Neural networks, particularly message-passing neural networks (MPNNs), are increasingly used as heuristics for hard combinatorial optimization problems. Yet many learning-based methods rely on supervision, reinforcement learning, or gradient estimators, causing high computational cost, unstable training, or limited guarantees. Classical approximation algorithms provide worst-case guarantees but are non-differentiable and cannot adapt to structure in natural input distributions. We study this tradeoff through Uniform Facility Location (UniFL), a problem with applications in clustering, summarization, logistics, and supply chains. We propose a fully differentiable MPNN that incorporates approximation-algorithmic principles without solver supervision or discrete relaxations. The model has provable approximation guarantees and empirically improves on standard approximation algorithms, narrowing the gap to integer linear programming.

翻译：神经网络，特别是消息传递神经网络（MPNN），正越来越多地被用作解决困难组合优化问题的启发式方法。然而，许多基于学习的方法依赖于监督、强化学习或梯度估计器，导致计算成本高、训练不稳定或保证有限。经典近似算法提供了最坏情况下的保证，但不可微且无法适应自然输入分布中的结构。我们通过均匀设施选址（UniFL）问题研究这一权衡，该问题在聚类、摘要、物流和供应链中具有应用。我们提出了一种完全可微的MPNN，它融入了近似算法原理，无需求解器监督或离散松弛。该模型具有可证明的近似保证，并且在经验上优于标准近似算法，缩小了与整数线性规划的差距。