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：本文利用半定规划（SDP）的强大算法工具，设计了一类可用于获得大规模组合优化问题最优近似解的图神经网络架构。具体而言，我们在唯一博弈猜想假设下证明了：对于最大约束满足问题，多项式规模的消息传递算法能够表示最强大的多项式时间算法。基于该结果，我们构建了高效的图神经网络架构OptGNN，在最大割和最大独立集等标志性组合优化问题上获得了高质量近似解。该方法在广泛真实数据集与合成数据集上，相较于神经网络基线模型和经典算法均取得了优异实证结果。最后，我们利用OptGNN捕捉凸松弛的能力，设计了一种算法，可从其学习到的嵌入中生成最优性对偶证书（最优解的上界/下界）。