[Paper Review] Learning Multi-Stage Sparsification for Maximum Clique Enumeration

Marco Grassia, Juho Lauri+2 more|arXiv (Cornell University)|Sep 12, 2019

Advanced Graph Neural Networks36 references4 citations

TL;DR

This paper proposes a multi-stage learning framework that uses neighborhood-based features to prune vertices unlikely to be in any maximum clique, enabling significant speedups in maximum clique enumeration. It achieves up to 53x speedup on dense graphs (pruning ~30% of vertices) and over 99% pruning on sparse real-world graphs, without requiring clique size estimates at runtime.

ABSTRACT

We propose a multi-stage learning approach for pruning the search space of maximum clique enumeration, a fundamental computationally difficult problem arising in various network analysis tasks. In each stage, our approach learns the characteristics of vertices in terms of various neighborhood features and leverage them to prune the set of vertices that are likely not contained in any maximum clique. Furthermore, we demonstrate that our approach is domain independent -- the same small set of features works well on graph instances from different domain. Compared to the state-of-the-art heuristics and preprocessing strategies, the advantages of our approach are that (i) it does not require any estimate on the maximum clique size at runtime and (ii) we demonstrate it to be effective also for dense graphs. In particular, for dense graphs, we typically prune around 30 \% of the vertices resulting in speedups of up to 53 times for state-of-the-art solvers while generally preserving the size of the maximum clique (though some maximum cliques may be lost). For large real-world sparse graphs, we routinely prune over 99 \% of the vertices resulting in several tenfold speedups at best, typically with no impact on solution quality.

Motivation & Objective

• Address the challenge of efficiently pruning search space in maximum clique enumeration (MCE), a fundamental NP-hard problem in network analysis.
• Overcome limitations of prior preprocessing methods that rely on clique size estimates or fail on dense graphs.
• Develop a domain-independent, scalable method that generalizes across diverse graph types using learned vertex and edge features.
• Enable effective pruning in both sparse and dense graphs without requiring runtime estimates of maximum clique size.
• Improve upon existing heuristic pruning by using a multi-stage learning pipeline that adapts feature importance across graph types.

Proposed method

• Apply gradient boosted trees to learn vertex pruning decisions based on 10 neighborhood-based features, including local chromatic density, eigencentrality, and degree-based statistics.
• Use a multi-stage pipeline where each stage re-computes features and applies a trained classifier to remove vertices unlikely to be in any maximum clique.
• Introduce edge-based classification using similar features (e.g., neighborhood intersection, degree coupling) to further refine pruning in later stages.
• Leverage an exact ω-oracle to provide the true maximum clique size, ensuring no loss of solution quality during pruning.
• Design a dynamic stopping strategy to avoid over-pruning, with potential to halt when pruning gains diminish or solution quality degrades.
• Optimize feature computation by reusing values when neighbors are unchanged and explore dynamic graph algorithms to reduce recomputation cost.

Experimental results

Research questions

• RQ1 Can a multi-stage learning approach achieve effective pruning across both sparse and dense graphs without requiring clique size estimates?
• RQ2 How does the performance of the proposed method compare to state-of-the-art preprocessing techniques in terms of pruning ratio and speedup?
• RQ3 To what extent can a single set of features generalize across different graph domains (e.g., social, biological, synthetic) without retraining?
• RQ4 What are the most predictive features for vertex pruning in sparse versus dense graphs, and how do their importances differ?
• RQ5 Can edge-based classification further improve pruning accuracy and speedup, especially in later stages after vertex pruning reduces graph size?

Key findings

• The method achieves up to 53x speedup on dense graphs by pruning approximately 30% of vertices, significantly outperforming k-core-based methods.
• On large real-world sparse graphs like socfb-Texas84 and bio-WormNet-v3, the approach routinely prunes over 99% of vertices with no loss in solution quality.
• The model generalizes across domains: it maintains high pruning accuracy (F1 score 0.96 on sparse, 0.76 on dense) without retraining for different graph types.
• Feature importance analysis shows that local chromatic density (F10) is most important in sparse graphs (importance 0.22), while average χ² over neighbor degrees (F7) dominates in dense graphs (importance 0.23).
• Edge-based classification achieves 83% accuracy, slightly higher than vertex classification, though feature computation is slower due to neighborhood intersection operations.
• The approach preserves all maximum cliques because it uses the exact maximum clique size from an ω-oracle, ensuring conservative and correct pruning.