Geometric Algorithms for Neural Combinatorial Optimization with Constraints
Nikolaos Karalias, Akbar Rafiey, Yifei Xu, Zhishang Luo, Behrooz Tahmasebi, Connie Jiang, Stefanie Jegelka
TL;DR
This paper tackles self-supervised learning for combinatorial optimization with discrete constraints by introducing an end-to-end differentiable framework that leverages Carathéodory-based polyhedral decomposition. The neural network outputs a point in the convex hull of feasible solutions, which is then decomposed into a sparsely supported distribution over feasible corner sets, enabling an exact, differentiable extension loss $F(\boldsymbol{x}) = \mathbb{E}_{S \sim \mathcal{D}_{\mathcal{C}}(\boldsymbol{x})}[f(S)]$ and a rounding strategy with guarantees. The approach is instantiated via a GLS-type decomposition for several constraint families (cardinality, partition matroids, spanning trees, independent sets) and shown to perform well on large-scale cardinality-constrained problems, with broad applicability to other CO tasks. Empirical results on Maximum Coverage demonstrate competitive or superior performance against neural and traditional baselines, while ablations highlight the importance of the decomposition design and inference strategies. Overall, the work provides a principled integration of convex geometry and neural solvers, offering scalable, differentiable handling of discrete constraints with rigorous rounding guarantees.
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.