Soft Computing

, Volume 21, Issue 21, pp 6453–6469 | Cite as

Two metaheuristics for solving the connected multidimensional maximum bisection problem

  • Zoran Maksimović
  • Jozef Kratica
  • Aleksandar Savić
Methodologies and Application
  • 107 Downloads

Abstract

In this paper, a connected multidimensional maximum bisection problem is considered. This problem is a generalization of a standard NP-hard maximum bisection problem, where each graph edge has a vector of weights and induced subgraphs must be connected. We propose two metaheuristic approaches, a genetic algorithm (GA) and an electromagnetism-like metaheuristic (EM). The GA uses modified integer encoding of individuals, which enhances the search process and enables usage of standard genetic operators. The EM, besides standard attraction–repulsion mechanism, is extended with a scaling procedure, which additionally moves EM points closer to local optima. A specially constructed penalty function, used for both approaches, is performed as a practical technique for temporarily including infeasible solutions into the search process. Both GA and EM use the same local search procedure based on 1-swap improvements. Computational results were obtained on instances from literature with up to 500 vertices and 60,000 edges. EM reaches all known optimal solutions on small-size instances, while GA reaches all known optimal solutions except for one case. Both proposed methods give results on medium-size and large-scale instances, which are out of reach for exact methods.

Keywords

Genetic algorithms Electromagnetism-like approach Evolutionary computation Graph bisection Combinatorial optimization 

References

  1. Birbil ŞI, Fang SC (2003) An electromagnetism-like mechanism for global optimization. J Gobal Optim 25(3):263–282MathSciNetCrossRefMATHGoogle Scholar
  2. Filipović V (2003) Fine-grained tournament selection operator in genetic algorithms. Comput Inform 22(2):143–161MathSciNetMATHGoogle Scholar
  3. Filipović V (2011) An electromagnetism metaheuristic for the uncapacitated multiple allocation hub location problem. Serd J Comput 5(3):261–272Google Scholar
  4. Fischetti M, Lodi A (2003) Local branching. Math Program 98(1–3):23–47MathSciNetCrossRefMATHGoogle Scholar
  5. Garey MR, Johnson DS, Stockmeyer L (1976) Some simplified NP-complete graph problems. Theor Comput Sci 1(3):237–267MathSciNetCrossRefMATHGoogle Scholar
  6. Ghosh A, Tsutsui S (2012) Advances in evolutionary computing: theory and applications. Springer Science & Business Media, Berlin, HeidelbergMATHGoogle Scholar
  7. Hansen P, Mladenović N, Urošević D (2006) Variable neighborhood search and local branching. Comput Oper Res 33(10):3034–3045CrossRefMATHGoogle Scholar
  8. Hendrickson B, Leland R (1995) An improved spectral graph partitioning algorithm for mapping parallel computations. SIAM J Sci Comput 16(2):452–469MathSciNetCrossRefMATHGoogle Scholar
  9. Kartelj A (2013) Electromagnetism metaheuristic algorithm for solving the strong minimum energy topology problem. Yu J Oper Res 23(1):43–57MathSciNetCrossRefMATHGoogle Scholar
  10. Kratica J (1999) Improving performances of the genetic algorithm by caching. Comput Artif intell 18(3):271–283MATHGoogle Scholar
  11. Kratica J (2012) An electromagnetism-like approach for solving the low autocorrelation binary sequence problem. Int J Comput Commun Control 7(4):688–695MathSciNetCrossRefGoogle Scholar
  12. Lim TY (2014) Structured population genetic algorithms: a literature survey. Artif Intell Rev 41(3):385–399MathSciNetCrossRefGoogle Scholar
  13. Maksimović Z (2015a) A connected multidimensional maximum bisection problem. Preprint. arXiv:1512.00614
  14. Maksimović Z (2015b) A multidimensional maximum bisection problem. Preprint. arXiv:1506.07731
  15. Neri F, Cotta C (2012) Memetic algorithms and memetic computing optimization: a literature review. Swarm Evol Comput 2:1–14CrossRefGoogle Scholar
  16. Shi J, Malik J (2000) Normalized cuts and image segmentation. IEEE Trans Patt Anal Mach Intell 22(8):888–905CrossRefGoogle Scholar
  17. Słowik A, Białko M (2006) Partitioning of vlsi circuits on subcircuits with minimal number of connections using evolutionary algorithm. In: Rutkowski L, Tadeusiewicz R, Zadeh LA, Zurada J (eds) Artificial intelligence and soft computing—ICAISC 2006, Springer, Berlin, Heidelberg pp 470–478Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Zoran Maksimović
    • 1
  • Jozef Kratica
    • 2
  • Aleksandar Savić
    • 3
  1. 1.University of DefenceMilitary AcademyBelgradeSerbia
  2. 2.Mathematical InstituteSerbian Academy of Sciences and ArtsBelgradeSerbia
  3. 3.Faculty of MathematicsUniversity of BelgradeBelgradeSerbia

Personalised recommendations