Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Matters Arising
  • Published:

Modern graph neural networks do worse than classical greedy algorithms in solving combinatorial optimization problems like maximum independent set

Nature Machine Intelligence volume 5, pages 29–31 (2023)Cite this article

Subjects

Matters Arising to this article was published on 30 December 2022

The Original Article was published on 21 April 2022

arising from: Martin J. A. Schuetz et al. Nature Machine Intelligence https://doi.org/10.1038/s42256-022-00468-6 (2022).

The recent work by Schuetz et al.1 ‘Combinatorial optimization with physics-inspired graph neural networks’ introduces a physics-inspired unsupervised graph neural network (GNN) to solve combinatorial optimization problems on sparse graphs. To test the performances of these GNNs, the authors show numerical results for two fundamental problems: the maximum cut and the maximum independent set (MIS), concluding “that the graph neural network optimizer performs on par or outperforms existing solvers, with the ability to scale beyond the state of the art to problems with millions of variables”. Here we show that a simple greedy algorithm, running in almost linear time, can find solutions for the MIS problem of much better quality than the GNN in a much shorter time. In general, many claims of superiority of neural networks in solving combinatorial problems are at risk of being not solid enough, as we lack standard benchmarks based on really hard problems. We propose one of such hard benchmarks, and we hope to see future neural network optimizers tested on these problems before any claim of superiority is made.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Comparison of the GNN and a simple DGA in computing ISs in d-RRGs.

References

  1. Schuetz, M. J. A., Brubaker, J. K. & Katzgraber, H. G. Combinatorial optimization with physics-inspired graph neural networks. Nat. Mach. Intell. https://doi.org/10.1038/s42256-022-00468-6 (2022).

  2. Cappart, Q. et al. Combinatorial optimization and reasoning with graph neural networks. Preprint at https://arxiv.org/abs/2102.09544 (2021).

  3. Kotary, J., Fioretto, F., van Hentenryck, P., & Wilder, B. End-to-end constrained optimization learning: a survey. In Proceedings of the 30th International Joint Conference on Artificial Intelligence (Ed. Zhou, Z-H.) 4475–4482 (IJCAI, 2021).

  4. Bengio, Y., Lodi, A. & Prouvost, A. Machine learning for combinatorial optimization: a methodological tour d’horizon. Eur. J. Oper. Res. 290, 405–421 (2021).

    MATH  Google Scholar 

  5. Selsam, D. et al. Learning a sat solver from single-bit supervision. In Proc. 7th International Conference on Learning Representations (ICLR, 2019).

  6. Cameron, C., Chen, R., Hartford, J. & Leyton-Brown, K. Predicting propositional satisfiability via end-to-end learning. In Proc. AAAI Conference on Artificial Intelligence Vol. 34, 3324–3331 (AAAI, 2020).

  7. Toenshoff, J., Ritzert, M., Wolf, H. & Grohe, M. Graph neural networks for maximum constraint satisfaction. Front Artif. Intell. 3, 98 (2021).

    Google Scholar 

  8. Boppana, R. & Halldórsson, M. M. Approximating maximum independent sets by excluding subgraphs. BIT Numer. Math. 32, 180–196 (1992).

    MATH  Google Scholar 

  9. Karp, R. M. & Sipser, M. Maximum matching in sparse random graphs. In 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981) 364–375 (IEEE, 1981).

  10. Wormald, N. C. Differential equations for random processes and random graphs. Ann. Appl. Probab. 5, 1217–1235 (1995).

    MATH  Google Scholar 

  11. McKay, B. D. Independent sets in regular graphs of high girth. Ars Combinatoria 23, 179–185 (1987).

    MATH  Google Scholar 

  12. Barbier, J., Krzakala, F., Zdeborová, L. & Zhang, P. The hard-core model on random graphs revisited. J. Phys. Conf.Ser. 473, 012021 (2013).

    Google Scholar 

  13. Angelini, M. C. & Ricci-Tersenghi, F. Monte Carlo algorithms are very effective in finding the largest independent set in sparse random graphs. Phys. Rev. E 100, 013302 (2019).

    Google Scholar 

  14. Coja-Oghlan, A. & Efthymiou, C. On independent sets in random graphs. Random Struct. Algorithms 47, 436–486 (2015).

    MATH  Google Scholar 

  15. Boettcher, S. Nat. Mach. Intell. (2022).

  16. Levinas, I. & Louzoun, Y. Planted dense subgraphs in dense random graphs can be recovered using graph-based machine learning. J. Artif. Intell. Res. 75, 541–568 (2022).

    MATH  Google Scholar 

  17. Deshpande, Y. & Montanari, A. Finding hidden cliques of size \(\sqrt{N/e}\) in nearly linear time. Found. Comput. Math. 15, 1069–1128 (2015).

    MATH  Google Scholar 

  18. Angelini, M. C. Parallel tempering for the planted clique problem. J. Stat. Mech. 2018, 073404 (2018).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Fisica, Sapienza Università di Roma, Rome, Italy

    Maria Chiara Angelini & Federico Ricci-Tersenghi

  2. Istituto Nazionale di Fisica Nucleare, Rome, Italy

    Maria Chiara Angelini & Federico Ricci-Tersenghi

  3. Institute of Nanotechnology (NANOTEC) - CNR, Rome unit, Rome, Italy

    Federico Ricci-Tersenghi

Authors
  1. Maria Chiara Angelini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Federico Ricci-Tersenghi
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally to this manuscript.

Corresponding author

Correspondence to Maria Chiara Angelini.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Machine Intelligence thanks the anonymous reviewers for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Angelini, M.C., Ricci-Tersenghi, F. Modern graph neural networks do worse than classical greedy algorithms in solving combinatorial optimization problems like maximum independent set. Nat Mach Intell 5, 29–31 (2023). https://doi.org/10.1038/s42256-022-00589-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s42256-022-00589-y

This article is cited by

Search

Advanced search

Quick links

Nature Briefing AI and Robotics

Sign up for the Nature Briefing: AI and Robotics newsletter — what matters in AI and robotics research, free to your inbox weekly.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing: AI and Robotics