Drawing (Complete) Binary Tanglegrams

Hardness, Approximation, Fixed-Parameter Tractability
  • Kevin Buchin
  • Maike Buchin
  • Jaroslaw Byrka
  • Martin Nöllenburg
  • Yoshio Okamoto
  • Rodrigo I. Silveira
  • Alexander Wolff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5417)

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(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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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Kevin Buchin
    • 1
  • Maike Buchin
    • 1
  • Jaroslaw Byrka
    • 2
    • 3
  • Martin Nöllenburg
    • 4
  • Yoshio Okamoto
    • 5
  • Rodrigo I. Silveira
    • 1
  • Alexander Wolff
    • 2
  1. 1.Dept. Computer ScienceUtrecht UniversityThe Netherlands
  2. 2.Faculteit Wiskunde en InformaticaTU EindhovenThe Netherlands
  3. 3.Centrum voor Wiskunde en Informatica (CWI)AmsterdamThe Netherlands
  4. 4.Fakultät für InformatikUniversität KarlsruheGermany
  5. 5.Grad. School of Infor. Sci. and EngineeringTokyo Inst. of TechnologyJapan

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