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.

翻译：最稠密k-子图问题旨在寻找包含k个顶点且其间边数最多的子图。本文研究了引入对角加载项的二元二次DkS问题的连续重构形式。研究表明，该非凸连续松弛在一定对角加载参数范围内具有紧致性，并分析了对角加载参数对优化景观的影响。在算法层面，本文分别基于Frank-Wolfe方法和显式约束参数化提出了两种无投影算法来解决松弛问题。实验表明，两种算法相较于现有技术均具有优势，其中基于Frank-Wolfe的算法在子图稠密度、计算复杂度以及处理超大规模数据集的可扩展性方面表现尤为突出。