On the Parameterized Complexity of Computing Balanced Partitions in Graphs

René van Bevern, Andreas Emil Feldmann, Manuel Sorge, Ondřej Suchý

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

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 VertexBisection 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 VertexBisection 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 nO(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.

Original languageEnglish
Pages (from-to)1-35
Number of pages35
JournalTheory of Computing Systems
Volume57
Issue number1
DOIs
StatePublished - 8 Jul 2015
Externally publishedYes

Keywords

  • Bisection
  • Cliquewidth
  • NP-hard problems
  • Problem kernelization
  • Treewidth reduction

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics

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