The complexity of finding uniform sparsest cuts in various graph classes

  • Authors:

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

  • Affiliations:

  • Humboldt Universität zu Berlin, Computer Science Department, Unter den Linden 6, 10099 Berlin, Germany;School of Engineering and Computing Sciences, Durham University, Science Laboratories, South Road, Durham DH1 3LE, UK;School of Engineering and Computing Sciences, Durham University, Science Laboratories, South Road, Durham DH1 3LE, UK;School of Engineering and Computing Sciences, Durham University, Science Laboratories, South Road, Durham DH1 3LE, UK

  • Venue:
  • Journal of Discrete Algorithms
  • Year:
  • 2012

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Abstract

Given an undirected graph G=(V,E), the (uniform, unweighted) sparsest cut problem is to find a vertex subset S@?V minimizing |E(S,S@?)|/(|S||S@?|). We show that this problem is NP-complete, and give polynomial time algorithms for various graph classes. In particular, we show that the sparsest cut problem can be solved in linear time for unit interval graphs, and in cubic time for graphs of bounded treewidth. For cactus graphs and outerplanar graphs this can be improved to linear time and quadratic time, respectively. For graphs of clique-width k for which a short decomposition is given, we show that the problem can be solved in time O(n^2^k^+^1), where n is the number of vertices in the input graph. We also establish that a running time of the form n^O^(^k^) is optimal in this case, assuming that the Exponential Time Hypothesis holds.