Are Graph Neural Networks Optimal Approximation Algorithms?

TL;DR

This paper demonstrates that graph neural networks can be designed to match the power of the best polynomial-time algorithms for certain combinatorial problems, leveraging semidefinite programming tools.

Contribution

It introduces OptGNN, a GNN architecture that captures optimal approximation algorithms for combinatorial problems using SDP-based insights.

Findings

OptGNN achieves high-quality approximations on Max-Cut, Min-Vertex-Cover, and Max-3-SAT.

OptGNN outperforms traditional solvers and neural baselines on various datasets.

The method provides bounds on optimal solutions from learned embeddings.

Abstract

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.