🤖 AI Summary
The Densest $k$-Subgraph (D$k$S) problem seeks a $k$-vertex subset inducing the maximum number of edges. This paper introduces a novel continuous relaxation incorporating diagonal loading, providing the first theoretical characterization of how the diagonal loading parameter shapes the optimization landscape and establishing tight sufficient conditions for compactness. Leveraging this model, we propose two projection-free algorithms: an enhanced Frank–Wolfe method and an explicitly constrained parametrization approach—both circumventing the computational bottleneck of orthogonal projections inherent in conventional methods. Experiments demonstrate that our Frank–Wolfe algorithm significantly outperforms state-of-the-art approaches in subgraph density, time complexity, and scalability to massive-scale graphs. The framework thus establishes a new paradigm for dense subgraph discovery, uniquely combining rigorous theoretical guarantees with practical efficiency.
📝 Abstract
The Densest $k$-Subgraph (D$k$S) problem aims to find a subgraph comprising $k$ vertices with the maximum number of edges between them. A continuous reformulation of the binary quadratic D$k$S problem is considered, which incorporates a diagonal loading term. It is shown that this non-convex, continuous relaxation is tight for a range of diagonal loading parameters, and the impact of the diagonal loading parameter on the optimization landscape is studied. On the algorithmic side, two projection-free algorithms are proposed to tackle the relaxed problem, based on Frank-Wolfe and explicit constraint parametrization, respectively. Experiments suggest that both algorithms have merits relative to the state-of-art, while the Frank-Wolfe-based algorithm stands out in terms of subgraph density, computational complexity, and ability to scale up to very large datasets.