The Complexity Status of Problems Related to Sparsest Cuts

  • Paul Bonsma
  • Hajo Broersma
  • Viresh Patel
  • Artem Pyatkin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6460)


Given an undirected graph G = (V,E) with a capacity function \(w : E \longrightarrow \mathbb{Z}^+\) on the edges, the sparsest cut problem is to find a vertex subset S ⊂ V minimizing ∑  e ∈ E(S,V ∖ S) w(e)/(|S||V ∖ S|). This problem is NP-hard. The proof can be found in [16]. In the case of unit capacities (i. e. if w(e) = 1 for every e ∈ E) the problem is to minimize |E(S,V ∖ S)|/(|S||V ∖ S|) over all subsets S ⊂ V. While this variant of the sparsest cut problem is often assumed to be NP-hard, this note contains the first proof of this fact. We also prove that the problem is polynomially solvable for graphs of bounded treewidth.


NP-hardness sparsest cut densest cut MSSC bounded treewidth 


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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Paul Bonsma
    • 1
  • Hajo Broersma
    • 2
  • Viresh Patel
    • 2
  • Artem Pyatkin
    • 2
  1. 1.Computer Science DepartmentHumboldt Universität zu BerlinBerlinGermany
  2. 2.School of Engineering and Computing Sciences, Science LaboratoriesDurham UniversityDurhamU.K.

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