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

Nature Machine Intelligence volume 5, pages 26–28 (2023)Cite this article

The Original Article was published on 30 December 2022

reply to S. Boettcher Nature Machine Intelligence https://doi.org/10.1038/s42256-022-00587-0 (2022).

We provide a comprehensive reply to the Comment written by Stefan Boettcher1 and argue that the comment singles out one particular non-representative example problem, entirely focusing on the maximum cut (MaxCut) problem 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 data, thereby refuting the Comment’s original performance statements. Furthermore, it has already been shown that physics-inspired graph neural networks (PI-GNNs) can outperform greedy algorithms, particularly on hard, dense instances. We also argue that the internal (parallel) anatomy of GNNs is very different from the (sequential) nature of greedy algorithms, and (based on their usage at the scale of real-world social networks) point out that GNNs have demonstrated their potential for superior scalability compared with existing heuristics such as extremal optimization. We conclude by highlighting the conceptual novelty of our work and outline some potential extensions.

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Fig. 1: Relative cut size for 3-regular graphs.

References

  1. Boettcher, S. Inability of a graph neural network heuristic to outperform greedy algorithms in solving combinatorial optimization problems. Nat. Mach. Intell. https://doi.org/10.1038/s42256-022-00587-0 (2022).

  2. Schuetz, M. J. A., Brubaker, J. K. & Katzgraber, H. K. Combinatorial optimization with physics-inspired graph neural networks. Nat. Mach. Intell. 4, 367–377 (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: Inability of a graph neural network heuristic to outperform greedy algorithms in solving combinatorial optimization problems. Nat Mach Intell 5, 26–28 (2023). https://doi.org/10.1038/s42256-022-00588-z

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