Exploring the Subgraph Density-Size Trade-off via the Lov\'asz Extension

TL;DR

This paper introduces a novel convex relaxation for the NP-hard Densest-k-Subgraph problem using the Lovász extension, enabling scalable algorithms that produce high-quality solutions with provable density bounds.

Contribution

It reformulates DkS as a submodular minimization problem with a closed-form Lovász extension, and develops a scalable ADMM-based algorithm for effective approximation.

Findings

Outperforms existing methods on real-world graphs

Produces solutions within 65-80% of optimal density

Offers a scalable approach for large graph instances

Abstract

Given an undirected graph, the Densest-k-Subgraph problem (DkS) seeks to find a subset of k vertices such that the sum of the edge weights in the corresponding subgraph is maximized. The problem is known to be NP-hard, and is also very difficult to approximate, in the worst-case. In this paper, we present a new convex relaxation for the problem. Our key idea is to reformulate DkS as minimizing a submodular function subject to a cardinality constraint. Exploiting the fact that submodular functions possess a convex, continuous extension (known as the Lov\'asz extension), we propose to minimize the Lov\'asz extension over the convex hull of the cardinality constraints. Although the Lov\'asz extension of a submodular function does not admit an analytical form in general, for DkS we show that it does. We leverage this result to develop a highly scalable algorithm based on the Alternating Direction Method of Multipliers (ADMM) for solving the relaxed problem. Coupled with a pair of fortuitously simple rounding schemes, we demonstrate that our approach outperforms existing baselines on real-world graphs and can yield high quality sub-optimal solutions which typically are a posteriori no worse than 65-80\% of the optimal density.