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

TL;DR

This paper provides a rigorous theoretical analysis of a non-convex relaxation for the Densest k-Subgraph problem, proving landscape properties and finite-step convergence of a saddle-escaping algorithm.

Contribution

It establishes the tightness of the relaxation, characterizes the optimization landscape, and proves finite-step convergence of a novel algorithm.

Findings

Relaxation is tight, matching the original problem's maximum.

All integral stationary points are local maxima, non-integral are strict saddles.

Proposed algorithm converges exactly in finite steps.

Abstract

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.