Drawing (Complete) Binary Tanglegrams

  • Authors:
  • Kevin Buchin;Maike Buchin;Jaroslaw Byrka;Martin Nöllenburg;Yoshio Okamoto;Rodrigo I. Silveira;Alexander Wolff

  • Affiliations:
  • Dept. Computer Science, Utrecht University, The Netherlands;Dept. Computer Science, Utrecht University, The Netherlands;Faculteit Wiskunde en Informatica, TU Eindhoven, The Netherlands and Centrum voor Wiskunde en Informatica (CWI), Amsterdam, The Netherlands;Fakultät für Informatik, Universität Karlsruhe, Germany;Grad. School of Infor. Sci. and Engineering, Tokyo Inst. of Technology, Japan;Dept. Computer Science, Utrecht University, The Netherlands;Faculteit Wiskunde en Informatica, TU Eindhoven, The Netherlands

  • Venue:
  • Graph Drawing
  • Year:
  • 2009

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Abstract

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 that the problem is hard even if both trees are complete binary trees. For this case we give an O(n 3)-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.