In this work we design graph neural network architectures that can be used to obtain optimal approximation algorithms for a large class of combinatorial optimization problems using powerful algorithmic tools from semidefinite programming (SDP). Concretely, we prove that polynomial-sized message passing algorithms can represent the most powerful polynomial time algorithms for Max Constraint Satisfaction Problems assuming the Unique Games Conjecture. We leverage this result to construct efficient graph neural network architectures, OptGNN, that obtain high-quality approximate solutions on landmark combinatorial optimization problems such as Max Cut and maximum independent set. Our approach achieves strong empirical results across a wide range of real-world and synthetic datasets against both neural baselines and classical algorithms. Finally, we take advantage of OptGNN's ability to capture convex relaxations to design an algorithm for producing dual certificates of optimality (bounds on the optimal solution) from the learned embeddings of OptGNN.

翻译：在本文中，我们设计了一种图神经网络架构，利用半定规划（SDP）的强大算法工具，能够为一类广泛的组合优化问题获得最优近似算法。具体而言，我们证明：在唯一博弈猜想的假设下，多项式规模的消息传递算法可以表示最大约束满足问题中最强大的多项式时间算法。我们利用这一结果构建了高效的图神经网络架构OptGNN，该架构在最大割（Max Cut）和最大独立集（Maximum Independent Set）等里程碑式组合优化问题上获得了高质量的近似解。我们的方法在涵盖神经网络基线方法和经典算法的广泛真实与合成数据集上均取得了强劲的实证结果。最终，我们借助OptGNN捕捉凸松弛的能力，设计了一种从其学习嵌入中生成最优性对偶证书（最优解边界）的算法。