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A.: Drawing (complete) binary tanglegrams: Hardness, approximation, fixedparameter tractability. Arxiv report
, 2008
"... Abstract. A binary tanglegram is a pair 〈S, T 〉 of binary trees whose leaf sets are in onetoone correspondence; matching leaves are connected by intertree edges. For applications, for example in phylogenetics, it is essential that both trees are drawn without edge crossings and that the intertre ..."
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Abstract. A binary tanglegram is a pair 〈S, T 〉 of binary trees whose leaf sets are in onetoone correspondence; matching leaves are connected by intertree edges. For applications, for example in phylogenetics, it is essential that both trees are drawn without edge crossings and that the intertree edges have as few crossings as possible. It is known that finding a drawing with the minimum number of crossings is NPhard and that the problem is fixedparameter 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 2approximation and a new and simple fixedparameter 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 0.878approximation. 1
Generalized binary tanglegrams: Algorithms and applications
 In Proc. of BICOP 2009, volume 5462 of LNCS
, 2009
"... Abstract. Several applications require the joint display of two phylogenetic trees whose leaves are matched by intertree edges. This issue arises, for example, when comparing gene trees and species trees or when studying the cospeciation of hosts and parasites. The tanglegram layout problem seeks ..."
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Abstract. Several applications require the joint display of two phylogenetic trees whose leaves are matched by intertree edges. This issue arises, for example, when comparing gene trees and species trees or when studying the cospeciation of hosts and parasites. The tanglegram layout problem seeks to produce a layout of the two trees that minimizes the number of crossings between the intertree edges. This problem is wellstudied for the case when the mappings between the leaves of the two trees is onetoone. However, in typical biological applications, this mapping is seldom onetoone. In this work we (i) define a generalization of the tanglegram layout problem, called the Generalized Tanglegram Layout (GTL) problem, which allows for arbitrary interconnections between the leaves of the two trees, (ii) provide efficient algorithms for the case when the layout of one tree is fixed, (iii) discuss the fixed parameter tractability and approximability of the GTL problem, (iv) formulate heuristic solutions for the GTL problem, and (v) evaluate our algorithms experimentally. 1
Untangling Tanglegrams: Comparing Trees by their Drawings ∗
, 2009
"... A tanglegram is a pair of trees on the same set of leaves with matching leaves in the two trees joined by an edge. Tanglegrams are widely used in biology – to compare evolutionary histories of host and parasite species and to analyze genes of species in the same geographical area. We consider optimi ..."
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A tanglegram is a pair of trees on the same set of leaves with matching leaves in the two trees joined by an edge. Tanglegrams are widely used in biology – to compare evolutionary histories of host and parasite species and to analyze genes of species in the same geographical area. We consider optimizations problems in tanglegram drawings. We show a linear time algorithm to decide if a tanglegram admits a planar embedding by a reduction to the planar graph drawing problem. This problem was also studied by Fernau, Kauffman and Poths (FSTTCS 2005). A similar reduction to a graph crossing problem also helps to solve an open problem they posed, showing a fixedparameter tractable algorithm for minimizing the number of crossings over all dary trees. For the case where one tree is fixed, we show an O(n log n) algorithm to determine the drawing of the second tree that minimizes the number of crossings. This improves the bound from earlier methods. We introduce a new optimization criterion using Spearman’s footrule distance and give an O(n 2) algorithm. We also show integer programming formulations to quickly obtain tanglegram drawings that minimize the two optimization measures discussed. We prove lower bounds on the maximum gap between the optimal solution and the heuristic of Dwyer and Schreiber (Austral. Symp. on Info. Vis. 2004) to minimize crossings. 1