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Drawing (Complete) Binary Tanglegrams: Hardness, Approximation, Fixed-Parameter Tractability

  • Buchin, Kevin
  • Buchin, Maike
  • Byrka, Jaroslaw
  • Nöllenburg, Martin
  • Okamoto, Yoshio
  • Silveira, Rodrigo I.
  • Wolff, Alexander
Publication Date
Sep 16, 2010
Submission Date
Jun 05, 2008
arXiv ID: 0806.0920
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A \emph{binary tanglegram} is a drawing of a pair of rooted binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example, in phylogenetics, it is essential that both trees are drawn without edge crossings and that the inter-tree edges have as few crossings as possible. It is known that finding a tanglegram with the minimum number of crossings is NP-hard and that the problem is fixed-parameter tractable with respect to that number. We prove that under the Unique Games Conjecture there is no constant-factor approximation for binary trees. We show that the problem is NP-hard even if both trees are complete binary trees. For this case we give an $O(n^3)$-time 2-approximation and a new, simple fixed-parameter algorithm. We show that the maximization version of the dual problem for binary trees can be reduced to a version of MaxCut for which the algorithm of Goemans and Williamson yields a 0.878-approximation.

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