The complexity of finding uniform sparsest cuts in various graph classes

P.S. Bonsma, Haitze J. Broersma, Viresh Patel, A.V. Pyatkin

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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 View the MathML source. 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(n2k+1), where n is the number of vertices in the input graph. We also establish that a running time of the form nO(k) is optimal in this case, assuming that the Exponential Time Hypothesis holds.
Original languageEnglish
Pages (from-to)136-149
Number of pages14
JournalJournal of discrete algorithms
Volume14
DOIs
Publication statusPublished - 2012
Externally publishedYes

Fingerprint

Graph Classes
Computational complexity
Polynomials
Decomposition
Graph in graph theory
Linear Time
Clique-width
Cactus
Outerplanar Graph
Bounded Treewidth
Interval Graphs
Exponential time
Undirected Graph
Polynomial-time Algorithm
NP-complete problem
Decompose
Unit
Subset
Vertex of a graph

Keywords

  • Treewidth
  • Unit interval graph
  • Parameterized complexity
  • Clique-width
  • Sparsest cut

Cite this

Bonsma, P.S. ; Broersma, Haitze J. ; Patel, Viresh ; Pyatkin, A.V. / The complexity of finding uniform sparsest cuts in various graph classes. In: Journal of discrete algorithms. 2012 ; Vol. 14. pp. 136-149.
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The complexity of finding uniform sparsest cuts in various graph classes. / Bonsma, P.S.; Broersma, Haitze J.; Patel, Viresh; Pyatkin, A.V.

In: Journal of discrete algorithms, Vol. 14, 2012, p. 136-149.

Research output: Contribution to journalArticleAcademicpeer-review

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T1 - The complexity of finding uniform sparsest cuts in various graph classes

AU - Bonsma, P.S.

AU - Broersma, Haitze J.

AU - Patel, Viresh

AU - Pyatkin, A.V.

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N2 - Given an undirected graph G=(V,E), the (uniform, unweighted) sparsest cut problem is to find a vertex subset S⊂V minimizing View the MathML source. 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(n2k+1), where n is the number of vertices in the input graph. We also establish that a running time of the form nO(k) is optimal in this case, assuming that the Exponential Time Hypothesis holds.

AB - Given an undirected graph G=(V,E), the (uniform, unweighted) sparsest cut problem is to find a vertex subset S⊂V minimizing View the MathML source. 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(n2k+1), where n is the number of vertices in the input graph. We also establish that a running time of the form nO(k) is optimal in this case, assuming that the Exponential Time Hypothesis holds.

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