Fast and Simple Densest Subgraph with Predictions

TL;DR

The paper addresses densest subgraph problems under a learning-augmented framework, aiming to harness a predictor that identifies most members of the optimal subgraph. It shows that, given a $(1-\epsilon)$-accurate partial solution, a simple linear-time augmentation yields a near-optimal density, with $(1-3\epsilon)$-approximation for undirected and $(1-O(\epsilon))$-approximation for directed cases, plus extensions to a clique-density notion. The authors provide concrete algorithms, rigorous guarantees, and empirical evidence on Twitch ego-networks demonstrating improvements over Charikar’s peeling and predictor-only outputs. This work offers practical, fast, and theoretically sound methods for graph densest-subgraph problems in real-world datasets where partial predictions are available. Overall, the approach combines simple augmentation with strong guarantees, enabling scalable, prediction-informed graph analysis with meaningful impact for network science and data mining tasks.

Abstract

We study the densest subgraph problem and its variants through the lens of learning-augmented algorithms. For this problem, the greedy algorithm by Charikar (APPROX 2000) provides a linear-time $ 1/2 $-approximation, while computing the exact solution typically requires solving a linear program or performing maximum flow computations.We show that given a partial solution, i.e., one produced by a machine learning classifier that captures at least a $ (1 - ε) $-fraction of nodes in the optimal subgraph, it is possible to design an extremely simple linear-time algorithm that achieves a provable $ (1 - ε) $-approximation. Our approach also naturally extends to the directed densest subgraph problem and several NP-hard variants.An experiment on the Twitch Ego Nets dataset shows that our learning-augmented algorithm outperforms Charikar's greedy algorithm and a baseline that directly returns the predicted densest subgraph without additional algorithmic processing.

Figures (6)

• Figure 1: Suppose the this is the densest subgraph and the predictor identifies the nodes $y_1,\ldots,y_5$. Returning $y_1,\ldots,y_5$ will yield an infinitely bad solution with density 0 while technically the model has recalled $5/7 \approx 71\%$ of the nodes in the densest subgraph.
• Figure 2: The approximation guarantee comes from the observation that the number of edges induced by $H^{\ast} \setminus S$ is small, and the greedy approach ensures that we added at least as many edges to the solution as $|E(H^{\ast} \setminus S, H^{\ast} \cap S)|$.
• Figure 3: An example densest subgraph in Twitch's ego-networks.
• Figure 4: Output density of Algorithm \ref{['alg:1']} vs. output density of the subgraph given by the predictor.
• Figure 5: Output density of Algorithm \ref{['alg:1']} vs. output density of Charikar's greedy algorithm.

Theorems & Definitions (10)

• Theorem 1: Main result 1
• Theorem 2: Main result 2
• Theorem 3: Main result 2
• Proposition 1
• proof
• proof : Proof of Theorem \ref{['theorem:greedy-alg-for-simple-variant-of-dsp']}
• Proposition 2
• proof
• proof : Proof of Theorem \ref{['theorem:greedy-alg-for-directed-variant-of-dsp']}
• proof : Proof of theorem \ref{['theorem:greedy-alg-for-simple-variant-of-dsp-different-density']}