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Drawing (Complete) Binary Tanglegrams

Hardness, Approximation, Fixed-Parameter Tractability


A 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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Correspondence to Martin Nöllenburg.

Additional information

Work started at the 10th Korean Workshop on Computational Geometry, Dagstuhl, Germany, 2007. A preliminary version [5] of this paper was presented at the 16th International Symposium on Graph Drawing (GD’08).

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Buchin, K., Buchin, M., Byrka, J. et al. Drawing (Complete) Binary Tanglegrams. Algorithmica 62, 309–332 (2012). https://doi.org/10.1007/s00453-010-9456-3

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  • Binary tanglegram
  • Crossing minimization
  • NP-hardness
  • Approximation algorithm
  • Fixed-parameter tractability