The Densest $k$-Subgraph (D$k$S) is a fundamental combinatorial problem known for its theoretical hardness and breadth of applications. Recently, Lu et al. (AAAI 2025) introduced a penalty-based non-convex relaxation that achieves promising empirical performance; however, a rigorous theoretical understanding of its success remains unclear. In this work, we bridge this gap by providing a comprehensive theoretical analysis. We first establish the tightness of the relaxation, ensuring that the global maximum values of the original combinatorial problem and the relaxed problem coincide. Then we reveal the benign geometry of the optimization landscape by proving a strict dichotomy of stationary points: all integral stationary points are local maximizers, whereas all non-integral stationary points are strict saddles with explicit positive curvature. We propose a saddle-escaping Frank--Wolfe algorithm and prove that it achieves exact convergence to an integral local maximizer in a finite number of steps.

翻译：最密集k-子图（DkS）是一个基础组合优化问题，以其理论复杂性和广泛的应用范围而著称。最近，Lu等人（AAAI 2025）提出了一种基于惩罚项的非凸松弛方法，取得了良好的实证性能；然而，对其成功机理的严格理论理解尚不明确。在本工作中，我们通过提供全面的理论分析来弥合这一差距。我们首先建立了松弛的紧致性，确保原始组合问题与松弛问题的全局最优值一致。随后，我们通过证明平稳点的严格二分性揭示了优化景观的良性几何结构：所有整数平稳点均为局部极大值点，而所有非整数平稳点均为具有显式正曲率的严格鞍点。我们提出了一种逃离鞍点的Frank-Wolfe算法，并证明该算法能在有限步数内精确收敛至一个整数局部极大值点。