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：在这项工作中，我们利用半定规划（SDP）的强大算法工具，设计了可用于大规模组合优化问题获得最优近似算法的图神经网络架构。具体而言，我们证明：在唯一博弈猜想成立的前提下，多项式规模的消息传递算法能够表示用于最大约束满足问题的最强多项式时间算法。基于此结果，我们构建了高效的图神经网络架构OptGNN，该架构能够在Max Cut和最大独立集等标志性组合优化问题上获得高质量的近似解。我们的方法在涵盖广泛真实场景与合成数据集的实验中，相较于神经网络基线方法和经典算法均取得了优异的实证结果。最后，我们利用OptGNN捕捉凸松弛的能力，设计了一种算法——通过从OptGNN习得的嵌入中生成最优性对偶证书（即最优解边界）。