Journal of Mathematical Modelling and Algorithms

, Volume 6, Issue 3, pp 319–344 | Cite as

A New Method, the Fusion Fission, for the Relaxed k-way Graph Partitioning Problem, and Comparisons with Some Multilevel Algorithms



In this paper a new graph partitioning problem is introduced, the relaxed k-way graph partitioning problem. It is close to the k-way, also called multi-way, graph partitioning problem, but with relaxed imbalance constraints. This problem arises in the air traffic control area. A new graph partitioning method is presented, the Fusion Fission, which can be used to resolve the relaxed k-way graph partitioning problem. The Fusion Fission method is compared to classical Multilevel packages and with a Simulated Annealing algorithm. The Fusion Fission algorithm and the Simulated Annealing algorithm, both require a longer computation time than the Multilevel algorithms, but they also find better partitions. However, the Fusion Fission algorithm partitions the graph with a smaller imbalance and a smaller cut than Simulated Annealing does.


Graph partitioning Multilevel Metaheuristic Fusion Fission 

Mathematics Subject Classifications (2000)

68R10 90C27 90B20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alpert, C.J., Huang, J.-H., Kahng, A.B.: Multilevel circuit partitioning. In: DAC, pp. 530–533 (1997)Google Scholar
  2. 2.
    Bichot, C.-E., Alliot, J.-M. : A theoretical approach to defining the European Core Area. Technical report, LOG–ENAC/CENA (2005)Google Scholar
  3. 3.
    Dhillon, I., Guan, Y., Kullis, B.: Kernel k-means, spectral clustering, and normalized cuts. In: ACM International Conference on Knowledge Discovery and Data Mining (2004)Google Scholar
  4. 4.
    Diekmann, R., Monien, B., Preis, R.: Using Helpful Sets to Improve Graph Bisections. American Mathematical Society, Providence, RI (1995)Google Scholar
  5. 5.
    Eurocontrol: The impact of fragmentation in European ATM/CNS. Technical report, Eurocontrol (2006)Google Scholar
  6. 6.
    FAA: Air Traffic ControL: FAA Order 7110.65K. Federal Aviation Administration (U.S. Department of Transportation) (1997)Google Scholar
  7. 7.
    Fiduccia, C.M., Mattheyses, R.M.: A linear-time heuristic for improving network partitions. In: ACM Design Automation Conference, pp. 175–181 (1982)Google Scholar
  8. 8.
    Garey, M., Johnson, D., Stockmeyer, L.: Some simplified NP-complete graph problems. Theor. Comp. Sci. 1(3), 237–267 (1976)MATHCrossRefGoogle Scholar
  9. 9.
    Greene, W.A.: Genetic algorithms for partitioning sets. Int. J. Artif. Intell. Tools 10(1–2), 225–241 (2001)CrossRefGoogle Scholar
  10. 10.
    Hallgren, A.: Restructuring European airspace: functional airspace blocks. Skyway, 20–22 (2005)Google Scholar
  11. 11.
    Hendrickson, B., Leland, R.: The chaco users guide. Sandia National Laboratories, 2.0 edition (1995a)Google Scholar
  12. 12.
    Hendrickson, B., Leland, R.: A multilevel algorithm for partitioning graphs. In: ACM/IEEE Conference on Supercomputing (1995b)Google Scholar
  13. 13.
    Johnson, D.S., Aragon, C.R., McGeoch, L.A., Schevon, C.: Optimization by simulated annealing: an experimental evaluation; Part I, graph partitioning. Oper. Res. (Society of America) 37(6), 865–892 (1989)MATHGoogle Scholar
  14. 14.
    Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1998a)CrossRefGoogle Scholar
  15. 15.
    Karypis, G., Kumar, V.: METIS : a software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices. University of Minnesota, 4.0 edition (1998b)Google Scholar
  16. 16.
    Kernighan, B.W., Lin, S.: An efficient heuristic procedure for partitioning graphs. Bell Syst. Tech. J. 49(2), 291–307 (1970)Google Scholar
  17. 17.
    Kirkpatrick, S., Gelatt, C., Vecchi, M.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)CrossRefGoogle Scholar
  18. 18.
    Kuntz, P., Layzell, P., Snyers, D.: A colony of ant-like agents for partitioning in VLSI technology. In: The Fourth European Conference on Artificial Life, pp. 417–424 (1997)Google Scholar
  19. 19.
    Langham, A.E., Grant, P.W.: A multilevel k-way partitioning algorithm for finite element meshes using competing ant colonies. In: The Genetic and Evolutionary Computation Conf., vol. 2., pp. 1602–1608. Orlando, FL (1999a)Google Scholar
  20. 20.
    Langham, A.E., Grant, P.W.: A multilevel k-way partitioning algorithm for finite element meshes using competing ant colonies. In: ACM GECCO (1999b)Google Scholar
  21. 21.
    Pellegrini, F.: Schotch and LibScotch. ENSEIRB–LaBRI, Universit de Bordeaux I, 4.0 edition (2006)Google Scholar
  22. 22.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22(8), 888–905 (2000)CrossRefGoogle Scholar
  23. 23.
    Soper, A., Walshaw, C., Cross, M.: A combined evolutionary search and multilevel optimisation approach to graph-partitioning. J. Glob. Optim. 29, 225–241 (2004)MATHCrossRefGoogle Scholar
  24. 24.
    Talbi, E.G., Bessiere, P.: A parallel genetic algorithm for the graph partitioning problem. In: Proceedings of the ACM International Conference on Supercomputing. ACM, Cologne (1991)Google Scholar
  25. 25.
    Walshaw, C.: The serial JOSTLE library user guide. University of Greenwich, 3.0 edition (2002)Google Scholar
  26. 26.
    Walshaw, C.: Multilevel refinement for combinatorial optimisation problems. Ann. Oper. Res. 131, 325–372 (2004)MATHCrossRefGoogle Scholar
  27. 27.
    Walshaw, C., Cross, M., McManus, K.: Multiphase mesh partitioning. Appl. Math. Model. 25, 123–140 (2000)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.LOG (DSNA–ENAC)ToulouseFrance

Personalised recommendations