# Approximation Ratios of Graph Neural Networks for Combinatorial Problems

arXiv: https://arxiv.org/abs/1905.10261 · HTML: https://mlsanity.com/paper/1905.10261/

## Key facts

- Cited by 69 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 May 2019.

## Built on by (most cited first, 30 of 69)

- [Graph Neural Networks with Learnable Structural and Positional Representations](https://mlsanity.com/paper/2110.07875/) (2021)
- [Combinatorial optimization and reasoning with graph neural networks](https://mlsanity.com/paper/2102.09544/) (2021)
- [How Powerful are Spectral Graph Neural Networks](https://mlsanity.com/paper/2205.11172/) (2022)
- [DropGNN: Random Dropouts Increase the Expressiveness of Graph Neural Networks](https://mlsanity.com/paper/2111.06283/) (2021)
- [Breaking the Limits of Message Passing Graph Neural Networks](https://mlsanity.com/paper/2106.04319/) (2021)
- [Theory of Graph Neural Networks: Representation and Learning](https://mlsanity.com/paper/2204.07697/) (2022)
- [Provably expressive temporal graph networks](https://mlsanity.com/paper/2209.15059/) (2022)
- [Optimization of Graph Neural Networks: Implicit Acceleration by Skip Connections and More Depth](https://mlsanity.com/paper/2105.04550/) (2021)
- [How does over-squashing affect the power of GNNs?](https://mlsanity.com/paper/2306.03589/) (2023)
- [Graph Neural Networks with Local Graph Parameters](https://mlsanity.com/paper/2106.06707/) (2021)
- [KerGNNs: Interpretable Graph Neural Networks with Graph Kernels](https://mlsanity.com/paper/2201.00491/) (2022)
- [A Theoretical Comparison of Graph Neural Network Extensions](https://mlsanity.com/paper/2201.12884/) (2022)
- [GraphGDP: Generative Diffusion Processes for Permutation Invariant Graph Generation](https://mlsanity.com/paper/2212.01842/) (2022)
- [Generalization in Graph Neural Networks: Improved PAC-Bayesian Bounds on Graph Diffusion](https://mlsanity.com/paper/2302.04451/) (2023)
- [On Representing Linear Programs by Graph Neural Networks](https://mlsanity.com/paper/2209.12288/) (2022)
- [Equivariant Polynomials for Graph Neural Networks](https://mlsanity.com/paper/2302.11556/) (2023)
- [Provably Powerful Graph Neural Networks for Directed Multigraphs](https://mlsanity.com/paper/2306.11586/) (2023)
- [The Expressive Power of Graph Neural Networks: A Survey](https://mlsanity.com/paper/2308.08235/) (2023)
- [Sign Language Translation with Hierarchical Spatio-TemporalGraph Neural Network](https://mlsanity.com/paper/2111.07258/) (2021)
- [Learning Efficient Positional Encodings with Graph Neural Networks](https://mlsanity.com/paper/2502.01122/) (2025)
- [On Representing Mixed-Integer Linear Programs by Graph Neural Networks](https://mlsanity.com/paper/2210.10759/) (2022)
- [Going beyond persistent homology using persistent homology](https://mlsanity.com/paper/2311.06152/) (2023)
- [Bridging the Gap between Spatial and Spectral Domains: A Unified Framework for Graph Neural Networks](https://mlsanity.com/paper/2107.10234/) (2021)
- [Going Deeper into Permutation-Sensitive Graph Neural Networks](https://mlsanity.com/paper/2205.14368/) (2022)
- [Deep Reinforcement Learning for Combinatorial Optimization: Covering Salesman Problems](https://mlsanity.com/paper/2102.05875/) (2021)
- [Rethinking the Capacity of Graph Neural Networks for Branching Strategy](https://mlsanity.com/paper/2402.07099/) (2024)
- [Are Graph Neural Networks Optimal Approximation Algorithms?](https://mlsanity.com/paper/2310.00526/) (2023)
- [Graph neural networks extrapolate out-of-distribution for shortest paths](https://mlsanity.com/paper/2503.19173/) (2025)
- [Learning Graph Algorithms With Recurrent Graph Neural Networks](https://mlsanity.com/paper/2212.04934/) (2022)
- [Lumos: Heterogeneity-aware Federated Graph Learning over Decentralized Devices](https://mlsanity.com/paper/2303.00492/) (2023)

## Abstract (arXiv)

In this paper, from a theoretical perspective, we study how powerful graph neural networks (GNNs) can be for learning approximation algorithms for combinatorial problems. To this end, we first establish a new class of GNNs that can solve a strictly wider variety of problems than existing GNNs. Then, we bridge the gap between GNN theory and the theory of distributed local algorithms. We theoretically demonstrate that the most powerful GNN can learn approximation algorithms for the minimum dominating set problem and the minimum vertex cover problem with some approximation ratios with the aid of the theory of distributed local algorithms. We also show that most of the existing GNNs such as GIN, GAT, GCN, and GraphSAGE cannot perform better than with these ratios. This paper is the first to elucidate approximation ratios of GNNs for combinatorial problems. Furthermore, we prove that adding coloring or weak-coloring to each node feature improves these approximation ratios. This indicates that preprocessing and feature engineering theoretically strengthen model capabilities.