## A.: Drawing (complete) binary tanglegrams: Hardness, approximation, fixedparameter tractability. Arxiv report (2008)

Citations: | 6 - 1 self |

### BibTeX

@MISC{Buchin08a.:drawing,

author = {Kevin Buchin and Maike Buchin and Jaroslaw Byrka and Martin Nöllenburg and Yoshio Okamoto and Rodrigo I. Silveira and Er Wolff and Fakultät Für Informatik and Universität Karlsruhe},

title = {A.: Drawing (complete) binary tanglegrams: Hardness, approximation, fixedparameter tractability. Arxiv report},

year = {2008}

}

### OpenURL

### Abstract

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 constantfactor 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. 1

### Citations

938 | Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
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- 1995
(Show Context)
Citation Context ...(see full version [2]) by reducing TL⋆ to a constrained version of the MaxCut problem, which can be approximately solved with the semidefinite programming rounding algorithm of Goemans and Williamson =-=[8]-=-. Theorem 4. There exists a 0.878-approximation for the TL ⋆ problem. 4 Fixed-Parameter Tractability We consider the following parameterized problem. Given a complete binary TL instance 〈S, T 〉 and a ... |

343 |
Method for visual understanding of hierarchical system structures
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(Show Context)
Citation Context ...aph drawing the so-called two-sided crossing minimization problem (2SCM) is an important problem that occurs when computing layered graph layouts. Such layouts have been introduced by Sugiyama et al. =-=[17]-=- and are widely used for drawing hierarchical graphs. In 2SCM, vertices of a bipartite graph are to be placed on two parallel lines (layers) such that vertices on one line are incident only to vertice... |

233 | On the power of unique 2-prover 1-round games
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(Show Context)
Citation Context ...s. We start by showing that binary TL is essentially as hard as the MinUncut problem. This relates the existence of a constant-factor approximation for TL to the Unique Games Conjecture (UGC) by Khot =-=[11]-=-. The UGC became famous when it was discovered that it implies optimal hardness-of-approximation results for problems such as MaxCut and VertexCover, and forbids constant factorapproximation algorithm... |

127 | The Unique Games Conjecture, integrality gap for cut problems and embeddability of negative type metrics into l1
- Khot, Vishnoi
- 2005
(Show Context)
Citation Context ...how that binary TL is essentially as hard as the MinUncut problem. If the (widely accepted) Unique Games Conjecture holds, it is NP-hard to approximate MinUncut—and thus TL—within any constant factor =-=[12]-=-. This motivates us to consider complete binary TL. It turns out that this special case has a rich structure. We start our investigation by giving a new reduction from Max2Sat that establishes the NP-... |

85 |
Edge crossing in drawings of bipartite graphs
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(Show Context)
Citation Context ...ed that edges are drawn as straight-line segments. In one-sided crossing minimization (1SCM) the order ofDrawing Binary Tanglegrams 3 the vertices on one of the layers is fixed. Even 1SCM is NP-hard =-=[6]-=-. In contrast to TL, a vertex in 1SCM or 2SCM can have several incident edges and the linear order of the vertices in the non-fixed layer is not restricted by the internal structure of a tree. The fol... |

36 | On distances between phylogenetic trees
- DasGupta, He, et al.
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(Show Context)
Citation Context ... the minimum number of crossings for visualization purposes. The number is not intended to be a tree-distance measure. Examples for such measures are nearest-neighbor interchange and subtree transfer =-=[3]-=-. Related problems. In graph drawing the so-called two-sided crossing minimization problem (2SCM) is an important problem that occurs when computing layered graph layouts. Such layouts have been intro... |

34 | Optimal upward planarity testing of single-source digraphs
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- 1998
(Show Context)
Citation Context ...of the tanglegram can be directed from one root to the other. Thus the existence of a planar drawing can be verified using a linear-time upwardplanarity test for single-source directed acyclic graphs =-=[1]-=-. Later, apparently not knowing these previous results, Lozano et al. [13] gave a quadratic-time algorithm for the same special case, to which they refer as planar tanglegram layout. Holten and van Wi... |

21 |
Disparate rates of molecular evolution in cospeciating hosts and parasites
- Hafner, Sudman, et al.
- 1994
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Citation Context ...d Japan Society for the Promotion of Science.2 K. Buchin et al. (a) arbitrary drawing (b) drawing of our 2-approximation Fig. 1: A binary tanglegram showing two evolutionary trees for pocket gophers =-=[9]-=-. visually compare pairs of trees arises in applications such as the analysis of software projects, phylogenetics, or clustering. In the first application, trees may represent package-class-method hie... |

16 | Visual comparison of hierarchically organized data
- Holten, Wijk
(Show Context)
Citation Context ...ater, apparently not knowing these previous results, Lozano et al. [13] gave a quadratic-time algorithm for the same special case, to which they refer as planar tanglegram layout. Holten and van Wijk =-=[10]-=- presented a visualization tool for general tanglegrams that heuristically reduces crossings (using the barycenter method for 1SCM on a per-level base) and draws inter-tree edges in bundles (using Béz... |

15 | A simplified NP-complete MAXSAT problem - Raman, Ravikumar, et al. - 1998 |

13 | Optimal leaf ordering for two and a half dimensional phylogenetic tree visualization
- Dwyer, Schreiber
- 2004
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Citation Context ...um number of crossings in any 2-layer drawing of the given graph that respects the vertex order of the fixed layer. The O ⋆ (·)-notation ignores polynomial factors. Previous work. Dwyer and Schreiber =-=[5]-=- studied drawing a series of tanglegrams in 2.5 dimensions, i.e., the trees are drawn on a set of stacked twodimensional planes. They considered a one-sided version of TL by fixing the layout of the f... |

11 | Comparing trees via crossing minimization
- Fernau, Kaufmann, et al.
- 2005
(Show Context)
Citation Context ...her tree, and (c) the number of crossings among the additional edges is minimized. As in the bioinformatics literature (e.g., [13, 16]), we call this the tanglegram layout (TL) problem; Fernau et al. =-=[7]-=- refer to it as two-tree crossing minimization. Note that we are interested in the minimum number of crossings for visualization purposes. The number is not intended to be a tree-distance measure. Exa... |

10 |
An improved bound on the one-sided minimum crossing number in two-layered drawings
- NAGAMOCHI
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Citation Context ...s not restricted by the internal structure of a tree. The following is known about 1SCM. The median heuristic of Eades and Wormald [6] yields a 3-approximation and a randomized algorithm of Nagamochi =-=[14]-=- yields an expected 1.4664-approximation. Dujmovič et al. [4] gave an FPT algorithm that runs in O ⋆ (1.4664 k ) time, where k is the minimum number of crossings in any 2-layer drawing of the given gr... |

9 | Fixed parameter algorithms for one-sided crossing minimization revisited
- Dujmović, Fernau, et al.
(Show Context)
Citation Context ...owing is known about 1SCM. The median heuristic of Eades and Wormald [6] yields a 3-approximation and a randomized algorithm of Nagamochi [14] yields an expected 1.4664-approximation. Dujmovič et al. =-=[4]-=- gave an FPT algorithm that runs in O ⋆ (1.4664 k ) time, where k is the minimum number of crossings in any 2-layer drawing of the given graph that respects the vertex order of the fixed layer. The O ... |

6 | Drawing binary tanglegrams: an experimental evaluation
- Holten, Nöllenburg, et al.
- 2008
(Show Context)
Citation Context ...r algorithm can also process general binary tanglegrams— without guaranteeing any approximation ratio. It works very well in practice and is quite fast when combined with a branch-and-bound procedure =-=[15]-=-. Next we consider a dual problem: maximize the number of edge pairs that do not cross. We show that this problem (for general binary trees) can be reduced to a version of MaxCut for which the algorit... |

5 | Seeded tree alignment and planar tanglegram layout
- Lozano, Pinter, et al.
(Show Context)
Citation Context ...ree is connected by an additional edge to the corresponding leaf in the other tree, and (c) the number of crossings among the additional edges is minimized. As in the bioinformatics literature (e.g., =-=[13, 16]-=-), we call this the tanglegram layout (TL) problem; Fernau et al. [7] refer to it as two-tree crossing minimization. Note that we are interested in the minimum number of crossings for visualization pu... |