Abstract
The Turán number ex(n, F) represents the maximum number of edges in an F-free graph on n vertices. Determining these numbers for general graphs is a long-standing and challenging open problem. Ideally, one aims not only to compute these numbers exactly but also to understand their asymptotic behavior, although such results are currently limited to specific cases. This article introduces new results for \(ex(n,C_{2k})\) and the associated parameter f(n, 2k), achieved through the application of reinforcement machine learning techniques. Specifically, the RLS algorithm, initially proposed by Zhou et al. for vertex coloring in graphs, has been adapted to address certain Turán problems involving edge coloring.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Bengio, Y., Lodi, A., Prouvost, A.: Machine learning for combinatorial optimization: a methodological tour d’horizon. Eur. J. Oper. Res. 290(2), 405–421 (2021)
Caccetta, L., Vijayan, K.: Maximal cycles in graphs. Discret. Math. 98, 1–7 (1991)
Clapham, C.R.J., Flockhart, A., Sheehan, J.: Graphs without four-cycles. J. Graph Theory 13(1), 29–47 (1989)
Dzido, T.: Proving correctness of some Turáan’s problem results obtained by the reinforcement learning technique, submitted for publication
Dzido, T.: Applying reinforcement learning to Ramsey problems. In: Mikyška, J., de Mulatier, C., Paszynski, M., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M. (eds.) ICCS 2023. LNCS, vol. 10476, pp. 589–596. Springer, Cham (2023)
Füredi, Z.: On the number of edges of quadrilateral-free graphs. J. Combin. Theory (B) 68, 1–6 (1996)
Füredi, Z., Gunderson, D.S.: Extremal numbers for odd cycles. Comb. Probab. Comput. 24(4), 641–645 (2014)
Gasarch, W.: https://www.cs.umd.edu/$~$gasarch/TOPICS/ramsey/ramsey.html. Accessed 29 Nov 2023
Gosavi, A.: Reinforcement learning: a tutorial survey and recent advances. INFORMS J. Comput. 21, 178–192 (2009)
Hendry, G.R.T., Brandt, S.: An extremai problem for cycles in Hamiltonian graphs. Graphs Combinator. 11, 255–262 (1995)
Mnih, V., Kavukcuoglu, K., Silver, D., et al.: Human-level control through deep reinforcement learning. Nature 518, 529–533 (2015)
Li, B., Ning, B.: Exact bipartite Turán numbers of large even cycles. J. Graph Theory 97(4), 642–656 (2021)
Silver, D., Schrittwieser, J., Simonyan, K., et al.: Mastering the game of Go without human knowledge. Nature 550, 354–359 (2017)
Woodall, D.R.: Maximal circuits of graphs I. Acta Math. Acad. Sci. Hungar. 28, 77–80 (1976)
Yow, K.S., Luo, S.: Learning-Based Approaches for Graph Problems: A Survey. https://arxiv.org/abs/2204.01057. Accessed 4 Jan 2023
Yuansheng, Y., Rowlison, P.: On graphs without 6-cycles and related Ramsey numbers. Utilitas Mathematica 44, 192–196 (1993)
Yuansheng, Y., Rowlinson, P.: On extremal graphs without four-cycles. Utilitas Mathematica 41, 204–210 (1992)
Zhou, Y., Hao, J.-K., Duval, B.: Reinforcement learning based local search for grouping problems: a case study on graph coloring. Expert Syst. Appl. 64, 412–422 (2016)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Dzido, T. (2024). New Results for Some Turán Problem Instances Obtained Using the Reinforcement Learning Technique. In: Nguyen, N.T., et al. Computational Collective Intelligence. ICCCI 2024. Lecture Notes in Computer Science(), vol 14811. Springer, Cham. https://doi.org/10.1007/978-3-031-70819-0_16
Download citation
DOI: https://doi.org/10.1007/978-3-031-70819-0_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-70818-3
Online ISBN: 978-3-031-70819-0
eBook Packages: Computer ScienceComputer Science (R0)