Are Graph Neural Networks Optimal Approximation Algorithms?
This work addresses the challenge of efficiently solving combinatorial optimization problems for researchers and practitioners, though it is incremental as it builds on existing algorithmic tools like semidefinite programming.
The paper tackles the problem of designing graph neural networks that capture optimal approximation algorithms for combinatorial optimization problems, achieving strong empirical results on benchmarks like Max-Cut and Max-3-SAT against solvers and neural baselines.
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.