There has been a growing interest in using neural networks, especially message-passing neural networks (MPNNs), to solve hard combinatorial optimization problems heuristically. However, existing learning-based approaches for hard combinatorial optimization tasks often rely on supervised training data, reinforcement learning, or gradient estimators, leading to significant computational overhead, unstable training, or a lack of provable performance guarantees. In contrast, classical approximation algorithms offer such performance guarantees under worst-case inputs but are non-differentiable and unable to adaptively exploit structural regularities in natural input distributions. We address this dichotomy with the fundamental example of Uniform Facility Location (UniFL), a variant of the combinatorial facility location problem with applications in clustering, data summarization, logistics, and supply chain design. We develop a fully differentiable MPNN model that embeds approximation-algorithmic principles while avoiding the need for solver supervision or discrete relaxations. Our approach admits provable approximation and size generalization guarantees to much larger instances than seen during training. Empirically, we show that our approach outperforms standard non-learned approximation algorithms in terms of solution quality, closing the gap with computationally intensive integer linear programming approaches. Overall, this work provides a step toward bridging learning-based methods and approximation algorithms for discrete optimization.

翻译：近年来，利用神经网络（特别是消息传递神经网络MPNN）启发式求解复杂组合优化问题的研究日益增多。然而，现有基于学习的复杂组合优化方法通常依赖于监督训练数据、强化学习或梯度估计器，导致显著的计算开销、训练不稳定或缺乏可证明的性能保证。相比之下，经典近似算法在最坏情况输入下能提供此类性能保证，但其不可微分且无法自适应地利用自然输入分布中的结构规律性。我们以均匀设施选址（UniFL）这一基础范例来应对这种二分困境——该问题是组合设施选址问题的变体，在聚类、数据摘要、物流和供应链设计等领域具有应用价值。我们开发了一种完全可微分的MPNN模型，该模型嵌入了近似算法原理，同时避免了求解器监督或离散松弛的需求。我们的方法能够对远大于训练所见规模的实例提供可证明的近似性能和规模泛化保证。实证研究表明，我们的方法在求解质量上优于标准的非学习型近似算法，缩小了与计算密集型整数线性规划方法之间的差距。总体而言，这项工作为离散优化领域基于学习的方法与近似算法之间的融合迈出了重要一步。