Gumbel-Softmax Optimization: A Simple General Framework for Combinatorial Optimization Problems on Graphs

  • Jing Liu
  • Fei Gao
  • Jiang ZhangEmail author
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 881)


Many problems in real life can be converted to combinatorial optimization problems (COPs) on graphs, that is to find a best node state configuration or a network structure such that the designed objective function is optimized under some constraints. However, these problems are notorious for their hardness to solve because most of them are NP-hard or NP-complete. Although traditional general methods such as simulated annealing (SA), genetic algorithms (GA) and so forth have been devised to these hard problems, their accuracy and time consumption are not satisfying in practice. In this work, we proposed a simple, fast, and general algorithm framework called Gumbel-softmax Optimization (GSO) for COPs. By introducing Gumbel-softmax technique which is developed in machine learning community, we can optimize the objective function directly by gradient descent algorithm regardless of the discrete nature of variables. We test our algorithm on four different problems including Sherrington-Kirkpatrick (SK) model, maximum independent set (MIS) problem, modularity optimization, and structural optimization problem. High-quality solutions can be obtained with much less time consuming compared to traditional approaches.


Gumbel-softmax Combinatorial optimization problems Sherrington-Kirkpatrick model Maximum independent set 


  1. 1.
    Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103. Springer (1972)Google Scholar
  2. 2.
    Mézard, M., Parisi, G., Virasoro, M.: Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications, vol. 9. World Scientific Publishing Company (1987)Google Scholar
  3. 3.
    Newman, M.E.J.: Modularity and community structure in networks. Proce. Natl. Acad. Sci. 103(23), 8577–8582 (2006)CrossRefGoogle Scholar
  4. 4.
    Timme, M.: Revealing network connectivity from response dynamics. Phys. Rev. Lett. 98(22), 224101 (2007)CrossRefGoogle Scholar
  5. 5.
    Casadiego, J., Nitzan, M., Hallerberg, S., Timme, M.: Model-free inference of direct network interactions from nonlinear collective dynamics. Nat. Commun. 8(1), 2192 (2017)CrossRefGoogle Scholar
  6. 6.
    Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Davis, L.: Handbook of Genetic Algorithms (1991)Google Scholar
  8. 8.
    Boettcher, S., Percus, A.: Nature’s way of optimizing. Artif. Intell. 119(1–2), 275–286 (2000)CrossRefGoogle Scholar
  9. 9.
    Andrade, D.V., Resende, M.G.C., Werneck, R.F.: Fast local search for the maximum independent set problem. J. Heuristics 18(4), 525–547 (2012)CrossRefGoogle Scholar
  10. 10.
    Khalil, E., Dai, H., Zhang, Y., Dilkina, B., Song, L.: Learning combinatorial optimization algorithms over graphs. In: Advances in Neural Information Processing Systems, pp. 6348–6358 (2017)Google Scholar
  11. 11.
    Li, Z., Chen, Q., Koltun, V.: Combinatorial optimization with graph convolutional networks and guided tree search. In: Advances in Neural Information Processing Systems, pp. 539–548 (2018)Google Scholar
  12. 12.
    Kipf, T.N., Welling, M.: Semi-supervised classification with graph convolutional networks. In: 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, 24–26 April 2017, Conference Track Proceedings (2017).
  13. 13.
    Jang, E., Gu, S., Poole, B.: Categorical reparameterization with gumbel-softmax. In: 5th International Conference on Learning Representations, ICLR 2017, Conference Track Proceedings, Toulon, France, 24–26 April 2017. (2017)Google Scholar
  14. 14.
    Maddison, C.J., Mnih, Teh, A.W.: The concrete distribution: a continuous relaxation of discrete random variables. In: 5th International Conference on Learning Representations, ICLR 2017, Conference Track Proceedings, Toulon, France, 24–26 April 2017 (2017)Google Scholar
  15. 15.
    Sherrington, D., Kirkpatrick, S.: Solvable model of a spin-glass. Phys. Rev. Lett. 35(26), 1792 (1975)CrossRefGoogle Scholar
  16. 16.
    Fortunato, S.: Community detection in graphs. Phys. Rep. 486(3–5), 75–174 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Newman, M.E.J.: Fast algorithm for detecting community structure in networks. Phys. Rev. E 69(6), 066133 (2004)CrossRefGoogle Scholar
  18. 18.
    Duch, J., Arenas, A.: Community detection in complex networks using extremal optimization. Phys. Rev. E 72(2), 027104 (2005)CrossRefGoogle Scholar
  19. 19.
    Brandes, U., Delling, D., Gaertler, M., Görke, R., Hoefer, M., Nikoloski, Z., Wagner, D.: Maximizing modularity is hard. arXiv preprint physics/0608255 (2006)Google Scholar
  20. 20.
    Eusuff, M.M., Lansey, K.E.: Optimization of water distribution network design using the shuffled frog leaping algorithm. J. Water Resour. Plann. Manag. 129(3), 210–225 (2003)CrossRefGoogle Scholar
  21. 21.
    Boyce, D.E., Farhi, A., Weischedel, R.: Optimal network problem: a branch-and-bound algorithm. Environ. Plann. A 5(4), 519–533 (1973)CrossRefGoogle Scholar
  22. 22.
    Wainwright, M.J., Jordan, M.I., et al.: Graphical models, exponential families, and variational inference. Found. Trends® Mach. Learn. 1(1–2), 1–305 (2008)CrossRefGoogle Scholar
  23. 23.
    Paszke, A., Gross, S., Chintala, S., Chanan, G., Yang, E., DeVito, Z., Lin, Z., Antiga, L., Lerer, A.: Automatic differentiation in PyTorch. In: NIPS-W, Alban Desmaison (2017)Google Scholar
  24. 24.
    Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., Devin, M., Ghemawat, S., Irving, G., Isard, M., et al.: Tensorflow: a system for large-scale machine learning. In: 12th {USENIX } Symposium on Operating Systems Design and Implementation ({ OSDI} 2016), pp. 265–283 (2016)Google Scholar
  25. 25.
    Williams, R.J.: Simple statistical gradient-following algorithms for connectionist reinforcement learning. Mach. Learn. 8(3–4), 229–256 (1992)CrossRefGoogle Scholar
  26. 26.
    Kingma, D.P., Ba, J.: Adam: a method for stochastic optimization. In: 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, 7–9 May 2015, Conference Track Proceedings (2015).
  27. 27.
    Boettcher, S.: Extremal optimization for sherrington-kirkpatrick spin glasses. Eur. Phys. J. B-Condens. Matter Complex Syst. 46(4), 501–505 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.School of Systems ScienceBeijing Normal UniversityBeijingChina

Personalised recommendations