  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.
