Drawing (Complete) Binary Tanglegrams
Buchin, Kevin and Buchin, Maike and Byrka, Jaroslaw and Nöllenburg, Martin and Okamoto, Yoshio and Silveira, Rodrigo I. and Wolff, Alexander (2009) Drawing (Complete) Binary Tanglegrams. In: Graph Drawing 16th International Symposium, GD 2008, September 21- 24, 2008, Heraklion, Crete, Greece , pp. 324-335 (Official URL: http://dx.doi.org/10.1007/978-3-642-00219-9_32).
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A binary tanglegram is a pair S, T of 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 drawing 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 general binary trees. We show thatthe problem is hard even if both trees are complete binary trees. For this case we give an O(n3 )-time 2-approximation and a new and simple ﬁxed-parameter algorithm. We show that the maximization version of the dual problem for general 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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