Tight Relaxation

Abstract Given an unweighted, undirected, and simple graph, the Densest k-Subgraph (DkS) problem aims to find a subgraph of k vertices that has the maximum average induced degree. In this paper, we consider an equivalent reformulation of the DkS problem via diagonal loading. On relaxing the combinatorial constraint of the reformulated problem, we show that the resulting non-convex, continuous relaxation is tight under certain conditions by leveraging an extension of the Motzkin-Straus theorem. We utilize two projection-free approaches to solve the relaxed problem: one based on the Frank-Wolfe algorithm and the other on explicit constraint parameterization. We compare their performance to state-of-the-art baselines across various benchmarks. Our empirical results show that the Frank-Wolfe-based algorithm proposed in this paper outperforms existing baselines in terms of subgraph density and computational complexity.