Which Algorithms Can Graph Neural Networks Learn?

Solveig Wittig, Antonis Vasileiou, Robert R Nerem, Timo Stoll, Floris Geerts, Yusu Wang, Christopher Morris
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:135232-135307, 2026.

Abstract

In recent years, there has been growing interest in understanding neural architectures’ ability to learn to execute discrete algorithms, a line of work often referred to as neural algorithmic reasoning. The goal is to integrate algorithmic reasoning capabilities into larger neural pipelines. Many such architectures are based on (message-passing) graph neural networks (MPNNs), owing to their permutation equivariance and ability to deal with sparsity and variable-sized inputs. However, much existing work is either largely empirical and lacks formal guarantees or it focuses solely on expressivity, leaving open the question of when and how such architectures generalize beyond a finite training set. In this work, we propose a general theoretical framework that characterizes sufficient conditions under which MPNNs can learn an algorithm from a training set of small instances and provably approximate its behavior on inputs of arbitrary size with worst-case guarantees. Our framework applies to a broad class of algorithms, including single-source shortest paths, minimum spanning trees, and general dynamic programming problems, such as the $0$-$1$ knapsack problem. In addition, we establish impossibility results for a wide range of algorithmic tasks, showing that standard MPNNs cannot learn them and derive more expressive MPNN-like architectures that overcome these limitations. Finally, we refine our analysis for the Bellman–Ford algorithm, yielding substantially smaller required training sets and significantly extending the recent work of Nerem et al., 2025 by allowing for a differentiable regularization loss. Empirical results largely support our theoretical findings.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-wittig26a, title = {Which Algorithms Can Graph Neural Networks Learn?}, author = {Wittig, Solveig and Vasileiou, Antonis and Nerem, Robert R and Stoll, Timo and Geerts, Floris and Wang, Yusu and Morris, Christopher}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {135232--135307}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/wittig26a/wittig26a.pdf}, url = {https://proceedings.mlr.press/v306/wittig26a.html}, abstract = {In recent years, there has been growing interest in understanding neural architectures’ ability to learn to execute discrete algorithms, a line of work often referred to as neural algorithmic reasoning. The goal is to integrate algorithmic reasoning capabilities into larger neural pipelines. Many such architectures are based on (message-passing) graph neural networks (MPNNs), owing to their permutation equivariance and ability to deal with sparsity and variable-sized inputs. However, much existing work is either largely empirical and lacks formal guarantees or it focuses solely on expressivity, leaving open the question of when and how such architectures generalize beyond a finite training set. In this work, we propose a general theoretical framework that characterizes sufficient conditions under which MPNNs can learn an algorithm from a training set of small instances and provably approximate its behavior on inputs of arbitrary size with worst-case guarantees. Our framework applies to a broad class of algorithms, including single-source shortest paths, minimum spanning trees, and general dynamic programming problems, such as the $0$-$1$ knapsack problem. In addition, we establish impossibility results for a wide range of algorithmic tasks, showing that standard MPNNs cannot learn them and derive more expressive MPNN-like architectures that overcome these limitations. Finally, we refine our analysis for the Bellman–Ford algorithm, yielding substantially smaller required training sets and significantly extending the recent work of Nerem et al., 2025 by allowing for a differentiable regularization loss. Empirical results largely support our theoretical findings.} }
Endnote
%0 Conference Paper %T Which Algorithms Can Graph Neural Networks Learn? %A Solveig Wittig %A Antonis Vasileiou %A Robert R Nerem %A Timo Stoll %A Floris Geerts %A Yusu Wang %A Christopher Morris %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-wittig26a %I PMLR %P 135232--135307 %U https://proceedings.mlr.press/v306/wittig26a.html %V 306 %X In recent years, there has been growing interest in understanding neural architectures’ ability to learn to execute discrete algorithms, a line of work often referred to as neural algorithmic reasoning. The goal is to integrate algorithmic reasoning capabilities into larger neural pipelines. Many such architectures are based on (message-passing) graph neural networks (MPNNs), owing to their permutation equivariance and ability to deal with sparsity and variable-sized inputs. However, much existing work is either largely empirical and lacks formal guarantees or it focuses solely on expressivity, leaving open the question of when and how such architectures generalize beyond a finite training set. In this work, we propose a general theoretical framework that characterizes sufficient conditions under which MPNNs can learn an algorithm from a training set of small instances and provably approximate its behavior on inputs of arbitrary size with worst-case guarantees. Our framework applies to a broad class of algorithms, including single-source shortest paths, minimum spanning trees, and general dynamic programming problems, such as the $0$-$1$ knapsack problem. In addition, we establish impossibility results for a wide range of algorithmic tasks, showing that standard MPNNs cannot learn them and derive more expressive MPNN-like architectures that overcome these limitations. Finally, we refine our analysis for the Bellman–Ford algorithm, yielding substantially smaller required training sets and significantly extending the recent work of Nerem et al., 2025 by allowing for a differentiable regularization loss. Empirical results largely support our theoretical findings.
APA
Wittig, S., Vasileiou, A., Nerem, R.R., Stoll, T., Geerts, F., Wang, Y. & Morris, C.. (2026). Which Algorithms Can Graph Neural Networks Learn?. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:135232-135307 Available from https://proceedings.mlr.press/v306/wittig26a.html.

Related Material