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Reply to: 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 32–34 (2023)Cite this article

The Original Article was published on 30 December 2022

reply to: Maria Chiara Angelini & Federico Ricci-Tersenghi Nature Machine Intelligence https://doi.org/10.1038/s42256-022-00589-y (2022).

We provide a comprehensive reply to the comment written by Angelini and Ricci-Tersenghi1 and argue that the comment singles out one particular non-representative example problem, entirely focusing on the maximum independent set (MIS) on sparse graphs, for which greedy algorithms are expected to perform well. Conversely, we highlight the broader algorithmic development underlying our original work2, and (within our original framework) provide additional numerical results showing sizable improvements over our original results, thereby refuting the comment’s performance statements. We also provide results showing runtime scaling superior to the results provided by Angelini and Ricci-Tersenghi. Furthermore, we show that the proposed set of random d-regular graphs does not provide a universal set of benchmark instances, nor do greedy heuristics provide a universal algorithmic baseline. Finally, we argue that the internal (parallel) anatomy of graph neural networks (GNNs) is very different from the (sequential) nature of greedy algorithms and emphasize that GNNs have demonstrated their potential for superior scalability compared with existing heuristics such as parallel tempering. We conclude by discussing the conceptual novelty of our work and outline some potential extensions.

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Fig. 1: Size of the independent set for random d-regular graphs, as a function of the number of nodes n, reported as relative approximation ratio with respect to the known theoretical upper bounds.

References

  1. 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. https://doi.org/10.1038/s42256-022-00589-y (2022).

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

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Authors and Affiliations

  1. Amazon Quantum Solutions Lab, Seattle, WA, USA

    Martin J. A. Schuetz, J. Kyle Brubaker & Helmut G. Katzgraber

  2. AWS Center for Quantum Computing, Pasadena, CA, USA

    Martin J. A. Schuetz & Helmut G. Katzgraber

Authors
  1. Martin J. A. Schuetz
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  2. J. Kyle Brubaker
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  3. Helmut G. Katzgraber
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Contributions

All authors contributed to the ideation and design of the research. M.J.A.S. and J.K.B. developed and ran the computational experiments, as well as wrote the initial draft of the the manuscript. H.G.K. supervised this work and revised the manuscript.

Corresponding authors

Correspondence to Martin J. A. Schuetz, J. Kyle Brubaker or Helmut G. Katzgraber.

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Competing interests

M.J.A.S., J.K.B. and H.G.K. are listed as inventors on a US provisional patent application (no. 7924-38500) on combinatorial optimization with graph neural networks.

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

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Schuetz, M.J.A., Brubaker, J.K. & Katzgraber, H.G. Reply to: Modern graph neural networks do worse than classical greedy algorithms in solving combinatorial optimization problems like maximum independent set. Nat Mach Intell 5, 32–34 (2023). https://doi.org/10.1038/s42256-022-00590-5

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