A linear-time algorithm to find a separator in a graph excluding a minor

  • Authors:

  • Bruce Reed;David R. Wood

  • Affiliations:

  • McGill University, Montréal, Canada and Centre National de la Recherche Scientifique, Sophia-Antipolis, France;The University of Melbourne, Melbourne, Australia

  • Venue:
  • ACM Transactions on Algorithms (TALG)
  • Year:
  • 2009

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Abstract

Let G be an n-vertex m-edge graph with weighted vertices. A pair of vertex sets A, B ⊆ V(G) is a 2/3-separation of order |A ∩ B| if A ∪ B = V(G), there is no edge between A − B and B − A, and both A − B and B − A have weight at most 2/3 the total weight of G. Let ℓ ∈ Z+ be fixed. Alon et al. [1990] presented an algorithm that in O(n1/2m) time, outputs either a Kℓ-minor of G, or a separation of G of order O(n1/2). Whether there is a O(n + m)-time algorithm for this theorem was left as an open problem. In this article, we obtain a O(n + m)-time algorithm at the expense of a O(n2/3) separator. Moreover, our algorithm exhibits a trade-off between time complexity and the order of the separator. In particular, for any given ε ∈ [0,1/2], our algorithm outputs either a Kℓ-minor of G, or a separation of G with order O(n(2−ε)/3 in O(n1 + ε+m) time. As an application we give a fast approximation algorithm for finding an independent set in a graph with no Kℓ-minor.