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Performance analysis of evolutionary algorithm for the maximum internal spanning tree problem

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Abstract

The maximum internal spanning tree (MIST) problem is to find a spanning tree with maximum number of internal node for an undirected graph. It is a variation of the well-known minimum spanning tree problem and is NP-hard. Evolutionary algorithms (EAs) have been successfully applied to solve many NP-hard combinatorial optimization problems in numerical empirical studies. However, researchers know little about their performance guarantees from theory aspect. This paper designs a valid fitness function to guide the well-studied evolutionary algorithm, \((1+1)\;\mathrm{EA}\), to optimize the MIST problem, and presents theoretical analysis to show that it can obtain a performance guarantee of 2 and 5/3 in expected runtime \(O(nm^2)\) and \(O(nm^4)\), respectively. Moreover, this paper proves that the (1+1) EA can achieve a performance guarantee of 3 for a variation of the MIST problem, where each vertex is associated with a weight. In addition, comparison analyses on two family instance graphs are presented to show that \((1+1)\;\mathrm{EA}\) is better than two local search algorithms. This theoretical study provides insight into the process of \((1+1)\;\mathrm{EA}\) constructing a performance guarantee solution for the MIST problem.

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References

  1. Al-Janabi S, Mohammad M, Al-Sultan A (2020) A new method for prediction of air pollution based on intelligent computation. Soft Comput 24(1):661–680. https://doi.org/10.1007/s00500-019-04495-1

    Article  Google Scholar 

  2. Al-Janabi S, Alkaim A, Al-Janabi E, Aljeboree A, Mustafa M (2021) Intelligent forecaster of concentrations (PM2. 5, PM10, NO2, CO, O3, SO2) caused air pollution (IFCsAP). Neural Comput Appli 33(21):14199–229. https://doi.org/10.1007/s00521-021-06067-7

    Article  Google Scholar 

  3. Binkele-Raible D, Fernau H, Gaspers S, Liedloff M (2013) Exact and parameterized algorithms for max internal spanning tree. Algorithmica 65(1):95–128. https://doi.org/10.1007/s00453-011-9575-5

    Article  MathSciNet  MATH  Google Scholar 

  4. Bjórklund A, Kamat V, Kowalik Ł, Zehavi M (2017) Spotting trees with few leaves. SIAM J Discret Math 31(2):687–713. https://doi.org/10.1137/15M1048975

    Article  MathSciNet  MATH  Google Scholar 

  5. Casel K, Dreier J, Fernau H, Gobbert M, Kuinke P, Villaamil FS, Schmid ML, van Leeuwen EJ (2020) Complexity of independency and cliquy trees. Discret Appl Math 272:2–15. https://doi.org/10.1016/j.dam.2018.08.011

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen ZZ, Lin G, Wang L, Chen Y, Wang D (2019) Approximation algorithms for the maximum weight internal spanning tree problem. Algorithmica 81(11–12):4167–4199. https://doi.org/10.1007/s00453-018-00533-w

    Article  MathSciNet  MATH  Google Scholar 

  7. Doerr B, Neumann F (2021) A survey on recent progress in the theory of evolutionary algorithms for discrete optimization. ACM Trans Evolut Learn Optim 1(4):1–43. https://doi.org/10.1145/3472304

    Article  Google Scholar 

  8. Doerr B, Neumann F, Sutton AM (2015) Improved runtime bounds for the (1+1) EA on random 3-CNF formulas based on fitness-distance correlation. In: Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation, pp 1415–1422, https://doi.org/10.1145/2739480.2754659

  9. Eiben AE, Smith JE (2015) Introduction to evolutionary computing. Springer. https://doi.org/10.1007/978-3-662-44874-8

  10. Huang Z, Zhou Y, Xia X, Lai X (2020) An improved (1+ 1) evolutionary algorithm for k-median clustering problem with performance guarantee. Physica A 539:122992. https://doi.org/10.1016/j.physa.2019.122992

    Article  MathSciNet  Google Scholar 

  11. Knauer M, Spoerhase J (2015) Better approximation algorithms for the maximum internal spanning tree problem. Algorithmica 71(4):797–811. https://doi.org/10.1007/s00453-013-9827-7

    Article  MathSciNet  MATH  Google Scholar 

  12. Lai X, Zhou Y, Xia X, Zhang Q (2017) Performance analysis of evolutionary algorithms for steiner tree problems. Evol Comput 25(4):707–723. https://doi.org/10.1162/evco_a_00200

    Article  Google Scholar 

  13. Li W, Cao Y, Chen J, Wang J (2017) Deeper local search for parameterized and approximation algorithms for maximum internal spanning tree. Inf Comput 252:187–200. https://doi.org/10.1016/j.ic.2016.11.003

    Article  MathSciNet  MATH  Google Scholar 

  14. Li X, Zhu D (2014) Approximating the maximum internal spanning tree problem via a maximum path-cycle cover. In: International symposium on algorithms and computation, pp 467–478, https://doi.org/10.1007/978-3-319-13075-0_37

  15. Li X, Zhu D, Wang L (2021) A 4/3-approximation algorithm for the maximum internal spanning tree problem. J Comput Syst Sci 118:131–140. https://doi.org/10.1016/j.jcss.2021.01.001

    Article  MathSciNet  MATH  Google Scholar 

  16. Muhlenbein H (1992) How genetic algorithms really work: I. mutation and hillclimbing. In: Proceeding of 2nd International Conference on Parallel Problem Solving from Nature, 1992, Elsevier, pp 15–25

  17. Neumann F, Wegener I (2007) Randomized local search, evolutionary algorithms, and the minimum spanning tree problem. Theoret Comput Sci 378(1):32–40. https://doi.org/10.1016/j.tcs.2006.11.002

    Article  MathSciNet  MATH  Google Scholar 

  18. Neumann F, Witt C (2010) Bioinspired computation in combinatorial optimization. Natural Computing Series, Springer. https://doi.org/10.1007/978-3-642-16544-3

  19. Ozeki K, Wiener G, Zamfirescu CT (2020) On minimum leaf spanning trees and a criticality notion. Discret Math 343(7):111884. https://doi.org/10.1016/j.disc.2020.111884

    Article  MathSciNet  MATH  Google Scholar 

  20. Pourhassan M, Roostapour V, Neumann F (2020) Runtime analysis of rls and (1+ 1) EA for the dynamic weighted vertex cover problem. Theoret Comput Sci 832:20–41. https://doi.org/10.1016/j.tcs.2019.03.003

    Article  MathSciNet  MATH  Google Scholar 

  21. Prieto E, Sloper C (2003) Either/or: Using vertex cover structure in designing FPT-algorithms - the case of k-internal spanning tree. In: Algorithms and data structures, international workshop, Wads, Ottawa, Ontario, Canada, July 30-august, https://doi.org/10.1007/978-3-540-45078-8_41

  22. Salamon G (2009) Approximating the maximum internal spanning tree problem. Theoret Comput Sci 410(50):5273–5284. https://doi.org/10.1016/j.tcs.2009.08.029

    Article  MathSciNet  MATH  Google Scholar 

  23. Salamon G (2010) A survey on algorithms for the maximum internal spanning tree and related problems. Electron Notes Discret Math 36:1209–1216. https://doi.org/10.1016/j.endm.2010.05.153

    Article  MATH  Google Scholar 

  24. Salamon G, Wiener G (2008) On finding spanning trees with few leaves. Inf Process Lett 105(5):164–169. https://doi.org/10.1016/j.ipl.2007.08.030

    Article  MathSciNet  MATH  Google Scholar 

  25. Sangdani M, Tavakolpour-Saleh A, Lotfavar A (2018) Genetic algorithm-based optimal computed torque control of a vision-based tracker robot: simulation and experiment. Eng Appl Artif Intell 67:24–38. https://doi.org/10.1016/j.engappai.2017.09.014

    Article  Google Scholar 

  26. Shi F, Neumann F, Wang J (2021) Runtime performances of randomized search heuristics for the dynamic weighted vertex cover problem. Algorithmica 83(4):906–939. https://doi.org/10.1007/s00453-019-00662-w

    Article  MathSciNet  MATH  Google Scholar 

  27. Shi F, Neumann F, Wang J (2021) Time complexity analysis of evolutionary algorithms for 2-hop (1, 2)-minimum spanning tree problem. Theor Comput Sci 893:159–75. https://doi.org/10.1016/j.tcs.2021.09.003

    Article  MathSciNet  MATH  Google Scholar 

  28. Sundhararajan M, Gao XZ, Nejad HV (2018) Artificial intelligent techniques and its applications. J Intell Fuzzy Syst 34(2):755–760. https://doi.org/10.3233/JIFS-169369

    Article  Google Scholar 

  29. Williamson DP, Shmoys DB (2011) The design of approximation algorithms. Cambridge University Press. https://doi.org/10.1017/CBO9780511921735

  30. Xia X, Peng X, Liao W (2021) On the analysis of ant colony optimization for the maximum independent set problem. Front Comp Sci 15(4):154329. https://doi.org/10.1007/s11704-020-9464-7

    Article  Google Scholar 

  31. Xie Y, Neumann A, Neumann F, Sutton AM (2021) Runtime analysis of RLS and the (1+1) EA for the chance-constrained knapsack problem with correlated uniform weights. In: Chicano F, Krawiec K (eds) Proceedings of the 2021 Annual Conference on Genetic and Evolutionary Computation Conference, ACM, pp 1187–1194, https://doi.org/10.1145/3449639.3459381

  32. Zehavi M (2017) Algorithms for k-internal out-branching and k-tree in bounded degree graphs. Algorithmica 78(1):319–341. https://doi.org/10.1007/s00453-016-0166-3

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhou Y, Zhang J, Wang Y (2015) Performance analysis of the (1+ 1) evolutionary algorithm for the multiprocessor scheduling problem. Algorithmica 73(1):21–41. https://doi.org/10.1007/s00453-014-9898-0

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhou ZH, Yu Y, Qian C (2019) Evolutionary learning: advances in theories and algorithms. Springer. https://doi.org/10.1007/978-981-13-5956-9

Download references

Acknowledgements

This work was supported by Zhejiang Province Public Welfare Technology Application Research Project of China (LGG19F030010), Industry-University Cooperation Collaborative Education Project of Ministry of Education (202002315052), National Natural Science Foundation of China (62162063, 61703183, 61773410, 61906069), Science and Technology Program of Guangzhou (202002030260), YMUN-research funding (yy2020bsky050), and Science and Technology Planning Project of Guangxi (2021AC19182).

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Authors and Affiliations

  1. College of Information Science and Engineering, Jiaxing University, Jiaxing, 314001, Zhejiang, People’s Republic of China

    Xiaoyun Xia

  2. Laboratory of Computer Science and Mathematics, Youjiang Medical University for Nationalities, Baise, 533000, Guangxi, People’s Republic of China

    Zhengxin Huang

  3. School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou, 510665, Guangdong, People’s Republic of China

    Xue Peng

  4. School of Artificial Intelligence, Sun Yat-sen University, Zhuhai, 519082, People’s Republic of China

    Zefeng Chen

  5. School of Software Engineering, South China University of Technology, Guangzhou, 510006, Guangdong, People’s Republic of China

    Yi Xiang

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Xia, X., Huang, Z., Peng, X. et al. Performance analysis of evolutionary algorithm for the maximum internal spanning tree problem. J Supercomput 78, 11949–11973 (2022). https://doi.org/10.1007/s11227-022-04342-5

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