Skip to main content
SpringerLink
Log in

Enhanced discrete dragonfly algorithm for solving four-color map problems

  • Published:
Applied Intelligence Aims and scope Submit manuscript

Abstract

The classic combinatorial optimization problem of graph coloring is one of the most famous NP-complete problems. One example of the graph coloring problem is the four-color map problem. There have been many applications of swarm intelligence optimization algorithms to this problem, but to date, such algorithms can only solve the four-color map problem with fewer than 100 regions. This article proposes an enhanced discrete dragonfly algorithm (EDDA) for four-color map problems. We use global and local discrete alternate search strategies-when there is at least one adjacent dragonfly around the i-th dragonfly, a global search is performed; when there are no other dragonflies around, a local search is performed. A greedy strategy, local differential cross strategy, and single-point switching strategy are then used to solve the problem of conflicts among adjacent nodes. Finally, six real-life maps are colored to verify the effectiveness of the proposed algorithm. The experimental results show that the proposed EDDA algorithm can solve the four-color map problem with more than 100 regions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33

Similar content being viewed by others

References

  1. Mirjalili S (2016) Dragonfly algorithm: a new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems. Neural Comput & Applic 27:1053–1073

    Article  Google Scholar 

  2. Gaurav D, Vijay K (2018) Emperor penguin optimizer: A bio-inspired algorithm for engineering problems. Knowl-Based Syst 159:20–50

    Article  Google Scholar 

  3. Zhong L, Zhou Y, Luo Q, Zhong K (2021) Wind driven dragonfly algorithm for global optimization. Concurrency and Computation: Practice and Experience 33:e6054

    Article  Google Scholar 

  4. Hussain K, Salleh M, Shi C, Shi Y (2018) Metaheuristic research: a comprehensive survey. Artif Intell Rev 5:1–43

    Google Scholar 

  5. Kennedy J, Eberhart R (1995) Particle swarm optimization. Proceedings of the IEEE International Conference on Neural Networks 4:1942–1948. https://doi.org/10.1109/ICNN.1995.488968

  6. Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61

    Article  Google Scholar 

  7. Arora S, Singh S (2019) Butterfly optimization algorithm: a novel approach for global optimization. Soft Comput 23:715–734

    Article  Google Scholar 

  8. Holland JH (1992) Genetic algorithms. Sci Am 267(1):66–72

    Article  Google Scholar 

  9. Storn R, Price K (1997) Differential evolution-A simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11:341–359

    Article  MathSciNet  MATH  Google Scholar 

  10. Adleman L (1994) Molecular computation of solutions to combinatorial problems. Science 266:1021–1024

    Article  Google Scholar 

  11. Macato P, Norman M, (1992) A memetic approach for the traveling salesman problem implementation of a computational ecology for combinatorial optimization on messade-passing systems. https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.50.1940&rep=rep1&type=pdf

  12. Jiao L, Wang L (2000) A novel genetic algorithm based on immunity. IEEE Transactions on Systems, Man & Cybernetics: Part A 30:552–561

    Article  Google Scholar 

  13. Van P, Aarts E (1987) Simulated annealing. In: Simulated annealing: theory and applications. Springer, Dordrecht, pp 7–15

    MATH  Google Scholar 

  14. Bayraktar Z, Komurcu M, Werner DH (2010) Wind Driven Optimization (WDO): A novel nature-inspired optimization algorithm and its application to electromagnetics. Antennas & Propagation Society International Symposium IEEE, 11-17 July 2010, Toronto

  15. Mirjalili S (2016) SCA: A sine cosine algorithm for solving optimization problems. Knowl-Based Syst 96:120–133

    Article  Google Scholar 

  16. Anita YA, Kumar N (2020) Artificial electric field algorithm for engineering optimization problems. Expert Syst Appl 149:113308

    Article  Google Scholar 

  17. Lam AYS, Li VOK (2010) Chemical-reaction-inspired metaheuristic for optimization. IEEE Trans Evol Comput 14:381–399

    Article  Google Scholar 

  18. Rao RV, Savsani VJ, Vakharia DP (2011) Teaching–learning-based optimization: A novel method for constrained mechanical design optimization problems. Computer Aided Design 43:303–315

    Article  Google Scholar 

  19. Qamar A, Younas I, Saeed M (2020) Political optimizer: A novel socio-inspired meta-heuristic for global optimization. Knowl-Based Syst 195:105709

    Article  Google Scholar 

  20. Esref B, Beyhan S (2020) Adolescent identity search algorithm (AISA): A novel metaheuristic approach for solving optimization problems. Appl Soft Comput 95:106503

    Article  Google Scholar 

  21. Bouchekara REHH (2020) Most valuable player algorithm: a novel optimization algorithm inspired from sport. Oper Res 20:139–195

    Google Scholar 

  22. Thompson KA, Dowslandj M (2005) Ant Colony optimization for the examination scheduling problem. J Oper Res Soc 56:426–438

    Article  MATH  Google Scholar 

  23. Brezonik L, Fister I, Podgorelec V (2018) Scrum Task Allocation Based on Particle Swarm Optimization. In: Korošec P, Melab N, Talbi EG (eds) Bioinspired Optimization Methods and Their Applications. BIOMA 2018. Lecture notes in computer science, vol. 10835. Springer, Cham

    Google Scholar 

  24. Meriem B, Guidoum N, SaiDouni DE (2015) A new and fast evolutionary algorithm for strict strong graph coloring problem. Procedia Computer Science 73:138–145

    Article  Google Scholar 

  25. Sarma SS, Mondal R, Seth A (1995) Some sequential graph coloring algorithms for restricted channel routing. Int J Electron 77(1):81–93

  26. Naz S, Akram M (2019) Novel decision-making approach based on hesitant fuzzy sets and graph theory. Comput Appl Math 38:7

    Article  MathSciNet  MATH  Google Scholar 

  27. Barman S, Pal M, Mondal S (2019) An optimal algorithm to find minimum k -hop dominating set of interval graphs. Discrete Mathematics Algorithms and Applications 11:1950016

    Article  MathSciNet  MATH  Google Scholar 

  28. Garey MR, Johnson DS, (1979) Computers and intractability: a guide to the theory of NP-completeness, W. H. Freeman. & Co. Subs. of Scientific American, Inc., New York

  29. Leite N, Fernandes CM, Melício F, Rosa AC (2018) A cellular memetic algorithm for the examination timetabling problem. Comput Oper Res 94:118–138

    Article  MathSciNet  MATH  Google Scholar 

  30. Leighton FT (1979) A graph coloring algorithm for large scheduling problems. Journal of Research of the National Bureau of Standards (United States) 84(6):489–506

    Article  MathSciNet  MATH  Google Scholar 

  31. Gamst A (2006) Some lower bounds for a class of frequency assignment problems. IEEE Trans Veh Technol 35:8–14

    Article  Google Scholar 

  32. Chow FC, Hennessy JL (1990) The priority-based coloring approach to register allocation. ACM Transactions on Programming Languages & Systems 12:501–536

    Article  Google Scholar 

  33. Assi M, Halawi B, Haraty RA (2018) Genetic algorithm analysis using the graph coloring method for solving the university timetable problem. Procedia Computer Science 126:899–906

    Article  Google Scholar 

  34. Akers BS Jr (1974) Fault diagnosis as a graph coloring problem. IEEE Trans Comput 23:706–713

    Article  MATH  Google Scholar 

  35. Zufferey N, Amstutz P, Giaccari P (2008) Graph colouring approaches for a satellite range scheduling problem. J Sched 11:263–277

    Article  MathSciNet  MATH  Google Scholar 

  36. Woo TK, Su SYW, Newman-Wolfe R (2002) Resource allocation in a dynamically partitionable bus network using a graph coloring algorithm. IEEE Trans Commun 39:1794–1801

    Article  Google Scholar 

  37. Israel RR, Graña M (2014) An empirical evaluation of gravitational swarm intelligence for graph coloring algorithm. Neurocomputing 132:79–84

    Article  Google Scholar 

  38. Dror M, Finke G (1999) On the complexity of a restricted list-coloring problem. Discret Math 195:103–109

    Article  MathSciNet  MATH  Google Scholar 

  39. Agrawal J, Agrawal S (2015) Acceleration based particle swarm optimization for graph coloring problem. Procedia Computer Science 60:714–721

    Article  Google Scholar 

  40. Fidanova S, Pop P (2016) An improved hybrid ant-local search algorithm for the partition graph coloring problem. J Comput Appl Math 293:55–61

    Article  MathSciNet  MATH  Google Scholar 

  41. Mahmoudi S, Lotfi S (2015) Modified cuckoo optimization algorithm (MCOA) to solve graph coloring problem. Appl Soft Comput 33:48–64

    Article  Google Scholar 

  42. Marc D, Ekim T, Bernard R, Cerasela T (2015) On some applications of the selective graph coloring problem. Eur J Oper Res 240:307–314

    Article  MathSciNet  MATH  Google Scholar 

  43. Kai Z, Zhu W, Liu J, He J (2015) Discrete particle swarm optimization algorithm for solving graph coloring problem. Springer, Berlin Heidelberg

    Google Scholar 

  44. Wang R, Zhou Y (2015) Local greedy flower pollination algorithm for solving planar graph coloring problem. J Comput Theor Nanosci 12:1–10

    Article  Google Scholar 

  45. Zhou Y, Hao J, Duval B (2016) Reinforcement learning based local search for grouping problems: A case study on graph coloring. Expert Systems with Application 64:412–422

    Article  Google Scholar 

  46. Chen K, Kanoh H (2016) A discrete artificial bee Colony algorithm based on similarity for graph coloring problems. In: 5th International Conference on the Theory and Practice of Natural Computing (TPNC), 2016-12-12 to 2016-12-13. https://doi.org/10.1007/978-3-319-49001-4_6

  47. Mosa MA, Hamouda A, Marei M (2017) Graph coloring and ACO based summarization for social networks. Expert Syst Appl 74:115–126

    Article  Google Scholar 

  48. Galán SF (2017) Simple decentralized graph coloring. Comput Optim Appl 66:163–185

    Article  MathSciNet  MATH  Google Scholar 

  49. Yüceoglu B, Güvenc S, Van Hoesel SPM (2017) A column generation based algorithm for the robust graph coloring problem. Discrete Applied Mathematics, Part 2:340–352

    Article  MathSciNet  MATH  Google Scholar 

  50. Mirsaleh MR, Meybodi MR, Michigan A (2017) Memetic algorithm for solving the vertex coloring problem. Journal of Computational Science 24:389–401

    Article  MathSciNet  Google Scholar 

  51. Denis C, Furini F, Malaguti E (2017) Solving vertex coloring problems as maximum weight stable set problems. Discret Appl Math 217:151–162

    Article  MathSciNet  MATH  Google Scholar 

  52. Chen K, Kanoh H (2017) A discrete firefly algorithm based on similarity for graph coloring problems. IEEE SNPD 2017, June 26-28, 2017, Kanazawa, Japan

  53. Aragón Artacho FJ, Campoy R (2018) Solving graph coloring problems with the Douglas-Rachford algorithm. Set-Valued and Variational Analysis 26:277–304

    Article  MathSciNet  MATH  Google Scholar 

  54. Xu J, Qiang X, Zhang K, Zhang C, Yang J (2018) A DNA computing model for the graph vertex coloring problem based on a probe graph. Engineering 4:61–77

    Article  Google Scholar 

  55. Zhou Y, Beatrice D, Hao J (2018) Improving probability learning based local search for graph coloring. Appl Soft Comput 65:542–553

    Article  Google Scholar 

  56. Marappan R, Sethumadhavan G (2018) Solution to graph coloring using genetic and Tabu search procedures. Arab J Sci Eng 43:525–542

    Article  Google Scholar 

  57. Moalic L, Gondran A (2018) Variations on memetic algorithms for graph coloring problems. J Heuristics 24(1):1–24

    Article  Google Scholar 

  58. Taha M, Khiyabani FM, Navimipour NJ (2019) A systematic study on meta-heuristic approaches for solving the graph coloring problem. Comput Oper Res 120:104850

    MathSciNet  MATH  Google Scholar 

  59. Meraihi Y, Ramdane-Cherif A, Mahseur M, Achelia D (2019) A chaotic binary Salp swarm algorithm for solving the graph coloring problem. In: Proceedings of the 5th International Symposium, MISC 2018, December 16–18, 2018, Laghouat

  60. Goudet O, Duval B, Hao JK (2020) Population-based gradient descent weight learning for graph coloring problems. Knowl-Based Syst 212:106581

    Article  Google Scholar 

  61. Karim B et al (2019) Solving graph coloring problem using an enhanced binary dragonfly algorithm. International journal of swarm intelligence research 10:23–45

    Article  Google Scholar 

  62. da Silva AF, Rodriguez LGA, Filho JF (2020) The improved ColourAnt algorithm: a hybrid algorithm for solving the graph colouring problem. International Journal of Bio-Inspired Computation 16(1):1–12

    Article  Google Scholar 

  63. Yassine, M, Mahseur, M, Acheli, D (2020) A modified binary crow search algorithm for solving the graph coloring problem. Int J Appl Evolu Comput (IJAEC) 11(2):1–19

  64. Oe A, Te B, Zct B (2021) An exact cutting plane algorithm to solve the selective graph coloring problem in perfect graphs. Eur J Oper Res 291:67–83

    Article  MathSciNet  Google Scholar 

  65. Zhao R, Wang Y, Liu C, Hu P, Li H (2020) Discrete selfish herd optimizer for solving graph coloring problem. Appl Intell 50:9–1656

    Article  Google Scholar 

  66. Abhirup B, Dhar AK, Basu S (2020) Graph coloring: a novel heuristic based on trailing path-properties, perspective and applications in structured networks. Soft Comput 24:603–625

    Article  MATH  Google Scholar 

  67. Dokeroglu T, Sevinc E (2021) Memetic teaching-learning-based optimization algorithms for large graph coloring problems Eng Appl Artif Intell. https://doi.org/10.1016/j.engappai.2021.104282

  68. Qin AK, Huang VL, Suguanthan PN (2009) Differential Evolution Algorithm With Strategy Adaptation for Global Numerical Optimization. IEEE trans. On Evolutionary Computations 13:398–417

    Article  Google Scholar 

  69. Wei J Guangbin L, Dong L (2008) Elite particle swarm optimization with mutation. In: 2008 Asia Simulation Conference - 7th International Conference on System Simulation and Scientific Computing, 10-12 October 2008, Beijing

  70. Singh A, Singh R (2015) Enhanced PSO for graph coloring problem. International Research Journal of Engineering and Technology 8:2395–0072

    MathSciNet  Google Scholar 

  71. Marappan R, Sethumadhavan G (2021) Solving graph coloring problem using divide and conquer-based turbulent particle swarm optimization[J]. Arab J Sci Eng:1–18

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant No. U21A20464, 62066005, and by the Project of Innovation Project of Guangxi Graduate under Grant No. YCSW2021157. Project of the Guangxi Science and Technology under Grant No. AD21196006, and.

Program for Young Innovative Research Team in China University of Political Science and Law, under Grant No.21CXTD02. We thank Stuart Jenkinson, PhD, from Liwen Bianji, Edanz Group China (www.liwenbianji.cn/ac), for editing the English text of a draft of this manuscript.

Author information

Authors and Affiliations

  1. College of Artificial Intelligence, Guangxi University for Nationalities, Nanning, 530006, Guangxi, China

    Lianlian Zhong, Yongquan Zhou & Qifang Luo

  2. Guangxi Key Laboratories of Hybrid Computation and IC Design Analysis, Nanning, 530006, Guangxi, China

    Lianlian Zhong, Yongquan Zhou & Qifang Luo

  3. Department of Science and Technology Teaching, China University of Political Science and Law, Beijing, 100088, China

    Guo Zhou

Authors
  1. Lianlian Zhong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongquan Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Guo Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Qifang Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Lianlian Zhong: Methodology, Writing - original draft. Yongquan Zhou: Writing - review & editing. Guo Zhou: Experimental results testing. Qifang Luo: Experimental results analysis, Software.

Corresponding author

Correspondence to Yongquan Zhou.

Ethics declarations

Competing interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhong, L., Zhou, Y., Zhou, G. et al. Enhanced discrete dragonfly algorithm for solving four-color map problems. Appl Intell 53, 6372–6400 (2023). https://doi.org/10.1007/s10489-022-03791-y

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10489-022-03791-y

Keywords

Search

Navigation