Primal-Dual Graph Neural Networks for General NP-Hard Combinatorial Optimization

Yu He, Ellen Vitercik

Keywords: neural algorithmic reasoning, graph neural networks, combinatorial optimization

TL;DR: We propose a general GNN-based framework using primal-dual approximation algorithms to approach NP-hard combinatorial optimization problems, advancing neural algorithmic reasoning beyond the polynomial-time-solvable domain.

Abstract:

Neural algorithmic reasoning (NAR) seeks to train neural networks, particularly Graph Neural Networks (GNNs), to simulate and generalize traditional algorithms, enabling them to perform structured reasoning on complex data. Previous research has primarily focused on algorithms for polynomial-time-solvable problems. However, many of the most critical problems in practice are NP-hard, exposing a critical gap in NAR. In this work, we propose a general NAR framework to learn algorithms for NP-hard problems, built on the classical primal-dual framework for designing efficient approximation algorithms. We enhance this framework by integrating optimal solutions to these NP-hard problems, enabling the model to surpass the performance of the approximation algorithms it was initially trained on. To the best of our knowledge, this is the first NAR method explicitly designed to surpass the performance of the classical algorithm on which it is trained. We evaluate our framework on several NP-hard problems, demonstrating its ability to generalize to larger and out-of-distribution graph families. In addition, we demonstrate the practical utility of the framework in two key applications: as a warm start for commercial solvers to reduce search time, and as a tool to generate embeddings that enhance predictive performance on real-world datasets. Our results highlight the scalability and effectiveness of the NAR framework for tackling complex combinatorial optimization problems, advancing their utility beyond the scope of traditional polynomial-time-solvable problems

Primary Area: learning on graphs and other geometries & topologies

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Submission Number: 5426