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Combinatorial optimization with physics-inspired graph neural networks

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Matters Arising to this article was published on 30 December 2022

Matters Arising to this article was published on 30 December 2022

A preprint version of the article is available at arXiv.

Abstract

Combinatorial optimization problems are pervasive across science and industry. Modern deep learning tools are poised to solve these problems at unprecedented scales, but a unifying framework that incorporates insights from statistical physics is still outstanding. Here we demonstrate how graph neural networks can be used to solve combinatorial optimization problems. Our approach is broadly applicable to canonical NP-hard problems in the form of quadratic unconstrained binary optimization problems, such as maximum cut, minimum vertex cover, maximum independent set, as well as Ising spin glasses and higher-order generalizations thereof in the form of polynomial unconstrained binary optimization problems. We apply a relaxation strategy to the problem Hamiltonian to generate a differentiable loss function with which we train the graph neural network and apply a simple projection to integer variables once the unsupervised training process has completed. We showcase our approach with numerical results for the canonical maximum cut and maximum independent set problems. We find 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.

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Fig. 1: Schematic of the GNN approach for combinatorial optimization presented in this work.
Fig. 2: Flow chart illustrating the end-to-end workflow for the proposed physics-inspired GNN optimizer.
Fig. 3: Example solution to MaxCut for a random 3-regular graph with n = 100 nodes.
Fig. 4: Numerical results for MaxCut.
Fig. 5: Numerical results for the MIS problem.

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Data availability

The data necessary to reproduce our numerical benchmark results are publicly available at https://web.stanford.edu/~yyye/yyye/Gset/. Random d-regular graphs have been generated using the open-source networkx library (https://networkx.org).

Code availability

An end-to-end open source demo version of the code implementing our approach has been made publicly available at https://github.com/amazon-research/co-with-gnns-example116.

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Acknowledgements

We thank F. Brandao, G. Karypis, M. Kastoryano, E. Kessler, T. Mullenbach, N. Pancotti, M. Resende, S. Roy, G. Salton, S. Severini, A. Urweisse and J. Zhu for fruitful discussions.

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Authors and Affiliations

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

    Martin J. A. Schuetz & Helmut G. Katzgraber

  2. AWS Intelligent and Advanced Compute Technologies, Professional Services, Seattle, WA, USA

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

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

    Martin J. A. Schuetz & Helmut G. Katzgraber

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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 and also wrote the initial draft of the the manuscript. H.G.K. supervised this work and revised the manuscript.

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

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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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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). https://doi.org/10.1038/s42256-022-00468-6

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