Mathematical Programming

, Volume 81, Issue 2, pp 229–256 | Cite as

The node capacitated graph partitioning problem: A computational study

  • C. E. Ferreira
  • A. Martin
  • C. C. de Souza
  • R. Weismantel
  • L. A. Wolsey


In this paper we consider the problem ofk-partitioning the nodes of a graph with capacity restrictions on the sum of the node weights in each subset of the partition, and the objective of minimizing the sum of the costs of the edges between the subsets of the partition. Based on a study of valid inequalities, we present a variety of separation heuristics for cycle, cycle with ears, knapsack tree and path-block cycle inequalities among others. The separation heuristics, plus primal heuristics, have been implemented in a branch-and-cut routine using a formulation including variables for the edges with nonzero costs and node partition variables. Results are presented for three classes of problems: equipartitioning problems arising in finite element methods and partitioning problems associated with electronic circuit layout and compiler design. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.


Branch-and-cut algorithm Clustering Compiler design Equipartitioning Finite element method Graph partitioning Layout of electronic circuits Separation heuristics 


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  1. [1]
    D. Applegate, R.E. Bixby, V. Chvátal, W. Cook, Finding cuts in the TSP, DIMACS Technical Report, 1994, pp. 95–05.Google Scholar
  2. [2]
    F. Barahona, A. Casari, On the magnetisation of the ground states in two dimensional Ising spin glasses, Computer Physics Communications 49 (1988) 417–421.Google Scholar
  3. [3]
    F. Barahona, M. Grötschel, M. Jünger, G. Reinelt, An application of combinatorial optimization to statistical physics and circuit layout design, Operations Research 36 (1988) 493–513.Google Scholar
  4. [4]
    F. Barahona, A. Mahjoub, On the cut polytope, Mathematical Programming 36 (1986) 157–173.Google Scholar
  5. [5]
    N. Boissin, Optimisation des Fonctions Quadratiques en Variables Bivalentes, Thèse de Doctorat, Conservatoire National des Arts et Métiers, Paris, 1994.Google Scholar
  6. [6]
    L. Brunetta, M. Conforti, G. Rinaldi, A branch-and-cut algorithm for the resolution of the equicut problem, Working Paper no. 361, IASI-CNR, Rome, 1993.Google Scholar
  7. [7]
    S. Chopra, M.R. Rao, On the multiway cut polyhedron, Networks 21 (1991) 51–89.Google Scholar
  8. [8]
    C.E. Ferreira, A. Martin, C.C. de Souza, R. Weismantel, L.A. Wolsey, The node capacitated graph partitioning problems: formulations and valid inequalities, Mathematical Programming 74 (1996) 247–267.Google Scholar
  9. [9]
    C.E. Ferreira, A. Martin, R. Weismantel, A cutting plane based algorithm for the multiple knapsack problem, SIAM J. on Optimization 6 (1996) 858–877.Google Scholar
  10. [10]
    C.M. Fiduccia, R.M. Mattheyses, A linear time heuristic for improving network partitionings, in: Proceedings of the 19th Design Automation Conference, Las Vegas, 1982, pp. 175–181.Google Scholar
  11. [11]
    M. Grötschel, Y. Wakbayashi, A cutting plane algorithm for a clustering problem, Mathematical Programming Series B 45 (1989) 59–96.Google Scholar
  12. [12]
    S. Holm, M.M. Sorensen, The optimal graph partitioning problem, OR Spektrum 15 (1993) 1–8.Google Scholar
  13. [13]
    D.S. Johnson, C.R. Aragon, L.A. McGeoch, C. Schevon, Optimization by simulated annealing: an experimental evaluation: Part I, Graph partitioning, Operations Research 37 (1989) 865–892.Google Scholar
  14. [14]
    E. Johnson, A. Mehrotra, G.L. Nemhauser, Min-cut clustering, Mathematical Programming 62 (1993) 133–152.Google Scholar
  15. [15]
    M. Jünger, A. Martin, G. Reinelt, R. Weismantel, Quadratic 0/1 optimization and a decomposition approach for the placement of electronic circuits, Mathematical Programming 63 (1994) 257–279.Google Scholar
  16. [16]
    W. Kernighan, S. Lin, An efficient heuristic procedure for partitioning graphs, Bell Systems Technical Journal 49 (2) (1970) 291–307.Google Scholar
  17. [17]
    T. Lengauer, Combinatorial Algorithms for Integrated Circuit Layout, Wiley, New York, 1990.Google Scholar
  18. [18]
    M.W. Padberg, G. Rinaldi, A branch and cut algorithm for the resolution of large-scale symmetric traveling salesman problems, SIAM Review 33 (1991) 60–100.Google Scholar
  19. [19]
    H.L.G. Pina, An algorithm for frontwidth reduction, International Journal on Numerical Methods in Engineering 17 (1981) 1539–1546.Google Scholar
  20. [20]
    C.C. de Souza, The graph equipartition problem: optimal solutions, extensions and applications, Doctoral Thesis, Faculté des Sciences Appliquées, Université Catholique de Louvain, Louvain-la-Neuve, Belgium, 1993.Google Scholar
  21. [21]
    C.C. de Souza, R. Keunings, L.A. Wolsey, O. Zone, A new approach to minimising the frontwidth in finite element calculations, Computer Methods in Applied Mechanics and Engineering 111 (1994) 323–334.Google Scholar
  22. [22]
    F. Vanderbeck, Decomposition and column generation for integer programs, Doctoral Thesis, Faculté des Sciences Appliquées, Université Catholique de Louvain, Louvain-la-Neuve, Belgium, 1994.Google Scholar
  23. [23]
    R. Weismantel, Plazieren von Zellen: Theorie and Lösung eines quadratischen 0–1 Optimierungs-problem, Dissertation, Technische Universität, Berlin, 1992.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1998

Authors and Affiliations

  • C. E. Ferreira
    • 1
  • A. Martin
    • 2
  • C. C. de Souza
    • 3
  • R. Weismantel
    • 2
  • L. A. Wolsey
    • 4
  1. 1.Universidade de São PauloPerdizBrazil
  2. 2.Konrad-Zuse-Zentrum für Informationstechnik BerlinGermany
  3. 3.Universidade Estadual de CampinasBrazil
  4. 4.CORE, Université Catholique de LouvainBelgium

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