In this work we design graph neural network architectures that capture 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, Min-Vertex-Cover, and Max-3-SAT. Our approach achieves strong empirical results across a wide range of real-world and synthetic datasets against solvers and neural baselines. Finally, we take advantage of OptGNN's ability to capture convex relaxations to design an algorithm for producing 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，设计了能够捕捉一类大规模组合优化问题最优逼近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最强大的多项式时间算法。基于此结果，我们构建了高效的图神经网络架构OptGNN，该架构在Max-Cut、Min-Vertex-Cover和Max-3-SAT等标志性组合优化问题上获得了高质量的近似解。我们的方法在涵盖真实世界数据和合成数据的广泛数据集上，相较于求解器和神经基线方法均取得了强实证结果。最后，我们利用OptGNN捕捉凸松弛的能力，设计了一种从OptGNN学习嵌入中生成最优解边界的算法。