Mathematical Programming

, Volume 74, Issue 3, pp 247–266 | Cite as

Formulations and valid inequalities for the node capacitated graph partitioning problem

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


We investigate the problem of partitioning the nodes of a graph under capacity restriction on the sum of the node weights in each subset of the partition. The objective is to minimize the sum of the costs of the edges between the subsets of the partition. This problem has a variety of applications, for instance in the design of electronic circuits and devices. We present alternative integer programming formulations for this problem and discuss the links between these formulations. Having chosen to work in the space of edges of the multicut, we investigate the convex hull of incidence vectors of feasible multicuts. In particular, several classes of inequalities are introduced, and their strength and robustness are analyzed as various problem parameters change.


Clustering Graph partitioning Equipartition Knapsack Integer programming Ear decomposition 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    E.H. Aghezzaf, “Optimal constrained rooted subtrees and partitioning problems on tree graphs,” Doctoral Thesis, Université Catholique de Louvain, Louvain-la-Neuve, Belgium (1992).Google Scholar
  2. [2]
    F. Barahona, “The max-cut problem in graphs not contractible toK 5Operations Research Letters 2 (1983) 107–111.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    F. Barahona and A.R. Mahjoub, “On the cut polytope,”Mathematical Programming 36 (1986) 157–173.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    S. Chopra and M.R. Rao, “The partition problem,”Mathematical Programming 59 (1993) 87–116.CrossRefMathSciNetGoogle Scholar
  5. [5]
    M. Conforti, M.R. Rao and A. Sassano, “The equipartition polytope I,”Mathematical Programming 49 (1990) 49–70.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    M. Conforti, M.R. Rao and A. Sassano, “The equipartition polytope II,”Mathematical Programming 49 (1990) 71–90.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    C.E. Ferreira, A. Martin, C.C. de Souza, R. Weismantel and L.A. Wolsey, “The node capacitated graph partitioning problem: A computational study, submitted.Google Scholar
  8. [8]
    M. Grötschel and Y. Wakabayashi, “Facets of the clique partitioning polytope,”Mathematical Programming 47 (1990) 367–387.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    M. Grötschel and Y. Wakabayashi, “A cutting plane algorithm for a clustering problem,”Mathematical Programming 45 (1989) 59–96.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    E. Johnson, A. Mehrotra and G.L. Nemhauser, “Min-cut clustering,”Mathematical Programming 62 (1993) 133–152.CrossRefMathSciNetGoogle Scholar
  11. [11]
    T. Lengauer,Combinatorial Algorithms for Integrated Circuit Layout (Wiley, New York, 1990).zbMATHGoogle Scholar
  12. [12]
    L. Lovasz and M.D. Plummer, “Matching theory,” inAnnals of Discrete Mathematics 29 (North-Holland, Amsterdam, 1986).Google Scholar
  13. [13]
    C.C. de Souza, “The graph equipartition problem: Optimal solutions, extensions and applications,” Doctoral Thesis, Université Catholique de Louvain Louvain-la-Neuve, Belgium (1993).Google Scholar
  14. [14]
    C.C. de Souza and M. Laurent, “Some new classes of facets for the equicut polytope,”Discrete Applied Mathematics 62 (1995) 167–191.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    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
  16. [16]
    R. Weismantel, “Plazieren von Zellen: Theorie and Lösung eines quadratischen 0–1 Optimierungsproblem,” Technical Report TR 92-3, Konrad-Zuse-Zentrum für Informationstechnik, Berlin (1993).Google Scholar
  17. [17]
    H. Whitney, “Non-separable and planar graphs,”Transactions of the American Mathematical Society 34 (1932) 339–362.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Mathematical Programming Society. Inc. 1996

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 PauloSão PauloBrazil
  2. 2.Konrad-Zuse-Zentrum für InformationstechnikBerlinGermany
  3. 3.Universidade Estadual de CampinasBrazil
  4. 4.COREUniversité Catholique de LouvainLouvain-la-NeuveBelgium

Personalised recommendations