The complexity status of problems related to sparsest cuts

  • Authors:

  • Paul Bonsma;Hajo Broersma;Viresh Patel;Artem Pyatkin

  • Affiliations:

  • Humboldt Universität zu Berlin, Computer Science Department, Unter den Linden 6, Berlin;School of Engineering and Computing Sciences, Durham University, Science Laboratories, Durham, U.K.;School of Engineering and Computing Sciences, Durham University, Science Laboratories, Durham, U.K.;School of Engineering and Computing Sciences, Durham University, Science Laboratories, Durham, U.K.

  • Venue:
  • IWOCA'10 Proceedings of the 21st international conference on Combinatorial algorithms
  • Year:
  • 2010

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Abstract

Given an undirected graph G = (V, E) with a capacity function w : E → Z+ on the edges, the sparsest cut problem is to find a vertex subset S ⊂ V minimizing Σe∈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.