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.

我们设计了图神经网络架构，利用半定规划的强大算法工具，来获得一类组合优化问题的最优逼近算法。通过研究，我们证明了多项式大小的消息传递算法可以表示 Max Constraint Satisfaction Problems 的最强多项式时间算法（基于 Unique Games Conjecture）。我们利用这一结果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经典组合优化问题上获得了高质量的近似解。我们的方法在各种真实和合成数据集上取得了强有力的实证结果，超过了神经基准和传统算法。最后，我们利用 OptGNN 捕捉凸松弛的能力，设计了一个从 OptGNN 学习的嵌入中产生对优解的对偶证明的算法。

图神经网络是否最优逼近算法？