On Densest $k$-Subgraph Mining and Diagonal Loading: Optimization Landscape and Finite-Step Exact Convergence Analysis

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.

💡 Research Summary

The paper addresses the Densest k‑Subgraph (D k S) problem, a classic NP‑hard combinatorial optimization task that seeks a vertex subset of fixed size k with maximum induced edge density. Building on the recent penalty‑based relaxation introduced by Lu et al. (AAAI 2025), the authors provide a rigorous theoretical foundation for its empirical success.

First, they reformulate the binary quadratic program max ½ xᵀAx subject to ∑x_i = k, x_i∈{0,1} by adding a diagonal loading term λI, yielding the objective ½ xᵀ(A+λI)x. Relaxing the binary constraints to the convex hull C_{n,k} = { x∈