The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지
DOI QR코드

DOI QR Code

A DEGREE REDUCTION METHOD FOR AN EFFICIENT QUBO FORMULATION FOR THE GRAPH COLORING PROBLEM

  • Received : 2023.07.14
  • Accepted : 2023.12.16
  • Published : 2024.02.28
Download PDF
⟨ Previous Next ⟩

Abstract

We introduce a new degree reduction method for homogeneous symmetric polynomials on binary variables that generalizes the conventional degree reduction methods on monomials introduced by Freedman and Ishikawa. We also design an degree reduction algorithm for general polynomials on binary variables, simulated on the graph coloring problem for random graphs, and compared the results with the conventional methods. The simulated results show that our new method produces reduced quadratic polynomials that contains less variables than the reduced quadratic polynomials produced by the conventional methods.

Keywords

Acknowledgement

This work was supported by TSHS R&E grant at Daegu Gyeongbuk Institute of Science and Technology (DGIST). (grant number 2022030163).

References

  1. Leighton, F.: A Graph Coloring Algorithm for Large Scheduling Problems. Journal of Research of the National Bureau of Standards 84 (1979), no. 6, 489-506. https://doi.org/10.6028/jres.084.024 
  2. Garey, M., Johnson, D. & Stockmeyer, L.: Some simplified NP-complete graph problems. Theoretical Computer Science 1 (1976), 237-267. https://doi.org/10.1016/0304-3975(76)90059-1 
  3. Chaitin, G.: Register Allocation & Spilling via Graph Coloring. SIGPLAN Not. 17 (1982), 98-101. https://doi.org/10.1145/872726.806984 
  4. Apolloni, B., Carvalho, C. & De Falco, D.: Quantum stochastic optimization. Stochastic Processes and Their Applications 33 (1989), 233-244. https://doi.org/10.1016/0304-4149(89)90040-9 
  5. Garey, M. & Johnson, D.: Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., 1990. 
  6. Kadowaki, T. & Nishimori, H.: Quantum annealing in the transverse ising model. Phys. Rev. E. 58 (1998), 5355-5363. https://doi.org/10.1103/PhysRevE.58.5355 
  7. Freedman, D. & Drineas, P.: Energy minimization via graph cuts: settling what is possible. 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05). vol. 2 (2005), 939-946. https://doi.org/10.1109/CVPR.2005.143 
  8. Balasundaram, B. & Butenko, S.: Graph Domination, Coloring and Cliques in Telecommunications. Handbook of Optimization in Telecommunications (2006), 865-890. https://doi.org/10.1007/978-0-387-30165-5-30 
  9. Ishikawa, H.: Transformation of General Binary MRF Minimization to the First-Order Case. IEEE Transactions on Pattern Analysis and Machine Intelligence 33 (2011), 1234-1249. https://doi.org/10.1109/TPAMI.2010.91 
  10. Sipser, M.: Introduction to the Theory of Computation. Course Technology, 2013. 
  11. Venegas-Andraca, S., Cruz-Santos, W., McGeoch, C. & Lanzagorta, M.: A cross-disciplinary introduction to quantum annealing-based algorithms. Contemporary Physics 59 (2018), 174-197. https://doi.org/10.1080/00107514.2018.1450720 
  12. Fang, Y. & Warburton, P.: Minimizing minor embedding energy: an application in quantum annealing. Quantum Information Processing 19 (2020), 191. https://doi.org/10.1007/s11128-020-02681-x 
  13. Silva, C., Aguiar, A., Lima, P. & Dutra, I.: Mapping graph coloring to quantum annealing. Quantum Machine Intelligence 2 (2020), 16. https://doi.org/10.1007/s42484-020-00028-4 
  14. Tabi, Z., El-Safty, K., Kallus, Z., Haga, P., Kozsik, T., Glos, A. & Zimboras, Z.: Quantum Optimization for the Graph Coloring Problem with Space-Efficient Embedding. 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) (2020), 56-62. https://doi.org/10.1109/QCE49297.2020.00018