2008/06/05 by Kevin Buchin, Buchin, Kevin, Maike Buchin +11
Computer Science · #Computational Complexity (cs.CC) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #cs.CC #cs.CG
paper · pdf · doi:10.48550/arxiv.0806.0920
Journal version
arxiv created 2010/09/16 · arxiv updated 2010/09/17
A binary tanglegram is a drawing of a pair of rooted 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 tanglegram 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 binary trees. We show that the problem is NP-hard even if both trees are complete binary trees. For this case we give an O(n3)-time 2-approximation and a new, simple fixed-parameter algorithm. We show that the maximization version of the dual problem for binary trees can be reduced to a version of MaxCut for which the algorithm of Goemans and Williamson yields a 0.878-approximation.