Genetic local search the graph partitioning problem under cardinality constraints

  • Yu. A. KochetovEmail author
  • A. V. Plyasunov


For the graph partitioning problem under cardinality constraints, a genetic local search method is developed. At each iteration of the method, there is a set of local optima of the problem. This set is used to search for new local optima with a smaller error. The local search problem with certain polynomially searchable neighborhoods is proved to be tight PLS-complete. It is shown that, in the worst case, number of local improvements can be exponentially large for any pivoting rule. Numerical experiments are performed in the special case of edge weights equal to unity, when local search is a polynomial-time procedure. The results of the experiments indicate that the method is highly efficient and can be applied to large-scale problems.


graph partitioning problem tight PLS-completeness local search genetic algorithms 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. Ausiello, P. Crescenzi, G. Gambosi, et al., Complexity and Approximation: Combinatorial Optimization Problems and their Approximability Properties (Springer-Verlag, Berlin, 1999).zbMATHGoogle Scholar
  2. 2.
    O. Goldschmit and D. S. Hochbaum, “Polynomial Algorithm for the k-Cut Problem,” Proceedings of 29th Annual IEEE Symposium on Foundations of Computer Science (IEEE Comput. Soc., 1988), pp. 444–451.Google Scholar
  3. 3.
    V. V. Vazirani, Approximation Algorithms (Springer-Verlag, Berlin, 2001).Google Scholar
  4. 4.
    H. Saran and V. V. Vazirani, “Finding k-Cuts within Twice the Optimal,” SIAM J. Comput. 24, 101–108 (1995).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    N. Guttmann-Beck and R. Hassin, “Approximation Algorithms for Minimum k-Cut,” Algorithmica 27(2), 198–207 (2000).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Yu. A. Kochetov, “Computational Bounds for Local Search in Combinatorial Optimization,” Comput. Math. Math. Phys. 48, 747–763 (2008).MathSciNetCrossRefGoogle Scholar
  7. 7.
    Yu. A. Kochetov, M. G. Pashchenko, and A. V. O. Plyasunov, “On the Complexity of Local Search in the p-Median Problem,” Diskret. Anal. Issled. Operat., Ser. 2. 12(2), 44–71 (2005).MathSciNetGoogle Scholar
  8. 8.
    E. Alekseeya, Yu. Kochetov, and A. Plyasunov, “Complexity of Local Search for the p-Median Problem,” Eur. J. Operat. Res. 191, 736–752 (2008).CrossRefGoogle Scholar
  9. 9.
    C. Walshaw, “Graph Partitioning Archive”
  10. 10.
    E. A. Nurminskii, “Decomposition and Parallelization of Computing Processes Based on Fejer Processes with a Small Perturbation,” Proceedings of 14th Baikal International Workshop on Optimization Methods and Applications (ISEM, Sib. Otd. Ross. Akad. Nauk, Irkutsk, 2008), Vol. 1, pp. 128–137.Google Scholar
  11. 11.
    S. Chopra and M. R. Rao, “The Partition Problem,” Math. Program. 59, 87–115 (1993).MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    F. Rendl and H. Wolkowicz, “A Projection Technique for Partitioning the Nodes of a Graph,” Ann. Operat. Res. 58, 155–179 (2005).MathSciNetCrossRefGoogle Scholar
  13. 13.
    T. Bertold, “Automatic Detection of Orbitopal Symmetries,” Abstracts of OR-2008 Conference (Augsburg, 2008), p. 198.Google Scholar
  14. 14.
    M. Peinhardt, “Breaking Model Symmetry for Graph Partitioning,” Abstracts of OR- (2008).Google Scholar
  15. 15.
    J. Dreo, A. Petrowski, Siarry P., and Taillard E. Metaheuristics for Hard Optimization (Springer-Verlag, Berlin, 2006).zbMATHGoogle Scholar
  16. 16.
    B. W. Kernighan and S. Lin, “An Effective Heuristic Procedure for Partitioning Graphs,” Bell Syst. Tech. J. 49, 291–307 (1970).zbMATHGoogle Scholar
  17. 17.
    C. M. Fiduccia and R. M. Mattheyses, “A Linear-Time Heuristic for Improving Network Partitions,” Proceedings of 19th Design Automation Conference (IEEE Comput. Soc. Press, Los Alamitos, CA, 1982), pp. 175–181.Google Scholar
  18. 18.
    G. Laszewski, “Intelligent Structural Operators for the k-Way Graph Partitioning Problem,” Proceedings of 14th International Conference on Genetic Algorithms (1991), pp. 45–52.Google Scholar
  19. 19.
    A. Moraglio, Y.-H. Kim, Y. Yoon, and B-R. Moon, “Geometric Crossovers for Multiway Graph Partitioning,” Evolutionary Comput. 15, 445–474 (2007).CrossRefGoogle Scholar
  20. 20.
    F. Glover and M. Laguna, Tabu Search (Kluwer Academic, Boston, 1997).zbMATHCrossRefGoogle Scholar
  21. 21.
    M. Yannakakis, “Computational Complexity,” in Local Search in Combinatorial Optimization (Wiley, Chichester, 1997), pp. 19–55.Google Scholar
  22. 22.
    Yu. Kochetov, “Facility Location: Discrete Models and Local Search Methods,” Combinatorial Optimization Methods and Applications (IOS, Amsterdam, 2011), pp. 97–134.Google Scholar
  23. 23.
    J. N. Hooker, Integrated Methods for Optimization (Springer-Verlag, New York, 2007).zbMATHGoogle Scholar
  24. 24.
    M. A. Posypkin and I. X. Sigal, “Application of Parallel Heuristic Algorithms for Speeding up Parallel Implementations of the Branch-and-Bound Method,” Comput. Math. Math. Phys. 47, 1464–1476 (2007).MathSciNetCrossRefGoogle Scholar
  25. 25.
    V. A. Garanzha, A. I. Golikov, Yu. G. Evtushenko, and M. X. Nguen, “Parallel Implementation of Newton’s Method for Solving Large-Scale Linear Programs,” Comput. Math. Math. Phys. 49, 1303–1317 (2009).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematics, Siberian BranchRussian Academy of SciencesNovosibirskRussia

Personalised recommendations