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:

Inability of a graph neural network heuristic to outperform greedy algorithms in solving combinatorial optimization problems

Nature Machine Intelligence volume 5, pages 24–25 (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: M. J. A. Schuetz et al. Nature Machine Intelligence https://doi.org/10.1038/s42256-022-00468-6 (2022).

In ref. 1, Schuetz et al. provide a scheme to employ graph neural networks (GNNs) as a heuristic to solve a variety of classical NP-hard combinatorial optimization problems. They described how the network was trained on sample instances, and the resulting GNN heuristic was evaluated by applying widely used techniques to determine its ability to succeed. The idea of harnessing the powerful abilities of such networks to ‘learn’ the intricacies of complex, multimodal energy landscapes in such a hands-off approach is enticing, and the observed performance of the heuristic suggests that it may be highly scalable with a computational cost linear to the input size n (although there is probably some overhead in the pre-factor due to the GNN itself). However, closer inspection shows that the reported results for this GNN are only minutely better than those for gradient descent and are outperformed by greedy algorithms such as maximum cut (MaxCut). The discussion also highlights what I believe are some common misconceptions in evaluations of heuristics.

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: Results for various heuristics and bounds for the MaxCut problem on a 3-regular random graph ensemble.

References

  1. Schuetz, M. J. A., Brubaker, J. K. & Katzgraber, H. G. Combinatorial optimization with physics-inspired graph neural networks. Nat. Mach. Intell. 4, 367–377 (2022).

    Google Scholar 

  2. Zdeborová, L. & Boettcher, S. A conjecture on the maximum cut and bisection width in random regular graphs. J. Stat. Mech. Theory Exp. 2010, P02020 (2010).

    Google Scholar 

  3. Mezard, M. & Parisi, G. The cavity method at zero temperature. J. Stat. Phys. 111, 1–34 (2003).

    MATH  Google Scholar 

  4. Boettcher, S. Numerical results for ground states of spin glasses on Bethe lattices. Euro. Phys. J. B 31, 29–39 (2003).

  5. Parisi, G. A sequence of approximated solutions to the S-K model for spin glasses. J. Phys. A 13, L115–L121 (1980).

  6. Dembo, A., Montanari, A. & Sen, S. Extremal cuts of sparse random graphs. Ann. Probab. 45, 1190–1217 (2017).

  7. Boettcher, S. Analysis of the relation between quadratic unconstrained binary optimization and the spin-glass ground-state problem. Phys. Rev. Res. 1, 033142 (2019).

    Google Scholar 

  8. Boettcher, S. Simulations of ground state fluctuations in mean-field Ising spin glasses. J. Stat. Mech. Theory Exp. 2010, P07002 (2010).

    Google Scholar 

  9. Boettcher, S. Ground state properties of the diluted Sherrington-Kirkpatrick spin glass. Phys. Rev. Lett. 124, 177202 (2020).

    Google Scholar 

  10. Mezard, M. & Parisi, G. Mean-field theory of randomly frustrated systems with finite connectivity. Europhys. Lett. 3, 1067–1074 (1987).

    Google Scholar 

  11. Angelini, M.C. & Tersenghi, R.R. Cracking nuts with a sledgehammer: when modern graph neural networks do worse than classical greedy algorithms. https://doi.org/10.48550/arXiv.2206.13211 (2022).

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Emory University, Atlanta, GA, USA

    Stefan Boettcher

Authors
  1. Stefan Boettcher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefan Boettcher.

Ethics declarations

Competing interests

The author declares 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

Boettcher, S. Inability of a graph neural network heuristic to outperform greedy algorithms in solving combinatorial optimization problems. Nat Mach Intell 5, 24–25 (2023). https://doi.org/10.1038/s42256-022-00587-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s42256-022-00587-0

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