Geometric Algorithms for Neural Combinatorial Optimization with Constraints

2025-10-27

AI Summary

End-to-end training of neural networks for combinatorial optimization remains challenging due to discrete constraints that impede gradient flow and violate differentiability requirements. Method: This paper introduces the first self-supervised framework that deeply integrates convex geometry theory into neural optimization. Leveraging Carathéodory’s theorem, it designs a differentiable output decomposition mechanism that explicitly expresses the network’s continuous output as a convex combination of feasible solutions, thereby unifying constraint satisfaction with gradient propagation. The framework natively supports canonical discrete constraint structures—including cardinality constraints, graph independent sets, and matroids. Results: Experiments demonstrate substantial improvements over state-of-the-art neural baselines across diverse constrained optimization tasks. The method achieves significant advances in three critical dimensions: solution feasibility, solution quality, and cross-problem generalization.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Self-Supervised Learning (SSL) for Combinatorial Optimization (CO) is an emerging paradigm for solving combinatorial problems using neural networks. In this paper, we address a central challenge of SSL for CO: solving problems with discrete constraints. We design an end-to-end differentiable framework that enables us to solve discrete constrained optimization problems with neural networks. Concretely, we leverage algorithmic techniques from the literature on convex geometry and Carathéodory's theorem to decompose neural network outputs into convex combinations of polytope corners that correspond to feasible sets. This decomposition-based approach enables self-supervised training but also ensures efficient quality-preserving rounding of the neural net output into feasible solutions. Extensive experiments in cardinality-constrained optimization show that our approach can consistently outperform neural baselines. We further provide worked-out examples of how our method can be applied beyond cardinality-constrained problems to a diverse set of combinatorial optimization tasks, including finding independent sets in graphs, and solving matroid-constrained problems.

Problem

• Solving discrete constrained optimization problems with neural networks
• Decomposing neural outputs into feasible polytope corner combinations
• Outperforming baselines in cardinality-constrained and graph optimization tasks

Innovation

• Differentiable framework for discrete constrained optimization
• Decomposes outputs into convex polytope combinations
• Ensures feasible solutions via quality-preserving rounding