Theory of Computing Systems

, Volume 57, Issue 1, pp 1–35 | Cite as

On the Parameterized Complexity of Computing Balanced Partitions in Graphs

  • René van Bevern
  • Andreas Emil Feldmann
  • Manuel SorgeEmail author
  • Ondřej Suchý


A balanced partition is a clustering of a graph into a given number of equal-sized parts. For instance, the Bisection problem asks to remove at most k edges in order to partition the vertices into two equal-sized parts. We prove that Bisection is FPT for the distance to constant cliquewidth if we are given the deletion set. This implies FPT algorithms for some well-studied parameters such as cluster vertex deletion number and feedback vertex set. However, we show that Bisection does not admit polynomial-size kernels for these parameters. For the Vertex Bisection problem, vertices need to be removed in order to obtain two equal-sized parts. We show that this problem is FPT for the number of removed vertices k if the solution cuts the graph into a constant number c of connected components. The latter condition is unavoidable, since we also prove that Vertex Bisection is W[1]-hard w.r.t. (k,c). Our algorithms for finding bisections can easily be adapted to finding partitions into d equal-sized parts, which entails additional running time factors of n O(d). We show that a substantial speed-up is unlikely since the corresponding task is W[1]-hard w.r.t. d, even on forests of maximum degree two. We can, however, show that it is FPT for the vertex cover number.


Bisection NP-hard problems Problem kernelization Cliquewidth Treewidth reduction 



René van Bevern and Manuel Sorge gratefully acknowledge support by the DFG, research project DAPA, NI 369/12. Ondřej Suchý is also grateful for support by the DFG, research project AREG, NI 369/9. Part of Ondřej Suchý’s work was done while with TU Berlin.

The authors thank Bart M. P. Jansen, Stefan Kratsch, Rolf Niedermeier and the anonymous referees for helpful suggestions.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • René van Bevern
    • 1
  • Andreas Emil Feldmann
    • 2
  • Manuel Sorge
    • 1
    Email author
  • Ondřej Suchý
    • 3
  1. 1.Institut für Softwaretechnik und Theoretische InformatikBerlinGermany
  2. 2.Combinatorics, OptimizationUniversity of WaterlooWaterlooCanada
  3. 3.Faculty of Information TechnologyCzech Technical University in PraguePragueCzech Republic

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