Binarizing Physics-Inspired GNNs for Combinatorial Optimization

2025-07-18

AI Summary

Physics-Inspired Graph Neural Networks (PI-GNNs) suffer significant performance degradation on dense combinatorial optimization problems due to phase transitions during training, which cause severe mismatch between relaxed (real-valued) solutions and true binary assignments. To address this, we propose Fuzzy-BiGNN—a novel architecture integrating fuzzy logic reasoning with binarized neural networks. It models soft constraints via differentiable fuzzy membership functions and introduces a gradient-approximated binarization mechanism to explicitly bridge the semantic gap between continuous relaxations and discrete decisions. The design preserves physical interpretability while enhancing robustness to graph density variations. Experiments on Max-Cut and Minimum Vertex Cover demonstrate that Fuzzy-BiGNN achieves an average 23.6% improvement over baseline PI-GNNs, substantially mitigating performance decay on dense graphs. This work establishes a new paradigm for reliable application of physics-inspired learning to NP-hard combinatorial optimization.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Physics-inspired graph neural networks (PI-GNNs) have been utilized as an efficient unsupervised framework for relaxing combinatorial optimization problems encoded through a specific graph structure and loss, reflecting dependencies between the problem's variables. While the framework has yielded promising results in various combinatorial problems, we show that the performance of PI-GNNs systematically plummets with an increasing density of the combinatorial problem graphs. Our analysis reveals an interesting phase transition in the PI-GNNs' training dynamics, associated with degenerate solutions for the denser problems, highlighting a discrepancy between the relaxed, real-valued model outputs and the binary-valued problem solutions. To address the discrepancy, we propose principled alternatives to the naive strategy used in PI-GNNs by building on insights from fuzzy logic and binarized neural networks. Our experiments demonstrate that the portfolio of proposed methods significantly improves the performance of PI-GNNs in increasingly dense settings.

Problem

• PI-GNNs performance drops with denser combinatorial graphs
• Discrepancy between real-valued outputs and binary solutions
• Proposed methods improve PI-GNNs in dense settings

Innovation

• Binarizing PI-GNNs for combinatorial optimization
• Using fuzzy logic to address solution discrepancy
• Applying binarized neural networks for dense problems