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(n
3)‐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.
